CONTENTS

I CELLULAR MOBILE COMMUNICATION

1. MOBILE SYSTEMS 8

1.1 The concept 9

1.2 Cell Signal Encoding 9

1.3 Frequency Reuse 10

2 FEATURS OF A CELLULAR NETWORK 13

2.1 Directional Antennas 13

2.2 Sectorization 14

2.3 Broadcast messages and paging 14

2.4 Movement from cell to cell and handover/handoff 15

3 MOBILE NETWORK : A CELLULAR NETWORK 16

3.1 Example of a cellular network: the mobile phone network 16

3.2 GSM 17

3.3 Handoff in mobile networks 18

4 GSM Frequency Bands 19

4.1 Coverage Comparison of different frequencies 19

5 Cellular Traffic 21

5.1 Quality of Service targets 21

5.2 Traffic Load and Cell size 22

5.3 Traffic capacity vs Coverage 23

5.4 Channel Holding time 24

6 GSM Architecture 26

6.1 Base Station Subsystem 26

6.2 Network Station Subsystem 28

7 Channel Reuse and Signal Strength 31

7.1 Channel Reuse 31

7.2 Cell Phone tower power Emission 32

7.3 Signal Strength 32

8 Reasons for weak signals 34

8.1 Rural Areas 34

8.2 Building Construction Material 34

8.3 Building Size 35

8.4 Multipath Interference 35

8.5 Diffraction and general attenuation 35

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8.6 Different operating frequencies 36

9 Code Division Multiple Access 37

9.1 CDMA basics 37

9.2 Steps in CDMA Modulation 38

9.3 Code Division Multiplexing(Synchronous CDMA) 38

10 The Future of Mobile Networking 41

10.1 Future Evolution: (Broadband )4G 41

10.2 Comparison to Similar Systems 41

II MICROWAVE ENGINEERING

1 Introduction 45

1.1 Microwave Frequencies 45

1.2 Microwave History Of Device 47

1.3 Microwave Applications 48

2 Waveguide -1 50

2.1 Modes of Operation 50

2.2 Rectangular Vs Circular Waveguides 50

2.3 Rectangular Waveguide 51

3 Waveguide Components -1 53

3.1 Attenuators 53

3.2 Electronically Controlled Attenuators 53

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3.3 Phase Shifters 54

3.4 Directional Couplers 55

4 Waveguide Components-2 57

4.1 Hybride Junctions 57

4.2 Hybride Ring or Rat Race 58

5 Microwave Propagation In Ferrites 59

5.1 Microwave Devices Employing Faraday rotation 59

5.2 Gyrator 59

5.3 Isolator 60

5.4 Other Ferrite Devices 61

6 Microwave Tubes 62

6.1 Klystron Amplifier 62

6.2 Two Cavity Klystron 63

6.3 Reflex Klystron 64

6.4 Cavity Magnetron 65

6.5 Other Types Of Microwave Tubes 66

7 Gunn Effect And Its Applications 68

7.1 The Gunn Effect 68

7.2 Gunn Diode 68

7.3 Gunn Diode Theory 68

7.4 Appications 70

8 Transmission Lines And Characteristic Impedance 72

8.1 Impedance Matching 73

8.2 Dielectric Constant And Effective Dielectric Constant 73

8.3 Lumped Elements Vs Distributed Elements 75

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8.4 VSWR And Return Loss 76

9 Microwave Measurements 78

9.1 Band Widths 78

9.2 Frequency Conventions 79

9.3 Harmonic Frequencies 79

9.4 Decibels 79

III DIGITAL COMMUNICATION

1 Basics of Digital communication 82

1.1 Introduction 82

2 Pulse code modulation 85

2.1 Modulation 85

2.2 Limitations 88

3 Digital modulation techquie-I 91

3.1 Amplitude shift keying 92

3.2 Phase shift keying 92

3.3 Differential phase shift keying 93

3.4 Binary phase shift keying 94

3.5 Quaternary phase shift keying 94

3.6 Frequency shift keying 97

4 Delta modulation 98

4.1 Principles 99

4.2 Adaptive delta modulation 99

5 Information theory101

5.1 Source theory 102

5.2 Information rate 102

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5.3 Channel capacity 102

6 Shannon Hartley theorem 104

6.1 Statement of theorem 104

6.2 Nyquist rate 105

6.3 Noisy channel coding theorem 105

6.4 Shannon Hartley theorem 106

7 Linear block codes 108

7.1 Formal definations 108

7.2 Hamming codes 109

7.3 Hadamard codes 110

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CELLULAR

MOBILE

COMMUNICATION

Chapter 1

Mobile Systems

Limitations of conventional mobile telephone systems:

One of the many reasons for developing a cellular mobile telephone systems and deploying it

in many cities is the operational limitations of conventional mobile telephone system

1. Service capability

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2. Poor service performance

3. Frequency utilization.

A mobile network is a radio network distributed over land areas called cells, each

served by at least one fixed-location transceiver known as a cell site or base station. When

joined together these cells provide radio coverage over a wide geographic area. This enables

a large number of portable transceivers (mobile phones, pagers, etc) to communicate with

each other and with fixed transceivers and telephones anywhere in the network, via base

stations, even if some of the transceivers are moving through more than one cell during

transmission.

Mobile networks offer a number of advantages over alternative solutions:

increased capacity

reduced power usage

larger coverage area

reduced interference from other signals

An example of a simple non-telephone Mobile system is an old taxi driver's radio system

where the taxi company has several transmitters based around a city that can communicate

directly with each taxi.

1.1 The concept

In a Mobile radio system, a land area to be supplied with radio service is divided into

regular shaped cells, which can be hexagonal, square, circular or some other irregular shapes,

although hexagonal cells are conventional. Each of these cells is assigned multiple

frequencies (f1 - f6) which have corresponding radio base stations. The group of frequencies

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can be reused in other cells, provided that the same frequencies are not reused in adjacent

neighboring cells as that would cause co-channel interference.

The increased capacity in a cellular network, compared with a network with a single

transmitter, comes from the fact that the same radio frequency can be reused in a different

area for a completely different transmission. If there is a single plain transmitter, only one

transmission can be used on any given frequency. Unfortunately, there is inevitably some

level of interference from the signal from the other cells which use the same frequency. This

means that, in a standard FDMA system, there must be at least a one cell gap between cells

which reuse the same frequency.

In the simple case of the taxi company, each radio had a manually operated channel

selector knob to tune to different frequencies. As the drivers moved around, they would

change from channel to channel. The drivers know which frequency covers approximately

what area. When they do not receive a signal from the transmitter, they will try other

channels until they find one that works. The taxi drivers only speak one at a time, when

invited by the base station operator (in a sense TDMA).

1.2 Cell signal encoding

To distinguish signals from several different transmitters, frequency division multiple

access (FDMA) and code division multiple access (CDMA) were developed.

With FDMA, the transmitting and receiving frequencies used in each cell are different from

the frequencies used in each neighbouring cell. In a simple taxi system, the taxi driver

manually tuned to a frequency of a chosen cell to obtain a strong signal and to avoid

interference from signals from other cells.

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The principle of CDMA is more complex, but achieves the same result; the distributed

transceivers can select one cell and listen to it. Other available methods of multiplexing such

as polarization division multiple access (PDMA) and time division multiple access (TDMA)

cannot be used to separate signals from one cell to the next since the effects of both vary with

position and this would make signal separation practically impossible. Time division multiple

access, however, is used in combination with either FDMA or CDMA in a number of systems

to give multiple channels within the coverage area of a single cell.

1.3 Frequency reuse

The key characteristic of a cellular network is the ability to re-use frequencies to increase

both coverage and capacity. As described above, adjacent cells must utilise different

frequencies, however there is no problem with two cells sufficiently far apart operating on the

same frequency. The elements that determine frequency reuse are the reuse distance and the

reuse factor.

The reuse distance, D is calculated as

D=(√K)R

where R is the cell radius and N is the number of cells per cluster. Cells may vary in radius in

the ranges (1 km to 30 km). The boundaries of the cells can also overlap between adjacent

cells and large cells can be divided into smaller cells .

The frequency reuse factor is the rate at which the same frequency can be used in the

network. It is 1/K (or K according to some books) where K is the number of cells which

cannot use the same frequencies for transmission. Common values for the frequency reuse

factor are 1/3, 1/4, 1/7, 1/9 and 1/12 (or 3, 4, 7, 9 and 12 depending on notation).

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In case of N sector antennas on the same base station site, each with different

direction, the base station site can serve N different sectors. N is typically 3. A reuse pattern

of N/K denotes a further division in frequency among N sector antennas per site. Some

current and historical reuse patterns are 3/7 (North American AMPS), 6/4 (Motorola

NAMPS), and 3/4 (GSM).

If the total available bandwidth is B, each cell can only utilize a number of frequency

channels corresponding to a bandwidth of B/K, and each sector can use a bandwidth of

B/NK.

Code division multiple access-based systems use a wider frequency band to achieve

the same rate of transmission as FDMA, but this is compensated for by the ability to use a

frequency reuse factor of 1, for example using a reuse pattern of 1/1. In other words, adjacent

base station sites use the same frequencies, and the different base stations and users are

separated by codes rather than frequencies. While N is shown as 1 in this example, that does

not mean the CDMA cell has only one sector, but rather that the entire cell bandwidth is also

available to each sector individually.

Depending on the size of the city, a taxi system may not have any frequency-reuse in

its own city, but certainly in other nearby cities, the same frequency can be used. In a big city,

on the other hand, frequency-reuse could certainly be in use.

Chapter 2

Basic Components of a mobile network

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2.1 Directional antennas

Fig 2.1Cellular telephone frequency reuse pattern.

Although the original 2-way-radio cell towers were at the centers of the cells and were

omni-directional, a cellular map can be redrawn with the cellular telephone towers located at

the corners of the hexagons where three cells converge. Each tower has three sets of

directional antennas aimed in three different directions and receiving/transmitting into three

different cells at different frequencies. This provides a minimum of three channels for each

cell. The numbers in the illustration are channel numbers, which repeat every 3 cells. Large

cells can be subdivided into smaller cells for high volume areas.

2.2 Sectorisation

By using directional antennae on a base station, each pointing in different directions,

it is possible to sectorise the base station so that several different cells are served from the

same location. Typically these directional antennas have a beamwidth of 65 to 85 degrees.

This increases the traffic capacity of the base station (each frequency can carry eight voice

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channels) whilst not greatly increasing the interference caused to neighboring cells (in any

given direction, only a small number of frequencies are being broadcast). Typically two

antennas are used per sector, at spacing of ten or more wavelengths apart. This allows the

operator to overcome the effects of fading due to physical phenomena such as multipath

reception. Some amplification of the received signal as it leaves the antenna is often used to

preserve the balance between uplink and downlink signal

2.3 Broadcast messages and paging

Practically every cellular system has some kind of broadcast mechanism. This can be

used directly for distributing information to multiple mobiles, commonly, for example in

mobile telephony systems, the most important use of broadcast information is to set up

channels for one to one communication between the mobile transceiver and the base station.

This is called paging.

The details of the process of paging vary somewhat from network to network, but

normally we know a limited number of cells where the phone is located (this group of cells is

called a Location Area in the GSM or UMTS system, or Routing Area if a data packet session

is involved). Paging takes place by sending the broadcast message to all of those cells. Paging

messages can be used for information transfer. This happens in pagers, in CDMA systems for

sending SMS messages, and in the UMTS system where it allows for low downlink latency in

packet-based connections.

2.4 Movement from cell to cell and handover

In a primitive taxi system, when the taxi moved away from a first tower and closer to

a second tower, the taxi driver manually switched from one frequency to another as needed.

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If a communication was interrupted due to a loss of a signal, the taxi driver asked the base

station operator to repeat the message on a different frequency.

In a cellular system, as the distributed mobile transceivers move from cell to cell

during an ongoing continuous communication, switching from one cell frequency to a

different cell frequency is done electronically without interruption and without a base station

operator or manual switching. This is called the handover or handoff. Typically, a new

channel is automatically selected for the mobile unit on the new base station which will serve

it. The mobile unit then automatically switches from the current channel to the new channel

and communication continues.

The exact details of the mobile system's move from one base station to the other

varies considerably from system to system (see the example below for how a mobile phone

network manages handover).

Chapter 3Mobile Network: a cellular network

3.1 Example of a cellular network:

The most common example of a cellular network is a mobile phone (cell phone)

network. A mobile phone is a portable telephone which receives or makes calls through a cell

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site (base station), or transmitting tower. Radio waves are used to transfer signals to and from

the cell phone.

Modern mobile phone networks use cells because radio frequencies are a limited, shared

resource. Cell-sites and handsets change frequency under computer control and use low

power transmitters so that a limited number of radio frequencies can be simultaneously used

by many callers with less interference.

A cellular network is used by the mobile phone operator to achieve both coverage and

capacity for their subscribers. Large geographic areas are split into smaller cells to avoid line-

of-sight signal loss and to support a large number of active phones in that area. All of the cell

sites are connected to telephone exchanges (or switches) , which in turn connect to the public

telephone network.

In cities, each cell site may have a range of up to approximately ½ mile, while in rural areas,

the range could be as much as 5 miles. It is possible that in clear open areas, a user may

receive signals from a cell site 25 miles away.

Since almost all mobile phones use cellular technology, including GSM, CDMA, and AMPS

(analog), the term "cell phone" is in some regions, notably the US, used interchangeably with

"mobile phone". However, satellite phones are mobile phones that do not communicate

directly with a ground-based cellular tower, but may do so indirectly by way of a satellite.

There are a number of different digital cellular technologies, including: Global System for

Mobile Communications (GSM), General Packet Radio Service (GPRS), Code Division

Multiple Access (CDMA), Evolution-Data Optimized (EV-DO), Enhanced Data Rates for

GSM Evolution (EDGE), 3GSM, Digital Enhanced Cordless Telecommunications (DECT),

Digital AMPS (IS-136/TDMA), and Integrated Digital Enhanced Network (iDEN).

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3.2 GSM

Structure of a 2G cellular network

A simple view of the cellular mobile-radio network consists of the following:

A network of Radio base stations forming the Base station subsystem.

The core circuit switched network for handling voice calls and text

A packet switched network for handling mobile data

The Public switched telephone network to connect subscribers to the wider telephony

network

This network is the foundation of the GSM system network. There are many functions that

are performed by this network in order to make sure customers get the desired service

including mobility management, registration, call set up, and handover.

Any phone connects to the network via an RBS in the corresponding cell which in turn

connects to the MSC. The MSC allows the onward connection to the PSTN. The link from a

phone to the RBS is called an uplink while the other way is termed downlink.

Radio channels effectively use the transmission medium through the use of the following

multiplexing schemes: frequency division multiplex (FDM), time division multiplex (TDM),

code division multiplex (CDM), and space division multiplex (SDM). Corresponding to these

multiplexing schemes are the following access techniques: frequency division multiple access

(FDMA), time division multiple access (TDMA), code division multiple access (CDMA),

and space division multiple access (SDMA).

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3.3 Cellular handover/handoff in mobile phone networks

As the phone user moves from one cell area to another cell whilst a call is in progress, the

mobile station will search for a new channel to attach to in order not to drop the call. Once a

new channel is found, the network will command the mobile unit to switch to the new

channel and at the same time switch the call onto the new channel.

With CDMA, multiple CDMA handsets share a specific radio channel. The signals are

separated by using a pseudonoise code (PN code) specific to each phone. As the user moves

from one cell to another, the handset sets up radio links with multiple cell sites (or sectors of

the same site) simultaneously. This is known as "soft handoff" because, unlike with

traditional cellular technology, there is no one defined point where the phone switches to the

new cell.

In IS-95 inter-frequency handovers and older analog systems such as NMT it will typically be

impossible to test the target channel directly while communicating. In this case other

techniques have to be used such as pilot beacons in IS-95. This means that there is almost

always a brief break in the communication while searching for the new channel followed by

the risk of an unexpected return to the old channel.

If there is no ongoing communication or the communication can be interrupted, it is possible

for the mobile unit to spontaneously move from one cell to another and then notify the base

station with the strongest signal. Cellular frequency choice in mobile phone networks

Chapter 4

GSM frequency bands

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Fig 4.1

The effect of frequency on cell coverage means that different frequencies serve better for

different uses. Low frequencies, such as 450 MHz NMT, serve very well for countryside

coverage. GSM 900 (900 MHz) is a suitable solution for light urban coverage. GSM 1800

(1.8 GHz) starts to be limited by structural walls. UMTS, at 2.1 GHz is quite similar in

coverage to GSM 1800.

Higher frequencies are a disadvantage when it comes to coverage, but it is a decided

advantage when it comes to capacity. Pico cells, covering e.g. one floor of a building, become

possible, and the same frequency can be used for cells which are practically neighbours.

Cell service area may also vary due to interference from transmitting systems, both within

and around that cell. This is true especially in CDMA based systems. The receiver requires a

certain signal-to-noise ratio. As the receiver moves away from the transmitter, the power

transmitted is reduced. As the interference (noise) rises above the received power from the

transmitter, and the power of the transmitter cannot be increased any more, the signal

becomes corrupted and eventually unusable. In CDMA-based systems, the effect of

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interference from other mobile transmitters in the same cell on coverage area is very marked

and has a special name, cell breathing.

One can see examples of cell coverage by studying some of the coverage maps provided by

real operators on their web sites. In certain cases they may mark the site of the transmitter, in

others it can be calculated by working out the point of strongest coverage.

Chapter 5

CELLULAR TRAFFIC

This article discusses the mobile cellular network aspect of telegraphic measurements.

Mobile radio networks have traffic issues that do not arise in connection with the fixed line

PSTN. Important aspects of cellular traffic include: quality of service targets, traffic capacity

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and cell size, spectral efficiency and sectorization, traffic capacity versus coverage, and

channel holding time analysis.

Teletraffic engineering is a necessary field in telecommunications network planning to ensure

that network costs are minimized without compromising the quality of service delivered to

the user of the network. This field of engineering is based on probability theory and can be

used to analyze mobile radio networks, as well as other telecommunications networks.

A mobile handset which is moving in a cell will record a signal strength that varies. Signal

strength is subject to slow fading, fast fading and interference from other signals, resulting in

degradation of the carrier-to-interference (C/I) ratio. A high C/I ratio yields quality

communication. A good C/I ratio is achieved in cellular systems by using optimum power

levels through the power control of most links. When carrier power is too high, excessive

interference is created, degrading the C/I ratio for other traffic and reducing the traffic

capacity of the radio subsystem. When carrier power is too low, C/I is too low and QoS

targets are not met.

5.1Quality of Service targets

At the time that the cells of a radio subsystem are designed, Quality of Service (QoS) targets

are set, for: traffic congestion and blocking, dominant coverage area, C/I, dropped call rate,

handover failure rate, overall call success rate.

5.2 Traffic load and cell size

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The more traffic generated, the more base stations will be needed to service the customers.

The number of base stations for a simple cellular network is equal to the number of cells. The

traffic engineer can achieve the goal of satisfying the increasing population of customers by

increasing the number of cells in the area concerned, so this will also increases the number of

base stations. This method is called cell splitting (and combined with sectorization) is the

only way of providing services to a burgeoning population. This simply works by dividing

the cells already present into smaller sizes hence increasing the traffic capacity. Reduction of

the cell radius enables the cell to accommodate extra traffic. The cost of equipment can also

be cut down by reducing the number of base stations through setting up three neighbouring

cells, with the cells serving three 120° sectors with different channel groups.

Mobile radio networks are operated with finite, limited resources (the spectrum of

frequencies available). These resources have to be used effectively to ensure that all users

receive service, that is, the quality of service is consistently maintained. This need to

carefully use the limited spectrum brought about the development of cells in mobile

networks, enabling frequency re-use by successive clusters of cells. Systems that efficiently

use the available spectrum have been developed e.g. the GSM system. Walke defines spectral

efficiency as the traffic capacity unit divided by the product of bandwidth and surface area

element, and is dependent on the number of radio channels per cell and the cluster size

(number of cells in a group of cells):

Where Nc is the number of channels per cell, BW is the system bandwidth, and Ac is Area of

cell.

Sectorization is briefly described in traffic load and cell size as a way to cut down

equipment costs in a cellular network. When applied to clusters of cells sectorization also

reduces co-channel interference, according to Walke. This is because the power radiated

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backward from directional base antennas (directional) is greater than that of omnidirectional

antennas by a factor which is the number of sectors per cell (or cell cluster).

5.3 Traffic capacity versus coverage

Cellular systems use one or more of four different techniques of access (TDMA, FDMA,

CDMA, SDMA). See Cellular concepts. Let a case of Code Division Multiple Access be

considered for the relationship between traffic capacity and coverage (area covered by cells).

CDMA cellular systems can allow an increase in traffic capacity at the expense of the quality

of service.

In TDMA/FDMA cellular radio systems, Fixed Channel Allocation (FCA) is used to allocate

channels to customers. In FCA the number of channels in the cell remains constant

irrespective of the number of customers in that cell. This results in traffic congestion and

some calls being lost when traffic gets heavy.

A better way of channel allocation in cellular systems is Dynamic Channel Allocation (DCA)

which is supported by the GSM, DCS and other systems. DCA is a better way not only for

handling bursty cell traffic but also in efficiently utilising the cellular radio resources. DCA

allows the number of channels in a cell to vary with the traffic load, hence increasing channel

capacity with little costs. Since a cell is allocated a group of frequency carries (e.g. f 1-f7) for

each user, this range of frequencies is the bandwidth of that cell, BW. If that cell covers an

area Ac, and each user has bandwidth B then the number of channels will be BW/B. The

density of channels will be. This formula shows that as the coverage area Ac is increased, the

channel density decreases.

5.4 Channel holding time

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Important parameters like the carrier to interference (C/I) ratio, spectral efficiency and reuse

distance determine the quality of service of a cellular network. Channel Holding Time is

another parameter that can affect the quality of service in a cellular network, hence it is

considered when planning the network. It must be mentioned that it is not an easy task to

calculate the channel holding time. (This is the time a Mobile Station (MS) remains in the

same cell during a call). Channel holding time is therefore less than call holding time if the

MS travels more than one cell as handover will take place and the MS relinquishes the

channel. Practically, it is not possible to determine exactly the channel holding time. As a

result, different models exists for modelling the channel holding time distribution. In

industry, a good approximation of the channel holding time is usually sufficient to determine

the network traffic capability.

One of the papers in Key and Smith defines channel holding time as being equal to the

average holding time divided by the average number of handovers per call plus one. Usually

an exponential model is preferred to calculate the channel holding time for simplicity in

simulations. This model gives the distribution function of channel holding time and it is an

approximation that can be used to obtain estimates channel holding time. The exponential

model may not be correctly modelling the channel holding time distribution as other papers

may try to prove, but it gives an approximation. Channel holding time is not easily

determined explicitly, call holding time and user's movements have to be determined in order

to implicitly give channel holding time. The mobility of the user and the cell shape and size

cause the channel holding time to have a different distribution function to that of call duration

(call holding time). This difference is large for highly mobile users and small cell sizes. Since

the channel holding time and call duration relationships are affected by mobility and cell size,

for a stationary MS and large cell sizes, channel holding time and call duration are the same.

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Chapter 6

GSM Architecture

The GSM architecture consists of three subsystems

1. Base station subsystem

2. Network Station subsystem

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3. Operating station Subsystem

6.1 Base station subsystem(BSS)

The base station subsystem (BSS) is the section of a traditional cellular telephone network

which is responsible for handling traffic and signaling between a mobile phone and the

network switching subsystem. The BSS carries out transcoding of speech channels, allocation

of radio channels to mobile phones, paging, transmission and reception over the air interface

and many other tasks related to the radio network.

Base transceiver station

The base transceiver station, or BTS, contains the equipment for transmitting and receiving

radio signals (transceivers), antennas, and equipment for encrypting and decrypting

communications with the base station controller (BSC). Typically a BTS for anything other

than a picocell will have several transceivers (TRXs) which allow it to serve several different

frequencies and different sectors of the cell (in the case of sectorised base stations).

A BTS is controlled by a parent BSC via the "base station control function" (BCF). The BCF

is implemented as a discrete unit or even incorporated in a TRX in compact base stations.

The BCF provides an operations and maintenance (O&M) connection to the network

management system (NMS), and manages operational states of each TRX, as well as

software handling and alarm collection.

The functions of a BTS vary depending on the cellular technology used and the cellular

telephone provider. There are vendors in which the BTS is a plain transceiver which receives

information from the MS (mobile station) through the Um (air interface) and then converts it

to a TDM (PCM) based interface, the Abis interface, and sends it towards the BSC. There are

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vendors which build their BTSs so the information is preprocessed, target cell lists are

generated and even intracell handover (HO) can be fully handled. The advantage in this case

is less load on the expensive Abis interface.

The BTSs are equipped with radios that are able to modulate layer 1 of interface Um; for

GSM 2G+ the modulation type is GMSK, while for EDGE-enabled networks it is GMSK and

8-PSK.

Antenna combiners are implemented to use the same antenna for several TRXs (carriers), the

more TRXs are combined the greater the combiner loss will be. Up to 8:1 combiners are

found in micro and pico cells only.

A TRX transmits and receives according to the GSM standards, which specify eight TDMA

timeslots per radio frequency. A TRX may lose some of this capacity as some information is

required to be broadcast to handsets in the area that the BTS serves. This information allows

the handsets to identify the network and gain access to it. This signalling makes use of a

channel known as the broadcast control channel (BCCH).

6.2 CELL SITE

A cell site is a term used to describe a site where antennas and electronic communications

equipment are placed on a radio mast or tower to create a cell in a cellular network. A cell

site is composed of a tower or other elevated structure for mounting antennas, and one or

more sets of transmitter/receivers transceivers, digital signal processors, control electronics, a

GPS receiver for timing (for CDMA2000 or IS-95 systems), regular and backup electrical

power sources, and sheltering.

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A synonym for "cell site" is "cell tower", although many cell site antennas are mounted on

buildings rather than as towers. In GSM networks, the technically correct term is Base

Transceiver Station (BTS), and colloquial British English synonyms are "mobile phone mast"

or "base station". The term "base station site" might better reflect the increasing co-location

of multiple mobile operators, and therefore multiple base stations, at a single site. Depending

on an operator's technology, even a site hosting just a single mobile operator may house

multiple base stations, each to serve a different air interface technology (CDMA or GSM, for

example). Preserved treescapes can often hide cell towers inside an artificial tree or preserved

tree. These installations are generally referred to as concealed cell sites or stealth cell sites.

Operation Range

The working range of a cell site - the range within which mobile devices can connect to it

reliably is not a fixed figure. It will depend on a number of factors, including

The frequency of signal in use (i.e. the underlying technology).

The transmitter's rated power.

The transmitter's size.

The array setup of panels may cause the transmitter to be directional or omni-

directional.

It may also be limited by local geographical or regulatory factors and weather

conditions.

Generally, in areas where there are enough cell sites to cover a wide area, the range of each

one will be set to:

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Ensure there is enough overlap for "handover" to/from other sites (moving the signal

for a mobile device from one cell site to another, for those technologies that can

handle it - e.g. making a GSM phone call while in a car or train).

Ensure that the overlap area is not too large, to minimize interference problems with

other sites.

In practice, cell sites are grouped in areas of high population density, with the most potential

users. Cell phone traffic through a single cell mast is limited by the mast's capacity; there is a

finite number of calls that a mast can handle at once. This limitation is another factor

affecting the spacing of cell mast sites. In suburban areas, masts are commonly spaced 1-2

miles apart and in dense urban areas, masts may be as close as ¼-½ mile apart. Cell masts

always reserve part of their available bandwidth for emergency calls.

Objects intruding into the fresnel zone between radio transmitters and receivers can greatly

affect signal strength.

The maximum range of a mast (where it is not limited by interference with other masts

nearby) depends on the same circumstances. Some technologies, such as GSM, have a fixed

maximum range of 40km (23 miles), which is imposed by technical limitations. CDMA and

iDEN have no built-in limit, but the limiting factor is really the ability of a low-powered

personal cell phone to transmit back to the mast. As a rough guide, based on a tall mast and

flat terrain, it is possible to get between 50 to 70 km (30-45 miles). When the terrain is hilly,

the maximum distance can vary from as little as 5 kilometres (3.1 mi) to 8 kilometres (5.0 mi)

due to encroachment of intermediate objects into the wide center fresnel zone of the signal.

Depending on terrain and other circumstances, a GSM Tower can replace between 2 and 50

miles of cabling for fixed wireless networks.

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Chapter 7Channel Reuse and signal Strength

7.1Channel reuse

The concept of "maximum" range is misleading, however, in a cellular network. Cellular

networks are designed to create a mass communication solution from a limited amount of

channels (slices of radio frequency spectrum necessary to make one conversation) that are

licensed to an operator of a cellular service. To overcome this limitation, it is necessary to

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repeat and reuse the same channels at different locations. Just as a car radio changes from one

local station to a completely different local station with the same frequency when you travel

to another city, the same radio channel gets reused on a cell mast only a few miles away. To

do this, the signal of a cell mast is intentionally kept at low power and many cases tilting

downward to limit its area. The area sometimes needs to be limited when a large number of

people live, drive or work near a particular mast; the range of this mast has to be limited so

that it covers an area small enough not to have to support more conversations than the

available channels can carry. Due to the sectorized arrangement of antennas on a tower, it is

possible to vary the strength and angle of each sector depending on the coverage of other

towers in view of the sector.

A cellphone may not work at times, because it is too far from a mast, but it may also not work

because the phone is in a location where there is interference to the cell phone signal from

thick building walls, hills or other structures. The signals do not need a clear line of sight but

the more interference will degrade or eliminate reception. Too many people may be trying to

use the cell mast at the same time, e.g. a traffic jam or a sports event, then there will be a

signal on the phone display but it is blocked from starting a new connection. The other

limiting factor for cell phones is the ability of the cell phone to send a signal from its low

powered battery to the mast. Some cellphones perform better than others under low power or

low battery, typically due to the ability to send a good signal from the phone to the mast.

The base station controller (a central computer that specializes in making phone connections)

and the intelligence of the cellphone keeps track of and allows the phone to switch from one

mast to the next during conversation. As the user moves towards a mast it picks the strongest

signal and releases the mast from which the signal has become weaker; that channel on that

mast becomes available to another user.

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7.2 Cell phone tower power emission

The U.S. government agency, the FCC, says:

"For example, measurement data obtained from various sources have consistently indicated

that "worst-case" ground-level power densities near typical cellular towers are on the order of

1 µW/cm2 or less (usually significantly less)."

That is 0.01 Watt per square meter. There is no temptation to use more power. The entire idea

of a "cell" phone system is to create small "cells" that don't interfere with each other.

The average energy received over the entire earth is about 250 Watts per square meter over a

24 hour day, ignoring clouds. So, on a day with no clouds, the average electromagnetic

energy received from the Sun is 25,000 times that received near a cell phone tower.

7.3 Signal strength

In telecommunications, particularly in radio, signal strength refers to the magnitude of the

electric field at a reference point that is a significant distance from the transmitting antenna. It

may also be referred to as received signal level or field strength. Typically, it is expressed in

voltage per length or signal power received by a reference antenna. High-powered

transmissions, such as those used in broadcasting, are expressed in dB-millivolts per metre

(dBmV/m). For very low-power systems, such as mobile phones, signal strength is usually

expressed in dB-microvolts per metre (dBµV/m) or in decibels above a reference level of one

milliwatt (dBm). In broadcasting terminology, 1 mV/m is 1000 µV/m or 60 dBµ (often

written dBu).

Examples

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100 dBµ or 100 mV/m: blanketing interference may occur on some receivers

60 dBµ or 1.0 mV/m: frequently considered the edge of a radio station's protected

area in North America

40 dBµ or 0.1 mV/m: the minimum strength at which a station can be received with

acceptable quality on most receivers

Chapter 8

Reasons for weak signal

8.1 Rural areas

In many rural areas the housing density is too low to make construction of a new base station

commercially viable. In these cases it is unlikely that the service provider will do anything to

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improve reception, due to the high cost of erecting a new tower. As a result, the only way to

obtain strong cell phone signal in these areas is usually to install a home cellular repeater. In

flat rural areas the signal is unlikely to suffer from multipath interference, so will just be

heavily attenuated by the distance. In these cases the installation of a cellular repeater will

generally massively increase signal strength just due to the amplifier, even a great distance

from the broadcast towers.

8.2 Building construction material

Some construction materials very rapidly attenuate cell phone signal strength. Older

buildings, such as churches, which use lead in their roofing material will very effectively

block any signal. Any building which has a significant thickness of concrete or amount of

metal used in its production will attenuate the signal. Concrete floors are often poured onto a

metal pan which completely blocks most radio signals. Some solid foam insulation and some

fiberglass insulation used in roofs or exterior walls has foil backing, which can reduce

transmittance. Energy efficient windows and metal window screens are also very effective at

blocking radio signals. Some materials have peaks in their absorption spectra which

massively decrease signal strength.

8.3 Building size

Large buildings, such as warehouses, hospitals and factories, often have no cellular reception

further than a few meters from the outside wall. Low signal strength is also often the case in

underground areas such as basements and in shops and restaurants located towards the centre

of shopping malls. This is caused by both the fact that the signal is attenuated heavily as it

enters the building and the interference as the signal is reflected by the objects inside the

building. For this reason in these cases an external antenna is usually desirable.

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8.4 Multipath interference

Even in urban areas which usually have strong cellular signals throughout, there are often

dead zones caused by destructive interference of waves which have taken different paths

(caused by the signal bouncing off buildings etc.) These usually have an area of a few blocks

and will usually only affect one of the two frequency ranges used by cell phones. This is

because the different wavelengths of the different frequencies interfere destructively at

different points. Directional antennas are very helpful at overcoming this since they can be

placed at points of constructive interference and aligned so as not to receive the destructive

signal. See Multipath interference for more.

8.5 Diffraction and general attenuation

The longer wavelengths have the advantage of being able to diffract to a greater degree so are

less reliant on line of sight to obtain a good signal, but still attenuate significantly. Because

the frequencies which cell phones use are too high to reflect off the ionosphere as shortwave

radio waves do, cell phone waves cannot travel via the ionospohere.

8.6 Different operating frequencies

Repeaters are available for all the different GSM frequency bands, some repeaters will handle

different types of network such as multi-mode GSM and UMTS repeaters however dual- and

tri-band systems cost significantly more. Repeater systems are available for certain Satellite

phone systems, allowing the satphones to be used indoors without a clear line of sight to the

satellite.

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Chapter 9

CODE DIVISION MULTIPLE ACCESS

Code division multiple access (CDMA) is a channel access method utilized by various radio

communication technologies. It should not be confused with the mobile phone standards

called cdmaOne and CDMA2000 (which are often referred to as simply CDMA), which use

CDMA as an underlying channel access method.

9.1 CDMA basics

One of the basic concepts in data communication is the idea of allowing several transmitters

to send information simultaneously over a single communication channel. This allows several

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users to share a bandwidth of different frequencies. This concept is called multiplexing.

CDMA employs spread-spectrum technology and a special coding scheme (where each

transmitter is assigned a code) to allow multiple users to be multiplexed over the same

physical channel. By contrast, time division multiple access (TDMA) divides access by time,

while frequency-division multiple access (FDMA) divides it by frequency. CDMA is a form

of spread-spectrum signaling, since the modulated coded signal has a much higher data

bandwidth than the data being communicated.

An analogy to the problem of multiple access is a room (channel) in which people wish to

communicate with each other. To avoid confusion, people could take turns speaking (time

division), speak at different pitches (frequency division), or speak in different languages

(code division). CDMA is analogous to the last example where people speaking the same

language can understand each other, but not other people. Similarly, in radio CDMA, each

group of users is given a shared code. Many codes occupy the same channel, but only users

associated with a particular code can communicate.

9.2 Steps in CDMA Modulation

CDMA is a spread spectrum multiple access technique. A spread spectrum technique is one

which spreads the bandwidth of the data uniformly for the same transmitted power. Spreading

code is a pseudo-random code which has a narrow Ambiguity function unlike other narrow

pulse codes. In CDMA a locally generated code runs at a much higher rate than the data to be

transmitted. Data for transmission is simply logically XOR (exclusive OR) added with the

faster code. The figure shows how spread spectrum signal is generated. The data signal with

pulse duration of Tb is XOR added with the code signal with pulse duration of Tc. (Note:

bandwidth is proportional to 1 / T where T = bit time) Therefore, the bandwidth of the data

signal is 1 / Tb and the bandwidth of the spread spectrum signal is 1 / Tc. Since Tc is much

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smaller than Tb, the bandwidth of the spread spectrum signal is much larger than the

bandwidth of the original signal. The ratio Tb / Tc is called spreading factor or processing gain

and determines to a certain extent the upper limit of the total number of users supported

simultaneously by a base station.

Each user in a CDMA system uses a different code to modulate their signal. Choosing the

codes used to modulate the signal is very important in the performance of CDMA systems.

The best performance will occur when there is good separation between the signal of a

desired user and the signals of other users. The separation of the signals is made by

correlating the received signal with the locally generated code of the desired user. If the

signal matches the desired user's code then the correlation function will be high and the

system can extract that signal. If the desired user's code has nothing in common with the

signal the correlation should be as close to zero as possible (thus eliminating the signal); this

is referred to as cross correlation. If the code is correlated with the signal at any time offset

other than zero, the correlation should be as close to zero as possible. This is referred to as

auto-correlation and is used to reject multi-path interference.

In general, CDMA belongs to two basic categories: synchronous (orthogonal codes) and

asynchronous (pseudorandom codes).

9.3 Code division multiplexing (Synchronous CDMA)

Synchronous CDMA exploits mathematical properties of orthogonality between vectors

representing the data strings. For example, binary string 1011 is represented by the vector (1,

0, 1, 1). Vectors can be multiplied by taking their dot product, by summing the products of

their respective components. If the dot product is zero, the two vectors are said to be

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orthogonal to each other (note: if u = (a, b) and v = (c, d), the dot product u·v = ac + bd).

Some properties of the dot product aid understanding of how W-CDMA works. If vectors a

and b are orthogonal, then and:

Each user in synchronous CDMA uses a code orthogonal to the others' codes to modulate

their signal. An example of four mutually orthogonal digital signals is shown in the figure.

Orthogonal codes have a cross-correlation equal to zero; in other words, they do not interfere

with each other. In the case of IS-95 64 bit Walsh codes are used to encode the signal to

separate different users. Since each of the 64 Walsh codes are orthogonal to one another, the

signals are channelized into 64 orthogonal signals. The following example demonstrates how

each users signal can be encoded and decoded.

9.4 USES

A CDMA mobile phone

One of the early applications for code division multiplexing is in GPS. This predates

and is distinct from cdmaOne.

The Qualcomm standard IS-95, marketed as cdmaOne.

The Qualcomm standard IS-2000, known as CDMA2000. This standard is used by

several mobile phone companies, including the Globalstar satellite phone network.

CDMA has been used in the OmniTRACS satellite system for transportation

logistics.

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Chapter 10

The Future of Mobile Networking

10.1 Future evolution: Broadband Fourth generation (4G)

The recently released 4th generation, also known as Beyond 3G, aims to provide broadband

wireless access with nominal data rates of 100 Mbit/s to fast moving devices, and 1 Gbit/s to

stationary devices defined by the ITU-R 4G systems may be based on the 3GPP LTE (Long

Term Evolution) cellular standard, offering peak bit rates of 326.4 Mbit/s. It may perhaps

also be based on WiMax or Flash-OFDM wireless metropolitan area network technologies

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that promise broadband wireless access with speeds that reaches 233 Mbit/s for mobile users.

The radio interface in these systems is based on all-IP packet switching, MIMO diversity,

multi-carrier modulation schemes, Dynamic Channel Assignment (DCA) and channel-

dependent scheduling. A 4G system should be a complete replacement for current network

infrastructure and is expected to be able to provide a comprehensive and secure IP solution

where voice, data, and streamed multimedia can be given to users on a "Anytime, Anywhere"

basis, and at much higher data rates than previous generations. Sprint in the US has claimed

its WiMax network to be "4G network" which most cellular telecoms standardization experts

dispute repeatedly around the world. Sprint's 4G is seen as a marketing gimmick as WiMax

itself is part of the 3G air interface. The officially accepted, ITU ratified standards-based 4G

networks are not expected to be commercially launched until 2011.

10.2 Comparison to similar systems

Car phone  A type of telephone permanently mounted in a vehicle, these often have more

powerful transmitters, an external antenna and loudspeaker for hands free use. They usually

connect to the same networks as regular mobile phones.

Cordless telephone (portable phone) 

Cordless phones are telephones which use one or more radio handsets in place of a

wired handset. The handsets connect wirelessly to a base station, which in turn

connects to a conventional land line for calling. Unlike mobile phones, cordless

phones use private base stations (belonging to the land-line subscriber), which are not

shared.

Professional Mobile Radio 

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Advanced professional mobile radio systems can be very similar to mobile phone

systems. Notably, the IDEN standard has been used as both a private trunked radio

system as well as the technology for several large public providers. Similar attempts

have even been made to use TETRA, the European digital PMR standard, to

implement public mobile networks.

Radio phone 

This is a term which covers radios which could connect into the telephone network.

These phones may not be mobile; for example, they may require a mains power

supply, or they may require the assistance of a human operator to set up a PSTN

phone call.

Satellite phone 

This type of phone communicates directly with an artificial satellite, which in turn

relays calls to a base station or another satellite phone. A single satellite can provide

coverage to a much greater area than terrestrial base stations. Since satellite phones

are costly, their use is typically limited to people in remote areas where no mobile

phone coverage exists, such as mountain climbers, mariners in the open sea, and news

reporters at disaster sites.

IP Phone 

This type of phone delivers or receives calls over internet, LAN or WAN networks

using VoIP as opposed to traditional CDMA and GSM networks. In business, the

majority of these IP Phones tend to be connected via wired Ethernet, however

wireless varieties do exist. Several vendors have developed standalone WiFi phones.

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Additionally, some cellular mobile phones include the ability to place VoIP calls over

cellular high speed data networks and/or wireless internet.

II

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ENGINEERING

Chapter 1

INTRODUCTION

1.1 Microwave Frequencies

The descriptive term microwaves is used to describe electromagnetic waves with

wavelengths ranging from 1 cm to 1 m. The corresponding frequency range is 300 MHz up to

30 GHz for 1-cm-wavelength waves. Electromagnetic waves with wavelengths ranging from

1 to 10 mm are called millimeter waves. The infrared radiation spectrum comprises

electromagnetic waves with wavelengths in the range 1 am (10 6 m) up to 1 mm. Beyond the

infrared range is the visible optical spectrum, the ultraviolet spectrum, and finally x-rays.

Several different classification schemes are in use to designate frequency bands in the

electromagnetic spectrum. These classification schemes are summarized in Tables 1.1 and

1.2. The radar band classification came into use during World War II and is still in common

use today even though the new military band classification is the recommended one. In the

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UHF band up to around a frequency of 1 GHz, most communications circuits are constructed

using lumped-parameter circuit componenta. In the frequency range from 1 up to 100 GHz.

lumped circuit elements are usually replaced by transmission-line and waveguide

components. Thus by the term microwave engineering we shall mean generally the

engineering and design of information-handling systems in the frequency range from 1 to 100

GHz corresponding to wavelengths as long as 30 cm and as short as 3 mm. At shorter

wavelengths we have what can be called optical engineering since many of the techniques

used are derived from classical optical techniques. The characteristic feature of microwave

engineering is the short wavelengths involved, these being of the same order of magnitude as

the circuit elements and devices employed.

The Table 1.1 shows the various bands of Microwave frequencies, their designations

and general applications of each and every band in modern field of Science and Technology

Table 1.1 Different bands of Frequencies and their applications

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Microwave freq.

The short wavelengths involved in turn mean that the propagation time for electrical

effects from one point in a circuit to another point is comparable with the period of the

oscillating currents and charges in the system. As a result, conventional low-frequency circuit

analysis based on Kirchhoffs laws and voltage-current concepts no longer suffices for an

adequate description of the electrical phenomena taking place. It is necessary instead to cany

out the analysis in terms of a description of the electric and magnetic fields associated with

the device. In essence, it might be said, microwave engineering is applied electromagnetic

fields engineering. For this reason the successful engineer in this area must have a good

working knowledge of electromagnetic field theory.

Table 1.2 represents modern labelling of frequency bands according to IEEE

standards.

Table 1.2 IEEE Frequency band Designation

There is no distinct frequency boundary at which lumped-parameter circuit elements

must be replaced by distributed circuit elements. With modern technological processes it is

possible to construct printed-circuit inductors that are so small that they retain their lumped-

parameter characteristics at frequencies as high as 10 GHz or even higher. Likewise, optical

components, such as parabolic reflectors and lenses, are used to focus microwaves with

wavelengths as long as 1 m or more. Consequently, the microwave engineer will frequently

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employ low-frequency lumped-parameter circuit elements, such as miniaturized inductors

and capacitors, as well as optical devices in the design of a microwave system.

1.2 Microwave History of Development

The great interest in microwave frequencies arises for a variety of reasons. Basic

among these is the ever-increasing need for more radio-frequency-spectrum space and the

rather unique uses to which microwave frequencies can be applied. When it is noted that the

frequency range 109 to 1012 Hz contains a thousand sections like the frequency spectrum from

0 to 109 Hz, the value of developing the microwave band as a means of increasing the

available usable frequency spectrum may be readily appreciated.

In more recent years microwave frequencies have also come into widespread use in

communication links, generally referred to as microwave links. Since the propagation of

microwaves is effectively along line-of-sight paths, these links employ high towers with

reflector or lens-type antennas as repeater stations spaced along the communication path.

Such links are a familiar sight to the motorist traveling across the country because of their

frequent use by highway authorities, utility companies, and television networks. A further

interesting means of communication by microwaves is the use of satellites as microwave

relay stations. The first of these, the Telstar, launched in July 1962, provided the first

transmission of live television programs from the United States to Europe.

At the present time most communication systems are shifting to the use of digital

transmission, i.e., analog signals are digitized before transmission. Microwave digital

communication system development is progressing rapidly. In the early systems simple

modulation schemes were used and resulted in inefficient use of the available frequency

spectrum. The development of 64-state quadrature amplitude modulation (64-QAM) has

made it possible to transmit 2,016 voice channels within a single 30-MHz RF channel. This is

competitive with FM analog modulation schemes for voice.

1.3 Microwave Applications

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Even though such uses of microwaves are of great importance, the applications of

microwaves and microwave technology extend much further, into a variety of areas of basic

and applied research, and including a number of diverse practical devices, such as microwave

ovens that can cook a small roast in just a few minutes

Waveguides periodically loaded with shunt susceptance elements support slow waves

having velocities much less than the velocity of light, and are used in linear accelerators.

These produce high-energy beams of charged particles for use in atomic and nuclear

research. The slow-traveling electromagnetic waves interact very efficiently with charged-

particle beams having the same velocity, and thereby give up energy to the beam. Another

possibility is for the energy in an electron beam to be given up to the electromagnetic wave,

with resultant amplification

Sensitive microwave receivers are used in radio astronomy to detect and study the

electromagnetic radiation from the sun and a number of radio stars that emit radiation in this

band. Such receivers are also used to detect the noise radiated from plasmas (an

approximately neutral collection of electrons and ions, e.g., a gas discharge). The information

obtained enables scientists to analyze and predict the various mechanisms responsible for

plasma radiation. Microwave radiometers are also used to map atmospheric temperature

profiles, moisture conditions in soils and crops, and for other remote-sensing applications as

well.

The development of the laser, a generator of essentially monochromatic (single-

frequency) coherent-light waves, has stimulated a great interest in the possibilities of

developing communication systems at optical wavelengths. This frequency band is

sometimes referred to as the ultra-microwave band. With some modification, a good deal of

the present microwave technology can be exploited in the development of optical systems.

For this reason, familiarity with conventional microwave theory and devices provides a good

background for work in the new frontiers of the electromagnetic spectrum.

The domestic microwave oven operates at 2,450 MHz and uses a magnetron tube with

a power output of 500 to 1000 W. For industrial heating applications, such as drying grain,

manufacturing wood and paper products, and material curing, the frequencies of 915 and

2,450 MHz have been assigned. Microwave radiation has also found some application for

medical hyperthermia or localized heating of tumours

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Chapter 2

WAVEGUIDES –I

2.1 Modes of Propogation

For a large variety of waveguides of practical interest it turns out that all the boundary

conditions can be satisfied by fields that do not have all components present. Specifically, for

transmission lines, the solution of interest is a transverse electromagnetic wave with

transverse components only, that is, Ez = Hz = 0, whereas for waveguides, solutions with Ez =

0 or Hz = 0 are possible. Because of the widespread occurrence of such field solutions, the

following classification of solutions is of particular interest.

1. Transverse electromagnetic (TEM) waves. For TEM waves, Ez = Hz = 0. The electric

field may be found from the transverse gradient of a scalar function *(x,y), which is a

function of the transverse coordinates only and is a solution of the two-dimensional

Laplace equation.

2. Transverse electric (TE), or H, modes. These solutions have Ez = 0, but Hz ¥= 0. All the

field components may be derived from the axial component Hz of magnetic field.

3. Transverse magnetic (TM), or E, modes. These solutions have Hz = "» but Ez ¥= 0. The

field components may be derived from Ez.

In some cases it will be found that a TE or TM mode by itself will not satisfy all the

boundary conditions. However, in such cases linear combinations of TE and TM modes may

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be used, since such linear combinations always provide a complete and general solution.

Although other possible types of wave solutions may be constructed, the above three types

are the most useful in practice and by far the most commonly used ones.

The appropriate equations to be solved to obtain TEM, TE, or TM modes will be

derived below by placing E, and Hz, Ez, and Hz. respectively, equal to zero in Maxwell's

equations.

2.2 Rectangular vs Circular Waveguides

Hollow-pipe waveguides do not support a TEM wave. In hollow-pipe waveguides the waves

are of the TE and TM variety. The waveguide with a rectangular cross section is the most

widely used one. It is available in sizes for use at frequencies from 320 MHz up to 333 GHz.

The WR-2300 waveguide for use at 320 MHz has internal dimensions of 58.42 in by 29.1 in

and is a very large duct. By contrast, the WR-3 waveguide for use at 333 GHz has internal

dimensions of 0.034 in by 0.017 in and is a very miniature structure. The standard WR-90 X-

band waveguide has internal dimensions of 0.9 in by 0.4 in and is used in the frequency range

of 8.2 to 12.5 GHz. The rectangular waveguide is widely used to couple transmitters and

receivers the antenna. For high-power applications the waveguide is filled with j inert gas

such as nitrogen and pressurized in order to increase the voltage breakdown rating.

Circular waveguides are not as widely used as rectangular waveguides but are

available in diameters of 25.18 in down to 0.239 in to cover tn frequency range 800 MHz up

to 116 GHz.

2.3 Rectangular Waveguides

Fig 2.3 Rectangular Waveguide

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The rectangular waveguide with a cross section as illustrated in Fig. 3.36 is an example

of a wave guiding device that will not support a TEM wave. Consequently, it turns out that

unique voltage and current waves do not exist, and the analysis of the waveguide properties

has to be carried out as a field problem rather than as a distributed-parameter-circuit problem.

In a hollow cylindrical waveguide a transverse electric field can exist only if a time-

varying axial magnetic held is present. Similarly, a transverse magnetic field can exist only if

either an axial displacement current or an axial conduction current is present, as Maxwell's

equations show. Since a TEM wave does not have any axial field components and there is no

center conductor on which a conduction current can exist, a TEM wave cannot be propagated

in a cylindrical waveguide.

The types of waves that can be supported (propagated) in a hollow empty waveguide are

the TE and TM modes discussed in Sec. 3.7. The essential properties of all hollow cylindrical

waveguides are the same, so that an understanding of the rectangular guide provides insight

into the behavior of other types as well. As for the case of the transmission line, the effect of

losses is initially neglected. The attenuation is computed later by using the perturbation

method given earlier, together with the loss-free solution for the currents on the walls.

The essential properties of empty loss-free waveguides, which the detailed analysis to follow

will establish, are that there is a double infinity of possible solutions for both TE and TM

waves. These waves, or modes, may be labeled by two identifying integer subscripts n and m,

for example, TEmn.

The integers n and in pertain to the number of standing-wave interference maxima

occurring in the field solutions that describe the variation of the fields along the two

transverse coordinates. It will be found that each mode has associated with it a characteristic

cutoff frequency fc „m below which the mode does not propagate and above which the mode

does propagate

TE10 mode is the most dominant mode in Rectangular waveguides

For a rectangular waveguide with a width a equal to twice the height ft, the maximum

bandwidth of operation over which only the dominant TE10 mode propagates is a 2:1 band.

For some system applications it is necessary to have a waveguide that operates with only a

single mode of propagation over much larger bandwidths. A transmission line supporting

only a TEM mode can fulfill this requirement but must then have cross-sectional dimensions

that are small relative to the shortest wavelength of interest. A coaxial transmission line will

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support higher-order TE and TM modes in addition to the TEM mode. Thus, to avoid

excitation of a higher-order mode of propagation, the outer radius must be kept small relative

to the wavelength. The small cross section implies a relatively large attenuation; so some

other form of waveguide is needed. The ridge waveguide illustrated in Fig. 3.46 was

developed to fulfill this need for a single-mode waveguide capable of operating over a very

broad band. Physically, it is easy to understand why the ridge waveguide has a very large

frequency band of operation. The center section of width W and spacing S functions very

much like a parallel-plate transmission line and consequently the ridge waveguide has a much

lower cutoff frequency for the same width and height as does the conventional rectangular

waveguide. Operation over bandwidths of 5 to 10 times more is possible.

Chapter 3

WAVEGUIDE COMPONENTS-1

3.1 ATTENUATORS

Attenuators may be of the fixed or the variable type, The first is used only if a fixed

amount of attenuation is to be provided. For bridge setups used to measure transmission

coefficients, the variable attenuator is used. There are many ways of constructing a variable

attenuator; only one type, the rotary attenuator, is considered

A simple form 0f consists of a thin tapered resistive card, of the type used for mat t

whose depth of penetration into the waveguide is adjustable

Perhaps the most satisfactory precision attenuator developed for AI • the rotary

attenuator, which we now examine in some detail. The h components of this instrument

consist of two rectangular-to-circular wa\ guide tapered transitions, together with an

intermediate section of circular waveguide that is free to rotate,. A thin tapered resistive card

is placed at the output end of each transition section and oriented parallel to the broad walls

of the rectangular guide. A similar resistive card is located in the intermediate circular-guide

section. Tlie incoming TE;o mode in ike rectangular guide is transformed into the TEn mode

in the circular guide with negligible reflection by means of the tapered transition. The

polarization of the TEn mode is such that the electric field is perpendicular to the thin res

card in the transition section. As such, this resistive card has a negligible effect on the TEn

mode. Since the resistive card in the center section can be rotated, its orientation relative to

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the electric field of the incoming TEn mode can be varied so that the amount by which this

mode is attenuated: adjustable.

3.2 Electronically Controlled Attenuators

For applications in various microwave systems, it is desirable to have an attenuator

whose attenuation can be controlled by the application of a suitable signal, such as a dc

voltage or a bias current. Two devices thai are suitable for use in an electronically controlled

attenuator are the PIN dioc and a field-effect transistor. These devices can be used as variable

resi=toi whose resistance is controlled by the applied signal.

3.3 PHASE SHIFTERS

A phase shifter is an instrument that produces an adjustable change in the phase angle of the

wave transmitted through it. Ideally, it should perfectly matched to the input and output lines

and should produce zero attenuation. These requirements can be met to within a reasonable

deg of approximation. There are a variety of designs for phase shifters mechanically

adjustable type. The rotary phase shifter is the best in class

Phase shifters are used to change the transmission phase angle (phase of S21) of a network.

Ideal phase shifters provide low insertion loss, and equal amplitude (or loss) in all phase

states. While the loss of a phase shifter is often overcome using an amplifier stage, the less

loss, the less power that is needed to overcome it. Most phase shifters are reciprocal

networks, meaning that they work effectively on signals passing in either direction. Phase

shifters can be controlled electrically, magnetically or mechanically. Most of the phase

shifters described on this web site are passive reciprocal networks; we will concentrate

mainly on those that are electrically-controlled.

While the applications of microwave phase shifters are numerous, perhaps the most important

application is within a phased array antenna system (a.k.a. electrically steerable array, or

ESA), in which the phase of a large number of radiating elements are controlled to force the

electro-magnetic wave to add up at a particular angle to the array. The total phase variation of

a phase shifter need only be 360 degrees to control an ESA of moderate bandwidth. Networks

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that stretch phase more than 360 degrees are often called line stretchers, and are constructed

similar to the switched line phase shifters to be described below.

Analog versus digital phase shifters

Phase shifters can be analog or digital. Analog phase shifters provide a continuously variable

phase, perhaps controlled by a voltage. Electrically controlled analog phase shifters can be

realized with varactor diodes that change capacitance with voltage, or non-linear dielectrics

such as barium strontium titanate, or ferro-electric materials such as yttrium iron garnet. A

mechanically-controlled analog phase shifter is really just a mechanically lengthened

transmission line, often called a trombone line. Analog phase shifters are a mere side-show

and will not be covered here in depth at this time. If you are interested in more information

on any of these analog phase shifter topics, let us know and we will try to accommodate you.

Most phase shifters are of the digital variety, as they are more immune to noise on their

voltage control lines. Digital phase shifters provide a discrete set of phase states that are

controlled by two-state "phase bits." The highest order bit is 180 degrees, the next highest is

90 degrees, then 45 degrees, etc, as 360 degrees is divided into smaller and smaller binary

steps. A three bit phase shifter would have a 45 degree least significant bit (LSB), while a six

bit phase shifter would have a 5.6 degree least significant bit. Technically the latter case

would have a 5.625 degree LSB, but in the microwave world it is best to ignore precision that

you cannot obtain. If you can't comprehend this point, you might want to consider a different

career such as accounting.

The convention followed for phase shifters is that the shortest phase length is the reference or

"off" state, and the longest path or phase length is the "on" state. Thus a 90 degree phase

shifter actually provides minus ninety degrees of phase shift in its "on" state.

Applications of phase shifters

Frequency translators

Phased arrays

Residual phase noise measurement

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3.4 DIRECTIONAL COUPLERS

A directional coupler is a four-port microwave junction with the properties discussed below.

With reference to Fig. 3.1, which is a schematic illustration of a directional coupler, the ideal

directional coupler has the property that a wave incident in port 1 couples power into ports 2

and 3 but not into port 4. Similarly, power incident in port 4 couples into ports 2 and 3 but

not into port 1. Thus ports 1 and 4 are uncoupled. For waves incident in port 2 or 3, the

power is coupled into ports 1 and 4 only, 80 that ports 2 and 3 are also uncoupled. In

addition, all four ports are matched. That is. if three ports are terminated in matched loads,

the fourth port appears terminated in a matched load, and an incident wave in this port suffers

no reflection.

Fig 3.1 Directional Coupler as a 4 port device

Directional couplers are widely used in impedance bridges for microwave measurements and

for power monitoring. For example, if" a radar transmitter is connected to port 1, the antenna

to port 2. a microwave crystal detector to port 3. and a matched load to port 4. the power

received in port 3 is proportional to the power flowing from the transmitter to the antenna in

the forward direction only. Since the reflected wave from the antenna, if it exists, is not

coupled into port 3, the detector monitors the power output of the transmitter.

If the coupler is designed for 3-dB coupling, then it splits the input power in port 2 into equal

powers in ports 2 and 3. Thus a 3-dB directional coupler serves as a power divider.

Directional couplers with 3-dB coupling are also called hybrid junctions and are widely used

in microwave mixers and as input, and output couplers in balanced microwave amplifier

circuits. There are many available designs and configurations for directional couplers, hybrid

junctions, and power dividers.

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Fig 3.2 Directional Couplar showing input and out put ports

Chapter 4

WAVEGUIDE COMPONENTS _II

4.1 HYBRID JUNCTIONS - Magic T

A waveguide hybrid junction, known as a magic T, is illustrated in Fig. 6.32. When a

TE10 mode is incident in port 1, the electric field within the junction is like that sketched in

Fig. 6.326. This electric field has even symmetry about the midplane and hence cannot excite

the TE1U mode in arm 4 since this mode must have an electric field with odd symmetry

(shown dashed in Fig. 6.326). Thus there is no coupling between ports 1 and 4. The coupling

between ports 1 and 2, and 1 and 3, is clearly the same, as may be seen from the symmetry

involved.

For a TE10 mode incident in arm 4. the electric field within the junction is sketched in

Fig. 4.1. Symmetry again shows that there is no coupling into port 1 (this is required by

reciprocity as well). The coupling from port 4 into ports 2 and 3 is equal in magnitude but

180" out of phase. since S,3 = S13, So,, = -S34, from symmetry.

Matching elements that do not destroy the symmetry of the junction may be placed in

the E-plane and //-plane arms so as to make S,, = S44 = 0. For a lossless structure we may then

show that the unitary properties of the scattering matrix require that S.J2 = S,;i = 0, so that all

ports are matched. In addition, S23 = 0; so ports 2 and 3 as well as ports 1 and 4 are

uncoupled. The hybrid T now becomes a directional coupler with 3-dB coupling, and is often

called a magic T, even though there is nothing magic about its operation. The magic T is

commonly used in waveguide balanced mixers and in bridge networks.

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Fig 4.1 Magic Tee

The Scattering matrix for a magic T is

0 1 1 0

1 0 0 1

S= 1 0 0 -1

0 1 -1 0

4.2 Hybrid ring or Rat Race

Applications of rat-race couplers are numerous, and include mixers and phase shifters.

The rat-race gets its name from its circular shape, shown below. The circumference is 1.5

wavelengths. For an equal-split rat-race coupler, the impedance of the entire ring is fixed at

1.41xZ0, or 70.7 ohms for a 50 ohm system. For an input signal V in, the outputs at ports 2 and

4 are equal in magnitude, but 180 degrees out of phase. The coupling of the two arms is

shown in the figure below, for an ideal rat-race coupler centered at 10 GHz (10,000 MHz).

An equal power split of 3 dB occurs at only the center frequency. The 1-dB bandwidth of the

coupled port (S41) is shown by the markers to be 3760 MHz, or 37.6 percent.

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Fig 4.2 Hybrid Ring

Chapter 5

MICROWAVE PROPAGATION IN FERRITES

5.1 MICROWAVE DEVICES EMPLOYING FARADAY ROTATION

The development of ferrite materials suitable for use at microwave frequencies has

resulted in a large number of microwave devices. A number of them have nonreciprocal

electrical properties; i.e., the transmission coefficient through the device is not the same for

different directions of propagation

5.2 Gyrator

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A gyrator is defined as a two-port device that has a relative difference in phase

shift of 180° for transmission from port 1 to port 2 as compared the phase shift for

transmission from port 2 to Port1.

The phase shift may be obtained by employing the nonreciprocal property of

Faraday rotation. It consists of a rectangular guide with a 90° twist connected to a circular

guide, which in turn connected to another rectangular guide at the other end. The two

rectangular guides have the same orientation at the input ports. The circular guide

contains a thin cylindrical rod of ferrite with the ends tapered to reduce reflections. A

static axial magnetic field is applied so as to produce 90' Faraday rotation of the TE,,

dominant mode in the circular guide. Consider a wave propagating from left to right. In

passing through the twist the plane of polarization is rotated by 90° in a counter-

clockwise direction. If the ferrite produces an additional 90° of rotation, the total angle of

rotation will be 180°, as indicated in Fig. 6.45. For a wave propagating from right to left.

the Faraday rotation is still 90° in the same sense. However, in passing through the twist,

the next 90° of rotation is in a direction to cancel Faraday rotation. Thus, for transmission

from port 2 to port 1, there net rotation of the plane of polarization. The 180° rotation for

transmission from port 1 to port 2 is equivalent to an additional 180° of phase shift since

it reverses the polarization of the field. It is apparent, then, that the device just described

satisfies the definition of a gyrator.

If the inconvenience of having the input and output rectangular guides oriented at 90°

can be tolerated, a gyrator without a 90" twist section can be built. With reference to Fig.

6.46, it is seen that if the ferrite produces 90° of rotation and the output guide is rotated by

90" relative to the input guide, the emerging wave will have the right polarization to

propagate in the output guide. When propagation is from port 2 to port 1, the wave arriving in

guide 1 will have its polarization changed by 180°, as shown in Fig. 6.46. Hence a

differential phase shift of 180° is again produced.

The solution for wave propagation in a circular guide with a longitudinal magnetized cylinder

placed in the center can be carried out exactly.t However, the solution requires a great deal of

algebraic manipulation, and it is very laborious to compute numerical values from the

resultant transcendental equations for the propagation constants. The solution does verify that

Faraday rotation takes place as would be expected, by analogy with propagation in an infinite

ferrite medium.

5.3 ISOLATOR

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The isolator is similar to the gyrator in construction except that it employs a 45° twist

section and 45° of Faraday rotation. In addition, thin resistive cards are inserted in the input

and output guides to absorb the field that is polarized, with the electric vector parallel to the

wide side of the guide, as shown in Fig. 6.47. The operation is as follows: A wave

propagating from port 1 to port 2 has its polarization rotated 45° counter clockwise by the

twist section and 45° clockwise by the Faraday rotator. It will emerge at port 2 with the

correct polarization to propagate in the output guide. A wave propagating from port 2 to port

1 will have its plane of polarization rotated by 90c and will enter the guide at port 1 with the

electric field parallel to the resistance card, and hence be absorbed. Without the resistance

card, the wave would be reflected from port 1 because of the incorrect

polarization, which cannot propagate in the guide constituting port 1. However, multiple

reflections within the isolator will lead to transmission in both directions, and this makes it

necessary to use resistance cards in both the input and output guides for satisfactory

performance. Typical performance figures for an isolator are forward transmission loss of

less than l dB, reverse attenuation of 20 to 30 dB, and bandwidth of operation approaching 10

percent-

5.4 OTHER FERRITE DEVICES

The devices utilizing ferrites for their operation described in the preced" sections

represent only a small number of the large variety of devices th-! have been developed. In

addition to the above, there are other forms of isolators, both reciprocal and nonreciprocal

phase shifters, electronically controlled (by varying the current in the electromagnet that

supplies the static biasing field) phase shifters and modulators, electronic switches and power

limiters, etc. The nonlinear property of ferrites for high signal levels has also been used in

harmonic generators, frequency mixers, and parametric amplifiers. A discussion of these

devices, together with design considerations, performance data, and references to the original

literature, contained in the book by Lax and Button, listed in the references at the end of this

chapter. The recent article by Rodriquez gives a good survey of the present status of ferrite

devices.

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Chapter 6

MICROWAVE TUBES

Microwave tubes are the prime signal sources in high-power radar systems. The

magnetron is the tube most frequently used and can provide many kilowatts of continuous-

wave (CW) output power and a megawatt or more of peak power with pulsed operation.

Magnetrons are also used for industrial heating applications and in microwave ovens for

consumer use. The traveling-wave-tube amplifier with power outputs up to 10 W or more is

the workhorse in satellite communications. The klystron tube can function as an oscillator or

as an amplifier. It can be designed for either low or high output power applications. In low-

power applications the klystron was once widely used as the local oscillator in microwave

receivers but has now been replaced by solid-state oscillators. Solid-state oscillators are

replacing n crowave tubes in many low-power transmitter applications also. Even thoug

many of the applications for microwave tubes have been taken over solid-state devices, the

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requirements for high power can only be me microwave tubes, so they are an essential device

for many systems.

Conventional low-frequency tubes, such as triodes, fail to operate at microwave

frequencies because the electron transit time from the cat the grid becomes an appreciable

fraction of the period of the sinu ^ signal to be amplified. In other words, propagation time

becomes short , and the same limitations that are inherent in low-frequencies are present in

low-frequency tubes also. Microwave tubes must be to utilize the wave-propagation

phenomena to best advantage.

Broadly speaking, there are two basic types of microwave tubes, one that employ

electromagnetic cavities (klystrons and some magnetron) and the other those that apply

those that employ slow-wave circuits (travelling-wave tubes)

Both types of tubes utilize an electron beam on which space-charge waves and cyclotron waves can be excited.

The space-charge waves are primarily longitudinal oscillations of the electrons and interact with the

electromagnetic fields in cavities and slow-wave circuits to give amplification.

6.1 Klystron Amplifier

A klystron is a specialized linear-beam vacuum tube (evacuated electron tube).

Klystrons are used as amplifiers at microwave and radio frequencies to produce both low-

power reference signals for super heterodyne radar receivers and to produce high-power

carrier waves for communications and the driving force for modern particle accelerators.

Klystron amplifiers have the advantage (over the magnetron) of coherently amplifying a

reference signal so its output may be precisely controlled in amplitude, frequency and phase.

Many klystrons have a waveguide for coupling microwave energy into and out of the device,

although it is also quite common for lower power and lower frequency klystrons to use

coaxial couplings instead. In some cases a coupling probe is used to couple the microwave

energy from a klystron into a separate external waveguide.

The name klystron comes from the stem form κλυσ- (klys) of a Greek verb referring to

the action of waves breaking against a shore, and the end of the word electron

Working

Klystrons amplify RF signals by converting the kinetic energy in a DC electron beam

into radio frequency power. A beam of electrons is produced by a thermionic cathode (a

heated pellet of low work function material), and accelerated by high-voltage electrodes

(typically in the tens of kilovolts). This beam is then passed through an input cavity. RF

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energy is fed into the input cavity at, or near, its natural frequency to produce a voltage which

acts on the electron beam. The electric field causes the electrons to bunch: electrons that pass

through during an opposing electric field are accelerated and later electrons are slowed,

causing the previously continuous electron beam to form bunches at the input frequency. To

reinforce the bunching, a klystron may contain additional "buncher" cavities. The RF current

carried by the beam will produce an RF magnetic field, and this will in turn excite a voltage

across the gap of subsequent resonant cavities. In the output cavity, the developed RF energy

is coupled out. The spent electron beam, with reduced energy, is captured in a collector.

6.2 Two Cavity Klystron

Fig 6.1 Two Cavity Klystron

In the two-chamber klystron, the electron beam is injected into a resonant cavity. The

electron beam, accelerated by a positive potential, is constrained to travel through a

cylindrical drift tube in a straight path by an axial magnetic field. While passing through the

first cavity, the electron beam is velocity modulated by the weak RF signal. In the moving

frame of the electron beam, the velocity modulation is equivalent to a plasma oscillation.

Plasma oscillations are rapid oscillations of the electron density in conducting media such

as plasmas or metals.(The frequency only depends weakly on the wavelength). So in a quarter

of one period of the plasma frequency, the velocity modulation is converted to density

modulation, i.e. bunches of electrons. As the bunched electrons enter the second chamber

they induce standing waves at the same frequency as the input signal. The signal induced in

the second chamber is much stronger than that in the first. The figure below is that of a two

cavity klystron amplifier.

6.3 Reflex klystron

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Fig 6.1 Reflex Klystron

In the reflex klystron (also known as a 'Sutton' klystron after its inventor), the electron beam

passes through a single resonant cavity. The electrons are fired into one end of the tube by

an electron gun. After passing through the resonant cavity they are reflected by a negatively

charged reflector electrode for another pass through the cavity, where they are then collected.

The electron beam is velocity modulated when it first passes through the cavity. The

formation of electron bunches takes place in the drift space between the reflector and the

cavity. The voltage on the reflector must be adjusted so that the bunching is at a maximum as

the electron beam re-enters the resonant cavity, thus ensuring a maximum of energy is

transferred from the electron beam to the RF oscillations in the cavity. The voltage should

always be switched on before providing the input to the reflex klystron as the whole function

of the reflex klystron would be destroyed if the supply is provided after the input. The

reflector voltage may be varied slightly from the optimum value, which results in some loss

of output power, but also in a variation in frequency. This effect is used to good advantage

for automatic frequency control in receivers, and in frequency modulation for transmitters.

The level of modulation applied for transmission is small enough that the power output

essentially remains constant. At regions far from the optimum voltage, no oscillations are

obtained at all. This tube is called a reflex klystron because it repels the input supply or

performs the opposite function of a klystron.

There are often several regions of reflector voltage where the reflex klystron will oscillate;

these are referred to as modes. The electronic tuning range of the reflex klystron is usually

referred to as the variation in frequency between half power points—the points in the

oscillating mode where the power output is half the maximum output in the mode. The

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frequency of oscillation is dependent on the reflector voltage, and varying this provides a

crude method of frequency modulating the oscillation frequency, albeit with accompanying

amplitude modulation as well.

Modern semiconductor technology has effectively replaced the reflex klystron in most

applications.

6.4 Cavity Magnetron

Fig 6.3 Magnetron

The cavity magnetron is a high-powered vacuum tube that

generates microwaves using the interaction of a stream of electrons with a magnetic field.

The 'resonant' cavity magnetron variant of the earlier magnetron tube was invented

by Randall and Boot in 1940. The high power of pulses from the cavity magnetron made

centimetre-band radar practical. Shorter wavelength radars allowed detection of smaller

objects. The compact cavity magnetron tube drastically reduced the size of radar sets so that

they could be installed in anti-submarine aircraft and escort ships. At present, cavity

magnetrons are commonly used in microwave ovens and in various radar applications.

The basic structure of a magnetron is a number of identical reason arranged in a

cylindrical pattern around a cylindrical cathode, as show, °*' Fig. 9.19. A permanent magnet

is used to produce a strong magnetic fi T normal to the cross section. The anode is kept at a

high positive voltage v relative to the cathode. Electrons emitted from the cathode are

accelerator toward the anode block, but the presence of the magnetic field BQ produce a force

~evrB„ in the azimuthall direction which causes the electron trajectory to be deflected in the

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same direction. If the cathode radius is a and the anode radius is 6, the potential at any radius

r is V(r) = V0Un(r/a)]/lln(b/a)].

6.5 OTHER TYPES OF MICROWAVE TUBES

In addition to the main types of microwave tubes already discussed the are a variety of

others as well. In one form of travelling-wave tube the resistance-wall amplifier, the helix is

replaced by a circular guide lined with a resistive material. The resistive lining enables a slow

wave to propagate in the guide, a wave that is highly attenuated in the absence of a beam If

an electron beam is present, amplification takes place with a growth constant aB large enough

to offset the attenuation due to the resistive lining. Thus a net overall amplification is

obtained,

In another form of travelling-wave tube, the double-stream amplifier. two parallel

electron beams are used. In this tube one of the beams provides the slow-wave structure, or

circuit, for the other beam.

It is also possible to amplify the space-charge waves directly by passing the beam

through a succession of accelerating and decelerating regions. This type of tube is called a

velocity-jump amplifier because the beam velocity v{l is periodically changed, or jumped, to

new values.

For both the O-type and M-type travelling-wave tubes, it is possible to adjust the beam

velocity so that it is equal to the phase velocity of any one of the spatial harmonics making up

the Bloch wave that can propagate along the periodic structure used for the slow-wave

circuit. In particular, interaction between the beam and one of the backward-propagating

spatial harmonics is possible

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Chapter 7

THE GUNN EFFECT AND ITS APPLICATIONS

7.1 The Gunn Effect

In some materials (III-V compounds such as GaAs and InP), after an electric field in

the material reaches a threshold level, the mobility of electrons decrease as the electric field

is increased, thereby producing negative resistance. A two-terminal device made from such a

material can produce microwave oscillations, the frequency of which is primarily determined

by the characteristics of the specimen of the material and not by any external circuit. The

Gunn Effect was discovered by J. B. Gunn of IBM in 1963.

7.2 The Gunn Diode

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In certain semiconductors, notably GaAs, electrons can exist in a high-mass low

velocity state as well as their normal low-mass high-velocity state and they can be forced

into the high-mass state by a steady electric field of sufficient strength. In this state they form

clusters or domains which cross the field at a constant rate causing current to flow as a series

of pulses. This is the Gunn effect and one form of diode which makes use of it consists of an

epitaxial layer of n-type GaAs grown on a GaAs substrate. A potential of a few volts applied

between ohmic contacts to the n-layer and substrate produces the electric field which causes

clusters. The frequency of the current pulses so generated depends on the transit time

through the n-layer and hence on its thickness. If the diode is mounted in a suitably tuned

cavity resonator, the current pulses cause oscillation by shock excitation and r.f. power up to

1 W at frequencies between 10 and 30 GHz is obtainable.

7.3 Gunn Diode Theory

The Gunn diode is a so-called transferred electron device. Electrons are transferred

from one valley in the conduction band to another valley. In order to understand the nature of

the transferred electron effect exhibited by Gunn diodes, it is necessary to consider the

electron drift velocity versus electric field (or current versus voltage) relationship for GaAs

(seeFigure 2). Below the threshold field, E th , of approximately 0.32 V/mm, the device acts

as a passive resistance. However, above E th the electron velocity (current) decreases as the

field (voltage) increases producing a region of negative differential mobility, NDM

(resistance, NDR). This is the essential feature that leads to current instabilities and Gunn

oscillations in an active device and is due to the special conductance band structure of direct

band gap semiconductors such as GaAs

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Fig 7.1 Gunn Effect

The energy-momentum relationship contains two conduction band energy levels, G

and L (also known as valleys) with the following properties:

l In the lower G valley, electrons exhibit a small effective mass and very high

mobility, µ 1 . l In the satellite L valley, electrons exhibit a large effective mass and very low

mobility, µ

2 . l The two valleys are separated by a small energy gap, D E, of approximately 0.31

eV. In equilibrium at room temperature most electrons reside near the bottom of the lower G

valley. Because of their high mobility (~ 8000 cm 2V -1s-1), they can readily be accelerated

in a strong electric field to energies in the order of the G -L intervalley separation, D E.

Electrons are then able to scatter into the satellite L valley, resulting in a decrease in the

average electron mobility, µ, as given below:

µ = (n 1 µ 1 + n 2 µ 2 ) / (n 1 + n 2 ( where n 1 = electron density in G valley, n 2 =

electron density in L valley

Above the high field, E H , most electrons reside in the L valley and the device

behaves as a passive resistance (of greater magnitude) once again. In a practical Gunn diode,

electrons are accelerated from the cathode by the prevailing electric field. When they have

acquired sufficient energy, they begin to scatter into the low mobility satellite valley and

slow down.

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The question of exactly how the NDR phenomenon in GaAs results in Gunn-

oscillations can now be answered with the aid of Figure 4. A sample of uniformly doped n-

type GaAs of length L is biased with a constant voltage source V=0

The electrical field is therefore constant and its magnitude given by E . 0 =V 0 /L.

From the bottom graph in Figure 4 it is clear that the electrons flow from cathode to anode

with constant velocity v3

7.4 Applications

Gunn diodes are reliable, relatively easy to install and the lower output power levels fall well

below the safety exposure limits. They are ideally suited for use in low noise sources such as

local oscillators, locking oscillators, low and medium power transmitter applications and

motion detection systems. Higher power varieties can be used in phase-locked oscillators or

as reflection amplifiers in point-to-point communication links and telemetry systems.

Microwave sources have the advantages over ultrasonic detectors of size and beam width,

and over optical systems of working in dusty and adverse environments. The low voltage

requirements of Gunn oscillators mean that battery or regulated mains supplies may be used,

(battery drain can be reduced by using low current devices or by operation in a pulsed

mode). However, microwaves are reflected from metal surfaces and partially reflected from

many others e.g. brick, Tarmac and concrete, and they are attenuated by oxygen, water or

water vapour. The range of application of Gunn sensors for industrial and commercial use is

extensive and the following is only a brief list:

Collision avoidance radar

Vehicle ABS

Traffic analyser sensors

`Blind spot' car radar

Pedestrian safety systems

Elapsed distance meters

Automatic identification

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Presence/absence indicators

Movement sensors

Distance measurements

Chapter 8

TRANSMISSION LINES AND CHARACTERISTIC IMPEDANCE

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Transmission line is any conducting structure that supports an electromagnetic wave

"in captivity". Most transmission lines use two conductors, where one is considered ground.

This includes coax (the outer conductor is ground), micro strip and strip line. The

transmission line that does not use a pair of conductors is waveguide. By the way, we are

talking about lossless transmission lines here, or at least near-lossless.

When microwave engineers talk about a "fifty-ohm system", what does that mean? A

common misconception is that if you placed an ohmmeter across the ground and conductor of

a fifty-ohm coax cable, you would always read 50 ohms. This is not the case, here's what

we're talking about: transmission lines have two important properties that depend on their

geometry, their inductance per unit length, and their capacitance per unit length. The

"characteristic impedance" of a system is calculated from the ratio of these two:

Z=sqrt(L'/C')

Where L' is the inductance per unit length and C' is the capacitance per unit length. Note that

higher inductance translates to higher impedance, and higher capacitance translates to lower

impedance. Notice also that the units of length don't matter, since they are "lost in the sauce".

The units of inductance and capacitance must be self-consistent, such as Pico-henries/foot

and Pico-farads/foot.

Let's start with coax cable. The inductance per unit length is mainly attributed to the

diameter of the centre conductor. Decrease this diameter (keeping everything else the same)

and you will increase the inductance. This also raises the characteristic impedance, referring

to the equation above. Filling the cable with a material of higher relative dielectric raises the

unit capacitance, and lowers the line impedance.

Another example: micro strip. Here unit capacitance and inductance are inexorably linked

together; widening the micro strip line decreases its inductance while it increases it

capacitance. Hence, wide lines are always lower in impedance than narrow lines for a given

substrate height. As with coax, the dielectric constant of the substrate has a big effect on

capacitance; using a higher dielectric substrate will yield a lower impedance line, all other

things being equal. So it is important not to mix up your Rogers Droid materials, once your

circuit is etched it is pretty hard to judge the dielectric constant from color and texture alone!

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8.1Impedance matching

Impedance matching of source and load is important to get maximum power transfer. If you

have a 75 ohm load, you don't want to drive it with a 50 ohm source, because it is inefficient.

You can learn more about the simple math behind maximum power transfer by clicking here.

Simple impedance transformation can be done using quarter wave transformers. Click here to

go to our main page on quarter-wave tricks!

8.2 Dielectric constant and effective dielectric constant

"Dielectric constant" is another way to say "relative permittivity". Check out our separate

page on permittivity for more info on this subject. Although some people use the phrase

"relative dielectric constant", this is incorrect, akin to saying "deja vu again".

Remember back to your physics class, when you learned that dielectric constant is used to

calculate the value of a capacitor? The higher the dielectric constant, the higher the capacitor

value. For an ideal parallel plate capacitor, the capacitance is calculated by:

C=( 0x RxA)/D

where 0 is the permittivity of free space (thanks, Maarten!), R is the relative permittivity

(the dielectric constant) of the material between the plates, A is the area of the parallel plates,

and D is the distance they are separated. Technically for this expression to be 100% accurate,

the material surrounding the plates must be of the same relative dielectric constant R, but

this induces only a small error in the calculation under most circumstances. 0 is equal to

8.854x10-12 Farads per meter (you should commit this to memory). Most often it is the

dielectric constant R that is most important in microwaves.

For electromagnetic radiation, the permittivity of the medium that the wave is propagating in

is equal to R 0. In a vacuum or in dry air, R is equal to unity, and the signal travels at the

speed of light. All electromagnetic energy, from 60 Hertz power that your electric company

sells you, to signals that the latest Mars satellite returns to earth, travels really, really fast. In a

vacuum, the speed of light, denoted "c" in textbooks, is 2.998 x 1010 centimeters/second

(thanks, Jared!) , or 2.998 x 108 meters per second, or about 186,000 miles per second, which

puts the moon about 1.5 seconds away by radio.

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The dielectric constant of a material can be used to quantify how much a material "slows" an

electromagnetic signal. The velocity of the signal within any transmission line that is 100%

filled with a material of dielectric constant R is computed by:

v=c/sqrt( R)

So if your strip line or coax transmission line is fabricated on a material with dielectric

constant 2.2, the velocity of propagation is only 67% of the speed of light in free space.

Similarly, because wavelength is proportional to velocity, the length of a quarter-wave

transformer is also 67% of what it would be in free space. Thus one of the tricks of reducing

the size of microwave components is revealed; by using materials of higher dielectric

constant, distributed structures can be made smaller. One of the advantages of using GaAs for

microwave ICs (known in the industry as MMICs) is its dielectric constant of 12.9, which is

appreciably higher than ceramics such as alumina, and most soft substrates.

A very good rule of thumb is that electromagnetic radiation in free space travels about

one foot in one nanosecond; a more exact value is 0.983571 feet per nanosecond. This slows

to about 8 inches per nanosecond for coax cables filled with PTFE (almost all coax cables are

filled with PTFE, or a combination of PTFE and air.) For more information please see our

discussion of group delay.

This brings us to the subject of "effective dielectric constant". In transmission lines

realized in micro strip media, most of the electric fields are constrained within the substrate,

but a fraction of the total energy exists within the air above the board. The effective dielectric

constant takes this into account. The effective dielectric constant of a fifty-ohm transmission

line on ten mil alumina is a number somewhere around 7, which is less than the dielectric

constant of the substrate bulk material (9.8). Another example of an effective dielectric

constant is if you were to create a strip line circuit using two sheets of substrates with

different dielectric constants. To a first order, the effective dielectric constant would be the

average of the two materials' dielectric constants. A third example is coplanar waveguide

transmission lines with air above the substrate. Here the effective dielectric constant is very

nearly the average of the substrate dielectric constant and one (the dielectric constant of

air=1). Thus the effective dielectric constant of CPW circuits on GaAs ( R=12.9) is

approximately

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8.3Lumped elements versus distributed elements

When the behaviour of a resistor, capacitor, or inductor can be fully described by a

simple linear equation, microwave engineers refer to it as a lumped element. For example, a

50-ohm resistor at low frequencies will obey Ohm's law (V=IxR). Put five volts across it and

it will draw 100 milliamps of current. "Lumped element hood" is restricted to components

that are operate at frequencies where they are physically much smaller than a quarter-

wavelength. For example, axial-leaded components perform well up to 10s of MHz, but at

one GHz, chances are that an axial-leaded resistor is closer to an open circuit, or a lousy

inductor, rather than an ideal resistor. This is why you will rarely be asked the resistor color

code as a microwave engineer!

At microwave frequencies, other factors must be considered. To accurately calculate the

behaviour of that same 50-ohm resistor, you need to consider its length, width, and thickness

of metal (due to the skin effect), and its proximity to the ground plane. This is when we must

consider it as a distributed element.

By designing really tiny parts, you can often consider them lumped elements, even at

microwave frequencies. You must keep the critical dimensions (such as length and width of a

thin-film resistor) small compared to an electrical quarter wavelength. For example, if you

are designing a 50 ohm micro strip load resistor at X-band, on an alumina substrate (dielectric

constant 9.8), a quarter wavelength is approximately 120 mils. You'd better keep both the

length and width of the resistor to less than 40 mils, or you else you have to spend some time

with a EDA simulation tool such as Agilent ADS or Eagleware Genesis evaluating the

performance. Where else but microwave engineering can you make a project out of designing

a stupid fifty-ohm resistor?!

At low frequencies, the metal that connects components together is treated as an ideal

connection, with no loss, no characteristic impedance, and no transmission phase angle.

When interconnects become an appreciable fraction of the signal wavelength, these

interconnections themselves must be treated as distributed elements or transmission lines. An

extreme example of the need to consider the distributed properties of transmission lines is

when we are dealing with a quarter-wavelength. At this electrical length (90 degrees), an

open circuit is transformed to a short circuit, and a short-circuit is transformed to an open

circuit! Think about this: a short-circuited 90 degree "stub" hanging in shunt off of a

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transmission line will be invisible to signals propagating down the transmission line, while an

open circuited 90 degree stub shunting a transmission line will cause a short circuit and the

propagating signal will get hosed! A whole lot of microwave engineering exploits this

concept, so you'd better understand it.

One "classic" distributed element is the quarter-wave transformer (we've written an entire

chapter on this and other quarter wave tricks! The quarter wave transformer is used to shift

the impedance of a circuit by the following simple formula:

Z2=sqrt(Z0ZL)

where Z2 is the characteristic impedance of the transformer, ZL is the load impedance, and Z0

is the characteristic impedance of the system you are trying to maintain. Do you detect a

pattern? Most of the equations on this page use the square-root function... perhaps they put

that button on your Casio calculator for a reason!

8.4 VSWR and return loss

VSWR stands for voltage standing wave ratio. It is a measure of how well a network is

matched to it's intended characteristic impedance (Z0), which is almost always 50 ohms in

microwave engineering. Return loss is just another way to express the same thing. Both are

used in microwave engineering, that's just to keep you on your toes.

VSWR dates back to the days when a "standing wave meter" was an important piece of lab

equipment. Long before you could buy s network analyzer for measuring how well a part is

impedance matched, the standing wave meter was used by engineers to evaluate the same

problem. A small probe was inserted into a waveguide, the output of which was rectified,

producing a current or voltage proportional to the electric field with the waveguide. The

engineer would pull the probe longitudinally along the waveguide, in search of local maxima

and minima readings. These are due to the standing wave within the transmission line. The

ratio of the maximum to the minimum voltage recorded was known as the voltage standing

wave ratio (VSWR). To this day VSWR is often used to quantify how well a part is

impedance matched. Always expressed as a ratio to unity, a VSWR of 1.0:1 indicates

perfection (there is no standing wave). A VSWR of 2:1 means the maxima are twice the

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voltage of the minima. A high VSWR such as 10:1 usually indicates you have a problem,

such as a near open or near short circuit.

Chapter 9

MICROWAVE MEASUREMENTS

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9.1Bandwidth

Bandwidth is a measure of how much spectrum your microwave system can respond to.

Bandwidth is often given in megahertz or gigahertz, calculated from from a low frequency FL

to an high frequency FH, the bandwidth is given by (FU-FL). Bandwidth is expressed a number

of other ways, which we will define here:

Three-dB bandwidth: for a network that has a non-ideal frequency response (which includes

all physical networks), the three-dB bandwidth is where the transmission coefficient S21 falls

off from its highest peak by three dB. Similarly, you could describe a network by its two-dB

or one-dB bandwidths.

Percentage bandwidth: for a system that works from a low frequency FL to an high

frequency FH, the percentage bandwidth is given by 100%x(FH-FL)/FC. FC is the center

frequency, equal to (FH+FL)/2. Note that it is possible to have more than 100% bandwidth by

this definition; an amplifier that works from 100 MHz to 10 GHz has a bandwidth of 200%.

Instantaneous bandwidth: this is a measure of how wide a spectrum a system can respond

to, without any type tuning. Using the analogy of radio, the IF bandwidth in an American FM

receiver is about 200 kHz, which is necessary to pass the full spectrum of a broadcast FM

signal. The demodulator processes this bandwidth to obtain the approximately 18 kHz

baseband bandwidth. The "dispreading" effect of this processing results in the superior signal

to noise ratio enjoyed by FM transmission. (Thanks for the correction, Miles!)

Tuneable bandwidth: tuneable bandwidth is a measure of how wide a spectrum a system

can respond to with the user allowed to change settings such as local oscillator frequency. For

a receiver, the tuneable bandwidth is almost always more than the instantaneous bandwidth.

An AM radio has a tuneable bandwidth of 540 kHz to 1600 kHz, or over one MHz of

bandwidth. This is about 100X its instantaneous bandwidth.

What does octave bandwidth mean? It implies that the the upper frequency of operation is

double the lower frequency of operation, for example, an amplifier that works from 2 to 4

GHz has one octave bandwidth. The origin of the word octave goes back to music theory,

where an octave is an interval of eight notes in the major scale. For reference, the interval

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from middle C to high C on a piano is an octave; high C is double the audio frequency of

middle C.

A device with an octave bandwidth always has 67% bandwidth (do the math for homework!)

9.2 Frequency conversion

A fundamental problem in electromagnetics is that for a signal to be radiated into free space,

an antenna must be on the order of 1/10 or more of a wavelength. Thus transmitting voice

without some type of upconversion would require a 30 kilometer antenna for a 10 kHz signal!

Thus, baseband signals need to ride on carrier waves, which are at RF and microwave

frequencies. Mixers are the devices that are used to convert from one frequency to another.

Upconversion means you are increasing the frequency of your signal, and downconversion

means you are decreasing it.

9.3 Harmonic frequencies

A harmonic frequency is 2X, 3X, 4X, etc. the frequency of a signal. Why is it called a

harmonic? Because in music, harmonic frequencies of 2X, 3X, 4X sound good together (they

are harmonious, like the Del Vikings). 2X and 4X frequencies are octaves, 3X is an octave

plus a perfect fifth.

A sub harmonic frequency is one that is 1/2, 1/3, 1/4 of a signal.

9.4 DECIBLES

This is simply the same logarithmic calculation but instead of comparing two power levels to

each other, you are comparing one power level to 1 mill watt. 10 dBm is the same at 10 mW,

20 dBm is the same as 100 mw, 30 dBm is the same as 1000 mw (or one watt).

How do you "think" in decibels compared to linear units? Just remember a few key

conversions and you will be all set to impress your friends with quick approximations of

some heavy multiplication and division (that is, if they are easily impressed). By the way, we

rounded these off so they will be easier to remember, if you need an exact answer, get a

calculator!

30 dB is an increase of 1000X in power

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20 dB is an increase of 100X in power

dB is an increase of 10X in power

6 dB is an increase of 4X in power

3 dB is an increase of 2X in power

2 dB is an increase of 1.6X in power

1 dB is an increase of 1.25X in power

0 dB is no increase or decrease in power

-1 dB is a decrease of 20% in power

-2 dB is a decrease of 37% in power (roughly a decrease of 1/3)

-3 dB is a decrease of 50% in power

-6 dB is a decrease of 75% in power

-10 dB is a decrease of 90% in power

-20 dB is a decrease of 99% in power

-30 dB is a decrease of 99.9% in power

When you input a 5 milliwatt signal into a power amplifier that has 12 dB of gain, the output

is 80mW. You can easily do the math in your head. Break down the 12 dB into 6 dB + 6 dB,

and remember that each 6 dB increases power by 4X, so you have an increase of 16X ( equal

to 4x4). Sixteen times five is eighty.

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III

DIGITAL

COMMUNICATION

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Chapter 1

BASICS OF DIGITAL COMMUNICATION

1.1 Introduction

In the simplest form, a transmission-reception system is a three-block system,

consisting of a) a transmitter, b) a transmission medium and c) a receiver. If we think of a

combination of the transmission device and reception device in the form of a ‘transceiver’

and if (as is usually the case) the transmission medium allows signal both ways, we are in a

position to think of a both-way (bi-directional) communication system. For ease of

description, we will discuss about a one-way transmission-reception system with the implicit

assumption that, once understood, the ideas can be utilized for developing / analyzing two-

way communication systems. So, our representative communication system, in a simple

form, again consists of three different entities, viz. a transmitter, a communication channel

and a receiver.

Fig 1.1

A digital communication system has several distinguishing features when compared with an

analog communication system. Both analog (such as voice signal) and digital signals (such as

data generated by computers) can be communicated over a digital transmission system. When

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the signal is analog in nature, an equivalent discrete-time-discrete-amplitude representation is

possible after the initial processing of sampling and quantization. So, both a digital signal and

a quantized analog signal are of similar type, i.e. discrete-time-discrete-amplitude signals.

A key feature of a digital communication system is that a sense of ‘information’, with

appropriate unit of measure, is associated with such signals. This visualization, credited to

Claude E. Shannon, leads to several interesting schematic description of a digital

communication system. For example, consider Fig.1.1 which shows the signal source at the

transmission end as an equivalent ‘Information Source’ and the receiving user as an

‘Information sink’. The overall purpose of the digital communication system is ‘to collect

information from the source and carry out necessary electronic signal processing such that the

information can be delivered to the end user (information sink) with acceptable quality’. One

may take note of the compromising phrase ‘acceptable quality’ and wonder why a digital

transmission system should not deliver exactly the same information to the sink as accepted

from the source. A broad and general answer to such query at this point is: well, it depends on

the designer’s understanding of the ‘channel’ (Fig. 1.1) and how the designer can translate his

knowledge to design the electronic signal processing algorithms / techniques in the ’Encoder’

and ‘decoder’ blocks in Fig. 1.1. We hope to pick up a few basic yet good approaches to

acquire the above skills. However, pioneering work in the 1940-s and 1950-s have

established a bottom-line to the search for ‘a flawless (equivalently, ‘error-less’) digital

communication system’ bringing out several profound theorems (which now go in the name

of Information Theory) to establish that, while error-less transmission of information can

never be guaranteed, any other ‘acceptable quality’, arbitrarily close to error-less

transmission may be possible. This ‘possibility’ of almost error-less information transmission

has driven significant research over the last five decades in multiple related areas such as, a)

digital modulation schemes, b) error control techniques, c) optimum receiver design, d)

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modelling and characterization of channel and so forth. As a result, varieties of digital

communication systems have been designed and put to use over the years and the overall

performance have improved significantly.

It is possible to expand our basic ‘three-entity’ description of a digital communication

system in multiple ways. For example, Fig. 1.1 shows a somewhat elaborate block diagram

explicitly showing the important processes of ‘modulation-demodulation’, ‘source coding-

decoding’ and ‘channel encoding – decoding’. A reader may have multiple queries relating to

this kind of abstraction. For example, when ‘information’ has to be sent over a large distance,

it is a common knowledge that the signal should be amplified in terms of power and then

launched into the physical transmission medium. Diagrams of the type in Figs. 1.1 and 1.2

have no explicit reference to such issues. However, the issue here is more of suitable

representation of a system for clarity rather than a module-by-module replication of an

operational digital communication system.

Fig 1.2

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Chapter 2

PULSE CODE MODULATION

Pulse-code modulation (PCM)

PCM is a method used to digitally represent sampled analogy signals, which was invented

by Alec Reeves in 1937. It is the standard form for digital audio in computers and

various Blu-ray, Compact Disc and DVD formats, as well as other uses such as

digital telephone systems. A PCM stream is a digital representation of an analog signal, in

which the magnitude of the analogue signal is sampled regularly at uniform intervals, with

each sample being quantized to the nearest value within a range of digital steps.

PCM streams have two basic properties that determine their fidelity to the original analog

signal: the sampling rate, which is the number of times per second that samples are taken; and

the bit depth, which determines the number of possible digital values that each sample can

take.

Fig 2.1 PCM Block Diagram

2.1Modulation

In the diagram, fig 2.2 a sine wave (red curve) is sampled and quantized for pulse

code modulation. The sine wave is sampled at regular intervals, shown as ticks on the x-axis.

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For each sample, one of the available values (ticks on the y-axis) is chosen by some

algorithm. This produces a fully discrete representation of the input signal (shaded area) that

can be easily encoded as digital data for storage or manipulation. For the sine wave example

at right, we can verify that the quantized values at the sampling moments are 7, 9, 11, 12, 13,

14, 14, 15, 15, 15, 14, etc. Encoding these values as binary numbers would result in the

following set of nibbles: 0111, 1001, 1011, 1100, 1101, 1110, 1110, 1111, 1111, 1111, 1110,

etc. These digital values could then be further processed or analyzed by a purpose-

specific digital signal processor or general purpose DSP. Several Pulse Code Modulation

streams could also be multiplexed into a larger aggregate data stream, generally for

transmission of multiple streams over a single physical link. One technique is called time-

division multiplexing, or TDM, and is widely used, notably in the modern public telephone

system. Another technique is called Frequency-division multiplexing, where the signal is

assigned a frequency in a spectrum, and transmitted along with other signals inside that

spectrum. Currently, TDM is much more widely used than FDM because of its natural

compatibility with digital communication, and generally lower bandwidth requirements.

There are many ways to implement a real device that performs this task. In real systems, such

a device is commonly implemented on a single integrated that lacks only the clock necessary

for sampling, and is generally referred to as an ADC (Analog-to-Digital converter). These

devices will produce on their output a binary representation of the input whenever they are

triggered by a clock signal, which would then be read by a processor of some sort.

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Fig 2.2

Demodulation

To produce output from the sampled data, the procedure of modulation is applied in

reverse. After each sampling period has passed, the next value is read and a signal is shifted

to the new value. As a result of these transitions, the signal will have a significant amount of

high-frequency energy. To smooth out the signal and remove these

undesirable aliasing frequencies, the signal would be passed through analog filters that

suppress energy outside the expected frequency range (that is, greater than the Nyquist

frequency fs / 2). Some systems use digital filtering to remove some of the aliasing,

converting the signal from digital to analog at a higher sample rate such that the analog filter

required for anti-aliasing is much simpler. In some systems, no explicit filtering is done at all;

as it's impossible for any system to reproduce a signal with infinite bandwidth, inherent losses

in the system compensate for the artifacts — or the system simply does not require much

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precision. Thesampling theorem suggests that practical PCM devices, provided a sampling

frequency that is sufficiently greater than that of the input signal, can operate without

introducing significant distortions within their designed frequency bands.

The electronics involved in producing an accurate analog signal from the discrete data are

similar to those used for generating the digital signal. These devices are DACs (digital-to-

analog converters), and operate similarly to ADCs. They produce on their output

a voltage or current (depending on type) that represents the value presented on their inputs.

This output would then generally be filtered and amplified for use.

2.2 Limitations

There are two sources of impairment implicit in any PCM system:

Choosing a discrete value near the analog signal for each sample leads to quantization

error, which swings between -q/2 and q/2. In the ideal case (with a fully linear ADC) it

is uniformly distributedover this interval, with zero mean and variance of q2/12.

Between samples no measurement of the signal is made; the sampling

theorem guarantees non-ambiguous representation and recovery of the signal only if it has

no energy at frequency fs/2 or higher (one half the sampling frequency, known as

the Nyquist frequency); higher frequencies will generally not be correctly represented or

recovered.

As samples are dependent on time, an accurate clock is required for accurate reproduction. If

either the encoding or decoding clock is not stable, its frequency drift will directly affect the

output quality of the device. A slight difference between the encoding and decoding clock

frequencies is not generally a major concern; a small constant error is not noticeable. Clock

error does become a major issue if the clock is not stable, however. A drifting clock, even

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with a relatively small error, will cause very obvious distortions in audio and video signals,

for example.

Extra information: PCM data from a master with a clock frequency that can not be influenced

requires an exact clock at the decoding side to ensure that all the data is used in a continuous

stream without buffer underrun or buffer overflow. Any frequency difference will be audible

at the output since the number of samples per time interval can not be correct. The data speed

in a compact disk can be steered by means of a servo that controls the rotation speed of the

disk; here the output clock is the master clock. For all "external master" systems like DAB

the output stream must be decoded with a regenerated and exact synchronous clock. When

the wanted output sample rate differs from the incoming data stream clock then a sample rate

converter must be inserted in the chain to convert the samples to the new clock domain.

Digitization as part of the PCM process

In conventional PCM, the analog signal may be processed (e.g., by amplitude compression)

before being digitized. Once the signal is digitized, the PCM signal is usually subjected to

further processing (e.g., digital data compression).

PCM with linear quantization is known as Linear PCM (LPCM).[1]

Some forms of PCM combine signal processing with coding. Older versions of these systems

applied the processing in the analog domain as part of the A/D process; newer

implementations do so in the digital domain. These simple techniques have been largely

rendered obsolete by modern transform-based audio compression techniques.

DPCM encodes the PCM values as differences between the current and the predicted

value. An algorithm predicts the next sample based on the previous samples, and the

encoder stores only the difference between this prediction and the actual value. If the

prediction is reasonable, fewer bits can be used to represent the same information. For

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audio, this type of encoding reduces the number of bits required per sample by about 25%

compared to PCM.

Adaptive DPCM (ADPCM) is a variant of DPCM that varies the size of the

quantization step, to allow further reduction of the required bandwidth for a given signal-

to-noise ratio.

Delta modulation is a form of DPCM which uses one bit per sample.

In telephony, a standard audio signal for a single phone call is encoded as 8,000 analog

samples per second, of 8 bits each, giving a 64 kbit/s digital signal known as DS0. The

default signal compression encoding on a DS0 is either μ-law (mu-law) PCM (North America

and Japan) or A-law PCM (Europe and most of the rest of the world). These are logarithmic

compression systems where a 12 or 13-bit linear PCM sample number is mapped into an 8-bit

value. This system is described by international standard G.711. An alternative proposal for

a floating point representation, with 5-bit mantissa and 3-bit radix, was abandoned.

Where circuit costs are high and loss of voice quality is acceptable, it sometimes makes sense

to compress the voice signal even further. An ADPCM algorithm is used to map a series of 8-

bit µ-law or A-law PCM samples into a series of 4-bit ADPCM samples. In this way, the

capacity of the line is doubled. The technique is detailed in the G.726 standard.

Later it was found that even further compression was possible and additional standards were

published. Some of these international standards describe systems and ideas which are

covered by privately owned patents and thus use of these standards requires payments to the

patent holders.

Some ADPCM techniques are used in Voice over IP communications.

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Chapter 3

DIGITAL MODULATION TECHNIQUES I

There are three major classes of digital modulation techniques used for transmission

of digitally represented data:

Amplitude-shift keying  (ASK)

Frequency-shift keying  (FSK)

Phase-shift keying (PSK)

All convey data by changing some aspect of a base signal, the carrier wave (usually

a sinusoid), in response to a data signal. In the case of PSK, the phase is changed to represent

the data signal. There are two fundamental ways of utilizing the phase of a signal in this way:

By viewing the phase itself as conveying the information, in which case

the demodulator must have a reference signal to compare the received signal's phase

against; or

By viewing the change in the phase as conveying information

— differential schemes, some of which do not need a reference carrier (to a certain

extent).

A convenient way to represent PSK schemes is on a constellation diagram. This shows the

points in the Argand plane where, in this context, the real and imaginary axes are termed the

in-phase and quadrature axes respectively due to their 90° separation. Such a representation

on perpendicular axes lends itself to straightforward implementation. The amplitude of each

point along the in-phase axis is used to modulate a cosine (or sine) wave and the amplitude

along the quadrature axis to modulate a sine (or cosine) wave.

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In PSK, the constellation points chosen are usually positioned with uniform angular spacing

around a circle. This gives maximum phase-separation between adjacent points and thus the

best immunity to corruption. They are positioned on a circle so that they can all be

transmitted with the same energy. In this way, the moduli of the complex numbers they

represent will be the same and thus so will the amplitudes needed for the cosine and sine

waves. Two common examples are "binary phase-shift keying" (BPSK) which uses two

phases, and "quadrature phase-shift keying" (QPSK) which uses four phases, although any

number of phases may be used. Since the data to be conveyed are usually binary, the PSK

scheme is usually designed with the number of constellation points being a powerof 2.

3.1 Amplitude-shift keying (ASK) is a form of modulation that represents digital data as

variations in the amplitude of a carrier wave.

The amplitude of an analog carrier signal varies in accordance with the bit stream

(modulating signal), keeping frequency and phase constant. The level of amplitude can be

used to represent binary logic 0s and 1s. We can think of a carrier signal as an ON or OFF

switch. In the modulated signal, logic 0 is represented by the absence of a carrier, thus giving

OFF/ON keying operation and hence the name given.

Like AM, ASK is also linear and sensitive to atmospheric noise, distortions, propagation

conditions on different routes in PSTN, etc. Both ASK modulation and demodulation

processes are relatively inexpensive. The ASK technique is also commonly used to

transmit digital data over optical fiber. For LED transmitters, binary 1 is represented by a

short pulse of light and binary 0 by the absence of light. Laser transmitters normally have a

fixed "bias" current that causes the device to emit a low light level. This low level represents

binary 0, while a higher-amplitude lightwave represents binary 1.

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3.2 Phase-shift keying (PSK) is a digital modulation scheme that conveys data by changing,

or modulating, the phase of a reference signal (the carrier wave).

Any digital modulation scheme uses a finite number of distinct signals to represent digital

data. PSK uses a finite number of phases, each assigned a unique pattern of binary digits.

Usually, each phase encodes an equal number of bits. Each pattern of bits forms

the symbol that is represented by the particular phase. The demodulator, which is designed

specifically for the symbol-set used by the modulator, determines the phase of the received

signal and maps it back to the symbol it represents, thus recovering the original data. This

requires the receiver to be able to compare the phase of the received signal to a reference

signal — such a system is termed coherent (and referred to as CPSK).

Alternatively, instead of using the bit patterns to set the phase of the wave, it can instead be

used to change it by a specified amount. The demodulator then determines the changes in the

phase of the received signal rather than the phase itself. Since this scheme depends on the

difference between successive phases, it is termed differential phase-shift keying (DPSK).

DPSK can be significantly simpler to implement than ordinary PSK since there is no need for

the demodulator to have a copy of the reference signal to determine the exact phase of the

received signal (it is a non-coherent scheme). In exchange, it produces more erroneous

demodulations. The exact requirements of the particular scenario under consideration

determine which scheme is used.

3.3 Differential Phase-shift keying (DPSK) is a digital modulation scheme that

conveys data by changing, or modulating, the phase of a reference signal (the carrier wave).

Any digital modulation scheme uses a finite number of distinct signals to represent digital

data. PSK uses a finite number of phases, each assigned a unique pattern of binary digits.

Usually, each phase encodes an equal number of bits. Each pattern of bits forms

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the symbol that is represented by the particular phase. The demodulator, which is designed

specifically for the symbol-set used by the modulator, determines the phase of the received

signal and maps it back to the symbol it represents, thus recovering the original data. This

requires the receiver to be able to compare the phase of the received signal to a reference

signal — such a system is termed coherent (and referred to as CPSK).

Alternatively, instead of using the bit patterns to set the phase of the wave, it can instead be

used to change it by a specified amount. The demodulator then determines the changes in the

phase of the received signal rather than the phase itself. Since this scheme depends on the

difference between successive phases, it is termed differential phase-shift keying (DPSK).

DPSK can be significantly simpler to implement than ordinary PSK since there is no need for

the demodulator to have a copy of the reference signal to determine the exact phase of the

received signal (it is a non-coherent scheme). In exchange, it produces more erroneous

demodulations. The exact requirements of the particular scenario under consideration

determine which scheme is used.

3.4 BPSK(Binary Phase shift keying)

BPSK (also sometimes called PRK, Phase Reversal Keying, or 2PSK) is the simplest form of

phase shift keying (PSK). It uses two phases which are separated by 180° and so can also be

termed 2-PSK. It does not particularly matter exactly where the constellation points are

positioned, and in this figure they are shown on the real axis, at 0° and 180°. This modulation

is the most robust of all the PSKs since it takes the highest level of noise or distortion to

make the demodulatorreach an incorrect decision. It is, however, only able to modulate at 1

bit/symbol (as seen in the figure) and so is unsuitable for high data-rate applications when

bandwidth is limited.

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In the presence of an arbitrary phase-shift introduced by the communications channel, the

demodulator is unable to tell which constellation point is which. As a result, the data is

often differentially encoded prior to modulation.

3.5 QPSK

Sometimes this is known as quaternary PSK, quadriphase PSK, 4-PSK, or 4-QAM.

(Although the root concepts of QPSK and 4-QAM are different, the resulting modulated radio

waves are exactly the same.) QPSK uses four points on the constellation diagram, equispaced

around a circle. With four phases, QPSK can encode two bits per symbol, shown in the

diagram with gray coding to minimize the bit error rate (BER) — sometimes misperceived as

twice the BER of BPSK.

The mathematical analysis shows that QPSK can be used either to double the data rate

compared with a BPSK system while maintaining the same bandwidthof the signal, or

to maintain the data-rate of BPSK but halving the bandwidth needed. In this latter case, the

BER of QPSK is exactly the same as the BER of BPSK - and deciding differently is a

common confusion when considering or describing QPSK.

Given that radio communication channels are allocated by agencies such as the Federal

Communication Commission giving a prescribed (maximum) bandwidth, the advantage of

QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth

than BPSK does - at the same BER. The engineering penalty that is paid is that QPSK

transmitters and receivers are more complicated than the ones for BPSK. However, with

modern electronicstechnology, the penalty in cost is very moderate.

As with BPSK, there are phase ambiguity problems at the receiving end, and differentially

encoded QPSK is often used in practice.

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Fig 3.1 QPSK Transmitter

Fig 3.2 QPSK Reciever

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3.6 Frequency Shift Keying - FSK

The two binary states, logic 0 (low) and 1 (high), are each represented by an analogue

waveform. Logic 0 is represented by a wave at a specific frequency, and logic 1 is

represented by a wave at a different frequency. 

.

Fig 3.2 FSK Representation

With binary FSK, the centre or carrier frequency is shifted by the binary input data.

Thus the input and output rates of change are equal and therefore the bit rate and baud

rateequal. 

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Chapter 4

DELTA MODULATION

Delta modulation (DM or Δ-modulation) is an analog-to-digital and digital-to-analog

signal conversion technique used for transmission of voice information where quality is not

of primary importance. DM is the simplest form of differential pulse-code

modulation (DPCM) where the difference between successive samples is encoded into n-bit

data streams. In delta modulation, the transmitted data is reduced to a 1-bit data stream.

Its main features are:

the analog signal is approximated with a series of segments

each segment of the approximated signal is compared to the original analog wave

to determine the increase or decrease in relative amplitude

the decision process for establishing the state of successive bits is determined by

this comparison

only the change of information is sent, that is, only an increase or decrease of the

signal amplitude from the previous sample is sent whereas a no-change condition

causes the modulated signal to remain at the same 0 or 1 state of the previous

sample.

To achieve high signal-to-noise ratio, delta modulation must use oversampling techniques,

that is, the analog signal is sampled at a rate several times higher than the Nyquist rate.

Derived forms of delta modulation are continuously variable slope delta modulation, delta-

sigma modulation, and differential modulation. The Differential Pulse Code Modulation is

the super set of DM. The block diagram of Delta modulation is given below in diagram 4.1

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Fig 4.1 Block Diagram of Delta modulation

4.1Principle

Rather than quantizing the absolute value of the input analog waveform, delta

modulation quantizes the difference between the current and the previous step, as shown in

the block diagram in Fig. 4.1

The modulator is made by a quantizer which converts the difference between the

input signal and the average of the previous steps. In its simplest form, the quantizer can be

realized with a comparator referenced to 0 (two levels quantizer), whose output is 1 or 0 if the

input signal is positive or negative. It is also a bit-quantizer as it quantizes only a bit at a time.

The demodulator is simply an integrator (like the one in the feedback loop) whose output

rises or falls with each 1 or 0 received. The integrator itself constitutes a low-pass filter.

4.2 Adaptive delta modulation

Adaptive delta modulation (ADM) or continuously variable slope delta modulation (CVSD)

is a modification of DM in which the step size is not fixed. Rather, when several consecutive

bits have the same direction value, the encoder and decoder assume that slope overload is

occurring, and the step size becomes progressively larger. Otherwise, the step size becomes

gradually smaller over time. ADM reduces slope error,at the expense of increasing quantizing

error.This error can be reduced by using a low pass filter.

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Chapter 5

INFORMATION THEORY

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Coding theory is one of the most important and direct applications of information theory. It

can be subdivided into source coding theory and channel codingtheory. Using a statistical

description for data, information theory quantifies the number of bits needed to describe the

data, which is the information entropy of the source.

Data compression (source coding): There are two formulations for the compression

problem:

1. lossless data compression : the data must be reconstructed exactly;

2. lossy data compression : allocates bits needed to reconstruct the data, within a

specified fidelity level measured by a distortion function. This subset of Information

theory is called rate–distortion theory.

Error-correcting codes (channel coding): While data compression removes as

much redundancy as possible, an error correcting code adds just the right kind of

redundancy (i.e., error correction) needed to transmit the data efficiently and faithfully

across a noisy channel.

This division of coding theory into compression and transmission is justified by the

information transmission theorems, or source–channel separation theorems that justify the

use of bits as the universal currency for information in many contexts. However, these

theorems only hold in the situation where one transmitting user wishes to communicate to

one receiving user. In scenarios with more than one transmitter (the multiple-access channel),

more than one receiver (the broadcast channel) or intermediary "helpers" (the relay channel),

or more general networks, compression followed by transmission may no longer be

optimal. Network information theory refers to these multi-agent communication models.

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5.1 Source theory

Any process that generates successive messages can be considered a source of information. A

memoryless source is one in which each message is an independent identically-distributed

random variable, whereas the properties of ergodicity and stationarity impose more general

constraints. All such sources are stochastic. These terms are well studied in their own right

outside information theory.

5.2 Information Rate

Information rate is the average entropy per symbol. For memory less sources, this is merely

the entropy of each symbol

5.3Channel Capacity

Communications over a channel—such as an Ethernet cable—is the primary motivation of

information theory. As anyone who's ever used a telephone (mobile or landline) knows,

however, such channels often fail to produce exact reconstruction of a signal; noise, periods

of silence, and other forms of signal corruption often degrade quality. How much information

can one hope to communicate over a noisy (or otherwise imperfect) channel?

Consider the communications process over a discrete channel. A simple model of the process

is shown below:in fig 5.1

Fig 5.1 Channel Capacity

Here X represents the space of messages transmitted, and Y the space of messages received

during a unit time over our channel. Let p(y | x) be the conditional probability distribution

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function of Ygiven X. We will consider p(y | x) to be an inherent fixed property of our

communications channel (representing the nature of the noise of our channel). Then the joint

distribution of X and Y is completely determined by our channel and by our choice of f(x), the

marginal distribution of messages we choose to send over the channel. Under these

constraints, we would like to maximize the rate of information, or the signal, we can

communicate over the channel. The appropriate measure for this is the mutual information,

and this maximum mutual information is called the capacity and is given by:

Chapter 6

SHANNON – HARTLEY THEORM

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In information theory, the Shannon–Hartley theorem (also known as Shannon's law)

is an application of the noisy channel coding theorem to the archetypal case of a continuous-

time analog communications channel subject to Gaussian noise. The theorem establishes

Shannon's channel capacity for such a communication link, a bound on the maximum amount

of error-free digital data (that is, information) that can be transmitted with a

specified bandwidth in the presence of the noise interference, assuming (a) the signal power

is bounded; (b)the Gaussian noise process is characterized by a known power or power

spectral density. The law is named after Claude Shannon and Ralph Hartley.

6.1Statement of the theorem

Considering all possible multi-level and multi-phase encoding techniques, the

Shannon–Hartley theorem states the channel capacity C, meaning the theoretical tightest

upper bound on the information rate (excluding error correcting codes) of clean (or arbitrarily

low bit error rate) data that can be sent with a given average signal power S through an analog

communication channel subject to additive white Gaussian noise of power N, is:

where

C is the channel capacity in bits per second;

B is the bandwidth of the channel in hertz (passband bandwidth in case of a

modulated signal);

S is the total received signal power over the bandwidth (in case of a modulated signal,

often denoted C, i.e. modulated carrier), measured in watt or volt2;

N is the total noise or interference power over the bandwidth, measured in watt or

volt2; and

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S/N is the signal-to-noise ratio (SNR) or the carrier-to-noise ratio (CNR) of the

communication signal to the Gaussian noise interference expressed as a linear power

ratio (not as logarithmic decibels).

6.2 Nyquist rate

In 1927, Nyquist determined that the number of independent pulses that could be put through

a telegraph channel per unit time is limited to twice the bandwidth of the channel. In symbols,

where fp is the pulse frequency (in pulses per second) and B is the bandwidth (in hertz). The

quantity 2B later came to be called the Nyquist rate, and transmitting at the limiting pulse

rate of 2B pulses per second as signalling at the Nyquist rate. Nyquist published his results in

1928 as part of his paper "Certain topics in Telegraph Transmission Theory."

6.3 Noisy channel coding theorem and capacity

Claude Shannon's development of information theory during World War II provided

the next big step in understanding how much information could be reliably communicated

through noisy channels. Building on Hartley's foundation, Shannon's noisy channel coding

theorem (1948) describes the maximum possible efficiency of error-correcting

methods versus levels of noise interference and data corruption.[5][6] The proof of the theorem

shows that a randomly constructed error correcting code is essentially as good as the best

possible code; the theorem is proved through the statistics of such random codes.

Shannon's theorem shows how to compute a channel capacity from a statistical description of

a channel, and establishes that given a noisy channel with capacity C and information

transmitted at a line rate R, then if

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there exists a coding technique which allows the probability of error at the receiver to be

made arbitrarily small. This means that theoretically, it is possible to transmit information

nearly without error up to nearly a limit of C bits per second.

The converse is also important. If

the probability of error at the receiver increases without bound as the rate is increased. So no

useful information can be transmitted beyond the channel capacity. The theorem does not

address the rare situation in which rate and capacity are equal.

6.4 Shannon–Hartley theorem

The Shannon–Hartley theorem establishes what that channel capacity is for a finite-

bandwidth continuous-time channel subject to Gaussian noise. It connects Hartley's result

with Shannon's channel capacity theorem in a form that is equivalent to specifying the M in

Hartley's line rate formula in terms of a signal-to-noise ratio, but achieving reliability through

error-correction coding rather than through reliably distinguishable pulse levels.

If there were such a thing as an infinite-bandwidth, noise-free analog channel, one could

transmit unlimited amounts of error-free data over it per unit of time. Real channels,

however, are subject to limitations imposed by both finite bandwidth and nonzero noise.

So how do bandwidth and noise affect the rate at which information can be transmitted over

an analog channel?

Surprisingly, bandwidth limitations alone do not impose a cap on maximum information rate.

This is because it is still possible for the signal to take on an indefinitely large number of

different voltage levels on each symbol pulse, with each slightly different level being

assigned a different meaning or bit sequence. If we combine both noise and bandwidth

limitations, however, we do find there is a limit to the amount of information that can be

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transferred by a signal of a bounded power, even when clever multi-level encoding

techniques are used.

In the channel considered by the Shannon-Hartley theorem, noise and signal are combined by

addition. That is, the receiver measures a signal that is equal to the sum of the signal

encoding the desired information and a continuous random variable that represents the noise.

This addition creates uncertainty as to the original signal's value. If the receiver has some

information about the random process that generates the noise, one can in principle recover

the information in the original signal by considering all possible states of the noise process. In

the case of the Shannon-Hartley theorem, the noise is assumed to be generated by a Gaussian

process with a known variance. Since the variance of a Gaussian process is equivalent to its

power, it is conventional to call this variance the noise power.

Such a channel is called the Additive White Gaussian Noise channel, because Gaussian noise

is added to the signal; "white" means equal amounts of noise at all frequencies within the

channel bandwidth. Such noise can arise both from random sources of energy and also from

coding and measurement error at the sender and receiver respectively. Since sums of

independent Gaussian random variables are themselves Gaussian random variables, this

conveniently simplifies analysis, if one assumes that such error sources are also Gaussian and

independent.

Chapter 7

Linear Block Codes

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In coding theory, a linear code is an error-correcting code for which any linear

combination of codewords is another codeword of the code. Linear codes are traditionally

partitioned into block codes andconvolutional codes, although Turbo codes can be seen as a

hybrid of these two types[1]. Linear codes allow for more efficient encoding and decoding

algorithms than other codes (cf. syndrome decoding).

Linear codes are used in forward error correction and are applied in methods for transmitting

symbols (e.g., bits) on a communications channel so that, if errors occur in the

communication, some errors can be detected by the recipient of a message block. The "codes"

in a linear block code are blocks of symbols which are encoded using more symbols than the

original value to be sent. A linear code of length n transmits blocks containing n symbols. For

example, the "(7,4)" Hamming code is a linear binary code which represents 4-bit values each

using 7-bit values. In this way, the recipient can detect errors as severe as 2 bits per block.

[2] As there are 16 distinct 4-bit values expressed in binary, the size of the (7,4) Hamming

code is sixteen.

7.1 Formal definition

A linear code of length n and rank k is a linear subspace C with dimension k of

the vector space   where   is the finite field with q elements. Such a code with

parameter q is called a q-ary code (e.g., when q = 5, the code is a 5-ary code). If q = 2

or q = 3, the code is described as a binary code, or a ternary code respectively.

7.2Generator matrix and parity check matrix

Because the linear code could be considered as a linear subspace C of   (and therefore a

codeword is a vector in this linear subspace), any codeword   could be represented as a

linear combination of a set of basis vectors   such

that 

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,

where   is the message and   is thegenerator matrix.

On another hand, for any linear subspace  , there is a dimension n − k null

space   such that  . The basis vectors of the null

space  form another matrix   such that  , where   is

called parity check matrix.

7.2Hamming codes

As the first class of linear codes developed for error correction purpose, famous Hamming

codes has been widely used in digital communication systems. For any positive

integer  , there exists a [2r − 1,2r − r − 1,3]2 Hamming code. Since d = 3, this Hamming

code can correct 1-bit error.

Example : The linear block code with the following generator matrix and parity check matrix

is a [7,4,3]2 Hamming code.

  : 

7.3 Hadamard codes

Hadamard code is a [2r,r,2r – 1]2 linear code and is capable of correcting many errors.

Hadamard code could be constructed column by column : the ith column is the bits of the

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binary representation of integer I, as shown in the following example. Hadamard code has

minimum distance 2r – 1 and therefore can correct 2r – 2 – 1 errors.

Example : The linear block code with the following generator matrix is a [8,3,4]2 Hadamard

code:  .

Hadamard code is a special case of Reed-Muller code If we take the first column (the all-zero

column) out from  , we get [7,3,4]2 simplex code, which is the dual code of Hamming

code. Let  be the parity check matrix of C, then the code generated by   is called

the dual code of C.

Reference

1. W.C.Y. Lee, Mobile cellular communications, Tata McGraw Hill, 2nd

Edition, 2006.

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2. Gordon L.stuber, Principles Of Mobile Communications, Springer

international 2nd edition,2007

3. Samuel Y.Liao - Microwave Devices And Circuits, 3rd Edition 1994.

4. Herbert J.Reich, J.G Skalnik, P.F.Ordung And H.L. Krauss – Microwave

Principles, CBS Publishers And Distributors, New Delhi,2004

5. Simon Haykin, John Wiley – Digital Communication,2005

6. H.Tabu And D.Schilling – Principles Of Communication Systems, THM,

2003

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