IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY

(UNIVERSITY OF LONDON)

DEPARTMENT OF ELECTRICAL ENGINEERING.

The Propagation of Millimetric Radio Waves through the Clear

Atmosphere.

by

Michael Raymond Inggs

A Thesis Submitted for the Degree of Doctor of Philosophy in the

Faculty of Engineering, University of London.

July, 1979.

ABS TRRCT.

Millimetre—wave propagation through a tenuous, random medium is

investigated by means of practical observations and a computer

simulation. The measurements obtained are in agreement with a theory

of propagation based on the degradation of coherence of the propagating

wave.

The practical measurements were taken on a 12 km link operating at

38 GHz. The receiver was designed as a variable spacing interferometer,

allowing measurements of the spatial structure of the phase and amplitude

of the perturbed wavefront. These fluctuations proved to be small and

a great deal of care was required in the design and implementation of

the experiment. The important aspects of this equipment are discussed

in detail.

A large amount of data is produced by this type of investigation,

ani.-an efficient data analysis system was devised. This involved a.

purpose—built, multi—channel device, as well as micro— and minicomp-

uters. The impact of microprocessor technology on data. collection,

experimental control and other aspects of laboratory work is assessed

in the light of practical experience with a number of systems.

The computer simulation is based on Fejer's. slab model of propagation.

The field over successive planes is obtained by ..a numerical implementa-

tion of Fresnel diffraction. At each plane the field is perturbed by

computer—generated psuedo—random numbers, correlated using a novel

technique. This phase screen represents the effect of the irregular-

ities of the previous slab.

This simulation is fully multiple scatter and can deal simply with

anisotropic irregularities. The results are presented graphically

as plots of the phase and amplitude of the field over successive planes.

The degradation of the field coherence is clearly visible. This simU-

lation technique shows great promise, and is directly applicable to

related fields such as sound propagation in the atmosphere or ocean.

CONTENTS..

ABSTRACT

1

CONTENTS

2

ACKNOWLEDGEMENTS

10

INTRODUCTION

12

Appendix I1 Current Earth/Satellite Millimetre—wave

Propagation Experiments

21

REFERENCES FOR INTRODUCTION

22

CHAPTER ONE THEORETICAL ASPECTS OF RANDOM PROPAGATION 24

1.1 General Definitions 25

1.1.1 Coordinate Systems 25

1.1.2 Vector Functions 25

113 Maxwell's Equations 25

1.1.4 The Wave Equation 26

1.1.5 Duality 26

1.2 Solutions of the Wave Equation 27

1.2.1 Infinite Plane Wave 27

1.2.2 The Magnetic Field 27

1.2.3 A More General Solution 28

1.2.4 Determining the Angular Spectrum 28

1.2.5 The Far Field 29

1.2.6 Transverse Electromagnetic Fields 30

1.2.7 Polarization 30

1.2.8 The Poynting Vector 31

1.2.9 The Intensity 31

1.2.10 Standard Defintions of Aperture Antenna 32

Performance

1.2.11 Coupling between Antennas and Plane Waves 33

1.2.12 Directive Gain 35

3

1.2.13 Effective Area of an Antenna 36

1.2.14 Power Gain 36

1.2.15 Coupling between Two Antennas 36

1.3 A Slab Model of Propagation 38

1.3.1 The Model 38

1.3.2 A Single Refractive Inhomogeneity 39

1.3.3 The Equivalent Statistics of a Slab 41

1.4 A Random Phase Screen 43

1.4.1 The Average Angular Spectrum 44

1.5 Coherence Theory 46

1.5.1 Mathematical Formulation 47

1.5.2 The Lateral Coherence Function for a 48

Random Phase Screen

1.5.3 The Coherence Function for Multi—slab 50

Propagation

1.6 Coupling between Antennas Immersed in Random Media 52

1.7 Statistics of the Fluctuations 54

APPENDICES FOR CHAPTER ONE 57

A1.1 Coordinate Systems 57

A1.2 Definitions of Antenna Performance 58

A1.3 Antenna/Plane Wave Coupling 60 .

A1.4 The Effective Area of an Antenna 63

A1.5 Coupling between Antennas 64

A1.6 The Gaussian Antenna 66

A1.7 The Characteristic Function 71

A1.8 - The-van Cittert—Zernike Theorem 72

A1.9 Equivalent Planar Statistics of a Slab 74

A1.10 A Solution for Many Slabs 76

A1.11 The Equivalence of the Fejer and Bramley models 77

78 REFERENCES FOR CHAPTER ONE

4

CHAPTER TWO THE PHYSICS OF THE LOWER TROPOSPHERE 80

2.1 The Radio Refractive Index Refractivity 81

2.2 Statistical Description of the Random Refractivity 82

2.2.1 Basic Definitions 83

2.2.2 Stationarity 85

2.2.3 Structure Functions 85

2.2.4 Refractivity Wavenumber Spectrum 87

2.2.5 Forms of the Autocovariance Function 87

2.3 Elements of Turbulent Flow 89

2.3.1 Spectral Decomposition and Taylor's 89

Hypothesis

2.4 Turbulence in the Lower Troposphere 92

2.4.1 Energy Sources 93

2.4.2 Static Stability 93

2.4.3 Mixing Processes 94

2.4.4 Wind Shear as a Mixing Process 96

2.4.5 Surface Roughness as a Mixing. Process 96

2.4.6 Convective Mixing 99

2.5 Refractivity Measurements 101

2.5.1 The General Situation 102

2.5.2 Surface Values of Refractivity 104

Fluctuations

2.5.3 Refractivity Fluctuations in an 104

Urban Environment

2.5.4 Spectra when the Scale Sizes have a 110

Distribution

2.5.5 Stationarity of the Refractivity 111

Fluctuations

2.6 Conclusions 112

5

APPENDICES FOR CHAPTER TWO 115

A2.1 Other Processes Affecting Millimetre—waves 115

A2.1.1 Rainfall 115

A2.1.2 Fog 117

A2.1.3 Snow 118

A2.1.4 Attenuation by Atmospheric Gases 118

and Vapours

A2.1.5 Multipath Propagation 120

REFERENCES FOR CHAPTER TWO 122

CHAPTER THREE AN EXPERIMENTAL 38 GHZ LINK IN AN 125

URBAN ENVIRONMENT

3.1 Description of the Path and Planning 125

3.2 Transmitter and Receiver Design 125

3.3 Receiver Performance 135

3.3.1 Determination of the Detector Power Law 135

3.3.2 Short Term Amplitude Stability 13B

3.3.3 Phase Difference Measurements 138

3.3.4 Receiver Mounting Arrangements 141

3.4 Installation and Preliminary Results 141

3.4.1 Pre—installation Calibrations 142

3.4.2 Installation of Link. 142

3.4.3 Early Results 148

3.4.4 Transmitter Antenna Wind Loading 149

3•5 Environmental Sensors and Data Recording System 149

3.5.1 Synoptic Recording System 149

3.5.2 Environmental Sensors 151

3.6 Analysis of Synoptic Data 152

3.7 The Self Oscillating Mixer 153

3.7.1 Waveguide Oscillators 155

3.7.2 The Self Oscillating Mixer 155

6

3.8 Propagation Experiment Equipment: Some Suggestions 158

APPENDICES FOR CHAPTER THREE 162

A3.1 The Receiver 162

A3.2 Automatic Frequency Control: an Analysis 171

A3.3 Crystal Palace Transmitter 179

A3.4 Local Oscillator 186

A3.5 The Influence of the Transmitter Antenna Movement 187

on Amplitude and Phase Difference Measured on a

Distant Plane

A3.5.1 General 187

A3.5.2 Tower Movement 187

A3.5.3 Antenna Panning Frame Movement 188

A3.5.4 Amplitude Effects of Antenna Movement 190

A3.5.5 Phase Difference Measurements of 191

Antenna Movement

A3.6 Synoptic Recording System 193.

A3.7 Wind Vane 194

A3.8 Electronic Thermometer 196

A3.9 Rain Gauge 197

A3.9.1 Circuit Operation 197

A3.10 Analysis of Synoptic Data 201

A3.11 Clock 204

A3.12 Published Work 208

REFERENCES FOR CHAPTER THREE 208

CHAPTER FOUR EXPERIMENTAL DATA 209

4.1 Phase Effects: Theoretical Predictions 209

4.1 .1 Phase Difference 210

4.1 •2 Calibration of the Interferometer 212

4.1.3 Practical Measurements 214

4.1.4 Results from Other Workers 214

7

4.2 Millimtere—wave Angle of Arrival Fluctuations 218

4.2.1 The Magnitude of Angle of Arrival 218

4.2.2 Antenna Performance during Angle of 220

Arrival

4.3 Amplitude Scintillation 221

4.3.1 Theoretical Predictions 221

4.3.2 Practical Measurements 225

4.3.3 Comparison with other Experimental Data 227

4.4 Signal Conditioning 229

4.5 Amplitude Fluctuation Frequency Spectra 240

4.5.1 Theoretical Predictions 240

4.5.2 Measured Scintillation Spectra 242

4.6 Gain Reduction in the Practical Situation 245

4.7 Scintillation Spectra during Rain 247

' 4.7.1 Practical Measurements 247

4.7.2 A Possible Explanation 247

4.8 Conclusions 251

APPENDICES FOR CHAPTER FOUR 253

A4.1 Determination of the Phase Centre of 253

an Aperture

A4.2 Statistics of the Field Received during 255

Angle of Arrival Fluctuations

A4.3 More Exact Analysis of Angle of Arrival 258

Statistics

REFERENCES FOR CHAPTER FOUR 259

CHAPTER FIVE A COMPUTER SIMULATION OF PROPAGATION 262

THROUGH A RANDOM MEDIUM

5.1 Theoretical Considerations 263

5.1.1 Practical Difficulties with the 263

Fejer and Bramley Model

8

5.1.2 Basic Theory of the Simulation 265

5.2 Model Details 267

5.2.1 The Gaussian Beam 267

5.2.2 Numerical Techniques: Scaling' 268

5.2.3 Numerical Techniques: the..-Angular Spectrum 271

-5.3 Random Number Generators 275

5.3.1 Generation of Normally Distributed Numbers 276

5.3.2 The -Generation of Uniformly Distributed 277

Random Numbers

5.4 The Generation of Correlated Random Numbers 279

5.4.1 Adjacent Summing Algorithm 280

5.5 Conclusions 284

APPENDIX FOR CHAPTER FIVE 287

A5.1 Convergence to a Gaussian Autocovariance 287-

Function

REFERENCES FOR CHAPTER FIVE 289

CHAPTER SIX SIMULATION RESULTS 290

6.1 Introduction 290

6.2 Program SIMUL 290

6.3 Program PLOTN 292

6.4 Model Tests: Deterministic Case 294

6.5 Simulation Runs 297

6.5.1 Strong Perturbation versus Free Space 304

Conditions

6.5.2 The Effect of the Medium Scale Size 310

6.5.3 Strong Fluctuation Case 321

APPENDICES FOR CHAPTER SIX-. 327

A6.1 Program SIMUL (listing) 327

A6.2 Input Data for SIMUL 331

A6.3 Output Data from SIL 333 MU

A6.4 Program PLOTN (listing) 334

9

REFERENCES FOR CHAPTER SIX 340

CHAPTER SEVEN MICROPROCESSORS IN THE LABORATORY 341

7.1 Microprocessor versus Microcomputer 342

7.2 A Typical Microcomputer 345

7.3 Applications 349

7.3.1 Computing Spectra 349

7.3.2 Transient Recorder 351

7.3.3 Statistical Analysis 351

7.4 Microcomputer Networks 352

7.5 Microcomputers as Intelligent Terminal Device 354

7.6 Microcomputers as an Aid to Numerical Analysis 358

7.7 Conclusions 360

APPENDICES FOR CHAPTER SEVEN 361

A7.1 Program which Samples and Stores Data 361

A7.2 Statistical Analysis Program 362

A7.3 Gaussian Integration Program 363.

A7.4 General Purpose Plotting Program 364

REFERENCES FOR CHAPTER SEVEN 365

CHAPTER EIGHT CONCLUSIONS AND FUTURE WORK 366

REFERENCES FOR CHAPTER EIGHT 374

10

ACKNOWLEDGEMENTS.

The ideas which arise from discussion and arguement with

colleagues forms an important part of any one person's work. I am thus

grateful to past members of the Electromagnetic Waves Applications Group:

H. Tosun, P. Fowles, A. Trivedi, H. Jairam, M. Nicolaides,

J. Kanellopoulos, G. Mitsoulis, C. Buie, S. Bose.

I am indebted to members of, staff of the Department for

information they provided in the form of lectures and discussions,

especially R. Dyott, G. Burrows, J. Roberts, J. Couzens.

The success of many aspects of the equipment constructed

was due to the competence of David Sage, the Group Technician.. Gordon

Rowe and the late Bill Salmon provided the most efficient and helpful

Component Stores in my experience.

I would like to thank Professor John Brown for affording me

the opportunity to work as a research assistant in the Department.

My co—worker, Ismail Mashhour, was exceptionally

helpful over the years, especially in terms of the long hours he spent

developing the computer programs to analyse the data.

Finally, I cannot thank my supervisor, Richard Clarke,

enough for his support in so many different aspects of the project. I

always felt that my ideas and work received careful attention and the

most constructive criticism. This is especially true of this thesis.

11

This work is dedicated to Trish, for her support and

encouragement during its execution.

12

INTRODUCTION.

In the last two years there has been a remarkable growth

of interest in millimetre—wave systems. Millimetre—wave propagation has

been researched for many years, largely out of academic interest: the

growing commercial interest now makes the results of this research very

important.

The intensive research into microwave power generation

resulting from the 1939-45 War produced klystrons which could readily

be modified to produce millimetre—waves. In many ways millimetre—wave

systems suffered the same fate as the early laser ie. the applications

need was not strong enough to draw sufficient research capital. This

idea probably inspired the comment in Microwave Systems News,

"Millimetre—waves: A solution that's finally found a problem."

The above remark is a paraphrase of a comment made about the laser.

The principal reason for the current interest in milli-

metre—waves is that the maturing of the• active:davice.technology now- - makes

compact, reliable and reasonably low cost circuits a reality. This

means that the inherently higher resolution of millimetre—waves may be

exploited: this also means that a millimetre—wave antenna system is

much more compact than its microwave equivalent.

At present, the biggest use of millimetre—waves is for

military applications. It is estimated that the World now spends one

million dollars a minute on arms production and develop ment1'. Some of

these applications are

• Radar (especially terrain mapping types)

• Radio Communications

• Target Designators

• Tracking

• Radiometers for a variety of purposes.

13

Even ignoring military applications, spectrum crowding in

commercial trunk microwave (satellite and terrestrial) systems has

produced an interest in millimetre—waves for communications uses.2'3,4,5,6

This thesis is confined primarily to the influence of the

clear, but slightly random, atmosphere on millimetre—wave systems.

The scope of the thesis is discussed later in this introduction.

The technology of millimetre—waves has hinged on the

development of active devices for genegv.Uipm and detection. Vacuum tube

sources (such as klystrons) have been available for 30 years, but are

unsuitable for modern, compact equipment. The invention of the Gunn,

IMPATT and TRAPATT diodes has virtually placed millimetre—wave sources

on a par with lower frequency equivalents. The development of the

Gyratron7'9'10 has achieved a megawatt pulsed source suitable for power-

ful; high resolution radars.

Passive components have not, in general been a problem.

The scaling of lower frequency devices produces acceptable performance in

most cases. The very small sizes of the waveguides used makes the

production of components very expensive. This has spurred some work

into cheaper guiding structures. Here, dielectric waveguides seem to

provide most of the answers. For large—scale production, microstrip

is most widely used at the lower end of the band. Above 40 GHz, severe

difficulties are encountered in the packaging and screening of such

circuits, and the fin—line or dielectric waveguide technologies are

more applicable.

Detector and varactor diodes have followed the progress of

solid—state sources. At present, the supercooled Josephson Junction

offers the best performance for low noise detection. PIN diode modu-

lators offering up to 1 Gbit/sec are available.

Terrestrial precipitation is undoubtedly the major problem

associated with millimtre—wave systems. A great deal of work is in

14

progress to investigate ways of overcoming this effect using space

diversity of reception and transmission. Appendix I1 gives some further

information about current experimental work on the effect of the lower

troposphere on earth—satellite communications links.

Military weapons systems are less concerned with the effect

of precipitation, since important offensives are rarely lauched in

bad weather; aerial systems either cannot become airborne or are

operating above the cloudbase anyway.

The gas and vapour absorption bands are troublesome, but

can be completely specified from accurate laboratory measurements.

There are a number of systems which deliberately use the 60 GHz oxygen

absorption band as a means of realizing covert communication. The gas

absorption of about 15 dB/km ensures a rapid attenuation of stray signal.

This may also be exploited to allow frequency re—use in congested urban

environments requiring extensive spectrum for data links.

It is argued that the optical fibre will fulfill the growing

need for communications channels within urban areas. The chief drawback

is that optical fibres require, as do cables, expensive ducts. The

situation is made more difficult by the presence of concrete and tarmac

in urban areas. The author feels that narrow beam millimetre—wave

systems working in the absorption bands might provide a cost—effective

solution.

Similar arguments may be applied to the use of millimetre—

waves for trunk communications within a country. Precipitation will

undoubtedly not allow any millimetre—wave link to operate with acceptable

"outage time" statistics. However, the very rapid change to digital data

transmission must be borne in mind. This in turn allows Packet Switching

techniques to be used. Data is broken into "packets", prefixed with

destination information. Thus, if any particular path is closed, the

supervising computer is easily able either to find an alternative open

15

path (space diversity) or to wait until the channel is open once

again. More reliable, and thus more valuable, spectrum at lower

frequencies can be used for essential data which cannot be delayed even

for short periods.

The use of millimetre—waves is only dawning, and the above

discussion points to only some of the vast potential of the frequency

band. Most tropospheric millimetre—wave systems operate for more than

99% of the time in clear atmospheric conditions; it is the aim of this

thesis to investigate in some detail the problems which can arise.

When-writing .a thesis, there is a strong motivation to

present mostly original work, since the, thesis is in part fulfilment

of a higher degree. On the other hand, there is a requirement that the

work, which is published, should also provide intelligible information

to a wider, not so specialist reader.

Besides some original contributions (detailed in the latter

part of this introduction), this thesis is more of a "systems" inves-

tigation: sources, antennas, antennas and their interrelationship.

The angular plane wave spectrum approach to •antennas and propagation

which is presented, is both rigorous and based on physical concepts which

are easily visualized. At all times the theory developed is related to

millimetre—wave problems.

In an attempt to be as general as is possible, when applied

to millimetre—waves, many results are not derived explicitly. It is

felt that enough detail is present to enable the last few specific steps

to be simply made. For example, details are given of the angular plane

wave spectrum representation of antenna performance. Further, the

influence of the random medium on the angular spectrum of waves propa-

gating through it is presented. The final step, determining the, perform-

ance of a particular antenna at a particular frequency in a random medium

with particular properties is not presented specifically.

16

This thesis also reports practical observations of a

millimetre—wave propagation experiment. The design, construction and

developmeRt,of the experimental equipment occupied considerable time

and effort. Most important difficulties and some solutions are pre-

sented in the hope that they will be of use to other workers in the

field. To this end, many circuit diagrams are presented in appendices.

Perhaps the most promising investigation reported in this

thesis is the implementation of a computer simulation of propagation

through a random medium. The simulation reported is of millimetre—wave

propagation through a random atmosphere, but the theoretical develop-

ment shows that it is widely applicable to related fields such as sound

propagation through a random medium.

A synopsis of this thesis is now given. This is most.

naturally presented in the order in which the chapters occur.

The first chapter is principally theoretical, and is where

many of the results required to explain practical propagation data are

derived. The theoretical approach to propagation chosen is that of the

angular plane wave spectrum. This treatment has the advantage of being

founded on rigorous mathematical priciples, but is also very easy to

visualize.

An antenna is shown to produce a spectrum of plane waves

which propagate through the atmosphere. Under "free space" conditions,

the angular spectrum of energy is shown to be invariant with distance

from the transmitter. A medium with random refractive index fluctuations

causes phase perturbations of the plane waves making up this spectrum.

This causes a redistribution of the energy amongst the different angles

of propagation.

To determine the performance of an aperture antenna under

such random conditions, it is first necessary to be able to describe

its performance in "free space" conditions. To this end, some time

17

is devoted to applying the angular spectrum- approach to the definitions

of antenna performance, as defined by the IEEE12.

Amongst the topics covered is the coupling of an angular

spectrum of waves to a receiving aperture. Once this is done, the effect

of the random medium is, in principle, simple to specify. The trans -

mitter sets up a spectrum of waves, which is modified by the random

atmosphere. This modified spectrum is coupled to the receiver and the

power received may be compared to the power which would be coupled under

non—random conditions. What is missing at this stage is the exact

modification which the random medium imposes on the propagating angular

spectrum.

• This latter problem is approached via a slab model of the

atmospheric propagation path. Provided the statistical properties of

the medium are known, it is shown that the modification to the angul—

ar spectrum can be calculated. The propagation path is divided into many

parallel slabs and the spectrum modification after each slab is derived.

This work was first done by Fejer14. Bramley 15 then showed that

Fejer's result could simply be obtained by combining the many slabs

into one, and considering the total random phase path through the whole

medium.

The chapter continues to show the reason why these two

models produce the same result for the emergent angular spectrum. This

is achieved most simply by introducing the lateral coherence function

approach to propagation. It is shown that the angular power spectrum

and the lateral coherence function are simply related via the Fourier

transform. There is much in common between this result and that of

the relationship between the temporal autocovariance function and the

frequency power spectrum familiar from Communications Engineering.

This is in fact another advantage of the angular spectrum approach, in

that the Fourier transform relationship on which it is founded is

18

familiar to many Engineers.

The slab model of propagation is investigated in some

detail, especially with application to millimetre-wave propagation

through the weakly random atmosphere. The theory is by no means

restricted to weakly random media, and can be applied to most situ-

ations where the scalar wave equation applies; acoustic propagation

in the random ocean (Clarke16). The choice fo the size of the slabs

must be approached with some care, but this is taken up in some detail.

With the theoretical tools developed, a short example is

given of the application of the results derived town antenna immersed

in a random medium.

The theory developed so far requires a knowledge of the spa-

tial and temporal nature of the random medium. As is well known, this

manifests itself through minute fluctuations in the refractive index of

the air. Chapter two is concerned with filling the details of the

behaviour of the troposphere, especially the lower troposphere.

The initial part of the chapter is concerned with a summary

of the processes which are important in causing random refractive index

fluctuations. The latter part of the chapter reports some practical

measurements of the refractive index fluctuations. made in an urban

environment. There is a fairly detailed discussion of the form of the

refractive index autocovariance function in the light of Comstock's17

work. This puts forward the idea that the autocovariance function

may be thought of as the sum of mMyYT,.oatocovarianee functions, each

one due to a different type of atmospheric process, and each having a

different characteristic scale size. The scale sizes thus have a

distribution. This concept seems very plausible and powerful.

As mentioned earlier, this thesis is also a report of

an experimental 11,6 km propagation path at 38 GHZ. Chapter 3 describes

the setting up of this experiment. The minute size of the amplitude

19

and phase fluctuations induced by the atmosphere required a great deal

of time and effort expended on refining the receiving equipment. Much

of this equipment is described, and will be of possible use to workers

undertaking propagation experiments.

Initially the self—oscillating mixer (SOr) was considered

for the receiver. Although not used in the final equipment, it is

shown to be of wide application, especially where low cost is impor-

tant and extreme sensitivity is not required.

The monitoring of propagation experiments makes a heavy

demand on data storage and analysis. The system used is described.

The initial system was fairly crude and inflexible, but still produced

acceptable results. The advent of the microprocessor has revolutionized

the possibilties available for experimental control and data

collection. This is discusad at some length in chapter 7.

Chapter 4 is an exposition of some aspects of the data:

taken on the experimental link. In all cases, a comparison is made

with the theoretical predictions, and where possible, with results

taken by other workers.

One of the principal difficulties of practical measurements

is the impossibilty of completely measuring the properties of the whole

propagation path. Point measurements of atmospheric variables are

assumed to be representative of the whole path. This, together with

Fejer's multi—slab model inspired a computer simulation of propagation

through-.a random medium.

In chapter 5 this model is described. The path to be

simulated is divided into a number of not necessarily identical slabs.

The statistical properties of these slabs may be provided by means of

computer—generated random numbers. This kind of experiment has the

potential for providing a much better understanding of the theory,

since the "atmosphere" may be controlled at will.

20

The chapter contains details of the method of implementing

the model, as well as ways in which it may be applied to different

propagation paths. Methods of generating suitable models of the random

atmosphere are discussed in some detail. This includes a fast method

of generating random numbers with a Gaussian autocovariance function.

Chapter 6 gives details of the programs used, with listings

provided in appendices. The bulk of the chapter is taken up with'

graphical examples of propagation through various model atmospheres.

The amount of analysis which can be done on the data from such a model

study is vast. The data from the simulations done is available on

magnetic tape, and could form the basis of some future work.

It is difficult to write about the impact of micropro-

cessors without becoming involved with cliches. Chapter 7 attempts to

relate the impact of these devices to propagation experiments. On a

broader front, there is also some discussion about their usefulness in

the laboratory, when incorporated in small "microcomputers". These

may be simply linked to larger machines or with other small machines.

The discussion is in the light of practical experience with a number of

small systems.

The thesis concludes with chapter 8, which provides a

summary of the work presented, but is principally a presentation of

some ideas for future work in the field of millimetre—wave propagation.

It must be reiterated that the use of millimetre—waves is only just in

its infancy and that not all of the necessary work has been done. No

doubt the growing use of systems will highlight certaih aspects and

inspire further work. There still exists a basic reluctance to use

millimetre—waves which will only finally be overcome by the publi-

cation of much more data, and when most important difficulties can

be quantified to sufficient accuracy.

21

APPENDIX I1 Current Earth/Satellite Millimetre—wave Propagation Experiments.

Satellite Related Experiments.

ATS-5 Uplink on 31,65 GHz

Downlink on 15,3 GHz

ATS-6

(NASA Exp.)

9—tone comb radiated at 20 and 30 GHz.

Transmits linearly polarized signal at 20 GHz.

ATS-6

(COMSAT Exp.)

About 23 ground stations transmit data in the

13 and 18 GHz bands, which is returned in the

4 GHz band to a single terminal.

Comstar 19 GHz beacon, linearly polarized and switched

between two orthogonal directions at 1 kHz rate.

28 GHz beacon, vertically polarized, modulated

with 264 MHz sidebands.

Japan ETS 34 GHz propagation experiment.

All the above experiments are operational, and the data acquired will,

no doubt, be available in the next few years.

22

REFERENCES.

1

"Soviets Push Radar Developement" 1978. Electronic

Warefare and Design Engineering, July 1978, p49.

Staff Report. 1975 U.K. Digital Microwave Relay Systems

above 10 GHz." Microwave Systems News, Aug./Sept. 1975,

5, No.4 p48.

3 Baccetti, B, Corazza, F and Valdoni, F "Millimetric

Waveguide System Developement for Communications in Italy."

Microwaves Systems News, Aug./Sept. 1975, No.4 p51.

4 "An Analysis of the Future" 1978. Microwave Systems News,

Vol.8 No.1 1978.

5 "The Italian Connection:. MM Waveguide Research at Marconi

Institute." 1976. Microwave Systems News, Aug./Sept.

1976, 6, No.4 p48EE.

6 Cuccia, L, Quam, W, and Hellman, C. 1977. "Above

10 GHz. Satcom Bands Spur New Earth. Terminal Developement."

Microwave Systems News, March 1977, 7, No.3 p37.

7 Hamilton, R.J. and Long, S. 1977. "Gunn Amps Fill the

K—Band Gap." Microwave Systems News, August 1977, 7,

No.8 p49.

8 Davis, R (Editor) 1977. "Millimetre—Waves: A Solution

that's Finally Found a Problem." Microwave Systems News,

September 1977, 7, No.9 p63.

9 Godlove, T.F. and Granatstein, V.L. 1977. "Gyrotron:.

Reborn Tube is a Millimetre Powerhouse." Microwave Systems

News, November 1977, 7, No.11 p75.

10 Dyott, R.B. and Davies, M.C. 1966. "Interaction

Between an Electron Beam of Periodically Varying Diameter

and E.M. Waves in a Cylindrical Guide."

23

IEEE Trans. E.D. Vol. ED-13 No.3.

11 "Towards collision: World Military spending takes off."

To the Point, November 1978, 7, No. 44.

12 Antenna Standards Committee of the IEEE Antennas and

Propagation Group, "IEEE Standard Definitions of Terms for

Antennas", IEEE Standard 145-1973.

14 Fejer, J.A. 1953. "The Diffraction of Waves in Passing

through an Irregular Refracting Medium."

Proc. Roy. Soc. Series A, 220, pp455-471.

15 Bramley, E.N. 1954. "The Diffraction of Waves by an

Irregular Diffracting Medium."

Proc. Roy. Soc. Series A, 225, pp515-518.

16 Clarke, R.H. 1973. "Theory fo Acoustic Propagation in a

Variable Ocean." J. Sound and Vibration, 34(4), pp457-454

17 Comstock, C. 1964. "On the Autocorrelation of Random

Inhomogeneities." J.A.S.A., 36, No.8, p1534.

24

CHAPTER ONE THEORETICAL ASPECTS OF RANDOM PROPAGATION.

The theoretical approach taken in this thesis is based on

the concept of the degradation of coherence of an electromagnetic wave

propagating through a medium containing random fluctuations in refractive

index.

Initially, a link is established between widely used

definitions of antenna performance and their performance in generating

an angular spectrum of plane waves. This allows the power coupled between

antennas to be calculated.

It is then shown that a slab model of the atmosphere due to

Fejer may be used to calculate the modification imposed by the random

atmosphere on a spectrum of plane waves propagating through it. The fact

that the atmospheric slabs may be considered as equivalent thin, phase

changing screens is justified.

Just as switches between the frequency and time domains are

made for convenience in communications engineering analysis, the link

between angular spectrum analysis and coherence theory is introduced.

This allows the effect of propagation through multiple slabs to be more

easily investigated.

Using these complementary analyses, a theory of random

propagation is possible which is fully multiple scatter and is able to

predict, amongst other things, the power coupled between antennas

immersed •-in random media and the statistics of the perturbed field.

Some of these aspects are discussed.

25

1.1 General Definitions.

This section consists of a number of unrelated topics

required as a basis for further analysis.

1.1.1 Coordinate Systems.

Generally rectangular and polar coordinate systems are

sufficient. The forms chosen are given in Appendix A1.1, as well as

convenient transforms between the two systems.

Vectors are denoted by the overbar, unit vectors with a

cap. Often coordinates are implicit ie.

Tr(x,y,z) = P(r, 0 , 4 ) 1.1.1

1.1.2 Vector Functions.

The electric field, for example, is a vector function ie.

w A

= E(x,y,z,)x + E(x,y,Z)y + EZ(x,y,z)z 1.1.2

In addition, the field quantities are due to monochromatic source of

frequency W . The term exp(it) is suppressed in the following

analysis. Time dependence becomes important when, for example, a

random medium imposes temporal fluctuations on the field. The components

then become 4 dimensional;

ie. Ex Ex(x,y,z,t) 1.1.3

It is usually clear from the context when this is so.

1.1.3 Maxwell's Equations.

These may be written as

•E=Pt/E.

~•B =0

Vx1+ ae= Ō at

QX 3— 1 OE= Jm c2

1.1.4

26

where pt is the total free charge density (C m-3)

. is the permittivity of the medium (F m-1)

IL is the permeability of the medium (H m-1)

c is the velocity of light

m is the free current density (A m-2)

For ohmic conductors,

5f =0'E

1.1.5

where 0 is the material conductivity (S m-1)

1.1.4 The Wave Equation.

The nonhomogenuous wave equation for E may be derived for

homogenuous media from 1.1.4 ie.

2E — 94a 21= 0 R+ 2J t2 E /u ā tf

For a source free medium, this becomes

t7 c.11 2

-7

at The solution of this equation is the subject of section 1.2.

1.1.6

1.1.7

1.1.5 Duality.

An interesting property of Maxwell's equations is seen if

the following substitution is made ie.

E' =—a8=—apH 1.1•8

H = aD = aS E

where a is a constant.

When substituted into Maxwell's equations, they are found to yield the

same set of equations, with E' for E, 8' for B. This means that if

a solution has been found for E and N, a "dual" solution exists, with

the electric and magnetic fields interchanged according to 1.1.8.

27

1.2 Solutions of the Wave Equation.

This section investigates solutions to the wave equation

1.1.7 for linear, isotropic, source free media.

1.2.1 Infinite Plane Wave.

A solution of 1.1.7 is

E{k,r) = Ā(k) exp(—ik•r) 1.2.1 ,

where lki = 2rr = w,r'

A is the wavelength

This is an:.infinite plane wave which is travelling in the k direction.

Since the wave is monochromatic, Aka is fixed. Since

A A A k = kxx + kyy + kzz

it is seen that knowledge of any two of kx, ky,

find the third.

Further, the medium is source—free:

0 • E = 0

1.2.3 and 1.2.2 together yield

1.2.2 is sufficient to

1.2.3

kx Ax(k) + ky Ay(k) + kz Az (IT)= 0 1.2.5

Again, only two components of A(k) need be specified and the third is

given by 1.2.5.

1.2.2 The Magnetic Field. The solution presented so far only concerns the electric field,

A magnetic field wave equation may be derived and solved. It is simpler,

however., to use Maxwell's equations for the source—free medium to show

A N(x,y,z) = V X E(x,y,z) = k X E 1.2.6

z 0

Zo is the wave impedance of the medium.

28

1.2.3 A More General Solution.

Since the wave equation 1.1.7 is linear, a general

solution for Ē and R may be constructed from a linear combination of

terms like 1.2.1 ie.

T(T) _ A.(k )•exp(-ik.•r) ,.1 1 1

Here the field at r is synthesized from an ensemble of A

infinite plane waves propagating in directions ki. The amplitude āi

may be suitably chosen. The uniqueness and validity of such a decompo-

sition will be be expanded upon, but the analogy with the Finite

Fourier transform is obvious.

In a manner similar to the way the continuous Infinite

Fourier transform may be derived from the Finite transform, it will be

stated without proof that

E(r) = ((aĀ(k).exp(—ik•r).dkx k •d y 1.2.8 -p

In this integration, Ā(k) is known as the Plane Wave spectrum or

Angular Spectrum which constitutes the field at r. k has become a

continuous variable and each k corresponds to a plane wave travelling

A in the k direction.

The integration is twofold, since eqtn. 1.2.5 will allow

A z

and dk z

to be written in terms of the x and y components.

The magnetic field may be obtained from eqtn. 1.2.8

using eqtn. 1.2.6 ie.

R(r) = 1 ~ k X Ā(k)•exp(—i7.7)•dkdky x

W~l JJ 1.2.9

1.2.4 Determining the Angular Spectrum.

Suppose the tangential electric field is known over a plane

at z = zo ie.

- E(xly•zo) = E0(x,y) = Eoxx + Eoyy 1.2.10

1.2.7

29

Substituting into eqtn. 1.2.8 and inverting the transform gives:

0 Ax(kl_ _ = exp(ik_z _ ) IEox(x,y)•exp(ikxx + ikyy)dx•dy

.40

a

Ay(c)1 z• = z 0

exp(ikzzo) Eoy(x,y)•exp(ikxx + ikyy)dx•dy

4 n Z

1'2•11

This formal inversion and transformation is, of course, subject to

the theorems of the Fourier transform, which are in most cases not

restrictive;. It may be loosely stated that the sufficiency conditions

of the Fourier transform are related principally to the amount of "energy"

associated with the function to be transformed. Practically occuring

field distributions are from necessity finite in energy content.

The advent of the Fast Fourier Transform (FFT) algorithm

has considerably simplified the procedure for obtaining transforms

on modern digital computers.

The important step that has been taken here is that the

formula now exists for finding the (()) necessary -to synthesize the field

at a point in space. The function 'Co may be thought of as an aperture

antenna and Ē(r)is the field measured at a point away from this aper-

ture. In principle Eo may now be manipulated to yield some desired form

for t(7); this is the principle behind antenna design.

1.2.5 The Far Field.

Eqtn. 1.2.8 has an interesting form when kr j,]1. The

method of stationary phase (Heading 1'1 ) may be used to show

lim Ē(r) = i•2Tr exp(—ikr)•Ā(170)

ks'~oo kzOI 71 •

where -1( = k r 0

This result is clearly identical to that obtained by

conventional diffraction theory. It predicts that the field falls off

4 11 2

1.2.12

30

as 1/r and provides an important insight into the physical meaning

of 1(1). Egtn. 1.2.12 predicts that the field in any direction r is

proportional to the component of the angular spectrum in that direction.

Thus in the far field region, the electric field is seen to be essenti-

ally a locally plane wave; in general it is spherical. The angular

spectrum function is thus a measure of the directional properties of a

radiating aperture.

A further important property of eqtn. 1.2.12 is that it

may be shown that the radial component of Ē(r)vanishes and the electric

and magnetic fields become transverse.

1.2.6 Transverse Electromagnetic Fields.

Suppose a plane electromagnetic (e.m.) wave propagates

A in the z direction such that the Ē vector always lies in the xy plane.

This means Ez 0. Egtn. 1.2.6 predicts that in general, Hz ` 0

This wave is said to be a transverse electric wave (TE). Similarly,

a transverse magnetic (TIS) wave may be configured.

In all the analysis which follows, TE waves will be

implied, unless otherwise stated. This is no limitation, since Duality

(section 1.1.5) may be used to transfer from one mode to the other.

1.2.7 Polarization.

Until, now, no attention has been paid to the form of

the field functions Ē,H. The simplest form for Ē is

E(x,y,z) = x E(x,y,z)

1.2.13

This is a field which is said to be linearly polarized in the il'

direction. In general, complex combinations such as

E(x,y,z) = x Ex(x,y,z) + Y`E(x,y,z) exp(i/S) 1.2.14

are possible, leading to elliptically or circularly polarized fields

31

(and waves

It is important to realize that any polarization may be

synthesized with suitable combinations of linear (eqtn. 1.2.13) polari-

zations.

There is thus no limitation if future analysis is only

Concerned with linearly polarized TE waves resulting from aperture fields

of the form

A = x E(x,y) 1.2.15

1.2.8 The Poynting Vector.

For a plane wave, the Poynting vector measures the power

flux (Wm 2) at any point, averaged over one cycle. By definition this is:

av =1.-EX H (Wm-2) 1.2.16

where * denotes conjugation.

In free space, for example, eqtn. 1.2.6 relates the elec-

tric and magnetic fields and

av = (EX ic) X Ē 1.2.17

2 Z 0

Using the far field value for Ē derived from eqtn. 1.1.12 gives

Lim av 2

Tt 2 I w,..)12.? 1.2.18

2 2 Z kr - .0 k r zo 0

where again, ko = k r

This latter equation is useful in the next section, when the radiant

intensity is calculated.

1.2.9 The Intensity.

The intensity is defined as the power radiated per unit

solid angle by a luminous (radiating) object, and is measured in

W Steradian-1 ie.

Source area A

P

Surface S of

rz

32

R(r) = Ps(r) 1.2.19

SL

Here, Ps(r) is the power flowing in the solid angle IL.

This is shown diagramatically in fig. 1.2.1.

fig. 1.2.1 The Intensity.

In a lossless, non—random medium, the intensity is

independent of the distance from the source P.

Calculating the intensity from the Poynting vector is

straight forward. Clearly,

IR(r) = I J2I.ilav(r)1 1•2.20

For the far field, eqtn. 1.2.19 may be used in eqtn. 1.2.19 to obtain

lim IR(r) = A 2 1 ~A(ko) 2 1.2.21

kC-s~ 2Z 0

1.2.10 Standard Definitions of Aperture Antenna Performance.

During the evolution of antenna engineering', terms have

arisen to describe the performance of aperture type antennas as both

radiators and receivers of e.m..energy.

As is often the case, confusion has arisen over some of

these terms. In order to establish some order, the IEEE of America has

published a standard 1.2

. Appendix A1.2 contains most of these terms,

together with a few which are necessary to take into account factors

33

which degrade antenna performance. These are due to Hill .

In this thesis, degradation of performance due to random

inhomogeneities in the propagation path will be investigated. The next

few sections will show how these definitions may be couched in angular

spectrum form.

1.2.11 Coupling between Antennas and Plane Waves.

Until now the performance of an aperture antenna has been

investigated in the transmit mode. In the far field, it is seen to give

rise to locally plane waves, whose amplitudes depend on the angular

spectrum.

Practical microwave radio systems will usually consist of

a receiver aperture receiving, in the far field, the essentially plane

waves due to a distant transmitter. The purpose of this section is to

calculate the performance of the receiver when receiving these plane waves,

in terms of its performance as a transmitter.

A rigorous, development of this was presented by Brown 4' 1 ■5

A further reference due to Paris bis useful, since he relates

the results of Angular spectra to the technique of probe—compensated.

near—field measurements. The growing use of near—field measurement

facilities highlights the importance of the angular spectrum approach.

Later it will be shown that a random medium disturbs a

single plane wave and causes a slight redistribution of energy-amongst

a small cone of angles about the direction of propagation. Thus, if the

coupling of the receiver to each of these plane waves is known, the

total received field can be obtained by integrating.

A simple analysis of coupling, based on that of Brown,

is given in Appendix A1.3. The geometry is shown in fig.1.2.2.

Using the Lorentz Reciprocity theorem the appendix

shows that the ratio of the received field in the feed waveguide to the

Reference

plane

Feed waveguide

34

Screened transmitter/ receiver

radiating Po

fig. 1.2.2 Aperture receiving plane waves.

35

radiated field due to the transmitter is

c = 2 ē. • Ā(kn) i i2ZP

0

1 •2.22

where Po is the power in the feed waveguide.

ēi is the incident plane—wave field

A is the angular spectrum of the antenna used as a

transmitter.

Physically this result means that the received power is proportional to

the component of the angular spectrum of the receiver aperture, which is

collinear with the incoming plane—wave direction.

The scalar product in eqtn. 1.2.22 takes into account any

differences between the polarization of the incoming wave and the angular

spectrum. This becomes an arithmetic product if the angular spectrum

(or vector pattern function) and the incoming wave are polarization matched.

1.2.12 Directive Gain.

This is essentially a measure of the directional properties

of the radiation from an antenna. It says nothing about the efficiency

with which the antenna radiates the power supplied by an attached trans-

mitter.

Suppose that the antenna actually radiates power P .

Definition (v) of A1.2 states

D(k) = I R (k)

1.2.23

is

where Iis is the equivalent isotropic intensity:

Iis = Po/ 4Ti

In the far field this is simple to evaluate:

lim D(k) = 2n X 2F(k)1 2 1.2.24 kribO Z P

0

36

1.2.13 Effective Area of an Antenna.

This is covered by definition (vii), A1.2. A more detail-

ed development of this concept is given in Appendix A1.4. Physically

the effective area is that area which would "scoop", from an incoming plane

wave, the power which appears in the feed waveguide of the antenna in

question.

If the antenna and received radiation are polarization

matched, the effective area is

Ae(k) %2 o(k) 1.2.25 411"

As expected, the effective area is a function of the direction from which

the incident radiation is coming, via the directive gain function.

1.2.14 Power Gain. (definition(viii), A1.2)

The power gain is similar to the directive gain, except

that the power considered to be radiated is that which flows to the right

of the reference plane of fig.1.2.2. This is not necessarily all radi-

ated, since the antenna structure may exhibit loss due to a variety of

causes. Let the power accepted by the transmitter antenna-.from the

transmitter at the feed reference plane be Pa. The power gain may then

be shown to be

g(k) = 4TC 1 rj2 I Sav(r)1 1 •2.26

P a

where IS av

its the power flux measured at some point r.

This equation may be evaluated for the fār'field situation.

1.2.15 Coupling between Two Antennas.

A fundamental requirement of radio transmission is that

the power coupled between two antennas can be calculated. The calcula-

tion has been deferred until now, since it is more easily written in

terms of directive gain and effective aperture.

37

A more detailed derivation and the geometric arrangement

are presented in Appendix A1.5, but is is stated here that

1im PR = DT(k) AR ( k)

kt-b pT 4TT

r2 0

Here, it is assumed that:

1) The antennas are polarization matched.

2) PR is the power available at the feed waveguide.

3) P1. is the power accepted by the transmit antenna.

4) DT is the directive gain of the transmit antenna.

5) AR is the effective area of the receiver antenna.

1.2.27

At this stage, the performance of aperture type antennas

has been investigated in some detail. The medium of propagation has,

however, been homogenuous and non—random. The next few sections continue

to modify the equations presented in previous sections, to take into a

account propagation through a medium containing random fluctuations in

refractive index.

To consolidate the concepts developed so far, Appendix A1.6

calculates the performance of the so—called Gaussian antenna. Although

perhaps idealistic in performance, it is a good approximation to the

narrow beam, low sidelobe antennas which are possible at millimetre—wave

frequencies. It also forms a basis on which to check a simulation presen-

ted in chapters 5 and 6.

to

c7

38

1.3 A Slab Model of Propagation.

This section will introduce a model of propagation through

an extended, random medium. By random, it is meant that the propa-

gation path is filled with.lumefluctuations in refractive index, n.

These will be called refractivity fluctuations, where the refractivity

N = (n-1) 106 1.3.1

How these arise is taken up in detail in chapter 2.

The idea of a slab model is appealing, since it is easily

visualized. It will further be shown that the model is reasonably

rigorous and certainly accurate enough for millimetre—wave frequencies.

Essentially, the model will consist of resolving the propa-

gation path into a number of slabs perpendicular to the direction of

wave propagation. The volume irregularities within each slab will

be assumed to be equivalent to an infinitesimally thin phase—changing

screen at the exit plane of the slab. It may seem surprising that such

a simplistic model works, but analysis will be presented to justify

and establish the limits for this procedure.

1.3.1 The Model.

fig. 1.3.1 Multi—slab modal of propagation.

39

The volume between the transmitter plane (z = 0) and the

receiver plane is resolved into n slabs, each of thickness Az = zo/n.

Each slab contains volume irregularities which will perturb the plane

wave which propagates from z=0 to z=zo.

Fejer was the original proponent of a similar model.

He calculated the angular spectrum after the single incident plane wave

had been scattered by the volume irregularities of the first slab. This

spectrum was allowed to be scattered by irregularities of the next slab,

and so on through the whole medium. An expression for the emergent

angular spectrum resulted.

Shortly afterwards Bramley 2showed that Fejer's result

could simply be obtained by considering only the random phase shift along

a number of paths parallel to PQ in fig.1.3.1. The reason for this

surprising result will be given in section 1.5, where a convenient

and powerful technique (Lateral Coherence Functions) will be presented.

Fejer's model is clearly multiple scatter, since the per

turbed spectrum is further perturbed at each slab.

Instead of calculating the volume scattering within each

slab, it is useful to investigate the scattering due to a single refrac-

tive inhomogeneity in simple terms.

1.3.2 A Single Refractive Inhomogeneity.

The slabs of fig.1.3.1 are composed of arbitrarily shaped

refractivity irregularities. Before considering the ensemble behaviour

of such irregularities, it is fruitful to examine the diffractive

behaviour of a single representative one.

The inhomogeneity of fig. 1.3.2 is characterized by a

dimension 10 and & N = N2 — N1, the refractivity difference being

assumed small.

The inhomogeneity will cause energy from the incident

Incide nt

40

plane wave

Observ

plane

tion

fig. 1.3.2 A Single Inhomogeneity.

plane wave to be scattered over a cone with semi—vertex angle

ed = x /lo 1.3.2

where i1 is the wavelength.

The scattered energy which propagates at this angle will be Tr out of

phase with the unscattered wave at a distance

LR —2/ 1.3.3

(It is interesting to note the similarity with the Rayleigh distance

1!8 used when discussing 'aperture antennas.) de Wolf "considers such

phenomenological conditions in a useful paper.

Evidently, however, a single refractive inhomogeneity

only becomes effective in causing substantial amplitude ripple on the

unscattered plane wave at this distance LR. Prior to this, it may

be thought of as a pure phase modulation. The phase shift will be

60=2 Tr lo Sn 1.3.4

The inhomogeneity may be thought of as a "phase object", which is a

term used in microscopy(Zernike' .).

41

It would thus seem that provided a slab is thinner than

the Rayleigh distance LR associated with the characteristic size of the

inhomogeneities, the effect of the slab is to provide a phase mod-

ulation only.

The next subsection will develop techniques for handling

the ensemble effect of many irregularities.

1.3.3 The Equivalent Statistics of a Slab.

If the autocovariance function of the refractivity fluc-

tuations is known, it is possibie to calculate the statistics of the

random phase shift induced by propagation across the slab.

n y

AL

fig.1.3.3 Phase shift due to single slab.

Appendix A1.9 calculates the phase shift across many paths parallel to

AB in fig.1.3.3 and shows that

Oa 2 BitS(u,v,"C ) z=z+aL = 10-12 k AL Ç 8 (u,v ,w, ) dw

-6°1 •3.5

where BN is the refractivity autocovariance (see chapter 2)

k=2Tr/X

42

The conditions necessary for the form 1.3.5 are:

i) N(x,y,z,t) is a stationary process

ii) o• = Bys(0,0,0) <<1 1.3.6

iii) wo AL, where wo is a typical scale size in the z direction.

Intrinsic with these assumptions is the fact that the phase

process of 1.3.5 will approach normality even if the N process is not

normal.

There is no requirement that the N process is isotropic.

The integral 1.3.5 can deal with oblate shaped irregularities.

Suppose the refractivity autocovariance is given by

BN(w,t) = — — W

— oN exp —Iu iv 2

t (t 0 l 0 0 0 .I

1.3.7

where aN is the refractivity variance.

Hence, using eqtns 1.3.5 and 1.3.7

i(u v,t)\ W10-12' z=z+ A. L ~ 10 ; 12 k2 d~L w 0'N •

exp{—t161 — v L 2"

1.3.8

The appendix also shows that this latter form is physically plausible.

Although egtn.1.3.7 seems an idealistic choice, chapter 2

shows that most practically measured forms of BN may be synthesized

from suitable Gaussian—shaped autocovariances, where the scale sizes,

uo and so on, are drawn from probability distributions.

The importance of this subsection, however, is that the

situations under which a slab may be converted into a plane screen have

been formulated. If this may be done, it is possible to avoid calcu-

lating volume scattering as Fejer did, and work only in terms of the

scattering due to the equivalent phase screen.

43

A simple numeric example shows that this model suffices for

millimetre—wave propagation. Chapter 2 will show that a typical scale

size in the atmosphere is 10,.10m. For X = 10-2m

LR = (10)2 / 10-2 = 104 m 1.3.9

Thus for millimetre—wave problems, large slab thicknesses are feasible,

with the proviso that the random phase shift induced is small. (This

is only necessary, in fact,to satisfy the approximations needed to simpli-

fy egtn.1.3.5.)

Again, chapter 2 will show that 2 A110 2 typically.

The equivalent phase variance is (from egtn.103.6)

= B (O,Q,O) =177/10-12 2Tī 2 104 10 .10-2

10-2

= 7,40 10-4

or o = 2,6410-2 rad = 1,52. deg.

where a slab of thickness ,Q4 meters has been used. Clearly the induced

phase is sufficiently small for the phase screen model to be used.

1.4 A Random Phase Screen.

The analysis of the previous section has established that

a sufficiently accurate model of millimetre—wave propagation may be

obtained by assuming that the medium may be replaced by an equivalent

thin phase screen. This section will examine the effect of thin phase

screens on an incident field distribution.

Incident plane wave

fig.1.4.1 Thin phase modulator.

44

An aperture radiates from the xy plane towards positive z.

The radiated field is incident on an infinite simally thin screen at

z=z 0 . Let the field just before the screen be

(x,y) 1.4.1

The screen is very thin and modulates the phase of the field only.

The field just to the right of the screen is thus

E+(x,y) = E (xry) exp(igx,y)) 1.4.2

The angular spectrum may now be evaluated for the emergent field and is

A() = exp(ikz(z-z0)) \ Ç(x,y)exp(i(x,y))

4 TT 2 -m exp(ikxx + ikyy) dx dy

1.4.3

at some point z,

zo. A similar expression for Ay(k) exists.

1.4.1 The Average Angular Spectrum.

The phase modulation fi(x,y) will be supposed to be a .:

random function of x and y. Formal descriptions of random quantities

are given in chapter 2.

Returning to eqtn. 1.4.3 and taking its expectation

0o

Ax(k)> = eXp(ikz(z-z0)) Ex(x,y) C.exp(igx,y))3>

4 it 2

1.4.4

Ex is deterministic and is removed from the averaging. Appendix A1.7

shows that

exp(ii(x,y)) > = exp(-0-2 2) 1.4.5

provided $(x,y) is a zero-mean, normally distributed variate , with

variance o~ys. Using eqtn. 1.4.5 in'1.4.4,

< Ax(k)> = exp(—o./2)

Axo(k) 1.4.6

•exp(ikxx + iky) dx dy

45

where AXO(k) is the undisturbed value. More generally,

A(k) = exp(—Q;/ 2) Ao() 1.4.7

Again, Ao(k) is the undisturbed value.

Physically, this means the average spectrum after perturb-

ation by the screen is the same as the unperturbed spectrum, but is

reduced by an-exponential factor which is a function of the intensity of

the phase screen fluctuations.

Amplitude scintillation of the field measured at one point

is caused by the interaction of the reduced (eqtn. 1.4.7) mean spectrum

and the scattered spectrum. Generally, energy in phase with the mean

spectrum is termed "coherent" and the rest, "incoherent".

To determine the nature of the incoherent spectrum,

knowledge-of the spatial characteristics of the phase fluctuations - is

required. This is taken up by Clarke1.10,1.11 ,

in the context of sound

propagation in a random ocean, but this may be modified slightly for the

troposphere and millimetre—waves.. Chapter 2 investigates the statis-

tical description of the troposphere in some detail.

It is instructive to see how this phase screen model may

be applied to antenna performance considerations. If a probe antenna

were to receive the power radiated from an antenna under test, but

immersed in a random medium, it would receive both coherent and inco-

herent contributions. It is thus possible to specify a mean, doherent

directive gain as

o(k) (-2- 2 r A2 A(701 2 exp(-cr .) 1.4.8 Z P 0

Using definition xii of A1.2, the realised radiation efficiency meas-

ured for the coherent power would be

1 RR _ exp(—ci)

1.4.9

46

Thus, in summary, it is seen that a simple slab model

coupled with a well defined set of antenna performance criteria,

produces a clear method of evaluating radio system performance under

random propagation conditions.

1.5 Coherence Theory.

The theoretical exposition so far has been in terms of

Fourier transforms between field distributions and angular plane—wave

spectra. In conventional temporal/frequency domain analysis of wave-

forms, switches between using spectra and temporal correlation func-

tions are often made. This is easily achieved using the well known

result

()) 18f( ~ )1 1.5.1 ,

(w) is the temporal power spectrum of f(t)

Bf(t) is the autocorrelation function of f(t)

It will now be shown that a similar change from angular spectrum

forms to a dual function known as the coherence function can be made.

The conditions of the transformation are laid down in the van Cittert-

Zernike theorem. This will in fact show that the angular plane—wave .

power spectrum and the spatial correlation function are related by a

Fourier transform.

Coherence theory has a number of advantages. Principally,

it deals directly with spatial and temporal correlation functions of the

field. Further, the effect of the random medium is calculated from

knowledge of the correlation (autocovariance in most cases) functions

of the medium. Chapter 2 will show that this is in fact one of the

simplest forms in which information about the troposphere can be gather-

ed.

47

It will be also shown that coherence theory provides a

simple justification for multi—slab model presented in section 1.3

and also shows why Fejer and Bramley's models are equivalent. (appendix

A 1 .11)

The definitions of antenna performance (Appendix A1.2) are

usually in terms of the angular power spectrum. Thus, if the coherence

function of a wave emerging from a random medium is known, a simple

Fourier transform will give the angular power spectrum and the perform-

ance of the receiver or transmitter may be suitably derived.

In fact, the coherence theory presented here will be a

simplified form; a more detailed and general form is available

(Papoulls1.12 and a special issue of the IEEE Trans. Antennas and Propag-

ationl'13

)

1.5.1 Mathematical Formulation.

The lateral coherence function (abbreviated to "coherence")

is 60

r (ū,'t) _ ~ E(T,t)•E*(r+ū,t+'~ ) dx dy dz dt 1.5.2

where it is assumed that E(r,t) is spatially and temporally stationary

and

T = r(x,y,z)

Under the statistical conditions assumed, it is valid to write the

self—convolution of expressions such as eqtn. 1.5.2 as an expectation

ie.

)• E*(%1J , t+Z ) dx dy dz dt

< E(r,t)•E*(r+u,t+Z )> 1.5.3

It is usually clear from the context when this is being done in the

analysis which follows.

48

Time is included since the random refractivity is a time

varying quantity and will induce temporal variations in the field.

It is convenient to suppose that the electric field con-

sists of separable random and deterministic parts ie.

E(r,t) = Er(r,t) Ed(r,t) e( ) 1.5.4

where Er is the random part (fluctuating)

Ed is the deterministic part

A is a polarization unit vector.

How the field becomes random will be taken up later. Naturally,

similar equations may derived for the magnetic field.

Appendix A1.8 shows that the average angular power spectrum

of the field described by eqtn. 1.5.4 is

oa A2(k,W). = 1

16 IT ō@~ u,v,'t ), exp (—i(wt—k)(x—kyy))

•dx dy dt 1.5.5

where rx(u,v, ) = rxd(u,v, t) rxr(.u,v, 2 )

r xd = < Ed(x,y,t) Ed(x+u,y+v,t+t )).

r xr — < Er(x,y,t) Er(x+u,y+v,t+t) >

This is the van Cittert—Zernike theorm which shows the relationship

FT 1 (k , W )1 2 r ) 1.5.6

FT where -a. means "are related by a Fourier transform".

Appendix A1.8 also shows that r'(u,'C) is an invariant under free space

(non—random) conditions ie. it is identical for whichever xy plane it is

calculated. This is not the case for propagation through a medium which

is itself random, as will shortly be demonstrated.

1.5.2 The Lateral Coherence Function for a Random Phase Screen.

Following the arrangement in section 1.4, the field

emerging after propagation through an infinitessimally thin phase-

49

changing screen is

E(x,y,t) = Ei(x,y,t) exp(ii(x,y,t))

1.5.7

Ei is the incident field and i is a random function of x,y,t

The coherence function of E is split into deterministic

and random parts. In the analysis which follows, x,y,z subscripts

will be dropped from the coherence functions. The implication is usually

clear enough. Thus the coherence function of the emergent field is

(i (u,v,zo,t) = rd(u,v,zo,'f) h(uvz,t) 1.5.8

zo is a constant denoting the position of the screen. 1-1d is the coherence

function of the incident field and (', applies to the random screen.

If another screen were located further down at z=z1,

with free space intervening, the r of egtn. 1.5.8 would replace ('d and

the emergent coherence function would be

r (u,v,z1, 1 ) = r(u, v , zo ) r'(u,v,z1

This multi—screen approach may be generalized as

1-5-9

Ī'i(u,v,z=z., t) = ri-1(u,v,z=zi-1 , t) P(1) (u,v,z=z., lc)

r. 1 is the coherence function after the ith screen.

Before continuing, some substance will be given to the

Ī'i. function. It will be supposed that[' can be split into the

product of three functions ie.

P c(u,v, 'C) = ru ry i 1 .5.11

If the $ functions are normally distributed and stationary,

r ?s(u,v,t) = exp[.— iBls(0,0,0) — B/(u,v, t )1 ]

1 •5.12

Bi is the phase autocovariance function, which is defined in chapter 2.

The importance of this latter equation is that it incorp-

orates, via Bi, the spatial characteristics of the screen fluctua-

50

tions. The next subsection continues to apply coherence theory to the

previously developed multi—slab model (section 1.3).

1.5.3 The Coherence Function for Multi—slab Propagation.

Subsections 1.3.2 and 1.3.3 established accurate conditions

under which volume irregularities could be condensed into thin "phase

objects". Principally, the requirement was that each phase path through

a slab should contain many inhomogeneities, but that the induced phase

variance should be small. Further, the irregularity scale size should be

large enough that the Fresnel distance became large. This allowed thick

slabs to be used. The above criteria were seen to be satisfied for most

millimetre—wave situations.

A principal drawback of the angular spectrum approach to

the multi—slab model is that at the interface between each slab, the

spectrum has to be inverse transformed to find the field, phase modula-

tion applied, and the field once again transformed to find the spectrum.

This can be partially alleviated by using thick slabs, providing the

above constraints are net violated.,

The coherence theory approach provides a simple method of

dealing with multi'-slab propagation. The propagation path is divided

into many slabs, the only condition being that they are much smaller

than the Fresnel length (eqtn. 1.3.3) associated with the irregularities.

Eqtn. 1.5.10 may now be applied successively, and the emergent field

coherence is

Ī' e(u,v,zo, 15 ) =#1-- r(~)(u,v,z., t) ro(u,v, t ) 1.5.13 i=1

where N is the number of not necessarily identical slabs.

r is the original, unperturbed field coherence.

r (1) is the phase coherence of the ith slab.

51

The solution of eqtn. 1.5.13 may not be simple, but it is

more general than the equivalent results using the angular spectrum

approach. If the statistics for the medium are homogenuous but not

necessarily isotropic, the solution of eqtn. 1.5.13 becomes straight

forward and is derived in Appendix A1.10.' It is assumed that the refrac-

tivity autocovariance is that given by eqtn. 1.3.7. Thus

r(u,v,'~ )\z=z = r(u,v, t ) z_0 exp` —o'yS L

2 +

( o j

v 2 +(t7 • (.11j7/"'Io V O~ I ]

where o'is = Bil(0,0,0)» 1

Ç

a

B~ = k z BN(u,v,w,'C) dw

1-5-14

1 •5.15

=oo The importance of this result is that the cf3 of eqtn. 1.5.15 can be

large: in this case it is essential that it is large for eqtn. 1.5.14

to hold. Atmore general form of eqtn. 1.5.14 is

r (u,v ,t) z=z= r'(u,v, t) z=0 exp —(B (0) — (ū)~ D ~ .1 1.5.16

Here u = ū(u,v,'t )

B;' is calculated from eqtn. 1.6.15

Eqtn. 1.5.14, which applies to propagation through an

extended random medium, predicts that, for example, the temporal

spectrum is broadened by a factor o4 over that of the medium fluctuations.

This can be obtained by applying the van Cittert-Zernike theorem (eqtn.

1.5.6) ie.

P OC exp \' — "CO2 W2 1.5.16

2

where P(w) is the output field-temporal power spectrum.

At this stage, the theory has been carried sufficiently

far to be able to predict some of the important characteristics of

aperture antennas in random propagation conditions.

52

The medium is divided into slabs and the effect of each

slab may be calculated. The criteria for choosing the slab thickness

have been stated. Using an iterative process, the coherence function

modification can be calculated for the field finally output from the

random medium.

Some of the advantages of this model are:-

i) It is fully multiple scatter.

ii) It deals with inhomogeneous propagation paths (by a suitable

choice of smaller, homogeneous slabs).

iii) It treats anisotropic irregularities straight—forwardly.

iv) It may treat extended propagation paths where only scattered energy

is left ie. the deterministic field contribution (in the form of

of the coherent part) has disappeared.

The next section will apply some of the results of the work

presented so far to practical situations. -

1.6 Coupling between Antennas Immersed in Random Media.

In far field conditions, the power coupled between

two aperture antennas may be calculated in a straight—forward manner.

In. fact, eqtn. 1.2.26 is

lim PR = DT(k) AR( k)

krō 0o P 2 T 47T r 0

1.6.1

where the symbols used have been defined.

In the free space situation, the coupling mechanism is

clear ie. the transmitter produces a locally plane wave which is inci-

dent on the receiver aperture.

The effect of a random medium has been shown to be the

production of a reduced plane—wave called the coherent part, plus

53

a narrow spectnum of incoherent plane waves (section 1.4).

A parallel analysis in terms of coherence functions has

shown that the emergent field autocovariance function has been modi-

fied by the medium. A Fourier transform of this modified function

would yield the same angular power spectrum as would be obtained using

the angular spectrum. It should be noted that although the coherence

function method is simpler to implement in general, the phase inform-

ation is lost ie. a power spectrum is produced. The angular spectrum

method (ie. applying successive phase screens to the field distribution

obtained by transformation of the angular spectrum), is used as the

basis of a simulation of propagation in chapter 5.

The effect of the random atmosphere may be characterized

in 3 obvious ways:

i) The medium has reduced the directive gain of the transmitter

only, or

ii) The medium has reduced the effective area of the receiver

only, or

iii) The transducers perform as in free space, but the medium has

an effect which may be characterized by a multplicative

"filter function" in eqtn. 1.6.1.

Physically, approach(iii)is the closest to reality, but the other

two may have their uses. Eqtn. 1.4.9 has shown how (i) might be

implemented, by assuming that the transmitter antenna has a realized

efficiency ~ RR'

In the case df propagation through tenuous random media

(ie. millimetre—waves in the clear atmosphere) all energy is essentially

forward scattered. The total power received by an aperture will thus

be virtually the same as under-non—random conditions. It must be

remembered, however, that the power received consists of coherent

and incoherent contributions.

For communications systems, it is the coherent power

54

which is important. This has been taken into account in the defini-

tion of eqtn. 1.4.8, for example.

The results discussed above naturally assume that the

scale size of the irregularities is much larger than the wavelength

of the e.m. wave in question. This is sometimes a contentious point,

but the discussion in chapter 2 will present the point of view that

in the troposphere, the intensity of the refractivity fluctuations

associated with the small fluctuations is minute compared to that

of the larger scale sizes.

It is evident that the interaction of the coherent and

the incoherent fields leads to fluctuations in the received siga±.

This is taken up in the next section. The reader is also referred to

chapter 4, where the theory presented so far is used in the analysis

of practical measurements taken on a 38 GHz link in an urban environ-

ment.

1.7 Statistics of the Fluctuations.

This section seeks to apply some simple, physical argu-

ments to obtain estimates of the nature of the fluctuations in power

received after propagation through a tenuous random medium. In this

situation, the results of section 1.4 may be applied.

All comments from now on should be seen in the context of

scattering by large (relative to the wavelength)irregularities.

Evidently (eqtn. 1.4.7) the expectation of the angular

spectrum is

<A(k)> = exp(—o' /2) Ao(k) 1.7.1

1.7.1

Ao is the unperturbed value.

The field which. is now propagating in random directions is thus

A — <A) The corresponding,-power.will be scattered over a range

Reduced by scatter '

Scattered field

55

of angles which is determined by

e d, ct d °G A /10 1.7.2

where 10 is a characteristic irregularity scale size. Since 10 is large,

compared to the wavelength, most of the energy is forward scattered.

Thus, if the initial field was a Gaussian distribution, the field at

a plane in the medium might be sketched as in fig. 1.7.1.

Unperturbed

fig. 1.7.1 Fields due to weak scatter.

To estimate the statistics of the field measured at a

point, consider the phasor diagram fig. 1.7.2, which is drawn

relative tothe, phase of the coherent power.

' Resultant random

i

phasor

z Est, exp t- 4121

fig.1.7.2 Summation of coherent and incoherent fields.

56

In general, Einc

has real and imaginary parts that are

jointly Gaussian, independent and of unequal variance. The resultant

of E. and

Ecoh is known as a Hoyt vector, with a complicated

4 distribution. This matter is discussed by Beckmann and Spizzichino

The situation is further complicated when the receiver

aperture is sufficiently large, so that the beam angle becomes smaller

than the scattering angle a d. This means, of course, that the aperture is larger than the medium irregularities. In this case, only-a portion

of the scattered, incoherent power is received, and the signal will

appear to fluctuate less than for a smaller aperture. This is known

as the aperture averaging effect.

The above calculations can be done more exactly. Firstly,

the angular spectrum of the scattered field would be calculated,

using the phase screen model and knowledge of the medium. For a

particular receiver aperture, the received incoherent field could then

be obtained by integrating the product of the incoherent spectrum and

the receiver pattern function over all angles.

It is seen that the detailed statistics of the received

signal is directly dependent on the nature of the medium. A prediction

of the statistical behaviour of the signal depends on an exact speci-

fication of the medium statistics. It could be argued that measurement

of the signal statistics could be used to deduce the medium statistics,

but further work on the uniqueness of the problem would have to be

undertaken.

/

P(x)'1)1) P(r)e3 c)

A

i

57

APPENDICES FOR CHAPTER ONE.

A1.1 Coordinate Systems.

The rectangular coordinate system is constructed for the orthogonal

triad of unit vectors x,y,z. The polar coordinates are oriented as shown

in fig.1.

The interrelationship is trivial and may be derived with the aid of

the figure.

x = r sin8

y = r cos8sin0 al • 1.1

z. -. r cos8cos0

=xx +yy +zz

a1.1.2

A A A A r = sinB x + cos8 sini y + cos8 cost z or X + .y + Q z

and Y2

+ + oct =

APPENDIX A1.2

Definitions of Antenna Performance.

The following set of definitions is in roughly alphabetic order.

) Antenna Efficiency for Aperture Type Antenna. le

maximum effective area of antenna

aperture area

ii)Aperture of Antenna- a

A surface, near or on an antenna, on which it is convenient to make

assumptions regarding field values for the purpose of computing fields at

external points.

iii) Aperture Illumination.

The field over the aperture as described by amplitude, phase'and

polarization distributions.

iv) Aperture Illumination Efficiency. 1

For a plane aperture

directivity

directivity when the illumination is uniform

v) Directive Gain 0(r)

For a given direction r

D(r) = 41c radiation intensity in the direction r total power radiated-by the antenna

vi) Directivity D

The value of the directive gain in the direction of its maximum

value.

vii) Effective Area of an Antenna Ae(r)

In a specified direction r it is A

Ae(r) = Power available at terminal of receiver antenna

Power per. unit-area- of plane`waua incident from diēctioh r'

58

1.2

59

viii) Power Gain g (r).

In a given direction r,

g(r) = 4 Irradiation intensity in direction r

net power accepted by.the antenna from a connected transmitter.

Note that this does not include reflection losses arising from mismatch

of impedance. A very common error is to confuse directive gain with gain.

ix) Radiation Efficiency I R

= total power radiated by antenna

total power accepted by the antenna.

x) Radiation Intensity IR(r)

In a given direction r, the power radiated from an antenna per unit

solid angle.

xi) Realized Gain g

The power gain of an antenna in its environment, reduced by losses

due to mismatch of the antenna input impedance to a specified impedance

and the non plane wave nature of the received radiation. The latter situa-

tion would arise after propagation through a random medium.

xii) Realized Radiation. Efficiency r7, RR The efficiency of an antenna in its environment reduced by all losses

suffered by it, including:

a) ohmic losses

b) mismatch losses

c) feedline losses

d) radome losses

e)random perturbations of the incoming plane wave.

xiii) Realized Directivity 0R

The directivity of an antenna reduced by the non plane—wave nature of

the received radiation.

The latter two definitions are not found in the IEEE Standard, but

are important when considering the performance of a receiving antenna

when the incoming plane wave has been distorted by the addition of a

radome, or propagation through a random medium,

1R

60

APPENDIX A1.3 Antenna/plane—wave Coupling.

Fig. Geometric arrangement to calculate coupling.

Lorentz's Lemma states that for the arrangement above,

(Er xTh - EiXHr). n ds = 0 a1.3.1

The r subscripts refer to fields radiating from the antenna, and the i

subscripts to the incident plane wave. The surface S is arbitrary,

but a sphere of radius R is convenient. It must of course be a closed

surface and contain no sources.

A screened waveguide feeder and transmitter deliver power

to the reference plane. The match is perfect and all the power Po

propagates in a single, lossless mode to the right of the reference

plane.

The radiated field is thus

= a1(waveguide field pattern)

a1 F2 (x dx dy a1.3.5

2Z 2

61

or Er =.a1 (Fg(x ~ Y))•• a1.3.2

The reference plane is chosen such that a is a real constant.

On reception, since only one mode is supported, the field

in the guide is •

= a2 g(x,y) al • 3.3

The magnetic field may be calculated from the appropriate Maxwell

equation.

The power fed by the perfectly matched transmitter is Po,

and so, evaluating the Poynting vector for the waveguide,

Z is the guide impedance.

The contribution to a1.3.1 over the guide reference plane

is thus 4 Po a2/a1.

The only other important contribution to a1.3.1 is that

over the surface of the sphere S. The radius R will be chosen large

enough such that the far field result eqtn. 1.2.12 hold. The transmitted

field .on this surface becomes

Er(r) = i2TT() exp(—ikr) a1.3.6 kr

The -incoming field at the surface, referenced to 0, is

A Ei(r) = ei exp(ikr k•r)

• a1.3.7

a1.3.7 and a1.3.6 are now substituted into a1.3.1. A complex integra-

tion results'wliich may be simplified by stationary phase to yield the

contribution due to the spherical surface as

al •3•8

Z___ 0

This means the contribution is essentially due to the component of the

angular spectrum collinear with the incoming plane wave direction,

i 2112 ēi • A(kn

)

a1

a2 = A -ei • A(k) i2Z P 0 0

c!1 •3• g

62

which is physically plausible.

Hence, a1.2.1 becomes

This is the plane wave to antenna coupling equation.

If the antenna and field are polarization matched, then

A(k))2 = 17.12 \A(

0) 2

Ae (k) = x2

1 2 D(k) 4W

4T' P. 21T A2 )A(k)J Z

a1.4.5

63

APPENDIX A1.4 The Effective Area of an Antenna.

The same geometry as fig. 1 of A1.3 is considered. The

power density of the incident plane wave is

= 1

n .0)1 = Ie~l 2

2Z 0

al•4.1

The power delivered to the left of the reference plane in the feed

waveguide is given by a1.3.9.

Pr (k ) = (a2)2 pi

a1

A 4 Pi A(k))2

4Z2 P2 o

al-4•2

a1-4-3

A straight application of vii of A1.2 is possible. The effective

area is

Ae(k) = Prec(k) — A4 I ei12(012 2Zo

4Z2 P. eij2

o i

4 1—A0012

2Z P. o 1 This latter eqtn. may be written

al•4.4

according to eqtn. 1.2.23 which defines the directive gain.

64

APPENDIX A1.5 Coupling between Antennas.

The coupling between an antenna and a plane wave (a1-3)

provides the basis for this analysis. The geometry is shown..in fig. 1.

fig.1 Power coupled between two antennas.

The aperture Tx gives rise to an angular spectrum of

plane waves at To in the region to the right of the transmitter aper-

ture. This spectrum is given by eqtn. 1.2.8. One of these waves is

dT(ro) = AT(k) exp(—i k•r) dkx dky a1.5.1

AT is the Tx angular spectrum.

Using the coupling equation 1.2.21, the field coupled into the

receiver feed guide is

dc = d a2 = h2 dE(ro) • AR(—k) a1.5.2 i2Z PT

AR is the receiver angular spectrum

PT is the radiated power.

The total field coupled may be obtained by'integrating a1.5.2 over all

wavenumbers. Ih the far field this integral may be solved by stationary

phase

lim c = A2 exp(—ikro) AT • AR a1.5.3

kro eo ZPTkro

65

If there is polarization match, AT •

is = A

T ARand the power coupled

lim c2 = %` 6

( kro-9.60 4Tr2 Z2 PT o ro

R)2

= 2 1T #\ 2 AT Ā AR

Z PT 2 Z PT 41r2

DT(k) A R(_k)

4 W r2

4

a1.5.4

In principle, of course, it is not necessary to go to the far field;

the power coupled may be calculated for any separation once the required

integration of a1.5.•2 has been done. The far field is simple, since

the radiator is producing locally plane waves which can be used in a

simple way via the aperture/plane—wave coupling result.

66

APPENDIX A1.6 The Gaussian Antenna.

In this analysis, the Gaussian antenna will have an

aperture field of-the form

E(x,Y) = Eo exp — x2 + y2

x 2 0

A x al•6.1

This is polarized in the x direction with a uniform phase distribution

over the aperture. For simplicity, a circularly symmetric form has been

chosen (ie. the "waist" is x in both the x and y directions.

Using eqtns 1.2.11 and 1.2.5 to transform a1.6.1

m

Ax(k) = 417 2

EO exp — x2+y2 exp i(kxx+kyY)J

x -co 0

= Eo x2 exp — x2(k2+k2 )

a1.6.2 .1

41T C %+

Ay(k) = 0 a1.6.3

Az(k) = —kx Ax(k) a1.6.4

k

The angular spectrum is now known on the z=0 plane. The field due to this

antenna at a distant point r (distant meaning kr >71 ) is obtained from

the far field result eqtn. 1.2.12

lim E(r) = i 2W kzo A(ko) exp[ ikr] a1-6.5

kr 4,60 r A

Here ko = kr = k (sin B x + cos8 sino y + cos 6 cos~ z

Hence,

E(T) = ikx2 case cost!) exp ikrr •

Zr exp[ k2xn (cos 8 sink) 2 + sin2 e I •

4

L x - sine z a1.6.6

It cas be seen that this is a transverse em field perpendicular to r.

Letting r = z, a1.6.6 may be written, taking magnitudes

dx dy

67

rE(F)1 IT'x (on axis only) a1-6-7

Clearly the field decays as 1/r and is proportional to the "area" (x2)

of the aperture, as expected.

Now consider the geometric arrangement of fig.1

t=0

fig.1 Geometry for finding field on a parallel plane.

The Gaussian antenna on the z=0 plane radiates into the

postive z halfspace. Suppose the field distribution over a plane at

z=zo is required. Each component of the spectrum A(k) (eqtns a1.6.2 etc.)

undergoes a phase shift exp[—i k • rlie.

A(k) z=z = expj ik•r 1 A(k)I z=0 a1.6.8

I o L

As a simplification, let 8 , C' <1/6 rad a1.6.9

Now r = f u2 + v2 + zō'2 a1.6-10

Thus i k•r ti ikzi1 + k2 + ky o

2k2.

ikzo + izo(k2+k2) a1•6•11

2 k

where a1.6.9 has been used to simplify. a1.6.8 may now be written:

A(k) 1

z=z = expj—ikzo exp (—iz °(k2+ L L 2k

Eqtn. a1.6.2 now becomes

A(k) = E°x2 exp —ikz° exp —w2(zo)(kX+ky)1 ~ 1 C 4

4

a1-6•12

a1.6-13

E x(u,v,z0 ) i1Tx0 exp{—ikz0J exp\—ik" u2+v2 L 2z0 )z

0

68

where w2(zo) =" xo — i2z o

/k al •6.14

Eqtn. 1.2.8 allows the field on the ku,v) plane to be calculated as 00

Ex(u,v,zo) = Ax(k)1 o LL exi(ukx + vky )J dkX dky

``

= exppkzoJ E xō 0 exp — u2+v2 L w2(zo) w2(z0)

al •6.15

Now 1/w2(z0) = k(kx2 + i2z)

k2 4 x0 + 4z2

This may be split into real and imaginary parts:

1/w2(zo) = 1/xō(zo) + i/R2(zo) al •6.16

with R2(zo) = k2x4 + 4zō

02kz

a1.6.18

x2(z0) = k2xo + 4zo

k2 x2 0

The field on the (u,v) plane is thus

Ex(u,v,zo) = Eo x2 o

exp( —ikzo

wz(Eo) L

a1.6.1B

exp —i(u2+v2) exp — u2+v2

R2(zo) x2(zo)

a1.6.19

Whenz0 ~kx2, the following limits hold

2 (R z0) •--~• 2z0/k

x0(zo) ---> 2z/kxo

x2/w2(zo) ikx2 o /2z0

al•6.20

Thus, for z » kx2, the x component of the field is

exp — T~x0(u2+v2 )

X2 z2

al •6.21

=~Lp

fig.2 Diffraction of a Gaussian aperture field.

69

This is recognizable as a spherical wave, centre of

phase (0,0,0), with a waist w(zo) which increases linearly with zo.

This wave is sketched in fig.2.

The derivation of the z

complex, but it may be shown, using

2 Ez(u,v,zo) = 2Eoxo

component of the

eqtn. a1.6.4 as

field is

a basis,

—u2+v2

more

that.

i īiu+1 exp

J kw3(z ) 0

w(zo

) w2((.,i

a1•6-22

which vanishes for u=0 and zo >~ 1. In the far field, a1.6.19 and a1.6622

are in agreement with the far field result derived in a different manner

in a1.6.6.

It is now possible to measure the performance of the

Gaussian antenna in terms of the definitions of appendix A1.2. For

simplicity, far field forms will be given. Further, it will be assumed

that the antennas will be large when measured in wavelengths, which in

turn means that they will have narrow beamwidths. This will allow small

angle approximations to be used (a1.6.9).

The directive gain is, by egtn.1.2.4

D(E3, 1) = 2 tr A2 A(e ,4 )12 ZP ` t 0

A2x4 (1 -0)2 exp —2 Ti2x2 (92+ 1)2 ) 0 8TfZ X2 a1.6.23

70

where the power radiated has been normalized to unity.

The directivity is the maximum value of this function ie.

D =)xō/B1TZ a1.6.24

Egtn.1.2.24may be used to find the effective area of the Gaussian

antenna used for reception, by substituting from a1.6.23. Similarly,

the power coupled between two Gaussian apertures may be calculated

using eqtn. 1.2.27 with a1.6.23,24.

71

APPENDIX A1.7 The Characteristic Function.

Suppose / is a random variate with probability density

p(f)= expl

al•7.1

2 TTFs

M 2cr1

The function

Gts(u) D < exp(iiu) > =

ao exp(i/u) p(/) d/

-b0 al•7.2

is known as the characteristic function. When p(i) has the form a1.7.1

(normal) then

2 Gis(u) = exp(-2ois u2 )

~A more comprehensive account is given by Seckmann .

al•7.3

The expectation of the power spectrum of a1.8.1 is 00

,Ax(k,W)1 2> 1

16 1T4 \\\\\ <E*(xYt) Ex(x',Y',t')> r

A change of variables is made ie.

x' = x + u

Y' =Y +v

t' = t +

Substituting this change into a1.8.2 oo

< lAx(k,U..))\

2> 4 cccU

16 TV —bo

exp[ —i(W (t—t')—kx(x—x')—ky(Y—Y')1

L dx dy dt dx' dy' dt' a1.8.2

al•8.3

E*(x,y,t) E(x+u,y+v,t+'t)>

72

APPENDIX A1.8 The van Cittert—Zernike Theorem.

The electric field over a plane is assumed to consist

of both deterministic and random parts, as, for example, in eqtn.

1.5.4. The angular spectrum may be calculated using eqtn. 1.2.11.

Ax(k,W) = exptikzzo, \E(xYt) exp i(wt—kxx—kyY)J

4T dx dy dt al • 8.1 —60

expr —i(cu -kxu—kyv)1 dx dy dt du dv d.'t

L al •8.4

By definition,

rx (u,v,t ) x,y,t) E,,(x+u,y+v,t+'C ))).dx dy dt

al•8.5

Thus a1 •8.4 is co

Ax(k'w ) I 2`..-

—16 'T1

[ l x(u,v,T

) eXpt i(w1—kxu—kyv)1•

du dv dt

al .5•6

73

This latter equation shows the formal relationship

41,1Ax(k,W)I 2> < F.T. > r x(u,v,'r, ) al-8-7

which is the required result.

It should be noted that in making the step from ..a1.8.1 to

a1.8.2, the conjugation process removes any z dependence from the power

spectrum. This means that in free space conditions, the coherence

function independent of z and is an invariant. This result may seem

surprising, since it is known that the form of the electric field is

changing with distance from an aperture. If, however, the auto—

covariance is calculated for a number of successive planes, it will be

seen to be constant.

74

APPENDIX A1.9 Equivalent Planar Statistics of a Slab.

Since the slab is being treated as a phase object, the

phase distribution of the field just emerging from the slab will now

be derived by considering many paths parallel to AB of fig. 1.5.3.

Suppose that the refractivity autocovariance is BN(u,v,w,'t )

and that the axes are oriented as in fig. 1.5.3. Then

j AL

y(x,y,t) = 10-6 k N(x,y,z,t) dz

0 where n is the refractivity.

The planar phase autocovariance is thus

a1.9.1

)\z=LSL < 6(x,y,t) j6(x+u,y+v,t+'t) > a1.9.2

Using a transform of the domain of integration of the resulting double

integral (Papoulin .5 ), having substituted from a1.9.1,

L1L. 2

't )1Z= AL = 10-12 5N(u,v,w,1:) dw L1L'

al-9.3 It is not necessary to assume the phase process is normal. The Central

Limit theorem (Beckmann5 5) may also be used to show that ) will tend

to normality when the path such as AB intersects enough independent

refractive inhomogeneities. The following argu anent will make this a

little clearer: let wo be a typical scale size measured in the z

direction. The number of irregularities intersected by a path such as

AB is

n a1.9-4

w 0

If wo 4< AL, n will be large and the Central-Limit theorem may be

applied. Further, if wo« AL, then a1.9.3 may be written as

Bi (u'v'% )1z= AL = 1 0

-12 k2 L BN(u,'~) dw a1.9.5

_m

A further constraint used to derive a1.9.3 is that

(uv

75

814(0,0,0) «. 1 al•9.6

In practice a compromise must be made in choosing a slab thickness AL

which allows n (a1.9.4) to be large and yet keeping the phase variance

small. When the refractivity variance is very small, care must be

taken not to violate eqtn. 1.5.3 ie.focussing effects become important.

For the phase autocovariance given by eqtn. 1.5.7, it is

2 interesting to calculatean estimatefor Qi = 8(0,0,0) .

The phase variance is

2 o•PS ^' n 10-12 2 2

wo a- al•9.7

where n is given by a1.9.4. Hence

o- ^, 10-12 k2 aLwoc-N2 al•9.8

which agrees within an order of magnitude with the rigorous result

given by eqtn. 1.5.8.

76

APPENDIX A1.10 A Solution for Many Slabs.

The stationary but not necessarily isotropic medium is

resolved into n slabs of width 4.L. The slab thickness has been

chosen carefully such that

uō , vo ~> AL

X a = Bi(0,0,0) C: 1 rad

Z

a1 •10.1

al •10.2

ie. the energy is essentially forward scattered and the induced vari-

ance per slab is small. If the medium is homogeneous, then eqtn.

1.5.13 may be written as

r(u,v,'t) z=z = (r (u,v, t))n r (u,v,'t =0 a1.10.3

where n = zo/AL

Provided the slab thickness is such that

OL 1 w

>>_ 0

the phase process for each slab becomes normal and eqtn. 1.5.16 applies

(Ī's(u,u., ))n = expr , .t —n(Bis(0,0,0)—yu,v,I )1

L

= exp —8(0,D,O)—B (u,v,'C )J a1 •10.5

where 60

10-12 zo BN(u,v,w, t) dw

—o0

al •1Q•6

since n L1L = z_.

77

APPENDIX A1•11 The Equivalence of the Fejer and Bramley Models.

The fact that Fejer's multi—slab model yields identical

results to merely tracing the phase through the whole medium is

physically puzzling. An explanation is fairly simple when the sit-

uation is viewed in terms of the degredation of coherence. This has

been shown by Clarke1•10, 1.11•

Fejer calculated the angular spectrum after each slab

and allowed each spectrum to be perturbed again before entering the

next slab, and so on through the medium. Thisiled to a differential

equation for the angular spectrum, which could be solved to find the

final spectrum emerging form the medium.

Clearly,, at,the exit plane of each slab the van Cittert-

Zernike theorem could have been applied to find the coherence function,

and the problem solved in the dual space.

This is exactly the process which has been done in the

previous appendix (A1..10). Eqtn. a1.10.6 gives the autocovariance of

the phase screen which is equivalent to the whole medium. A more

complex result is possible if the medium is not homogenuous. Thus it

is seen that it is valid to consider the whole medium as one slab, as

Bramley did.

The physical reason for the, result depends on the fact

that the inhomogeneities are large compared to the wavelength, which is

in turn the basis of the slab model. This was discussed in subsection

1.3.2.

78

CHAPTER ONE REFERENCES.

1.1 Heading, J. 1962. "An Introduction to Phase—Integral

Methods". Methuen and Co., London. p141.

1.2 Antenna Standards Committee of the IEEE Antennas and

Propagation Group, "IEEE Standard Definitions of Terms

for Antennas", IEEE Standard 145-1973.

1.3 Hill, J.E. 1976. "Gain of Directional Antennas",

Watkins—Johnson Co. Tech—Notes, Vol.3, No.4, July/

August 1976.

1.4 Brown, J. 1958. "A Generalized Form of the Aerial Recip-

rocity Theorem", Proc. ICE, Part C, 105, pp472-475.

1.5 Brown, -J. 1958. "A Theoretical Analysis of Some Errors

in Aerial Measurements", Proc. IEE, Part C, 105,

pp 343-351.

1.6 Paris, D.T., Leach,W.M. and Joy, E.B. 1978. "Basic

Theory of Probe—Compensated Near—Field Measurements"

IEEE Trans. Antennas and Propagat. Vol. AP-26, No.3,

May 1978, pp373-378.

1.7 Lorrain, P and Corson D. 1970. "Electromagnetic Fields..

and Waves" W.H. Freeman and Co., San Francisco.

1.8 de Wolf, D.A. 1974. "Waves in Turbulent Air: A

Phenomenological Model" Proc. IEEE, Vol.62, Nov. 1974.

1.9 Zernike, F. 1935. "Das Phasenkontrastverfahren bei der

Mikroskopischen Beobachtung" Z. Tech. Phys, 16, p454.

1.10 Clarke, R.H. 1974. "Sound Propagation in a Variable

Ocean" J. Sound and Vibaration, 34(4), pp457-477.

1.11 Clarke, R.H. 1973. "Theory of Acoustic Propagation in

a Variable Ocean" NATO SACLANTCEN Memorandum SM-28,

79

October 1973. SACLANT ASW Research Centre, Viale

San Bartolomeo 400, I-19026—La Spezia, Italy.

1.12 Papoulis, A "Optical Systems and Transforms"

__McGraw—Hill Book Company.

1.13 IEEE Trans. Antennas and Propagat. Vol. AP-15, January

1967.

1.14

Beckmann, P and Spizzichino, A 1963 "The Scattering

of Electromagnetic Waves from Rough Surfaces."

Pergamon Press, London.

1.15 Papoulis, A 1965 "Probability, Random Variables and

Stochastic Processes." McGraw—Hill Book Co.

80

CHAPTER TWO. THE PHYSICS OF THE LOWER TROPOSPHERE.

The previous chapter has developed a theoretical basis

for the investigation of wave propagation through a medium which intro-

duces random spatial and temporal phase perturbations in the propagating

wave. This chapter will attempt to analyse how these phase perturbations

might arise.

The phase perturbations will be assumed to be due to

refractive index fluctuations, in time and space, about a mean value.

The chapter is thus an investigation of the spatial and temporal fluc-

tuations of the radio refractive index in the lower Troposphere.

It must be pointed out that the physics of the lower Tropo-

sphere is complex, partially unresolved and posseses a vast literature.

This chapter must, therfore, present a somewhat simplistic overview.

To do this, only simple and physically reasonable models will be presen-

ted. For a comprehensive treatment, Kerr2,1, Bean and Dutton2.2 are

good sources of discussion on the implications of t1eteorolgy for Radio

Propagation. Scorer 3 is a useful description of the physics of the lower

Troposphere, but not directly from the Propagation point of view.

Because the formalism of Chapter 1 requires it, and

because it is mathematically simpler, the description of the atmospheric

variables is statistical. Formal definitions of the commonly used

statistical.. functions are given in section 2.2.

The time scale of the atmospheric refractivity fluctuations

varies from fractions of a second to hours for large frontal activity.

Although all fluctuations are ultimately important from the systems point

of view, this chapter concentrates on short time—scale or dynamic processes.

However, for completeness' sake, an appendix is included which discusses

the importance of precipitation and longer time—scale fluctuations on

system performance.

81

2.1 The Radio Refractive Index: Refractivity.

The radio refractive index, n, at a point r in a medium

is

n(T,t) = co/c(r,t) 2.1.1

where c is the em wave speed at r, time t

co is the free space wave speed.

The assumption is that the medium is isotropic and the wave velocity is

a scalar quantity, independent of the wave polarization.

Generally co is replaced with <c>, the average for the

medium. This is discussed in section 2.2.

Real perturbations of the refractive index are minute in

the clear troposphere, and so, following normal practice, the

refractivity N is defined as:

N(r,t) = (n(r,t)-1) 106 2.1.2

where both are measured at a point T, time t.

Experimentally it has been shown that (Bean and Dutton )

N(r,t) = 77,6 P(r,'t) + 4810 eC,t) 2.1 •3 T(r,t) T(r,t)

T is the absolute temperature in kelvins

P is the dry air pressure in millibars (mb)

e is the partial pressure of water vapour (mb).

It has been shown also, that the refractivities of water vapour, dry

air and oxygen have less variablility with frequency in the range 9 to

72 GHz than the variations in the measured formulae. Hence egtn.2.1.3

may be treated as non—dispersive over this frequency band.

Egtn.2.1.3 may be partially differentiated to find the

refractivity variations with respect to the variables T,P,e. (In

what follows, the functional T,t dependence will be dropped, but is

implicit.)

82

Differentiating partially,

= a QT + bae + cAP 2.1.4

At sea level (N=319 T=15C P=1013mb e=10,2mb ie. 60% relative humid-

ity) the constants are:

a = —1,27 K

b = 4,50 mb-1

c = 0.27 mb-1.

Egtn.2.1.3 is useful as a rough guide when evaluating the possible

importance of T,P or e fluctuations on the refractivity. Gossard2 4 has

developed tables of a,b,c and related quantities for various altitudes and

climatic regions.

Workers dealing with optical propagation have carried out

similar analyses; it is important to remember the optical and radio

refractivities are different, Generally, optical waves are less in—

fluenced,by water vapour than are radio waves.

Egtn.2.1.4 indicates that temperature and humidity offer

the greatest potential for causing refractivity fluctuations, when likely

values for T, e, P are considered. This matter is taken up in some

detail in sections 2.5 and 2.6.

The next section will discuss the statistical functions

which may be used to describe the random refractivity of the Troposphere.

2.2 Statistical Description of the Random Refractivity.

This section develops most of the formal definitions

required for the statistical description of random refractivity fields.

Useful references are: Papoulis2.18

, Bendat2•19, Tatarskii2.20,

Beckmann2.21, Ilelsa

2.22

The definitions will be in terms of the refractivity N,

with an implicit r,t dependence. Similar expressions exist, naturally,

83

for wind speed, ,temperature and so on.

2.2.1 Basic Definitions.

The refractivity measured at a point in time and space

may be thought of as the resultant of an infinitely large number of

random functions, differing in detailed structure, but having the

same probability density function. The set of functions is called the

"ensemble".

The "ensemble average" or "expectation" is formed by

.taking the average of all members of the ensemble:. o

< N(r,t)) = E[N(r,t)J = LP(N) N dN where P(N) is the probability density function.

2.2.1

E is the "expectation operator".

Ensemble averages may betaken of any—function of N, eg.:

< N(r,t) N(r+ū,t+'G )»

In practice, it is not usually possible to calculate <N>

in the manner defined above. Consider the time dependence of N observed

at a point. Fig.2.2.1a shows that strictly the infinite sum or random

functions should be used to measure <N> .

Usually, however, the scheme shown in fig.2.2.1b is

adopted.. Time averages are calculated. It is assumed that if the process

is sampled long enough, then all members of the ensemble will be encoun-

tered. This is the ergodic assumption, and assumes that the function

in stationary ie. its statistical properties do not vary with time.

Further useful functions are:

The ensemble variance

o-N: = ( ( N(r, t) — <N> )2 >

The autocovariance

)),711 iN(12,t2)-4N72, )> }

2.2.2

84

(a) Infinite sum - of random functions.

(b) Sampling a random function.

Fig.2.2.1 Random functions.

85

or B(u,T ) = < (N— < N1> )(N2— 4N2) ))

where ū = 1-72

'c t 1_t2

2.2.3

and it is again assumed that the process is stationary. The geometry

of egtn.2.2.3 is shown in fig.2.2.2. It will be recalled that this last

equation is of great importance for determining the equivalent phase

screen which may replace the effect of a random medium, as in section 1.3.

2.2.2 Stationarity.

Before defining -stationarity, it is important to clarify

the experimental technique which is usually used to estimate the

ensemble average, variance etc. The refractivity, for example, is

sampled for a certain period of time; during this time,. samples are

taken at fixed time intervals. The sample average is estimated by

n 4 N) lim 1 'L Ni 2.2.4

AT n —j oc n i=1

where dT is the interval over which samples were taken.

If the ensemble averages and variances estimated in this way do not

change as the calculation is repeated over successive time intervals,

the refractivity N is said to be weakly stationary. An example of a

practical measurement is contained in section 2.5.5.

In practice, it is found that the ensemble average is

usually the first to show slight variations. This prompted the concept

of structure functions, as will now be discussed.

2.2.3 Structure Functions.

Some thought will show that the following function will

be invariant with linear changes in the average value

2 < 1N(r1,t1)-N(r2,t2), 2> 2'2.5

A

r ~ I I

- - -p'

_ v

A

X.

86

Fig.2.2.2 Samples taken at two points in space.

Fig.2.2.3 Fluctuation wavenumber spectrum.

u2 ,t2 0 0

BN(u,-C) = QN exp —ū•ū —T2 2.2.8

87

For a weakly stationary process

D (u,T ) 2 n 1-,N ( ,, t ) 1 PN = 13N j 0N 2.2.6

The structure function is used 'extensively in chapter 4, where practical

measurements of the phase difference measured on a 38 GHz experimental

path are presented.

2.2.4 Refractivity Wavenumber Spectrum.

A temporal function may be thought of, via the Fourier

transform, as a spectrum of sinusoidal waves. A generalization is to

consider spatial fluctuations as being constituted from a spectrum of

sinusoidal waves, with varying wavenumber.

A wavenumber spectrum may be defined as co

i(17,00) = 1 ÎÇBN(, ) expl i(w.t —K•ū)] du di;-

(2~)4 `~ L 2.2.7

where K zz

In most cases the exact form of ii(K,U)) is unknown.

approximate form is shown in fig.2.2.3. (K,w) is a measure of the

"intensity" associated with scale sizes 1, where 1=211r/IRI.

Fig.2.2.3 indicates that there is a large amount of energy

associated with large scale sizes (small K) structures.. These large

structures in turn cause smaller structures, and so on. This will be

examined in some detail in section 2.3. Before returning to 'this topic,

some more detail on the autocovariance function BN is desirable.

2.2.5 Forms of the Autocovariance Function.

Some commonly used forms are:—

BN(L,Pr ) = o-N exp —Iu~ - t uo to

2.2.9

B(ū) = 2 N

BB

(1E1)cl K 1' q

uo uo

2.2.10

2q q-1

where Kq is a Bessel function of the second kind with imaginary argu-

ment. The latter form is due to von Karman and is proposed for turbu-

lent media. The Gaussian form (egtn.2.2•8) is convenient because of

its mathematical tractability.

The work of Comstock 2. 2 .... is worth mentioning at this junc-

ture, since he shows that most forms of autocovariance function may be

obtained by combining the Gaussian form, with a probability function

associated with the uo ,1 o values.

•

As an example, suppose a process produces a Gaussian

autocovariance function, with correlation length uo.

BN(u) = expr—u2/uō I 2.2.11

If a number of such processes exist, each with a different

characteristic size uo, it is possible to find the composite autoco.-

variance function, provided the probability function from which the uo

values are drawn, is known.

Suppose

p(u0) = g(uo) expr u%um ] 2.2.12

then m

8N(u) = ,.p(u0) BN(uluo) duo

If g(uo) is constant, it can be shown that

2.2.14

BN(u) = exp— ~u(~u 2.2.15.

um is the variance of the scale sizes. With g(uo)=const., the scale

sizes are drawn from a Gaussian distribution with variance zum. Thus the

summation of random processes with Gaussian autocovariance and whose

correlation lengths are normally distributed, produces an exponential

form for the autocovariance function.

Similarly, if g(uo) = 2uolum, p(uo) is Rayleigh distrib-

89

uted and the resultant autocovariance is

BN(u) = H1 {ul

1

(1-1:1-1

um um

2.2.16

Here H1 is a Henkel function, order 1, with imaginary argu rnent.

In a second paper2 26 Comstock goes on to show that 2.2.16

is a special case of a family of autocovariances

BN (u)

(m?

(u) = H j u

2 1 um

For k an. odd integer ks-1

Hk (u ) e_u/um ( bi i=0

2.2.17

2.2.18

In section 2.5 some practical measurements will be shown which, support

these forms of the autocovariance function.

2.3 Elements of Turbulent Flow.

To fully describe the velocity field in the case of turbu-

lent flow over arbitrarily shaped surfaces, the Navier—Stokes equation

has to be solved. This is extremely difficult, except for a few ideal-

ized situations.

However, dimensional arguments allow approximate solutions

to many problems. The work of Kolmogorov in applying such arguments

to random flow in the atmosphere is described by Tatarskii2 20 .

Similar summaries are given by Strobehn 2.28 and Fante

2.27.

This section only quotes the results of this work and

develops some of the points necessary for some measurements presented in

section 2.5.

2.3.1 Spectral Decomposition and Taylor's Hypothesis.

In section 2.2, the refractivity fluctuations were described

by the autocovariance function BN(ū,'C ). Also introduced was its trans-

90

form , (1‹ U)). Associated with the wavenumbers 1171 were scale sizes

which characterized fluctuations with different spatial extents.

A rough functional form for l was introduced (fig.2.2.3),

is a measure of. the strength associated with the fluctuation wave--

numbers. Energy was input at large scale sizes and is transferred down-

wards in scale size, until_at very small scale sizes the energy is

removed by dissipation.

Two scale sizes, Lo and 10, the inner and outer limits

are shown in fig.2.2.3. Between K0 (=21T%L0) and Km (=2n /10) the

energy flow is lossless. This is known as the inertial subrange.

In fact, the energy transfer is bidirectional, but because of the

dissipation at the small scale sizes, the nett flow is downwards in

scale size.

The form of the spectrum is reasonably well known in the

inertial region. Kolmogorov used essentially dimensional arguments to

derive the shape of the spectrum-in this region. This work is summarized

by Tatarskii2 20. Experimental results show reasonable agreement with

the form of the spectrum predicted by Kolmogorov ie.

1(k) oC K-11/3 Km< K < K0 2.4.1

The "input region" (0 < K < Ko) is difficult to quantify.

Probably the most difficult problem is to assign 'a value for K 04.

This region is associated with the input of energy to the fluctuation

processes. It is certain that the value of Ko will vary considerably

with ambient conditions such as surface roughness, surface temperature,

path length , air temperature and so on.

On a warm day, when significant convective'updraughts are

established, . K0 is probably small (ie. L~ is of the order of hundreds

of metres). A cold day with strong wind speed is, however, probably

associated with much larger values for K0 (ie. L0''10m or less). The

assignment of KO values is investigated in some detail in section 2.5,

91

where probabilities are assigned to the presence of a particular scale

size.

The dissipative region is also a difficult theoretical

problem, and even more so for practice. Generally 10 is thought to be

between 1 and 10mm. For millimetre—wave propagation, this problem is

not of great importance, since the strength associated with these small

scale sizes is minute compared to the structures of the order of metres

in size.

Before examining some specific forms for f, it is worth

considering Taylor's "frozen turbulence" hypothesis.

Suppose that a turbulent medium has a mean flow velocity. If

the temporal spectrum of the turbulent flow contains only low frequency

components, the time taken for a new realization of the field to evolve

may be longer than the time taken for the mean velocity to sweep a

complete structure past a single sampling device. This means the man-

velocity carries a "frozen replica" of the turbulent field past a fixed

point. In general, smaller scale sizes are associated with the higher

frequency components of the frequency spectrum. Since these smaller

scale sizes are minute in intensity, millimetre—waves propagating

through the medium are not likely to be influenced by any failure of

Taylor's hypothesis. It wo 4ld'seem that Taylor's hypothesis may be used

in describing the troposphere fairly accurately as far as millimetre—

waves, are concerned.

Kolmogorov derived a relationship for the structure function

for irregularities in the inertial range:

DN(u) = CN u2/3

DN is the refractivity structure function

u=1ū1

CN is the refractivity structure constant.

2,4.2

92

Expressions fort N are quoted (Strobehn2.28) which are

extremely complex and have not been totally confirmed by practical

measurements. For example, Strobehri2.28 gives

N(~) 4C exp_ K2/Km~ (1 + K2L2)11/6

where Km = 5,92/10

oC = o2 L3 r(1 /66 ) C(KmLo)o

1ī r(1/3)

C(KmLo) = 1 + r(11/6) r(-1/3) r(1/3) r(3/2)

2.3.3

KmL0)-2/3 for K L )i1

m. o

Normalization of 2.3.5 requires

CN _ 1,9 cN Lo2%3 2.3.4

The spectra above refer to the 3—D forms. Tatarskii 2-20

explains the method of obtaining 2-0 and 1—D forms. The 2-0 spectrum

in the inertial subrange is

(K)eC.K 5/3 2.3•,5

2.4 Turbulence in the Lower Troposphere.

"Turbulence" is used loosely by radio scientists to denote

what is probably better called "fossil turbulence". Tdrbulent flow is.

per se associated with minute temperature and pressure fluctuations,

which are entirely negligible for millimetre—wave propagation. However,

as will be shown, the flow is responsible for the patchy distribution

of parcels of humid, warm or cold, air. These parcels or inhomo-

geneities are responsible for random perturbations of radio waves

propagating through the troposphere.

The problem is thus to isolate processes which might cause,

say, an excess of humidity and then, how this excess is likely to be

redistributed. Often the spatial and temporal characteristics of such

parcels can be determined from the dynamic process responsible for

93

redistribution. This section will examine the troposphere and attempt

to outline some of the more important processes for causing fluctuations.

2.4.1 Energy Sources.

The previous section indicated that energy is input from

large scale processes and transfers down in scale size until dissipation

removes it as heat.

On a global basis, due to differences in surface thermal

conductivity, temperature gradients exist in the atmosphere. These

temperature differences lead to pressure gradients, which in turn lead

to the nett displacement of parcels of air. These pressure/temperature

differences extend from tens of--metres tō many thousands of km in extent.

Once air movement has been established, the transport of

water vapour etc. may take place. The formation of clouds results,

which' leads to further temperature changes of the screened surface,

and so on.

The subject of planetary atmospheric dynamics is obviously

complex, and is making great strides since the advent of satellite

photography and radiometer measurements.

It is necessary to confine the discussion to those concepts

and. structures which are likely to perturb the relatively short propagation

paths envisaged for millimetre—wave systems. The descriptions below

follow the work of Scorer2-3'2.5 ~

The principal reason that planned use of millimetre—wave

systems is over short distances is due to precipitation. This is taken

up in the appendix to this chapter.

2.4.2 Static Stability.

Consider a parcel of air at a temperature T which is sudden-

ly moved upwards. Because of the normal upward pressure gradient,

94

unless the air above is much cooler, the parcel will tend to return

to its previous level underthe action .of hydrostatic pressure.

The sun inputs ejm energy between 0,3)Am and 4/m. The

earth radiates as a black body between 3 and 100 pm , with a mean at

10/u m. As a consequence, at night the ground is generally cooler than

the air above it. The air becomes stably stratified. In the next sec—

Dv i

tion it is shown that this stability is disturbed by wind shear, ,

v being the wind speed.

Locally warm surfaces give rise to convective updraughts

(section 2.4.6). These updraughts also. lead to perturbations of the

static stability.

2.4.3 Mixing Processes.

Section 2.1 showed the importance of temperature and humid-

ity fluctuations for causing changes in radio refractivity. Two generic

names are given to processes which disperse such inhomogeneities ie.

i)"natural" viscosity, conductivity and diffusion

ii)"eddy" viscosity, conductivity and diffusion.

The natural processes occur in conditions of- uniform wind

velocity, with a slow increase in velocity with height. Their

efficiency in smoothing out the fluctuations may be fairly accurately

calculated.

When the flow becomes unsteady, counterparts of these

processes take place, due to the eddies formed in the unsteady flow.

It is known that these eddy processes are more important than the

natural processes.

The flow over a not necessarily smooth surface is depicted

in fig.2.4.1, where the relative importance of eddies is shown. Most

terrestrial microwave links thus lie in the lower 200m, where inhomo-

geneities are efficiently dispersed by eddy processes. The atmosphere

ht

Outer Layer

of Frictional

Influence

Eddy Importance

200a.

ht

95

Fig.2.4.1 The develop ment and relative importance of eddies.

96

for these links may be thought of as being well mixed.

The thickness of the "turbulent boundary layer" depends on

the horizontal wind velocity.and surface roughness. An approximate

thickness of 30m is estimated for a wind speed of 5 ms-1. Chapter 3

of Kerr2 1 has a more detailed discussion.

2.4.4 Wind Shear as a Mixing Process.

Suppose that initially a laminar wind flow exists parallel

to a smooth surface. The velocity close to the surface will be reduced

due to drag at the boundary, and a vertical gradient 4.results, as

depicted in fig.2.4.2

Elementary dynamics shows that if such a gradient exists,

a parcel of air travelling at the lower speed is pushed upwards by the

air travelling even slower, below it. This shear force is propor-

tional to ā~. As shown in fig.2.4.2, this causes mixing to take place.

It is not necessary to have a surface to cause wind shear

it is often. possible to have two layers travelling at different speeds.

Shear and consequent mixing takes place at the boundaries of the layers.

Complicated velocity fields can result at such boundaries, varying

from slight ripples to "billows turbulence" as shown in fig.2.4.3.

Wind shear is hence not important for terrestrial links;

it is important at heights above 200m and should be taken into account

for earth—satellite paths. Section 2.5 will discuss some work due to

Lane and Paltridge2.23,2.24

who observed fairly strong refractivity

fluctuations due to wind shear at an elevated layer.

2.4.5 Surface Roughness as a Mixing Process.

Microwave line of sight paths usually involve the atmos-

phere close to the surface, and the surface can have a variety of

profiles. The flow over this sort of surface is turbulent, and .

involves substantial mixing (fig.2.4.4). .

f/ / If / / / / / / / / / Ī /

A

V V

Z

97

V // 1/ 1 1 1 1 1 1 1 1 1 1 1 I

Fig.2.4.2 Unsteady flow due to wind shear.

V1 Boundary Ripple -----

~— V2

V1

Billows Turbulence

V2

Fig.2.4.3 Unsteady flow developing at flow boundaries.

A

Section

vortex ring

turbulent flow t 1

where 5 for most structures

nR

It

98

Fig.2.4.4 Unsteady flow over an urban surface.

Fig.2.4•5 Generic shape of thermal plume, jet etc.

99

The intensity and spatial extent of this mixing is a

function of:

i) surface wind velocity

ii) spatial extent of surface roughness.

2.4.6 Convective Mixing.

During the day in nature and even at night in an, urban

environment, convection is the source of the mixing energy close to

the earth's surface.

Air in contact with a hot surface warms and buoyancy forces

cause the parcel of air to rise upwards. This process, repeated, sets

up a convective cell. The shape of these and similar (jets and plumes)

structures is intersting, having the generic shape given in fig.2.4.5.

They are stable, euen in fairly strong horizontal wind speed

conditions: fig.2•4.6 shows that their vortex ring structure allows

them to lean over in the wind. Scorer discusses this more exhaustively.

These structures are responsible for transporting clouds of

water vapour from surface evaporation, industrial processes etc. many

.hundreds of metres upwards into drier air. These moist parcels are

carried away by winds once they have reached neutral buoyancy. As was

discussed, eddy processes diminish in importance with altitude— fig.2.4.1-

and mixing takes place slowly by natural diffusion. The moist patchiness

• will persist for many hours, leading to strong radio refractivity

fluctuations, confined to a thin layer a few hundreds of metres from

the local surface.

The convective cells may be massive and extend over many

km. laterally and vertically. Industrial environments are likely to

produce smaller scale plumes and jets, which may, nevertheless, be

intense. A combustion or air conditioning vent is likely to have a

strong temperature and humidity difference with the surrounding air.

firma It( II/

Resultant

velocities add velocities

subtract

100

Temperature Vapour content Maximum AT

gm H2O/kg air

-17 1 0,1

— 8,5 2 0,2

O,5 4 0,4 .

10,5 8 0,8

21 16 1,6

32 32 3,2

Data for sea level P = 1000 mb

Table 2.4.1 Temperature difference due to vapour content difference.

C C

Fig.2.4.6 Behaviour of plume under horizontal wind flow,

101

Variations in water vapour pressure have an important side

effect in that they cause fluctuations in air temperature.

Water vapour is less dense than dry air:

J- water 0' 6 P air

If a parcel of humid air is to have the same density as the surrounding

air, it must have a different temperature, by a factor of 5

0,1 per gm of vapour per kg. of air.

ie QT (C) = 0,1 o(gm kg-1)

This effect is tabulated in Table 2.4.1 where the maximum

perturbation (dry versus saturated air) isLshown. The moist air is

carried aloft by thermal convection has also thus an implicit, refractivity

fluctuation due to this temperature difference.

The layers of humid air are more important for earth/satellite

links than for terrestrial links. Terrestrial links without good

clearance are more likely to be influenced by industrial and urban jets

and plumes.

2•S Refractivit Measurements.

Previous sections have given mathematical and physical form

to the idea of refractivity fluctuations in space and time. This section

will present practical measurements which have been made by various

workers and some of the author's results from an urban environment. It

is important to be able to place some quantitative measure on refractivity

fluctuations in order to be able to use the results of chapter 1.

Early workers'(eg.•Shephard 2.4 concentrated on measurements

of the mean lapse of refractivity. This was part of the search for

structures such as inversions, which were thought to be responsible

for abnormal propagation conditions such as ducting. The backscatter of

radiowaves by refractivity irregularities was not clearly detectable by

102

early radar.

The advent of modern, sensitive radars and troposcatter

communication systems has led to much interest in the vertical distrib-

ution of refractivity fluctuation strength. Sensitive doppler radars

are capable of measuring the velocity of irregularities many km upwards

into the atmosphere.

Optical astronomers were the first, naturally, to have such

an interest. Care, however, must be exercised in interpreting their

results, since the optical refractivity is not influenced by water

vapour fluctuations (see section 2.1 and Bufton et alt ?).

2.5.1 The General Situation.

Probably the most recent and comprehensive paper discussing

the height distribution of radio refractivity is that of Gossard 2-4

This paper develops a method of predicting the value of CN (egtn.2.4.2)

from measurements of CT, the temperature structure constant. This is

based on assumptions about the distance required for eddy processes

(section 2.4,) to disperse abnormalities in temperature or vapour pressure:

the so called "mixing length".

Fig.2.5.1 shows 3 plots of CN from Gossard, for 3 different

climatic regions. Perhaps the most interesting features are:-

i) the decrease with height of CN is slow for the first 5km in most

cases

ii) the rapid decrease of CN above 2km for Continental climates.

It must be emphasiszed that fig.2.5.1 represents calculated

values, but as shown by Crane (private communication reported by

Gossard), the values estimated from doppler radar returns are in satis-

factory general agreement, except for surface values. This is shown

in fig.2.5.2.

The above results are useful for the prediction of earth-

10'5 V.t icrw t0'14

103

Fig.2.5.1 Height distribution of structure constant. (Calculated)

Fig.2.5.2 Height distribution of structure constant from radar

returns.

104

satellite propagation performance. The situation seems to be that only

the lower 5km are important (relatively) in inducing random effects.

The situation for the lower 1km seems, however, to be

less clear. This is very important for the prediction of terrestrial

path performance. The next few sections are devoted to .a more detailed

analysis of the situation in the lower 1km.

2.5.2 Surface Ualuea_of.Refractivi:ty Fluctuations.

Two papers by Lane and Paltridge2.23, 2.24 .provide a good

indication of the situation in the lower 2km. The measurements of refrac-

tivity presented were made with a microwave cavity.refractometer, and

hence have an added plausibility.

Fig.2.5.3 shows some typical values obtained over a rural

area. The refractometer was carried by a balloon; the vertical plots are

thus not simultaneous, but the atmospheric•.stationarity should be such

that they are fairly representative of an instantaneous vertical

sounding.

The interesting feature is that strong fluctuations are

observed, usually at the base of elevated inversions, isothermals or

layers with pronounced wind shear. This latter point is in.te-resting in.

context of the mechanism outlined in section 2.5.6, where thin laYers

of humidity spread out from convective updraughts.

Since these measurements were made in a rural environment,

it was felt that an investigation of the urban situation was necessary.

The results and methods of analysis are presented in the next section.

2.5.3 Refractivity Fluctuations in an Urban Environment.

Crane's work (Gossard ) predicts a high strength for

surface refractivity fluctuations. An urban environment has potential

for strong fluctuations induced by processes such as exhausts from indus-

2.0

105

Mori%Q.t.( Vertical per par

1 1 1 •I k i

32S 1S 20 4 .G 9 10 ,o 5 0 S 10 T(C) wind

M15

Nt (km)

1.0

250 z • 300 N

10

ID

• .• ••

. O 0 0

I m0

0 ō

0 o 0

nor 1.g

0 o

\

.

100.0

(Lane and Paltridge2.23,2624)

rig:2.5.3 Refractivity measurements in a'r.ural environment.

c Scale Site distribution

N

i~u~o corratiaacc. (~ (K~ K value. cot

((K)= 11z41 x l

nwoxi►tieair fnrw► far 0,~k»1

t uo) cc

exp{ -*: 1

B )e

exp _ k

Ciax ='Iz

0-A4k = 2;,Ī3' _ t (k) > K

~Z K2 + i ~.

p `AA0)

.0 exp

oC

-- ,u 2 as,µ,

3( ) oc

e'`P ~' 1 µ

311(2 T:1M le&k)

4M&X = 0,L4.0

a,, K = D,1 1 1131O4

~ ,}<. K-2

vM k2 f 1

Fig. 2.5.7 Refractivity spectra when the scale sizes have a

distribution.

106

trial processes and localized hot—spots such as car parks and dark roofs.

A microwave refractometer similar to that used by Lane and

Paltridge was sited above a 60m building in central London. Fig.2.5.4

shoals a photograph of the site and fig.2.5.5 a diagram of the equipment

used for analysis. The refractometer consists of an open and closed Invar

cavities, excited by a 10GHz signal. The difference in refractive

index causes a difference signal which is suitably amplified and is

calibrated to produce 1/11 N,units/volt.

The output of the refractometer was further amplified and

input to a real time spectrum analyser with a digital integrator. This

signal was also sampled by a microcomputer, which produced continuous

estimates of the refractivity variance.

A synoptic recording system (see chapter 3) produced aver-

age measures of wind speed and air temperature.

The spectrum analyser calculates a 200 point spectrum•

with a resolution of 0,01 Hz. These spectra, each taking 8 secs. to

measure, are averaged in a digital memory. Fig.2.5.6 shows examples

of fluctuation power spectra, plotted to a logarithmic base.

Using the frozen turbulence hypothesis (section2.4),

these temporal spectra may be converted to spatial wavenumber spectra;

by dividing the frequency abscissae by• the mean wind speed.

The results for the spectra are in general agreement with

Lane and Paltridge2.23, 2.24 ie. the lower wavenumber components exhibit

•

grater variability than the higher, inertial range. The slope of the

inertial range was usually greater than the Kolmogorov —5/3; values

greater than 2,0 were not observed.

The variability of the low wavenumber components is probably

a manefestation of the non—stationarity of the "input" processes, as

was discussed previously.

The shape of the spectrum for low wavenumbers is also

107

Fig.2.5.4 Site used for refractivity measurements.

AMP

t

REF ELEcTgoNtCS OPEN

0

REAL TIME SPECTRUM

ANAL`(SER

AND INTEGRATOR

WIND SPEED DIR.

FORMgTTE R

PROCr CnLCuLAT.R

RDL

MICRO

CoMPQTER

108

PRINTER

SPECTRA

Yt4CtIfNCE S'iNdPT1C

Fig.2.5.5 Equipment used for refractivity measurements.

❑ . N Q O Cn

r

0

• 0

0 0

0

01 CD aD. cm a 03. c= N < 03 U

,109

Fig.2.5.6 Refractivity fluctuation power spectra.

o 0 N

I I

(8 P) . 111181.03c1S 83MOd

O . c+n

110

undoubtedly influenced by the failure of the frozen turbulence hypo-

thesis. This is because the time taken for a large inhomogeneity to

be swept past a measurement device is longer than the time taken for

some of the energy associated with this •inhomogeneity to have been

passed on to a.smaller scale size.

2.5.4 Spectra when the Scale Sizes have a Distribution.

The conventional technique for catering for different

synoptic conditions is to assign different. values of Ko and K

m in the

spectrum of the form 2.3.3. However, as was mentioned in section 2.2,.

Comstockrs work provides a more easily understood means of catering for

the situation where the scale sizes have a distribution, which may be

asymmetric-due to , for example,---.the measurement: technique enforcing

cutoff.

Two situations. will now be analysed in more detail. The -

autocovariance will be assumed to be the sum of a number of Gaussian --

shaped autocovariance functions with different scale sizes. In the

first instance, the scale sizes will be drawn from a normal distribution,

and from a Rayleigh distribution in the second case.

Following eqtns. 2.2.17, 2∎2.18, the resultant autoco-

variances functions are:—

BN(u) = expC—u/ua]

4(u) = Wexp —Oy ,

u

exponential

Laplacian

2.5.1

2•.5.2

where o is the variance of the scale size distributions.

To find the corresponding wavenumber spectra, the trans-

form relationship m

N(K) = BN(u) expV—iKul du 2.5.3

apo- L ..11

may be used. Ignoring constants, the transforms for eqtns. 2.5.1,2 are

111

ilN(K) = K /( c K + 1) 2.5.4

N

(K) sin(2 Tan-1(oK)) 2.5.5

(cūK2 + 1)

Fig. 2.5.7 tabulates the eSsen.tial differences of these two functions.

The first important point: is the large—K value slope.

As the underlying scale size distribution changes from the symmmetric

normal form to tha-asymmgtri-c.,Rayleigh form, the slope changes from

—1 to —2. The Kolmogorov —5/3 lies between the two. Practical obser—

vations tend to favour values near —2 (see section 2.5.2). This could be

due to either a cutoff for low K from the measurement technique or from

the basic asymmetry of the natural processes. Probably both are impor-

tant.

It is intereāting to substitute measured-valuasL into the- - -

spectral function 2.5.5. From fig.2.5.6, the half. power point is o.bser4-

ved for K^"5,0 m-1. Fig. 2.5.7 may be used to obtain

c- K = 1,39

for . this value of K, ie.

o7 = 1,39/5,0 = 0,28

The scale size variance is thus approximately 0,28m. The lower cutoff

K,ualue is obtained from

a-K = 0,17

ie. Ko = 0,17/0,28 0,61 m-1

The important wavenumbers thus lie on the range 0,61, 5,0] ie. the

the important scale sizes are on the range [10,2 , 1,0, metres. This

would seem to be intuitively correct.

2.5.5 Stationarity of the Refractivity Fluctuations.

In order to obtain an -idea of the period of time over which

the received signal could be analysed, a check of the refractivity

112

stationarity was thought to be useful.

The microcomputer was used to sample the refractivity at a

rate of 10 Hz. The standard deviation was calculated every90 secs (ie.

every 900 samples) and is plotted versus time in fig. 2.5.8.

The mean std. dev. of about 0,06N is observed to fluctuate

with peaks of nearly 50% of the mean std. dev.. It can be seen,

however, that periods of about 10 minutes may be taken to have constant

std. dev. and are stationary in the weak sense.

It was also thought interesting to observe the correlation of

the refractivity std. dev. with the local wind speed and air temperature.

Fig.2.5.9 shows the relevant data. In this case the variables were

sampled, every. 10 secs and the statistics computed every 10 minutes.

The data was taken from the early morning on a sunny day.

There is no short term correlation between the variables, but an obvi-

ous linear trend. exists over the whole sampling period. This is typical.

of the characteristics of calm, sunny days, ie. the refractivity.

variance increases towards midday, when the atmosphere becomes more

energetic due to heat input from the sun, via surface heating. A cor-

responding increase in signal variance is observed.

2.6 Conclusions.

A presentation of the physical properties of the lower

troposphere has been made. While lacking in mathematical rigour, it is

felt that sufficient data has been presented for calculations to be made

using the theory of chapter 1, of the spatial and temporal characteristics

of millimetre—waves propagating through the lower troposphere.

Another important consideration is the topology of the

propagation path. Again, processes which have potential for producing

strong refractivity fluctuations have been high-lighted. It will be

0,0q -

113

•

a

• 005

-o

a

ba

I 12, 14 20 24 28 32 34

time (minutes)

Fig.2.5.8 Refractivity standard deviation versus time.

Fig.2•5•9 Correlation of atmospheric variables.

1,0 -

40 44

wind speed

• -a t, co co

• ci

49-;

02

n

— •

•

—

/ +1'efractivity std. dev. 1-

I( 1 t t11( 1 , 101,00 fi koo

time (hours)---

•

temperat re

//// \\\*

NJ / f

114

shown in chapter 4 that two paths in the same locality can exhibit

vastly different scintillation characteristics, because of topology

differences.

This chapter has also presented a simple means of charac-

terizing the refractivity autocovariance as the sum of GQMssian auto—

covariancas, with a distribution of scale sizes. This method offers

great potential for synthesizing the refractivity autocovariance when

the fluctuations are known to come from certain, well categorized

sources, such as factory flues etc. '

Finally, some practical measurements of refractivity in

urban London are presented. These will be used in the simulation -

programs• described in chapter 6.

115

APPENDIX A2.1 Other Processes Affecting Millimetre—Waves.

The topics discussed in this appendix are strictly beyond

the stated scope of this thesis, but since they are very important

for millimetre—wave systems, it is important that they should be

recorded for completeness' sake.

Undoubtedly the biggest drawback to using the microwave

spectrum above 15 GHz is the effect of atmospheric precipitation. A

further disadvantage is the gas absorption, but this can be put to

good use, as was discussed in the introduction to this thesis.

A2.1.1 Rainfall.

Rainfall attenuation due to multiple scatter from drops

and dielectric heating loss within the drops. The attenuation is thus

a function of:—

i) Rainfall intensity

ii) Drop—size distribution

iii) (EM wavelength/drop—size) ratio

iv) spatial extent of the rain shower

,v) Polarization of the em wave and canting angle of the raindrops.

Rogers2 9 has published a useful review paper on the

statistical implication of rainfall on system design; Tahim2 1 has

implemented such considerations in a theoretical design of a 38 to 40 GHz

system.

The conclusion of work carried out to date seems to be that

practical terrestrial paths must be restricted to be not longer than 2km.

However, a point which should be investigated is the spatial extent

of rainfall rates.

Rainfall rate and drop —size distributions are a complex,.

function of climate and local geography. Links are designed to operate

in worst case situations, where the rainfall rate reaches certain

116

maximum values. This rate is assumed to be uniformly distributed

over the whole propagation path.

However, the spatial extent of rainfall rate seems to be

inversely proportional to the rainfall rate ie. more intense showers

are localized over a few square km. It seems thus that currently

designed sytems are based on fairly pessimistic assumptions.

In a qualitative way, the above remarks have been borne

out by the practical 12km path which is described in chapter 3. The

rainfall attenuation observed seems to reach a maximum value ie.

more intense showers are spatially confined and contribute to attenu-

ation over only, part of the path. The deepest fades were observed in

the fairly rare occurrence of widespread, light, frontal rain.

Rain scattering is important for radiometer measurements.

Zā 2*11 vody considers this in some detail. However, the major use of '

radiometers (besides radio—astronomy) is for estimating attenuation

for earth—satellite links, which is beyond>the scope of the present work.

For terrestrial links, a more important effect of rain is

the cross—polarization induced by the non—spherical nature of rain drops.

This effect has become important because of the current switch to

"frequency re—use" schemes. Here, two channels fo information are

radiated at the same carrier,frequency, but with orthogonal polariz-

ations. The rain is hence responsible. for coupling energy from one

polarization into the other until an unacceptable amount. of "crosstalk"

has been reached.

Evans and Troughton2.12, 2.13, have produced some.theo-

retical calculations, which give an indication of the the approach.

Practical measurements are also being made (Fimbel et al2 14). The'•

latter report that on a 53 km path at 13 GHz, a horizontally polarized

wave suffers greater attenuation. than a vertical one. This differ-

ential attenuation is greater than expected from.theory.

In summary, it would appear that differential attenuation

117

and depolarization in rain will pose serious problems to systems relying

on polarization to discriminate between adjacent channels. It can be

argued that the. 'large bandwidths available at millimetre—wavelengths .._

should, to a large extent, obviate the need for frequency re—use

schemes.

A2.1.2 Foq.

Since fog is essentially condensed water, .the attenuation

is most at 22 GHz, the water molecular resonance. The attenuation

depends on the liquid water content, which may be estimated by

"visibilty". Kerr 2-1 reports the table 2.1.

Visibility Attenuation in dB%km

m X = 1 , 25 cm.. X= 3,2 cm X = 10 cm

30 1,25 0,20 0,02

90 0,25 .. 0,04, _0,004 •

300 0,045 . 0,007 - 0,001

Visibilty

m

150

1000

Attenuation at X =.7,9 mm

dB/km

0,5

0,05

Table 2.1 Attenuation due to fog.

A temperature correction is necessary because the dielectric

constant of water varies with temperature. Data is also given in the

.

'Measured for T = OC

at 15C multiply attenuation by.0,6•

at 25C multiply attenuation by. 0,4

118

table for the experimental link described in chapter 3. The accuracy

of such readings is suspect due to the subjectivity of estimating

"visibility".

A2.1.3 Snow.

This is difficult to quantify because of the difficulty

in measuring "snowfall". Generally the effect of ice crstals is to cause

depolarization and differential phase shift of the wave. (Hendry

et al.2 15).

Also important for millimetre—wave systems is the effect of

snow and ice gathering on radomes and feeds of antennas.

A2.1.4 Attenuation by Atmospheric Gases and Vapours.

Here the absorption of wave energy is due to molecular

resonances. The theory is given in Kerr 1. The amount of absorp-

tion depends directly on the number of molecules in the volume in

question ie. the gas density.

In the range 0,1 to 80 GHz, the most important resonances

are those of water vapour (H20) and oxygen (02) which have peaks at

= 1,35 cm (22,235 GHz) (H20)

X = 0,5 cm (60 GHz) a series of lines (02)

The resonance mechanism has in fact a series of peaks,

and because of the subsequent vibrational decay, is dependent on

temperature and pressure.

Experimental measurements have shown good agreement

between theory and practice for 02, but H2O loss is greater than

explained by theory for frequencies above 50 GHz. Practical and

theoretical results are given in fig.A2.1.

10" Attenuation

119

Attenuation in dB per km

2 due to H2O

I due to 02 (assuming standard

vapour content of

7,75 gm/m3.)

24

0.01

aa

4005

QOW.

ROCf

40005

40002

10

5

2

1

R5

0.2

0.1

02 0,5 1 2 5 10 20 50 frequency (GHz)

101

to -3

0,81 1,5 2A {a004 6 8 10 Practical measurement of H2O vapour attenuation after Gunn and East.

'Fig. A2.1. Attenuation _due to gas absorption.

120

A2.1.5 iultipath Propagation.

This term is used rather loosely and interchangeably

with ducting. The term multipath here, is the geometric situation

described by fig.A2.2.

Two antennas are linked by the direct path A. Above the

link exists a sharp refractive index inversion. Energy associated with

a side lobe or off—axis gain of the transmitter is refracted towards

the receiver. If it arrives such that A — B = X/2, partial cancella-

tion occurs, leading to a fade of the received signal.

The problem is more significant for high bit—rate digital

systems, since the longer path B has then a time delay which is

significant when compared to the modulation rate.

The phenomenon is hence associated with: .

a) the presence of refractive abnormalities

b) broad beam antennas

c) long propagation paths.

Practical measurements have been made by Thompson et al.2.16

It is true to state that the problem of predicting system

performance lies more with the meteorological structures than calcu-

lating out the refraction problem with known antenna performance.

Since millimetre—wave paths will be restricted to a few

km by rainfall āttenuation, it seems unlikely that multipath will be

a problem. This is further reinforced if narrow beam antennas are

used. Thompson et al.2•16 measured time delays of less than 10

— sec.

for multipath phenomena, which means that modulation rates of about

6 GHz are feasible.

121

Fig. A2.2 Geometry of multipath propagation.

122

CHAPTER TWO REFERENCES

2.1 Kerr. Editor. "Propagation of Short Radio Waves".

Dover Edition.

2.2 Bean, B.R. and Dutton, E.J. 1966 "Radio Meteorology"

Dover Edition 1968.

2.3 Scorer, R.S. 1978 "Environmental Aerodynamics".

Ellis Norwood.

2.4 Gossard, E.E. 1977 "Refractive Index Variance and Its

Height Distribution in Different Air Masses."

Radio Science, 12, No.1, ppB9-1O5, Jan-Feb. 1977.

2.5 Scorer, R.S. "The Causes of Atmospheric Inhomogeneities"

Bulletin Astronomique, tome XXIV, fasc.2.

2.6 Sheppard, P.A. 1946. "The Structure and Refractive Index

of the Lower Atmosphere." Contribution to "Meteorological

Factors in Radio Wave Propagation" Conference Proceedings,

8th April, 1946. Physical Society, pp37-78.

2.7 Bufton, J.L., Minott, P.O., Fitzmaurice, M.W. and

Titterton, P.J. 1972. "Measurements of Turbulence

Profiles in the Troposphere." Journl. Optical Society of

America, 62, No.9, Sept. 1972, pp1068-1070.

2.8 Saxton, J.A. 1946. "The Dielectric Properties of Water

Vapour at very High Radio Frequencies". ibid. Sheppard.

2.9 Rogers, R.R. 1976. "Statistical Rainstorm Models:

Their Theoretical and Physical Foundations" IEEE Trans.

Antennas and Propagat. July 1976. pp547-566.

2.10 Tahim, K.S. 1975. Radio-Relay Systems Operating at

at Frequencies above 15 GHz. Propagation Characteristics

and System Design." MSc. Project Report, University

College, Dept. Electrical Engineering, London.

2.11 Zavody, A.M. 1974. "Effect of Scattering by Rain on

1 23

Radiometer Measurements at Millimetre Wavelengths"

Proc. IEE, 121, No.4, April 1974, pp257-263.

2.12 Evans, B.G. and Troughton, J. "Calculation of Cross-

Polarization due to Precipitation" IEE Conference on

Radio Systems above 10 GHz. p162.

2.13 ibid. p172.

2.14 Fimbel, J, Juy, M, 'Boithias, L. 1976. "Importants

Affaiblissements Differentiels dus a la Pluie mesures

sur une Liason de 53 km a 13 1GHz" Electronics Letters,

12, No.5, 4th March 1976.

2.15 .Hendry, A, McCormick, G.C., Barge, B.L. 1976.

" Ku-Band and S-Band Observations of Differential Propa-

gation Constant in Snow" IEEE Trans. Antennas and Propagat.

July 1976, pp521-525.

2.16 Hogg, D.C. 1968 "Millimetre Wave Communications through

the atmosphere" Science, 159 , No.38, pp39-46.

2.17 Thompson, M.C. et al. 1975. "Phase and Amplitude

Scintillations in the 10 to 40 GHz Band."

IEEE Trans Antennas and Propagat. AP-23, No.6, pp792-797.

2.18 Papoulis, A. 1965. "Probability, Random Variables

and Stochastic Processes". McGraw-Hill,' 1965.

2.19 Bendat, J.S. and Piersol, A.G. 1971. "Random Data:

Analysis and Measurement Procedures" Wiley-Interscience.

2.20 Tatarskii, U.I. 1971. "The Effects of the Turbulent

Atmosphere on Wave Propagation." Israel Program for

Scientific Translations, Jerusalem.

2.21 Beckmann, P. 1968. "Elements of Applied Probability

Theory" Harcourt, Brace and World, New York.

2.22 Meisel J and Cohn, D. 1978. "Decision and Estimation

Theory" McGraw-Hill, New York.

124

2.23 Lane, J.A. and Paltridge, G.W. "Small—Scale Variations

of Radio Refractive Index in the Troposhpere" Part 1.

Proc. IEE, 115, No.9, pp1227-1239.

2.24 ibid. Part 2.

2.25 Comstock, C. 1964. "On the Autocorrelation of Random

Inhomogeneities" J.A.S.A., 36, No.B, p1534.

2.26 Comstock, C. 1965. "On Wave Propagation in a Random

Medium" Journ. Geophysical Research, 70, No.9, p2263.

2.27 Fante, R.L. 1975. "Electromagnetic Beam Propagation in

Turbulent Media" Proc. IEEE,' 63, No.12, pp 1669-1691.

2.28 Strobehn, J.W. 1968. "Line—of —Sight Wave Propagation

through the Turbulent Atmosphere" Proc: IEEE, 56,

No.B, pp 1301-1318.

125

CHAPTER THREE AN EXPERIMENTAL 38 GHZ LINK IN AN URBAN ENVIRONMENT.

This chapter will describe the design and installation of a

3B GHz link in an urban environment. The design arose from the need to

study the spatial and temporal fluctuations of the signal after propa-

gating through a tenuous, random medium.

Also described is the data recording system built to allow

correlation to be made between the received signal statistics and. ambient

- weather-conditions:

3.1 Description of the Path and Planning.

The experimental path chosen was 11,6 km long, between the

BBC Crystal Palace Transmitter Tower and the roof of the Department of

Electrical Engineering, Imperial College, London.

This path affords good clearance of obstacles and essentially

links the north and south banks of the Thames Valley. The good clearance

is due to the height of both the transmitter and the receiver above the

intervening valley. Fig.3.1.1 shows the path profile, together with a

map of the region.

3.2 Transmitter and Receiver Design.

At the time of design (1973/1974) millimetre-wave components

were expensive and difficult to obtain. This situation is rapidly

improving, mainly due to military interest in these frequencies.

The equipment budget was not large, and thus the final design was

limited more by cost than strict experimental expediency.

A Gunn Diode oscillator of about 10mW power seemed the

cheapest, most reliable source at that time. Thus to boost the

effective radiated power (ERP), a high gain transmitter antenna was

chosen. This was also done to enable smaller receiving antennas to be

128

used. A further reason was to enable the concepts of narrow beam

propagation (chapter one) to be invoked in subsequent analysis. A low

gain, wide beamwidth antenna would have illuminated a substantial

portion of the path surface and the received signal would have contained

effects due to changes in surface reflectivity.

Ideally, high frequency pulse modulation would have been

implemented, giving a realistic measurement of system performance.

Unfortunately, only one supplier of a suitable modulator could be found,

and the price proved prohibitive. The final system was thus a simple

continuous wave (CW) transmitter.

The field received after propagation through the link medium

was expected to exhibit spatial and temporal fluctuations. In order to

investigate the spatial structure, a two channel receiver was thought

desirable. The theory developed in chapter one predicts largely phase

effects on the incoming wavefront. A variable baseline interferometer

seemed the best solution to the above two measurement requirements. The

tentative design is shown in fig.3.2.1.

After receiving quotations, the final equipment choice was

as shown in fig:3.2.2. The IF sections of the receiver were designed and

built. These-were housed in a suitable rack(fig.3.2.3). Comprehensive

circuits are given in appendix A3.1. A few points are worth elaborating.

The receivedsignal fluctuations are expected to be very

small. The signal also exhibits diurnal changes due to vapour and gas

concentration changes in the atmosphere. It was decided that the

diurnal changes would not be investigated in detail; it was thus

sufficient that the receiver exhibited good short—term stability only.

It was also important that a good signal to noise (S/N)

ratio existed to avoid convolution of the receiver noise spectrum with

the atmospheric fluctuation spectrum. Further, it was thought interest-

ing to examine the fluctuation spectrum during rain, when the mean level

\Coueie -p,~

• Golf

r Course

EL l~s ā1

} S ~''tiei

126

10 0 12

41

9.0-)N 4

Ē loo

Ō too

-8

Distance from Crystal Palace (km)

Fig.3.1.1 Experimental path geometry.

mixers

AZ multiplier used as phase detector.

127

commoa local oscillator

Fig.3.2.1 Interferometer.

129

1

Transmitter on 270' level of

3

BBC Crystal Palace Tower.

Fig3.2.2 Final equipment choice.

4

4

9 9

Key:

1 Plessey GD044 10 mW Gunn oscillator, 38,05 GHz.

2 Circulator, MESL type.

3 1m Cassegrain antenna. Gain 50 dB, beamwidth 0,4° C & S

4 25 cm paraboloid antennas. Gain 38 dB, beamwidth 6° Antennas

5 Balanced Mixers. Decca type MW19-22-38. Noise figure with

IF amp, 12 dB.

Level set attenuator, Flann Microwave.

7 Isolator, GEC Hirst Research Labs.

8 Gunn Oscillator. See section 3.7.1.

Decca type 120-20P 120 MHz IF amplifiers. 20 MHz 8W.

10 Purpose built receiver. See section 3.2.

Rain Guage Power Supplies Clock

AFC Nodule Receiver Modules

Anemometer

130

Data Formatter Thermometer

Phase Detector

Fig.3.2.3 Receiver and Sensor Rack.

131

was reduced.

A good 5/N ratio required a narrow detection bandwidth.

A good dynamic range also suggested a linear detector— in fact an

envelope detector was used. The effect of IF stability on short—term

stability will now be analysed.

Fig.3.2.4 shows an IF amplifier with a bandwidth B. It is

assumed that the gain—frequency function of the amplifier is roughly

parabolic in shape ie.

G(fif) 4/1-1) Ō (f fif

Thus DG = —8(1—W) fif / 82

?r. f

3.2

A fif 2B2JG/fi f(R —1) 3.3

From egtn.3.3 the maximum A fif permitted if the gain is to remain

within AG of its central value, may be estimated. Let LG = 1%,

B = 50 kHz, (fo fif) = 25 kHz. Then,

,,fif < 177 Hz

As can be seen, this is a stringent constraint on the IF stability.

This sort of stability can only be achieved with a phase locked IF

system. The simplest way out is to use a broader IF bandwidth B.

This approach was taken with the receiver used. The constraint on the IF

short—term stability is not so stringent asfif approaches fo, as can

be seen if eqtn. 3.3 is examined. Keeping close to the centre of the IF

amplifier passband requires a good long—term stability for the receiver.

A more complete analysis of the IF systems is given in appendix A3'2.

A further point to consider about the receiver configuration.

is the sensitivity of the phase difference measurement. It will now be

shown, that the phase difference at 38 GHz is the same as at any IF.

132

G

1,0 _

Fig.3.2.4 Assumed IF amplifier response.

W12

Fig.3.2.5 Model of phase detection system.

133

MYWINS POWER

V AIR INTAKE

>)

MTiI NS FtLTSR Temp.

Revlabr

w

D. C. S O??L (

D.C. FILTER

LOWER. CoNPARTMeNt

SENSOR UPPER COMPARTMENT

D• C. REGUL19rOa

GUNN

OI LLI4TOR

iso LA īOR `WNTF14NA

PRN

Fig.3.2.6 Transmitter block diagram.

134

Fig.3.2.5 is relevant. The three input waves are:

W1 = .A1(t) sin(w 1 t — /1(t))

W2 = A2(t) sin(0)2t — i2(t))

3.4 W3 A3(t) sin(W 3t — 03(t))

Each mixer produces the products:

= W1 W2 = A1 A2 (cos((ro1 — W2)t + (i1_02)) + cos( sum terms...) 2

W' W13 = W1W _ 3 Al

2 cos((W 1 — w3)t + (01-43)) + cos( sum terms ...)

3.5

The IF amplifiers only respond to, the W-2, 6)1— 3 frequencies.

Thus the output to the final mixer is, after limiting,

W12 = C cos((W 1 — W2)t + 1/1 - 02)

W13 = D cos((W1— W3)t + 01 — 03)

Ignoring constants, the output from the final mixer is

Vo W1211/13°C tos((w 3 2)t + 03(t) — 02(t) +,sum terms...

3.7

Since the receiver is illuminated by a point source, CO3 = W 2 and,

Vo oC cos(/3(t) — 02(t)) 3.8

proving that the phase difference is conserved.

The following section will give the final performance

figures for 'the receiver. The transmitter will now be discussed.

As mentioned previously, this consists of a Gunn Diode

oscillator feeding a 1m cassegrain antenna, with a gain of 50 dB.

Fig.3.2.6 shows a block diagram. Detailed circuits are given in

appendix A3.3. The Crystal Palace Transmitters radiate together

about 100 kW of VHF and UHF power, and so, initial difficulties were

experienced with spurious modulation of the Gunn Diode. This was resolved

by careful screening and filtration of the 240V mains supply.

3.6

135

The transmitter enclosure is lagged with a 5cm layer of

polystyrene foam. This ensures that the inside of the enclosure changes

very slowly in temperature and is not immediately affected by direct

sunlight. The output frequency (which is a function cif temperature)

is thus a slow function of time and is easily followed by the receiver

AFC circuit.

Similar ideas are used with the receiver local oscillator.

Details are in appendix A3.4. The local oscillator was designed by the

author. This is described in section 3.7, which also outlines some

work into usingGunn Diodes as self—oscillating mixers. It is felt

that this device offers great potential for non critical receiver

applications.

3.3 Receiver Performance.

A number of tests and calibrations were made of the receiv-

er in a laboratory environment to determine the base levels of receiver

induced fluctuations. These will be described.

3.3.1 Determination of the Detector Power Law.

The first test was to determine the response of the

detectors to changes in input microwave power. The experimental setup

is shown in fig.3.3.1. The attenuation A (a precision rotary vane

attenuator was adjusted and the detector output rrltages were noted.

The results are shown in table 3.3.1.

The analysis of the random medium dealt principally with

the field strength as the dependent variable. For convenience, the

analysis of the detector is in terms of the input fieldsEi1 or Ei2,

and the output voltages from the detectors, E01,

E02. Table3.3.1 is

thus modified and plotted in fig.3.3.2.

The linear relationships of these latter two curves suggests

136

20 dB horns

Fig.3.3.1 Laboratory test of receiver,

Attenuation A U1 U2

dB Norm. to maximum value

0 1,000 1,000

1 ,845 ,957

2 ,704 ,907

3 ,593 ,857

4 ,500 ,814

5 ,418 ,757

6 ,345 ,714

7 ,276 ,672

8 ,245- 9629

9 1204 ,600

1L ,184 ,543

12 ,138 ,486

14 ,097 ,400

16 ,066 ,329

18 ,041 ,271

20 ,026 ,229

Table 3.3.1 Receiver output versus input power.

s

0

1

2

5 a

6 0

Lu 7

10

11

13

14

15

16

137

E;o ( ) -: 1 2 3 4 S 9

10

Fig.3.3.2 Detector response.

138

a relationship of the form:

= Enj oj ij

Linear regression of these two curves shows:

n1 = 1,517

n2 = 0,525

Inversion of .egtn.3.9 allows the input field to be determined when the

output voltage from the detector has been measured.

The short—term fluctuations measured are of the order of-a

few percent, for normal link conditions, and so a polynomial approxi-

mation to egtn.3.9 may be used ie.

E1j = aoj + aljEoj + a2jEoj +

3.11

A least squares technique was used to calculate the coefficients for the

two channels. A cubic was found to give sufficient accuracy. Fig.3.3.3

shows the data of table 3.3.1 corrected using the polynomials.

3•3.2 Short Term Amplitude Stability.

The data analysis system was used to take samples of the

receiver output over periods of about 5 minutes (see section 3.5.1).

The sampling rate was about 2 Hz. The rms fluctuations were typically

less than 0,5% of the mean value. This was done using the experimental

setup of fig.3.3.1. Thus, 0,5% of the mean may be taken as the base

level of receiver fluctuations, for both channels.

3.3.3 Phase Difference Measurements.

A network analyser (manufactured by Hewlett Packard) was

used to monitor the phase difference between the two channels at 120 MHz

(the first IF). A Phase detector built by a co—worker was used to measure

the phase difference at the 8 MHz second IF.

3.9

3.10

Corr

ecte

d ou

tput

fie

ld.

1,0 01 8

139

0,2 0,4. 0,6

Input field

Fig.3.3.3 Linearized detector output.

Fig.3.3.3 Interferometer mount.

140

Fig.3.3.5 Front view of receiver site.

141

The transmitter was rotated on a turntable, about an axis

which was not the phase centre of the waveguide antenna. This yields a

phase perturbation of the wave received by the interferometer. This

phenomenon is analysed as an appendix of the next chapter. The phase

difference measured by the network analyser was used to calibrate the

low frequency phase detector.

Detailed results will not be presented here, but it was

4. 8.1 seen that the phase detector was linear over a range of — 90°

With the turntable stationary, a similar sampling technique

to that described in the previous section established the base level of

phase fluctuations to be less than 10 rms.

3.3.4 Receiver Mounting Arrangements.

The figs. 3.3.4 and 3.3.5 illustrate the method used to obtain

a stable, variable spacing of the antennas. The mounting frame possesses

considerable inertia (mass is about 150kg) and is mounted via building

jacks to the roof of the main building. The hut enclosure makes no

physical contact with the equipment, except via the signal and power

cables.

At present, to change the spacing, sections of the local

oscillator waveguides are removed or added and the antenna carriages

slid to the required spacing. The spacings obtainable are thus fixed by

the waveguide lengths.

3.4 Installation and Preliminary Results.

This section will describe the commissioning of the

experimental link, together with the preliminary results obtained.

These early results were important in assessing the shortcomings of the

equipment, and suggested some improvements which were in fact implemented.

142

3.4.1 Pre—installation Calibrations.

The transmitter and receiver antennas, because of their

narrow beamwidths, required sighting devices to align them. The

transmitter antenna was prov itad.with an astronomical sighting telescope

and the receiver antennas with a simple U-sight as an a rifle.

These sighting devices were aligned over a distance of some

200m, between two buildings at the Collge. The antennas were both aligned

until the maximum possible power was received. The sights were then

adjusted accordingly.

A further important measurement was made on this 200m range.

A power meter was used to measure the power received using one of the

25 cm dishes; the transmitter and 1 m cassegrain antenna were used as

the source. A distance of 200 m just satisfies the Rayleigh far field

criterion .(2a2/X) for this large antenna. The gain realized by the

large antenna should thus be within a few percent of its maximum.

Radiation patterns_ a-f the 1m antennaara }i.ven in fig.3.4.1. A 20 dB pyramidal horn was obtained and its gain calculated

• from its diimensions3 1. This was used as a standard to measure the gain

of the 25 cm.paraboloid accurately as

= 3705 dB — 0,2

Radiation patterns of the 25cm dish are given in fig.3.4.2. Using the

• simple experimental setup as in fig.3.4.3, a simple receiver was

calibrated in terms of input power versus output IF power, with a fixed

level of local oscillator power. This would enable the received field

strength to be measured when the link was operating.

3.4•2 Installation of Link.

The transmitter was hoisted to the 270' level of the BBC

Tower and bolted into place by the BBC staff. Photographs of the

1.•17•

£.5

TA

o .. Degrees

4' Cassegrain reflector at 38 GHz. Radiation pattern in H plane

-4 -3 - 2 1 0 Degrees

4' Cassegrain reflector at 38 GHz. Radiation pattern in E plane

to • LA •

• N N '

H—plane pattern of 25cm paraboloid.

048

-to 46

-2o d6

copolar response

crosspolar

--3o 46

20 30 40 10 20 10 30

i ■ 10

1 2.0

1 10

t I I I 50 40 30

0 AB

• CA

• N

-20 ,c1 }3

copolar response

crosspolar

E—plane pattern of 25cm paraboloid.

Power Meter

3 dB WAVE

Preci-ion Attenuator Power

Meter

14 dBm

49 dB= 1

11,6 km loss 145,5 dB

37,5 dB ± 0,2

147

local Oscillator

Fig.3.4.3 Receiver calibration.

Pr measured as —50, 8 dal' - 0,2

Theoretical value of Pr is 14,7. 49 + 377,5 - 145,5 s 44,3 d33m f 1,4

There is thus an excess loss of 6,5 dB t 1,6

Fig.3.4.4 Link power measurement.

148

transmitter in position are included in appendix A3.3. Once in place,

the sighting telescope was used to align the transmitter antenna.

Radio telephones were used to optimise this alignment to receive maximum

power. The panning frame adjustment thread proved to be too coarse and

hence a perfect alignment in both azimuth and elevation was never obtained.

Fig.3.3.4 shows the results of the power measurements.

There is an apparent excess loss of 6,5 dB. Taking a figure of 0,11 dB

per km as being the gas/vapour loss (see chapter 2) and including the

measured attenuation of the window glass as 3,4 dB, 4,7 dB of this

excess may be explained. This places the remaining loss within the error

bounds of the experiment.

Taking into account the receiver noise figure of about .12 dB,

the clear air signal ..to noise ratio of the link was about 40 dB.

3.4.3 Early Results. .

Early results began to confirm the feeling that the

atmospherically induced scintillations would be very difficult to

measure accurately, because of their small strength. Rainfall

attenuation manifested itself as the major effect on signal strength.

The magnitude of the scintillations was found to depend

strongly on wind strength, especially on gustiness. It was discovered

that the whole receiver hut moved during gusts; the interferometer base

was then supported on jacks attached to the main building roof. This

brought about an immediate improvement.

The movement of the front wall during strong wind continues

to induce signal fluctuations. These can probably be attributed to

reflections from the window frames. All measurements have thus to be

taken in low wind conditions.

The temporal spectra of the fluctuations were found to be

limited to less than a few Hz, as expected.

149

3.4.4 Transmitter Antenna Wind Loading.

The influence of wind gusts on the receiver structure led

to some speculation as to similar effectue on the transmitter antenna.

This is considered in some detail in appendix A3.5. Also considered here

is the effect on the phase difference received at a distant plane.

A conclusion which may be drawn from this analysis is that

measurements could only be undertaken in fairly low wind conditions

(less than 20 ms-1 say) and should definitely not be undertaken in

blustery conditions.

The analysis also highlights an important consideration

in the design of the mounting arrangements of the high gain, narrow

bandwidth antennas which are feasible in the millimetre—wave band, .ie.

the stability of such mounts.

3.5 Environmental Sensors and Data Recording System.

Since the effect of the lower troposphere on the milli-

metre—wave link was to be investigated, some thought was given as to

which atmospheric variables were important and could be used to typify .

certain conditions. This section will describe the data recording

system developed.

3.5.1 Synoptic Recording System.

In order to classify the weather conditions on a synoptic

basis, the following variables were thought to be important:

a) Wind Speed

b) Wind Direction

c) Air Temperature

d) Rainfall Rate

e) Time : of , Day.

The system depicted in fig.3.5.1 was designed to allow records to be

CC w J J 0 CC Z 0 U

150

wind speed A/D

wind direction

air temp

rain gunge

field strength

clock BCD

/ ASC

I IRS 23

2 s

cann

er/c

onv e

r ter

3analogue channels

FM TAPE RECORDER

fig. 3.5.1 The digital Synoptic recording System.

151

made for future analysis.

The variables a to a above are digitized and fed in parallel

to the formatter, using binary coded decimal (BCD) coding. Samples

are taken every ten seconds, this being controlled by the clock.

The formatter scans the parallel input lines according to a

stored program and converts the output to the serial ASCII RS232 standard.

This serial digital data plus three analogue channels can be

recorded onto a FIS magnetic tape recorder, or be analysed directly,

as will be described in the next section (3.6).

When data is stored on the recorder, the recorder is

controlled by the system clock. It is switched on for about 5 minutes

every hour.

Generally the 3 analogue channels would be the field

strengths of the two receiver channals..and the phase difference between

them.

This system was designed by the author and executed as an

undergraduate project. Some details of this project are contained in

appendix A3.6.

3'5.2 Environmental Sensors.

Wind speed is measured by a hot thermistor (N10D C).

The cooling is dependent on the wind speed. The thermistor is placed

in abridge and the amount of power required to rebalance the bridge is

an indication of wind speed. The device used was manufactured by

Prosser Scientific Instruments . It was necessary to interface this

device to drive the analog to digital converter (ADC).

The output is non—linear and has to be linearized. This

was simple to implement rising the analysis system which will be described

in section 3.6.

Wind direction is measured by a wind vane with digital output.

152

This device was designed and built by the author; circuits are presented

in appendix A3.7.

The anemometer probe was combined with the wind vane and

mounted on the end of a pole 3 m above the receiver hut roof. It can

be seen in the fig.2.5.4.

The electronic thermometer was commissioned as an under-

graduate project. The design which emerged as most suitable used a

transistor as the temperature sensitve element. Although having a long

time constant, it was robust and simple to calibrate. Appendix A3.8

has further details.

A drop—counting rainguage developed by the Appleton

Laboratory was used. The author developed a suitable counter for

integrating the output over 10 second intervals. The counter output

was in BCD and could be interfaced directly with the formatter. Circuits

are given in appendix A3.9.

The final channel of the recording system is designated as

"Field Strength" but is, in fact, a general purpose ADC which may be

used to record any data required on a synoptic basis.

The clock (appendix A3.10) was also designed and built by the

author. It has thumbwheels which allow the cluck to be preset. The

thumbwheels then act as a memory as to when the recording period began.

The clock only displays hours and minutes. A further pair

of thumbwheels allows a 2 digit "day number" to be set. This day informa-

tion is recorded by the formatter and allows for easy identification of

tapes.

3.6 Analysis of Synoptic Data.

The method by which the digitally recorded data was analysed

will now be discussed. The analysis of the analogue data is covered by

the next chapter.

Fig.1 of appendix A3.10 shows the sequence of ASCII

153

characters that represented the data gathered from the sensors feeding

the formatter. This sequence of characters was input to a TEK31

programmable calculator via a suitable interface. The calculator was

programmed to divide this string of characters into 6 channels. The

necessary linearizations were done, and the program then accumulated

sums and sums of squares for each channel. This was done for each

burst of data, every ten seconds.

On detecting a change in the time channel of greater than

10 minutes, the accumulated sums were stored as a block in the calculator

memory. When memory was full, the whole of the memory was dumped onto

cassette tape.

When a complete data tape had been read into the calculator

in this way, a new program was entered into the calculator and the

blocks of stored sums were output as suitable means and standard deviations

on the calculator printer. Samples are shown in fig.3.6.1.

3.7 The Self—Oscillating Mixer..

During the design of the receiver, the self—oscillating

mixer (SOM) was considered. The concept is simple, and is no more

difficult to implement than the manufacture of a Gunn diode oscillator.

The SOM has a great deal of potential for low cost, medium

sensitivity receivers.3.3' 3.4 . The author has published some information

about the use of the SOM for antenna measurements 5. This is repro-

duced in appendix A3.12.

An undergraduate project also proved the viability of a

working receiver at 38 GHz, which could be used to transmit data at

high rates, using simple FM modulation.3 6

The basis of the SOM is a waveguide cavity oscillator,

which is the subject of the next section.

154

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PERPENDICLLAR

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Fig.3.6.1 Sample printout from calculator.

155

3.7.1 Waveguide Oscillators.

Fig.3.7.1 shows a typical waveguide cavity oscillator.

The Gunn diode offers a low impedance, and thus the distance L between

the diode and the moveable choke forms a cavity and determines the

frequency of oscillation. This simple model is complicated by the

parasitic susceptances of the diode package and DC supply post.

Fig.3.7.2 shows a realistic design. The guide height has

been reduced in the cavity section, in order to better match the low

diode impedance. The guide width has been reduced to cut off lower

modes of oscillation. The diode is also mounted off—centre, to further

match the low diode impedance.

Fig.3.7.3 shows the performance of the final design,

compared to a high Q cavity (such as in fig.3.7.1, where the diode is

highly mismatched). Although the high Q version produces more peak

power, it has a narrow tuning range. A further problem with high Q

oscillators is the "switchon effect". Unless the plunger is tuned to a

precise value, the oscillator will not start when DC power is applied.

The final design is seen to be tunable over a wide range,

and only exhibits switchon problems at the extremes of the working

range.

An advantage of the high Q design, however, is that they

are capable of better spurious FM noise performance. This is an

important consideration for sensitive receiver applications.

3.7.2 The Self—Oscillating Mixer.

Fig.3.7.4 shows the essentials of the 5DM concept. The

diode oscillates at a frequency offset from the incoming wave frequency

by the IF. The diode is non—linear, and thus produces the required

beat frequency, which may be tapped from the DC supply by a suitable

filter.

156 DC Supply

C= c c C IW IW I4

diode ' cavity ength L adjustable tuning choke.

supply wires gold cap ceramic anulus insulator

wall to cap C

lead inductance

diode Z

GAs pellet

threaded heat sink

Typical diode package.

Diode position Anodized choke

Top VIEW

Micrometer

Return spring

tension spring Cavity made from brass and gold

plated.

SMA connector

SIDE VIEW

Equivalent circuit for diode package

fig• 3.7.1 Waveguide cavity Oscillator using Gunn diode.

Generally b* is chosen to give a cutoff frequency about 10%

below the centre frequency of the required oscillator.

a,b are the dimensions of the waveguide for the band in question.

fig. 3.7.2 Working design for Gunn Oscillator.

.,final design

high Q cavity 0--

35 36 37 3B 39 f (GHz)

40

final design

1 1 1 1

j6 17 1a 19 20 21

micrometer (mm)

Pow

er (

mW

) Fr

eque

ncy

( GHz

fig. 3.7.3 Performance of Cavity Oscillators.

158

The noise figure and conversion loss was found to vary

markedly from diode to diode, but noise figures between 15 and 20 d$

are common.

Upconversion is also possible (Spiwak. 7) ie. a signal

is injected via the DC choke. This signal may be frequency or amplitude

modululated. The author confirmed these observations, but no quanti-

tative measurements were done.

The uses of the S0(1 are many, especially for low cost

applications. Fig.3.7.5 shows some of these. The doppler radar appli-

cation seems particularly promising for anti—collision radars for

motor vehicles.

3.8 Propagation Experiment Equipment: Some Suggestions.

Some general comments about the design of propagation

experiments may be found useful in the light of the authorls experience

with the equipment described in this chapter..

Most research programs are restricted in terms of the budget

and it very important to assign a definite priority as to which aspect

is likely to be the most important.

In the case of millimetre—wave propagation through clear

air, the atmospherically induced scintillation is likely to be very

small. Receiver short—term stablity is thus important.

Gas absorption changes will require a transmitter/receiver

with good long term stability. In view of the downward trend in prices

for millimetre—wave equipment, it would seem to be prudent to use a

transmitter referenced to a stable crystal oscillator. This considerably

simplifies receiver design. If the receiver local oscillator may be

similarly referenced, an extremely stable receiver is possible. This

scheme is outlined in fig.3.8.1.

If access to the transmitter or receiver site is limited,

159

dc in ~----~~

fi

> if out

dc supply choke < ~ tuning choke

diode

circulator

Ar

if out

horn antenna A

dc 114 fig. " *7-'4 Principle of Self-oscillating Mixer.

IF Out AM Modulation AM Modulation IF out

a) Two way simplex data link.

Doppler frequency

b) Narrow beam Doppler Radar/ Intruder Alarm.

Raindrops (for example)

Doppler frequency

Spectrum

c) Measurement of movement velocity spectra.

fig. 3=75 '. Some potential uses of the SOM.

Nultiplier Chaia

Crystal oscillator

fo fif R

IF amplifier

160

fo Crystal oscillator

Multiplier chain.

Power amplifier

fig. 3.8.1 Preferred Receiver Configuration.

161

reliability and simplicity of the inaccessible equipment is important.

If possible, both transmitter and receiver sites should be readily

accessible.

The transmitter and receiver antennas should be well anchored

and impervious to wind buffeting. This is important and difficult to

achieve for large, high gain antennas. Alignment of antennas is

greatly facilitated if a fine panning adjustment is available.

The problem of data analysis has been considerably eased

by the advent of microcomputers. This aspect will be convered in some

detail in chapter 7.

I CrysfaI Osc.

—

1

to phase Mfereri de+ec+or

~} Fc Uv►it.

1

— T

I I;1,0;ter

Dctec+ea field siteltlA

162

APPENDIX A3.1 The Receiver.

Only one channel of the receiver will be described,

since the other channel is identical. The receiver components are

housed in a rack, which was shown in fig.3.2.3., as separate modules.

This is a convenient means of describing the receiver; an outline is

given in fig.1.

381OS

i dei+ical ditawnel . Decca 3i,R3 GHE

0,5 wi W

Decca V iZo MHK ► F lamp.

1- - — — 12$MHz g MKE ( I <

Local Oscillator

1?K

6X f'lul1' plier

Product- I Dclectr

D.C. n Illp

1 I

e

Fig.1 The receiver in modular form.

As can be seen from the figure, the receiver is a double

conversion superhet.. The front end consists of a waveguide balanced

mixer and matched 50dB gain 120 MHz IF amplifier, purchased from Decca

Radar1. These set the receiver noise figure at 12dB. The bandwidth

of the first IF was 201MHz.

163

The output"of the first IF amplifier is fed to a second

mixer, where a conversion to a second IF of 8 MHz is made. This IF

amplifier has a bandwidth of 500 kHz, which is also the detection

bandwidth. The components of this module are shown in fig.2.

The 128 MHz local oscillator (figs. 4 and 5) is extremely

Cumbersome, because of historical accident. Originally it was intended

that the 2nd. IF stages ...would be phase locked to a crystal oscillator

running at 8 11Hz. A crude 120 MHz discriminator would have kept the

first IF reasonably stable 0:1 MHz); the broad bandwidth of this

stage relaxes the requirement for a stable IF. The phaselocked second

If would have kept the IF very stable and alleviated problems of ampli-

tude drift due to IF drift. The detection bandwidth could be consider-

ably reduced, giving a wider dynamic range. This scheme is outlined

in fig.3.

Unfortunately, a stable 120 11Hz discrithinator.prov_ed

difficult to manufacture. This was principly because a satisfactory

120 MHz limiting amplifier could not be found. Eventually the simpler

circuit in fig.1 was evolved, with the crystal oscillator and multi.--

plier from the prototype circuit used to obtain a 128 MHz local oscilla-

tor. These two functions are shown in figs.4 and 5. The oscillator

circuit is conventional. The multiplier circuit uses a number of non-

linear amplifiers coupled by double—tuned transformers, tuned to the

2nd harmonic of the previous stage.

The detector/limiter module is given in fig.6. A high

gain limiting amplifier (CA3043) feeds a product detector, which is

also fed by the unlimited 8 MHz signal from the previous module in the

chain. This forms a synchronous detector which is linear in field

strength. The DC component of the product detector is suitably ampli-

fied for recording purposes. An output from the limited 8 MHz signal

is available for use with the phase detector, for phase difference

200a F

L1 13 toms 14" forw t L2 tut LR 2962 ferritt. colt

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•

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124 MH/ eAc.

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,w)41

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15v 1304.

HP 9082. M1XER

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1

ni 40 PF

2N5451

165

38 G-1-4t

L. 0.

n RFC Bioa2 B 120 MHz Amp. 120 MHz

DiSaiw+inator

120 MHz

128 M Nz-

8 MHt

1LX Mi4I+4- plier

SMHE ticsstal V. C. 0

$ M1-1z,

Phase iockeD loop a w,pli-'ier

Fig.3 Prototype super—stable receiver.

0 tISV

4 1SV

220n.

Lto

10K

3

10 nF

104.F .560 1OF

.1.512F

II s21)F

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=10nF

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in tiNG

9

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10

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220k

5Ok

3--,

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IZERO I

7.44 2 4.

3

220k Detector

D.C. Out

O

10nF

(OAF lOnF

0 O

0 ISV L4w4►W 8 MHz for 0

phase. Detector.

254

LIMITER ENVELOPE DETECTOR D.C. AMPLIFIER.

Fig.6 Detector/limiter module.

..." ~,

to . -..J

ID

:3 :t: N

a. ~,

CJl n Ii ~,

3 ~,

::J III rt­o Ii

n ~,

Ii n C ~,

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lit

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--------------~--------4_----------~----~------------__o-15V

170

measurements.

Fig.7 shows the 8 MHz discriminator which was used to

adjust the Gunn local oscillator in an AFC loop. The meter indicates

the magnitude of the feedback voltage, and is thus an indication of

the tuning of the Gunn oscillator. This is useful when setting up

the receiver. AFC loop amplification is provided by the 741 amplifier.

The only remaining component of the system is the Gunn

local oscillator, which is the subject of appendix A3.3.

REFERENCE.

1

"Handbook for 120 MHz pre—amplifier type 120-20P

fitted with Decca 1W19-22-38 Q Band Balanced Mixer."

Decca Radar Research Labs, Walton—on—Thames, Surrey.

171

APPENDIX A3.2 Automatic Frequency Control: an Analysis.

Automatic frequency control (AFC) is a commonly used servo

system in superhet. receivers to stabilize the IF against local

oscillator drift.

Fig.1 shows the concept in its simplest form. The discrim-

inator is a circuit which produces an output voltage proportional to the

deviation of an input RF voltage from some central frequency. Ideally it

does not respond to any amplitude fluctuations of the input signal.

The local oscillator is brought into alignment. by feeding back this

discriminator voltage to a frequency sensitive element of the local

-oscillator.

If fo > fl then the IF is fo — f1. Suppose that the

local oscillator drifts and increases slightly in frequency. fIF will

thus decrease, producing a decreasing voltage from the discriminator.

This in turn decreases the local oscillator (LO) frequency, tending to-

keep fIF constant. Naturally permutations of this circuit are used.

Some points about the operation of this system spring to

mind, such. as:

i) What stability of fIF can be achieved?

ii) How far from the condition fo—fl = fIF can f

1 be adjusted before

the loop "jumps out of lock"?

iii) How close does fl have to be to f in order for the loop to

"acquire lock"?

How much gain is needed in the loop and how does it affect the

previous three points?

• To answer these questions, a more detailed circuit is need-

ed, as in fig.2.

Referring to fig.2, the following may be derived:

fIF fo

v = c —U )- r

a3.2.1

a3.2.2

c1F

IF t4MP1 LIMITER .

Fig.1 Basic AFC system.

172

fL

VCO

Fig.2 AFC system to be analysed.

173

Al is the gain of the loop amplifier. Vr is a DC offset voltage whose

inclusion will be justified later.

v1 = Ad (f IF—fĪF) a3.2.3

HerefĪF is the desired value of IF..

The only remaining relationship needed is that between

fl and v; this will be specified as a function fl(vc).

This system of equations will be solved subject to the

condition that

āfIF/ 1)fo ..__j 0

This means that the error in IF due to drifts in fo must be as small as

possible'. It would be as simple to require 3fIF/ aft .- 0.

situations the latter is of course, the usual case.

Equation a3.2.1 gives

afIF/ ofo 1 afl/ āfo

but a f1/ of = afl/ a v avc/ o f

o c o

Egtns.. a3•2•2,3. give

ōvc/ a fo = Al ā fIF/ afo

Substituting into egtn.a3.2.4 gives finally

fIF/ fo = 1/(1+A Df, ,1v“)

n practical

a3.2.4

a3.2.5

a3.2.6

a3.2.7

a3.2•8

Thus for ()f IF/ a

f to tend to zero or to be within some O

design criterion,

A ?fl/ v = K where K is a large constant. c

Hence f = K vc/A = Ao vc is the desired function, which was

probably intuitively obvious. Ao is the VCO pulling factor, usually

measured in MHz/volt.

174

A simple—minded approach to design an AFC loop might be

as follows: A and Ad are usually not simple to alter at will, and

thus to achieve a required IF stability,. Al is increased indefinitely.

It is now prudent to see the effect of V , which could

characterize loop amplifier drift. It could also represent a fluctua-

tion due to an unwanted sensitivity of the discriminator to the level

of signal feeding it.

Some substitution shows

— )f IF//Vr Ao Al,

— K/Ad

The IF error is seen to depend on two elements ie.

a3.2.9

d fIF I K

VSV

r! + Ad ~ f

o 11+K

This is shown diagrammatically in fig.3. It can be seen that if

drifts in Vr occur, the only way to increase fIF stability is to

increase the discriminator gain Ad. To take a practical example,

suppose

Ad = 1.V MHz-1

Qfo = 1 MHz

QVr =10 mV

The minimum value of IF drift will occur when

Afo = KLSV r

or K ^' 10

1+K

Ad

For this value of K, A f IF N 44 kHz. Increasing Ad ten times allows K to be increased ten times, which reduces

the total error accordingly. Thus it is seen to be important to have a

stable, highly sensitive discriminator, which is insensitive to AM

fluctuation of the IF signal. There are many forms of discriminator

available. Most use the phase shift properties of tuned circuits. For

highest stability, crystal discriminators seem the best. As will be

seen presently, phase locked loop (PLL)techniques do offer alternatives,

fL(m4.9 vco RESPONSE LOOP AMPLIFIER RESPoNSE

01SC RIM LN Alt R RESPONSE.

175

Due to bVr

feedback factor

p41 < Adz < p~3

k

Curves for f ixee) nVr 4fo

Fig.3 IF error under different conditions.

AY-0

,tro (sat)

A.Ti(Sat)

Fig.4 Loop amplifier response.

Fig.5 Responses of elements of an AFC loop.

176

although more complex in implementation.

The range of operation of the AFC system will now be

investigated. Practical considerations will always result in the loop

having a finite range of operation, before saturation sets in. For

simplicity, the transducers will be assumed to have the linear rela-

tionships of the form shown in fig.4. Further, it is assumed that the

the saturation output voltage of the loop amplifier just drives the VCO

to its maximum deviation from its central value, as in fig.5.

Suppose that fo is increased monotonically downwards

towards fi.. When fo is far from fi, fl = fi, there is no output from

the discriminator and vo=O. When the point fo=fi+fIF(max)is reached,

the discriminator gives a positive output, which then drives the loop

amplifier output negative, bringing the loop into lock, with an error

in IF determined by a3.2.8. This process is indicated in fig.6.

Continuing to decrease fo gives an error voltage from the

discriminator which is amplified by the loop amplifier, which in turn

adjusts fl. However, when the loop amplifier (or the VCO transfer curve)

saturates, "lock" is lost.

A similar state of affairs occurs when fo is increased

upwards. f1—f3 (see fig.6) is called the "capture range" and f4—f2,

the"lock range" of the loop. Clearly the capture range depends entirely

on the discriminator range and th flock range is a function of the satura— ,

tion of the loop amplifier or VCO transfer curve.

The error in IF, QfIF, decreases from a maximum value

set by the loop gain, and decreases as fo is tuned towards fl.

It is also worth notirg.. the, .e-ffect of the "S"—shaped

discriminator curve on the acquisition process. As fo is decreased, the

discriminator first gives an increasing voltage, which tends to drive

the VCO further from fo. It is usually best in practice to be able to

switch in the AFC circuit after the required signal has been placed

scar TI NIN. I

I I

177

43 /CRPNRE RtINCrE

f, +4

ll i

K

VCO

CIF

LOCK RANGE

Fig.6 Lock and capture range of a receiver with AFC.

fo V

ofL Ming

LooP RMPUF1ER

MODVLATION ATE 0/p

DETECTOR

I IF

OafXaI OSG.

CIF

Fig.7 Phase locked IF.

Fig.B Variant of phase locked AFC.

178

within the capture range of the system.

It is possible to derive the time domain response of the

AFC servo system, but this has not been done, since in most applica,a

tions, AFC is arranged to have a very slow time constant, and only

follows the long term drift of the local oscillator in a receiver.

A method which may be used to obtain the most stable IF

is shown in fig.7. Here the IF is phase—locked to a stable crystal

oscillator. The full analysis of this circuit lies in the realms of the

PLL theoryA3.2.1.

Another variant of the PLL technique is shown in fig.8.

•In this case, the IF stability depends on the inherent stability of the

VCD. This may be crystal stabilized if necessary. This system is,

however, sensitive to changes in the input RF signal. It can be

shown that

Of IF hi N+1

The above processes are seen to involve multiplication of the VCO

frequency. If this is done digitally, there is potential for easily

tuned, digitally controlled, stable receivers.

This appendix has given a short exposition of the design

considerations of AFC loops. It is seen that a little care taken to

analyse even the simplest situation can produce the optimum performance

from the system.

REFERENCE

A3.2.1 Blanchard, A. 1976. "Phase Locked Loops" Wiley—

Iwter-science Publication, 1976..

179

APPENDIX A3.3 Crystal Palace Transmitter.

The transmitter consists of a CW oscillator and ancill-

ary equipment, feeding a 1 m cassegrain antenna. Fig.1 gives some

general views of the transmitter equipment under test and mounted in

position athe. .270 foot level of the BBC Crystal Palace Transmitter.

Fig.2 shows the oscillator enclosure with the equipment

tray withdrawn for test. The components are labelled in the diagram.

Fig.3 is a circuit diagram of the Gunn diode power supply

and mains filtration circuits. The oscillator is well protected again-

st equipment overvoltage by a zener diode and fuse.

Fig.4 shows the transmitter enclosure temperature control-

ler. The speed of a cooling fan is set by the air temperature sensed

by a bead thermistor in a bridge circuit. The internal temperature of the

enclosure is kept within 1 C rms of 40 C by this arrangement.

Power to the whole enclosure is supplied via a weatherproof

connector.

eneT 400J OLZ qe se44Tws 1

rn

Scalar feed and subreflector.

Antenna with radome removed.

182

Transmitter mounted to antenna.

• • i ' t

r

• • .r.• r„.∎ `i. = 'Or

w F Gunn oscillator

410 Circulator.

183

D.C. Regula

•

--41.11ft..: r lL

Temperature Controller

Fig.2 Transmitter enclosure under test.

oI P

W&22

C;iculntoc

PIcsSec~jj GDO,4 Osc; 114-tor

Ul+er //~'

l.oWei-

11

* R.Š. tye 218 407

t R.S. T iaw al Switan

1A

20vr~ tia As-Connie r

A B C

CANON Tin 10- CS (6 Ws) CONNECTOR

Fig. 3 ! CRYSTAL PALACE TRANSMITTER,

1H4ao4 6.~ k

pr 5W N.3V REGULATED

240V n~.

TIC 22 L D

0

185

Fig.4 •Temperature regulator.

AFC D C Regu lator

(FAN

D.0 Sut+li

rc,mp. Regulator

MAINS

1 86

APPENDIX A3.4 Local Oscillator.

The circuit for the local oscillator is given in fig.1

B } I C 14 plant T

Fig.1 Local oscillator.

The DC supply is regulated and applied to a Gunn diode

oscillator. This oscillator was manufactured according to the general

design characteristics given in section 3.7. The regulator circuit is

identical to that used in the transmitter supply, except that the AFC

correction voltage is applied via a 10 kfl.-resistor to pin 5 of the

)4A723 regulator IC. A 15 V signal applied to this resistor causes a

roughly '2'100 MHz frequency shift of the local oscillator.

The oscillator is fed via an isolator and padding attenu-

ator A to an H—plane tee piece. The attenuator is adjusted so that 0,5mW

is available from the two waveguide ports B and C. These feed the the

mixers of the two receiver channels.

The above circuitry is enclosed in a box which is kept at

constant temperature by a cooling fan and temperature sensing circuit,

again identical to that of the transmitter (previous appendix).

187

APPENDIX A3.5 The Influence of the Transmitter Antenna Movement on

Amplitude and Phase Difference Measured on a Distant Plane.

A3.5.1 General.

The transmitter antenna (Tx) is a 1 m cassegrain design

with a gain of 50 dB and beamwidth 0,48 deg. It is mounted via a pan-

ning frame to the BBC Transmitter Tower. The tower itself is of open

steel girder construction and stands on an open hillside.

There are thus three potential sources of antenna

movement, under varying wind conditions:

i) 3-D movement of the antenna on its panning frame,

ii) transverse movement of the whole tower,

iii) torsional movement of the tower.

The BBC have recently done extensive computer modelling of

the tower under various wind loads. The torsional movement'is thought-

to be minute, and will be entirely ignored in the following analysis.

The 270 foot level (where the Tx is mounted) is thought to move less than

2 inches in a 45 ms-1 wind, corresponding to less than 0,2 inches in a

20 ms 1 wind, which is a more likely occuring value.

The different physical configurations for tower movement

will now be considered, together with their corresponding amplitude and

phase effects on the field received on a distant plane.

A3.5.2 Tower Movement.

Fig.1 shows the antenna mounting position.

Sik Tb

Pav►nivuS f+awK adiu4w,a t

tauuuuuu.uu{lu€

Side Eleva}ioh

013►n

Top El evaliov,

188

Wind loading along the Tx axis will cause the Tx beam to be elevated or

depressed by an angle O given by fig.2.

1e- - ---

r -

1

wind velocity

MS—1 s mm

69

mrad

45 50

0,63

20

5 0,063

Fig.2 Pointing angle error due to tower movement.

The above figures are, of course, over and above any movement of the

antenna panning frame, which will now be examined. The resultant ampli-

tude effect will be estimated in section A3.5.4.

A3.5.3 Antenna Panning Frame Movement.

Fig.3 shows the panning frame arrangement.

elvmdiotA piJot

A

Fig.3 Antenna panning frame.

189

To simplify calculations, it will be assumed that when

when the antenna is buffeted, it moves in elevation and azimuth about

the same point which is estimated to be 0,3 m behind the focal plane of

the dish. This is illustrated in fig.4.

im

Fig.4 Axis of rotation of the antenna.

The following table may be constructed:

x 89 = Sx/0,5m

mm mrad.

0,01 0,02

0,05 0,10

0,10 0,20

0,50 1,00

1,00 2,00

1,50 3,00

2,00 4,00

2,50 5,00

3,00 6,00

4,00. 8,00

Table 1. Pointing angle error due to panning frame movement.

190

A3.5.4 Amplitude Effects of Antenna Movement.

The previous two sections have estimated the pointing

angle errors due to antenna movement. This section will estimate the

corresponding amplitude effects measured over a distant receiving plane.

It is assumed that the receiving antenna is essentially broad-beam, and

may be treated as a point receiver. This is true in practice, since the

receiving antennas have 6,5 deg. beamwidths.

The radiation pattern of the Tx is assumed to be

E(9 )oC E0 Sin(6)/B a3.5.1

The distribution which best fits the transmitter pattern is

E(x) oL sin(0,015x)/(0,015x) (x in metres) a3.5.2

which is the field which would be measured over a plane 11,6 km from '

the transmitter.

Suppose now that the transmitter antenna is moved a small

distance 6x, as in fig.4. The field distribution across the receiver

plane will be displaced a distance

x = L 56 where L = 11,6 km a3.5.3

and se is given by table 1. A point receiver will thus register a

decrease in power, and field. In table 2, this power and field change

is compared with the maximum power/field which can be received, ie. the

on-axis values.

6x mm

0,01

5e mrad

0,02

x=L5e m

0,232

Power dB

-1,76E-5

% Perturbation of field

0,0002 0,05 0,01 1,16 -4,38E-4 0,005 0,10 0,20 2,22 -1,75E-3 0,02 0,50 1,00 11,6 -4,39E-2 0,50 1,00 2,00 23,2 -0,18 2,01 1,50 3,00 17,4 -0,40 4,48 2,00 4,00 46,4 -0,713 7,88 2,50 5,00 58,0 -1,12 12,15 3,00 6,00 69,6 -1,64 17,20

Table 2. Received field fluctuation due to antenna movement.

191

It is seen in table 2 that a movement of only 1 mm of

the transmitter dish is sufficient to cause a fluctuation of 2% in

field strength. It was thus apparent that measurements in strong wind

conditions were likely to be contaminated by antenna movement effects.

In fig.2 it was estimated that the pointing error due to

movement of the whole tower was only about 0,06 mrad in a 20 ms-1 wind.

Table 2 predicts that the amplitude effect will be minimal (about 0,005%).

Perhaps the most important point to arise from this analysis

is the extreme difficulty associated with mounting large millimetre—wave

aperture antennas. A fluctuation of a few percent is irrelevant for a

communications link, but is extremely important if the antenna is being

used to probe the atmosphere.

A3.5.5 Phase Difference Measurements of Antenna Movement.

For this analysis it will be assumed that the transmit-

ting aperture has a Gaussian field distribution, as was analysed in

appendix A1.6. Further, it is assumed that the phase across the aperture

is uniform. The wave which radiates from such an aperture slowly expands

and becomes a spherical wave. The phase centre of the spherical wave

is at the centre of the aperture, but only when the field is sampled

over a plane a large distance away from the aperture. A measurement of

the field on the aperture would prediot the phase centre as being situated

infinitely far behind the aperture. The apparent phase centre moves from

infinity until it appears to be on the aperture, when the field is being

measured in the very far field.

It may be shown from the analysis of appendix A1-6 that the

radius of curvature of the wave as a function of distance from the aper-

ture is given by

r(L) = L (1 + (Tr a2A L)2)

where L is the distance from the aperture.

23.5,4

192

The variation of apparent radius of curvature with distance

from the aperture is tabulated in table 3.

L 1.00 )

11600 11614 10000 10015 8000 8020 6000 -6026 4000 4040 2000 2079 1000 1883

Table 3. Radius of curvature as a function of point of measurement.

It is seen from this table that the radius of curvature

predicted by measurements made on a plane 11600 m from the aperture.

would place the phase centre 14 m behind the physical aperture.

Appendix A4.1 analyses the situation where an antenna

is rotated about a point other than its phase centre. The analysis

shows that such a rotation leads to a phase difference between the

phase measured at two points symmetrically placed about the beam axis

(see fig.A4.1.1).

Assuming that the phase centre of the antenna is indeed.

14m behind the physical aperture and substituting the suspected worst

case antenna lateral movement, the phase difference due to antenna

movement may be ignored, if egtn.a4.1.4 is usedoto estimate the phase

error.

193

APPENDIX A3.6 Synoptic Recording System.

This system was developed as an undergraduate student

project1. A number of schemes were considered before the final system

of scanning the binary output from the sensors and converting to

ASCII code was chosen as being most flexible.

Detailed circuits will not be presented, since they are

available in reference 1. Two major advantages of this system are:

i) the sequence and frequency with which a...par.:ticular channel is

scanned is simply changed by altering the program stored in a

ROM on the Formatter boards.

the ASCII output may drive many terminals and computer inputs

directly.

The scanning program used and the method of analysing the

ASCII signal is described in Appendix A3.10.

REFERENCE

Aspinall, C.R. 1976. "The Developement of a Digital

Recording System" Third Year Project Report, Imperial

College Dept. Electrical Engineering, 1976.

cock() W ted

•

\\ ~1. Detectors

194

APPENDIX A3.7 Wind Vane.

Extensive enquiries revealed a difficulty in obtaining a

low—cost,• robust windvane which gave a binary output. The simple

device shown in fig.1 was devised.

The lamps shine onto a linear row of 5 photo—darlington

transistors. The disc has a pattern of concentric slots milled into it

which obscures the light from certain transistors as it rotates. The

slot configuration has been chosen to produce a BCD code. The 5 bit

pattern allows the BCD values 00 to 19 to be realized, giving 20

sectors (ie 18 deg. each). This is sufficient accuracy for the

synoptic nature of the equipment.

Fig.2 gives the pattern cut onto the disc, together

with a circuit of a typical detector.

Thermistor anemometer

Fig.1 Wind vane.

Pho}o Do ilhvion ccv+fiie5

195

no At A4 ns Bo P,G D code.

S 04 aboUe ciscuit use).

Fig.2 Windvane disc pattern and photo—detector circuit.

1 96

APPENDIX A3•B Electronic Thermometer.

A simple but robust device was developed as an under-

graduate project1. The requirement was for a device to work over a

range of —10 to 40 C.

As described in the reference, a transistor fulfills

the above requirements. The emitter—base voltage of a transistor

changes about 2 mV C. This voltage is suitably amplified and applied

to an analogue,_ta digital converter. The output BCD code is then fed

to the data formatter.

The time constant of this device is long (about 2 minutes)

• since the transistor package used has a fairly large thermal inertia.

This was not considered to be a disadvantage in this application.

REFERENCE.

1 Skelton, T. 1976. "A Transistor Atmospheric Temperature

Sensor" Third Year Project Report,. Imperial College

Dept. Electrical Engineering, 1976.

197

APPENDIX A3.9 Rain Guage.

A drop—counting rain gauge developed by the Appleton

Laboratory was adapted for use with the data formatter. The rain

gauge produces standard sized drops from a funnel/capillary tube

arrangement. These drops cause pulses when they interrupt the light

beam passing the end of the capillary tube. The circuit shown in fig.1

was developed by the author to act as a count store which could be

interrogated by the formatter.

A3.9.1 Circuit Operation.

a) REMOTE MODE. (SW1 in REMOTE position).

In this position, the output of G1 is always high. A

negative going pulse on the READ input causes an inverted pulse to be

presented to MS2. The negative edge of this pulse (positive edge of

READ) causes a 2 mS pulse which clocks the count value from the counters

into the latch and is thus displayed.

The negative edge of the pulse from MS2 then fires MS3,

which in turn resets the counters to zero via G3.

The inverted output of MS3 is also output as DATA pulse,

which means new data is now available in the latch formed by the 4

7474's (0—type flip flops). This Data and DATA pulse are available from

the module.

Timing Sequence.

READ ZJA

G1 _J—'

M52 cLocK

Q comir +t uisferre? +v alisp69 eta olio

© COUNT is set -i-o zero reaAJ

for new 40sec

period.

M53 i ~( G2 ~ ~j ,D1gTA

74 74 7444

CL IKL — D CL LK 0

P CK Q p

4 -4-) GHD C~

7444

RESET D ISPU9

CLOCK

UNIrS rENs

8 SCD LINES

Ft* FORMfTTE12

b

D1

c f

b D c n d

e.

7447

1 98

COORT

INPUT

60 IMP Rpf tK gig

Va.- D VCc- 124! 8 Ra C

/410

Fig1a. Raingauge counters, latches and display drivers.

ipMF rtE555 .

199

RAIN ro 'cLGXJ ON

1.)11-CH

1.20k

0'4y 'DAM/

20k NET EDO E

qct,

20k

r 3412t 'RE.SET

I 0$ C01)NTER

14t2I

~_ NC

r-C IRI

14214 i4 s

PULSE i- Q FROM ~ afal&E

RESET- PUSH Burro N

=

GI Vcc Cyt 43 OPI G3 G2 0P3 02 G4 01'2. CT4

0P4

3400 ALL NAND GATES 1400

3 couMTEtz' 20.5

21at N

2%6 Zr

if TO 12ESET DtSPLR

G4

1 12E14-DI REMOTE

v i >.-- LOCAL. • CAM 6PATE

IowS 40S

•

CfLIBRi4TbN PUMP

Fig.lb Raingauge monostables and switching circuit.

200

b)- LOCAL MODE (SW1 in LOCAL position)

The output of G1 consists of READ pulses supplied by

the 10 s astable multivibrator. The rest of the sequence is as above.

When the DATA pulse occurs, the output data is valid.

CALIBRATION MODE. (SW1 in CALIB position)

The circuitry is still in the local mode, except that the

calibration pump is now switched on. The display should read a count of

about 30 to 35 (rainfall rate of about 50 mm hr 1). If it is less,

it is possible that the filter of the raingaUge has become blocked.

e) RESET

Pressing RESET at any time clears the display and counters.

The oscillator is also restarted on a new 10 sec cycle.

REFERENCE.

1 "Appleton Lab. Rapid Response Raingauge", Appleton Lab.,

Ditton Park, Slough, Bucks.. 1971.

201

APPENDIX A3.10 Analysis of Synoptic Data.

The data formatter produces a string of serial ASCII codes.

This stream of data was input to a serial interface for a TEK31 program-

mable calculator manufactured by Tektronix Corp. of the USA.

It is possible to program the calculator to examine each

ASCII character presented to the interface and execute a certain set of

instructions, depending on which character was received.

The formatter was programmed to intersperse the data from

each channel by the ASCII characters CR and LF (carriage return and

line feed), The receipt of these characters signalled to the calculator

program the start of a new channel of data.

Fig.1 shows the sequence of characters output by the form-

atter. It is seen that each channel is preceded by a digit corresponding

to the channel number. The next characters are the value of the channel

information, terminated by CR/LF, as explained.

The whole sequence is preceded by "DLE,1", which is

interpreted by the calculator as a command to execute the program steps

beginning at label 1. The TEK31 allows "labels" to be inserted in

programs as entry points.

The program first identifies a channel and then stores the

sum, sum of squares and increments a sample counter. These totals are

stored in the calculator registers (256 for this machine). .A separate

_set of registers is associated with each channel of information. In

some cases (eg. wind speed) normalization of the value is first done

before accumulation.

Once the program recognized that the time registered by

the clock had changed by 1 hour from the time first registered, all the

accumulated data was shifted further along in the stack of registers.

A new set of sums etc. was then initiated.

202

When all registers were full, a dump of all registers to magnetic

tape casette was made.

Once sufficient data had been accumulated, a new program

was loaded. This program examined the registers and produced a statis-

tical analysis as a listing on the calculator printer. An example

was given. in fig . 3.6.1 .

The statistics reflect averages taken over an hourly

period, but in some cases the sampling program was altered such that

the sums.were accumulated over ten minute periods instead. The formatter

produces a scan every 10 secs, regardless of how often the data is

being accumulated.

203

DLE Interpreted by the TEK 31 as the command

"Execute the steps starting at label 1"

DI")MINUTES

D2

D3 1HOURS 04

D5 DAY NO.

D6

CR/LF

"9" channel marker

D1 WIND SECTOR

02

CR/LF

"S" channel marker

D3 UNITS

04 TENS

D5 HUNDREDS

D6 THOUSANDS

CR/LF

"7" channel marker

D7 UNITS

D8 TENS

D9 HUNDREDS

D10 THOUSANDS

CR/LF

"6"

channel marker

D11 UNITS

D12 TENS

D13 HUNDREDS

CR/LF

"5" channel marker

D12 UNITS

D13 TENS

D14 HUNDREDS

015 THOUSANDS

CR/LF

TIME INFORMATION

WIND SPEED FROM ADC

AIR TEMP. FROM ADC.

RAIN DROP COUNT

OUTPUT FROM SPARE ADC

(usually signal strength)

*Note: The "D.." refers to the ASCII code for a BCD value.

Fig.1 Sequence- of characters output by data formatter.

204

APPENDIX A3.11 Clock.

Digital clocks are available as single integrated circuits,

but in order to keep the number of pins used to a minimum, the output

time count is only available in multiplexed form. This is difficult

to interface with the formatter, which requires the data in parallel

form. A clock circuit was thus designed by the author from discrete

integrated circuits.

The design is shown in figs.1,2,3 and fulfills a number

of important functions. As well as the time, it provides a pulse every

ten seconds, which starts the data formatter on a measurement cycle.

It also provides a pulse once an hour to start the tape recorder.

Fig.1 shows the reference oscillator, which is a

phase locked loop referenced to the mains frequency. It is hence relative-

ly impervious to mains interference, since it has been given a long

time constant. The VCO output is divided by 3000 (50X60) to give

pulses every second and every minute.

Fig.2 shows the output and display counters which hold

the minutes and hours counts. NAND gates are used to check for 60

and increase the HOURS count by one. Similarly, the HOURS count is

reset at 24.

The 74192 counters used may be preset by BCD inputs to the

AIN,BIN etc. pins. A set of BCD coded thumbwheels are used to set the

start time of the recording period. A two digit DAYS thumbwheel is

also provided. This is fed directly to the formatter and identifies

a particular data run.

Fig.3 gives the monostable which is fired whenever the

MINS tens digit changes to 5; this only happens once an hour and is

used to start the tape recorder.. Also shown is a 10 counter which

fires- -a mōnostablē- every ten secōrrds; this-pulse is-used -to-start 'thefbrmatter

on a measurement cycle. Finally, the displays and supply regulator are

shown.

+5V

v M~x

7410

10

+5V

13V 240V

5k [FREQ1

13v

10k v

7490

410S/

V

3490

--10

imv+. - 8v

sec

1-4c12

205

Fig.1 Clock reference oscillator and =3000 counters.

+5Y

}220?

NE 565A

31;

4 k? vco

RESET

206

MINS TENS

Fig.2 Clock HOURS and MINUTES counters.

SO MINS DErecr I T.R . Sīr4r r MON OST ABLE

C

-!- EVERY HOUR

10 SEC COUNTER +S +S

SECS

10011

f EoRMwTrE>z

i4121 Vt,~s

TO TAPE TIMER

3410

Fig.3 Clock regulator, display, start monstables.

W. 74121

U TO FOR MriTTE2

tSv

207

Orr a crr

corNOM — IW4ODE

NP- NP.

cnr DP -Care -

a.

l71;

- coMMoM %NOOE - cor - NP - cnri - CRrc - Nc - ttar d

d. PP

Rs SecSwittat Display

ov 6C D to 7- Surt■it

4 0C about

10k SEC.. fir,

t511

180-f2.

. BC lOq

0V

SECS LED.

S V Regulet+or

LA gol SV

14V 1_ - 1 ..L

JwbF - foo, .F s

208

CHAPTER THREE REFERENCES.

3.1 Silver, S. 1-9:49 "Microwave Antenna Design"

McGraw—Hill Book Company, New York.

3.2 Handbook for Prosser Scientific Instruments Electronic

Anemometer. Prosser Ltd., Rayleigh, Essex.

3.3 Lazarus, M.J. et al. 1972 "Have a High—quality

V—Band Lind and Low Costs Too" Microwaves, 11, Nov. 1972.

3.4' Lazarus, M.J. 1972 "A Millimetre—wave Gunn mixer with

—90 dBm sensitivity, using a MOSFET/bipolar AFC circuit."

Proc. IEEE(letters) June, 1972.

3.5 Inggs, M.R. 1978 "Self—Oscillating Mixer Cuts Antenna

Test Costs" Microwaves, 17, No.4, April, 1978.

3.6 Keleher, N.M. 1977 "Prototype Compact Millimetre

Data/TV Link" Undergraduate Project Report, Dept.

Electrical Engineering, Imperial College, London SW7.

3.7 Spiwak, R.R. 1968 "Frequency Conversion and Ampli-

fication with an LSA Diode Oscillator", IEEE Trans.

on Electron Devices, ED-15, No.8, pp 614-615. August

1968.

APPENDIX A3.12 Published Work.

Reference 3.5 is bound at the end of this thesis.

209

CHAPTER FOUR EXPERIMENTAL DATA.

Some of the propagation data taken over a two year period

during which the 38 GHz link was in operation is discussed in this

chapter. The data shown was selected for the clarity in which it displayed

aspects of the wave/medium interaction. Where possible, comparison

is made with the results of other workers.

The theoretical treatment of propagation through tenuous

random media presented in chapter one predicts that the incoming wave

has a spatial and temporal phase modulation imposed on it by the medium.

These phase modulations lead, in turn, to amplitude fluctuations.

The amplitude of these phase and amplitude fluctuations may be predicted

from the properties of the medium refractivity fluctuations. The initial

part of this chapter discusses the measured phase and amplitude data and

shows that they are in good agreement with the theory.

Since the important atmospheric scale sizes are larger than

most receiving apertures likely to be used in the millimetre-wave band,

it is shown that-the atmospherically induced phase effects may be treated

as an angle of arrival fluctuation. The performance of aperture antennas

under these conditions is analysed.

Some fluctuation spectra were measured during rain conditions.

These spectra displayed peaks at higher frequencies. This phenomenon

is discussed in the light of a simple model of raindrop scattering.•

4.1 Phase Effects: Theoretical Predictions.

In this analysis, it will be assumed that the medium

has a Gaussian autocovariance function, as in eqtn. 1.3.7. Section

2.2.5 has shown how this simple model may be made more realistic, by

combining a number of Gaussian autocovariances. It will be shown,

however, that the simple approach gives good agreement with practice.

The medium is assumed to be equivalent to a single phase

210

screen, as was discussed in section 1.3. This phase screen has an

autocovariance function given in egtn.1.3.8, ie.

B/(u, v, T) = 10-12k2 L :wo TN exp —(.y. _ v ll2

_ \2

~uo vol ~~0 4.1-1

4.1.1 Phase Difference Measurements.

The receiver was configured as a variable baseline

interferometer, and so it is important to demonstrate that the phase

difference measurements obtained from this receiver are a measure of

the phase structure function. For an antenna spacing of d, the mean

square (m.s.) phase fluctuation is

a; (d) = <4(u) - gu2)) 2> where u1—u2 = d

For an isotropic phase structure

o2 (d) = 2 a- (1 — B (d))

4.1 •3

$~ 310- 4.1.4

since /(u1 ) (u2) = o; Bli(d)

B is given by egtn.4.1.1.

Since Bi4(d) ---s 0 as d , the general variation of m.s. phase

difference is of the form shown in fig.4.1.1.

Fig.4.1.1 General form of the phase structure function.

5 15 25 10 20 30

10 _ path length L

medium scale size

11,6 km

20 m 100 GHz

refractivity std. dev. 0,1 N

60 GHz

5

38 GHz

20 GHz

phas

e dif

fer e

nce (

deg

)

•

Fig.4.1.2 Phase difference as a function of spacing.

Antenna spacing (m)

212

In one dimension, using a Gaussian form for B1,

dpi = 2 e/ 1 - exp` 4.1 •5

For small receiver spacings, d « wo and the exponential in egtn.4'1.5

may be expanded as

exp{ —(d/wo)21 " 1 — (d/w0)2

Together, egtns.4.1.5 and 4.1.1 give

Ō~ F2 d or-AS

= 42 TT 4 k (L/w0)2 10-6

tTN d

where d.« 0

4.1 .6

4.1 •7

4.1.8

Initially, the rms phase difference increases linearly with receiver

spacing ie. when the receiver spacing is small compared to the medium scale

size.

It is further interesting to note that egtn.4.1.8 predicts

a decrease in rms phase difference with increase in scale size. This is

because the wave encounters effectively fewer inhomogeneities (N.'L/w0)

and the initial slope of the structure function is reduced by the

increased scale size.

Fig.4.1.2 shows plots of the rms phase difference predicted

by egtn.4.1.5 over a 11,6 km path. Before comparing the predictions of

this figure with practical measurements, the calibration of the phase

difference equipment will be described.

4.1.2 Calibration of the Interferometer.

The phase difference measuring apparatus was described in

chapter 3. This section describes the calibration of the phase detector

and the determination of the base level of phase fluctuations due to

the receiver itself. Also described is the method by which the receiver

spacing could be changed.

213

The calibration of the phase detector was undertaken using

the experimental arrangement shown in fig.4.1.3.

Tumwm i tter om

tth r►ntable.

S MNS.

Fig.4.1.3. Calibration of the phase detector.

The rotation of the transmit aperture about a point which is not the

phase centre of the aperture causes a phase difference between the signals

received by the two channels, as is shown in .appendix A4.1.

The 8 MHz phase detector was compared to the phase difference

measured by a Hewlett Packard Network Analyser connected as shown in

fig.4.1.3.

After this calibration, the transmitter was left fixed, and

the base level of phase fluctuation in the phase detector was sampled by

a microcomputer with an ADC. The base level was found to be about 0,3°rms.

There are two main contributions to this phase noise: IF instabilities

with differential dispersion of the IF chain and, noise in the multiplier

214

used as a phase detector.

In order to change the antenna spacing, it was necessary

to insert or remove extra lengths of waveguide which connected the mixers

to the local oscillator. Each change of spacing and 3 minute data

recording took about 10 minutes. To check the stationarity of the phase

difference over the 30 minutes required to take measurements at 0,8m,

2m and 3m, the equipment was left at a fixed spacing, receiving_ the

signal from the Crystal Palace transmitter. Fig.4.1.4 shows a 14 hour

period where the rms phase difference has been plotted. Each point was

calculated from 900 samples taken at 5 Hz. A slow decreasing trend is

seen, together with fluctuations of about 20% in the rms values. This

gives an estimate of the error bars associated with measurements taken

over different spacings.

4.1.3 Practical Measurements.

Fig.4.1.5 shows a number of sets of data for various spacings„

This data is typical of much of the data taken. The rms value was seen

to be a function of weather conditions, which set the strength and scale

size of the medium fluctuations.

The rms phase difference was almost always linearly

dependent on spacing, implying the scale size was much greater than 3m.

The magnitude of the phase difference measured fits the

predictions of fig.4.1.2 fairly well. The largest fluctuations .were

observed on warm, calm days and coincided with largest signal fluctuations,

as expected.

Plots of the temporal nature of the fluctuations are

given in figs.4.5.1 to 4.5.4 of section 4.5.2. They are also seen to be

closely correlated with the amplitude fluctuation behaviour.

4.1.4 Results from Other Workers.

The report published by Tolbert at al.4.1 has some

+

3 5cpcara4io i (m) 1

~SF.t wiII) Sr ea.

4 15'w15-

1000 7 JULY 1966 CLEAR 2-5 KNOT WINDS

f\i4 3d8•

SIGNAL POWER

H H5 sec

~Im SEPARATION

i Is.

PHASE DIFFERENCE

(a)

T

7m SEPARATION

215

3

bir

2-

I t I I c 1 I I I

12 24 3o 42 54 66 fi8 +; ow- C w; nufe6 ) —*

Fig.4.1.4 Stationarity of the phase difference measurements

Fig.4.1.5 Phase difference as a function of spacing (11,6 km link)

Fig.4.1.7 Phase difference at 94 GHz 4.1

216

interesting phase difference measurements taken at 9 and 35 GHz, on a

number of paths. An interferometer was used, with a variable spacing

available for the 9 GHz system. The 35 GHz system was fixed with about

a 2m spacing.

On a 10 mi (6km) path, the relevant data is given in

table 4.1..

Run rms phase difference

deg

rms amplitude rms field

dB %

1 2,3 0,1 1,2

2 4,2 0,4 4,7

3 6,1.. 0,7 8,4

Table 4.1 Phase difference and amplitude fluctuations 1.

These values compare facourably with the magnitudes measured on the

11,6 km path, taking into account the difference in geography and

path length. The path in question was well clear of the local surface

and urban areas.

Another set of interesting data is shown in fig.4.1.6,

which shows the range of rms phase difference measured on a 9 GHz,

3 mile (4,8km) path. The range of phase difference is seen to exhibit

two distinct peaks, and further, is still increasing with antenna

• spacing at 500' (170m). This would appear to be interesting evidence

of a range of scale sizes being present, at different times, in the

troposphere.

More recent work (Etchverry at al. ).exhibits some data

from an interferomter with spacings up to 7m. The propagation path was

19km at 94 GHz. The phase difference was.seen.to'increase with antenna

separation. Fig.4.1.7 shows some of the data. The most interesting

data, which is a number of plots of the receiver pattern response of a

i ""T)

1-" LO • .t:>-. -" • 1-0'1

"0 ::r 01 (/)

m

0. ~ 1-" -1) -1) m 1-:1 m ~ :::J t Cl m 5

N ~ ->

01 '-..; -.J eT·

1.0 -Er Cl b'<l :J: N • 4 .t:>-

• N

fOO '2.00 '!>oo MD 50D

218

4,6m dish, receiving the randomly perturbed signal, is discussed in

section' 4.3.

4.2 millimetre—wave Angle of Arrival Fluctuations.

The previous section has established theoretically and

practically, that for small spacings (<10m, say) the phase difference

measured by two small apertures increases linearly with spacing.

Fig.4.2.1 shows that physically, this is because the

spacing is much smaller than the phase scale size. Further, it is seen

that the incoming wave appears to have an angle of arrival fluctuation.

This section will investigate this viewpoint.

Fig.4.2.1 Phase fluctuations as angle of arrival.

4.2.1. The magnitude of Expected Angle of Arrival Fluctuations.

Fig.4.2.2 shows two small apertures a distance d apart,.

receiving a plane wave at an angle to the normal to AB

e

Fig.4.2.2 Phase dif-ference/angle of arrival relationship.

219

The phase difference between the waves recieived at A and B is

k91 = k s

where k = 27i /A

or Q~ = k d sin 9 ti k d 9 for small

cr = k d ore Thus

4.2.1

4.2.2

4.2.3

where a- is the rms angle of arrival fluctuation.

In the practical system, spacings of 0,8m, 1,9m and 2,9m may be

chosen. Table 4.2 shows the AS std. dev. predicted by egtn.4.2.3,

for given angle of arrival std. dev., o-8 .

.

d(m)

0,8 1,9 2,9

ce cr Li (deg) (deg)

0,01 0,64 1,51 2,31 0,03 1,91 4,53 6,92 0,10 6,36 15,10 23,10 0,15 9,54 22,67 34,60 0,20 ' 12,73 30,22 46,10

Table 4.2 Phase difference due to angle of arrival.

Given that the statistics of the phase modulation imposed

by the medium is known, it is simple to calculate the statistics of 0,

the angle of arrival fluctuation. Assuming a Gaussian autocovariance

(egtn4.1.1) and that d4K wo, egtn.4.1.8 may be used in eqtn 4.2.3:

k d or9 o0~ = 22 d a/ /w0

Or ore = 27 a./ /k wo 4.2.4

= 7' 2 22 (LA.). o-N . 10 6 4'2.5

Thus, in principle, for small aperture (d €Zwo) antennas, the medium

may be thought of as inducing random tilts to an incoming plane wave.

220

The next section will examine the performance of an antenna in

such conditions.

4.2.2 Antenna Performance during Angle of Arrival Fluctuations.

An antenna whose aperture dimension is much smaller than the

medium scale size, effectively sees the medium as inducing random

tilts to an incoming plane wave. Suppose the receiver pattern function

is Gaussian:

Er(8) = exp C—(8/ Go ) 2] 4.2.6

where G. = X /7ta, a being the aperture dimension.

The incoming wave is now assumed to be incident at angles 9 , and it

is assumed that B is normally distributed with variance 4 . It is

then possible to show (appendix A4.2), that the mean and variance of

the signal received, will appear to be

< E> = 1 — ab /Oo2 4.2 7

cr = 4 (re /eo )4 r

where it is assumed that a-0/00 « 1

4.2.8

It is, in.fact, possible to show in more general terms (appendix A4.3)

that

G Er> = exp [ crē P6,21

which holds even when a /6n is not small.

4.2.9

It must be stressed that this reduction in recieved field

is in excess of the exp[— o /2J by which scattering has reduced the

amplitude of the coherent wave after propagation through the random

medium. It will be shown in section 4.6, however, that this reduction

is likely to be negligible unless the transmitter or receiver antenna

are misaligned.

221

4.3 Amplitude Scintillation.

This section is in two parts; predictions of the

nature and magnitude of amplitude scintillations are made, followed

by a subsection presenting some of the observations made on the practical

link. The observed fluctuations proved to be very small, and a great

deal of care had to be taken with the recording and analysis of the

data.

Because of the small size of the fluctuations, recordings

of the mean signal strength were out of the question, since the known

fluctuations in equipment gain and so on were known to be larger, in

the long term, than the fluctuations in signal due to the medium. The

signal was thus conditioned before recording. This is taken up in

section 4.4.

4.3.1 Theoretical Predictions.

The theoretical approach taken is very simple. The analysis.

of chapter one has shown that each component of the coherent angular

spectrum of plane waves proagating through the random medium is reduced

by a factor ex o / p 2J . a., is the variance of the phase fluctuation

induced along the whole path. This assumes that the antennas used have

broad angular responses.

Although physically small apertures at millimetre—wave

frequencies have a narrow beamwidth, they will have-a much broader

response than the scattered spectrum of waves due to the large

refractive inhomogeneities. Thus nearly all the scattered energy may be

thought of as being received by the receiver.

To a good approximation, then, the coherent power

coupled between two perfectly aligned atennas is

c = PR/PT oG DT DR exp co 4.3.1

222

where DT and DR are the transmit and receive antenna directivities.

This is the non—random power which is coupled. Since most

of the scattered power is being received on the main lobe of the receiver

antenna, the ratio of the random power to the non—random power is

Pran~ < P> = = 1 — ex[_]

exp\ o-

= exp ~ — 1

= Q if cr K 1 its

The field fluctuation, in % is then

Y = Eran/(E> = 100 a ( %

4.3.2

4.3.3

For a Gaussian refractivity autocovariance, oro is given by egtn.4.1.1:

= n2 2 wo L 10-122 4.3.4

where k = 27T /A

L is the path length (m)

wo is the medium scale size (m)

PN is refractivity variance. (N2)

Figs.4.3.1 to, 4.3.3 show the % fluctuation of the received field for

various millimetre—wave frequencies, under different atmospheric

conditions. It is interesting to note, that for most atmospheric

conditions, the percentage field fluctuation, r , is only a few

percent.

The probable statistics of the received fluctuations are

discussed in chapter one. This matter is also taken up in more detail

by a co—worker, Mashhour8 1. The fluctuations may be described by the

distribution of a Hoyt vector ie. a strong non—random component plus

a small amplitude component in random phase relationship to the main

component. The random component itself may be resolved into two zero

14 -

223

4 -

2 —

S 6 3-

diyfaurG (km.)-->

Fig.4.3.1 % fluctuation versus path length.

4= too Gt{?

10

60

0.1 0.01

scale size. (na) at 1,0 10

Fig.4-'3.2 % fluctuation versus scale size.

f= IooGNi ZO

224

r-. 10

CCL

D.1 1 10 100

Refrarii vi+c stā. dev (TN (N)

Fig.4.3.3 '96 fluctuation versus refractivity std. dev.

225

mean, jointly normal vectors, which may. have equal• .vatiances.

4.3.2 Practical Measurements.

Initial recordings were made of the detector output on a

slow chart recorder (1"/hr). This was done for some months to obtain a

general idea of the behaviour of the link.

One of the first manifestations was the correlation of the

fluctuation intensity with wind speed. As mentioned in the previous

chapter, this was traced to hut movement and was almost cured by

re—engineering the interferometer supports. Measurements were, however, •

ruled out during strong or blustery wind.

Fig.4.3.4 shows a typical period from early to late

morning on a calm sunny day. The fluctuation intensity is seen to be

extremely small, picking up later in the morning. This increase is due

to surface heating by the sun. Localized hot surfaces, especially in

an urban environment, lead to warm parcels of air rising under

bouyancy forces. In low wind conditions, the eddy processes are

negligible and the warm parcels of air have a large scale size. The

fluctuation in field, , is linearly proportional to o-N and to the

root of the scale size (eqtns 4.3.3, 4.3.4). The prediction is thus

that calm, ; warm weather produces the most fluctuations.

The predictions of the previous paragraph were fully borne

out in practice. The largest fluctuations observed were during calm

weather when the sun warmed the surface which had been made wet by dew

or rain. The local heating in this case leads to clouds of warm, moist

air with large scale size. In these situations, p values of up. to

10% were recorded.

Fig.4.3.5 shows a chart recording during a period of frontal

activity characterized by rain squalls driven by strong wind. The onset

of attenuation is seen to be rapid, with a slower increase back to full

•

nr.oo

226

10"00

4(

S~Oo

tiW\~ (\\(5)

Fig.4"-3· 4 38 GHz s~~inal, morning to noon.

Field (aTb. \!V\i1~)

IShOO 13 hOD tZhOO U '-00

411i:ii<::----- iime (hf5)

Flg.4·~·5 38 GHz during rain.

'1"00

to" 00 'l~oo

227

level. Although rainfall attenuation was not investigated in detail,

the feeling obtained was that for paths of up to 10 km or so, rainfall

is the only major problem for millimetre—wave propagation.

It also became apparent that heavy rainfall rates were not

necessarily associated with the strongest fades; the opposite being

true, in fact. Roughly speaking, it appears that a maximum attenua-

tion of about 10 dB is reached, this being during widespread, light

rain. This phenomenon high—lights the necessity for taking into account

the spatial extent of rainfall when making attenuation predictions.

Unfortunately, at present little such data exists, but the growing

use of weather radars should change this situation. Until such time

as this is taken into account, the design of millimetre—wave

communications links will be unnecessarily pessimistic.

In-conclusion, it is felt that the simple practical

measurements above give some confidence in the theoretical treat-

ment and the atmospheric models used in the first two chapters.

4.3.3 Comparison with Other Experimental Data.

Some of-the earliest reported millimetre—wave propagation

work is due to Tolbert et al.. 1. On a 96 km path at 35 GHz, the field

was observed to fluctuate 18% rms, of the mean. The prediction of

fig.4.3.1 is 15%, assuming that the atmospheric parameters are similar.

A 16km path was also measured and yielded 5% field

fluctuations, against a prediction of 6% from fig.4.3.1.

Fig.4.3.6 shows results taken by Etchverry et al.4 2.

These are for a 19km path at 94 GHz. It can be seen that even with

spacings of up to 7m, the field fluctuations are well correlated, as

is predicted by the theory used here. This is confirmed by the

multiple scans of the receiver antenna, showing displacements of the

pattern which could be associated with angle of arrival fluctuations of

7m SEPARATION

RELA

TIVE PO

WER

228

0400 15 JULY 1966 CLEAR CALM

SIGNAL POWER

-H 5see

PHASE DIFFERENCE

(a)

0400 15 JULY 1966 CLEAR CALM

(b) (a) Amplitude and phase fluctuations, relatively quiet atmo-

sphere. (h) Superposed antenna patterns, relatively quiet atmosphere.

Fig.4.3.6 94 GHz signal behaviour

0700 15 JULY 1966 CLEAR CALM

1000 7 JULY 1966 CLEAR 2-5 KNOT WINOS

--i 1--5see CHANNEL I

d8 1.0

CHANNEL 2 ANTENNA SEPARATION 7m

dB

Log-amplitude fluctuations, 7-m separation.

-. f.-5see

k- 25 arc sec

(b) .

(a) Amplitude and phase fluctuations, turbulent atmosphere. (b) Superposed antenna patterns, turbulent atmosphere.

0900 HAZY CALM

I JULY 1966

18•

T Phase difference, quiet atmosphere, 1-m separation.

229

the incoming wave. This is further confirmation of the idea of large

scale sizes as being the most important in the atmosphere.

A number of papers have been produced describing observations

at. 36 and 110 GHz on a 4km path which, in fact, ises the receiver site

of this work for transmission. (References 4.17 to 4.21, 4.6).

Although the locality is similar, the path geometries are too dissimilar

for direct comparison The 4km path passes, with very little clearance,

through a number of large ventilation ducts. In general, the

fluctuations measured on this path are many times larger than on the

11,6km path, which has an average clearance of 30m.

4.4 Signal Conditioning.

The original experimental design had envisaged recording the

two amplitude and phase difference channels onto a magnetic FM tape—

recorder. It soon became apparent that this was not feasible, due to

the small size of the fluctuations compared to their mean values. The

tape recorder noise level would have seriously biased the analysis.

A simple offset circuit, with amplification, is unsuitable

since a small shift in mean value quickly leads to amplifier saturation.

A running mean circuit was thus devized. The first amplifier of fig.4.4.1

has a gain of 1 and a time constant of about 4 minutes. The second is

a difference amplifier, fed by the fluctuating signal and the averaged

version.. The gain of this second stage may be varied at will.

The circuit proved to be most useful in practice, and

allowed fluctuation measurements to be made during rain, when the mean

signal-had been much reduced. The output of this circuit was used for

recording purposes.

The temporal analysis of the data was carried out on a

PDP 15 computer with a 12 bit resolution ADC (ie 1 part in 4096).

The contribution of digitization noise is thus negligible, since

230

Fig.4.4.1 Running mean removing circuit.

240

the fluctuating signal was amplified to occupy the full dynamic range

of the AOC.

4.5 Amplitude Fluctuation Frequency Spectra.

The computation of the frequency spectra of fluctuating

signals has had a wide exposure in the literature; indepth details of

the methods used to produce spectra will thus not be given, but are

available 1. A useful reference is Bendat and Piersol 2.19

Spectra were produced in two ways: by digitally sampling

and computing using a FFT algorithm, or, using an analogue, real—

time spectrum analyser. This latter device was described in chapter 2.

The smoothing of the spectra produced by either method is

the usual trade off of smoothness versus resolution. In the case of

the digitally calculated spectra, the input time series of _sampled

data was broken up into 10 records. A transform was performed on each

record and an "average spectrum" was produced by averaging these 10

spectra.

4.5.1 Theoretical Predictions.

It is possible, in principle, to find the temporal

fluctuation spectrum in an exact form if the aūtocovariance (spatial

or temporal) function of the medium refractivity is known. If this

knowledge exists, then the field autocovariance function after

propagation through the medium can be calculated using the lateral

coherence -function approach developed in chapter one.

In chapter one it was shown that for a tenuous random

medium, the fluctuating component of the field was in fact the

autocovariance of the induced phase fluctuation ie.

rt. (u, z ) = r0(t, t 4.5.1

241

The Īto function can be evaluated, provided the statistical nature of

the medium is known. Idealistically then, the temporal scintillation

power spectrum is simply derived by taking the Fourier transform of

egtn.4.5.1 ie.

4.5.2

Usually rr may be split into a product of spatial and temporal component functions. This state of affairs is only true for zero wind speed

conditions. The effect of wind on the random refractivity was discussed

in chapter 2. Taylor's Hypothesis may be used (with due care) to show

that the spatial refractivity fluctuations are carried past the receiver

and become temporal fluctuations. The resultant temporal refractivity

function is thus composed of fluctuations due to the upward movement

under bouyancy or wind shear, plus the horizontal wind induced movement.

For wind speeds greater than a few ms-1, the vertical

movements may be neglected and only vw, the wind speed is considered.

A fluctuation of scale size wo will be carried past a receiver in a

characteristic time

o wo/vw 4.5.3

Thus, under frozen turbulence conditions, spatial autocovariance may

be transformed into a temporal function by egtn.4.5.3. For simplicity,

take a Gaussian autocovariance

1 r(u) = eXp —(u/wo)2 1 4.5.4

This is transformed to a temporal autocovariance

Ir r(ir) = expl7(vw1/w0)21 4.5.5

The fluctuation spectrum is thus of the form

Ito) oG exp C—(woW/vw)2]

4.5.6

Although the function chosen .was idealistic, the following predictions

are true for any monotonically decreasing r r(',) ie.

242

i) The spectrum broadens with increase in wind speed.

ii) The spectrum broadens as wo decreases.

Again, if a more realistic spectrum is required, a

number of Gaussians may be combined, as was outlined in section 2.6.4.

4.5.2 Measured Scintillation Spectra.

Fig.4.5.1 shows the spectra obtained on a calm, clear

morning. The atmospheric scale sizes are expected to be large, and as

predicted by the model in the previous section, the fluctuation

spectrum is narrow.

Later in the day, the air temperature increased, as did

the average wind speed. The increase in fluctuation energy due to

local heating and flow over the rough, urban surface indicates that

smaller scale sizes would also be important. The spectra shown in

fig.4.5.2 confirm this hypothesis., The amplitude fluctuation spectrum

is considerably wider than for the calm conditions. The average slopes,

however, seem to be essentially the same.

Fig.4.5.3 should be compared to fig.4.3.1. The air temp-

erature is not radically different to the first figure, but the wind

speed is much higher. The second figure is seen to have a much wider

spectrum.

Fig.4.5.4 shows spectra taken during light rain. The mean

signal was reduced, but the fluctuation was only a few percent of the .

mean; in fact, not radically different to clear air conditions.

The spectrum is observed to be much broader, however than

the clear air conditions. This matter is taken up in the next section,'

where spectra computed over a wider frequency range are reported.

It may be observed from these four figurest'that the

phase spectra are directly related to the fluctuation spectra, as

is expected from the analysis of the previous section. The phase

1

~r-

0i1 f (Hz) 1,0

-10 -

Wind velocity 6 ms

Air temp. 19C

-30 -

Pow

er s

pec

trum

(dB

)

-Zo

1,0 f' (Hz)

Phase Difference Amplitude

Pow

er s

pec

trum

Phase Difference

Fig.4•S)3 Fluctuation spectra.

Fig.4+5F•.4 Fluctuation spectra during rain.

0.1 f (Hz) 1,0 0.1 f (Hz) 1.0

0,1 f(Hz)

Phase Difference

a

0,1 f (Hz) 1,0

Pow

er s

pec

trum

(dB

) -10

-10

-30 -

-30 - Air temp. = 27C Wind velocity = 3 ms-1 •

Phase Difference Amplitude

-20 -

•

Air temp. = 21C Wind velocity = 5 ms -1

Fig.4•S•1 Fluctuation spectra.

Fig.4.5.2. Fluctuation spectra.

0,1 f (Hz) f (Hz)

245

"ripple" imposed by the medium is a "phase image" of the medium,

whick is,.—in-turn, being carried along by the mean wind speed. The

higher this speed, the faster are the expected fluctuations in phase

difference.

This interrelationship between the phase structure of

a wave, its amplitude scintillation and the medium scale size is

clearly demonstrated in chapter 6, where the results of a computer

simulation are presented. The plots of the spatial distribution of

phase and field shown in this chapter show up the importancs of the

medium scale size in determining the scale size of the wave fluctuations.

4.6 Gain Reductions in the Practical Situation.

Using the results of the previous few sections, it is

possible to estimate the performance of ••aperture,-antennas,- under

different atmospheric conditions. Rather than examine all possible

paths and frequencies, only a few, typical results will be presented..

The path parameters chosen are:

length L 5km

scale size w 0

20m

frequencies 20, 40, 60, 100 GHz

refractivity aN = -0 to 0,1 (N2)

The received.power reduction is in two parts:

exp oisl due to scattering

4.6.1

exp= 2 028 /80l due to angle of arrival

4.6.2

a2 is calculated from egtn.4.2.5

The total gain loss is thus

1(dB) = 10 Log[ e(1+(2/kw0e0)2 o;) 4.6.3

Fig.4.6.1 shows egtn.4.6.3 plotted for a range of medium

refractivity variance. This has been done for an antenna beamwidth of.

0,50. An unrealistic beamwidth of 0,01 is also shown to demonstrate

6~ 0 Fig.4.6.1 Gain reduction as a function of!,!refractivity std:. dev.

0,6° beamwid{G

01 01 ° brawl widtln.

PatL L = S km Scale site Wo = 2o111

3,0

g

r 2,0 •

11 0

247

the very small gain reduction due to angle of arrival.

It should be pointed out that although angle of arrival

may seem negligible, the situation becomes much more serious if the

receiver aperture is slightly off—axis. The small angle of arrival

fluctuations are amplified by the antenna response curve; this is,

of course, exploited by monpulse radars. The analysis of appendices

A4.2 and A4.3 should be repeated, with suitable changes to the mean

value of the incoming angle of arrival.

4.7 Scintillation Spectra during Rain.

The mean—removing circuit described in section 4.4

allowed recordings to be made during rainfall, when the mean signal

was much reduced. The next section will show :.same of these results

and explain them via a simple model.

4.7.7 Practical Measurements.

Fig.4.7•1 shows a number of spectra taken'during rain. The

real time spectrum analyser was used to produce these spectra. The most

obvious and interesting phenomenon is the appearance of higher frequency

peaks in the spectra. These peaks seem to shift to higher frequencies

as the wind speed increases. Unfortunately, there are not enough of

these measurements available to correlate the peak of the spectrum

definitely with the wind.

4.7.2 A Possible Explanation.

A simpIb-model will be used to try and correlate the

occurrence of a peak in the spectrum with wind speed. In fig.4.7.2,

a single drop is assumed to be carried horizontally by the component

of the wind speed perpendicular to the path.

The component of velocity along the line OR is

248

-{0

-10

-30 10 12. 14 ti. 18 Z0

-t0

-Z0

0

0 A0 20.

Fig.4.7.1 Spectra during rain.

249

Fig.4.7.1 Rain drop scattering

v(6) = vl sin 6 4.7.1

All drops along the line OR will scatter energy omni-directionally.

()%).> drop size): this energy will arrive at the receiver aperture with

a time varying phase compared to the unscattered energy. This will give

rise to a flux tiiation frequency of

= r = k v1 sine 4.7.2 dt

(k = Zit/A )

The energy = rē reined-- is -Obtlportionai-ta -the amplitude-of • the_ pat-tērn

function, A(9). Evidently then, the received fluctuation temporal.

spectrum will be

(w) cc A09) 4.7.3

where I.1 is the component of the fluctuation spectrum due

to the drop velocities perpendicular to the antenna axis. A(9) may

written as A(w) by using the functional relationship of egtn.4.7.2.

Similarly, the component of wind velocity parallel to

the antenna axis will give rise to fluctuation frequencies

CO = k v case 4.7.4

where vo is the parallel component of wind speed.

Now, suppose that the antenna pattern function is

250

• A(e) = exp[ (e/8,)2J 4.7.5

Since narrow beam antennas are being considered, small angle

approximations may be made for egtns.4.7.2 and 4.7.4. The two spectra

resulting are

and

exp 172/ 0e2 k2

in(u0 cC exp C — 2J{k vil —W 1/(k vii,]

4.7.6

4.7.7

These two spectra are sketched in fig.4.7•Z,

116

Fig.4.7••2 Spectra due to rain drop scattering.

The two frequencies for which the spectra are both 1/e are:

W2

- k vl OO 2

k vn (1— 02/2)

4.7.8

4.7.9

These are both narrow spectra. In practice, the wind direction is

random in time and varies at different parts of the path. The spectra

above will this be "smeared". A more realistic form is shown in

fig.4.•7.3, where a number—of realizations of v1, v/1 have occured during

the time taken to measure the spectrum.

The important point is that a peak in the spectrum is

predicted at

= k <v„> 4.7.10

251

Fig.4.7s3 Predicted spectrum during rain.

Here, 4 v11~ is the mean parallel component of the wind.

The terminal velocity of the raindrops has been neglected

in the analysis so far, but it will have the same effect as the

perpendibu ar• 'comppna ttsf_ • wind.•speed, v .

Suppose that the parallel component of the wind speed

is known. The peak in the spectrum should correlate with this value.

It is known that the wind was essentially perpendicular to the path

when the spectra were measured. If the parallel compone?ent was thus

only 0,1 ms-1, the peak in the spectrum should occur at

f = u1, /X i 0,1/10-2 ie. 10 Hz

This is of the right order. Unfortunately, the prevailing wind respon-

sible for most rain is almost exactly perpendicular to the path, and

data was thus difficult to obtain for a range of wind directions.

4.8 Conclusions.

This chapter has shown that the theory of chpater one,

together with quantitative values for atmospheric variables produces

realistic predictions of the behaviour of millimetric radio waves

propagating through the clear atmosphere.

252

It is further felt that the practical measurements taken

can only strengthen confidence in the use of millimetric radio waves.

It is seen that precipitation is the only serious hazard. Even here,

more practical measurements of the spatial distribution of precipitation

intensity are needed before realistic link performance can be

predicted.'

The measurements of this chapter confirm that in a tenuous,

random medium, millimetre—waves may be modelled as being essentially

the same as in free space conditions, but further, having a slight

random phase modulation. The scale size of these phase effects is the

same as the medium (ie 10 to 100m).

These phase effects lead to minute amplitude fluctuations;

the magnitude of these fluctuations may be simply estimated.

The temporal nature of the phase and amplitude effects is

also seen to be directly related to the medium refractivity fluctuations.

253

APPENDIX A4.1 Determination of the Phase Centre of an Aperture.

To calibrate the interferometer, a transmitting horn was

rotated about a point away from its phase centre. It will now be

shown that this forms the basis of a technique for finding the phase

centre of an aperture. Fig.A4.1.1 shows the experimental arrangement.

Fig.A4.1.1 Experimental arrangement to find phase centre.

Co is the phase centre of the transmitting antenna. The antenna has

been rotated about a point CR, which is a distance 1 from Co.

A and 8 are two receivers, a distance L from C', spaced a distance

2D apart.

The phase difference between signals received at A and 8 is

i = k (C08—00A) a4.1.1

If the angle 9 is small, and Lei 1, then

OE'" 1

Co D AIL

C A+C 8"0 2L -o 0

Hence,

CoB — CoA k 2D10 /L

or 1 = L Ai/2k0

a4.1.2

a4.1-3

a4.1.4

254

This analysis assumes AB is perpendicular to CRE and that CRE

bisects-AB.

The analysis may be repeated, this time assuming that

CRE is displaced a distance d upwards, as in fig.A4.1.2.

new aX1s

Fig.A4.1.2 Displacement of the axis of symmetry.

R

T i d 2D

Hence,

L1~ = . 2kDd/L + 2kDl8/L a4.1.5

In a practical situation, the exact phase centre of an

antenna is often unknown. It is a simple matter to use the above ..

analysis to a^curately determine the phase centre. The antenna is

placed on a turntable, a known distance from two receiving horns.

The turntable is rotated in the usual manner, except that the phase

difference of the two receivers is noted for two values of rotation

angle, 91

and 02. IfLli1 and L42 are the phase differences

measured, then

where

b 541 d + p i e.]

1/2 _ 1d +'lA 2 a4.1.6

= 2kD/L

It is not necessary to know the displacement of the axis of symmetry, d,

since,

1 = 1 1 A01 14512 13

01 02

a4.1.7

255

APPENDIX A4.2 Statistics of the Field Received during

Angle—of—Arrival Fluctuations.

The situation is shown graphically in fig.A4.2.1. The

object of this analysis is to predict the probability distribution

of the field received.

Fig.A4.2.1 Amplitude fluctuations caused by angle—of— arrival.

Generally, the angle of arrival fluctuations are minute,

and a parabolic approximation to the receiver antenna pattern function

is possible, ie.

E(e ) = exp(—(9/60)2) n. 1 — 2 a4.2.1

The probability distribution of B , the angle of arrival, will be

assumed to be normal. z

Pg P 1 ex L e .1

a4.2.2 Q =

—

The assumption that 9 is normal is reasonably realistic, as can be

seen in fig.A4.2.2. This shows histograms of the angle—of—arrival

measured by the interferometer described in chapters 3 and 4.

To find the probability distribution of E, the received

field, it is necessary to transform according to:

-43

CD

256

9-m

CD

— 4

- M

—

Fig.A4.2.2 Normalized histograms made of angle of arrival measurements

made on the 38 GHz link over 11,6 km.

PE(6) = Pes_) + Pe(62)

\Pei1 \P8(82),

Pe = (Pe) where

a4.2.3

257

and 01 , e 2 are the roots of E(8) = o.

This type of transformation is discussed by Papoulis1.

It is more convenient to work in terms of

e = 1 — E = e 2/A2 0

The probabability function pe is thus, using a4.2.3,

Pe(e) _s~ e 2 exp~ el e

a4.2.4

a4.2.5

Once Pe is known, the mean and variance follow fairly simply:

ao

e'> = S e pe _ (6) de J eo

{ • e' 1 (3/2) (2 0'g 3/2

= Cg 2/ eo2 82

ae var[e] = (e — Ge))2 pe de

4(Cpt

Transforming back to E(9), eqtns. a4.2.4,5,6,7 give

a4.2.6

a4.2.7

PE 1 80 1 exp{-0(1—E)1 a4.2•8

afr as W7=if L e

G E> = 1 — Qg2/8,2

Ē 4 ( \4 Bo

REFERENCE.

1 Papoulis, A. 1965. "Probability, Random Variables

and Stochastic Processes" McGraw—Hill, 1965.

• a4.2.9

a4.2.10

258

APPENDIX A4.3 More Exact Analysis of Angle—of—arrival Statistics.

The previous appendix assumed that the angle of arrival

fluctuationswas small compared to 00, the antenna beamwidth. This

appendix will relax this condition. The pattern function of the

antenna is still assumed to be Gaussian as in fig.A4.2.1.

Using the same transformation as a4.2.3 but making the

substitution

= ln E

it may be shown that

px(e) = —2rr ere x 2

exp L 2

at? x~

It is important to remember that the domain of Xis

—ōo < x.

a4.3.1

a4.3.2

a4.3•3

Thus:

and

(ao 0

4. X. > =pt = Cie. p, d

= •0 X) pa (-dX) 0

_ -c /90

Vat LL% Ī _ = CX —(xi )2 px dX. :1 ~

_ —4(261' )4

a4.3.4

a4.3.5

The above equations are exact and may be used even when the angle of

arrival fluctuations are large compared to the pattern function

beamwidth.

259

CHAPTER FOUR REFERENCES.

4.1

Tolbert, C.W., Fannin, B.M. and Straiton, A.W. 1956.

"Amplitude and Phase Difference Fluctuations of 8.6

Millimetre and 3.2 Centimetre Radio Waves on Line—of-Sight

Paths" University of Texas Electrical Engineering Research

Laboratory Report No.78.

4.2 Etchverry, R.D., Heidbreder, G.R., Johnson, W.A. and

Wintroub, H.J. 1967.• "Measurements of Spatial Coherence

in 3.2—mm Horizontal Transmission" IEEE Trans. Antennas

and Propagat. AP-15, No.1, Jan.1967.

4.3 Lee, R.W. and Harp, J.G. 1969 "Weak Scattering in

Random Media, with Applications to Remote Probing"

Proc. IEEE, 57, No.4, April 1969.

4.4 Roche, J.F. et al. 1970 "Radio Propagation at 27-40 GHz"

IEEE Trans. Antennas and Propagat., AP-18, No.4, July 1970.

4.5 Thompson, M.C. et al. 1975 "Phase and Amplitude

Scintillations at 9.6 GHz on an Elevated Path"

IEEE Trans. Antennas and Propagat. Nov. 1975.

4.6 Vilar, E and Matthews, P.A. 1971 "Measurement of Phase

Fluctuations on Millimetric Radiowave Propagation"

Electronics Letters, 7, No.18, Sept. 1971.

4.7 Lai—iun Lo et al. 1975 "Attenuation of 8.6 and 3.2 mm

Radio Waves by Clouds" IEEE Trans. Antennas and Propagat.

AP-23, No.6, Nov. 1975.

4.8 Barzilai, G. 1975 "Research on Statistical Aspects of

Tropospheric Propagation" Radio Science, 10, No.7.

4•g Mandics, P.A. et al. 1973 "Spectra of Short—term

Fluctuations of Line—of—Sight signals: Electromagnetic

260

and Acoustic" Radio Science, 8, No.3, March 1973.

4.10 Minott, P.O. 1972 "Scintillation in an Earth—to—Space

Propagatiōn Path." JOSA, 62, No.7, July 1972.

4.11 Alferness, R. 1974 "Self-induced Irradiance Fluctuations

of a Laser Beam in an Absorbing? Turbulent Medium"

JOSA, 64, No.12, December 1974.

4.12 Brusard, G. 1976 "A Meteorological Model for Rain

Induced Cross Polarzation" IEEE Trans. Antennas and

Propagat., AP-24, No.1, Jan.1976.

4.13 Troughton, J and Evans, B.G. 1976 "Rain Induced Deflection

of Microwave and Millimetre—Wave Radiowaves"

Electronics Letters, 12, No.3, Feb. 1976.

4.14 Liang—chi Shen 1970 "Remote Probing of Atmosphere and

Wind Velocity by Millimetre Waves" IEEE Trans. Antennas

and Propagat., AP-18, No.4, July 1970.

4.15 Grody, N.C. 1976 "Remote Sensing of Atmospheric Water

Content from Satellites using Microwave Radiometry"

IEEE Trans Antennas and Propagat., AP-24, No.2, March 1976.

4.16 Leung Tsang and Jin Au Kong 1976 "Microwave Remote

Sensing of a Two—Layer Random Medium" IEEE Trans. Antennas

and Propagat., AP-24, No.3, May 1976.

4.17 Mavrokoukoulakis, N.O. et al. 1977 "Observations of

Millimetre—Wave Amplitude Scintillations in a Town

Environment" Electronics Letters, 13, No.14, July 1977.

4.18 Ho, K.L. et al. 1978 "Determination of the atmospheric

refractive index structure parameter from refractivity

measurements and amplitude scintillation measurements at

36 GHz" J. Atmospheric and Terrestrial. Physics, 40.

4.19 Viler, E and Matthews, P.A. 1978 "Summary of Scintillation

Observations in a 36 GHz Link across London"

261

IEE Antennas and Propagation Conference 1978. •

Conference Publication No.169.

4.20 Ho, K.L. et al. 1977 "Wavelength. Dependence of Scintil-

lation Fading at 110 and 36 GHz" Electronics Letters,

13, No.7, March 1977.

4.21 Cole, R.S. et al. 1978 "The Effect of the Outer Scale

of Turbulence and Wavelength on Scintillation Fading at

Millimetre Wavelengths" IEEE Trans.. Antennas and Propagat.

AP-26, No.5, Sept 1978.

262

CHAPTER FIDE A COMPUTER SIMULATION OF PROPAGATION THROUGH

A RANDOM MEDIUM.

Chapter 1 established a theoretical basis for describing

propagation through a medium containing tenuous volume irregularities

in refractive index. This was done by means of a slab model, which

was shown to be sufficiently accurate for millimetre—wave propagation

problems.

The next chapter described the lower troposphere and gave

indications of the intensity of the refractivity fluctuations.

Further, a. statistical description was introduced, of the spatial

and temporal nature of the medium,., which could simply be incorpor-

ated

into the propagation model.

Chapters 3 and 4 showed that a practical propagation path

behaved in a manner. which may be related to the model introduced.

These chapters also high—lighted the large practical difficulties in

making definitive measurements of the spatial and temporal nature of

the incoming wave. The principal- difficulties may be associated with:—-

lack of detailed knowledge of the propagation medium— especially

its homogeneity and whether it is statistically stationary

ii) problems with building instrumentation stable enough to resolve

the minute signal fluctuations.

iii) the data gathered on one path is peculiar to a particular

climate and locality.

A computer simulation solves many of the above problems,

as will be shown in this and the following chapter.

263

5.1 Theoretical Considerations.

The work of Fejer and Bramley5.2

is an important

precursor to this simulation. The difference between their approaches

was discussed in chapter 1, and they were seen to be equivalent.

Essentially, Fejer divided the propagation path into slabs.

He calculated the scattering of an incident plane wave due to the

volume irregularities of the first slab. This produced an angular

spectrum of energy which was allowed to be perturbed by the next slab,

and so on through the medium.

This produced an emergent spectrum which had been multi-

ply scattered. Bramley showed that the same result could be obtained

by considering the random phase shift along a number of parallel paths

through the medium, and condensing the medium into one random phase

screen. The reason for this equivalence was demonstrated in•chapter 1,

both physically and theoretically. The model which will shortly be

presented is seen to be a mixture of these two approaches.

5.1.1 Practical Difficulties with the Fejer—Bramley Model.

The model presented by Fejer and Bramley deals with

the propagation of a single plane wave, which is difficult to recon-

cile with finite beam practical situations. Since the beam extent is

finite, larger scale irregularities may be thought of as be ing essen-

tially constant over the beam and are effectively "cut off". However,

as the beam diffracts and broadens, the "cut off" scale size becomes

larger.

Information about the field fluctuation distributions is

difficult to obtain. Fig.5•1•1 shows how the distribution is likely

to be different at various distances from the beam axis. This is due to

the change in the ratio of incoherent to coherent power. The scattered

power level has been exaggerated. The on—axis resultant vector does

E4.

incoherent

264

fig. 54'1.1 Field fluctuation characteristics as a function of distance from the centre of a beam.

265

not fluctuate much in power or phase, since the incoherent field is

small in comparison. This situation degrades until, at a point

completely off—axis, the field is completely fluctuating and the

phase becomes that induced by the medium and can take on all values.

5.1.2 Basic Theory of the Simulation.

This subsection will develop the mathematical basis of

the computer simulation. The physical model is, in fact, similar

to that of Fejer. The simulation will, however, be seen to rely

on the idea of equivalent phase screens. This idea has been fully

justified, and so the computer simulation is fully multiple scatter.

As was shown in chapter 1,. the field over a plane and its

angular spectrum are related by the Fourier transform.

E(x,y,z) F-F'T A(k) 5.1 .1

The initial field is assumed to be known. Using eqtn.

5.1.1, the angular spectrum may be deduced. Consider one plane wave

of that spectrum that propagates as in fig.5.1.2, to the observation

plane. Neglecting amplitude and the exp L—ikzo]phase changes, the

spectal component A(k) is seen to transform as

A(kx) A(kx) exp`ikx zo Cos 9]

(only 1 dimension will be considered throughout).

Thus in general,

e(x') = —1 A(k) expfikzoCJ

5.1.2

where C = Cos 8 L

Since the energy is essentially forward scattered and narrow beam—

width antennas are considered, a small angle approximation is possible:

C = 1 — Sin28 J 1 — 62/2

where S = Sin6

Apply random phase screen

Random case only

1

Define initial aerture field

Transform to find the Angular Spectrum

Apply Fresnel correction to find spectrum over. receiver plane.

1 Inverse transform the corrected spectrum to find the field over the receiver plane.

Repeat N times

fig. 5'1'3 Basis of the simulation program.

A x A l

266

fl L)

Equz -- Phasc (R(kx) )

2_=0 7__=20

fig. 5'1'2 Justification of the Fresnel correction.

CO5 e - zo)

k ia ®Z 2

for e << 1/4 raa

267

oo

Hence e(x') = exp ikzoS2I A(X) exp ix'k dkx 5.1.3

-o0 2

Now S = kx/k Qo

e(x') exp izokX A (kxC

) exp ix'kx] dkx 5.1.4

eo 2k

This approximation holds for kx/k <1/6 rad and is known as the

Fresnel approximation. This matter was taken up more rigorouely in

Appendix A1.6 where the Fresnel approxiamtion was derived (egtn.al•6.11)

to find the field over a surface near to a radiating Gaussian aperture.

The model itself may be formulated as the flow chart given

in.fig.5.1.3. The propagation path is divided into N slabs and the approx-

imation is applied N times to produce, in the final iteration, the

field at the output plane. There are, naturally, a host of practical

difficulties associated with this procedure, but these will be taken

up in the next section. .

The extension of this algorithm to simulate propagation

through a random medium hinges on the fact that the volume irregularities

of each slab may be condensed into an equivalent thin phase screen. As

shown in.fig.5.1.3, the inclusion of random effects is a simple matter.

A random phase modulation is applied to the output from each slab.

The practical problems of simulating a random phase screen with spatial

correlation is considered in section 5.3.

5.2 Model Details.

This section concentrates on details of the implementation

of the model just described. Attention is paid to practical limitations.

5.2.1 The Gaussian Beam.

The initial aperture field chosen was a Gaussian distrib-

ution ie.

268

e(x) = Eo exp—(x/wo) 2] 5.2.1

This choice has a number of advantages:

i) it is mathematically tractable.

ii) it is a reasonably good idealization of the pencil—beam antennas

which are possible at millimetre—wave frequencies.

iii) closed form expressions are available for the field distribution

in free space, which allows the model to be compared to theory.

The propagation of a Gaussian beam is summarized in fig.

5.2.1.. The expressions for the amplitude and phase are derived in

appendix A1.6.

e(x z) = Eo exp /(x,z)) exp C—(x/w(z))2j 5.2.2

where

w(z) = wo 11. + z

2 1/474/o

/(x,z) = kz —'t(z)

1r = Tan-1{. ) z/ Ti wo

5.2.3

5.2.4

5.2.5

When z•Tiw2/X , it is straight forward to show that

w(z) ----> A z/rcwo 5.2.6

and i(x,z) becomes that of a &phericel_wave with phase radius of

curvature equal to z.

These expressions will be used to check the simulation

program in the next chapter, in the case of propagation through free

space.

5.2.2 Numerical Techniques: ,Scaling,.

The forward and reverse transforms required by the model

algorithm are implemented using the efficient Fast Fourier Transform

(FFT), which has a large literature. In order not to fall foul of

I t•

zlx

t E—at XCEN W O

fig. 532~2, Computer storage or rield values.

NtIAX

x

269

e(x)

fig. 5.2.1 Propagation of a Gaussian Beam.

270

the usual considerations such as ūndersampling, it is necessary to

define the numerical problem.

Fig.5.2.2 gives a graphical representation of the sit-

uation. The abscissae are given as distance as well as the equivalent

array indices. The electric field is sampled at a regular interval

A. x and stored as real and imaginary parts (rectangular coordinates)

in two arrays XR, XI. I is the array index and the x coordinate is

thus

x = I Ax 5.2.7

It is important to note that x is constrained to be posi-

tive: this is a requirement of the FFT algorithm. The field is

centred at

I = ICEN (x = ICEN * DX) 5.2•B

Other important parameters are given in fig.5.2.1.

The choice of /fix is important. The, value of' Ax determines

the maximum wavenumber of the calculated angular spectrum, according'

to

kx(max) = 2 rr /I x 5.2.9

The choice of d x is a compromise between satisfying the sampling

criterion (5.2.9) and keeping down the number of points required for

computation. This is best demonstr.ated_by means of a practical example.

Suppose a Gaussian aperture field has an initial waist of

wo and wavelength X . Far from the aperture, the waist will be

w(zo) "- X zo/Trwo 5.2.10

If N points are being used to store the field values, it would seem

reasonable that at all times the beam (ie. the waist) should occupy

only 1/10 of these N points ie.

w(zo) N Ax/10 5.2.11

ie. Ax > 10X zo/Tr Nwo 5.2.12

271

This gives a minimum value for ilx.if the maximum value of z. is known.

•

0

for a particular simulation. N would be chosen on a requirement such

as computer time. An advantage of the FFT is that 'a doubling of N

would only double the computation time.

As an example, let

wo = 3,6

~► = 1 cm Qx ti 2,2 > cm

N = 4096

z 0

= 10 km

So far the spatial domain only has been considered. The

number of points required to specify the field is

IWO = wo/A x 5.2.13

As is usual with the Fourier transform, the number of points used to

specify the resultant spectrum will be

ITFM = 1/IWO 5.2.14

Thus care must be taken that the transform does not become Darrow

enough to be undersampled. A safe procedure would be to require that

the field and spectrum would occupy roughly the same number of points.

This is achieved if

w0/Ax = IWO N 5.2.15

The numerical example above is seen to have been well chosen, since

3,6/0,022 = 164'1-.3 4096

5.2.3 Numerical Techniques: the Angular Spectrum.

Two major difficulties will be examined in detail ie.

the implication of the shifted field symmetry and the application of

the Fresnel correction.

The FFT algorithm produces an angular spectrum sampled

at an interval

0 kx = 21T/N iSx 5.2.14

272

where N is the number of points. The spectrum produced is "folded",

as shown in.fig.5.2.3. The folding or maximum wavenumber is

kx(max) = Tr/Qx 5.2.15

The spectral values for kx > kx(max) are in fact the negative spectral

values ie. the spectrum is calculated over the range

kx ( < kx(max) 5.2.16

The initial field was defined over positive values of x only:— the

aperture is centred about an axis of symmetry

x = XCEN = xo

5.2.17

It is fairly simple to estimate the effect of this shift on the spectrum.

The Shift Theorem for Fourier transforms is

f(x—x0) = expixk] 5.2.18

Hence

A(kx) .= eXp — x—xo 2 = b exprixokx exp —(x 2 L x

w o 0

5.2.19

The spectrum is seen to be multiplied by an oscillatory term dependent

on xo. A large value of xo makes this term rapidly oscillatory. The

spectrum, however, is sampled at an interval Qkx given by egtn.5.2.14.

Since the spectrum must be retransformed to yield the field values, the

oscillations due to xo must not violate the sampling theorem applied

this time to the spectrum.

In fact this effect is of less importance than the possible

errors due to the Fresnel correction. It suffices to note that xo

should be made as small as is possible and yet contain sufficient

information about both skirts of the field distribution. In the simu-

lation, xo was generally chosen such that

ICEN = N/4 5.2.20

It is intersting to compare this form of reasoning with the standard

practice of "adding zeroes" when temporal spectra are being evaluated.

ANW

1 ~ r

1 ~ r

1 1 i r

I

273

fig. 5_!243 Folded output of FFT algorithm.

Fresnel phase correction

I

I

1

1

i I ,

symetric points also corrected

only this curve is computed

14x

fig. 5.2.4 Fresnel correction to computer arrays.

274

In order to examine the Fresnel correction, the shift

correction will be ignored. The aperture field chosen is

e(x) = exp —(W t o

5.2.21

with transform

A(kx) oC expi

``

—(w0kx/2) 21 5.2.22

The Fresnel correction is applied and

a(k) z exp izok2 exp —(wokx02k 2 I

5.2.23

Again, the function required to be transformed to find the field is

multiplied by a rapidly oscillating function, whose period depends

on z . 0 Some simple analysis is possible. Put

Akx) = zo k2/2k 5.2.24

The "sampling frequency" is

fs = 1/Tr kx 5.2.25

A form of the sampling theorem holds and it is reasonable to require

that

. fs/2 4 = 2 z0 kx/k 5.2.26 kx

It would be a very stringent condition to require that egtn.5.2.26 was

to hold for all kx, when zo is fixed. A more reasonable condition is

for egtn.5.2.26 to hold up until kx = 4/w0 ie. when the spectrum has

reduced to 1/e of its peak value. It may be shown that

z <. k wo N Lax/16 TT

5.2.27

Putting

w 0 = 3,6 m

k = 2TT/10

N = 4096

= 5 10-2

275

This condition (egtn.5.2.27) is borne out in practice.

A number of runs were made of the simulation program (see next chapter)

and large errors in predicted field patterns were observed with

zo = 10 000 m. Egtn.5.2.27 is in essence, a constraint on the maxi-

mum slab thickness which may be used when X, wo, N,Ax are fixed.

It is not a serious hindrance for practical simulations.

As indicated in fig.5.2.3, the spectrum is stored in the

arrays in "folded" form. Since the spectrum was not examined in general,

no computer time was spent in reformatting the data into a more

natural, symmetric form. In fact, the subroutine applying the

Fresnel correction was written to cater for the folded format.

Fig.5.2.4 indicates how simply this is achieved. A further saving in

computer time was made by only calculating one half of the correction

values and applying these values to both positive and the symmetric

"negative" spectral values.

5.3 Random Number Generators.

The simulation algorithm requires the perturbation of the

field by correlated random numbers. This section deals with the genera-

tion of uncorrelated, independent random numbers with a known distribu-

tion and moments. Section 5.4 will show how correlation may be intro-

duced to these independent numbers.

The refractivity fluctuations of the troposphere are the

resulatant of many random functions. The Central Limit theorem

(Beckmann 5.5 ) indicates that the sum of independent variates drawn

from "compact" distributions tends to produce variates which are norm—

ally distributed. The normal distribution is simple to produce and

hence it was chosen as the distribution of the simulated phase screen.

276

5.3.1 Generation of Normally Distributed Numbers.

A comprehensive guide to the methods available for the

computer generation of normally distributed numbers is given by .

Atkinson 6, together with timings when implemented on various

computer systems.

An overall view of the simulation process indicates that

the FFT involves the biggest expense in computer time; there is thus

no pressing need to use the fastest possible random number generator.

One of the fastest and simplest approximate methods for

generating normals is to sum uniformly distributed independent numbers

(uniforms) ie. N

X =. 2u. j 4.1 1

u. uniform over a,b 1 5.3.1

The generation of uniforms is taken up in the next subsection.

It is important to establish criteria by which numbers may

be tested for conformance to particular distributions. Following

Beckmann 5 the following quantities are relevant

i The kth initial moment

mk(X) D < Xk>

5.4.2

ii The kth central moment

Mk (X) 2 < (X x )k>

5.3.3

If the X are from a distribution which is symmetric about

its mean, it may be shown that all central moments of odd order should

vanish. Thus the dimensionless quantity

S = M3/ M2 5.3.4

should vanish for symmetric distributions. It is known as the coeffici-

ent of skewness. A distribution may be said to be positively or nega-

tively skewed.

Next, consider the 4th central moment of a normal distribu-

tion

277

00

o4 t4 exp`—t2 Ii dt = 3

4 = 3M2

21T L 2 _b

Thus, for a normal distribution with o- = 1,

5.3.5

0 D( )_ 3 =

M2 2

)k0 —> a flatter peak than for the normal

ō > D a sharper peak distribution

As a sample of these tests, consider table5.1. Here

uniform variates from the CDC Corporation Extended Fortran function

RANF were summed to form the normals. The number of uniforms, N, was

varied. The moments were computed over only 200 samples; even so,

the process seems well behaved in that the moments calculated are close

to the limits _ predicted by the theory above.

5.3•2 The„Generation of Uniformly Distributed Random Numbers.

Recourse to most books on computing (eg. Knuth 7) or

simulation will yield information on the generation of uniformly

distributed numbers. The two simplest are the multiplicative (M.0.)

or Additive (A.C.) Congruential methods.

The M.C. method consists of chossing a modulus m and

calculating the congruence

un+1 = a un (modulo m) 5.3.7

To start the process, the constant a and the initial value (or seed)

u 0

have to be chosen. Depending on the choice of these two values, a

sequence of numbers results which may be random, but will have a

finite cycle length, after which the same sequence of numbers will be

obtained from the process. It is possible to choose these values to

obtain the longest cycle length (Cenolly5.8).

The A.C. method is similar, and is of the form

un+1 = soon + a1un-1 } a2un-2 +(modulo m)

5.3.8

5.3.6

278

N

X (R) = X =f i

Run X(N) Var X (N) SxcN) Yx(+ )

—,0023 0,9916 ,0082 —,0311

—,0019 1,0004 ±,0464 —,0769

—,0607 0,9734 —,2305 ,0766

4 —,0192 1,0410 ,0757 —,0873

5 ,0077 0,9737 ,1097 —,1332

N = 12 for all runs

Moments computed for 200 samples in each run.

Table 5.1

Type

Tests on normal variates generated by summing-uniforms.

Examples Probability

1 All digits different 1234 0,504

2 1 identical pair, others 1471 0,432

different 2234

3 2 identical pairs 3344 0,027

mutually different 6776

4 3 identical digits,

1 different

2322 0,036

5 All digits the same 6666 0,001

Table 5.2 The four hand poker test for a sequence of digits.

279

This may be termed a F(m,N) process. Again, the choice of the ai, m,

N determines the cycle length. Potentially the A.C. method has the

longest cycle length. They are, however, less attractive in that

they require N additions and N multiplications. Conolly5.5 has

formalized some of the considerations determining cycle lengths.

Although examples of both types were tested, the Fortran

function RANF was used, principally because of speed. The RANF function

is a M.C. process which has the advantage of being programmed for the

CDC machine. The execution time is roughly 13 ps per number.

Besides the problem of periodicity, checks should be made

of the randomness of the digits making up each number. The "Four Hand

Poker Test" is an example. Any 4 sequential digits of a random number

are isolated. These 4 digits may be categorized as in table5.2.

Detailed tests of M.C. generators have been performed by Heng , who

concluded that performance was adequate. It has been assumed that

RANF behaves in a similar fashion.

5.4 The Generation of Correlated Random Numbers.

The normals generated so far are independent of each other.

They are not representative of the spatially sampled refractivity which

exhibits spatial and temporal correlation (chapter 2).

A number of methods exist for numerically introducing an

arbitrary correlation function to independent numbers.

The most general formalism is the matrix method, where a

vector with random components (independent) is multiplied by a

matrix whose elements are chosen to introduce correlation between the

components of the vector product formed. This method can be very

expensive in computer time.

Only a single method will be investigated. It will be called

280

the adjacent summing method, which is a reference to the algorithm

used. It will be seen to produce a Gaussian autocovariance function.

5.4.1 Adjacent Summing Algorithm.

Suppose N normally distributed, independent numbers

have been produced. A number of passes are made, where each

number is replaced by the sum of that number and the adjacent number.

This is most easily explained diagrammatically, as in fig.5.4.1.

It can be seen that

Yn = Xn + Xn+1

Zn

= 1 + 2 Xn + Xn-1 5.4.1

etc.

Clearly, the coefficients of the Xi become the Binomial coefficients

ie. if Xi is the ith number in the series after j passes have been

made of summing adjacents,

Xi — ,Ck Xi+k

5.4..2

where jCk = t ; /L j±

and a repeated index implies a summation

k = 0(1)j

Physically, the correlation process is evident in that

each number becomes dependent on the values around it. The "correl-

ation length" is dependent on j, the number of passes.

To calculate this correlation, smooth functions have to be

replaced by discrete, sampled functions, but in the spirit of the

definitions of chapter 2,

.19(1) =C Xi Xi+1> 5.4.3

jCk Ck+l Xk Xk+l>

2,C1+ j~ Xk Xk+l > 5.4.4

Fig.5.4.1 Adjacent summing correlating process.

1,0

Smooth curves are exp—(x2/N)

+ measured autocovariance for N=25

■ N=20

• N = 15

6 7 • 8~9 '0,0

0

t

0,8

1 0,6 cal

C . y

j 0,4

0,2 —

3 4

Displacem est --v 5

281

x, •

1st Pass

2nd Pass

EnsevAloie of iv►e)epty devtt rav,ōow, tnuwtk 5

Xa x3 X4 Xs

• • • •

4- 4- -4-

x° x3 x~x+ x~ xs xsi + -I- 4- -4- • • • •

X1 +Z4tX3 X2 42X0X4 X3i2X4+Xs X4t2z6+XL

4 •

X,tXz

• • •

etc. ( h ?asSe5 af SLL A M l .aā'ateut5

282

Since the Xi are independent,

Xk Xk+1%> = 6 2

k,k+l ak 5.4.5

where Si k,k+l is zero unless 1 is zero

o•X is VarrXi1

The variance of the correlated numbers is the value of the autoco-

variance function at 1=0 ie.

A' Jc

X = (0) = 2 ic k jCk o'X N3 k. o 5.4.6

Taking o•X=1, the normalized autocovariance function of the correlated

variables is,

8(1) = (1)/0 5.4.7

But (jCk)2 = 2jCj and 5.4.8 kilo

B(1) = 17 (1)/2jC j 5.4•9

'The discrete form of the autocovariance function (eqtn.

5.4.9) is difficult to visualize, but it can be shown that

B(1) — jCk JCk+1/2jCj

2jC 2jC j+l/ j

N expr 12/ j] for j>>1 5.4.10

(Appendix A5.1) LL

1

The variables thus are approximately Gaussian correlated

with a correlation length (B(1) = 1/e) of

1 = 5.4.11

Fig..5.4.2 shows the results of a practical implementation of this

algorithm, for various values of j. The agreement between theory

and practice is very good, even for small values of j. The RANF

function was used to generate the Xi.

The adjacent summing technique is easily thought of in

terms of the generalized matrix method, ie.

N X

5.4.12

283

1 jck ic 1 Jck JCj

X~

A

where X' is an N element column vector

N A is a j X N matrix

X is an N element column vector

The X components are, of course, independent. The rows of A are the

Binomial coefficients. The components of 70 exhibit the necessary

correlation..

The normalization process has important implications for

computer speed. The most direct normalization process is to divide

each variable by 2 after every correlating pass, ie.

Xi = (Xi-1 + Xj-1)

/2 5.4.13 i+1

The penalty is heavy in terms of the number of divisions

required. If N numbers are correlated by j passes, then N*j

divisions are required. (In binary arithmetic, this is easily

implemented by a shift operation, but this is not simply possible

in FORTRAN).

The approach taken is to rather complete j passes without

1 normalization, and to divide the correlated variates by [2jCj]2:

this reduces the number of divisions to N. This cannot be done indis-

criminately, since 2gCj

is rapidly divergent.

The problem is aggravated by the fact that if a correlation

length 1 is required, egtn.5.4.11 predicts that 12 passes must be

made. Hence, if long correlation lengths are required, a hybrid of

the latter normalization process is best ie. the correlation is done

over a number of passes, with a normalization at the end of each pass.

Yet another means is available to compute long correlation

284

lengths; the algorithm is indicated by fig.5.4.3. Suppose N

numbers are required with correlation length 1. Only N/2 numbers are

produced and correlated using (1/2)2 passes. These normalized num-

bars are then placed in alternate positions of a new array. The missing

elements are then set equal to the mean of the two adjacent elements.

This is shown in fig.5.4.3. The produces an approximation to the N

numbers, correlated-over-l. samples.

The saving in computer time is large. Ignoring normali-

zation, the conventional process requires (N*12) additions; the

new process requires only N (L)2 additions, which is an 8 times saving.

(2 \2J Fig.5.4.4 shows that the new method produces measured

autocovariances which approximate the theoretical values closely,

again even for small 1 values. The modified algorithm has been

incorporated in the simulation program described in the next chapter.

5.5 Conclusions.

Although this chapter discusses the theoretical and

practical limitations of the computer model, suggestions for improve-

ments will be postponed to the end of this thesis. Future applications

of the technique are.dscussed.there..

The model presented has been shown to be feasible and

powerful. Areas of application are wide, and the next chapter will

outline some of the uses to which the data may be put.

The latter portion of this chapter provides details of

the generation and control of the random phase screens required by the

model. Although only Gaussian autocovariance functions are discussed;

any other functional form can be synthesized from a sum.: of Gaussians,

as was shown in chapter 2.

The whole concept of a multiple screen model has been

0 • 1 2 3 4 5 6 7 8 9 10

Oisp lace wt-elet —3

1,0

0,8

50,6 "' C

exp[—(x2/48)]

ō 0,4 S • measured autocovariance for C N = 48 e

0,2 •

• I

''*1) 285

I

11111 ••••• • /

I

14

• Correlated random numbers in alternate cells

❑ (Xn + Xn_1)/2 inserted after correlated numbers

have been generated.

Fig.5•4.3 Economic means of producing long correlation lengths.

286

fully justified in the first chapter of this thesis. The simulation

may thus be similarly justified, most certainly when applied to

millimetre—wave situations. Realistic form may be given to the phase

screens used if the information presented in chapter 2 is used.

287

Appendix A5.1 Convergence to a Gaussian.Autocovariance Function.

As given in eqtn. 5.4.10, the general term of 8(1) is

given by

8(t) = iCk 3Ck+1 / 2J j

where k = 1(1)j

2jcj+1 / 2jcj

= Li / t1 Lill Now, let j',>1. Stirling's formula for large Li. is

L i 2tr j j j e—j

or in Li'. a + (j+4) In j — j

where a = 1n57

a5.1.1

a5.1.2

a5•.1.3

a5. t.4

25.1.5

Taking the logarithm of a5.:1.3

InB(1)^- 2.1n Li —in +i — in ZIA a5.1•6

= 2(j+i)ln j — (j+t+')ln(j+g)

ln( j 3) a5•,1.7

= 2(j+4)ln j — (j+`L+i)(ln j + ln(1+9./j))

— (j-9.+4)(1n j + ln(1-1/j)) a5.1.8

= — (j+t+*)ln(1+1/j) — —14)1n(1 —1/j) a5•,1.9

Now, ln(1+x) x — x2 + x3 + .... —1<x a5.1.10

2 3

Thus eqtn a5•1.9 may be simplified with the expansion a5•.1.•10

provided

a5.1.11

Hence In 8(4)'+- — (j+4+ )(q/j — z (t/j)2 + )

- (j+14)( Vi - '(t/j)2 .25.1.12

= 32+ /I2 — 2 t4 + terms of order (9./j)4

j 72 3 33 a5.1.13

288

It is thus apparent that

8(3).., exp1.121 3

a5.1.14

to an accuracy which can be determined from eqtn a5.1.13, since t

and j are known. At all times it must be remembered that j, the

number of passes must be large (in practice 10 or greater) and further,

that ./j < 1 (eqtn a5•1•11) if the above analysis is to hold.

The author is indebted to Prof. B.W. Connolly, Dept.

Mathematics, Chelsea College, London, for furnishing the above proof.

289

CHAPTER FIVE REFERENCES.

5.1 Fejer, J.A. 1953. "The Diffraction of Waves in Passing

through an Irregular Refracting Medium".

Proc. Roy. Soc. Series A, 220, pp455-471.

5.2 Bramley, E.N. 1954. "The Diffraction of Waves by an

Irregular Diffracting Medium".

Proc. Roy. Soc. Series A, 225, pp515-518.

5.3 Booker, H.G., Radcliffe, J.A., Shinn, D.H. 1950.

Phil. Trans. A, 262, p579.

5.4 Newish A,, 1951. Proc. Roy. Soc. Series A, 209, p81.

5.5 Beckmann, P. 1967.. "Elements of Applied Probablity

Theory". Harcourt, Brace and World Inc.

5.6 Atkinson, A.C. and Pearce, M.C. 1976. "The Computer

Generation of Beta, Gamma and Normal Random Variables."

Journ. Roy. Stat. Soc. Series A, 139, Part 4, pp431-467.

5.7 Knuth, D.E. "The Art of Computer Programming" Vol. 2.

5.8 Conolly, B. Private Communication.

5.9 Heng, B.K.E. 1975. "An Extension of the Central Limit

Theorem." Third year project report, Dept. Electrical

Engineering, Imperial College, London.

290

CHAPTER SIX SIMULATION RESULTS.

6.1 Introduction.

The mathematical basis of the simulation was established

in the previous chapter, as well as chapter one. This chapter is

mainly pictorial, since this was the medium chosen to represent the

output of the simulation program.

A description of the programs used is given, although

listings are relegated to an appendix. The main simulation program

SIIIUL,is written in a "standard" version of FORTRAN IV, and should

run on other machines with a minimum of adaption. The graphical

program, PLOTN, depends heavily on the graphics programs peculiar to

the Imperial College Computer Centre, and is thus less widely applicable.

Once the program SIMUL had been written, a number of tests

were made of the propagation under non—random, free—space conditions.

For Gaussian beam propagation, deterministic formulae for beam—width

as a function of distance from the radiating aperture exist (Appendix A1.6)

and comparisons could be made between the predicted values and those

measured from the simulation program. Once these checks proved the

program to be sound, the random case was investigated.

The graphical output from the simulation program under •

model atmospheres is presented as a series of figures, forming a large

part of this chapter.

6.2 Program SIMUL.

A full listing of the program is given in appendix A6.1.

The logic and component parts of the program are shown in fig.6.2.1.

It has been written to be as flexible as possible. Data is taken in two

stages. The first block of data sets up the details of the accuracy,

and so on, to which the simulation is to be carried. This includes the

generation of the initial aperture field, which is assumed to be Gaussian

in the listing appended. A suitable subroutine could be included to set

291

(sncr)

/1 ,1put

ma, r+ paravncitf5

ALE I NORT

Create. Ixiiia Rpcs tuft cset 4:

Real tlnc pa vomitus/

x: s lab

A

SIR C¢l AōF (FFT)

51R CoRR

> < > <

> <

> <

S

Mb4.T

S IR FR ES

> <

FILE DRTAI

Fig..6.2.1 Components of program SIIIUL.

292

up any other field distribution. It is important to take into account

the constraints on aperture width, sample interval, and so on which

were discussed in the previous chapter.

The second block of data is determined by the number of

slabs into which the propagation path has been split. The parameters

(thickness, rms refractivity, and so on) of each slab are read in

during execution. This allows inhomogenuous paths to be easily accomo-

dated. The details of this data are given in appendix A6.2.

The field found after each slab is output as arrays of

real and imaginary numbers. Only the central portion of the arrays

(where the field is mostly confined) are output. Details are given in

appendix A6.3. It should be pointed out that the output from each slab

is approximately 20 000 characters, and a multi—slab model will most

certainly require extended disc or tape storage facilities.

Since the data is expensive to create in terms of computer

time used, it is certainly worth preserving the output from simulation

runs on tape for further analysis.

The present set of programs uses the data purely for

graphical output, but more detailed statistical analysis is possible.

This point is taken up at the end of this chapter.

6.3 Program PLOTN.

This program takes data (real and imaginary parts) created

by SIMUL and produces graphical output. In common with all programs

using computer graphics, it is heavily dependent on software peculiar

to a particular computer installation. A few general points are worth

making.

The graphics programs provided by the Imperial College

Computer Centre (ICCC) are designed to be independent of the graphics

device (flatbed plotter, etc.) which they are driving. In this case,

293

the plot files generated by PLOTN were previewed on a VDU. If they

proved to be satisfactory, an instant hardcopy could be obtained from

a unit attached to the graphics VDU, or the plot - files were sent for

conversion to microfilm. The graphics presented in this chapter are

printed from the 35mm microfilms produced in this way.

The program was designed to enable plots of the field or

power across an aperture to be plotted. These plots can be scaled linear-

ly or logarithmically (ie. in dB field or, dB power). Other options

include setting the width of the aperture to be plotted, and the possi-

bility of normalizing all results to the maximum value of the first plot.

Phase is also plotted. This imposes some difficulty, since

the small phase modulation imposed by the medium is minute compared to

the spherical phase taper of the expanding beam. To overcome this

problem,a least squares routine was used to remove this trend before

plotting. Again, the range of the phase and aperture domains can

set at will.

A complete listing is given in appendix A6.4. Any subrout-

ines referenced but not listed are subroutines drawn from the ICCC graph-

ics library or the NAG5 library6.2.

The program does produce some simple quantitative results,

which are output to each of the plots. The source of these values is

given in fig.6.3.1.

In the case of strong medium fluctuations, care must be

exercised in interpreting the waist measurement, as is shown in fig.6.3.2.

Because of the presence of a strong null, a waist reading xo, which

is much narrower than the envelope value xe , results. This calculation

could be improved if a running mean was used.

Similarly, the rifts phase measurement is calculated using

only the points which are plotted and have been detrended. If the

plotted domain is small (only a few correlation lengths wide),

294

biased result is possible.

At this point, some explanation of the figures contained

in the next two sections is necessary. The computer-generated-plots

are produced on 35 mm film; these film have then to be printed for

viewing purposes.

The final size of these photographs was underestimated,

and it became apparent that the reprographic methods used to produce

this thesis were unable to fully reproduce the axes lettering with

sufficient definition. Fig.6.3.3 thus shows two typical plots of

phase- and amplitude.

In all amplitude plots, the.aperture axis is 100m and

the received field is plotted in dB, 0 to -52 dB. This is equivalent

to 0 to -104 dB power. All amplitudes are normalized to the first

aperture field maximum: "spread loss" due to propagation is not

included

The aperture axis for the phase plots is 10 m throughout.

The phase axis scale varies, but is not important.

6.4 Model Tests: Deterministic Case.

To test the accuracy of the modelling program, some runs

were made with no random phase perturbations. Since a Gaussian

initial aperture distribution was used, it is possible, using

egtn.al-•-6•:18, to predict the waist at any distance L from the _

initial aperture.

Fig.6.4.1(a to e) shows the result of a typical run..

The plots are logarithmic field values. The effects of rounding errors

can be seen for very small field values. Single precision arithmetic

was used in this simulation, but the-large 60 bit word of the CDC

machine ensures the small level of roundoff noise shown in these

figures.

VIG

E(x) Il xo

295

Fig.6.3.2 Possible errors is measured waist due to field fluctuations.

296

PHASE STD DEV 0.2728 Z (METRES) 10000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M) 0.25 PHASE STD DEV [RAD) 0.279 WAIST (M) 0.150

_ 4. •i.x •i.W 4.w

s

L

s

74.r - 4 -54. •t.M 4 Ō154 Pa RIO AP ITŪRf " " " " ~•~ ~•~

..W "

- 1 o f 2 3 4

w►cTfG5j~

(X

.7 fa

r Ao

wer

1

Z (METRES) WAVELENGTH (METRES) PHASE CORR LENGTH (M) PHASE STD DEV (RAO)

q

-.364

s -404

-444-

s -484

s -524

10000 0.010 0.25 0.279 0 1150 -8

_S<

-SO -4o -3o

:r.« 4.M •Ī..M ipsi a1 RLON8 Mfllfūllt 1m Am 4. 4. 4.

-ZO - (O 0 10 20 30

a.o. ..o. 8.0. ,L.w

40 So

Fig.6.3.3 Sample output

297

Table 6.4.1 shows the waist figures from fig.6.4.1

compared to those predicted by egtn.a1.6.1S. As can be seen, the

simulation program produces sufficient accuracy.

Tests were carried out on the random number generator,

but these were detailed in the previous chapter.

6.5 Simulation Runs.

This section contains the graphical output of a few

sample simulation runs. These runs have been chosen to illustrate

certain aspects of the beam/medium interaction.

The data is presented as a number of subsections. A

description of each run is followed by a number of pages of graphical

output obtained from program PLOTN.

The phase and amplitude over a plane is plotted at 2 km

intervals. Each page shows two sets of contrasting data at the same

distance from the transmitter.

In all the plots shown, a Gaussian aperture field of

the form

E(x)Iz_0 = exp[ (x/3,6)2J

(x in m) 6.5.1

was used. The wavelength was 1 cm (ie. 30 GHz).

In all models, 10 slabs, 1000m thick, were used to

simulate the propagation path. The total path length of 10 km is

realistic in terms of practical millimetre—wave systems.

298

1 PHASE STD DEV 2 Z

3 WAVEL

4 PHASE CORR

5 PHASE STD DEV

6 WAIST

Key:

1 Rms phase fluctuation measured after parabolic: trend has been

removed. Computed from the points plotted.

2 Distance from initial aperture of this frame.

3 Wavelength of the em. wave.

4 Correlation length of the phase modulation applied during this slab.

5 Rms. phase modulation applied to this slab. This is calculated from

the properties of the medium, such as oN, 10, and so on.

6 Distance from peak field at which the field first reaches 1/e of

the peak value.

Fig.6.3.1 Key to information supplied on plots.

z (m)

waist (experimental) (m)

waist (theory) ( m'

.

0 3,6 1000 3,75 3,75 2000 4,05 4,15 3000 4,55 4,55 4000 5,15 5,15 5000 5,85 5,85 6000 6,65 6,55 7000 7,45 7,35 8000 8,25 8,25 9000 9,15 9,05 10000 10,05 9,55

Table 6.4.1 Test of simulation program in free space conditions.

7 (METRES/ 1000 WAVELENGTH (METRES/ 0.010 PHASE CORR LENGTH IMI 0.05 PHASE STO DEV IRR01 0.000 WRIST (MI 3.750

7 (METRES/ 2000 WAVELENGTH (METRES! 0.010 PHASE CORR LENGTH (MI 0.05 PRASE STO DEV (RAD) 0.000 WAIST (MI 4.050

299

Fig.6.4.1a Propagation under free space conditions.

300

4 Z (METRES/ 3000 WAVELENGTH (METRES) 0.010

4 PHASE CORR LENGTH IM/ 0.05 PHASE STO DEV IRAO) 0.000

4 WAIST (M1 4.550

4

4

•!

4

4

a

4

4

.M -•.• 1uT'.ia a3 mince 4.

Fig.6.4.1b

Z (METRES) WAVELENGTH (METRES)

4000 0.010

4 PHASE CORR LENGTH (M1 0.05 PHASE STO 0EV (RA01 0.000 WAIST 1M/ 5.150

4

4

4

4

-4

i

4

4

'

' T , in

Z tMETRESI 5000 WAVELENGTH )METRES) 0.010 PHASE CORR LENGTH IM) 0.05 PHASE 5T0 DEV IRAN 0.000 WAIST II)) 5.550

Z (METRES) 8000 WAVELENGTH (METRES) 0.010 PHASE CORN LENGTH 111) 0.05 PHASE 5T0 DEV tRR01 0.000 WRIST (M) 6.650

301 '

Fig.6.4.1c

1~ AU prone

302

Fig.6.4.I d

Z METRES) 7000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M1 0.05 PHASE STO OEV (RA01 0.000 WAIST (M) 7.450

I (METRES) 3000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 0.05 PHASE ST0 0EV (RA0) 0.000 WAIST (MI 3.250

Z (METRES] 10000 WAVELENGTH (METRES( 0.010 PHASE CORR LENGTH (MI 0.06 PHASE STD DEV (RAW 0.000 WAIST (MI 10.050

303 "

i

!

i

i

!

!

!

Z METRES' 9000 WAVELENGTH IMETRESI 0.010 PHASE CORR LENGTH (MI 0.06 PHASE 510 0EV (RA01 0.000 WAIST (Ml 9.160

Fig.6.4.1e

304

6.5.1. Strong Perturbation versus Free Space Conditions.

The plots contained in this section are linear, in field.

The upper plot is of the beam propagating under free space conditions,

and is contrasted with the situation where intense .refractivity

fluctuations exist.

In the random case, each slab was modelled as having a

refractivity std. dev. of 10 N, and a scale size of 0,5 m. This

is an unrealistically intense fluctuation for millimetre—wave

conditions, but serves to show the difference between the random

and non—random cases.

The scale size used is smaller than is found in nature,

but was used to ensure that substantial amounts of energy were scat-

tered out of the main beam. This is seen to be the case, and the

degradation of coherence of the wave (beam) is clear.

305

Z METRES) 2000 WAVELENGTH (METRES! 0.01 PHASE CORR LENGTH 1M) 0.00 PHASE STO DEV (HAD) 0.00 WAIST (Ml 4.05

Z (METRES) 2000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH 1141 0.50 PHASE ST0 DEV (RAO) 0.27 WRIST (Ml ♦.95

.1 a

4-Tm31$ •L1 oaf's .a .a fa .a ta ... .a La

Fig.6.5.1a Free space versus strongly random conditions.

I (METRES) 4000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH (M1 0.00 PHASE STD DEV (RA01 0.00 WAIST 1M1 5.15

306

Fig.6.6.lb

I )METRES) 4000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH IMI 0.50 PHASE STD DEV (ARD) 0.27 WAIST [M1 0.25

Z (METRES] 6000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH IM1 0.00 PHASE STD 0EV (RAD) 0.00 WRIST 1M) • 6.65

.r.. ..a 41.0 ma

Z (METRES) 8000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH IMl 0.50 PHASE 5T0 DEV (RAD) 0.27 WAIST (Ml 0.35

J

d

r

307

Fig.6.5.1c

r. f

r

111

Z (METRES) 6000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH (M► 0.50 PHASE 510 OEY IRA01 0.27 WAIST IM) 0.35

-ma - ~ a.• .tia s.+ a.a -rr •'ēur'f#i w~.ē eeS~.r

308

'•

Z (METRES) 8000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH (M1 0.00 PHASE STD OEV (RAO) 0.00 WAIST (M) 8.25

.... .... , , .. .. - •1'., a ..a r.. .

Fig.6.5.1d

I (METRES) 1000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH (MI 0.00 PHASE STD DEV (RA01 0.00 WAIST IMI 10.0

I (METRES) 1000 WAVELENGTH (METRES) 0.01 PHASE CORR LENGTH (MI 0.50 PHASE 5T0 DEV (RA01 0.27 WAIST IMI 0.35

t r~ • 1ia'.f1 .42 rdiSw a. u~

\~Ī

309

Fig.6.5.1e

310

6.5.2 The Effect of the Medium Scale Size.

In the first five pages of this subsection, the amplitude

is plotted (figs.6.5.2a—e). The next five pages are plots of the

phase, with the spherical trend due to normal diffraction removed

(figs.6.5.2f—j).

In these runs, realistic values of refractivity std.

dev. have been used, ie. oil = 0,1 N. The phase screen equivalent to

to each slab was calculated according to the analysis of chapter one.

The upper plot on each page is when a scale size of 1m

has been used, .whereas the lower uses 0,25 m.

Two aspects are important. In the case, of the smaller

scale size, the buildup of scattered energy is much more rapid than

for the larger scale size. This is very apparent in the first few

plots.

The last few plots show quite clearly that the spatial

properties of the field are directly related to the medium scale size.

The large scale size medium gives rise to larger spatial correlation

of the field over an aperture.

The phase plots also show up clearly the effect of the

medium scale size in determining the spatial characteristics of the

phase fluctuations.

It can also be .seen that the main beam is undistorted: a

plot over a much smaller amplitude range would show very small

amplitude ripples.

Z !METRES) 2000 WAVELENGTH 1METRESI 0.010 PHASE CORR LENGTH (M1 1.00 PHASE STO DEV !RAO) 0.002 WAIST (M1 4.050

Z (METRES) 2000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH IMI 0.25. PHASE STD DEV IRA01 0.004 WRIST (MI 4.050

0

311

Fig.6.5.2a The effect of scale size.

Z (METRES) 4000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH 1M) 1.00 PHASE STD DEV (RAD) 0.002 WRIST (Ml 5.150

1 F 4

s.. i. •rue .. a m w '111412V ni42 N & aa aa

.a am a+ AM a.

Z (METRES) 4000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M) 0.25 PHASE STD 0EV (RAO) 0.004 WAIST (H) 5.150

1)4,

.L3 .dke as ,

i F

4

R

4

312

Fig.6.5.2b

2 (METRES) 6000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH 1111 1.00 PHASE 5T0 OEV 1RA01 0.002 WRIST (RI 6.650

313

Fig.6.5.2c

I (METRES) 6000 ' WAVELENGTH (METRES) 0.010

PHASE CORK 1ENGTH (MI 0.25 PHASE STD DEV IRA01 0.004 WAIST 1111 6.750

'I .141.441.

Z (METRES) 8000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH IMI 1.00 PHASE 5T0 0EV (RA01 0.002 WAIST (Ml 8.250

Z (METRES) 8000 WAVELENGTH (METRES/ 0.010 PHASE CORR LENGTH (MI 0.25 PHASE STD 0EV (RAO) 0.00♦ WAIST 111) 8.350

314

Fig.6.5.2d

a ra .ra -ra aa

-UM a.a -1. 1 4,1 m 11 4t„,

ga .a .a .a .a .a .a .a ia

I (METRES) 10000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M1 1.00 PHASE STO DEV IRA01 0.002 WRIST (M) 10.050

'111.4.1A ill miter r. ar r. w.a aa a

I (METRES) 10000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 0.25 PHASE STO 0EV (RAO( 0.004 WAIST (MI 10.050

t 4.1i.i.iū IKONS aeIT..

a

315

Fig.6.5.2e

PHASE 510 0EV 0.0018 Z (METRES/ 2000 WAVELENGTH /METRES] 0.010 PHASE CORR LENGTH (MI 1.00 PHASE 5T0 DEV (RA01 0.002 WAIST (MI 4.050

em AO 40

PHASE 510 0EV 0.0022 I (METRES/ 2000 WAVELENGTH (METRES/ 0.010 PHASE CORR LENGTH (MI 0.25 PHASE STO DEV (RADI 0.004 WAIST (MI 4.050

316

Fig.6.5.2f

PHASE STO DEV 0.0020 2 METRES/ 4000 WAVELENGTH METRES/ 0.010 PHASE CORR LENGTH (MI 0.25 PHASE STO 0EV (RA01 0.004 WRIST 1M1 S.ISO

a • a.. a._ .a ia

317

PHASE STD DEV 0.0024 Z METRES) 4000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH IMI 1.00 PHASE STO DEV IRADI 0.002 WRIST IH) S.150

Fig.6.5.2g

PHASE STD 0EV 0.00)• 2 (METRES] 6000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH IM) 1.00 PHASE ST0 0EV IRA01 0.002 WAIST (MI 6.650

PHASE STD 0EV 0.001• Z (METRES) 6000 WAVELENGTH (METRES) 0.010 PHASE CORR.LENGTN IMI 0.25 PHASE 5T0 0EV (RA01 0.00• WRIST (M) 6.750

mar ra'Z . iw

318

Fig.6.5.2h

PHASE STD DEV 0.0019 2 (METRES) 8000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M) 1.00 PHASE 5T0 DEV IRAQI 0.002 WAIST 1M1 8.250

PHASE STD DEV 0.0022 2 (METRES) 8000 WAVELENGTH (METRES! 0.010 PHASE CORR LENGTH IMI 0.25 PHASE STD DEV (RAQ) 0.004 WRIST (MI 8.350

319

Fig.6.5.21

PHASE STO DEV 0.0016 Z (METRES) (0000 NRVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 1.00 PHASE STO DEV (RAD) 0.002 MIST (M) 10.050

PHASE STO DEV 0.0021 Z (METRES) 10000 NAVELENOTH•(METRES) 0.010 PHASE CORR LENGTH (M) 0.25 PHASE STO DEV (RRO) 0.006 WAIST (M) 10.050

320

Fig.6.5.2j

321

6.5.3 Strong Fluctuation Case.

This is very similar to the previous subsection. The

same two scale sizes (0,25m and 1,Om) have been used, but a very

intense random perturbation has been imposed.

Each phase screen was given a phase variance of (0,279 rad)2,

which is about two orders of magnitude larger than that which the

atmosphere might be expected to impose. This intensity may, however,

be encountered by an underwater sound wave.

It is again interesting to note the obviously different

spatial nature of the fluctuations which develope. The field at 10 km

is seen to be very different for the two different scale sizes. The

depths, (peak to peak) of fluctuation are the same, but the larger

scale size situation seems to give rise to a wave which has broken

up into a number of "beams".

.411 .a ra .a •a •a .a

Z (METRES] 2000 WAVELENGTH (METRES/ 0.010 PHASE CORR LENGTH (M1 1.00 PHASE STD DEV TRAD) 0.279 WRIST 1M] 4.750

Z (METRES) 2000 WAVELENGTH (METRES/ 0.010 PHASE CORR LENGTH IM1 0.25 PHASE 5T0 DEV (RAO) 0.279 WAIST (N) 4.050

322

Fig.6.5.3a Strong fluctuation case.

Z (METRES) 4000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M1 1.00 PHASE STD DEV (RAO) 0.279 WAIST (MI 4.250

4

4

4

}

4

4 t t P 4 l

L

4 s aa •i •ia •Ya „.+a.

W[t RAY R{ilAt •a ,i . Na Na Na Na is Na 4a

I (METRES) 4000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 0.25 PHASE STD DEV (RAO) 0.279 WA ST (IMI 0.750

V"1

as yla .oa •ia .ia ba -ia Na- •i ŌIniz RYll,o.etI Sa Va ra

323

Fig.6.5.3b

Z (METRES) 6000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH IM) 1.00 PHASE 5T0 0EV (RAO) 0.279 WAIST (M) 7.250

Mm Am Wm Wm aw. .ti. 4 44.41- aa

324

Fig.6.5.3c

a.a a► aa as aa aa ~IA& wie minor aa aa a► aa a aa a.. aa .a

Z (METRES) 6000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (M) 0.25 HASE STO 0EV (RAO 0.279

T PM) 0.150

Z (METRES) 8000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 1.00 PHASE 5T0 0EV (RAD) 0.279 NRISj,(II) 0.450

r. a,r Aar as .ar sr '1■811.11 al weilte

-a-s• -rr -r.. a,r -MN „s *5 •+ ia 41.46 *a .s •r

s Z (METRES) 8000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 0.25 PHASE STO OFFV (RAO) 0.270 WAIST,(r) 1 . Ī 0,050

325

Fig.6.5.3d

E IMETRESI 10000 WAVELENGTH (METRES) 0.010 PHASE CORR LENGTH (MI 1.00 PHASE STO DEV (RAD) 0.279 RUST (1)) n 1.550

a Aim...

Am Aa Sim Am Am Am Am Aa Am

Z IMETRESI 10000 WAVELENGTH IMETRE51 0.010 PHASE CORR LENGTH IMI 0.25 PHASE STO OEV IRAQ) 0.279

5 1 0 SO

4

4

4

SI

4

-ra -"lama nit .ziPSer

Aa

326

Fig.6.5.3e

327

APPENDIX A6.1 PROGRAM SIMUL.

7B/10/13. 18.00.42. PROGRAM SIMUL 00050C***:*******************:****:**:**:**:**:**:****:**:**:*******:*****:***:**:**

00051C PROGRAM SIMUL 00052C**********************************************:**********:****:**:* 00053C PROGRAM TO SIMULATE PROPAGATION OF APERTURE FIELD THROUGH A MEDIUM 00054C FILLED WITH RANDOM IRREGULARITIES. THE PROGRAM PARAMETERS ARE ENTERED 00055C AS DATA ACCORDING TO SPECIFICATIONS GIVEN ELSEWHERE. 00056C 00057C**********************************************:***********:**:**:**:**:**:**:****

00059 NO LIST 00100 PROGRAM SIMUL(INPUT,OUTPUT,INDAT,DATA1,TAPE5=INPUT,TAPE6=OUTPUT, 00110+TAPE7=DATA1,TAPE8=INDAT) 00130 COMMON XR(4096),XI(4096),PHI(4096),UT(2048) 00140 LOGICAL IRAN 00150C NMAX IS THE NO OF POINTS USED FOR TRANSFORMS 00170 NMAX=4096 00172 PI=3.14159265 00174 M1=INT(ALOG(FLOAT(NMAX))/ALOG(2.0)+0.5)+1 00176C ICEN IS THE BEAM CENTRE INDEX FOR THE ARRAYS 00178 ICEN=1025 00180C READ INITIAL WAIST(M) 00185 READ(8,400)140 00190C READ WAVELENGTH (M) 00195 READ(8,400)AMDA 00200C READ SAMPLE INTERVAL (M) 00205 READ(8,400)DX 00210C READ NO OF SLABS TO BE USED 00215 READ(8,401)LPN 00220C IWO IS THE BEAM WAIST IN SAMPLE INTERVALS 00225 IWO=INT(W0/D(+0.5) 00230C I1110 IS THE INTERACTION RANGE 00235 IW10=10*IWO 00240C**************:************************************:********=*****:*********

00242C 00245C CLEAR ARRAYS AND CREATE INITIAL FIELD DISTRIBUTION 00250 DO 1 I=1,NMAX 00255 XR(I)=0.0 00260 1 XI(I)=0.0 00265 DO 2 I=ICEN-IW10,ICEN+IW10 00270 2 XR(I)=EXP(-FLOAT(I-ICEN)**2/FLOAT(IW0*IWO))

328

G0275C********************************************: ************************: ** 00280C 00282C NOW ITERATE, SLAB AT A TIME, READING CO,SCALE,DZ FROM INDAT 00283C Z IS DISTANCE FROM INITIAL APERTURE FIELD 00284 Z=0.0 00285 DO 200 LP=1,LPN 00290C READ CORR LENGTH(C0), REFRACTIVITY STD DEV(SCALE),SLAB THICKNESS DZ 00295 READ(8,402)CO,SCALE,DZ 00300 Z=Z+DZ 00302C L IS CORR LENGTH IN SAMPLE INTERVALS 00305 L=INT(CO/DX+0.5) 00310C A IS FRESNEL CORRECTION FACTOR USED TO CORRECT PHASE OF SPECTRUM 00315 A=PI*DZ*AMDA/(FLOAT(NMAX*NMAX)*DX*DX) 00320C CONVERT SCALE TO EQUIVALENT PHASE SHIFT 00325 SCALE=SCALE*SQRT(DZ/C0)*2.0*PI*1.OE-6/AMDA 00520C 00521C MODULATE APERTURE FIELD WITH RANDOM SCREEN 00522 CALL CORR(ICEN,IWO,SCALE,L) 00523 CALL MDLT(ICEN,IWO) 00525C 00530C EXECUTE TRANSFORM TO FIND THE ANGULAR SPECTRUM 00540 TRAM=.FALSE. 00550 CALL C06ABF(XR,XI,NMAX,TRAN,M1,UT) 00560C 00562C APPLY FRESNEL CORRECTION TO ANGULAR SPECTRUM 00570 CALL FRES(A,IWO,NMAX) 00600C 00610C EXECUTE INVERSE TRANSFORM TO FIND FIELD 00620 TRAM=.TRUE. 00630 CALL CO6ABF(XR,XI,NMAX,TRAN,M1,UT) 00640 CALL PLOT1(ICEN,4*IW0) 00642C 00650C OUTPUT PARAMETERS AND REAL,IMAGINARY ARRAYS 00660 WRITE(7,302)2.0*DX,FLOAT(ICEN)*DX,Z/1000.0,AMDA,CO,SCALE 00670 IS=ICEN-1024 00680 IF=ICEN+1023 00690 WRITE(7,301)(XR(I),I=IS,IF,2) 00700 WRITE(7,301)(XI(I),I=IS,IF,2) 00710C 00720 301 FORMAT(F9.6) 00730 302 FORMAT(6F9.6) 00732 400 FORHAT(F8.3) 00734 401 FORMAT(I5) 00736 402 FORMAT(3F10.3) 00740C 00750 200 CONTINUE 00760 STOP 00770 END

329

00772C*:*********:********* **** k***** k*:**** k*** k* k***:****:* k*Y ** k*

00773C 00774C SUBROUTINES 00775C 00776C************************s**:****:****:********:*** *:***** **:**:** 00780 SUBROUTINE CORR(ICEN1,I141,SCALE,L0) 00785C CORRELATION LENGTH L0 AND STD DEV=SCALE. THE NUMBERS ARE 00786C RETURNED IN ARRAY PHI, CENTRED ON ICEN. 00787C*************************************:**:****:**:****:*****

00790 COMMON XR(4096),XI(4096),PHI(4896),UT(2048) 00800, DO 700 I=1,2048 00805 PHI(I)=0.0 00807 700 UT(I)=0.0 00810 N=10*IU01+2 00820 DO 1 I=1,N 00830 UT(I)=0.0 00840 DO 2 J=1,12 00850 2 UT(I)=UT(I)+RANF(0) 00860 1 UT(I)=UT(I)-6.0 00870C 00880 L=(L0**2)/4 00890 DO 3 I=1,L 00900 DO 4 J=1,N-1 00910 4 UT(J)=UT(J)+UT(J+1) 00920 3 CONTINUE 00930C 00940C SCALE FOR UNITY VARIANCE 00950 S=1.0 00960 J=L+1 00970 DO 5 I=1,L 00980 S=S*FLOAT(J)/FLOAT(I) 00990 5 J=J+1 01000 S=SORT(S) 01010 DO 7 I=1,N 01020 7 UT(I)=UT(I)*SCALE/S 01030C 01040 J=1

01050 DO 8 I=ICEN1-10*IU01,ICEN1+10*I1101,2 01055 PHI(I+1)=(UT(J)+UT(J+1))/2.0 01060 PHI(I)=UT(J) 01070 8 J=J+1 01080 RETURN 01090 END 01091C 01093C*****:*******************:******:*:*****:*********:****:**:**

01100 SUBROUTINE MDLT(ICEN1,IW01) 01101C SUBROUTINE APPLIES MODULATION IN PHASE TO COMPLEX PAIR 01102C XR,XI: THE PHASE IS HELD IN ARRAY PHI 01103C 011040******************************:****************:*****:* 01110 COMMON XR(4096),XI(4096),PHI(4096),UT(2048) 01120 DO 1 I=ICEN1-10*IW01,ICEN1+10*IW01 01130 FS=SIN(PHI(I)) 01140 FC=COS(PHI(I)) 01150 C=XR(I) 01160 XR(I)=XR(I)*FC-XI(I)*FS 01170 1 XI(I)=C*FS+XI(I)*FC 01180 RETURN 01190 END 01191C

330

O1195C******************************_*********:********:****** 01200 SUBROUTINE PLOT1(ICEN1,IU01) O1201C LINE PRINTER PLOT OF MAGNITUDE OF XR,XI O1202C O1204C**********•******************************************* 01210 COMMON XR(4096),XI(4096),PHI(4096),UT(2048) 01220 K=ICENI-2*IN01-1 01230 DO 1 I=1,4*II101 01240 UT(I)=FLOAT(I) 01250 J=K+I 01260 1 PHI(I)=SORT(XR(J)*XR(J)+XI(J)*XI(J)) 01270 CALL GRAFIC(UT,PHI,4*IW91) 01280 RETURN 01290 END O1291C 01292C***************************************************** 01300 SUBROUTINE FRES(A,IIJ01,NMAX1) O1301C SUBROUTINE TO APPLY FRESNEL PHASE CORRECTION TO COMPLEX O1302C PAIR XR,XI. ARRAYS ARE ASSUMED TO BE IN FFT ORDER. O1303C O1304C**************************************************:*** 01310 COMMON XR(4096),XI(4096),PHI(4096),UT(2048) 01320 K=NMAX1+1 01330 DO 1 I=1,10*II41 01340 J=K-I 01350 FI=FLOAT(I)**2*A 01360 FS=SIN(FI) 01370 FC=COS(FI) 01380 COMM • 01390 XR(I)=XR(I)*FC-XIII)*FS• 01400 XI(I)=C*FS+XI(I)*FC 01410 C=XR(J) 01420 XR(J)=XR(J)*FC-XI(J)*FS 01430 1 XI(J)=C*FS+XI(J)*FC 01440 RETURN 01450 END O1452C***************************************************** READY.

331

Appendix A6.2 Input Data for SIMUL.

The data is read from TAPES, which is a file called INDAT.

The data format is:—

1 WO (f8.3)

2 AMDA (f8.3)

3 DX (f8.3)

4 LPN (I5)

then LPN sets of the following:

CO (f10.3), SCALE (f10.3), DZ (f10.3)

6 7

The following is a brief description of each parameter.

The quantities in brackets are the FORTRAN format required for each

parameter.

1) WO is the waist (metres) of the initial Gaussian aperture field

distribution.

2) AMDA is the wavelength (metres) of the e.m. wave

3) DX (metres) is the interval between sample points of the field

distributions of the simulation.

a) LPN (integer) is the number of slabs into which the propagation

path is to be .resolved.

Then follows LPN sets of parameters, one for each slab. These param-

eters are:-

5) CO is the correlation length (metres) of the refractivity fluctuations.

6) SCALE ( N units) is the rms value of the refractivity fluctuations.

7) DZ (metres) is the thickness of the slab.

For convenience, an interactive program (CRDAT) has been

written, which will request the above data and write all the data to the

332

file IND.4T in the correct format. A listing follows.

78/10/13. 18.06.05. PROGRAM CRDAT 00100 PROGRAM CRDAT(INPUT,OUTPUT,INDAT,TAPE5=INPUT,TAPE6=0UTPUT,TAPE7=INDAT) 0O11OC 00120C PROGRAM TO CREATE DATA FOR SIHUL ON "INDAT 0013OC 00140 WRITE(6,1) 00150 1 FORMAT(*BEAM WAIST IN METRES (F8.3)*) 00160 READ(5,2)C0 00165 WRITE(7,2)CO 00170 2 F0RMAT(F8.3) 00180 WRITE(6,3) 00190 3 FORMAT(*WAVELENGTH IN METRES (F8.3)*) 00200 READ(5,2)AMDA 00205 WRITE(7,2)AMDA 00210 WRITE(6,4) 00220 4 FORMAT(=SAMPLE INTERVAL IN METRES (F8.3)*) 00230 READ(5,2)DX 00235 WRITE(7,2)DX 00240 WRITE(6,5) 00250 5 FORMAT(*NUMBER OF SLABS (I5)*) 00260 READ(5,6)NUM 00265 WRITE(7,6)NUH 00270 6 FORMAT(I5) 00280 WRITE(6,7) 00282 X=123456.789 00284 WRITE(6,1O)X,X,X 00286 10 F0RMAT(2X,3F10.3) 00290 7 ' FORMAT(*CO3 STD DEV, SLAB WIDTH ACCORDING TO FOLLOWING COLUMNS*) 00300 DO 8 I=1,NUM 00310 READ(5,9)CO,SCALE,DZ 00320 WRITE(7,9)CO,SCALE,DZ 00330 9 FORMAT(3F1O.3) 00345 8 CONTINUE 00347 STOP 00350 END

333

Appendix A6.3 Output Data from SIIIOL.

The field distribution after each slab of the simulation

is written to a file DATA1 on TAPE7. The format of this data is now

described. There is one block of data for each slab simulated.

DX (f9.6), CEN (f9.6), Z'(f9.6), AMDA (f9.6), CO (f9.6), SCALE (f9.6)

1 2 3 4 5 6

XR

1024 points in f9.6

XI

1024 points in f9.6

The following details are relevant to the data fields:

1) DX (metres) is the distance between sample points of the XR and

XI arrays.

2) CEN (metres) is the distance from the leftmost sample point to the

beam axis ie. the maximum value of field in the unpeturbed case.

3) Z (km) is the distance from the original aperture to the field

values which are now being output.

4) AMDA (metres) is the e.m. wavelength.

5) CO (metres) is the medium refractivity correlation length.

6) SCALE (radians) is the equivalent phase shift induced by the slab

refractivity fluctuations— see line 00325 for the formula used.

XR and XI are the real and imaginary components of the field just imm-

erging from the slab in question, at a distance Z from the initial

distribution, sampled at an interval DX.

334

APPENDIVA6,■4---PROGRAM PLOTN.

00100 PROGRAM PLOTN(INPUT,OUTPUT,DATA1 ,PARAM,TAPE8=PARAN,TAPE=INPUT,TAPE6=OUTPUT, 00110+TAPE7=DATA1,TAPE62) 001200******************************************************** 00130C THIS PROGRAM PLOTS AMPLITUDE AND PHASE FROM DATA1 00140C IT HAS A VARIETY OF INTERACTIVE OPTIONS. 00150C******************************************************* **************** * 00160C 00170 COMMON X(1026),Y(1026),XR(1624),XI(1024),DX,XCEN,Z,AMDA,CO3SCL 00180 DIMENSION P(18),SI(25),W(1024) 00190 LOGICAL AMPO,LGLN,AUSE,RADE,FI0P,NH1L 00200C CLEAR ARRAYS 00210 DO 105 1=1,1026 00220 X(I)=0.0 00230 105 Y(I)=0.0 00240 DO 100 I=1,1024 00250 XR(I)=0.0 00260 100 XI(I)=0.0 00270 CALL START(2) 00280C 00290C READ FIRST BLOCK TO OBTAIN PARAMETERS OF RUN 00300 CALL REID 00310C 00320C WRITE OUT PARAMETERS 00330 WRITE(6,10) 00340 WRITE(8,10) 00350 WRITE(6,11)DX,Z,AMDA,CS,SCL 00360 WRITE(8,11)DX,Z,AMDA,CO,SCL 00370C 00380C ESTABLISH NO OF BLOCKS TO BE PLOTTED FROM DATA1 00390 WRITE(6,12) 00400 READ(5,13)NUM 00410C ESTABLISH AMPLITUDE PLT FEATURES 00420 WRITE(6,14) 00430 110 READ(5,15)A 00440 IF(A.E0.2HAM)AMPO=.TRUE. 00450 IF(A.E0.2HPO)AMPO=.FALSE. 00460 IF(A.EO.2HAM.OR.A.E1I.2HPO)G0 TO 120 00470 WRITE(6,16) 00480 GO TO 110 00490 120 WRITE(6,17) 00500 130 READ(5,15)A 00510 IF(A.EQ.2HLO)LGLN=.TRUE. 00520 IF(A.E0.2HLI)LGLN=.FALSE. 00530 IF(A.E0.2HLO.OR.A.EO.2HLI)G0 TO 140 00540 WRITE(6,16) 00550 140 URITE(6,18) 00560 150 READ(5,15)A 00570 IF(A.E0.2HAU)AUSE=.TRUE. 00580 IF(A.E0.2HSE)AUSE=.FALSE. 00590 IF(A.OR..NOT.A)G0 TO 160 00600 WRITE(6,16) 00610 GO TO 150 00620 160 IF(AUSE) GO TO 165 00630 WRITE(6,26) 00640 READ(5,22)AMX 00650 READ(5,22)AMN 00660 URITE(6,28) 00670 162 READ(5,15)A

335

00680 IF(A.EQ.2HYE)NM1=.TRUE. 00690 IF(A.E0.2HN0)NM1=.FALSE. 00700 IF(A.E0.2HYE.OR.A.EQ.2HNO)GO TO 165 00710 WRITE(6,16) 00720 GO TO 162 00730 165 WRITE(6,21)FLOAT(1024)*DX 00740 READ(5,22)XLNA O075OC O076OC ESTABLISH PHASE PLOT REQUIREMENTS 00770 WRITE(6,19) 00780 170 READ(5,15)A 00790 IF(A.EQ.2HRA)BADE=.TRUE. 00800 IF(A.E0.2HDE)RADE=.FALSE. 00810 IF(A.OR..NOT.A)G0 TO 180 00820 WRITE(6,16) 00830 GO TO 170 00840 180 WRITE(6,20) 00850 - 190 READ(5,15)A 00860 IF(A.EQ.2HAU)FIOP=.TRUE. 00870 IF(A.EQ.2HSE)FIOP=.FALSE. 00880 IF(A.OR..NOT.A)G0 TO 200 00890 WRITE(6,16) 00900 GB TO 190 00910 200 IF(FIOP)G0 TO 205 00920 IJRITE(6,27) 00930 READ(5,22)PMX 00940 READ(5,22)PHN 00950 205 WRITE(6,23)FLOAT(1S24)*DX 00960 READ(5,22)XLNP O0970C 00980C SET PARAMETERS FOR X DOMAINS 00990 CALL DOMAIN(NPA,ICENA,XFSTA,XLNA) 01000 CALL DOMAIN(NPP,ICENP,XFSTP,XLNP) 01010 10 FORMAT(* X INCREMENT Z (METRES) LAMDA (M) L0 (METRES) PHASE 01020+VAR (RAD)*) 01030 11 FORMAT(5F12.3) 01040 12 FORMAT(*NO. OF BLOCKS TO BE PLOTTED*) 01050 13 FORMAT(I2) 01060 14 FORMAT(*-AMPLITUDE-OR-POWER-PLOT*) 01070 15 FORMAT(A2) 01080 16 FORMAT(*WHAT? -TRY AGAIN*) 01090 17 FORMAT(*-LOG-OR-LINEAR-PLOTS*) 01100 18 FORMAT(*-AUTO-OR-SET-SCALING OF PLOTS*? 01110 19 FORMAT(*PHASE PLOTS IN-RADIANS-OR-DEGREES-*) 01120 20 FORMAT(*-AUTO-OR-SET-SCALING OF PHASE PLOTS*) 01130 21 FORMAT(*APERTURE DOMAIN LENGTH IN METRES, LT OR EQ*,F5.1) 01140 22 FORMAT(F5.1) 01150 23 FORMAT(*APERTURE DOMAIN LENGTH (M) FOR PHASE PLOTS, LT OR EQ TO *,F5.1) 01160 24 FORMAT(* Z (M) MAX FIELD *,A5,* WAIST (N) PHASE STD DEV*) 01170 25 FORMAT(F9.1,3F16.4) 01180 26 FORMAT(*SPECIFY DEPENDENT VARIABLE MAX, THEN MIN*) 01190 27 FORMAT(*SPECIFY PHASE RANGE, MAX THEN MIN*) 01200 28 FORMAT(*SHOULD ALL PLOTS BE REFERENCED TO EMAX=1.0-YES-OR-N0-*) O121OC O122OC PRINT MAX FIELD VALUE HEADINGS 01230 ALP=5H 01240 IF(.NOT.AMPO)ALP=5HPOWER 01250 WRITE(6,24)ALP 01260 URITE(8,24)ALP

336

01270C 01280C NOW PLOT AMPL/PHASE NUM TIMES 01290 DO 1000 IN=1,NUM 01300C CALCULATE POWER OR AMPLITUDE 01310 J=0 01320 IF(.NOT.AMPO)G0 TO 220 01330 DO 230 I=ICENA-NPA/2,ICENA+NPA/2 01340 J=J+1 01350 230 Y(J)=SORT(XR(I)*XR(I)+XI(I)*XI(I)) 01360 GO TO 240 01370 220 DO 250 I=ICENA-NPA/2,ICENA+NPA/2 01380 J=J+1 01390 250 Y(J)=XR(I)*XR(I)+XI(I)*XI(I) 01400 240 CALL MXMN(NPA,YMX) 01410 CALL WAIST(NPA,YMX,AMPO,UNVE) 01420C 01430C SCALE INTO DB IF NECESSARY 01435 YMX1=YMX 01440 IF(NM1)YMX=1.0 01450 IF(.NOT.LGLN)G0 TO 260 01460 DO 270 I=1,NPA 01465 IF(Y(I).LE.1.0E-10)Y(I)=1.0E-10 01470 270 Y(I)=10.0*ALOG10(Y(I)/YMX) 01480C 01490C APPLY LIMITING TO ARRAY IF -SET- IS SELECTED 01500 260 IF(.NOT.AUSE)CALL LNT(NPA,AMX,AMN) 01510C 01520C NOW GENERATE X ARRAY AND SCALE BOTH FOR PLOT 01530 XC=XFSTA 01540 DO 280 I=1,NPA 01550 X(I)=XC 01560 280 XC=XC+DX 01570 CALL SCALE(X,20.0,NPA,1) 01580 CALL SCALE(Y,15.0,NPA,1) 01590 ALP=10HFIELD 01600 IF(LGLN)ALP=18HFIELD (DB) 01610 IF(LGLN.AND..NOT.AMPO)ALP=10HPOWER (DB) 01620 CALL AXIS(0.0,0.0,ALP,10,15.0,90.0,Y(NPA+1),Y(NPA+2)) 01630 CALL HEAD(NPA,UNVE) 01640 CALL LINE(X,Y,NPA,1,0,1) 01650 CALL NEWPAGE 01660C 01670C NOW PLOT PHASE. FIRST SCALE FOR DEG OR RAD 01680 DR=1.0 01690 IF(.NOT.RADE)DR=180.0/3.14159265 01700C 01710C 01720C CALCULATE PHASE 01730 J=0 01740 DO 290 I=ICENP-NPP/2,ICENP+NPP/2 01750 J=J+1 01760 IF(XR(I).LE.1E-10)XR(1)=1.0E-10 01770 290 Y(J)=DR*ATAN(XI(I)/XR(I)l 01780C 01790C

337

01800C SCALE MODIFIED ARRAYS; CREATE X ARRAY 01810 XC=XFSTP 01820 DO 300 I=1,NPP 01830 X(I)=XC 01840 300 XC=XC+DX 01850C REMOVE PARABOLIC TREND FROM PHASE BEFORE PLOTTING. 01860 DO 310 I=1,NPP 01870 310 U(I)=1.0 01880 K1=3 01885 L=.FALSE. 01890 CALL E02ABF(NPP,X,Y,W,K1,NU,SI,P,L) 01900 DO 320 I=1,NPP 01910 320 Y(I)=Y(I)-(P(1)+X(I)*(P(2)+X(I)*P(3))) 01920C LIMIT Y ARRAY IF NECESSARY. 01930 IF(.NOT.FIOP)CALL LMT(NPP,PMX,PMN) 01940 CALL SCALE(X,20.0,NPP,1) 01950 CALL SCALE(Y,15.0,NPP,1) 01960C CALC PHASE STD DEV 01970 SUM=0.0 01980 SUM2=0.0 01990 DO 330 I=1,NPP 02000 SUM=SUM+Y(I) 02010 330 SUM2=SUM2+Y(I)*Y(I) 02020 OMN=SUM/FLOAT(NPP) 02030 SIGMA=SART(ABS(SUM2/FLOAT(NPP)-OMN*OHN)) 02040C

02050C PLOT AXES, HEADINGS ETC 02060 ALP=IOHPHASE-RAD- 02070 IF(.NOT.RADE)ALP=10HPHASE-DEG- 02080 CALL AXIS(0.0,0.0,ALP,10,15.0,90.0,Y(NPP+1),Y(NPP+2)) 02090 CALL HEAD(NPP,UNVE) 02100 CALL SYMBOL(0.5,14.0,0.25,13HPHASE STD DEV,0.0,13) 02110 CALL NUMBER(4.0,14.0,0.25,SIGMA,0.0,4) 02120 CALL LINE(X,Y,NPP,1,0,1) 02130 WRITE(6,25)Z,YMX1,UNVE,SIGMA 02140 WRITE(8,25)Z,YMX1,UNVE,SIGNA 02150 IF(IN.NE.NUM)CALL REID 02160 1000 CALL NEWPAGE 02170 CALL ENPLOT 02180 STOP 02190 END

338

02200C**************************************************** 02210C 02220C SUBROUTINES 02230C 02240C**********************:**********:**********_********** 02250 SUBROUTINE DOMAIN(NP,ICEN,XFST,XL) 02260C CALCULATES NO OF POINTS ETC NEEDED TO REALISE DOMAINS 02270C 02275C***********************************************:********:**:***:****:*** 02280 COMMON X(1026),Y(1026),XR(1024),XI(1024),DX,XCEN,Z,AMDA,CO,SCL 02290 NP=INT(XL/DX+0.5) 02300 ICEN=512 02310C THIS ASSUMES THE INPUT DATA IS SYMMETRIC ABOUT CENTRE OF ARRAY 02320 XFST=-XL/2.0 02330 RETURN 02340 END 02350C 02355C***********************:**:**:***********:**********************:****** 02360 SUBROUTINE MXMN(NP,YMX) 02370C FINDS YMX, THE MAX VALUE OF NP POINTS OF ARRAY Y 02380C 02385C******************************************:*:************:****:****:**** 02390 COMMON X(1026),Y(1026),XR(1024),XI(1024),DX,XCEN,Z,AMDA,CO,SCL 02400 YMX=0.0 02410 DO 100 I=1,NP 02420 100 IF(Y(I).GE.YMX)YMX=Y(I) 02430 RETURN 02440 END 02450C 02455C***:****:*******:*************************************:**:****:******:***** 02460 SUBROUTINE LMT(NP,BMAX,BMIN) 02470C LIMITS NP POINTS OF Y ARRAY TO LIE BETWEEN BMAX,BMIN 02480C 02485C****************************************:**:**:********************** 02490 COMMON X(1026),Y(1026),XR(1024),XI(1024),DX,XCEN,Z,AMDA,CO,SCL 02500 DO 100 I=1,NP 02510 IF (Y(I).GE.BMAX)Y(I)=BMAX 02520 100 IF(Y(I).LE.BMIN)Y(I)=BMIN 02530 Y(1)=BMAX 02540 Y(NP)=BMIN 02550 RETURN 02560 END 02570C 02575C***:****:***:****:****:****:**:******:*************************:************** 02580 SUBROUTINE REID 02590C READS HEADINGS AND 2*1024 POINTS FROM-DATA1- 02600C - 02605C******************************************:****:****:******_*****:******** 02610 COMMON X(1026),Y(1026),XR(1024),XI(1024),DX,XCEN,Z,AMDA,CO,SCL 02620 READ(7,10)DX,XCEN,Z,AMDA,CO,SCL 02630 READ(7,11)(XR(I),I=1,1024) 02640 READ(7,11)(XI(I),I=1,1024) 02650 Z=Z*1000.0 02660 RETURN 02670 10 FORMAT(6F9.6) 02680 11 FORMAT(F9.6) 02690 END 02700C

339

02705C***k#k*:k:kkt**0k*m***k*k*Y*k*k k*k#*mt.***kk**0***rnm*rml*I4k*Pt 02710 SUBROUTINE HEAD(NP,UNVE) 02720C PLOTS HEADINGS AND X AXIS 02730C 02735C******************************k*:** #:r•*:~ *:ti**rnu*1:*k*:~ *:k*1**:ti**tf:*+:*ir• 02740 COMMON X(1026),Y(1026),XI(1024),XR(1024),DX,XCEN,Z,AMDA,CO,SCL 02750 CALL SYMBOL(13.5,14.0,0.25,10HZ (METRES),0.0,1O) 02760 CALL SYMBOL(13.5,13.5,0.25,19HWAVELENGTH (METRES),0.0,19) 02770 CALL SYMBOL(13.5,13.0,0.25,21HPHASE CORR LENGTH (M),0.0,21) 02780 CALL SYMBOL(13.5,12.5,0.25,19HPHASE STD DEV (RAD),0.0,19) 02790 CALL SYMBOL(13.5,12.0,0.25,9HWAIST (M),0.0,9) 02800 CALL NUMBER(19.0,14.0,0.25,Z,0.0,-1) 02810 CALL NUMBER(19.0,13.5,0.25,AMDA,0.0,3) 02820 CALL NUMBER(19.0,13.0,0.25,C0,0.0,2) 02830 CALL NUMBER(19.0,12.5,0.25,SCL,0.0,3) 02840 CALL NUMBER(19.0,12.0,0.25,UNVE,0.0,3) 02850 CALL AXIS(0.0,0.0,23HDISTANCE ALONG APERTURE,-23,20.0,0.0,X(NP+1),X(NP+2)) 02860 RETURN 02870 END 02880C 02885C***********************************r***k***1:*::*►:*e*:f:*Ema*t t*****k**: : 02890 SUBROUTINE UAIST(NP,YMX,AMPO,UNVE) 02900C CALCULATES THE WAIST OF A GAUSSIAN DISTRIBUTION 02910C STORED IN ARRAY Y. AMPO IDICATES IF Y IS FIELD OR POWER 02915C 02917C******************************************:t:s**:+:**:r**:+:**:t*•*******:r•*** 02920 COMMON X(1026),Y(1026),XR(1024),XI(1024),DX,XCEN,Z,AMDA,CO,SCL 02930 LOGICAL AMPO 02940 IL0=1 02950 IHI=1024 02960 E=YMX*EXP(-1.0) 02970 IFI.NOT.AMPO)E=E*E 02980 I=50 02990 1 IF(Y(I).GE.E)G0 TO 2 03000 I=I+1 03010 IF(I.EO.NP)G0 TO 5 03020 GO TO 1 03030 2 ILO=I 03040 3 I=I+1 03050 IF(Y(I).LE.E)G0 TO 4 03060 IF(I.EO.NP)G0 TO 5 03070 GO TO 3 03080 4 IHI=I 03090 5 UNVE=(FLOAT((IHI—ILO)/2)+0.5)*DX 03100 RETURN 03110 END 03120C**************: ***:********ft***:+:*******k*****x*:+:***:r.** READY.

340

CHAPTER SIX REFERENCES.

6.1 Imperial College Computer Centre Graphics Software

Bulletins. ICCC, Mechanical Engineering Building,

Exhibtion Road, London SW7 2AZ.

6.2 Numerical Algorithms Group Mk 5F version.:: software.

NAG Central Office,, 7 Banbury Road, Oxford 0X2 6NN.

341

CHAPTER SEVEN MICROPROCESSORS IN THE LABORATORY.

Introduction.

For the research worker, the influence of the micropro-

cessor is most strongly apparent in two aspects of research,activity.

On the first level, it is extremely useful as a replacement for com-

plex logic circuitry, used, for example, to gather data automatically

from an experiment. Modifications to the method of operation of the

equipment incorporating such a device now involve a change of a small

program, rather than a radical rewiring of purpose built logic

circuits.

Perhaps the more important innovation brought about by the

microprocessor, is the configuration of a microprocessor and related

devices to form what are most usefully referred to as "microcomputers".

The fact that these microcomputers are far more powerful than many

computers available in the 1960's is incidental. The main revolution is

that of cost. These small systems may be obtained for as little as €1000.

In this chapter, the author will examine the possible con-

stituents of a small system, and high—light some of the uses to which

they may be put. This should be of interest to readers considering the

design of propagation experiments and related studies.

The description of work involving microprocessors is

particularly fraught with difficulties, since as a rapidly evolving

technology, there are no standard texts on the subject. It is impossi-

ble not to have to use at least some of the jargon which has come in

to use: explanation is made when the term first arises in the text.

A further difficulty. lies in the fact that advances in

integrated circuit manufacture are combining many previously separate

devices into one package. It was thus decided that the most effective

solution for describing something useful in a short space, was to con-

342

centrate on one system, which is representative of what is available

at the moment.

7.1 Microprocessor versus Microcomputer.

As can be seen from fig.7.1, the microprocessor is a part

of what is termed a microcomputer. It is perhaps the most important

part, but is totally dependent on a number of peripheral devices in

order to be useful as a system.

When used to replace a complex arrangement of logic,

the processor may be used virtually on its own. Usually only some extra

ROM (read only memory) is required.

The reality behind this apparently simple viewpoint is

somewhat complex. The final circuit may be very simple, But anyone

contemplating the design of such equipment must have available the.

following essentials;

a) The exact requirements of the final equipment.

b) Detailed knowledge of the microprocessor chosen to control the

circuit.

c) A detailed knowledge of the interfacing (especially timing). of

microprocessors to other circuitry.

d) The use of programs on a larger machine to convert the program

written in the mnemonic programming language of the micropro-

cessor to binary representations which have to be entered

into the ROM.

e) Equipment to enter the binary code into the ROM or some other

memory device.

The above requirements may seem rather daunting and con-

firm the viewpoint that attempting to use a microprocessor in its

simplest form requires large resources, which are not available to most

To

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343

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Fig.7.2 Microprocessor developement kit (for the Motorola 6800)

Ta TTY

344

researchers. This is especially true if the data collection or experi-

mental control aspect is only a small part of the overall project.

The situation is made a lot simpler by the availability

from microprocessor manufacturers of what are known as "develop ment

kits". These are not as comprehensive as the microcomputer shown in

fig.7.1, but consist of some of the more important devices. Such a

system is shown in fig.7.2. The essential advantage of these systems is

that the interfacing with the microprocessor is much simplified. The

loading of manually assembled programs is possible via the rudimentary

keyboard. Once entered, these programs may be stored on casette tapes

recorded on an audio taperecorder. All these complex functions are

fulfilled by programs written by the manufacturer and stored in a ROM.

This program is known as the monitor. The user is now faced with the

relatively simple task of interfacing with what us usually referred to as

the PIA (peripheral interface adaptor).

Even at this level, the process of setting up a system.

is laborious. Complex interval timing or numerical manipulation

require large, complex coding in the very rudimentary machine code.

There is a heavy overhead in time taken to become familiar with this

language.

The situation can be alleviated to some extent by the

availability, for some processors, of high level compilers such as

CORAL. These are implemented -on a large machine from a high level

language similar to FORTRAN. The large machine produces a listing in

the microprocessor language of the more complex, high level program.

Even the simplest of high level language codes will produce a vast

output in machine code, especially if numeric manipulation is required.

Clearly, although there are some difficulties, it is

possible to use a minimum of electronic packages to achieve a fairly

complex logic function. It is now useful to proceed to the next level

345

of sophistication, the microcomputer. At all times, it is very

important to weigh up carefully the advantages in cost of the simple

system against the simplicity of operation of the microcomputer which

will be described in the next section. It is the author's contention

that there are very few situations where the more complex system is not

ultimately more cost—effective.

7.2,, A Typical Microcomputer.

As mentioned previously, a particular system will be

described. It is representative of the trend which microcomputers seem

to be following. The particular system is based on the Polymorphic

Systems Inc. "Poly 88". This is in turn based on one of the most widely

used 8—bit microprocessors, ie., the INTEL 8080A.

A complete system diagram is given in fig.7.3. The most

important aspect of the system is that it is divided into boards, each

fulfilling one or many functions. In this case, the boards are linked

together by a common bus, known as the "S100" bus. The 5100 bus was

first introduced by IMSAI of the United States and is under consideration

by the IEEE for possible international standardization.

The idea of a bus structure is by no means new, but is

an important aspect to be considered by a prospective purchaser of a

machine! Some manufacturers produce very good equipment based on

their own unique bus structure. There is no reason not to use such a

machine, provided the manufacturer is able to offer all types of function

board-. that the user requires. Some of the functions which may be

required are:

a) Analog to Digital and Digital to Analog (ADC and DAC) Converters

of sufficient resolution.

b) General purpose digital input or output boards for interfacing

with other equipment.

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346

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Fig.7.3 Microcomputer based on the Poly 88 and the S100 bus.

347

c) Serial input/output boards conforming to one of the industry

standards such as the 85232 or 20 mA current loop type. This

allows links to be set up between computers or communication

with a device such as a serial printer.

Memory boards allowing the program capacity of the machine to

be simply increased.

Magnetic disc controllers. These control a small "mini—floppy"

disc drive, allowing about 106 characters of data or program

to be stored.

Hard—wired arithmetic boards. These do numeric operations such

as multiplication and division using hard—wired, but fast,

circuits. These boards often speed up the numerical capabili-

ties of a microcomputer by a cfactor of ten.

The system described in fig.7.3 is composed of boards from

a number of different manufacturers, but because they all conform to the

5100 standard, there are no problems with compatibility. This is not

to say that the S100 is the only or even the best bus structure. At

the moment it is perhaps the most widely used bus. This is mainly

because of the huge growth of the "hobby computing" market in the United

States. Amateur Radio enthusiasts pioneered the use of short—wave radio,

and it seems as if the amateur computer enthusiast has led industry in...

establishing some sort of bus standard.

The references for this chapter are from necessity mainly

manufacturers' data sheets and manuals. They do no reflect the best

solution in view of the rapid rate at which new and better devices

become available.

Until now, only the hardware aspects of the system have been

described. In many respects the software, ie, the programming languages

and programs which may be implemented on the machine is just as

348

important.

In view of the limited memory available to these machines,

interpreter languages are most easily implemented. These are high

level languages which are entered into the machine. The machine stores

the program exactly as entered, although this may be in an abbreviated

form. On being requested to execute the program, the machine Fgoes

through this same program, executing each operation as specified in the

program. BASIC is a typical language implemented. Interpreters differ

from the compiler approach taken on large machines. In a compiler based

machine, a number of passes are made through the source program, and

from this is produced a completely translated machine code representa-

tion of the original source program. An interpreter may be likened more

to the process an operator goes through when operating a simple electronic

calculator to evaluate a complicated expression.

Compilers require a large amount of storage, since a..

number of passes and translations have to be done. The fall in price

of disk—controllers and mini—floppy disc drives has enabled some manu-

facturers to. offer compilers for the more sophisticated end of the

microcomputer market. Implementations of FORTRAN and COBOL are available.

It is generally true that the versions of BASIC Fwhich are

implemented on the small machines is highly non—standard. This is mainly

to include instructions which allow the programmer to access the various

devices, such as DAC/ADC boards,- in a simple way. In this way it

becomes a trivial step to obtain a numeric representation of the voltage

input to the ADC as a BASIC variable. Once this has been done, complex

scaling and processing is possible, using the full set of mathematical

functions available in BASIC (ie. SIN, COS, and so on). The lineari-

zation of data is simple. An example of a comprehensive program using

non—standard BASIC is given in appendix A7.1. This program samples,

data, scales it and then plots the scaled data by driving an XY plotter

349

from is DAC board.

In a similar way, voltages may be output and used to

control an experiment. The machine may be used as a feedback control

unit. A voltage sensed may be compared with predetermined limits and

the machine will provide an output voltage which will complete the

necessary correction.

7.3 Applications.

In this section, a few examples will be given of how the

system Used in the propagation experiment described in chapters 3 and 4

was found useful in a number of applications. Although only available

for the latter portion of the experiment, it was possible to use the

machine for a variety of tasks.

The list of uses given is far from complete, but will

help to emphasise the simplicity with which such a system may be adapted

to execute a number of tasks.

7.3.1 Computing Spectra.

Using the ADC, an input waveform may be sampled, and the

samples stored. The sampling speed is dictated in the usual way by the

maximum frequency content of the waveform. In the Poly 88 system, the

setting of the sampling rate is particularly simple, in that the main

processor board has a special counter which is updated by the AC mains

frequency. It is thus a simple matter to write a program in BASIC which

examines this counter frequently, and takes a sample every time a certain

number of mains cycles has taken place.

With this technique, the maximum sampling rate is that of

the mains frequency, which is more than adequate for the narrow band-

width of propagation data, in most cases. The ADC is capable of sampling

rates of about 20 kHz. More sophisticated methods are available to time

350

these higher sampling rates, but these depend on the particular machine

being used.

The storage of the data may be approached in two ,•/ways.

The ADC used in this system has only an B bit resolution.. This is the

same width as the length of the system memory, and so the most efficient

technique is to store each sample in one word (byte) of memory. To

do this, the non standard BASIC is useful. Examination of the listing

given in appendix A7.1 will show that this has been done.

A difficulty with this method arises when the spectrum is

to be computed using a BASIC implementation of the FFT algorithm. In

such a program it is assumed that the data to be transformed is stored in

an array of BASIC variables. In principle it is possible to modify such

programs to work with the data stored in 1 byte locations, but this is

tedious and. inaccurate.

A more convenient technique of storing the data is to read

it directly into BASIC arrays of variables. This is expensive in memory

usage, since 8 digit accuracy BASIC will take about 5 bytes to store

each variable. For small machines, this approach will severely limit.

the. number of samples which may be stored.

For the 16K (16 X 1024) block of memory available to this

machine for storing BASIC ('ie the interpreter program). and the FFT

program, it was found that about 256 sample points was the maximum

which could be held.

The dynamic range of the ADC (8 bits) and the small

number of samples restricts the accuracy of the spectra which are

possible with the small machines, but they are invaluable for the

on—line monitoring of the behaviour of a signal. Section 5 of this

chapter will show how more sophisticated processing is possible if the

data collected is passed to a more powerful machine.

351

7•-3_•2 Transient Recorder.

The methods described in the previous section may be

used to sample an input waveform and store the samples in the machine's

memory. Again, the sampling rate may be accurately controlled. It is

also possible to control the time at which the recording starts and

stops. Once in memory, the samples are available for examination at

leisure.

A number of ways of displaying the samples are available.

If a DAC is available in the system, the samples may be 2converted

back into voltage and displayed on an oscilloscope or even a chart/

XY recorder. The rate at which the waveform may be replayed is easily

controlled by program. It is also possible to edit the data.

On this system, the resolution is limited to 8 bits, but

higher resolution ADC/DAC's are available.

The samples may, of course, be processed before dis-

playing to remove transducer non—linearities or other effects. It is

also possible to take into account transducer scaling factors and display

information in the fundamental units required. Appendix A7.1 is an

example of the implementation of some of these ideas.

7.3.3 Statistical Analysis.

Once again using the sampling techniques described in the

previous subsections, it is possible to examine the statistics of the

sampled waveform and calculate some of the moments such as mean, vari-

ance and so on. It is not necessary in this case to retain the samples,

since computational formulae for these moments only require sums of

products of the samples to be retained.

If the ADC is a multichannel device, it is possible to

sample a number of waveforms simultaneously. These samples may be used

to calculate the cross correlation between these channels of data.

352

This is particularly useful for propagation type experiments. Once

again it is not necessary4to retain the samples in memory, since the

sums of products and cross—products are sufficient to measure the

cross correlation coefficients.

It must be stated, however, that sometimes it is necessary

to retain the data samples and do numeric computation once the required

number of. samples has been taken. This is true when the sampling speed

required (to represent the data accurately) is too high for the machine

to complete the necessary arithmetic after each sample. It should be

remembered that these small machines do arithmetic by program rather

than by circuitry; this is inherently slow. A complex linearization

of data can be particularly slow. Appendix A7.2 is an example.

There are a number of diagrams in chapters 2,3 and 4,

showing the statistics of data analysed by the Poly 88 system. Again,

it must be stressed that the resolution of the ADC limits the accuracy

of the statistical quantities estimated, but the ability of being able

to analyse data on—line on a continuous basis is very useful.

The next two sections will investigate two further uses of.

a microcomputer in some detail.

7.4 Microcomputer Networks.

Large ("mainframe") computers have been linked together for

a number. of years. The postal authorities in various countries support

many methods for physically linking machines, and this represents one

of the largest growth areas in communications.

The linking together of a number of small machines is not

nearly ās.. complex as the virtually error—free methods devized for large

machines.

For the scientific user, the need to link two small

machines 'might arise in a number of applications. For example, con-

CONTROL

CoMPUTER

/\,y-( MODEM)

PARALLEL TRANSFER

353

sider an application where a number of channels of data must be sampled

at a remote site and transmitted via a single voice quality channel

back to a data analysis centre. A solution to this problem is shown

in fig.7.4.

_—}-- SEgIAL TRRNSFER

Fig. 7.4 Sampling and transmission of data from a remote site.

The low cost microcomputer is programmed to sample input

analog data. This data is transmitted via a modulator/demodulator

(modem) and is further processed by a receiving computer with a

suitable modem. This might well be another microcomputer.

A further level of sophistication is possible if the link

between the two machines is made bi—directional. The controlling

machine at the receiver site may then issue commands to the remote

machine,, which could then respond with specific data.

Unfortunately, a "Tower of Babel" situation exists for much

of the communication between māchines ie. many standards exist and are

totally incompatible. This is especially true of parallel data transfers.

Fortunately, the small machine manufacturers have tended to

stick to the RS232 or 20 mA current loop standard for serial communication.

354

This takes place at accurately controlled bit rates. The linking

together of two small machines is thus most easily achieved using a serial

RS232 interface, especially if an asynchronous mode is used. This means

that each character transferred has a clearly defined start and stop.

This method is robust in terms of data errors, but is not the most

efficient. Bit rates are usually kept to below 1200 bits per second

in this mode.

The ease with which machines may be linked also depends on

the power of the interpreter commands which control the Universal

Sychronous/Asynchronous Receiver/Transmitter (QSART) which handles the

physical data flow. It is often necessary to write small portions of

machine code to handle these transfers successfully.

.7.5 Microcomputer as an Intelligent Terminal Device.

An "intelligent terminal" is a generic name given to a

terminal (keyboard and display) connected to a larger "host" computer,

where the terminal is capable of executing a complex task when commanded

by either the user on the keyboard, or by the host machine.

A simple terminal transfers each character as it is entered

by the user, to the host machine; an intelligent device might have a

small block of memory available and store a line of characters before

transmitting them to the host. The terminal might also allow the user

to edit.the line for errors or additions before being commanded to

transfer the whole line to the host machine.

The above is a very simple example of an intelligent terminal.

By present day standards, this would not qualify for the description

of "intelligent terminal".

The Poly 88 system previously described, was used as a

terminal into the main Imperial College computer network. The program

required to do this on a very simple level is not complex; the logic

TRANSFER CHAR FROM KBD aUFF t TO VOV D1seute

KES

NO

COVE CHAR TO USART FOR TRANSM(S51oN

VDU

USPRT

TRANSFER CHMRILTER FROM

VSggT gOFFE i0 VDU.

355

of the program is shown in fig.7.5.

KEV60ARD

PARA LLE L TRANSFERS

fig.7.5 Microcomputer as terminal.

While using this type of system as a VDU terminal to

develop the programs implementing the propagation simulation described

in chapters 5 and 6, a fundamental limitation of the VDU type terminal

became apparent.

A typical VDU will only display between 12 and 24 lines of

text at any time. Usually there are between 32 and 80 characters on

each line. As each new line is entered by the user or host machine,

the upper line is "scrolled" off the display and is lost. If the program-

mer wishes to check any of the. lines lost in this way, he is forced to

request a re—listing from the host machine. However, it is necessary

356

for the user to know the exact line number of the program line he

wishes to see. If he requests to❑ many lines to be displayed, once

again some important part of the listing is lost. In the case of data

lines, the situation is far worse, since no line numbers exist for

the user to use in requesting a short listing. When reviewing data,

most users are forced to abandon the VDU and seek a printing type

terminal.

The above considerations led the author to devize a system

using the Poly 88. This idea was implemented as an undergraduate

project (Dale 7.1 ). The whole concept is indicated in fig.7.6.

VDU RcTS AS A 15 LINE

r WINDOW"

DATA ENTERED FROM

HOS1 CDMPuTER

TOP

END

Fig.7.6 Concept of a terminal buffer store.

As characters are received from the host machine, they are

stored sequentially into the microcomputer memory. To save on storage,

the characters generated by the user on the keyboard are not recorded.

357

The program written to do this operated on two levels.

In the normal mode, the program behaved exactly like a conventional

VDU type terminal, but in addition stored the characters as described

above.

However, on detecting a certain control character (CNTRL—A)

from the user, the program entered a "display mode". In this mode,

the machine responded to single character commands from the keyboard.

The character "R" caused the first 15 lines of the host

computer's response to be displayed on the VDU. Each succeeding "D"

moved this display one line downwards in the recorded data; a "U" had

the opposite effect. The letter "E" moved the VDU "window" to display

the last 15 lines, entered just before the display made was entered.

Thus, if the user had once requested a program or data

listing from the host machine, it was possible for the user to step

through the whole listing at leisure, using single keystrokes to do

this.

Only 15 lines were displayed, since on leaving the display

mode, (CNTRL—A again), the last line was left free for the user to enter

a line, using the display as a template. It is very useful in editing

to have the original line available as a template.

This controlling program was written in INTEL 8080 Assem-

bler Language, taking just short of 1024 bytes of program store to

implement. This code was entered into an erasable, programmable,

read-only. memory (EPROM) and plugged into a socket available on the main

processor board. The program is thus available immediately on switch-

ing on the machine.

In practice, the program and conbept worked very well.

A few improvements did suggest themselves, and would make the concept

even more useful. What is envisaged is a more powerful line editor.

The program would work as described previously, with the following

facility as an addition.

358

When in display mode, it would be possible for the user

to move a cursor character anywhere on the screen. Further, the data

lying adjacent to this cursor could then be altered at will, ie.,

characters could be inserted or deleted, and so on. When a satisfac-

tory version of the old line had been reconstructed, a single keystroke

would transmit this whole line to the host machine, as the new version.

The principal advantage of such a form of editing is that

most of the characters of the line being corrected would already be

present; most errors in programming involve changing only one or two

characters in a whole line.

The program described is more sophisticated than the simple

description given here. A number of useful methods were devised to

pack the information more efficiently into the limited memory available.

Because the memory available is finite, •the data file was

made "circular" ie. the oldest information was overwritten first when

the available memory was full. More details are available in the

following references: Dale7.1, Inggs and Sage7.2.

7.6 Microcomputers as an aid to Numerical Analysis.

In principle, all research workers have access to comput-

ing facilities, usually this takes the form of a large machine made

accessible via a network of terminals.

For complex programs, the large machine with high speed

input/output devices is essential. There are, however, a great

number of simple computations which do not quite warrant the effort of

coding and running on large machines. Often the set of commands necessary

to get the program running on a large system nearly equals•the program

in lengths

The microcomputer fills the gap admirably. On one level

359

it may be treated as a programmable calculator. It is programmed in

BASIC, which has distinct advantages over the very simple instruction

set of programmable calculators.

A research worker is very quickly able to build up a

library of simple programs which are useful for. his field of work.

These are stored on magnetic disc or tape and may be simply loaded and

run at any time, directly in the laboratory. Again, a large instal-

lation can do all of this; the small system is simpler to operate. An

example of a useful integration program is given in Appendix A7.3.

A more subtle advantage becomes apparent when finding

errors in program logic or syntax (debugging). With the microcomputer,

it is possible to halt the execution of a program at any moment, and

examine the contents of variables directly. The execution can then

resume and the process be repeated until the logic error has been traced.

On large, timesharing systems, debugging in this way is not possible.

Suspect variables have to be printed out by means of inserted program

code. This often results in large amounts of output just to trace a

small error in logic.

It is useful in some cases to examine the effect of vary-

ing certain parameters on a fairly complex mathematical expression.

Simple graphical output from microcomputers is possible, and makes the

above task fairly simple. For the Poly 88 machine described, this was

' achieved using an XY plotter attached to the DAC. A simple general—

purpose plotting program written in BASIC was produced to this end. A

listing is given in appendix A7.4. This program converts the user

supplied function into voltage levels suitable for driving the XY plotter.

The program listed is more sophisticated; it includes features such as

pen up/down control, scaling and so on.

Again, graphics is far more sophisticated when implemented

on large machines. The principal disadvantage is the expense of

360

graphics display units— a figure of £5000 is about the lowest at present

for a non—permanent VDU type. The advantage of the "hands on" control

of the small machine becomes very apparent when the user of a large

installation typically has to wait overnight for work to be processed

on a drum or flat—bed plotter. This reduces the user to correcting very

few errors during a day's running.

7.7 Conclusions.

This chapter has attempted to show some of the uses to

which-a microcomputers may be put. The microcomputer has been shown to

be a low cost device made possible by recent advances in microelectronics.

It is envisaged that the microcomputer will become an essen-

tial part of any laboratory, in the way in which the oscilloscope is at

present. As the technology advances, the microcomputer will be able to

do all the tasks at present done by purpose—built instruments. There is

already a trend toward this state of affairs, in that instrument manu-

facturers are now designing merle their devices with digital control

interfaces.

One of the most important advances in this field, has been

the IEEE-488 Standard? 4. This standard describes a general—purpose

16—line digital interface bus, specifically designed for the simple

implementation of digital control of devices. The microcomputer, as a

' controlling element, is thus already able to make routine measurements

of most quantities in a laboratory.

For example, using a programmable ADC, it is possible

for a fast microcomputer to become a very efficient storage oscilloscope.

The advantages become particularly apparent when long and tedious meas-

urements have to be made.

Undoubtedly, the next few years will see radical changes in

the form of general—purpose laboratory equipment.

361

APPENDIX A7.1 Program which samples and stores data.

5 REM********** PROGRAM SMPPLT ******************** 10 REM PROGRAM TO SAMPLE AND PLOT WAVEFORM INPUT TO CHAN 1 20 REM OF ADC, AFTER SAMPLING AT USER SET RATE, POINTS ARE 30 REM PLOTTED ON USER SET AXES— ONLY Y AXIS MAY BE SET. 40 REM THE TIME AXIS REPRESENTS 255/FSAMP SECONDS. 50 REM ************************************************** 60 INPUT"SAMPLING FREQUENCY (HZ)? "FF 1

70 T1=50/F 80 PRINT"***** RTC ON TO START SAMPLING *****" 90 FILL 3072v0

, 100 T=EXAM(3072)\IF T<10 THEN 100 110 PRINT"SAMPLING "

120 J=24000 • 130 FOR I=U TO 255 140 FILL 3072v0\Y=INP(49)\FILL JrY\J=Jf1 150 T=EXAM(3072)\IF T<T1 THEN 150 160 NEXT I 170 PRINT"***** SAMPLING OVER—RTC OFF *****" 180 INPUT"SCALING FACTOR IN UNITS/VOLT? ',Si 190 G1=2.56*S1/138 200 REM**************************************** 210 PRINT"CALIBRATE PLOTTEB:O=> BOTTOM LEFT"

. 220 PRINT" 1=> TOP RIGHT" 230 PRINT" 2``> CONTINUE TO PLOT" 240 C=EXAM(3085) 250 IF C=13 THEN 240 '360 C=EXAM(3085) 270 IF C<} 48 THEN 310 • 280 FOR 1=228 TO 128 STEP-1\OUT(49)I\OUT(50)I\NEXT 290 OUT(49)128\OUT(50)128\C=EXAM(3085) 300 IF C<>48 THEN 260 ELSE 290 310 IF C<> 49 THEN 350 320 FOR 1=27 TO 127\OUT(49)I\OUT(50)I`.NEXT 330 DUT(49)127\OUT(50)127\C=EXAM(3085) 340 IF C<>49 THEN 260 ELSE 330 250 FILL(3084)1 360 INPUT1"YMIN9 '012 37C' INPUT" YMAX? "0'1 . 380 S2=ABS((Y1—Y2)/255) 390 FOR I=1 TO 20\Y=EXAM(24000)\IF Y>127 THEN Y=Y-256 400 Y=Y*S1/S2 410 IF Y>127 THEN Y=127 420 IF Y<-128 THEN Y=-128 430 IF Y<0 THEN Y=Yf256 440 OUT49r128\OUT50vY\NEXT z 450 •J=24000 460 IF Y<138 THEN Y=128 470 K=24000 480 FOR I=0 TO 255\Y=EXAM(K)\IF '0127 THEN Y=Y-256 490 Y=INT(Y*S1/S2)\K=Kf1 500 IF Y>127 THEN Y=127 510 IF Y<-128 THEN Y=-128 520 IF Y<0 THEN Y=Yf256 .530 J=If128\IF J>255 THEN J=J-256 540 OUT49,J\OUT50,Y\DUT51p127 550 FOR L=1 TO 30\L=L\NEXT L 560 NEXT :I 570 OUT51,128 580 STOP 590 END 600 REM**************************************

---------

362

APPENDIX A7.2 Statistical analysis program.

5 REM**************** PROGRAM CONSMP ******************* 10 REM CALCULATES MEAN AND VARIANCE OF VOLTAGE ON CH1 OF ADC 20 REM ON A CONTINUOUS BASIS. SCALING IS DONE. 30 REM RESULTS OUTPUT TO PRINTER. 40 REM**************************************************** 50 REM 60 DIM A$(48) 70 REM PROGRAM **** SMPL1 **** 80 INPUT"DATE AND TIME? ',A$ 90 PRINT"INPT RCVR SPCNG IN MTRS" 100 INPUT W 110 PR%NT"SPCFY SMPLNG RATE" 120 INPUT S\R1=1/S\R=50*R1 130 PRINT"SPCFY REQDNO OF SMPLD PNTG" 140 INPUT N 150 INPUT"SCALING FACTOR—VOLTS/UNIT? ",C 160 PRINT~******* SWITCH RTC ON *******" 170 FILL 3072v0 180 Z=10\S1=0\S2=0\J=0 190 T=EXAM(3072)\IF T<Z THEN GOTD 190 200 PRINT "SAMPLING STARTS" 210 FILL 3072,0 220 T=EXAM(3072)\IF T<R THEN GQTO 22() 23{) %=INP(49) 240 IF )0127 THEN ){=%-256

250 S1=S14.X\S2=824-X*X 260 J=Jf1 270 IF J THEN GOTO 210 280 PRINT"** SAMPLING IS OVER ** SWITCH RTC OFF **" 290 A=2~56/128\A=A/C 300 S1"=S1*A\S2=S2*A*A ^

310 M=S1/N\V=S2/N—M*M 320 PRINT CHR$(15),"************************************" 330 PRINT"RECOQD: "rA$ 340 PRINT"SPACING: "r%5F1rW 350 PRINT Nv" SAMPLES TAKEN." 360 PRINT 370 PRINT' MEAN VALUE : "vM 380 V=SQRT(ABS(V)) 390 PRINT"STD~ DEV. :°rti 400 PRINT"***********************k************" 410 PRINT CHR$(19)

420 G[JTO 160

363

APPENDIX A7.3 Gaussian integration program.

5 ********************PROGRAM INTG********************* 10 REM EVALUATES INTEGRAL OF FNA(X) FROM A TO B USING 6 POINT 20 REM GAUSSIAN QUADRATURE. DOMAIN IS BROKEN INTO N SEGMENTS 30 REM AND THE ALGORITHM APPLIED TO EACH. 40 REM****************************************************** 50 REM 60 DATA ^17132449,~36076157,,46791393 70 DATA ~46791393v~360761577^17132449 ' BO DATA —~93246951v—~66120938,-423881918 90 DATA ~24861918,~66120938,~93346951 100 FOR I=1 T 6\READ G(I)\NEXT 110 FOR I=1 TO 6\READ H(I)\NEXT 120 INPOT"LOWER LIMIT? "vA 130 INPUT"UPPER LIMIT? "rB 140 INPUT"NO^ OF INTERVALS? ",N 150 D=#BS((B—A))/N\S1=0\B`:AfB 160 FOR J=1 TO N 170 S=0\C=(B—A)/2^0\D1=CfA

180 FOR I=1 TO 6\X=C*H(I)fD1lS=SfFNA(X)*G(I) 185 NEXT I 190 S=C*S\81=S14.S\A=AfD\B=BfD 200 NEXT J

' 210 PRINT\PRINT°INTEGRAL IS "/S1 220 STOP

230 REM*********************************************** 240 REM USER DEFINED FUNCTION. 250 DEF FNA(X)= 260 REM 270 REM**>K********************************************

364

APPENDIX A7.4 General purpose plotting program.

10 REM ************* PROGRAM PLOTTER ********************** 20 REM THIS PROGRAM PLOTS AN ARRAY OF 255 POINTS ON AN XY 30 REM PLOTTER. THE POINTS ARE CALCULATED BY A USER DEFINED 40 REM FUNCTION AT LABEL 1000. THIS FUNCTION SHOULD BE DEFINED 50 REM IN THE USUAL WAY AND MAY BE MULTILINED. 60 REM THE X VOLTAGE IS AT CH1, Y AT CH2 OF THE DAC. 70 REM PEN CONTROL IS ALSO SUPPLIED FROM CH3 80 REM FOLLOW INSTRUCTIONS AS SUPPLIED BY PROGRAM. WHEN RUNNING 90 REM WHEN USING THE '0',"1",'2' FACILITY TO SET THE PEN/PAPER 100 REM IT IS SUFFICIENT TO MERELY TYPE THE CHOICE KEY 110 REM**************************************** 120 PRINT"CALIBRATE PLOTTER:0=> BOTTOM LEFT' 130 PRINT' • - 1=> TOP RIGHT" 140 PRINT° 2=:r CONTINUE TO PLOT" 150 C=EXAM(3085) 160 IF C=13 THEN 150 170 C=EXAM(3085) 180 IF C•::::• 48 THEN 220 190 FOR I=228 TO 128 STEP-1\OUT(49)I\OUT(50)I\NEXT 200 OUT(49)128\OUT(50)128\C=EXAM(3085) 210 IF C<>48 THEN 170 ELSE 200 220 IF C•:::'•:• 49 . THEN 260 230 FOR 1=27 TO 127\OUT(49)I\OUT(50)I\NEXT 240 OUT(49)127\OUT(50)127\C=EXAM(3085) 250 IF C<>49 THEN 170 ELSE 240 260 FILL(3084)1 270 INPUT1'XMIN? ',X2 280 INPUT' XMAX? ",X1 290 INPUT1'YMIN? ',Y2 300 INPUT" YMAX? 310 S1=ABS(X1—X2)/255\S2=ABS(Y1—Y2)/255 320 X=X2 330 PRINT"PLOTTING STARTS" 340 FOR I=1 TO 30\Y=INT((FNA(X)—Y2)/S2)-128 350 IF - Y<-128 THEN Y=-128 360 IF Y:••127 THEN Y=127 370 IF Y•:0 THEN Y=Y+256 380 OUT49,128\OUT50,Y\NEXT I 390 OUT51,127\REM PEN DOWN 400 X=X2\FOR 1=128 TO 383.\Y=INT((FNA(X)—Y2)/82)-128 410 IF Y•(-128 THEN Y=-128 420 IF Y>127 THEN Y=127 430 IF Y•0 THEN Y=Y+256 440 J=I\ IF J>255 THEN J=J-256 450 0UT49,J\OUT 50,Y\OUT51,127 460 X=X+S1\NEXT I 470 FOR I=1 TO 100 480 0UT49,J\OUT50,Y\OU.T51,127\NEXT 490 OUT51,128\REM PEN UP 500 PRINT'PLOTTING IS COMPLETE" 510 INPUT"TYPE—I—FOR ANOTHER PLOT ON THE SAME AXES-0—TO ESCAPE ',C 520 IF C=1 THEN 320 530 REM************************************************************** 540 DEF FNA(X)= • 550 REM THIS IS THE USER DEFINED FUNCTION

365

CHAPTER SEVEN REFERENCES.

7.1 Dale, C. 1978 "The Interfacing of a Small Computer to

a Time—sharing Network." Undergraduate Project Report,

Dept. Electrical Engineering, Imperial College, London.

7.2 Inggs, 1, and Sage, D. 1978. "Instruction Manual for

the Poly 88 System." E.N. Waves Applications Group,

Imperial College, london.

7.3 Witten, I.H. 1979. "Computer buses"

Wireless World, 85, No.1518, February 1979. p32.

7.4 IEEE (United States) Standard 488-1975.

7.5 Adams, J.H. 1979. "A Scientific Computer"

Wireless World, 85, No.1520, April 1979, p44.

366

CHAPTER EIGHT. CONCLUSIONS AND FUTURE WORK.

This chapter is a summary of the important aspects of

the work contained in this thesis, together with indications of those

areas where, in the author's opinion, there is scope for further in-

vestigation. The discussion of material follows roughly the order in

which it has been presented in the previous chapters.

The angular plane wave spectrum/coherence function approach

to wave propagation through extended, tenuous, random media has been

shown to be directly applicable to millimetre—wave problems. The ap-

proach has the added advantages of being simple to visualize and based

on Fourier transform theory, which is familiar to most engineers.

A multi—slab model, simply related to the statistical

properties of the real medium was presented. From the above, it may

be concluded that a beam propagating through a tenuous medium may be-

resolved into two angular spectra, known as the coherent and incoherent

spectra.

The coherent, or mean, spectrum is reduced by exp`—o• \

where o-./ is related to the refractivity fluctuation intensity, scale

size, and so on. The incoherent spectrum is more complex. It consists

of real and imaginary components whose distribution is zero—mean,

jointly normal, but may be unequal in variance. It is felt that more

detailed analysis -of the statistical nature of the medium induced fluc-

tuations should be attempted for the millimetre—wave situation. Some

aspects of this are considered by I1ashhour 1.

The first chapter showed that the formal definitions of

antenna performance may be couched in angular spectrum form; this means

that the power coupled between two antennas can be calculated, given

the nature of the random medium in which they are immersed.

Some simple expressions for the average power coupled were

367

obtained, but there is scope for a more detailed analysis of the inter-

relation of the aperture size, medium scale size, refractivity fluc-

tuation intensity_and so on, when applied to millimetre—wave problems.

Again, Mashhour8 1 has investigated some of these aspects.

The angular spectrum approach is applicable to other fields

where the scalar wave equation describes a propagating wave field;

sound propagation in the random ocean is an example.

The slab model requires statistical information about the

medium refractivity fluctuations. Chapter 2 is a review of the physical

processes which are important :sources of refractivity fluctuations. It

is shown that most of the fluctuation energy is associated with scale

sizes between 5 and 30 m.

Practical measurements of refractivity fluctuations in urban

London are presented. It is seen from this work, and from others, that

surface values of a (refractivity variance) are likely to be less than

1 N2; 0,01 N2 being more typical. Auch more intense fluctuations are

observed in elevated layers; .here mixing processes are weak and take a

long time to smooth out refractivity fluctuations.

Different natural and man—made processes are responsible for

generating fluctuations with different scale sizes. The refractivity

autocovariance must thus be thought of as a sum of these source auto—

covariance functions. Comstock showed that most of the commonly observed

autocovariance functions may be synthesized from a summation of Gaussian .

autocovariances having different scale sizes which are drawn from a

distribution (Rayleigh, uniform, etc.). It isfelt that some further

work on this idea would be fruitful. Especially useful would be a

categorization of different atmospheric processes and the distribution

of scale sizes which they are likely to produce. It would thus be possible

to examine a prospective propagation path, isolate the important processes,

and then be able to construct a realistic refractivity autocovariance

368

function. This would be useful in the computer simulation of propagation

which will be discussed.

Chapters 3 and 4 describe the design. and execution of a

millimetre—wave propagation experiment. It became apparent that the

very small magnitude of the induced scintillation required careful

receiver design and construction.

The experimental link was.over nearly 12 km at 38 GHz.

From the communications viewpoint, it is very heartening that the received

field fluctuations were less that 10% of the mean value. Atmospheric

precipitation is thus the only serious problem associated with milli-

metre—wave systems.

Rainfall rate is not specially uniform and this was evident

form the experimental data. Very heavy rainfall at the receiver site was

usually associated with only small fades in signal strength. This is

because intense rain is localized into small cells only a few km. in

extent. It is the author's opinion that precedence should be given to

rain radar studies, in order that a reliable set of statistical data

may be accumulated for the spatial distribution of rainfall in various.

parts of the world.

The measurements of the amplitude and spatial phase fluc-

tuations are in good agreement with the theoretical treatment of the

first chapter. From the systems viewpoint,. the effect of the.medium is

to introduce angle of arrival fluctuations into the incoming wave.

Typically these are of the order of milliradians. Large aperture anten-

nas (>10m) will also be influenced by a loss of coupling efficiency,

since the atmospheric scale sizes are of the same order in size.

An extremely powerful method of probing the atmosphere is

possible if a large (about 200m) millimetre—wave array was constructed.

This array would have element spacings of about 10m. The phase differences

between the individual elements would allow the phase—front of the

369

incoming wave to be determined. From this information, the scale size

of the refractivity fluctuations could be determined, since the phase

front is a replica of the medium fluctuations., From knowledge of the

scale size, path length and the rms phase fluctuation, the refractivity

fluctuation strength can be calculated; it would have to be assumed that

the path was homogenuous, however.

Clearly the greatest difficulty with propagation experiments

is the measurement of the atmospheric variables. In .general, point

measurements have to be assumed to be representative of the whole medium.

At higher frequencies, technological problems arise with producing

suitable receivers. These considerations led to the idea of attempting

a computer simulation of propagation through a. tenuous random medium.

It was established in the first chapter that the effect of

a tenuous slab could be considered as being identical to a thin, phase—

changing screen. The conditions under which this was true were shown -

not to be restrictive.

The computer model thus consists of resolving the propa-

gation path into a number of parallel slabs. An angular spectrum,nof

waves representing a transmitting aperture was allowed to propagate

through the first slab as if through free space. On leaving the slab,

however, the electric field was calculated by inverse transforming the

angular spectrum. This field was then perturbed by a phase screen which

was equivalent to the whole slab. The angular spectrum was once again

obtained from the Fourier transform of this perturbed field distribution.

This process was repeated for all slabs constituting the propagation

path. 8°2

In order to limit the size of the computational arrays, it

is recommended that narrow beam problems only be considered. This also

means that the medium refractivity scale sizes should be much larger than

the wavelength. This ensures that the scattered energy does not spread

370

too far,' requiring large arrays to define the field adequately.

In cases where broad beam or small scale sizes need to be

considered, it is a relatively simple matter to increase the sampling interval

of the data every few iterations. This means that a constant number of

samples represents a larger spatial (or angular) domain.

A danger here, however, is that the sampling resolution

becomes too large compared to the scale sizes of the refractivity.

Clearly, smaller scale sizes lead to larger energy spread, requiring

larger sampling intervals, and leads rapidly to a contradiction. It

has been shown, however, that realistic millimetre—wave problems can

be modelled without difficulty.

It is felt that, provided efficient FFT algorithms are

used, there is not a great deal of improvement possible in the compu-

tational time taken to implement the forward and reverse transforms. If

the sampling intervals are kept constant, considerable savings in time

may be made by computing the sine/cosine values required by the FFT

algorithm and storing these in arrays for repeated use. This time saving

could be used to increase the number of points used, provided sufficient

machine storage was available.

The generation of the random phase screens simulating the

medium is interesting and has scope for further work. The generation of

the underlying uniformly distributed numbers is done efficiently using an

algorithm tailored to the COC machine used. It is the transformation

of these numbers to simulate spatial correlation which requires further

investigation.

In the work reported here, the uniforms were summed to

produce normally distributed variates; this is based on the Central

Limit Theorem, and was shown to prod'irce- variates_c4nfprming adequately to the

normal distribution. Correlation was introduced, by summing adjacent

numbers in a number of passes. This was shown both theoretically and

371

practically to produce a Gaussian autocovariance function. Techniques for

speeding up the correlating process by an order of magnitude were discussed.

The work reported in chapter 2 shows that a Gaussian auto—

covariance is not completely realistic. Using Comstock's idea, it is

possible, in principle, to synthesizet-.natu'a••lly occuring autocovariances

from a number of Gaussians with randomly chosen scale sizes. In practice,

this is likely to be very expensive in computer time.

There:is thus a strong motivation to perhaps examine the

possibility of designing a random number generator which produces

intrinsically correlated numbers. It may even be possible to control the

autocovariance at will.

Chapter 6 showed the results of a number of simulations in

graphical form. The potential of the simulation technique is very

apparent. There is scope for extending the simulation work to include

extensive statistical analysis of the field phase and amplitude fluc-

tuations. These results could be used to check theoretical predictions.

The results--presented in chapter 6 are of a single reali-

zation of the field at, say, the receiver plane. It is thus possible

to investigate the spatial statistics at will. It is also possible to

include temporal effects. Temporal effects are directly related to the

movement of the refractivity inhomogeneities. These movements are prin-

cipally due to wind.

Taylor's "frozen turbulence" hypothesis asserts that the time

taken for an inhomogeneity to be swept through a transmitting beam is

often shorter than the time taken for the inhomogeneity to decay. This

is especially true for millimetre—wave problems, since transmitting

beams are narrow and the important, intense refractivity fluctuations

have large scale sizes and long decay times.

It would thus be straight forward to simulate wind effects

by dong a number of passes through all slabs, but each time shifting the

individual phase screens laterally. The lateral displacements would be

372

selected according to the wind telocity being modelled and the time

interval between each run.

The field could then be sampled at a point and the temporal

fluctuations analysed. This could be done at a number of points, more

and more off-axis, and the change in statistics monitored.

The simulation method can also be simply extended to cater

for inhomogeneous distributions of refractivity fluctuation intensity

and wind speed.

The programs presented cater for- 2-d- simulations only;

the extension to 3-0 would be relatively straight-forward. There would,

however, be a large increase in computer time and storage. Some of

this can be offset by carefull analysis of the problem. Since most

antennas are circularly symmetric, there is probably a need to use cylin-

drical polar coordinates for the field transformations.

There is no'reason why the simulation method should not be

extended to related fields, where the angular plane-wave spectrum method

is applicable.

The thesis concludes with a chapter describing the impact

of microprocessors in the laboratory and on the design of experiments.

.It is shown that this has been principally a revolution in cost.

Formerly, automated instrumentation was only feasible for

large-scale production-line testing. The evolution of microelectronics

has changed this completely. It is felt that research workers should

decide on the microcomputer which they will use, and design the experi-

ment around this device and its peripherals.

Some discussion of the component parts of microcmputers

is presented; the most important conclusion is that it is almost always

best to choose a machine which is based on a universal bus structure and.

further, should be able to support a high levēl language such as BASIC.

The last few years has seen a growing interest in applications

373

of millimetre—waves. Since the principal problem is known to be

precipitation, most of these applications will be for the clear atmo-

sphere. It is felt that the angular spectrum approach gives a powerful

means of analysing propagation through the clear but random, atmosphere

in a simple, physical way. It has been validated by the experiments

reported here and elsewhere. Finally, it has been shown to lead to a

computer simulation technique which has great potential as, an investi-

gative tool.

374

CHAPTER EIGHT REFERENCES.

8.1

Ilashhour, I.A.M. 1979- "Coupling to a Receiving Aperture

from a Random Medium at flillimetric Wavelengths".

PhD Thesis, University of London, 1979.

8.2 Inggs, M.R. and Clarke, R.H. 1979 "A Computer

Simulation of Propagation through a Tenuous Random

ddium" Paper presented at 1979 URSI/AP—S Symposium

held in Seattle, Washington State, June 1979.

Experiments with self-oscillating Gunn-diode mixers reveal an economic means of taking millimeter-wave antenna readings. Sensitivity of —95 dBm is achieved with a 1-MHz IF bandwidth.

THE self-oscillating mixer, a milli-meter-wave component that uses a

single Gunn or Impatt device as an oscillating and mixing element, has the potential to lower the cost of making antenna pattern measurements in the 20-to-60-GHz range.

Antenna-pattern measurements in the millimeter-wave bands, often taken with the test rig shown in Fig. 1, are handicapped by the limitations of signal sources currently available at extra high frequencies. In particular, the low output power of the oscillators, typically less than 50 mW, establishes a stringent requirement on the sensitivity of the receiver used in the test set up.

The self-oscillating mixer (SOM) overcomes many of these source limitations. Used as a receiver, the SOM is signifi-cantly more sensitive than crystal or diode detectors. Although not quite as sensitive as the double-balanced mixer used in superheterodyne receivers, the SOM approach is considerably less. expensive.

Figure 2 describes a basic self-oscillating mixer.',2 The active element, either a Gunn or Impatt diode, oscillates at a frequency set by the cavity. The incoming signal mixes with the locally generated frequency because of the diode nonlinearity. The resulting IF is tapped off the DC supply post by means of a simple filter arrangement. Establish sensitivity needs

To investigate the sensitivity requirement of the test receiver, let's return to the test rig shown in Fig. 1. A transmitter is coupled to a standard-gain horn, which illuminates a large, narrow-beam antenna—the component under test. In order for the full gain of the antenna under test to be realized, the separation of transmitter and receiver, R, should satisfy the far-field, or Rayleigh criterion:

R = 4D2 (1)

where D is the aperture of the antenna under test and A is the wavelength of the test signal. The assumption here is that the aperture is uniformly illuminated in amplitude and phase; if Eq. 1 is satisfied, the situation should be accurate for more realistic tapers used in practice.

The gain of an aperture antenna can be estimated from:

G cc 10 logie 14'1-A J dB L J

(2)

Michael R. Inggs, Research Officer, Electromagnetic- Wave Applications Group, Department of Electrical Engi - neering, Imperial College, London SW7 2AZ, England.

1. Antenna under test is illuminated by a narrow-beam horn.

2. The basic, self-oscillating mixer uses a Gunn or Impart diode.

Table 1: Sample computations for a 38 GHz measurement

Receiving antenna gain (G)

Approximate aperture (D)

Rayleigh far field (R)

Maximum received powers

Maximum level above sensitivity2

dB meters meters dBm dB

10 0.008 0.03 2 47 15 0.014 0.10 -3 42 20 0.025 0.32 —8 37 25 0.045 1.01 —13 32 30 0.079 3.20 —18 27 35 0.141 10.12 —23 22 40 0.251 32.00 —28 17 45 0.447 101.18 —33 12 50 0.795, _ 319.96 —38 7 55 1.413 1011.81 —43 2 60 2.513 3199.62 —48 —3 65 4.469 10118.07 —53 —8

1. Source Power = +12 dBm; Source Antenna Gain = +15dB. 2. Tangential Sensitivity Of Diode Detector = —45 dBm.

100 MICROWAVES • April, 1978

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ti detecltcrti ijacdwrcfth is' $ H then tn .detectod;i real f r(se 'figure). ,•.;

11.0mai4rc ;the.bandvridtn B.is: itougyt siRasane,.signat pow P#: ;this'l . what thi :ttetecfo; would 9r diicato.

' s. :wduFc.not giscrtriitrr ta-00:0i:400 eSt ..!āfigt±(a) bc:

1+ 47sigif4'. is n;7w irtioct004. ithppiāer eqūāt.ti ?i ttien:: x ,lf'e pQ inete:~.wo i i%1.00te ari tt reas2°ot:3 dB.;T1tis. ;':rneihocf.- cs:rtsod' to.,.measutb ;;tf~s': :sgnatequals-norse.' :'cor:I'Lt i ii ttis.ēquwa1. >: `~5igna to no:se'ra[;ti`of;tir(ty outp.

ts -,iif::#tip.:i'~C~i+ et...-';.~;;:'`•;i;'`:.':r:<`s.,,;.n;.. There 'are:. nta►ty: c1Ofin itiop.s 4 :ito*e':

ti:e .fofir~v4~ `~ i#g.Izru#i€, - r;h ni;t:iō-tiōis rāt;ir4 .sf

where A is the effective area of the antenna. For ease of estimation, assume that:

A~w D2/4 (3) Finally, the loss due to antenna separation R is:

L = 20 logia r4'rR l dB (4) Lxi

Table 1 relates receiving antenna gain, aperture, and the Rayleigh distance required for full gain to be developed at 38GHz. The fourth column of Table 1 lists the maximum received noise power (power available to the receiver) for the specific case of a 12-dBm transmitter feeding a 15-dB horn. Calculations were made using Eqs. 2 to 4.

Choose a receiver

Three types of receivers are often considered for pattern measurements. The simplest is the crystal or diode detector (Fig. 3(a)). Here, the transmitter is amplitude modulated at a few kilohertz and the small currents induced in the crystal or diode are amplified and used as a measure of received power.

MICPOWAV7:S 9 April, 1978

The main advantage of the diode detector is tuning range: it- may be simply tuned over an entire waveguide band by means of stubs or an attached cavity.

The major disadvantage of this approach is poor sensi-tivity. The sensitivity of a Q-band diode detector, measured under signal-equals-noise conditions (see sidebar), is about —45 dBm for a detection bandwidth of about 1 MHz. Based on this —45-dBm figure, the fifth column of Table 1 shows the number of dB above tangential sensitivity that the maximum received power will be as a function of the gain of the antenna under test. It is obvious that dynamic range available for measurement is limited, especially for low-gain antenna measurements.

This situation may be improved by using a narrow detection bandwidth, or even synchronous detection (Fig. 3(b)). This sophistication may yield another 20 dB or so in signal-to-noise ratio, but adds cost and complexity to the test set up. Further problems may arise due to transmitter frequency drift: the detector has to be retuned a number of times during an experimental run.

A much more satisfactory, but expensive solution is the (e ,Brined or; nest ?urge)

101

OiSCRIVAMAT

rc: c.Nt?EC.T(ON _

Y SOURCE

SELF-OSCILLATING MIXER

(continued from p. 101) superheterodyne receiver (Fig. 3(c)). Without delving into the various definitions of noise figure', this class of receiver will have a signal-equals-noise output with an input of -100 dBm or lower (depending on IF bandwidth). Thus, it offers at least 50-dB more dynamic range than the crystal detector. Frequency drift is negated by a simple automatic-frequency-control system.

The main disadvantage to the superheterodyne approach is cost. If a range of frequencies must be measured, a range of local oscillator sources and mixers must be used. If a

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4. Three-piece construction includes an interchangeable wafer-mounted diode. A matched unit could be used as a signal source.

NATOR

3. Several receiver designs can be considered far anten-na pattern measurement. A simple detector design (a) is expensive but offers poor sensitivity. Synchronous detec-tors (b) improve sensitivity, but the receiver must be constantly retuned. Superheterodyne receivers (c) are an expensive way to buy high sensitivity. Self-oscillating mixers (d) offer —95 dBm sensitivity at a reasonable cost.

broadband mixer is used, sensitivity will probably be limited to —55 dBm—only about 10 dB more than a crystal detector.

Finally, consider a receiver based on the self-oscillating mixer (Fig. 3(d))--a single component that acts as mixer and local oscillator. The SOM is fairly simple to manufac-ture, being basically nothing more than a Gunn or Impatt source.3 Usually, the packaged diode is mounted in a waveguide backed with a plunger-set cavity. DC is supplied via an anodized choke, which also serves as an IF output.

A practical design is shown in Fig. 4. Here, the diode is mounted in a wafer sandwiched between a tapered section and a cylindrical cavity. To match the low diode impedance to that of the waveguide, reduced-height transmission line is used. When working at the top end of a waveguide band, it is a good idea to reduce the width as well to prevent spurious modes. The dimensions given in the diagram are for a source with a frequency between 35 and 40 GHz. The arrangement was found to be mechanically tunable over a range of about 6 GHz, but not without "switch-on" problems at the extremes of the range.

AZIMUTH Ohl

5. Measurement of a 25-cm paraboloid at 38-GHz reveals a dynamic range of 40 dB. Limiting factor in taking this E-plane data was reflection, not SOM sensitivity.

The wafers and diodes should be kept in matched pairs, one to act as a source and a second to operate as a self-oscillating mixer in the receiver. Two or three pairs could cover a complete waveguide band.

Experiments on SOM test receivers in the 30 to 40 GHz range and 60-MHz IF reveal a signal-equals-noise sensi-tivity of about —95 dBm with a detection bandwidth of 1 MHz. A measured pattern, shown in Fig. 5, indicates that the dynamic range of the measurements was more limited by range reflections than by sensitivity deficiencies. The noise performance of the . SOM was found to be roughly constant for intermediate frequencies between 10 MHz and 1 GHz..•

References

1. R. R. Spiwak, "Frequency Conversion and Amplification With An LSA Diode Oscillator," IEEE Trtmsoctions On Electron Devices, Vol. ED-15, No. 8, pp. 614-615, (August, 1968).

2. M.J. Lazarus, S.Novak, Y.C. Kin, "Have a High-Quality V-Band Link and Low Costs, Too," icroli ares, Vol. 11, No. 11, pp. 52-53, (November,1972)

3. W. Foster and F. A. Myers, -The Gunn Effect," Microunre Systems News. (June/July, 1974).

4. K. L. Smith, "Noise: Confusion In More Ways Than One," Wire/ors T4 5•f, . p.107, (March, 1975).

MICROWAVES • April, 1978