of - Cranfield University

11111111111111111111111111111111111111111 1401183931

CRANFIELD INSTITUTE OF TECHNOLOGY

DEPERTMENT OF ELECTRONIC SYSTEM DESIGN

PhD Thesis

Academic year 1989-1990

ALI ABBAS ALI

THEORY AND ASSESSMENT

OF AN

IMPROVED POWER SPECTRAL DENSITY ESTIMATOR

Supervisor : Prof. H.W.Loeb

June 1990

This Thesis is submitted in partial submission for the

degree of Doctor of Philosophy

To my wife Eman

my two dOti8hters Hind and Ra8ad

and my boy Ahmed

wi th love.

ABSTRACT

This thesis is concerned with the processing of time

domain signals received by a single sensor. An example of

such signals is the radar return, which is used in one way

or another to estimate the power spectral density a

frequency representation of the power of the signal in

order that we can pick up and track the moving targets.

since the POWER SPECTRAL DENSITY ESTIMATION is a fundamental

tool in digital signal processing, the theory of the

different approaches to PSDE is given in the Literature

review chapter.

The aim of this research is to develop a technique for

the Power Spectral Density Estimation (PSDE) of multiple

signals in white noise, which has high resolution capability

and less frequency estimation errors. Hence, the various

techniques mentioned above are tested for their detection,

resolution capabilities and performance.

Finally the different parameters affecting the resolution

and detection capabilities of the Eigen Vector Decomposition

Techniques (EVDT) for PSDE are studied in some depth.

ACKNOWLEDGEMENT

All the praises and thanks be to GOD,

beneficient, the most merciful.

the most

I would like to extend my sincere gratitude to my

supervisor Prof. H.W.LOEB for his constant advice and

encouragement during this research.

My thanks also to the Government and People of the

Republic of Iraq for the financial support during my stay in

England.

Many thanks also to the staff of the Department of

Electronic System Design and in particular to Dr. D.Russell

for reading the draft of this thesis, Mr. K.Bowdler for his

help with the mathematics and Mrs. M.B.Shield, Mrs.

J.S.Hornsby and Mr. G.M.Young for their help in getting the

laser copy of this thesis.

Finally I would like to thank my beloved wife Eman and my

lovely childrens Hind, Ragad and Ahmed for whom I own my

life.

ak

bk

a k ... app u~ e pn b pn B

EV

B WEV

B SEV

CIF

E[. ]

Ep f

I

Ln N

T

P(f)

Rxx(m)

Rxx

Rxx(llk)

x(t)

xn X(f)

Xm(f)

wn ~ (t)

~t

~f

w

LIST OF SYMBOLS

AR model parameters.

MA model parameters.

Prediction parameters.

Prediction error power.

Forward prediction parameters.

Backward prediction parameters.

Covariance Matrix with unity eigen values.

NCM with unity eigen values.

SCM with unity eigen values.

Frequency search vector.

Expectation operator.

Prediction error energy.

Linear frequency.

Identity matrix.

Symmetric data window.

Number of data samples.

Observation length; T=N~t.

PSD.

Autocorrelation function.

Covariance matrix.

Element of the covariance matrix.

Deterministic analoge signal.

Discrete (sampled) signal.

Fourier Transform of x(t).

Discrete Fourier Transform.

White Gaussian noise.

Dirac Delta function.

Sampling Interval.

Frequency separation.

Radian frequency.

AIC

AN

ANDGS

AR

ARMA

ASP

BT

CM

CN

DFT

DGS

DOA

DSP

EV

EVDM

EVDT

EVM

FIR

FT

FFT

FRL

GFC

GLDE

GTF

ICM

IIR

LDE

MA

HDL

HEM

HL

LIST OF ABREVIATIONS

Akaike Criterion.

Additive Noise.

Additive Noise DGS.

Auto Regressive.

Auto Regressive Moving Average.

Array Signal Processing.

Blackman-Tukey.

Covariance Matrix.

Convolutive Noise.

Discrete FT.

Data Generating System.

Direction Of Arrival.

Digital Signal Processing.

Eigen Values.

Eigen Vector Decomposition Method.

Eigen Vector Decomposition Technique.

Eigen Vector Hethod.

Finite Impulse response.

Fourier Transform.

Fast FT.

Fourier Resolution Limit.

General Form Criterion.

General LDE.

General Transfer Function.

Inverse Covariance Hatrix.

Infinite Impulse Response.

Linear Difference Equation.

Moving Average.

Schwartz and Ressanen Criterion.

Maximum Entropy Method.

Maximum Likelihood.

MLH

MLSE

MPCM

HUSIC

NCM

NDDGS

NEV

NFAP

NICM

NSS

PC

PCM

PER

PHD

PICM

PMFT

PSD

PSDE

SEV

SCM

SICM

SLF

SLS

SOT

SSP

TCM

TEV

TICM

Maximum Likelihood Method.

Maximum Likelihood Spectral Estimation.

Modified PCM.

Multiple Signal Classification.

Noise CM.

Noise Driven DGS.

Noise EV.

Number of Free Adjustable Parameters.

Inverse NCM.

Noise Subspace.

Principal Components.

Principal Components Method.

Periodogram.

Pisarenko Harmonic Decomposition.

Pover Inverse Constraint Method.

Parametric Model Fitting Technique.

Pover Spectral Density.

PSD Estimation.

Signal EV.

Signal CM.

Inver se SCM.

Source Location Finding.

Spontaneous Line Splitting.

Sub-Optimum Technique.

Signal Subspace.

Total CM.

Total EV.

Inverse TCM.

****************************************************************** : NOTE: The folloving vords and abreviations vhich appeared in S * * : the text and many figures are supposed to stand for: :

! 1) Exact Covariance Matrix for True Covariance Matrix. S * * : 2) Simulated Covar. Matrix for Estimated Covar. Matrix.~

* * * 3) EVDM for EVM. * * * * * * 4) TCM for CM. ~ :*****************************************************************

CONTENTS

I . ABSTRACT.

II. ACKNOWLEDGMENT.

III. LIST OF SYMBOLS.

IV. LIST OF ABREVIATIONS.

1.

2.

CHAPTER ONE: INTRODUCTION

1. 1. PROBLEM FORMULATION AND SOLUTION.

1.2. THESIS LAY OUT.

CHAPTER TWO: LITERATURE REVIEW

2.1. INTRODUCTION.

2.2. CONVENTIONAL PSDE mehods.

2.3. PARAMETRIC PSDE methods.

2.3.1. Modelling techniques:

PAGE

1-2

1-4

2-1

2-4

2-8

2-10

2.3.1.1. Auto Regressive Moving Average

model. 2-10

2.3.1.2. Auto Regressive model. 2-13

2.3.1.3. Moving Average model. 2-14

2.3.2. Estimation of the model spectra . 2-14 . 2.3.2.1. AR spectra. 2-15

2.3.2.2. ARMA spectra. 2-29

2.3.2.3. MA spectra. 2-33

2.3.2.4. Prony's method. 2-34

2.4. NON PARAMETRIC POWER SPECTRAL DENSITY ESTIMATION

methods. 2-39

2.4.1. Pisarenko Harmonic Decomposition

method. 2-40

2.4.2. MLM Spectral Estimation method. 2-44

3.

4.

2.4.3. Eigen Vector Eigen Value Decomposition

Techniques : 2-46

2.4.3.1. Principal Components method. 2-46

2.4.3.2. MUSIC Algorithim method. 2-50

2.4.3.3. Eigen Vector method. 2-50

2.5. MULTIDIMENSIONAL SPECTRAL ESTIMATION. 2-51

CHAPTER THREE:

3.1. INTRODUCTION.

PERFORMANCE TEST OF THE DIFFERENT PSDE APPROACHES

3.1.1. Detectability.

3.1.2. Resolution capability.

3.1.3. Estimation bias.

3.2. TEST PROCEDURE.

3.3. DETECTABILITY TEST:

3-1

3-1

3-1

3-2

3-2

3-3

3.3.1. Test example. 3-3

3.3.2. Estimators detection abilities. 3-7

3.4. RESOLUTION CAPABILITY TEST: 3-12

3.4.1. Test examples. 3-12

3.4.2. Estimators resolution capabilities. 3-15

3.5. ESTIMATION BIAS TEST: 3-16

3.5.1. Test examples.

3.5.2. Estimators performance.

3-16

3-28

CHAPTER FOUR: HIGH RESOLUTION PSDE ESTIMATORS

4.1. INTRODUCTION.

4.2. THEORETICAL MODEL.

4.3. MAXIMUM ENTROPY METHOD.

4-1

4-2

4-7

4.4. EIGEN VECTOR EIGEN VALUE DECOMPOSITION TECHNIQUE

FOR PSDE : 4-9

4.4.1. The Signal Sub-Space and the Noise

Sub-Space. 4-9

4.4.2. Eigen Vector Method. 4-11

4.4.3. MUSIC method. 4-13

5.

4.4.4. The New Proposed method.

4.5. PARTITIONING :

4-13

4-16

4-17

4-21

4.5.1. Wax and Kailath criterion.

4.5.2. The New proposed criterion.

CHAPTER FIVE: CAPABILITY OF THE HIGH RESOLUTION PSDE APPROACHES

TO ESTIMA TE AND RESOL VE SIGNAL FREQUENCIES

A comparison study

5.1. INTRODUCTION. 5-1

5-2 5.2. PERFORMANCE OF THE ESTIMATORS:

5.2.1. Using The True Covariance Matrix: 5-2

5.2.1.1. The effect of SNR variations.5-2

5.2.1.2. The effect of frequency

separation variations. 5-3

5.2.1.3. The effect of relative phase

variations. 5-4

5.2.2. Using The Estimated Covariance

Matrix • . 5.2.2.1. The effect of data length

variations.

5-6

5-6

5.2.2.2. The effect of SNR variations.5-B

5.2.2.3. The effect of frequency

separation variations. 5-9

5.2.2.4. The effect of relative phase

variations. 5-10

5.3. THE EFFECT OF A THIRD NEARBY STRONG SIGNAL. 5-11

5.4.

5.3.1. True Covariance Matrix Case. 5-11

5.3.2. Estimated Covariance Matrix Case. 5-12

RESOLUTION OF TWO CLOSELY SEPARATED SIGNALS OF

UNEQUAL POWERS.

5.4.1. True Covariance Matrix Case.

5.4.2. Estimated Covariance Matrix Case.

5-12

5-13

5-13

6.

7.

CHAPTER SIX: THE PREDICTION OF THE EVDT PERFORMANCE FROM THE BEHAVIOUR

OF THE EIGEN V ALUES OF THE COVARIANCE MATRIX

6.1. INTRODUCTION. 6-1

6.2. TEST PROCEDURE. 6-1

6.3. THE EFFECT OF OBSERVATION LENGTH VARIATIONS 6-3

6.3.1. Single Sinusoid Case. 6-3

6.3.2. Multiple Sinusoids Case. 6-4

6.4. THE EFFECT OF SNR VARIATIONS: 6-7

6.4.1. single Sinusoid Case. 6-7

6.4.2. Multiple Sinusoids Case. 6-13

6.5. THE EFFECT OF FREQUENCY SEPARATION

VARIATIONS. 6-18

6.6. THE EFFECT OF THE RELATIVE PHASE VARIATIONS. 6-23

CHAPTER SEVEN:

7.1. CONCLUSIONS.

CONCLUSIONS AND SUGGESTION FOR FURTHER WORK

7.2. SUGGESTION FOR FURTHER WORK.

7-1

7-5

APPENDIX ONE. DERIVATION OF THE OPTIMUM WEIGHT FOR CAPON

(MLM) FILTER.

APPENDIX TWO. REPRESENTATION OF THE COVARIANCE MATRIX IN

TERMS OF ITS EIGEN DATA.

APPENDIX THREE. LIST OF TWO DATA SAMPLES RECORD.

REFERENCES.

Chapter One

INTRODUCTION

CHAPTER ONE

INTRODUCTION

We are currently facing an industrial revolution of high

technology in which digital signal processing plays a

fundamental role. The final objective of this field - where

ideas and methodologies from system theory, statistics,

numerical analysis, computer science and very large scale

integrated circuits (VLSI) technology have been combined -,

is to process a finite set of data (time or space domain)

and to extract important information which is hidden in it.

Among the most fundamental and useful tools in digital

signal processing (DSP) has been the estimation of the Power

Spectral Density (PSD) of a discrete time deterministic and

stochastic process. The advances achieved so far in

communication, radar, sonar, speech, biomedical and image

processing systems are related to the expansion of new power

spectrum estimation techniques.

One of the earliest and most popular techniques for power

spectral density estimation is the Fourier Transform, which

became very efficient and more popular after the invention

of the Fast Fourier Transform (FFT) in the midsixties. But

the lack of resolution - which depends mainly upon the data

length - and the sidelobe-leakage, are the main limitations

to the use of this technique.

Over the last two decades, or so, there has been

considerable interest in so-called modern techniques for

PSDE -see Kay and Marple [33], Ulrych and Clayton [54],

1-1

Cadzow [7] and Haykin [23]-, their works and others form

good references for consultation. The new techniques offer

high resolution and less frequency bias.

Recent work shows a great interest in a group of

techniques which depend upon the Eigen vector Eigen value

Decomposition of the data covariance matrix and the

so-called signal subspace and noise subspace .This group of

techniques, pioneered by Pisarenko [45], offers the best

resolution achieved to-date.

So, the available power spectral density estimation

techniques may be considered in a number of separate classes

namely, Conventional Techniques (or 'Fourier type' ),

Modelling Techniques (Auto Regressive Moving Average (ARMA) ,

Auto Regressive (AR) and Moving Average (MA) modelling),

Nonparametric Techniques (such as Maximum Likelihood Method

(MLM) , Pisarenko Harmonic Decomposition (PHD), and Eigen

Vector Decomposition Techniques (EVDT). Research in this

area has extended to Multidimensional, Multichannel, and

Array Signal Processing problems.

Each one of the above mentioned approaches to power

spectral density estimation has certain advantages and

1 imi tations, not only in terms of estimation performance,

but also in

implementation,

capabilities.

terms of estimation complexity, cost of

finite data length effects and resolution

1.1 PROBLEM FORMULATION and SOLUTION:

Suppose we have a segment x(t), of a sample function

from a zero mean stationary random process and we wish to

generate an estimate of the power spectral density.

1-2

When it is desired to distinguish between sharply peaked

components of the spectrum at some minimal separation , then

the choice of a particular estimate is largely dependent on

the time of observation (data length T), -i. e the total

sampling time N~t, where N, is the number of samples and ~t

is the sampling interval-. Now when N is large and ~f (the

frequency separation between the two peaks to be resolved)

is larger than the resolution limit (l/N~t), then any of the

large number of schemes will achieve the desired resolution

with reasonably small estimate variance. In many situations

however, the observation interval (and hence the number of

samples N) is constrained to be relatively short (e.g. when

x(t) may only be considered stationary over a short time

interval) and one must choose an estimate subj ect to the

requirement

i.e ~f~(l/N~t) (1.1.1)

which is not satisfied using most of the available

approaches, and this will be the requirement upon which we

will depend in testing the different algoritms.

The solution to this problem is the use of Eigen Vector

Decomposition Techniques (EVDT) which were pioneered by

Pisarenko (1973) and Ligget (1973) and improved by Schmidt

(1979) and Bienvenu and Koop (1980). It is a high resolution

technique which is based on the underlying orthogonality

relation existing between the 'noise subspace', spanned by

the eigen vectors corresponding to the smallest eigen values

of the random process covariance matrix and the ' signal

subspace' spanned by the eigen vectors corresponding to the

largest eigen values.

1-3

1.2 THESIS LAY OUT:

The thesis is comprised of seven chapters in addition

to three appendices and an attachment. It is organized as

follows :

Chapter one is the Introduction chapter, and Chapter tvo

presents the Literature review -summary of the theory of the

different approaches to the power spectral density

estimation -.

Chapter three contains the simulation results of most of

the aforementioned approaches and an objective comparison

study to show the disability of most of the algorithms to

resolve closely separated sinusoids contaminated by white

Gaussian noise, when ~f is less than the resolution limit.

The theory of the high resolution techniques -Maximum

Likelihood Method (MLM), Maximum Entropy Method (MEM) and

Eigen vector Decomposition Technique (EVDT)- are given in

Chapter four, in which the new proposed algorithm is

developed. In addition, the Partitioning problem (or the

problem of separating the signal eigen values from the noise

eigen values) will be dealt with in this chapter, and a new

method for this process is suggested.

Chapter five contains the simulation results of the

proposed algorithm together with those of the most widely

used algorithms -Maximum Likelihood Method, Maximum Entropy

Method and Eigen Vector Method-, for the purpose of

comparison.

The Parameters affecting the resolution capability of the

Eigen vector Decomposition Technique are dealt with in

Chapter six, while conclusions and suggestions for further

work are given in Chapter seven.

1-4

Appendix One contains the representation of the

Covariance Matrix in terms of its Eigen vectors and Eigen

valueas, Appendix Tvo contains the derivation of the Optimum

weight for the HLH filter and Appendix Three contains two

data records as an example of the data generated and used to

test the different PSDE approaches.

The list of the Fortran 77 Programs and Subroutines,

written to simulate and test the different Power Spectral

density Approaches is given in Attachment One.

1-5

Chapter Two

LITERATURE REVIEW

CHAPTER TWO

LITERATURE REVIEW

2.1. INTRODUCTION : Spectral estimation has progressed through several

stages since FOURIER established the basis for defining a

spectrum of a function. Fourier analysis has played a

primary role in much of the earlier as well as more recent

efforts in spectral estimation of and frequency retrieval

from experimentally collected data.

The Fourier Transform (FT) is an excellent method of

obtaining an estimate of the spectrum of a time domain

signal. So, if we have x(t) as a deterministic analog

waveform, then its Fourier transform will be :

(X)

X(f) = J x(t)exp(-j2rrft)dt (2.1.1)

-(X)

and the power spectrum estimation at frequency f is :

(2.1.2)

Now, if the signal x(t) is sampled at constant rate of

~t's intervals to produce a discrete sequence x = x(n~t) for n

-oo~ n ~oo, then the sampled sequence can be obtained by

multiplying the original time function x(t) by an infinite

set of equispaced Dirac delta function 0 (t ). The Discrete

Fourier Transform (DFT) of this sampled sequence can be

written, using distribution

theory, [5], as :

2-1

X(f) = Jm[~_~X(t)5(t-nAt)AtJeXp(-j2rrft)dt] -00

00

- At~ xnexp(-j2rrfnAt) (2.1.3) -00

But in the practical spectral estimation problems, it is

desired to estimate the PSD with this estimate being based

on only a finite set of data samples (observations) N, and

the transform is discretized also for N values by taking

samples at the frequencies f=ml1f, for m=O,1,2, ...... ,N-1,

where I1f=l/Nl1t [33], then

N-1

Xm(f) - At~ xneXp(-j2rrmAfnAt) n=O

N-1

- At~ x nexp(-j2rrmn/N) n=O

(2.1.4)

equation (2.1.4) is the familiar discrete fourier transform

(DFT ).

Now, let us consider a more practical case, where it has

applications, such as in Radar, Doppler processing, Adaptive

filtering, Speech processing, Spectral estimation, Array

processing, .... ,etc, it is desired to estimate the

statistical characteristics of a wide-sence stationary,

stochastic process rather than a deterministic, finite

energy waveform. The energy of such process is infinite, so

that the quantity of interest is the Power Spectral Density

2-2

The Autocorrelation function of such process is given by,

where E and * denote the expectation operator and the

complex conjugate respectively.

Or 1

N-m-1

L n=O

(2.1.5) N-m

for m = O,l, ...... ,M, and M~N-1. Equation (2.1.5) is called

the unbiased estimator.

This autocorrelation function possesses the following

properties [44] :

(1 ) I R (m)1 ~ R (0) xx xx

* (2) R (-m) = R (m) xx xx

(3) R (0) = E[X2

] xx n

The first property states that Rxx(m) is bounded by its

value at the origin, the third property states that this

bound is equal to the mean squared value called the power in

the process.

The negative lag estimates are determined from the

positive ones in accordance with the conjugate symmetric

property (2) of the autocorrelation function.

2-3

Jenkins-Watts [26] and Parzen [42] and [43] provided

arguments for the use of the autocorrelation lag estimate

which tends to have less mean square error than the estimate

expressed by equation (2.1.5). This new estimate is called

the biased estimate and written as follows :

1 N-m-1 I Xn+m x~ n=O

(2.1.6) N

Current methods for PSDE can be classified into four

categories as follows :

(1) Conventional PSD estimation methods.

(2) Parametric PSD estimation methods.

(3) Non-Parametric PSD estimation methods.

(4) Multidimentional PSD estimation methods.

2.2. CONVENTIONAL PSDE methods:

In 1959, Blackman and Tukey [4] presented a generalized

procedure for estimating the PSD - see Fig( 2. 1) This

procedure involves

autocorrelation lags

data samples and (2)

estimates as follows :

H 1\ ......

two steps, (1) determining the

estimates Rxx(m) using the available

taking the Fourier Transform of these

P (f) BT -I LnRxx(m) exp(-j2rrfm~t) (2.2.1)

n=-H

where (-1/2~t)~f~(1/2~t), and Ln is a symmetric data window

that is chosen to achieve various desirable effects such as

side lobe reduction. This window is sometimes selected to be

rectangular in which case Ln=1.

2-4

This power spectral density estimate is in fact the

discrete-time version of the Wiener-Khinchine expression

which relates the autocorrelation function via the FT to the

PSD [33], which states that:

ex)

P (f) = J RxX(T) exp(-j2rrfT)dT (2.2.2)

-00

In Blackman-Tukey - Equ.(2.2.1) above - it is seen that

only a finite number of autocorrelation terms (2H+l) are

involved in the spectral estimate, which is a direct

consequence of the fact that only a finite set of

autocorrelation lag estimates are obtainable from the

observation set if a standard lag estimation method is used.

Alternatively the PSD can be calculated directly from the

data set xO' ........ ,xN- 1 through the Fourier Transform as

follows :

......

PpER (f) -1

Nilt

N-l

I bt ~ xn exp (-j2rrfnbt) n=O

for (-1/2Ilt)~f~(1/2Ilt)

2

(2.2.3)

Equation (2.2.3) is called the Periodogram expression (or

estimate) for the power spectral density estimation, -see

Fig(2.1) -, which is computationally inefficient - the same

can be said for BT estimate -. But the advent of the Fast

Fourier Transform (FFT) in the mid sixties popularised these

two methods. It permits the evaluation of equ. (2.2.3) at

the discrete set of N equally spaced frequencies f m= m~f Hz,

for m = 0,1, ...... ,N-l and ~f = l/N~t.

2-5

where

N-l

Xm(f) - At~ xn exp(-j2rrmn/N) n=O

2

(2.2.4)

(2.2.5)

The Periodogram by itself is not a good power spectral

density estimation since its variance does not satisfy

statistical criteria, and it can be viewed as a special case

of BT estimate since it will yield identical numerical

results to that of BT estimate when the biased

autocorrelation estimate is used and as many lags as data

samples (M=N-l) are computed.

2-6

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These approaches to PSD estimation normally suffer from

some inherent limitations. Such limitations are, first the

distortion caused by the side lobe effect -side lobes from

strong frequency components can mask the main lobe of weak

frequency components-, which is in turn caused by the tacit

windowing of the data, (the assumptions made about the data

outside the measurement interval to be equal to zero ). The

second limitation is that of frequency resolution, i.e its

ability to distinguish between two closely separated

signals. The resolution is always limited to the main lobe

width of the window transform [22], which is proportional to

the observation length (T=N~t).

Zero padding the data sequence before transforming will

not improve the resolution of the periodogram, but it will

smooth the appearance of its estimate by interpolating

additional PSD values between those that could be obtained

with a non-zero padding, see Fig.(2.2).

2.3. PARAMETRIC PSDE methods:

In this type of power spectral density estimation, the

observed data are considered to be the output of a model

whose parameters are sought as equivalent to the spectrum,

i.e we try to fit a model to the data in hand, then solving

for the model parameters. Hence this type of spectral

estimation becomes a three steps approach :-

a) Select the time series model.

b) Estimate the model parameters using the available

data samples or autocorrelation lags ( known or

estimated ).

c) Then as a last step, obtain the spectral estimate by

substituting the estimated model parameters into the

model theoretical PSD implied by the model.

2-8

f\J I

\0

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FIG. ( 2.2 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerenl PSDE melhods *

o c ~ "V C ~

> . > r

I

~ o ~

~ I

m M . '" , en ... " ~ z

CD I

> "V ,., I

CD o

w ... .. .. g

Now, in the following, each type of the Parametric Model

Fitting Technique (PMFT) will be discussed together with its

advantages and disadvantages starting with the General

Transfer Function (GTF).

2.3.1. MODELLING TECHNIQUES:

A common approach to characterising the spectrum of a

stationary random process is to model the process as the

output of a rational linear system excited by white Gaussian

noise. The model may be purely descriptive, or it may be

structurally identified with an actual system whose unknown

parameters are to be estimated; in either case, the model is

fully def ined by the locations of the system's poles and

zeros in the complex plane.

2.3.1.1. AUTO REGRESSIVE MOVING AVERAGE model:

In this model, the input driving sequence W , as we n

proceed, is a white Gaussian noise, of zero mean and

variance equal u 2, and the output sequence x, is the w n

observation sequence which needs to be modelled. These two

sequences are related to each other by the linear difference

equation as follows :

q

-L i=O

b.w . ~ n-~

p

-L a.x . J n-J (2.3.1)

j=l

This General Linear Difference Equation (GLDE) represents

the general rational filter -Infinite Impulse Responce (IIR)

filter-I see Fig. (2.3), whose transfer function is written

as :

2-10

H(z) -

where

B(z)

A(z)

q

B (z) - I b i i=O

-i z

(2.3.2)

(2.3.3)

which represents the transfeer function of the Finite

Impulse Response (FIR) filter, and

p

A (z) -I i=O

-i a. z ~

(2.3.4)

is the transfeer function of the recursive filter (some

times called all pole filter).

NOw, relating the output power of the filter, through the

TF, Equ.(2.3.2), to the power of the input driving process

Pw(z) as follows :

*' *' B (z) B (l/z )

*' *' Pw(z) A (z) A (l/z )

(2.3.5)

2-11

The power of the input driving process (white Gaussian

noise) is Pw(Z) = u~~tl hence the PSD formula -Equ.(2.3.5)

implied by the Auto Regressive Moving Average (ARMA) model

will be as follows :

B (z)

A (z)

2

(2.3.6)

Evaluating Equ.(2.3.6) along the unit circle z=exp(jw~t)

where w = 2rrf, and (-1/2~t)~f~(1/2rr~t), equation (2.3.6)

will be :

2

q 2

b o+ I bkexp (-j2rrfk~t)

k=l - u2~t

w P

1 + I akexp (-j2rrfk~t)

k=l

or simply ~2

A (f) I (2.3.7)

2-12

which represents the PSD of the ARHA model whose ak and bk

parameters need to be estimated.

2.3.1.2. AUTO REGRESSIVE model:

The Auto Regressive (AR) model can be easily

deduced from the ARHA model simply by assuming that all the

b i terms in equation (2.3.1), except b o=l, are zeros, see

Fig. (2.3),then :

p

xn - - ~ akxn - k + wn k=l

(2.3.8)

Inspecting equation (2.3.8), we can conclude that the

present value of the process equals the weighted sum of the

past values plus a noise term.

The Auto Regressive (AR) spectra can be deduced from tha

ARMA spectra -equation (2.3.7)- as well, so :

P

1 + ~ akexp (-j2rrfkdt) k=l

2 (2.3.9)

where the AR ak's parameters are needed to be estimated.

2-13

2.3.1.3. MOVING AVERAGE model:

NOw, let us assume that all the a. terms, except ~

ao=l, are zero, and see what will happen. Equation (2.3.1)

will become :

The resultant equation, equ. (2.3.10) above,

the Moving Average (HA) model -see Fig. (2. 3 )-.

can be deduced from equ. (2.3.7) to be equal:

2 2

P (f) - u ~t B (f) KA It"

2 - u ~t It"

q 2

1 + ~ b k exp(-j2rrfkdt)

k=l

(2.3.10)

represents

HA spectra

(2.3.11)

Again, in order to have an estimate to the HA spectra, we

need to estimate its model parameters, bk's.

2.3.2. ESTIMATION OF THE HODEL SPECTRA:

In all the three types of modelling approaches to

Power Spectral Density Estimation (PSDE) mentioned earlier

in this chapter, one need only to know the model parameters

and the noise variance in order to use either of the three

equations for the estimation of the process PSD. Hence a lot

of estimation methods have been developed so far to estimate

the model parameters, and one of the major motivations for

the current interest in the modelling approaches is the

higher frequency resolution they can achieve over those

2-14

which can be obtained using the Conventional Techniques

which were discussed previously.

2.3.2.1. AR spectra:

The process is said to be an AR (p) process if it

is generated (or can be modelled) using equation (2.3.8) and

its spectra can be estimated using equation (2.3.9).

So, the present task is to determine the (p+l) parameters

(ak'U~) of th AR model which can be achieved using the first

well known relationship between the AR parameters and the

autocorrelation function. This relationship is known as the

Yule-Walker normal equations, which can be derived simply by • • multiplying Equ. (2.3.8) by xn+k and tak1ng the expected

value as follows :

2-15

A. ARMA MODEL

n

+

B. MA MODEL

C. AR MODEL

U n

+

Fig.(2.3)

MODEL CONFIGURATIONS

-1 2

-1 2

-1 2

2-16

-1 2

-1 Z

-1 2

X n

Hi9her Orders

t t t t

X n

Higher Orders

.. .. .. t

Higher Orders

,. ,. ,. ,.

~ x

n ( -

for k>O

(2.3.12)

for k=O

Equation (2.3.12) can be put in a more compact form (matrix form) as follows :

R xx(O) Rxx(-l) Rxx(-p) 1 2 . . . . . . (1' v

Rxx(l) Rxx(O) . . . . . . R (-p+1) a 1 0 xx (2.3.13) • • • • -

• • • • •

• • • • Rxx(p) Rxx(p-1) ...... Rxx(O) a 0

Solving equation (2.3.13) with (p+1 ) estimated

autocorrelation lags R (0), ••••• , R (p), and using the xx xx fact R (-m)=R* (m) will allow the determination of the AR xx xx parameters ak and the noise variance (1';.

One of the most efficient algorithms to solve these

equations is known as Levinson-Durbin algorithm, which can

solve it with p2 operations [13] and [57].

An equivalent representation of equation (2.3.13) in

terms of the PSD as a function of frequency f is :

2-17

where

00

Px(f) - ~ Rxx(n)eXp(-j2rrfnAt) -00

p

_\ a R (n-k) L k xx k=l

(2.3.14)

for Inl~p

(2.3.15)

for Inl>p

From equation (2.3.15) above, it is easy to see that the

AR modelling preserves the known lags and recursively

extends the lags beyond the window of the known lags.

Equation (2.3.14) is identical to BT PSDE - Equ. (2.2.1) - up

to lag p, but continues with an infinite extrapolation of

autocovariance function rather than windowing it to zero. It

is for this reason, the AR modelling does not suffer from

the side lobe leakage effect, and the extrapolation implied

by equation (2.3.15) is responsible for the high resolution

property of the AR spectral estimation [33]. See Fig( 2. 1)

for the PSD estimate by Yule-Walker method.

The most popular approach for AR parameters estimation

with N data samples was introduced by Burg in 1967, [6].

Burg argued that the autocorrelation extrapolation should be

selected to yield positive definite autocovariance function

with maximum entropy. Thus the process with such an

autocovariance sequence would be the " most random " one

possible on knowledge of only the autocovariance lag values

from 0 to p.

2-18

The maximum entropy relationship to AR PSD assuming a

Gaussian random process is :

1/2At

J In Px(f)df (2.3.16)

-1/2At

where Px(f), representing the PSD of the time series xn

' can

be found by maximizing equation (2.3.16) subject to the

constraint that the (p+1) known lags satisfy the

Wiener-Khinchine theorem :

1/2At

J px (f)exp(-j2rrfnAt)df - Rxx(n)

-1/2At

(2.3.17)

where n = O,1, ........ ,p, and the solution is found, by the

use of Lagrange multipliers [33J, to be equivalent to the AR

PSD -Equ. (2.3.9) -as shown below:

P 2

1 + ~ apk exp(-j2rrfkAt)

k=1

(2.3.18)

where a a and (j2 are the pth order predl.· ctl.· on pk'·······, pp P parameters and prediction error power, respectivly.

Now, let us consider a more practical situation where one

has data rather than autocovariance lags. By operating

directly on the data without estimating the autocovariance

lags, it is possible to obtain better AR parameters estimate

2-19

and hence better AR spectral estimates.

Least squares prediction techniques are used in this

case, either forward only linear predictions for the

parameters estimate, or they employ the combination of the

forward and backward linear prediction., as Burg algorithm

works -see below-, so linear predictions and AR modelling of

a random process are intimately related to each other.

Now, if one wishes to predict xn on the basis of the

previous p samples [23J, then:

p

- -I apk k=l

x n-k

and the forward prediction error is :

p

-I apk k=O

x n-k

(2.3.19)

(2.3.20)

where apo

=l., by definition, and the prediction error energy

s simply :

2 p

-I I n k=O

a x pk n-k

2

(2.3.21)

Equation (2.3.21) can be written in matrix form as

follows :

E - XA (2.3.22)

2-20

The optimum value of the AR (prediction) parameters can

be obtained simply by equating the derivatives of

Equ. (2.3.21) to zero, the result will be :

, i=1,2, ... , P

and the minimum error energy is given by :

p

-L apk k=O

(2.3.23)

(2.3.24)

Equations (2.3.23), and (2.3.24) can be combined in a

single matrix equation form as follows :

( Ep 0 0 ....... 0 ) T

(2.3.25)

According to the summation limits for the error power Ep

which appeared in equations (2.3.23) and (2.3.24), equation

(2.3.25) will be regarded solved as covariance

equations, Yule Walker equations, previndoved linear

equations, or postvindoved linear equations [33}.

Now, if the process is stationary, the coefficients of

the backward prediction error filter will be identical to

those of the forward one.

Equation (2.3.20) represents the forward linear

prediction error of a wide sense stationary process. The

2-21

backward linear prediction error of such process can be

written as :

(2.3.26)

where p s n s N-l

Burg, in his attempt to estimate the prediction ( or AR )

parameters, minimizes the sum of the forward and backward

prediction error energies.

N-l

Ep - I n=p

e pn

2

+

N-l

I 2

(2.3.27)

n=p

To ensure a stable AR filter (i.e poles within the unit

circle), Burg constrains the AR parameters to satisfy the

Levinson recursion, so :

* apk - ap-1,k + app ap-1,p-k (2.3.28)

Thus using this constraint - Equ. (2.3.28) -, one needs

only to estimate a .. for i=1,2, .... ,p, which can be obtained ~~

simply by setting the derivatives of Ep - Equ. (2.3.27) -

w.r.t. a .. to zero, then the result will be : ~~

2-22

N-l

- 2 L k=i

a .. -~~

(2.3.29) N-l

L ( I b i - 1 , k-l

2

+ 2

ei-l~k ) k=i

It is obvious that a .. ~ 1 for all i. Equations (2.3.28) ~~

and (2.3.29) together will generate a stable all-pole

filter. See Fig(2.4) for the PSDE using Burg algorithm.

One of the difficulties associated with the AR modelling

is that the order p is not known a priori, so if the

computed order was too low, the obtained spectra will be

highly smoothed, on the other hand, if the computed order

was too high, it will introduce spurious frequencies in the

estimate.

2-23

N I

N ~

, ~ , •

~ 0

• ... \.J

~ Qc

~ 0 Q.

, ~ ,

• ~ ~

• ... ~ ~ o Q.

.'0-' 2.n PRONY ~thod , 2.08 ~ , 1.13 • '-50

I .37

~ ~ •

,-'4 ... \.J

0 •• ' UJ e;

0." Ck ~

0.4' 0 Q.

0.23

I 0.00' } . _ _ _ 0.00 0.50 '.00 '.50 2.00 2.50 3.00 3.50 1.00 ".50 ~.OO

)(10- 1

FflI4CT. OF SAIfPL ING Ff1ED.

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7 ••

'.33. ,.,.. .... 3

".011

~

~

'" ~ , a ~ .... -.J

~

~ ,

, .011 • q • • • ; 0 ; • ~ ~ 0 0 I 0.09 0.'" ,.os I." 2.0' 2.'4 3.03 3.'2 4.02 4." '.00

X'O-I FRACT. OF SAf1PlING FRED.

.. 83 P ISI4~NKO (PHD, .. thod

1.:19

, • I~

0.91

0.57

0.43

0.19

-o.~

-0.29

-0.~3

~ -4

~ .... ~ • ~

• ~ o ~

~ t

;

-0.771 0 0 Ito to • • • • I I :' 0.00 0.50 1.00 ,.~ 2.00 2.~ 3.00 3.~ 4.00 ".~ ~.OO •

)(10- 1 ~ FRACT. OF SAffPllNG FIlEO.

X'D', _________________ ~nL~n~.~~~t~r~.~ ________ i 0.40 •.

I -0.04

-0.48.

-0.'3

-, .,.,. -, .B2.

-2.25

-2.'" -3. I' -3.'9

-f.OfK e eo. e e; : , , eel 0.00 o.~ 1.00 ,.~ 2.00 2.~ 3.00 ,.~ 4.00 4.~ '.00

X'O-I FRACT. OF SAf1PlING FRED.

, .. .. v

c -N w • ;R

CD • ~

-w .., .. '"

FIG. ( 2.4 ) POWER SPECTRAL * For diFFerent

DENSITY ESTIMATES PSDE methods *

AR modelling, and Burg algorithm in particular, give good

spectral estimate with considerably higher frequency

resolution when compared with conventional and other

modelling estimates. On the other hands, it suffers from

many problems,

Splitting (SLS)

such as frequency biases and Spectral Line

-SLS is the occurrence of two or more

spectral peaks where only one peak must exist-, see

Fig( 2. 5). The latter problem i. e (SLS) was widely

studied by many researchers, for example Fougere et al [15},

who studied this phenomenon in detail, noted that the SLS

was most likely to occur when :

1) The signal-to-noise ratio is high.

2) The initial phase of the sinusoidal components

is some odd multiple of n/4 .

3) The data duration has an odd number of quarter

cycles of the sinusoidal components.

4) The number of estimated AR parameters is a large

percentage of the number of data samples.

Fougere [16} said that the cause of the SLS in the Burg

algorithm was due to the fact that the prediction error

power is not truly minimized, and he presented a rather

complicated minimization which will ensure convergence and

get rid of the SLS as well.

Many Least-Squares algorithms have been suggested so far

to correct the phenomenon of SLS in the AR estimation

methodes, for example, Ulrych-Clayton [54} and Nattal [41}

independently suggested a Least Square algorithm which

minimizes the prediction error power Ep which can be

effectively performed by equating the derivatives of E p w.r.t. all the prediction parameters apk and not just aii (as in the case of Burg algorithm). A fast computational

algorithm has been developed [35} to solve the normal

2-25

equations obtained.

Recently, H. K. Ibrahim [24] proposed a solution to the

problem of SLS in Burg algorithm for the case of a single

sinusoid. In his modification a new estimate of the

first-order reflection coefficients was proposed, which was

obtained by minimizing the forward and the backward

prediction error energies of the second-order filter w.r.t.

a 21 and a 22 and then using the Levinson recursion. He

applied a generalization of this estimate to the weighted

Burg algorithm, where an improvement in the speed of the

Data-Adaptive Weighted Burg Technique (DAWBT) is achieved.

He then suggested [25] a modification to the optimum Tapered

Burg (OTB) algorithm, which was developed by Kaveh and

Lippert [31], the new improved technique gave spectral

estimates which exhibit no spontaneous line splitting (SLS)

and are independent of the initial phase in the case of

single sinusoid.

The effect of white noise on the AR spectra is to produce

a smoothed spectrum [32]. This smoothing or loss of

resolution has been shown to be due the fact that the

estimated AR poles are drawn into the origin of the Z-plane

due to the introduction of spectral zeros due to the noise.

So the high resolution ability of the AR spectral estimation

decreases as the SNR decreases, [36] and [37], and the all

poles model assumed is no longer valid in the presence of

observation noise, and the solution for this is contained in

the Auto Regressive Moving Average (ARMA) model, as follows

Assume Yn defines an AR process xn corrupted by

observation noise v n ' then

Y - x + v n n n

2-26

N I

N ...,J

X101 ~ 0.59 YULE-tiALKER wtpthod

j

en " -0.06 E.t. Frl' Ie 0.155

'" ~ Q.. , ~ -..,J

~

~ ,

-0.71 E.t. Fr-t ~,. O.~4~

- I .36

-2.01

-2.S7

-3.32

-3.97

-1.S2

-5.27

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XIO- 1


AP • Nal..., .1...,.ld

MtAX • ~

SA ... FIt • • 1.000

NR.I T\D.J • 1.00

FRE05. • 0.2!OD

l"ll.",.... 0.17

!Nt. ( dB ,. !O.OOO

STANO.OEY •• 0.0031821778801884

FIG. ( .2.5 ) POWER SPECTRAL DENSITY ESTIMATES SLS in Auto Regressive Modelling

o c .... '" c ~

> . > r

• ~ o -i

~ , •

m )(

• '" ~ f\J UI ... ~ z

-tf I -n m m • ~

o UI .. .. f\J

Xn and wn are assumed to be uncorrelated, and wn has zero

mean and variance equals u~ .Then the power spectra of the

overall process is :

A(z) I + u~ At

where u~ is the convolutive (input) noise variance.

A(z) 2] At

or 2

(2.3.30)

A(z)

which indicates that the PSD of Y n is no longer

characterized by the all pole model. Equation (2.3.30) has

zeros as well as poles (ARMA model) and the estimation of P Y

using purely AR technique is equivalent to an approximation

to the more general ARMA technique, [3D).

One last point is that the AR modelling is appropriate to

Noise Driven Data Generation Systems (NDDGS) and is

inappropriate to the Additive Noise Data Generation System

(ANDGS). However the distinction between these two types is

important. There are, for the purpose of distinction, two

ways of incorporating noise into Data Generation Systems

(DGS), these are as follows :

a) As INPUT -Convolutive Noise (CN).

b) As output -Additive (or Measurement) Noise (AN).

See Fig(2.6) for theblock diagrams of the two types.

2-28

2.3.2.2.ARMA spectra:

The process is said to be an ARMA(p,q) process if

it is generated according to (or modelled by) the Linear

Difference Equation (LDE) - Equ. (2.3.1) -, and so its power

spectral density can be estimated using equation (2.3.7).

The poles a k are assumed to be within the unit circle of the

z-plane (to ensure a stable filter), whereas the zeros bk

may lie any where in the z-plane.

Our task in this section is to determine the values of

the a k and b k parameters of this model in order to be able,

then, to estimate the PSD of the process. Many techniques

have been proposed to estimate the ARMA parameters. The

problem in using these techniques is that, they involve many

matrix computations and iterative optimization operations

[33]. If a best least squares modelling is desired, it is

then found that the generation of the optimal ak , bk parameters involves the least mean square solution of the

highly non linear Yule-Walker equations which is

computationally inefficient and normally not practical for

real time processing. So, in order to provide a linear

solution for the ARMA model's AR parameters, many

researchers proposed the use of the sub-Optimum Technique

(SOT) which generally estimate the AR and MA parameters

separately rather than jointly. One such techniques [ 8] ,

which was proposed by J. A. Cadzow in 1979, is called the

Extended Yule-Walker, which can be represented in matrix

form as follows :

2-29

IN NO ON

U n .. t ..

PUT j

ISE LY

Y = FY + U n+1 n n

Y n

.. UNIT DELAY .. ---.. .. OBSERV ED DATA

F I .. ..

A. CONVOLUTIVE MODEL; INPUT NOISE ONL~

NOISE FREE SIGNAL (S~st'M IMpulse RtSponcf) COHVOLVED UITH THE NOISE.

FI SEQ

IN COHDI

ON

HITE UENCE )( n .. ---..

PUT !IONS LV

Y = 2 + U n n n

.. t UNIT DELAY •

F

Y 2 n n .. + .. .. ..

j OBSERV ED DATA

L.. r'"

U n r1EASUREnEHTS N OISE

E. ADDITIVE MODEL; MEASUREMENTS NOISE

ONL~ • NOISE FREE SIGNAL (2 ) IS ADDED TO THE NOISE.

n

Fig.(2.6) THE TWO WA~S OF INCORPORATING NOISE INTO DATA GENERATING S~S.

2-30

• •

R (q+p-1) xx

RXX (q-1) ............ .

RXX (q ) · ........ · .. .

R (q-p+1) xx

R (q-p+2) xx

RxX(q+p-2) ........... Rxx(q)

RXX (q+1)

RXX (q+2)

R (q+p) xx

a

(2.3.31)

An algorithm requiring (p2) operations has been developed

by Zohar [57] to solve these equations. The ARHA model's AR

parameters can then be found simply by solving the set of

linear equations :

A(z) - 1 + -k z (2.3.32)

This is equivalent to applying the ARHA process -time

series xn

to the pthorder non recursive filter with

transfer function A(z) whose coefficients correspond to the

AR parameters obtained upon solving equation (2.3.31). This

fil tering procedure produces the so-called Residual time

series as shown in Fig(2.7) below:

2-31

Bq(Z) Yn ~ .. ... Ap(Z) ,

Ap(Z)

Fig. (2.7) Filtering the ARMA process with the all-zero

filter A(z).

Another technique~ [21] was developed by D.Group~

D.J.Krouse and J.B.Moor~ which equates the impulse response

of the ARMA filter, whose parameters are sought, to the AR

filter impulse response with infinite number of parameters as follows :

where

B(z)

A(z)

1

C(z)

C(z) -

OJ

1 + L ck k=1

-k z

(2.3.33)

Thus ck

can be estimated using any of the previously

mentioned techniques and then relating them to the ARMA

parameters - Equ. (2.3.33) -

As a third method, a least squares input output

identification technique has been proposed to estimate the

ARMA parameters, which involves the estimation of the

unknown cross correlation between the input and the output.

This unknown cross correlation will cause the normal

equations to be again, nonlinear. In practice the excitation

2-32

noise process is estimated from the time series itself by a

boot-strap approach, as with the lattice filter

configuration [17] for example, and hence the cross

correlation can be estimated then, which leads to the

estimation of the ARMA parameters.

2.3.2.3. HA spectra:

As stated in section (2.3.1.3) the HA process is

the one that can be obtained as the output of an all zero

filter driven by a white noise process.

W n-k

where bk

, k=Oll, ... , q are the

coefficients) and w is the U 2

E[Wn]=OI and E[Wn +k wn ] = ~wok'

(2.3.34)

HA model parameters (filter

driving white noise with

The auto-correlation function of such a process is

defined, [33]1 by :

q-k

cr! L for k - 0 1 1 1 ", .q

i=O (2.3.35)

o I for k > q

and the PSD estimate can be defined, [7] and [33], as :

2-33

q

P~(f) - ~ Rxx(k) exp(-j2rrfkdt) k=-q

(2.3.36)

Hence, if only an estimate to the MA spectra is required

and if (q+l) lags of the autocorrelation function are

available, then the use of equation (2.3. 36) can achieve

that. But if the MA model parameters are required, then we

need to solve the nonlinear set of equations

-Equ. (2.3.35)-, and so, we need to determine the model

order.

Chow [101 suggested (when only the data samples are

available) the use of the unbiased estimate - Equ. (2.1.5) -

for the autocorrelation lags. He stated that the MA model

order is that for which the autocorrelation lags approaches

zero rapidly. So having obtained the model order, we can use

equation (2.3.11) -repeated here- to compute the moving

average model spectra.

P (f) - (j21lt KA w

q

1 + ~ bk exp(-j2rrfkdt)

k=l

2.3.2.4. PRONY'S method:

2

(2.3.37)

Though Prony's method is not a spectral estimation

technique in the usual sense, a spectral interpretation can

be provided for it. Originally it is a technique for

modelling data of equally spaced samples by a linear

combination of exponentials ( P exponentials, each has

arbitrary amplitude Ak , phase 8k , frequency fk and damping

factor cxk

).

2-34

T Let X = xO,xl' ..... ,xN_l be, as before, the observation

-or data samples- vector to which Prony's method tries to

fit the model is given by :

p "" L bk

n xn - zk for 0::5 n ::5 N-l (2.3.38)

k=l

where b k AkeXp(jBk ) -

and zk - exp [ (ak + j2rrfk ) At ] (2.3.39)

Equation (2.3.38) represents a set of nonlinear equations

in the unknown bk

parameters. In matrix form, it can be

written as . •

"" X - ~B (2.3.40)

"" JT where X - [ Xo xl x 2 . . . . . x N- 1

1 1 · . . . . . . . . . 1

zl z2 · . . . . . . . . . Z

~ P -

N-l N-l N-l zl z2 · . . . . . . . . . zp

and B - [ bl

b 2 b 3 . . . . . b ] T P

In order to find the exponential model parameters ( Ak ,

B f and a ), we need to minimize the squared error c, k' k k

defined as :

2-35

N-1

C = L I Xn

2

(2.3.41)

n=O

which is a difficult nonlinear least squares minimization.

There are a lot of methods to do this job, such as that

suggested by McDonough and Huggins [39]. But Prony~s method

is simpler and provides satisfactory solution though it

doesn't minimize equation (2.3.41). It can be developed as

shown below;

Let ~(z) be a polynomial defined as :

p l/J(z) - n ( z - zk )

k=l

p

-L i=O

p-i a. z ~

(2.3.42)

Using equation (2.3.38), the (n-m) sample estimates can

be written as :

x n-m

p

-L b i 1=1

(2.3.43)

Multiplying both sides by am and summing over the past

(p+1) products gives:

p

L m=O

p

-L b i 1=1

for ps n sN-1

p

L m=O

n-m am z1

2-36

(2.3.44)

N b b ' n-m n-p p-m ow, y su st1tuting z z z equat' (2 3 44) 1 - 1 1 1 1on. . can be written as :

p

L a x m n-m m=O

p

L a x m n-m - 0

m=O

p

or L a x m n-m m=l

p \ a zp-m L ml m=O

) t/J(z) o z=z

1

for p~ n ~N-1

Define en as the estimation error, i.e

e - x - x n n n

and sUbstitute Equ. (2.3.46) in Equ. (2.3.47) we get:

p

- -L a x m n-m a e m n-m m=l m=O

(2.3.45)

(2.3.46)

(2.3.47)

(2.3.48)

As in Pisarenko Harmonic Decomposision (PHD) -see section

(2.4.1)-1 equation (2.3.48) represents a special

ARMA(p+1 I P+1) model with identical MA and AR parameters, but

2-37

unlike PHD, the a. coefficients are not constrained to ~

produce unit modulus roots.

To establish the extended Prony method, one needs to

define the last summation term in equation (2.3.48) as E,

i. e :

P

E = I a e m n-m (2.3.49) m=O

Substitute Equ. (2.3.49) in Equ.(2.3.48) and rewrite, we get :

p

--I a x m n-m + E (2.3.50) m=l

Thus Prony's method sub-optimally minimizes Ln:~ll En 12 instead of the true optimum minimization of LN

-1 1 e 12 which

n=p n leads to a set of nonlinear equations that are difficult to

solve.

Careful inspection of equation (2.3.50) leads to the fact

that the parameters estimation is now reduced to an AR

linear prediction parameters estimation which has been dealt

with previously in this section .

Thus Prony's extended method can be summarized by the

following four steps :

1) Determine the a. parameters ~

estimate of equation (2.3.50).

2-38

by least squares

and

where

2) Determine the z i roots by rooting the polynomial

equation (2.3.42).

3) Determine the b parameters by a least square m ..... minimization of Llx-xI2. A well known solution for

this minimization is given by :

4) Compute the parameters of the exponential model as follows :

a. Amplitude A. - 1 b. 1 ~ ~

b. Phase B. - tan-1 [ Im(b. )/Re(b.) ] ~ ~ ~

c. Frequency f. - tan-1 [ Im(z. )/Re(z.) ] /2rrflt

~ ~ ~

d. Damping Factor cx. - lnl zil2/flt ~

..... ..... 12 e. The PSD P(f) - 1 X(f)

P

X(f) - ~ Am eXp(j8m)

m=l

See Fig(2.4) for the PSDE using this method.

2.4. NON PARAMETRIC SPECTRAL ESTIMATION METHODS:

Unlike the Parametric Technique (PT)I no model

parameters are implicitly computed in estimating the PSD

using these approaches. This category includes Pisarenko

Harmonic Decomposition (PHD) approach, Maximum Likelihood

Method (MLM J., as well as the Eigen Vector Decomposition

(EVD) approaches.

2-39

2.4.1. PISARENKO HARMONIC DECOMPOSITION method:

Pisarenko Harmonic Decomposition (PHD) method is used

for estimating frequencies of sinusoids corrupted by

additive white noise. The main key to this method is the

determination of the smallest eigen vector of the data

covariance matrix. The algorithm is developed as follows :

A deterministic process consisting of p real sinusoids of

the form sin(2rrfi l1t) can be represented as 2pth order

difference equation of real coefficients of the form :

2p

--L a x m n-m (2.4.1)

m=l

In this case, the am are coefficients of the symmetric

polynomial I/I(z)

Z2p+ 2p-1 I/I(z) - a 1 z + ......... + a 2pz + 1 (2.4.2)

Assuming unit modulus roots of the form zi=exp(j2rrfi l1t),

where f. are arbitrary frequencies between -1/2I1t and 1/2I1t, ~

the polynomial equation can be written,[33], as :

p

I/I(z) -L i=l

* (z-z. )(z-z.) ~ ~

(2.4.3)

For sinusoids in additive white noise, the random process

will be :

2-40

2p

--I a x + v rn n-rn n (2.4.4)

rn=l

where Yn is the noisy process, xn -as above- is the

deterministic process and v is the white Gaussian noise, n uncorrelated with the sinusoids, hence;

and

Now,

gives :

substituting x - Y -v into equation n-rn n-rn n-rn

2p

-I a v rn n-rn rn=O

(2.4.4)

(2.4.5)

which has the structure of a special ARMA(p, p) process in

which the MA and AR parameters are identical.

In matrix form, equation (2.4.5) can be written as :

(2.4.6)

where yT - [ Yn Yn- 1 · . . . . . . . . Yn- 2p ]

AT - [ 1 a1 a 2 · . . . . . . . . a 2p ]

wT - [ V v n-l · . . . . . . . . v n-2p ] n

2-41

Multiplying both sides of equation (2.4.6) by Y,

substituting Y = X +W in the right hand side and taking the n n n expectation gives :

But

and

R (0) yy . . . . . . .

. . . . . . . R (0) yy

(2.4.7)

(2.4.8)

(2.4.9)

where R is the covariance matrix of the random process. ¥y U w is the noise variance.

and I is the identity matrix.

Using Equ. (2.4. 8) and (2.4.9), equation (2.4.7) can be

written as :

R A yy 2

- Uw A (2.4.10)

Thus, if the autocorrelation function Ryy(k) is known,

then the ARHA parameters can be found as the solution of the

eigen equation -Equ. (2.4.10)- in which u~ is the smallest

eigen value and A is the corresponding eigen vector.

Equation (2.4.10) forms the basis of the harmonic

decomposi tion approach developed by Pisarenko [46], which

gives the exact frequencies and powers of p real sinusoids

in white noise.

2-42

Determination of the ARMA parameters vector A will

provide the evaluation of the roots of the polynomial

equation -Equ. (2.4.2)- which gives the exact frequencies.

The autocorrelation lags and the sinusoids power are

related to each other as follows :

p

+ L Pi i=l

cos(2rrf.kllt) ~

(2.4.11)

for k :I; 0

Or in matrix form :

,

where ,

and

F -

R (1) yy

P -

(2.4.12)

p

cos(2rrf1

11t) ........... COS(2rrfpllt)

cos(2rrf1pllt) ........... COS(2rrfppllt)

Thus, the sinusoids power can be computed using equation

(2.4.12) and the noise power using equation (2.4.11). See

Fig(2.4) for the PSDE using PHD method.

2-43

2.4.2. MAXIMUM LIKELIHOOD SPECTRAL ESTIMATION method:

One of the most popular techniques for power spectral

estimation which possesses high resolution capability and

exhibts less variance, is the Maximum Likelihood Method

(MLM). It is originally developed by Capon [9}1 in 1969, for

frequency wavenumber analysis. In MLMI one estimates the PSD

by effectively measuring the power out of a set of

narrow-band filters, or we can say it is a "sliding"

band-pass filter which adjusts itself to the random process

under consideration in such a way that the spectral estimate

at one frequency is least affected by the spectral

components of other frequencies. These filters are Finite

Impulse Response (FIR) type with k weights (taps).

NOw, if x2

...... xk

} represents the T

A = [ a 1 a 2 ...... a k} be the observation vector,

weights vectorl then :

(2.4.13)

represents the output of the aforementioned set of filters.

In matrix form, equation (2.4.13) can be written as :

(2.4.14)

The average power can be computed by taking the

expectation of equation (2.4.14)1 i.e

P - E[ * y y ] (2.4.15)

where H denotnes the complex conjugate transpose and Rxx-

E[XX*] 1 as before, is the covariance matrix.

2-44

If, we now constrain the gain of the system to the

signals at particular frequencies to be unity by defining a

frequency vector e as follows :

(2.4.16)

where e - col [ 1 e jw e j2W ....... e j (k-1)W]

Then using Lagrange method, we can minimize the average

output power subject to this constraint by defining a cost

function H(w) as shown below

H(w) = P + ex ( 1 - ATe) (2.4.17)

where ex is an arbitrary constant. The minimization can be

achieved by differentiating equation (2.4.17) w. r. t. the

weights vector A, and equating the derivative to zero, -see

Appendix Two-, we will have;

A = opt

where A is the optimum weight. opt

(2.4.18)

The Maximum Likelihood Spectral Estimate (HLSE) -Pn(w)

as a function of frequency w, is then given by :

1 (2.4.19)

2-45

Thus, we can see from equation (2.4.19) that in order to

compute the HLSE, one needs only to estimate the covariance

matrix Rxx of the observation vector. See Fig(2.4) for the

PSDE using HLM.

2.4.3. EIGEN VECTOR DECOMPOSITION TECHNIQUES • . The Eigen Vector Decomposition Techniques (EVDT),

which has been developed originally for use in Array Signal

Processing (ASP), has a wide range of applications in both

the space and time domain. In this section an overview is

presented of the most important algorithms where eigen

vectors of correlation type matrices are used.

2.4.3.1. PRINCIPAL COMPONENTS method:

The first area where eigen vectors of correlation

type matrices have been used is the Principal Components

(PC) analysis.

Let V I V , ..••..•• I V 1 2 H

be the orthonormalized eigen

vectors of the covariance matrix Rxx such that V1

corresponds to the largest eigen value A1

, V2

the second

eigen vector corresponds to the second largest eigen value

A , and so on, in other words ; 2

> . . . . . . . .- A ~ 0 H

Then, the eigen vectors of Rxx are defined by the

property :

R V. = A.V. xx ~ ~ ~ I i - l.l2.1 ........ ,H (2.4.20)

where R is estimated from the data samples using equation xx

(2.1.6) after subtracting the samples mean X, where

2-46

1 -X -N

N

I i=l

X. ~

(2.4.21)

The J.th 1 .. sca ar pr1nc1pal component of X is then defined

[29},as :

(2.4.22)

where X=col [X1X

2 • •••••••• x

N] is the data samples vector.

Now, the method of principal components is used to find

the principal component 11. that has, on average, large J

variance. So if X represents p sinusoidal signals in white

Gaussian noise, then [27] :

Rxx

or Rxx

where

P 2

I I H - (Tw + A.V.V. ~ ~ ~

i=l

- Rww + Rss

Rww is the noise covariance matrix.

Rss is the signal covariance matrix.

(2.4.23)

The p largest eigen vectors corresponding to the second

term in the above equation -Equ. (2.4.23)- are called the

signal subspace and the (M-p) eigen vectors corresponding to

the (M-p) smallest eigen values -they normally have the same

value which equals to (T;- constitute an orthogonal subspace

called the noise subspace.

2-47

NOw, suppose that in one way or another, we can separate

the two covariance matrices mentioned above (see Chapter 4),

and if we use the signal covariance matrix Rss instead of

the whole covariance matrix R in computing the HL spectra, xx we still have a resonable estimate.

- 1 -1

i.e P PC = [ c! R s s C ] (2.4.24)

where C, as defined in the previous section, is a frequency

vector" -see Fig. (2. B) for the Principal Components

estimate-.

2-48

N , ~ \0

Xl0l ~ 0.34

~ -0.03

" -0.4'

~ -0.78 Cl , -1 .' G

~ ..... -I .~3 .... ...J -, .9' ~ -2.28 ~

~ -2.66 , -3.03

PCN $ptaCt,ra ~

~ " ~ Cl , ~ "" .... ..,J

~ ~

~ ,

X'O' f1US I C lW't,hod 0.77,

-0.08

-0.92.

-, .76

-:/.60

-3.4~

-4.29.

-!Y. '3

-'.97.

-6.82.

o C -4 'V C -4

>

~

• ~ o -4

~ , - 3.40 1 "" , e h' : • , , e , e .. ~ '.. 'c "" .. '" u rl(, " "",I

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 -7·66'1 •• ",,, ..c:::, .. ,' e "'" see soc e 5 ,,;::",4. """"". "",;::,..... 'U s ss,sel :'

0.00 O.SO 1.00 '.50 2.00 2.50 3.00 3.50 4.00 f.!JO '.00 ~ Xl0- 1

FRACT. OF SAf1PlING FREO.

~ X'O' NfN sppct,ra O.63~, ______________________ ~ ______________ _

~ -0.06

" -0.16 t:) \n _, .f, Cl ,

-2.14 t:) ~ -2.83. ..... ...J

~ ~ ~ ,

-3.~2

-f.22.

-f.91

-~.60

-G .291 e.. 'f "US U ,D ,n, , e. e, c. 'I"""""'" .::.r..t

X'O-1 FRACT. OF SAI'fPlING FRED.

~ x'o' EVOI'f 5~C t,ra O.82~. ______________________ ~ ______________ ~

~ -0.08

" -0.98. t:)

~ -1.88. , -:/.79.

Cl ~ -3.69. .... ..., ~ § ,

-f.'9,

-'.f9.

-6.39.

-7.29

-B. '91." u ,e 0 eeoe us. ,; eO,n n,.. n , .hU: e h ... "e" $ ,eo ::::..t 0.00 O.!JO 1.00 '.!JO 2.00 2.::JO 3.00 3.'0 f.OO ".'0 '.00

XIO- 1 0.00 O.::JO 1.00 I.::JO 2.00 2.::JO 3.00 3.::JO f.OO 'f.::JO '.00

)('0-' FRACT. OF SANPL ING FREO. FRACT. OF SAI'fPlING FREO.

FIG. ( 2.8 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerent PSDE methods -

.. , m .. ...,

~ z

w , ~ lU , ~

N -~ o

2.4.3.2. MUSIC ALGORITHM method:

One of the recent eigen vector decomposition

approaches which has superior resolution capabilities is the

MUltiple SIgnal Characterization (MUSIC) algorithm developed

by Schmidt [51] in 1979.

Recall the ML spectral estimate - Equ.(2.4.19) - :

P KL

= 1

(2.4.25)

Now, if we use a specially defined matrix B instead of WE v

the whole covariance matrix R in the equation above, it xx can be rewritten as :

P MUSIC

where BWEV

-

M

L i=p+l

1

H V.V. ~ ~

(2.4.26) C

is the noise covariance matrix with

the noise eigen values set to the same value (taken here as

unity), -see Fig. (2.8) for the PSDE using this method-.

2.4.3.3. EIGEN VECTOR method: The Eigen Vector (EV) approach to power spectral

density estimation developed by D.H.Johnson and DeGraaf [27J

differs slightly from that of Schmidt. The only difference

is that the noise covariance matrix is used without any

constraint on its eigen values.

2-50

1

H

where -1 \

RWV - .L ~=p+l

1

A. ~

H V.V. ~ ~

(2.4.27)

(2.4.28)

is the inverse of the noise covariance matrix computed from

the (M-p) noise eigen vectors. Fig. (2.8) shows the PSDE

using this algorithm.

2.5. MULTIDIMENSIONAL SPECTRAL ESTIMATION:

There are many situations where the signals are

inherently multidimensional Such situations, which can be

found in radar, sonar, radio astronomy, .. ,etc, present many

theoretical and practical difficulties that need to be

tackled [38}. Most of the one-dimensional spectral

approaches, such as the DFT, MLM, Burg algorithm, AR, and

Pisarenko methods, are used in the m-dimensional spectra. A

detailed study can be found in ref. [38}, where the different

approaches mentioned above are derived for the m-dimensional

spectral estimation. A particular emphasis was given to MEM

for its high resolution performance. Unlike the

l-dimensional case where HEM and AR were equivalent, in the

m-dimentional case the true HE estimate is distinctly

different from the spectra derived by AR modeling. In fact

the computation of the m-dimensional HE spectra appears to

require the solution of a nonlinear equation problem.

A topic of current interest is that of Bispectrum and

Trispectrum Estimation [40}. The general motivations behind

the the use of bispectrum were the deviation from normality,

2-51

phase estimation, and detection and characterization of non

linear mechanisms. that generate time series.

2-52

Chapter Three

PERFORMANCE TEST OF THE

DIFFERENT PSDE APPROACHES

CHAPTER THREE

PERFORMANCE TEST OF THE

DIFFERENT PSDE APPROACHES

3.1. INTRODUCTION:

In this chapter the different PSDE approaches,

discussed in the previous chapter, are tested and compared

for their performance capabilities and limitations. Three

criteria are used to evaluate the performances of the above

mentioned estimators, these are :

a) Detectability.

b) Resolution Capability.

c) Estimation Bias.

In the following a brief explanation will be given for

each of these criteria.

3.1.1. DETECTABILITY: This is defined as the ability of the estimator to

detect the signals, i.e the degree to which the side lobes

are small so that they are not confused with peaks

corresponding to the signals. Thus, detection analysis

assesses the conditions under which the number of signals

present in the random process can be determined accurately.

3.1.2. RESOLUTION CAPABILITY: Resolution is defined as the ability of the

estimator to resolve two closely separated signals. However,

3-1

resolution becomes very difficult to be achieved as signals

become more and more closely separated, since as we

mentioned earlier in Chapter Two, in the real situations,

only finite data samples are normally available.

If two signals are separated in frequency by a sufficient

amount, their frequencies will be resol ved l in which case

the estimator exhibits two distinct maxima (peaks). On the

other hand, the estimator may fail to resolve the signals

frequencies, in this case, the estimator displays a single

maximum in some intermediate frequency.

3.1.3. ESTIMATION BIAS:

Finally, an estimator can both detect and resolve

signals but yields inaccurate estimate of their frequencies.

Thus Estimation bias can be defined as the amount of

deviation between the estimated frequency and the true one.

3.2. TEST PROCEDURE : A Fortran 77 subroutine was written for each of the

different PSDE approaches mentioned in Chapter Two together

with two main driving programs" whose flow charts are given

in Fig.(3.1) to Fig. (3.3) and Fortran 77 listing in

Attachment One, to allow the above mentioned tests to be

done. The test data is generated according to the formula :

p

-I i=l

where A. ~

sinusoid,

(i. e the

A.exp[j(nw·+<Pi)] ~ ~ . + w ,

n i=O"l" ... 1 N-1 (3.2.1)

is the amplitude and <Pi is the phase of the ith

w . = 2rrf., and f. is the normal ised frequency, ~ ~ ~

frequency of the ith sinusoid divided by the

3-2

sampling frequency), v is a zero mean white Gaussian noise n wi th variance equal to 0'2, and N is the number of data

v samples. The Signal/Noise Ratio (SNR) is computed from the

formula :

A~ ~

2 O'~ ) dB (3.2.2)

Then the resulting spectral estimate is normalized v.r.t its peak value and transformed in dB. Thus the quantity

presented in the figures is the normalized PSD in dB. that

is :

PSD PSD(normal. ) - 10 log10( )

PSD max.

dB (3.2.3)

3.3. DETECTABILITY TEST:

3.3.1. TEST EXAMPLE:

In performing the detection test on the different

PSD approaches, we used, AS A TEST EXAMPLE, one sinusoidal

signal of unit amplitude A, normalized frequency £=0.25, and

phase ~=o (Degrees) contaminated by a white Gaussian noise.

The SNR used was 10 dBs and the noise variance O'~ was

calculated as follows :

2 O'v -

2 X 10o.txSNR

and the data

Equ. (3. 2. 1 ).

samples were

3-3

(3.3.1)

generated according to

SURT

READ or GENERATE RANDOM PROCESS

DAtA SA~PLES (Ilock 1)

(HiOSE ONE OF THE P D EST U'ATORS

READ 'IANS' 1.IT. i.PERIOD. 3.YU. 4.IURG. S.PISAREHl<O.

HO

'.AfF. '.PRON~.

Fig. (3.1.)

FLO~ CHART OF PROGRAM HIND ~HICH COMPUTE THE CONVENTIONAL AND THE PARAMETRIC PSD ESTIMATES

CALL SUBROUTIHE BLl<MT

CALL SUBROUTIHE PERIODFFT

CALL SUBROUtINE YULUKR

CALL SUBROUtI HE BURGT

CALL SUBROUTINE PISARENKO

CALL SUBROUtI NE ATFT

CALL SUBROUTINE PRONYT

3-4

CALL SUBROUtI HE PEAKS

COMPo EST. FREQS.

CO~PUT NORMALISED PSD

(or 109 nOrMil.)

~R IrE OR PLOT RESULTS

STOP

YES

r-+

( STA~T )

• ~IAD or 6EHE~AtE AHDO~ PROCESS DATA SA~PLES

(Block 1)

• CO"PUTE THE COVARIANCE "ATRIX (TRUE or ESTI~.)

t DECOMPOSE THE

COVARIANCE "ATRIX INTO ITS ~ I V

f SET

L1 : 1 L2 : 12

l CHOOSE ONE OF THE

PSD ESTIMATORS I READ ' I ANS'

1. PCM 2."PCM 3.MEM

4.MUSIC 5.nL" 6.EVM

FIG. (3.2) FLO.... C H ART FOR PRO GR A M A H M ED ~HICH COMPUTES THE EIGENVALUE DECOMPOSITON TECHNIQUE ESTIMATES

L READ FREQUENCY ..-------.... "XIS RANGE

~i~ CALL ~AUEHU~BER TO ESTI ~ATE

• No. OF SIGNALS iOMPUTE THE FRE UEHCY SEARCH

U CtOR E (II)

HS I ----. S YES

.. IANS:1 .. ?

, NO IS YES .. IANS:2 r

?

NO IS YES

IANS:3 .. --.

?

t NO IS YES

L IANS:4 .. ?

~ HO

IS YES .. IANS:5 .. ?

NO NO IS YES

IANS=6 .. r

?

3-5

V I ~ i i

~

SET L2:NS COMPUTE S I C~

• SET L2:N5 CO~PUTE

~ODIFIED HIC"

+ COMPUTE TIC"

• SET L1:NS+1 SET ALL t=1

CO~PUTE N C~

+ CO~PUTE 1st

COLU~ OF TICM

+ SET L1:NS+1

PERFORM ~~-.--~.. VECTOR-MATRIX

-r "UL TIPL I CAT IONS

I~

CALL SUBROUTI NE PEA}(S

EVAL. EST. FREOS.

COMPUT HORMALISED PSD

(or 10i nOrMAl.)

, YR IfE OR PLOT

RESULTS

YES IS

THERE -~ ANY MRE

EST.RE ?

NO

( StOP

2

READ NOISE SD

GENERAtE WHItE GAUSSIAN NOISE

NO YES

tYPE OF RANDOM PROCESS

l.Sinus.onllJ RP ~.NOist onllJ RP R2~6s~IAA~~~·RP

1 OR 3

SD CONPUtE NOISE SD

ADD NOISE to SINUSO. DATA

CONPUTE THE TRUE

COVARIANCE nAtRI~

READ 1. N I NS 2. Am I 3.Fr<I) l:l,NS 4.PHI(1) 5.SHR(1) GENERAtE SIN.DAtA

NO YES

SET

N:1eee

1

NO

NO

YES

NO

YES

READ ' H'

READ Y ,n:1, ••. , N n

H:n

CONPUTE tHE ESTIMAtED COVARIANCE

nAtRIx .. R:H YY 1

I I

Fig. (3.3) FLO'"' CHART OF BLOCJ< (1.) ,",H I CH READS OR

GENERATES DATA AND COMPUTES EITHER THE TRUE OR

ESTIMATED COVARIANCE MARTIX FOR PROGRAMS HIND RAGAD AND AHMED

3-fS

3.3.2. ESTIMATORS DETECTION ABILITIES:

The conventional PSD estimators have a common

problem, that is they suffer from the ambiguities that arise

due to the side lobe leakage, -the side lobe of a strong

signal can mask the main lobe of a nearby veak signal-.

Thus they cannot detect (assess) the signals that actually

exist in the random process and are very sensitive to SNR

variations, see Fig. (3.4) for the Periodogram and Blackman-Tukey PSD estimates.

Modeling approaches to PSD estimation have higher

detection ability with better side lobe supression when

compared with the conventional PSD estimators if the correct

model is chosen to model the time series. AR modeling has

the highest detectability among the modeling approaches,

-see Fig(3.4) for the estimate of Yule-Walker and Burg

methods as examples of AR modeling-.

Walker [33] was the first to consider the problem of

estimating the AR parameters of an AR series corrupted by

additive noise. He evaluated the asymptotic efficiency and

variance of the parameter estimates, upon which the

performance of the AR model ing depends. Pagon [7] proves

that the correct model for an AR series plus noise is the

ARMA model and through the use of nonlinear regression

methods develops strongly consistent efficient estimates.

In Chapter Two it was mentioned that the ARMA process can

be modeled by an AR model with infinite number of

coefficients which is obvious from Fig. (3.5) -another

example of two sinusoids in white Gaussian noise is used

which shows that the Burg algorithm estimator was incapable

of detecting the two signals (resolving them as two separate

peaks) when a small number (4 and 8) of coefficients were

used and they were resolved when this number increased to

3-7

'vol I

CD

X'O' PER[ODOGRAN m~thod ~ 0.9ITI------------------~=_~~~ ______________ 1

~ -0.09

" -1 .09

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~ ,

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E.C.. Fr('" o.~

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" -0.61

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~

~

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~ ..... -' ~ ~ ~ ,


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XIO-' FRACT. OF SAffPLING FREO.

FIG. ( 3.4 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerenl PSOE melhods *

, en .. v

~ z

N ..., • TI

m m I

CD o

--..., . . N W

VI ...,

'" ~ \",

~ Q.. , ~ ,.." .... ..... ~ ~ ~ ,

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-0.95

-, .921 I \ Poles = 4

-2.70

-3.~7

- ..... 4

-~.31

-6.19

-7.OS

-7.93 .. --~ .......... -•••• -_ .. - .. -w- ..................... _ .... _ ........................ -......... -.............................. -.- ........ - ............................................ -. 0.09 0.57 1.06 1.55 2.05 2.54 3.03 3.52 4.02 1.515.00

XIO-l FRACT. OF SANPLING FREO.

AI' • Na'.., sl""",'d

.... AX I 14

SA" .FIt. • I .000

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.... n.Phe... 0.00 0.00

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STAtCJ.(~y. • 0.1000000000000000

RE50... linn 10.0'S

o c ~ 'lJ c -4

> . > r-

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!1 , I

m JC o ,.. w , en .. XIO' BURG Algorithm m~thod X'O ' 0.80 '" 0.82, BlRG Algor i thllt .. thod I ~

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-0.99,

-I .8~ ~ -1.89, Q..

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-2.13 ,

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Poles = 8 t:l -3.61 ~ -3.10

Poles = 16

.... -<f.!W ..... -<f.61

-'.38 ~ -'.~I ~

-6.27 ~ -6. <f 1 , -1. " -1.32

-9.031,. eei' eo". 'h" Ii auSi OJ h" D,e u e., "ue $ , e cu. ~".~.". " .1 0.08 0.~7 1.06 I." 2.0~ 2.,i 3.03 3.~2 i.02 i.~' '.00

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XIO-' X'O-' FRACT. OF SANPLING FREO. FRACT. OF SAf1PI...lNG FRED.

FIG. ( 3.5 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerent PSDE methods *

~ z

-..., '" I :J > ~ I

CD o

.A .0

g .. g

'vol I ~

0

XIOI '" 0.62 YlA...E-tiALKER mt-thod

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XIO-I FRACT. OF SANPLING FREO.

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XIO-I


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o c: ~

~ -4

> • > ,... , ~ o -4

~ , , m )(

• " ~ en ... ...,

~ z .. .... , 5 I

~

IV

co W .a.

16. When a higher SNR value was used and due to the

interpolation of the autocorrelation lags outside the

observation window involved in the computation of Burg

algorithm, it was capable of detecting these two signals at

smaller number of coefficients whereas Yule-Walker algorithm

could not resolve them, see Fig. (3.6).

A major problem with AR modeling is that it exhibits

Spectral Line Spl i tting (SLS ). This problem, as mentioned

earlier in chapter 21 was studied by many researchers who

found that SLS is due to many factors such as the high SNR,

the number of coefficients is a large percentage of data

samples, etc. See Fig. (3.7)1 for the estimate of the same

signal l in which the SLS phenomenon is very clear.

X'O ' rULE-~ALI(ER method " 0 .59 __ ---------=..;:.=.::.-:~.:..:.::..-~---_,

~ -0.06

'" -0.71 9 ~ -1.36

, -2.01

~ -2.67 .... ...J

~

~ ,

-3.32

-3.97

-".62

-5.27

EsC.. Frl,,. tJ.7SS

-5.93 57 I 06 1.55 2.05 2.51 3.03 3.52 ".02 ".51 5.00 0.08 O. • Xl0- 1

FRACT. OF SANPLING FREQ.

Fig. (3.7) Spontaneous Line Splitting in AR Power

Spectral Density Estimation.

3-11

Non parametric spectral estimation methodes possess the

best detection ability though some of them, such as

Pisarenko and MUSIC algorithms, need a prior knowledge of

the number of signals existing in the random process,

whereas MLM and MEMI -which can be regarded to have the best

performances-I do not need such information. The Principal

Components Method possesses the worst detection ability due

to the large side lobes associated, which nearly have the

same signal power. See Fig.(3.8) and Fig.(3.9) for the

estimates by these methods.

Finally Eigen Vector Decomposition Technique, and MUSIC

algori thm in particular, can be regarded to have the best

detectabili ty among all the PSDE methods. Fig. (3. 9) shows

the spectra of these two methods in which the effect of

setting the smallest eigen values to unity in MUSIC on

whitening those portions of the spectra which are not at the

signal frequency is obvious.

3.4. RESOLUTION TEST :

3.4.1. TEST EXAMPLE:

Three experimental tests were carried out to assess

the resolution capabilities of the different algorithms. The

random process used was composed of two equipower sinusoidal

signals corrupted by white Gaussian noise. The data samples

were generated according to equation (3.2.1). The signals

amplitudes were A1

=A2

=1, with the normalized frequencies

sets were as follows (a) f1=0.15 and f 2=0.20 1 (b) and (c)

f 1=0.151

and f 2=0.17 being close to each other. The data

length was (64) samples for tests a and b l and (25) samples

for test c, whereas the initial phases were set to zero and

the SNR used was 30 dBs for all the three tests.

3-12

'" ~

" ~ a.. , ~ "" ..... -.J

~

~ ,

w I '" ~

~ W

" ~ a.. , Cl

~ ..... -.J

~ ~ <:: ,

x.o· 0.66

-0.07

-0.79

-, .5' -2.24

-2.96

-J.GO

-4.11

-~.~

-6.59 0.00 0.50

X'O' 0.39.

-O.Of

-0.41.

-0.9'

-I.3f

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-2.21

-2.5f

-3." -3.~ . .

0.00 0.'0

NEff spt!>C t,,.tJ

E.I.. Frl,,. D.~

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO- 1

Ff?ACT. OF SA11PL ING Ff?EO.

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Ed. Frl'" D."'"

'.00 ,.!jO 2.00 2.'0 3.00 3.!jO f.OO f.!jO '.00 XIO-'

Ff?ACT. OF SAI1PLING Ff?EO.

,., • MDI ... 1 ....... ld

..... • I.

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IU..,.DEY •• O."'117~'81311

IIftlLATRJ C..wl....:. "' ......

FIG. ( 3.8 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *

g ~ -4

> • > r-

• ~ o -i

~ • m M • " , ., .. ..,

~ --.. • ~ lIU I

~

-fIJ .. .. ~

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~ ~ ~ ,

w I "'" ......

~ llo

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~ ..... -.J

~

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XI02 0.12

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~ -1.12. Q. , -2.'4

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;t -'.84 , -6.66

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" C -4

~ . ~ , ~ o -4

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FI1ACT. OF SAf1PI...lNG FlIED.

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~ -0.08

" -, .01 Cl V) -1.94 Q. ,

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-4.12.

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-6.~1.

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o .. fIJ CD


3.4.2. ESTIMATORS RESOLUTION CAPABILITIES:

As the usual measure of resolution is the Fourier

Resolution Limit (FRL) which equals (l/N~t), where N~t=T is

the data length, all the PSDE approaches were capable of

resolving the two signals when the frequency separation

(0.05) between the two signals was larger than the

resolution limit (0.01563), see Fig(3.10) to Fig(3.12). The

two signal frequencies then separated by (0.02£), -test s experiment b-, which is still larger than the FRL, most of

the estimators were capable of resolving the two signals and

the only approaches which were incapable of the resolution,

-see Fig(3.13) to Fig(3.15)-, were MLH, for which the SNR

used was less than the threshold SNR value required to

resolve these signals, and PCH which gave

instead due to the high minimum frequency

a broad peak

separation it

-see Chapter 5 for requires to resolve multiple signals,

more details-.

T.Srinavsan, D.C.Swanson and F.W.Symons [52] presented a

relationship between model order and data length for the

ARMA time series model to resolve two closely separated

sinusoids in white Gaussian noise. They stated that higher

order ARMA models are required as the data becomes shorter

due to the fact that the autocorrelation estimates become

poor to achieve the required resolution.

The high resolution ability of the AR modeling is highly

affected by the low levels of SNR, that is due to the

smoothing caused by the noise, -see Chapter 2-. This effect

can be clearly noticed in Fig. (3.16) which shows that the

Burg and Yule-Walker algorithms were incapable of resolving

the two signals when the SNR was as low as 5dB and they

resolved them, -see Fig. (3.17)-, when the SNR increased to

as high as 10dB.

3-15

In addition to what has been said in section 3.3.21

about

the conventional and modeling PSDE approaches, they share

another major problem, that is the poor resolution

obtainable from processing short data records which can be

easily noticed by the .incapability of these approaches to

resolve the two signals when the frequency separation (0.02)

between the two signals was less than the resolution limit

(0.04) which leads us to the argument that these estimators

are sensitive to short data records. Fig(3.1B) to Fig(3.20)

represent the results of the third test experiment performed

using 25 data samples.

Although the Principal Components Method (PCM) is

incapable of resolving the two signal freuencies separated

by less than Fourier resolution limit (FRL) and gave a broad

spectrum peak instead, the Eigen Vector Decomposition

Technique (EVDT) lies at the top of the nonparametric

approaches to PSDE in terms of its superior performance. It

is least sensitive to finite averaging (short data record)

and requires the least SNR to detect and resolve two closely

separated signals, -this matter viII be studied further in

Chapter 5-. Fig. (3.15) shows the estimates of this group

from which it is clear that MUSIC [51] approach has the same

resolution capability as that of the EV Method [27] though

their spectra have different appearances.

3.5. ESTIMATION BIAS TEST:

3.5.1. TEST EXAMPLES:

The test examples which had been used earlier in

testing both the detection and resolution capabilities of

the different PSDE approaches were used here for testing the

Estimation Biases of the above estimators. In addition

another test example performed for the purpose of this test,

3-16

w I ~ ...J

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FIG. ( 3.10 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerent PSDE methods -

4rI , m .A

" ~ Z

..., , i , ~

-~ -~ .. ~

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..... Q)

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F~ACT. OF SANPLING F~ED.

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FIG. ( 3.11 ) POWER SPECTRAL DENSITY ESTIMATE ~ For diFFerenl PSDE melhods -

o c ~

~ -4

:. • :. r

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~ , , m J( . ~

crt , en ~ ..,

~ z -.... S , ~

-'" w w .. ~

~

~ " f;l Q.. , ~ "" .... ...J

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~ ,

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-2.18.

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-3.'9.

-4.29.

-'.00 -'.70

-S.fl . ~ . 0.00 O.~

PeN SpPC tr(J

E.,. ~,,. II.''''''

E.,. F~ I,. II.1tJtJtJ

1.00 1.50 2.00 2.50 3.00 3.50 1.00 ".50 5.00 .10- 1

F1?ACT. OF SAffPL ING FIlED.

f1U51C Ittt-t,hod ~

E.,. FrO" II.'!JDO ~ " E.,. ~ I ,. II.1tJtJtJ ~ Cl.. , re '" .... ...., ~ § ,

1.00 I.~ 2.00 2.~0 3.00 3.~ ".00 ... ~ ~.OO )(10- 1

F1?ACT. OF SANPL ING FREO.

"" ...... ., .lftUeOld

..... • I •

~.Rt. • '.000

...... TOOI • '.00 '.00

Fltf05. • O.I!OD 0.2OGO

.. 'til....... 0.00 0.00

M. ( cit ,. 30.000 30.000

ITA..,.nn •• 0.03'52211&90'8131

III:ICI.. .linn '0.0'"

'''0.'''10 Cowerlertce "'''1.

X'O' 0.79. EVOff s~trG

-0.08. E.,. I'rl'" tI. ,,,,.,

-D.9f E.,. 1Irl1" II.1fItIO

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-2.S7.

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-4.fO.

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XIO-I F1?ACT. OF SAI1Pl ING F1?EQ.

FIG. ( 3.12 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerent PSDE methods *

~ -4

~ ..... > . > ,-

• ~ o .... ~ , •

m JC • " '" , '" .A

" ! z .. ~ • S • ~

'" . . M o ..

w I

N o

XIOI '" 0.82. PERIODOGRAff ~lhod

~ -0.09

" -0.98 Cl V) -t.89 «l ,

-2.78 I:) ~ -3.68 .... ..,J

~ ~ ~ ,

-4.59

-~.48

-8.39

-1.27

Ed. ~,,. e.'7'. E.l. ~I" e.,,,f

X,O' '" 0.(;3 YlA..E-flAlKER .. thod

i

~ -0.06

" -0.75. Cl '" -I.f6 Q.. ,

-2.16 a ~ -2.'" .... ..,J

~ ~ ~ ,

-3."

-f.25

-f.~

-'.6f

E.t. ~,,. o.'~

'.t. IIWI" o.'~ @ ~

~ -4

~ • ~

• ~ o -4

~ , •

-8. '71 "e •• ~ e; e $ • • us e e ,e cue", e " e • so $ e ". I 0.09 0.51 1.06 1.55 2.05 2.51 J.03 3.52 1.02 1.515.00

XIO- I -6.3~t.;,:, 0.5/1'.03 ,'.'3''2.02 i.,; 3.02 i~f t;,'j":,'.,o ltJtJ I ~

XIO-I " , FRACT. OF SAIfPlING FIlED.

'" ~IOI BlACKffAN-TUKEY ".lhod O.'f •• ____________________________________ ~

Q) " -0.05

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"" .... ..,J -3.0f ~

~ -3.';4

-f.24 , -f.Bf

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-,. f31 0 $ U eo' U " S e. eo. cue .1

FRACT. OF SAI1PI..ING FRED.

'" x,o' 0.B3 BURG Algoritt- .. "hod

i

Q) ~ -0.08.

" -0.99 t:)

~ -1.90. , a

-2.B2.

~ -3.73. .... ..... ~

~ ,

-f.';f

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'.t. 1Irl,,. 0.'''' ht. flWl" 0.'7' ••. I •. J

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XIO- I 0.06 0.'7 1.06 I." 2.05 2.,f 3.03 3.'7 f.07 f." '.00

XIO-I FRACT. OF SAIfPlING FRED. FRACT. OF SAI1PI..ING FlIED.

FIG. ( 3.13 ) POWER SPECTRAL • For diFFerent

DENSITY ESTIMATES methods -PSDE

.. .A ..,

! z .. UI , 5 , ~

-.. --w -~

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~ ~

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~ ~

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x,o' 0.",

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NEff sppctro

Ed. Frl,,. tI. "",

E.t.. Frl1" tI.',;,o

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 XIO- 1

Ff?ACT. OF SAffPL IN(; FI?EQ.

f1lff sppctro

E.'. Frl',. O.'SOD

1.00 ,.~ 2.00 2.'0 3.00 3.'0 'f.00 ".'0 '.00 X'O-I

Ff?ACT. OF SAffPL IN(; FI?EO.

,,,,. IIDI.., elnueold

.... • I.

.... Rt. • •• 000

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Iftlt........ 0.00 0.00

~. Cell,. 30.000 30.000

l'AtG.nEY •• 0.03'17777810'1131

~5OL.ll"I' 10.0'"

IUU .. ATEO Cowerlenc. "'''1.

FIG. ( 3. 14 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *

~ ~ ~

:. • :. r

• ~ o ~

~ •

m J(

• " '" , en .. ..,

~ z

.... • ~ lU • ~

~ 4>

~

~

~

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~ ~ ~ ,

~

w I ~ t\)

t\)

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~ .... -.J

~ § ,

XIO' 0.11

-0.01

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-1.07 ,.---------.-----0.00 0.50

x,o' 0.68.

-0.07

-0.82.

-, .!f7.

-2.32.

-3.08

-3.83

-4.58

-'.33

-6.09.

-6.8f 0.00 0.:50

PCN s~ct.r(J

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 X10- 1

FllACT. OF SANPL ING FllEO.

NUS 1 C ",~ t. hod ~

E.,. h-(,,. tI. ,,,., ~

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~ ~ ~ ,

'.00 ,.~ 2.00 2.'0 3.00 3.!f0 f.OO f.!fO '.00 XIO-'

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"". No'''' .'....0'41 ..... • I.

IA".~. ".000 ..... ITdJI • 1.00 1.00

~O!. • 0.'''''' 0.'''''' '''n .. ".... 0.00 0.00

_. ( dB" 30.000 30.000

5TAJCJ.0EY •• 0.OllS11JJ880I8R3I

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-7. '6

-B·~lh" " ....... Cueh au e .... H_ • e "cucu " .. _ ::;: c 2 hi """I 0.00 O.~ '.00 ,.~ 2.00 2.!JO 3.00 3.!f0 f.OO f.!JO '.00

XIO-' FllACT. OF SAf1PL.ING FRED.

FIG. ( 3.15 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerent PSDE methods *

~ .... " c ~

~ . ~ .-• ~ o .... ~ , m x • " UI , en ~ ..,

! z .. -..... , ~ ,., , ~

.. A o .. N

'"

w I

I\)

W

.'0' ~ 0.13 YULE-tiALKER trtpthod

• ~ -0.01 E.,. Fro('" II. 'S3

"- -0.~2

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-I .~7

~ -1.91

"" .... -oJ

~ ~ ~ ,

~

~ "-

~ Q. , Cl

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~ § ,

-2.42

-2.89

-3.37

-3.81

-4.32, e e,,, , 0" cu. U h.""." S hi" '''Uhlrl~ hue I 0.01 0.51 '.03 '.53 2.02 2.52 3.02 3.5' 1.01 1.50 5.00

.10-' FRACT. OF SA11PL ING FRED.

x,o' BtmG Ai gor j thllt fItEI thod 0."8.

-0.0'-1 r\ E.,. ~,,. II. '.f' -0.'7

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-3. ,g

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FRACT. OF SAf1PL ING FRED.

... MDI .... 1...,..,141

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.... ITWI • 1.00 1.00

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IUMJ.DEY •• o .sn4' ",.,.,.:147 IIfIOl. lI"" to.O'"


@ -4

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-.. -.. ..., -8

w I

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~ -0.05 E.t. Fri,,. D. "13 ........... 1...,.ld

" ~ Q.. , ~ '" ...... ...J

~ ~

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~ § ,

-o.s. E.t. Fril" ".'~ ... AX • I.

-, .17 IMP .At. ., .000

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-2.95 RtEG!. • O.'!GO 0.,7GO

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-3.97 _. ( dB" '0.000 '0.000

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-5.09 , u. ,ee, e .." .""'" .,"""""" "Ie sus " ".,,,,,,,..* """", 0.04 0.54 1.03 I.S3 2.02 2.52 3.02 3.51 4.01 1.50 5.00 ~SG.. U"1t '0.0'" .'0-'

FRACT. OF SAffPllNG FREO.

XIO' 8~ Algorithm m~thod 0.1;2.

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XIO- 1

FRACT. OF SAf1PLING FREO.

FIG. ( 3. 17 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerenl PSDE melhods -

~ .... ~ ... > • > .-• ~ o .... ~ , m M . " w , ." .&

" ~ z

-~ , i , CD o

-A

.& A .. N

'"

w , I\J U1

X'O I "" 0.50 PERIODOGRAff ~t.hod

j

~ -0.05

" -0.60 Cl \Il -I.IG Q.. ,

-1.71 Cl ~ -2.2& ..... -oJ

~ ~ ~ ,

-2.82

-3.37

-3.93

-1.18

E.'. ~,,. II. ,,.,

x,02 BtRG Algor-it.,. .. t.hod "" O.,f I i

~ -0.01

" -0. I' Cl ~ -0.32 ,

-0.48. f:l ~ -0.G3 .... -oJ

~ a:: ~ ,

-0.18

-0.9f

-1.09

-I.~

•• t. ~,,. II.''''

~ ~

~ -4

:. :. r

• ~ o -4

~ , -~ .03 1 " 0 u e " • • e us • e. 0" su ... ," • Sf eo $ ,'" ~ 0 ,

O.'G O.G1 1.12 I.GI 2.09 2.58 3.0G 3.551.03 1.52 5.00 XIO-I

-I.f°J.2~"t.G8 ,'.IG' ,'.6'. 2.;2 2'.6;;;.08 'j.56~':~f f'.~: ;'.00 IE XIO-' CJI ,

FIlAC T. OF SAI1PI...ING FIlED.

"" x,o' BLACKNAN- TUKEY ~thod O.fO~ ......................................... ______________________________ ~

~ -O.Of

" -0.f8

~ Q.. -0.92. , -, .31.

a ~ -, .. , ..... ..... -oJ -2.2' ~ § -1.59

-3.If , -3.~

Ed. ~,,. II. '.f'

-f.O~t.G u , U' ,; ". , GO' U ,.::=::, ctel

FIlACT. OF SAffPllNG FIlED.

'" x'o' YUlE-fiALKEfI .. t.hod 0.f9~ ......................................... ~~~~~ __ ~~ ____ ..... ....

~ -O.~ " -0.~9. t:)

~ -1.13. , -I.S7.

Cl ~ -2.21 ..... -oJ

~ § ,

-2.~

-3.29.

-3.B3.

-4.37

Ed. ~,,. II.""

-f.9't::::. u sus • ceq 0 $ C ; ,ui e. :tt.. £.1 0.08 0.~1 1.06 I.~~ 2.0~ 2.~f 3.03 3.~2 f.02 f.~1 ~.OO

XIO-I 0.08 0.~1 1.06 I.~~ 2.~ 2.~f 3.03 3.~2 f.02 f.~' ~.OO

XIO-I FIlACT. OF SAf'fPlING FIlED.

FIG. ( 3.18 ) POWER SPECTRAL ~ For diFFerent

FIlACT. OF SA1fPI..ING FIlED.

DENSITY ESTIMATES methods )IE PSOE

1\01 (II ..,

! z

-..... • S • ~

-.. (II Ut .. ~

'" ~ " ~ Q.. , ~ "" ..... ..... ~ ~ ~ ,

'" w 1 ~ N

0\ " ~ Q.. , Cl

~ ..... ..... ~ § ,

'''0' 0.58

-0.0&

-0.70

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-2.61

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-S.16

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X'o' O.~o.

-o.~

-O.~9.

-'.'f -, .68.

-'.'3.

-'.n -3.32.

-..... , -... ~

0.00 O·.~

NEff s~t,rtl

E.l. Fro('" II. 'S'"

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO- I

F~IcCT. OF SIcf1PL ING F~O.

I1t.. ff Sppc t,rtl

E.l. ~,,. ". ,,,,,,,

1.00 I.~ 2.00 2.~ 3.00 3.~0 f.OO f.~ '.00 X'O-I

F~IcCT. OF SIcf1PlING F~O.

"". MDlev .1,..,.141

...ax • 2!

SA" .At. • I .000

.... ,T\IJt • 1.00 1.00

Rtf 05 • • O.'!DO 0.1710

I",'" .Phe... 0.00 0.00

_. ( cit,. 30.000 30.000

STAtIO.on •• O.Oll.)2"IIGI8131

~SOL.ll"I' 10.0+00

SI .... A'EO C.,....I~ ttl",.

FIG. ( 3.19 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerent PSDE methods -

~ ~

~ -4

> . > ,... , ~ o ~

~ • m JC • '" ~

~ v

~ ....

! • ~

.. tJI ..., -VI

W I ~ ....,J

.'0' "'" 0 .1' PeN SfN>C t.ra j

~ -0.01 ~

-o.~

~ Q. -0.95 ,

- t .41 Cl ~ -1.87 ~

-.I

~ ~ ~ , -3.23

-3.69

-4. 151 "" ,es'h, e" sa" u ~ .. ~. 0; eeou ........ "0 : • )at". e "I

"'" CfJ ~

~ Q. , Cl ~ ~

-.I

~ ~ ~ ,

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00

X'O' 0.6'

-0.06

-0.7f

-'.fl

-2.09

-2.76

-3.f3

-f.I'

-f.78

-3.f6

-6.13 0.00 o.~o

FRACT. OF SAf1PL ING FRED.

f1USIC ",~t.hod

.10- 1

E.,. Frl I,. 11.1.'" E.,. ~ I,. II.''''''

"'" ~ " ~ Q. , Cl ~ ~

-.I

~

~ ,

1.00 '.!W 2.00 2.~0 3.00 3.~ ".00 ... ~ ~.OO XIO- I

FRACT. OF SAf1Pt..ING FRED.

'II"~ ""'" .'ftUMld

... AX • n

~.At. '.GaO NIIl.TdJI • t.OO t.OO

RlEO!. • O.t!OO 0.1'"

.ft.l....... 0.00 0.00

_. ( dB,. 30.000 30.000

"AJIO.OEY •• 0.031 81177110 I 1131

IfIOl.l,n" ~.04G0

.,fU.AlED C..er'~ .........

X'D' 0.73. EVON 5PKtra

-0.07. E.,. J1W,,. II. ,.,., -0.88.

E.t. IIW,,. II.''''' -1.69.

-2.~.

-3.30.

-f.II

-f.92.

-7.34 10 • au e Ie e. _-. wn sa n _-." , e $ , • ;;- h .1 0.00 o.~ 1.00 I.!W 2.00 2.!W 3.00 3.!W 'f.00 ".!W 5.00

XIO-I

FRACT. OF SAf1Pf..ING FRED.

FIG. ( 3.20 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerent PSDE methods ~

~ -4

~ -4

> . > r-

• ~ o .... ~ , m M • " ~

~ ...,

! z .. .... , j , ~

-.. tJI UI .. ..

in which the two equipower signals were assigned normalised

frequencies of (0.15) and (0.18).

3.5.2. ESTIHATORS PERFORMANCES:

It is of importance for the estimator which detects

and resolves two closely separated signals to estimate their

frequencies correctly. The incorrectness of estimation

, which is called Estimation bias" can be defined as the

amount by which the estimated frequencies deviate from the

true ones and which differs alot from one estimator to

another. It is the purpose of this section to assess this

amount of deviation.

All the PSD estimators, -Fig. (3.4), Fig(3.8) and

Fig. (3.9)-" yield an unbiased estimates when there is only

one signal present in the random process. On the other hand

the estimates are generally biased when there are two

signals or more in the random process. The amount of biases

that the different estimators will have, depends normally

upon how much the two signals frequencies are apart and upon

their SNR. So, inspecting Fig(3.10) to Fig(3.12) and

Fig(3.21) to Fig(3.23) shows that the conventional and the

parametric approaches

signals whatever large

means that they are

gave biased estimates for the two

the frequency separation was which

biased estimators, whereas the

nonparametric approache gave less biase (HEM) or unbiased

estimates (HLH"PCH"HUSIC, and EVH) when the two signals were

sufficiently appart and gave biased estimates when the two

signals were separated by less amount, -see Fig( 3.21) to

Fig( 3. 23 )-.

As a result, we can say that Eigen Vector Decomposition

Technique possesses the best performance among the different

approaches to PSDE because, as we have just seen, it has the

3-28

best detection and resolution capabilities and exhibits less

estimation biases.

3-29

W 1

W o

X102 PERIOOOGRAN m~thod ~ 0.12~i------------------~~~~~--~----____ -'

~ -0.01

" -0.14

~ a... -0.21

, -0.40 Cl ~ -O.~1 .... ..,J -0.S7

~ ~ ~ ,

-0.90

-0.93

-1.06

f\f E.l. ~r('" fl. 'flH

E.l. 1%-(1,. tJ.,~

x,02 Br.RG Algorittw .. thod ~ 0.". I

~ -0.0'

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-0.38. Q ~ -O.!JO .... ..,J

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-0.52 .

-0."'.

-0.99

E.l. Frf,,. fI.'m"!J

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X'O-I F~ACT. OF SANPL ING FREO.

XJ.o;, BLACKffAN- TUKEY Iftt'thod • ~

~ -O.~

" -0.5' Cl ~ -1.15 Q. ,

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-3.95.

-'f."

E.,. 1%-('" tJ. , •• E.'. 1%-(1" II. ,f.f

FRACT. OF SANPLING FRED.

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~ -0.05

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-2.00.

~ -2.5'f .... ..,J

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-3.93.

-'f.~.

-'.23.

'.t. I'rf'" tJ.'" '.1. ""'1" tJ.''''

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X'O-I FRACT. OF SANPLING FREO. FRACT. OF SAI'fPL ING FRED.

FIG. ( 3.21 ) POWER SPECTRAL * For diFFerenl

DENSITY ESTIMATES melhods • PSDE

v: en .A

"

~ -..., , 5 , ~

N -o ..., .. ~

'" ~ '" ~ Q. , ti1 "" .... '" ~ ~ ~ ,

w '" I ~ w

..... '" ~ Q. , Cl

~ .... '" ~ ~ ~ ,

"10' 0.59

-O.OS

-0.71

-I .~

-2.00

-2.6~

-3.29

-3.94

-4.59

-5.23

-5.98 . ---.---- .. -----0.00 0.50

X'D' 0.49,

-0.'"

-0.'9

-,. '2

-, .66

-2.20

-2.73,

-3.27,

-3.8'

-4.3f

-4.BB 0.00 o.~o

NEff sppct.ra

E.'. Frl,,. tI.'InO

E.t. ~". tI. 'f~

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 "10- 1

FRACT. OF SANPL ING FRED.

f1l11 spt-Ct.ra

E.,. ~,,. tI.,,,.,

'.00 I.~ 2.00 2.~0 3.00 3.~0 4.00 4.~ ~.OO XIO- 1


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-- • 14

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Chapter Four

HIGH RESOLUTION PSD ESTIMATORS

CHAPTER FOUR

HIGH RESOLUTION PSD ESTIMATORS

4.1. INTRODUCTION:

Several techniques, such as HLH and HEH, which were

developed originally for spectral estimation in the time

domain have been employed in array signal processing, -for

the solution of Direction Of Arrival (DOA) and Source

Location finding (SLF) problems-, because of their high

resolution capabilities. But the significantly degraded

directional spectrum estimates they gave, due to the

correlation between the emitting sources or the existence of

coloured noise, created the neccessity for new methods

possessing higher resolution and, or lower computational

burden.

Eigen Vector Decomposition Technique (EVDT) for power

spectral density estimation is the most promising approach.

It is mainly used for the processing of signals received by

spatially distributed arrays of sensors, which attracted

considerable attention over the past twenty years.

The theory of the HLH, HEM and EVDT algorithms, which

were mentioned briefly in the literature review chapter, are

presented here in order to give the neccessary background

for the development of the proposed new algorithm.

Finally, the partitioning of eigen vectors into two

subspaces, the Signal Subspace (SSP) and the Noise Subspace

(NSS) is presented in the last section, where a new

criterion for separating the eigen values of these two

subspaces is proposed and tested using both the true and the

estimated covariance matrices.

4.2. THEORETICAL HODEL:

~t ~ = [ ] X 1X2······ .. XN represents the data vector of a discrete, wide sense stationary random process,

where

p

Xn - L A. exp[j(nw·+~·)l + Vn (4.2.1) ~ ~ ~

i=l

P

- L s. + w ~n n

i=l

The random process is assumed to consist of p(soo)

sinusoids contaminated by a white Gaussian noise. Each

sinusoid has unknown amplitude

(Osw.srr) and phase ~. which ~ ~

distribution over [O,2rr].

A., ~

has

normalised frequency w. ~

a uniform probability

Now, let us suppose that we filtered this discrete time

series through a filter, whose output gain, at a particular

frequency (or frequencies), is constrained to unity.

. AHC=l ~.e (4.2.2)

where A = col (a o a 1 ....... aH- 1 ) represents the filter

coefficients (weights) vector, H denotes complex conjugate

transpose and C is the constraint vector (sometimes called

the frequency search vector), the elrments of which at any

frequency wn

' n=O, ..... ,N-l, are given by :

i - O,l, ....... ,H-l (4.2.3)

4-2

The optimized filter coefficients subject to this

constraint -see section (2.4.2. )-, will be :

R- 1 CH

xx (4.2.4)

and the Maximum Likelihood (HL) PSD estimate [9] and [34] is

given by :

= [ cH (4.2.5)

where R;~is the Inverse Covariance Matrix (ICM).

So the task at this stage is to compute the theoretical

covariance matrix. Taking the expectation of equation

(4.2.1) gives:

Rxx(l"k) - E[ II-x1xk ] 1 - k - 1"2,, . . . . . "M

p p

- E[( L sil + V 1 ) ( L S:k + "': )] i=l i=l

P P P (4.2.6)

- E[( L Sill ( L s:k )] + E[ L sil"': ] i=l i=l i=l

P

] + E[ ] E[ "'lL s:k II-+ v1wk

i=l

4-3

NOw, let us assume that the signals are independent, i.e

uncorrelated between each other, and they are uncorrelated

with the white Gaussian noise. Also, the noise is assumed to 2 have zero mean and variance equals uv.Hence,

(4.2.7)

o for 1 * k

(4.2.8)

for 1 - k

and

p p

- E[( ~ Ail exp[j(lwi+~i)]) ( ~ A:k eXp[-j(kWi-~i)])J

4-4

p

- E[ L AilA;k eXP[j(l-k)Wil] i=l

exp[j(l-k)w.] ~

where Pi is the power of signal i.

for 1 - k

(4.2.9)

for 1 ;t: k

Using equations (4.2.7) to (4.2.9), equation (4.2.6) can be reduced to :

p

L P. exp[j(l-k)w.] for 1 ;t: k ~ ~

i=l

Rxx(l,k) - (4.2.10)

p

L P. + u; for 1 - k ~

i=l

which means that the covariance matrix of the random

process can be considered as the sum of the signal

covariance matrix and the noise covariance matrix.

i.e (4.2.11)

4-5

where RXX is the Total random process Covariance Matrix (TCM ).

Rss is the Signal Covariance Matrix (SCM).

Rss(lll)

Rss(211)

•

RSS(112) ........ Rss(lIH)

Rss(212) ........ Rss(2IH)

•

Rss(llk) can be computed -using Equ. (4.2.9)- as follows:

p

L P. exp[j(l-k)w.] for 1 ~ k ~ ~

i=1

Rss(llk) - (4.2.12)

P

L P. for 1 k ~ -i=l

R = (T2 I is the Noise Covar. Matrix and I is the Identity ww It'

matrix.

Substituting equation (4.2.11) in equation (4.2.5)1 we

get :

orusing theory of matrices, equation(4.2.5)canbewritten as

(4.2.13)

4-6

which means that the MLSE of the total random process is in

fact the sum of the MLSE of the signals only process and the

HLSE of the noise only process, -ve vill return to it

later-.

This power spectral density estimator -Equ. (4.2.5) and so

Equ. (4.2.13)- is unable to resolve two closely separated

sinusoids when the separation is less than the reciprocal of

the observation time T - N~t, i.e less than Fourier

Resolution Limit.

4.3. MAXIMUM ENTROPY METHOD:

Maximum entropy method can be thought of as a

discrete filter which adjusts itself to be least-disturbed

by power at frequencies different from those to which it is

tuned, [26]. This operation of the ME filter may be

considered as minimizing the output power subject to the

constraint

(4.3.1)

where vector Z is the same vector C in equation (4.2.2).

Since there is a great flexibility in defining the

constraint vector Z, [20] and [27], it is useful to define

1't as :

ZT - [ 1, 0, 0, ....... , 0 ] (4.3.2)

which means that the constraint vector Z will force the

weight vector A to have the first element equal to one.

Using this definition of the constraint vector

-Equ. (4.3.2)-, the optimum weight, subject to the constraint

specified in equation (4.3.1), will be reduced to

4-7

-1 RXX (1,1)

(4.3.3)

-1 where Rxx (1,1) is the first diagonal element in the

inverse covariance matrix. The output power of this filter [18], will be :

But

-1 if -2

Rxx Z

rfI --1

Rxx (1,1)

-1 if 1 Rxx Z equals the first column of R;x

N-l M

~ ~ c.(w )R- 1 (i,l) L L ~ n xx n=O i=l

(4.3.4)

(4.3.5)

where c . (w ), n=O, .... N-l, as stated before, is the ~ n

• th ~

element of the constraint vector C at frequency wn '

NOTE: It is also possible [50], to define the constraint

vector as ZT= (0,0, ...... ,1) which in turn means that we fix

the end element weight to be unity. Then the output power

will be given by :

-2

PKE

- [R~~ (M,M) ]2 -1

Rxx (M-i+l,M) (4.3.6)

4-8

which gives the same spectral estimate as that obtained by equation (4.3.5).

This method is sometimes called "Power Inversion

Constraint Method (PICM)" I and is useful when the wanted

signal is below the noise level.

4.4. EIGEN VECTOR DECOMPOSITION TECHNIQUES for PSDE :

There is a precise analogy between Direction Of

Arrival (DOA) estimation in the space domain and the

Frequency Estimation (FE) in the time domain, which allows

almost all the array processing approaches, developed for

bearing problems, to be used for frequency estimation

problems.

This problem -frequency estimation or retrieving-I which

is highly related to the PSD estimation, has been treated by

many researchers so far. Among the most popular techniques

is that developed by Pisarenko in 1969 -see chapter 2- l which

is based on the use of the smallest eigenvector of the

observed process.

4.4.1. The SIGNAL SUBSPACE and the NOISE SUBSPACE:

The eigen values and eigen vectors of matrix Rare xx usually obtained by utilizing some standard methods of

numerical analysis. However, eigen data can be obtained

directly from the data samples by using the adaptive

algorithms which can recursively update the eigen vector

estimate using incoming new samples directly [29}. Using the

theory of matrices [23}1 the eigen vectors can be defined by

the property :

4-9

R V. - A.V. xx ~ ~ ~ i-I, ...... , H (4.4.1)

where V., A. are the eigen vectors and the associated eigen ~ ~

values respectively.

Using this identity the covariance matrix can be

represented by its eigen data -see Appendix B- as follows :

M

-I i=l

H A.V.V. ~ ~ ~

(4.4.2)

For the ideal case, where the covariance matrix is known

and assuming, as before, that the signals are uncorrelated

between each other and with the noise, equation (4.4.2) can

be written as follows • •

P R I H u 2

I (4.4.3) - A.V.V. + xx ~ ~ ~ w i=l

But, as we know, in the actual situation the covariance

matrix is normally estimated from the data samples, and the 2

smallest eigen values will not have the same value (uw). So

Equ. (4.4.3) will be as below:

p

I i=l

H A.V.V. ~ ~ ~

M

+ I i=p+l

H A.V.V. ~ ~ ~

(4.4.4)

where A1~ A2~ A3~ ...... ~ AM' which means that we have p

largest eigen values represent the p signals and (M-p)

smallest eigen values represent the noise signal.

4-10

Inspecting equation (4.4.4), the first RHS term spanned

by the p eigen vectors, corresponding to the p largest

(signal) eigen values, is called the Signal Subspace (SSP),

whereas the second term spanned by the remaining (M-p) eigen

vectors, corresponding to the (M-P) smallest (noise) eigen

values, is called the Noise SubSpace (NSP).

The separation of the signal subspace from the noise

subspace -which will be dealt with in section (4.5)- is

called Partitioning. The accuracy of the EVDTs depends

mainly upon this partitioning, which is normaly difficult

and not obvious.

4.4.2. EIGEN VECTOR method:

Let us assume that C - BF can maximize the resolution

of the estimator of equation (4.2.5), where B is a matrix to

be found, then

(4.4.5)

Matrix B must have the property that those elements of

the frequency vector F lying in its null space correspond

only to the signal frequencies which actually exist in the

random process.

Let B = B be the matrix which consists of the sum of WEV

the outer products of the noise eigen vectors, then

M

BWEV L H

- V.V. ~ ~

(4.4.6)

i=p+l

4-11

NOw, due to the orthogonality relationship between the p

largest eigen vectors -signal subspace-, and the (H-p)

smallest eigen vectors -noise subspace-, we have :

B V. - 0 WEV ~

for i - 1." ..•••. ,p

Or in other words, the p largest eigen vectors lie in the

null space of the matrix B , and vector F represents any WEV

of these vectors, This is the only choice by which we can

obtain perfect resolution of multiple signals from the EVDT,

Thus :

BH WEV

-1 R B

XX WEV

H V.V. ~ ~

(4.4.7)

which represnts the noise inverse covariance matrix (NICH).

Equation (4.4.5) becomes:

Or

P EV

H

- [~( L i=p+1

1 H --V.V.

A. ~ ~ ~

) F ]-1 (4.4.8)

(4.4.9)

which is the same as the second RHS term of equation

(4.2.13).

Equation (4.4.9) represents the Eigen vector Method (EVH)

proposed by D.H.Johnson and S.R.DeGraaf [27], which allows

the computation of the HL spectra, with higher resolution

capability, by using only the Noise covariance Matrix (NCH)

instead of the Total Covariance Matrix (TCH) used in the

4-12

conventional MLM I-Equ. (4.2.13)-1 developed by Capon [9).

4.4.3. MUSIC method:

NOw, if we assume that in equation (4.4.8) all the

(M-p) smallest eigen values are set to the same value (A),

then the PSDE will be as follows :

Or

P MUSIC

P MUSIC

M

- [ K [ F' (L V i V~ (4.4.10)

i=p+1

= [ K( F F ) ]-1 (4.4.11)

where K = 1/A is a constant, which is normally taken as 1.

Equation (4.4.11) represents the MUltiple SIgnal

Classifications (MUSIC) algorithim developed by R.O.Schmidt

[51}1 which achieves the same degree of resolution as that

obtained by the Eigen vector method mentioned above.

4.4.4. The NEW PROPOSED method:

NOw, if we define matrix B I -the matrix to be

found-I in equation (4.4.5) as equal the noise covariance

matrix Rww' then;

B - Rww -

M

L H A.V.V. ~ ~ ~

i=p+1

(4.4.12)

and the matrix vector multiplication of equation (4.4.7)

becomes as follows:

4-13

Which means

H

- \ V.V~ L ~ ~ i=p+1

that this

- B WEV

matrix vector

represents a matrix consisting of the sum

products of the Noise Eigen Vectors (NEV).

(4.4.13)

multiplication

of the outer

Now, let us define another matrix B , which consists of SEV

the outer products of the largest eigen vectors ,-the Signal

Eigen Vectors (SEV)-, as follows:

B SEV

p

-I i=l

H v.v. ~ ~

(4.4.14)

The addition of the two

equations (4.4.13) and (4.4.14)

a new matrix B as shown below :

matrices B and B of WEV SEV

respectively will constitute

EV

P H

I H +I H B + B - v.v. v.v.

SEV WEV ~ ~ ~ ~

i=1 i=p+1

H

I H B (4.4.15) - V.V. -

~ ~ EV

i=l

But a property of the eigen vectors of any matrix is

that, the outer products of all its eigen vectors will

constitute an Identity Matrix, so :

BEV

- BSEV

+ B WEV - I

4-14

Or B - I - B WEV SEV

P - I - L V.V~ (4.4.16) ~ ~

i=l

substituting equations (4.4.13) and (4.4.16) in equation

(4.4.5) will constitute the new estimate as follows:

P 1 P - [ FI ( I - LVi v~ ) F]-mPC

i=l

P -1 - [ FIIF - FH( L ViV~ ) F ] (4.4.17)

i=l Or

p ) F ]-1 P [l-FI(L H - V.V.

mPC ~ ~

i=l (4.4.18)

- [ 1 - ~B F ]-1 SEV

which has a very high resolution ability to the closely

separated sinusoids in white Gaussian noise.

If we inspect carefully this new proposed estimator,

Equ.(4.4.18), we can see that the only computations needed

is to compute the outer products of the largest eigen

values, -the Principal Components (PC)- of the TCH after

decomposition, and then perform the frequency search by the

vector products, -see Fig.(4.1) for flow chart of how this

method works-. Further more, we can notice that these

computations are exactly those specified by the first RHS

term of the HLSE -Equ.(4.2.13) except that we have the

signals eigen values set to unity.

4-15

The spectral estimate represented by the first RHS of

equation (4.2.13) is called the Principal Components Method

(PCH) , because it uses the principal eigen vectors. It has

poor resolution and a lot of ambiguities due to the large

side lobes included. However it gives the same spectral estimate obtainable from the conventional Bartlett Estimate [34],which is . by gl.ven • •

1 P - d'R C

BT N

2 XX (4.4.19)

4.5. PARTITIONING:

Several techniques, such as those presented in the

previous sections, have been developed for the purpose of

spectral estimation and for determining the bearings of

acoustic sources, using either the smallest (signal) or

largest (noise) eigen vectors of different correlation type

matr ices. In these methods, as we have already mentioned,

the correlation matrix is first estimated from the available

samples and then decomposed to its eigen values and their

associated eigen vectors.

Now, if we suppose that we have a signal-only random

process consisting of p sinusoids as follows :

p

xn - ~ Aiexp[j(nwi+~n)l i=1

(4.5.1)

then its covariance matrix will have p non-zero eigen

values, so it has a rank of p. But, if this random process

is contaminated by white Gaussian noise, see Equ. (4.2.1),

then its covariance matrix will have H>p non-zero eigen

values as follows :

4-16

(4.5.1)

and hence, the rank of this new covariance matrix is H.

As is mentioned in several places in this thesis, the

white noise is assumed to be uncorrelated with the signals,

so it does not have any components along the signal

subspace, -i.e the contribution due to noise on these

projections must be zero-. In this case, it is easy to

distinguish between the signal eigen values and the noise

eigen values, specially when the exact covariance matrix is

known, see Table (4.1), or when the signal levels are above

the noise level. But, with low SNR, this distinction will be

difficult, so to partition the covariance matrix into signal

and noise subspaces, we need some more accurate and reliable

methods.

4.5.1. WAX and KAILATH method :

Many researchers have worked hard so far to develop

such a method for determining the number of the signals in

the random process under consideration. Among these methods

was the approach based on the observation that the number of

signals can be determined from the multiplicity of the

smallest eigenvalues of the covariance matrix of the random

process -see table (4.1)-. This approach was developed by

Wax and Kailath [55], which is based on the application of

Information Theoretic criteria (ITe) for model

identification introduced by Akaike [2], Schwartz [49] and

Rissanen [47]_

4-17

~. 1

( START

READ IN (GENER.) RANDOM PROCESS

DATA SAMPLES Y , n= 1 , 2, • • • , N n

COMPUTE THE * COVARIANCE "ATRI~ R=E[YY 1 (TRUE or ESTIM.)

EIGEN STRUCTURE DECOMPOS IT I ON

~ , V

, , U i

ESTIMATING THE FORMING THE HUMBER OF SIGNALS ...---.. ~ c~~~Hl~~EN~l~fiI~

, , P H

R = 1-R ,R = ~ V U w s s i=1 i i

FORMIHG THE COMPUT I HG THE PSD N~----1 FREQUENCY SEARCH

VECTOR

IS ~1T

YES

SEARCH FOR THE PEAJ<S

(EstiM. Frfqs.)

PLOT (or UR I TE> THE RESULTS ON SCREEN or AS HC

,

C STOP

NO ... r

INCREASE CP by 21T/H .(O)=-1T

Fig.(4.~) FLO~ CHART FOR THE PROPOSED ALGOR I THM - Modi f' i ed Pri nc i pa 1 COMponents Method (MPCM)

4-18

RP : Noisy sinusoid ~ 25 SAMP.FR. 1.000 AMPLITUDS 1.00 1.00 FREQS. 0.1500 0.1700 SNRs ( dB ): 20.000 20.000 STAND.DEV. : 0.1000000000000000 RESOL.LIMIT :0.0400

A. THE EIGEN VALUES OF THE TRUE COVARIANCE MATRIX ARE :

WR( 1)= WR( 2)= WR( 3)= WR( 4)= WR( 5)= WR( 6)= WR( 7)= WR( 8)= WR( 9)= WR(10)= WR(ll)= WR(12)=

0.9999999999999751D-02 WI( 1)= 0.9999999999999836D-02 WI( 2)= 0.9999999999999892D-02 WI( 3)= O.9999999999999932D-02 WI( 4)= 0.9999999999999961D-02 WI( 5)= O.9999999999999990D-02 WI( 6)= 0.1000000000000001D-Ol WI( 7)= O.1000000000000006D-01 WI( 8)= 0.1000000000000008D-Ol WI( 9)= O.1000000000000009D-01 WI(10)= 0.1107922627001691D+Ol WI(ll)= O.2291207737299831D+02 WI(12)=

O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO

B. THE EIGEN VALUES OF THE ESTIMATED COVARIANCE MATRIX ARE :

WR( 1)= WR( 2)= WR( 3)= WR( 4)= WR( 5)= WR( 6)= WR( 7)= WR( 8)= WR( 9)= WR(10)= WR(ll)= WR(12)=

O.4155386787660106D-Ol WI( 1)= O.5998899499997195D-Ol WI( 2)= O.6940042877011540D-Ol WI( 3)= O.7684392129689654D-Ol WI( 4)= O.8317660889306407D-Ol WI( 5)= O.1094884547944087D+OO WI( 6)= O.1422369873733004D+OO WI( 7)= 0.2192343936683020D+OO WI( 8)= O.3602294074841305D+OO WI( 9)= 0.6535003547944589D+OO WI(10)= O.3072095441910018D+Ol WI(ll)= O.2049222948047228D+02 WI(12)=


Table (4.1) THE EIGEN VALUES OF THE TRUE AND THE ESTIMATED COVARIANCE MATRICES OF A RANDOM PROCESS CONSISTING OF TWO SINUSOIDAL SIGNALS OF EQUIPOWERS IN WHITE GAUSSIAN NOISE.

4-19

It was shown [55], that the Akaike criterion (AIC) tends

to overestimate the number of the signals in the large

sample limit, so it gives inconsistent estimate, while the

criterion introduced by Schwartz and Rissanen (HDL) yields

consistent estimates.

We can represent these two criteria in a General Form

criterion (GFC) which is given by :

(4.5.2)

where

M N

n A. i=p+l ~

HL - M r-n (4.5.3)

[ 1

L A. M-p ~

i=p+l

and the other parameters will differ according to the

desired criterion as shown in Table (4.2) below. The Number

of Free Adjustable Parameters (NFAP) will depend upon the

model, and it is for our assumed model as indicated by the

table.

4-20

Variable AlC HDL

K 2 1 1

K 2 0.5 2

K 1 log N 3

NFAP p(2H-p)+1 p(2H-p)+1

Table (4.2) Table of values assigned to the GFC to act as AlC or MDL criterion

The number of the signals, i.e the rank of the SCH, is

taken as the value of p for which the chosen criterion is

minimized.

4.5.2. The NEW PROPOSED method: A new simpler and computationally efficient method

for determining the number of signals in the random process

is proposed in this research. It utilizes the same General

Form criterion (GFC) presented by equation (4.5.2), but with

different values for its variables. This new method is

summarized by the following steps :

1) Put the eigen values in a descending order.

2) Calculate the HL as follows :

HL - Ap - A I avo

1 H (4.5.4)

L where A = H A.

avo ~

i=l

4-21

3) Set the variables as follow . •

K 1

- -1

K - K = 1 2 3

NFAP - p(l+l/M)

4) Calculate the value of p which minimises the

value to the GFC. This value of p represents the

number of sinusoidal signals present in the

random process.

As an example, let us use the same random process that

used in evaluating the eigen values of table (4. 1) above.

Table (4.3) shows the values of GFC(p) for p=1,2, ...... ,M

from which we can see that this proposed criterion estimated

the number of signals correctly by assigning the minimum

value to GFC(p) at p=2 for both the true and the estimated

covariance matrix.

4-22

A. USING THE TRUE COVARIANCE MATRIX . . p- 1 GFC ( 1)- 4.1231819229383292 p- 2 GFC ( 2)- 2.0636117629184551 * p- 3 GFC( 3)- 3.9431474189785243 p- 4 GFC( 4)- 5.0264806728389979 p- 5 GFC( 5)- 6.1098141651180506 p- 6 GFC( 6)- 7.1931476573971035 p- 7 GFC ( 7)- 8.2764806728389979 p- 8 GFC ( 8)- 9.3598141651180506 p- 9 GFC ( 9)- 10.4431471805599452 p- 10 GFC(10)- 11.5264811496761563 p- 11 GFC(ll)- 12.6098141651180509 p- 12 GFC(12)- 13.6931481342342618

B. USING THE ESTIMATED COVARIANCE MATRIX . . p- 1 GFC ( 1)- 3.9944458412878881 p- 2 GFC( 2)- 2.1228164696604679 * p- 3 GFC( 3)-= 3.6294620663267149 p- 4 GFC ( 4)- 4.8956705956233632 p- 5 GFC ( 5)- 6.0562888033764101 p- 6 GFC ( 6)- 7.1794346665585731 p- 7 GFC( 7)- 8.2792317553389354 p- 8 GFC ( 8)- 9.3755997084024898 p- 9 GFC( 9)- 10.4620446306095103 p- 10 GFC(10)- 11.5490240222177811 p- 11 GFC(ll)- 12.6369473094765690 p- 12 GFC(12)- 13.7292121044005362

* Represents the rninumum GFC value.

able (4.3) TABLE OF THE COMPUTED VALUES OF THE PROPOSED CRITERION USING THE TRUE AND THE ESTIMATED COVARIANCE MATRICES.

4-23

Chapter Five

CAPABILITY OF THE

HIGH RESOLUTION PSDE APPROACHES

TO ESTIMATE AND RESOLVE SIGNAL FREQUENCIES

A Comparison Study

CHAPTER FIVE

CAPABILITY OF THE HIGH RESOLUTION PSDE APPROACHES TO

ESTIMA TE AND RESOLVE SIGNAL FREQUENCIES A comparison study

5.1. INTRODUCTION:

In chapter 3, we mentioned that resolution is one of

the main performance criteria -such as computational

complexity, detectability and estimation bias- by which any

PSDE method should be judged.

Resolution can be analyzed, however it is difficult, as

for the asymptotic case of infinite averaging, which means

that the "true" covariance matrix is assumed known, [30 J. But as we mentioned earlier, the real world situation allows

only a finite sequence of data samples, which in turn means

a limited or finite amount of averaging is possible and

hence the estimated covariance matrix is far from being good

enough to give the correct PSD or the standard of resolution

required.

In this chapter, the high resolution PSDE approaches

-MLM, MEM, EVM and the new proposed method (MPCM)- presented

in chapter four are studied further and a computer program

was used to compare their resolution capabilities with

respect to data length, SNR, frequency separation and

relative phase variations for the cases of true and

estimated covariance matrices. The effect of a third nearby

strong signal on the resolution capabilies was investigated

as well.

5-1

5.2. PERFORMANCE OF THE ESTIMATORS :

5.2.1. USING TRUE COVARIANCE MATRIX:

We have just mentioned that the infinite averaging

assumes that the true covariance matrix is known, and it is

of interest to study the behaviour of the above mentioned

estimators using this covariance matrix rather than the

estimated one. In this section, the performance of Maximum

Likelihood (ML), Maximum Entropy (ME), Eigen Value

Decomposition (EVD) , and the Modified Principal Components

(MPC) methods to resolve two closely separated signals of

equal powers is studied. Plots of the PSDE of these methods

for the different situations are presented as well.

5.2.1.1. THE EFFECT OF SNR VARIATIONS:

A useful measure of the resolution capability of

an estimator is the signal-to-noise ratio it requires to

resolve two closely separated signals of equal powers

contaminated by white Gaussian noise. The two signals used

were of the normalized frequencies 0.15 and 0.17 being close

to each other and the true covariance matrix was calculated

according to Equ. (4. 2. 10). The SNR was

until each estimator was unable to

sinusoids.

reduced gradually

resolve the two

Fig( 5.1 )shows the PSD estimates of the four estimators

for the case of high SNR which indicates the ability of all

of them to detect and resolve the two signals. When the SNR

was reduced to the level where the two signals were just

resol ved by the MLM, the other estimators were able to

resolve them -see Fig( 5. 3)-. This SNR value is called the

threshold SNR value of MLM, which is higher than the

threshold SNR values of the remaining three estimators which

are computed in the same way and listed in Table (5.1)

below. MLM gave biased estimate when the SNR reduced to

5-2

20dB and could not resolve the two signals when the SNR

reached lower values as shown in Fig(S.S). The plots for the

PSDE of the other three estimators at the threshold SNR

value of HEM are presented by Fig(S.6).

The new proposed method has the same threshold SNR value

as that for EVM proposed by Johnson and DeGraaf [27]1 which

seems to be common for all the EVDT methods. Fig( S. 1) to

Fig(S.9) show the effect of SNR variations on the different

PSD estimators.

Threshold SNR values (dB) No. Estimator True Estimated

Covariance Matrix Covariance Matrix

fr.set 1 fr.set 2 IJ.15/.17 0.15/0.18

1. MLM 21 >90 >90

2. MEM 9 >90 3

3. EVM -90 S 0

4. MPCM -90 S 0

Table(S.l) Threshold SNR values for the four approaches to

PSDE using the true and estimate covar. matrices.

5.2.1.2. THE EFFECT OF FREQUENCY SEPARATION VARIATIONS:

The true covariance matrix was generated in the

same way as explained in the previous section with the two

signals being apart by (0.03fs ) in which case all the four

estimators were capable of resolving the two signal

frequencies 1 -see Fig( S. 10 )-. Then the second signal

frequency was moved towards the first one causing the

frequency separation to be less and less until situations

5-3

were reached where each assigned estimator was just able to

resolve the two frequencies. This value of the frequency

separation represents the minimum separation between the two

signal frequencies below which the assigned estimator will

be incapable of resolving them, -see Fig(5.11) to Fig(S.13).

Fig(S.14) shows that the Johnson and DeGraaf approch is

capable of resolving the two sinusoids at closer separation

(0.0020fs ). since the resolution itself is a function of SNR

it is obvious that these frequency separation limits are

function of SNR, -i.e when a lower SNR value was used, the

frequency separation at which the assigned estimator was

capable of resolving the two signals was bigger than that

when SNR was high-. Frequency separation of (0.0020f ) was s the minimum separation with which EVH can resolve the two

signals and when the frequency separation was reduced

further (O.OOlfs ' and O.OOOOOlfs )' this method was incapable

of resolving the two signals and it gave a single peak at

O.lSf only, see Fig(S.lS). So we can say that the point at s which the peaks merge into one is a practical limit beyond

which two components can not be easily separated from one

another using this technique and that can be predicted from

Table( 5.2) which shows that the covariance matrix of this

random process have only one very high eigen value.

S.2.1.3. THE EFFECT OF RELATIVE PHASE VARIATIONS:

Inspecting equation (4.2.10), repeated below for

simplicity, according to which the true covariance matrix is

generated indicates that the initial phases of the sinusoids

and hence the relative phase between them have no effect on

the detection and resolution capabilities of these

estimators because the two signals are assumed uncorrelated

and hence the term representing them is no longer present,

-see Fig(S.16) and Fig(S.17) for the case of 0.0 and

5-4

30.0degrees relative phases-, which gives no sign of any

change in the estimators performances.

p

~ Piexp[j(l-k)wi ] i=l

for l;ek

for l-k

A.} THE EIGEN VALUES OF THE TRUE COVARIANC MATRIX ARE :

WR( 1}= WR( 2}= WR( 3}= WR( 4}= WR( 5}= WR( 6)= WR( 7}= WR( 8}= WR( 9}= WR(10}= WR(ll}= WR(12}=

0.99999999999965700-03 WI( 1)= 0.99999999999983410-03 WI( 2}= 0.99999999999986630-03 WI( 3}= 0.99999999999989670-03 WI( 4)= 0.99999999999996970-03 WI( 5)= 0.99999999999999250-03 WI( 6}= 0.10000000000000060-02 WI( 7)= 0.10000000000000170-02 WI( 8)= 0.10000000000001320-02 WI( 9}= 0.10000000000001400-02 WI(10}= 0.10000028227069930-02 WI(ll)= 0.24000999997177290+02 WI(12}=

0.00000000000000000+00 0.00000000000000000+00 O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO O.OOOOOOOOOOOOOOOOD+OO

B.) THE EIGEN VALUES OF THE ESTIMATED COVARIANCE MATRIX ARE :

WR( 1)= WR( 2}= WR( 3}= WR( 4}= WR( 5)= WR( 6}= WR( 7)= WR( 8)= WR( 9)= WR(10}= WR(11)= WR(12)=

0.75448159931265090-01 WI( 1)= 0.87046283263452090-01 WI( 2)= 0.98450905756963830-01 WI( 3)= 0.11267303288446340+00 WI( 4)= 0.12935769756430230+00 WI( 5)= 0.16031177802384240+00 WI( 6)= 0.22373567056883390+00 WI( 7)= 0.3248858047320566D+00 WI( 8)= 0.54566267049445360+00 WI( 9)= 0.11134502298678800+01 WI(10)= 0.46657660938671340+01 WI(ll)= 0.40415432795836820+02 WI(12)=


TABLE No. ( 5.2) EIGEN VALUES OF THE TRUE ANO THE ESTIMATED COVARIANCE MATRICES OF A RANDOM PROCESS COMPOSED OF TWO VERY CLOSELY SEPARATED SIGNALS IN WHITE GAUSSIAN N:::::SE

5-5

(5.2.1)

5.2.2. USING THE ESTIMATED COVARIANCE MATRIX:

When the covariance matrix is estimated from a

finite number of observations, then it will be an

approximation to the true one [ 3]. The effect of this

approximation manifests itself as a perturbation on the

noise subspace, which is mainly used in the EVDT methods,

and consequently on their ability of frequency resolution.

Though the Eigen Vector Decomposition Techniques (EVDT)

become the more interesting work in high resolution, their

resolution capability,or precisely their ability to estimate

the exact locations of signal frequencies, is highly

affected by using the estimated covariance matrix, i.e the

peaks locations might be biased. In this section the effects

of the changes in the parameters mentioned earlier in the

introduction to this chapter on the performance of MLM, MEM,

EVM, and MPCM using the estimated covariance matrix are

investigated.

5.2.2.1. THE EFFECT OF DATA LENGTH VARIATIONS:

The data length has different effects on the PSDE

of the main features that specify approaches, and it is one

the estimator to have a good performance or not. The

detection and resolution capabilities of all the PSD

estimators improved as the data length increases until the

ideal case where the infinfte averaging is reached in which

case the true covariance matrix becomes known. On the other

hand, as the data length shortened the detection and

resolution abilities of these estimators become worse and

worse. In this section the effect of data length variations

on the performance of MLM, MEM, EVM, and MPCM was

investigated and we discovered that MLM was the most

affected by the data length variations and that MPCM was

the least affected estimator.

5-6

We started with (401) data samples which is quite a large

number and with it all the estimators were able of detecting

and resolving the two signals of normalized frequencies

0.15fs and 0.18fs ' but when the second signal frequency was

moved to 0.17£5 HLH was unable to resolve the two signals

and showed a single peak at the intermediate frequency

o. 16f 5 although the frequency separation was still larger

than Fourier resolution Limit (FRL), -see Fig( 5.18) and

Fig( 5.19), -. Then a data length of (201) samples was used

and all the estimators, except HLM, were able to detect and

resolve the two signal frequencies.

The data length was reduced further and further until it

reahed a value (41) samples where all the three

estimators,-MEM, EVM, and MPCH- resolved the two signals but

with a great bias this time. MEM possessed the least bias

which indicates that it was the best among the three in

locating the frequencies when the data length was large

enough. But when the data length (N) was assigned a value of

(25) samples and less HEM was unable of detecting and

resolving the two signals and gave a single peak at the mid

frequency (0.16fs

)' -see Fig(5.21) to Fig(5.24).

NOW, as N reduced more and more, the only technique that

capable of detecting and resolving the two signals was the

eigen vector decomposition technique (EVDT) represented by

EVH and the new proposed method (HPCH), Fig( 5.25), and the

latter was the best because it gave more distinction between

the two peaks, -i. e it gave sharper peaks than EVM-. The

most interesting observation here is that this approach was

capable of detecting and resolving the two signals when the

data length was very short, 4 or even 2 data samples only,

on a condition that the number of signals is known a priori,

-see Fig(5.26)-.

5-7

5.2.2.2. THE EFFECT OF SNR VARIATIONS:

In section (5.2.1.1) we studied the effect of the

SNR changes when we have the true covariance matrix. Now, in

this section we will study the effect of SNR changes also,

but using finite number of data samples (or the estimated

covariance matrix).

Fig(5.27) shows that HLH was unable to detect and resolve

the two closely separated signals whatever SNR value been

used, while the other three estimators detected and resolved

the two signals with the same degree of accuracy. The SNR

then reduced further and further until it reached a value of

3.0dB which represents the SNR threshold value of HEHI and

when it was reduced to 2.0dB 1 HEM was not able to resolve

the two signal frequencies l -see Fig( 5. 28) to Fig( 5.32)-.

These experimental tests were performed with the two signals

at normalized frequencies of O.lSf and 0.18f I then s s repeated with the two signals at the more closely spaced

frequencies of O.lSfs and 0.17fs from which we noticed that

HEM was unable to resolve the two signals whatever SNR value

used as shown in Fig(S.33). In order to check the threshold

SNR value for the EVDT, the SNR was reduced more and more

until it reahed a minimum value of S.OdB for this set of

signal frequencies, below which no estimator could resolve

the two signals, -see Fig(S.34) to Fig(S.36)-. The threshold

SNR values were as given in Table(S.l).

There are four points discovered from the two sets of

experimental tests performed, the first was that when the

SNR was high, the estimation biases for all the four

estimators were high and as we reduced SNR the accuracy of

resolution improved until we reached the threshold values

where no resolution below them can be achieved. The second

point was that the resolution of HPCH was the best. The

third point was that the threshold SNR values for any

5-8

estimator is a function of the frequency separation between

the two signals (i.e how much they are close to each other)

which confirm what we said in chapter three about the SNR

needed to resolve the two signals. The forth and last point

was that the threshold SNR value for any estimator to

resolve two closely separated signals using true covariance

matrix is lower than that of the same estimator to resolve

the same set of signals using the estimated covariance

matrix,-see Table(5.1)-.

5.2.2.3. THE EFFECT OF FREQUENCY SEPARATION VARIATIONS:

As a reality, any estimator will be capable of

detecting and resolving the two signals when their

frequencies are sufficently apart and there will be a

frequency separation limit for any estimator to be able to

resolve these two signals beyond which no resolution can be

achieved.

Fig(5.37) shows the four estimates of the four algorithms

under study which indicates that all of them were capable of

resolving the two signals -though the estimates were biased

when there was a separation between their frequencies of

(0. 05f s)' which is more than the fourier resolution limit

(FRL). But when this separation reduced to (0. 03f s) we

noticed that MLM was unable to detect and resolve these two

signals and it gave instead a single peak at f=0.165f s ' -see

Fig(5.38)-, so this frequency separation represents the dead

limit below which MLM is incapable of resolving the two

signals. The dead limit value of frequency separation for

HEM, which is indicated by Fig(5.39), was found to be equal

(0.02fs )·

The methods of EVDT were capable of resolving the two

signals with no limits to the frequency separation and this

5-9

feature was discovered when EVH and HPCH detected and

resolved the two signals having approximately the same

normalized frequencies 0.15fs and 0.150001fs (i.e. frequency

separation of o.OOOOOlfs )' and gave two peaks at the wrong

frequencies -see Fig(5.42)-, while these estimators gave one

peak located at (0.15f ) when the true covariance matrix was s used, -see section (5.2.1.2)-. This phenomenon can be

predicted by checking the eigen values of the two matrices,

-see Table (5.2)-, which shows that the estimated covariance

matrix possesses two high eigen values representing the two

peaks, where as the true covariance matrix possesses only

one very high eigen value which in turn represents the only

peaks that the estimates gave.

Finally, we performed two sets of experiments, one set

with SNR=40dB and the second set with SNR=10dB from which we

can see that the resolution of the EVDT methods improved at

the lower SNR value which confirms what we have said in

section (5.2.2.2).

5.2.2.4. THE EFFECT OF THE RELATIVE PHASE VARIATIONS:

The power spectral density estimates of all the

four estimators under consideration are presented in

Fig(5.43) for the case of 0 degrees relative phase. Again,

HLH can not resolve the two signals, but when the relative

phase between the two signals became 30 degreees the

resolution of all the estimators including HLH improved,

with HLH possessing the least estimate bias. The estimation

accuracy improved further when the relative phase was

assigned a value of 90 degrees, that is because the two

signals were orthogonal and had no components towards each

other, i.e they are completely uncorrelated. The resolution

became worse and worse as the relative phase increased

further and further until it reached the value of 180

5-10

degrees at which the estimates were exactly those when the

relative phase was equal to 0 degrees, and that obviously

because the two signals were in phase, see Fig( 5. 43) to

Fig( 5.49).

5.3. THE EFFECT OF A THIRD NEARBY STRONG SIGNAL :

The effect of the presence of a third strong signal on

the resolution of the two closely separated sinusoids was

investigated for both the true and the estimated covariance

matrix cases.

5.3.1. TRUE COVARIANCE MATRIX CASE:

It was seen that this third signal has no effect on

the detection and resolution capabilities of the Eigen

Vector Eigen Value Decomposition Technique (EVDT) as far as

it is located (in frequency) far enough, -more than the

minimum frequency separation limit mentioned in section

(S.2.1.2)-. Fig(S.50) to Fig(S.S3) shows the effect of this

third signal on the PSDE of the above mentioned estimators

as it approaches the other frequencies.

The PSD estimates of MLM is presented in these figures as

well, from which it is clear that the resolution capability

of this method is highly affected by the presence of such a

strong nearby signal. When this third signal was at 0.2fs '

-Fig( S. 51 )-, MLM was capable of resolving the two closely

separated signals which were at the normalized frequencies

o. lSf sand o. 17f s respectively, but when it became nearer

and nearer no resolution achieved because it entered the

minimum frequency separation limit with which this method

can not resolve the two signals, -see Fig(S.52) and

Fig(S.53)-.

5-11

5.3.2. ESTIMATED COVARIANCE MATRIX CASE:

Earlier we mentioned the effect of the estimated

covariance matrix upon the detection and resolution

capabilities of the different approaches to PSDE under study

due to the variations in the data legth, SNR, frequency

separation and relative phase. In this subsection we will

show how these capabilities are affected when a third strong

signal is present nearby taking the actual case of short

data length from which we estimated our covariance matrix.

Fig(5.54) shows that the power spectral density estimates

of the four estimators for the case of no such third signal

from which we can detect the inability of the MLH to resolve

the two signals because they were separated by less than

Fourier resolution Limit (FRL). Fig(5.55) to Fig(5.57) show

the effect of the third signal as it became closer and

closer from which we can say that the detection and

resolution capabilities of these estimators are highly

affected and the only estimator that was capable of

resolving the signals was the HEM.

5.4. RESOLUTION OF TWO CLOSELY SEPARATED

SIGNALS OF UNEQUAL POWERS :

We were concerned so far with the detection and

resolution of two closely separated signals of equal powers.

It is of interest that we test the abilities of the

different algorithms to detect and resolve the same signals

when they have unequal powers, which seemed to be highly

affected by the ratio of the signals powers. How much these

abilities were affected was depending upon the type of the

covariance matrix used, the signals SNR levels and the

individual estimators as well.

5-12

5.4.1. TRUE COVARIANCE MATRIX CASE:

Fig(5.58) to Fig(5.61) show the power spectral

density estimates of the four estimators for the cases that

the ratio (A2/A1 ) was equal (0.5, 0.25, 0.1, and 0.05)

respectively, from which we can see that the detection and

resolution capablities of EVM and MPCM estimators were

completely unaffected by this ratio. The reason why they did

not affected was that they possess a very low (-90dB or even

less) threshold SNR values. MLM was the most affected

estimator and MEM was the less affected and that can be

predicted easily by carefully checking their threshold SNR

values listed in Table(5.1) above.

Comparing Fig( 5.62) with Fig( 5.58) we can see that MLM

and MEM were unable to resolve the two signals, -Fig(5.61)-,

though the ratio of their amplitudes was (0.5) which was the

same as that used in obtaining the estimates of Fig(5.57),

but because the weak signal SNR was (7.5dB), -less than the

threshold SNR values-, and the strong signal SNR was (10dB)

which is just above the threshold SNR values of these two

estimators. So, not only the ratio of the two signals which

affect the resolution capabilities of MLM and HEM, but the

signals SNR levels as well.

5.4.2. ESTIMATED COVARIANCE MATRIX CASE:

The case was completely different when the estimated

covariance matrix was used, the two signals were unresolved

by MLM and MEM estimators for the case of (A2/A1=0.5) and

that is obvious because the SNR levels of the high power

signal, (Pl=40dB) , was less than the threshold SNR values of

these two estimators listed in Table(5.1) above.

EVM and MPCM capabilities of resolving the two signals

were highly affected by the use of the estimated covariance

5-13

matrix which can be easily seen by comparing Fig(5.63) and

Fig(5.58). Fig(5.64) and Fig(5.65) present the power

spectral density estimates of these two methods only, for

the remaining values assigned to the ratio P/P1

. It is

clear from Fig( 5. 65) that these two methods gave the same

levels of resolution for the two signals under test whatever

values were assigned to the amplitudes ratio when it is less

than (0.1) and this indicates that Eigen Vector

Decomposition Technique (EVDT) is the less affected

approach. The reason for this is that we still have two

highly distinct eigen values, see Table(5.3), upon which the

degree of resolution of this technique is mainly dependent.

A. ) THE EIGEN VALUES OF THE TRUE COVARIANCE MATRIX ARE :

WR( 1)= 0.9999999999999233D-03 WI ( 1) = O.OOOOOOOOOOOOOOOOD+OO WR( 2)= 0.9999999999999576D-03 WI( 2)= O.OOOOOOOOOOOOOOOOD+OO WR( 3)= 0.9999999999999671D-03 WI( 3)= O.OOOOOOOOOOOOOOOOD+OO WR( 4)= 0.9999999999999806D-03 WI ( 4) = O.OOOOOOOOOOOOOOOOD+OO WR( 5)= 0.9999999999999880D-03 WI( 5)= O.OOOOOOOOOOOOOOOOD+OO WR( 6)= 0.1000000000000006D-02 WI ( 6) = O.OOOOOOOOOOOOOOOOD+OO WR( 7)= 0.1000000000000033D-02 WI( 7)= O.OOOOOOOOOOOOOOOOD+OO WR( 8)= 0.1000000000000048D-02 WI ( 8) = O.OOOOOOOOOOOOOOOOD+OO WR( 9)= 0.1000000000000060D-02 WI( 9)= O.OOOOOOOOOOOOOOOOD+OO WR(10)= 0.1000000000000075D-02 WI(10)= O.OOOOOOOOOOOOOOOOD+OO WR(11)= 0.2178209400977524D-01 WI(11)= O.OOOOOOOOOOOOOOOOD+OO WR(12)= 0.1210021790330802D+02 WI(12)= O.OOOOOOOOOOOOOOOOD+OO

B. ) THE EIGEN VALUES OF THE ESTIMATED COVARIANCE MATRIX ARE :

WR( 1)= 0.1806311947238607D-01 WI( 1)= O.OOOOOOOOOOOOOOOOD+OO WR( 2)= 0.2242805280916115D-01 WI( 2)= O.OOOOOOOOOOOOOOOOD+OO WR( 3)= 0.2820321009721945D-01 WI( 3)= O.OOOOOOOOOOOOOOOOD+OO WR( 4)= 0.3087537397493176D-01 WI( 4)= O.OOOOOOOOOOOOOOOOD+OO WR( 5)= 0.3367876167066347D-01 WI( 5)= O.OOOOOOOOOOOOOOOOD+OO WR( 6)= 0.4152060915178024D-01 WI ( 6) = O.OOOOOOOOOOOOOOOOD+OO WR( 7)= 0.5960216536600802D-01 WI ( 7) = O.OOOOOOOOOOOOOOOOD+OO WR( 8)= 0.8667678201911521D-01 WI( 8)= O.OOOOOOOOOOOOOOOOD+OO WR( 9)= 0.1414019318795685D+00 WI( 9)= O.OOOOOOOOOOOOOOOOD+OO WR(10)= 0.2820312842278482D+00 WI(10)= O.OOOOOOOOOOOOOOOOD+OO WR(11)= 0.1193411068032746D+01 WI(ll)= O.OOOOOOOOOOOOOOOOD+OO WR(12)= 0.1027558900309252D+02 WI(12)= O.OOOOOOOOOOOOOOOOD+OO

TABLE No. ( 5.3 ) THE EIGEN VALUES OF THE TRUE AND THE ESTIMATED COVARIANCE MATRICES OF A RANDOM PROCESS CONTAINING TWO SIGNALS OF

UNEQUAL POWERS ( RELATIVE AMPLITUDES A2/A1=0.1 ) IN A WHITE GAUSSIAN NOISE

5-14

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-0.36. E.&.. Frl1,. O.'~

-0.69. !iNR , til). if 0 • DDDO

-1.02.

-I.~

-1.68

-2.01

o C -i

" C -i

> . > r

, • " r

-2.34

-2.67.

-3 .ool.." ... ", ... L,.~.~", ...... "" ...... , ................... " ..... ..1 Il' 0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.50 4.00 4.50 5.00 ~

o -4 N :I • ,

XIO-I

FRAC T. OF SANPL I NG FREQ.

XJ.tf.. NodiFied peN sppctra

J -0.01 E.&.. Frl I). 0.1100

-0.17. E.". Fr(1). O.I~

-0.33. !iNR (til). ifO. DDDO

-0.49

-0.65

-0.81

-0.97.

-1.13

-1.29

-1.441 us, c ,,.,.,;:;,,,,,,,,,,,,,,,,, .. i'i"'" .;;;....US 'I""" hi """"'''''". leu, .. " .1 0.00 O.~O 1.00 I.~O 2.00 2.~0 3.00 3.50 4.00 i.~O ~.OO

XIO-I FRACT. OF SANPL ING FREO.

N , N UI

"" ~ z

N en I :I > ;0 I

CD o

N

N .. ~

FIG. ( 5. 1 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *

"" ~

" ~ It ,

~ .... -' ~ ~ ~ ,

\,J1 I --" 0\ ""

~ " ~ It , ()

~ .... -' ~ ~ ~ ,

X101 NLN spttctra "" 0.68

-0.07 E.t. Frll,. 0.1700 ~ -0.81 "

E.t. Frll). O.'~ ~ -1 .55 It

SI#l (dl/)- 23.0000 , -2.30

-3.04 ~ "" ....

-3.78 -' ~

-4.52 ~ ~ -5.27 ,

-6.01

-6.751e255eeeee~u.e,u,z'i,uzeus.e,uiu"ui,ec,sDou",~ Dueesi",:,:,uu"C;.UDUluICSI.UU".,i'COSI,I,1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 1.50 5.00

XIO- I

FI?ACT. OF SANPL ING FRED. xlo2 NEN spttctra 0.12 "" -0.01 E.t. Frll,. O.I~ ~ -0. I'f " E.t. FrlV. 0.1700

~ -0.26 It SNIl (t./)- 23.0000 ,

-0.39

-0.'2 ~ "" ....

-0.6~ -' ~

-0.77 ~ -0.90

~ ,

-I. '-'1, """:;"' .. ,,,,,",, " 0 eo., DO,S e ~u, .....c=;" ... ' .... UU'h' .. , .. ,,''''''''' ..... ,,'',,'

0.00 0.'0 1.00 1.'0 2.00 2.~0 3.00 3.~O i.OO f.~O '.00 X10- 1

FI?ACT. OF SANPLING FRED.

x I 02 EVON SpttC tra 0.30)

-0.03 E.t. Frl I)· 0.1700

-0.36 E.t. Fr-(2" O.I~

-0.69. 5NII (t.". 23.0000

-1.02.

-1.3~

-1.68

-2.01

-2.3f ~

o C -4

" C -4

:>

> r

I • ;yl o -4

-2.67

- 3 .ool.."" ... ""L" ... ~"""""", """""""" """ .. " ..... ..1 ll' 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 f.OO f.50 '.00 •

~ • I

XIO-I ,.. '" , FRACT. OF SANPLING FREO.

Xlo2 NodiFi~d peN sp~ctra O./f.

-0.011 I I E.t. Fr-U). 0.1700

-0. 17. t II II E.t. Fr-ll). O.'~

-0.33. J II II SNR (.). 23.0000

-0.f9.

-0.65

-0.81

-0.97.

-/.13

-1.29

-I. 4.1"c"",,;e~ .. ,h'h ce'i"'" ""'''''''~hS US,"US" sl''''''h',,,, .. ", .. , .. ",,1 0.00 o.~o 1.00 1.50 2.00 2.50 3.00 3.'0 f.OO f.~O '.00

Xl0- 1 FI?AC T. OF SANPL I NG FRED.

'" UI ...,

~ z

~ I :I > ;0 I

CD o

'" '" 00 .. ~


~

~

" ~ tl , ~ "" ...... -J

~

~ ,

\J1 I ~

-.J ~

~

" ~ tl , Cl

~ ...... -J

~ ~ <! ,

)(10' 0.61

-0.06

-0.76

-1 .16

-2.16

-2.96

-3.56

-4.26

-1.96

-5.65

-6.35 0.00 0.50

Xl02 O. "

-0.01

-0.13

-0.2~

-0.36.

-0.48

-0.60

-0. "2.

-0.84

-0.9~

- I .07 0.00 O.~O

NLN speoctra ~

E.,. Frl 1 ,. O.I7(J() Vl " E.,. FrlV. O.I!JOt) ~ tl

SI#I (dll)- 21. DODO , ~ '" .... -J

~ ~ ~ ,

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 XIO-'


NEN speoctra ~

E.,. Frl". O. r!JOt) ~

" E.,. FrlV. O.I7(J() ~ tl

SI#I (til). 2 I • DODO , Cl

~ .... ...., ~ ~ ~ ,

1.00 I.~ 2.00 2.~0 3.00 3.~0 4.00 4.'0 ~.OO XIO- 1


XI02 0.30 EVON speoctra

-0.03. E.'. Fri". 0.1700

-0.36 E.t. Fr(2" O.,!JOt)

-0.69 SI#I (til'. 21.0000

-1.02.

-I.~

-1.68.

-2.01

-2.34 ,

o C -4 ~ C -4

> . > r

I • ~ o -1

-2.6".

-3.001. .. ..... .. L .. .. ~, " ........ "",, .. '" .... "" .. ,,"" .. " .. .I ~ 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 f.50 5.00 •

XIO-I '" t\J , FRACT. OF SANPL ING FRED.

NodiFi~d peN 5p~ctra Xl02 0.14

-0.01 E.t. Fril,. 0.1700

-0.1". E.t. Fr(2). O.I!JOt)

-0.33. SI#I (til,. 21. OOOD

-0.f9

-0.65

-0.81

-0.9".

-1.13.

-1.29

-1.44 1"''''''''''';;''''';;''1 ,eel' e""i""U"'~"""i""e""i"'''''''' • "".,,, ... eo.1 0.00 O.~O 1.00 1.'0 2.00 2.~ 3.00 3.'0 f.OO 4.'0 '.00


t\J til ...,

~ z

t\J en I

::I > ~ I

CD o

t\J .. w

til


"'" ~ '" t:l (I') ll.. \

~ .... ...,J

~ ~ ~ ,

\J"1 I ~

en "'" ~

'" ~ ll.. , t:l ~ .... ...,J

~ ~ ~ ,

X10' f1LN spectra xl02 EVON spectra 0.62 "'" 0.30

-0.06 Est. Fr( I)· O. I!J~ ~ -0.031 I I Est. Frl I)· 0.1700

-0.74 '" -0.36 Est. Fr(1). O. ,,;~

~ Est. Frl1" O. I!JOO

- I .43 -0.69 Q.. SNfI (dIU- 10.0000 SNII (ell). 20.0000 ,

-2. 11 -1.02

-2.79 ~ ""

-1.3' .... -3.47 ....,J -I.6B.

~

~ -4.15 ~ -Z'O'I -4.81 ~ -2.31 , -5.52 -2.61

-6 .20 i Oil 5:: "uG .. 0 , .,: : , """I"" "Ii i'''''' "";;;,.",,,,, ... : it "US ; Ii II D "',,: 'Ii" U; "Ii "",I -3.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 1.'0 5.00

Xl0- 1 XIO-I F~ACT. OF SANPLING FREO. FRACT. OF SANPLING FREO.

xl02 Xl02 N£N sppctra NodiFied peN spectra 0.10 "'" 0.14

-0.01 Est. Fr( 1 )- O.I!JOO ~ -0.01 Est. Frll,. 0.1700

-0.12 '" -0. 11 Est. Frl1" 0.' 700 ~ Est. Ft--U" O.I!JOO

-0.21 -0.33 ll.. SNfI (ell'· 1O.000D SNfI (ell'. 10.0000

-0.35 ,

-0.49

~ -O'SSl

-0.46 ~

JUt ....

-O.'B ....,J -0.81 ~

-0.69 ~ -0.97

-O.BI ~ -1.13. ,

-0.92 -1.29

-1.03 - 1.44 0.00 0.'0 1.00 1.'0 2.00 2.50 3.00 3.50 1.00 4.50 5.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00

XIO-I XIO-I F~ACT. OF SANPLING FREO. FRACT. OF SANPL ING FREO.


0 c .... ~ c .... :> · :> r-

, • ~ 0 -1 N ::J • • m )( . ,.. N , N UI ...,

~ z .. N en • ::J

:> ;10

• CD 0

N . . ~ .. ~

"" ~

"" ~ a... , @ "" ...... ..,J

~ ~

~ ,

\J1 I ~

\,()

"" ~ "" t:) (n

a... , t:)

~ ...... ..,J

~

~ ,

Xl0' 0.1& NLN spf5ct.ra ""

-0.05 E.t. Fr(')· 0.1600 ~ -0.5& "" - I .07

SIll (din- 10.0000 ~ a...

- I .57 ,

-2.08 51 "" ......

-2.59 ..,J

-3.10

-3.SI

~ ~

~

Xl02 0.30. EVDN spf5ct.ra

-0.03. E.C.. Frl,,· O. 1700

-0.36 E.C.. Frl2). 0.1500

-0.69. SNII t.). '0.0000

-1.02.

-1.3!1

-1.68.

-2.01

-2.34

o c .... " c ~

> • > r

I , ~ o ~

-1. I 2

-1.&3 .-------- .. - .... 0.00 0.50

,

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 XIO-l

-2.67.

~

-3 ·~:oo,o:~,: 5o~Sii"':;:OO"3:S'ii"f:o;, .';50' J 00 I E XIO-I

~ , I

FRACT. OF SANPLING FREO. X'O ' 0.64 NEN spf5ctra

-0.06 Est. FrCl). 0.1100

-0.77. EsC.. Fr(2)- 0.1500

- 1.47. SIll (.)- 10.0000

-2.18.

-2.88.

-3.'9.

-4.29

-'.00

-'.71

-6. "', lou.c.;, .:=>til"""; $ ."",,,e,,,, e;e.su;,:",...c;. .. ,,,,,U;h;'UU,,,,,;u, ' .. "",ue" u ,,, .. I

"" ~

"" ~ , Cl

~ ...... ...,J

~ ~ :t! ,

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 i.oo i.50 5.00 XIO- I



XJ.~ NodiFi~d peN sp~ctra I

-0.01 EsC.. Frl". 0.'700

-0.17. E.C.. Frt 2,. 0.1500

-0.33. SNIII t.). '0.0000

-0.49.

-0.65

-0.81

-0.97.

-1.13

-1.29.

-'.441"", "",..;;"",,,,,,,,,,,,,,,. cs""""";:;"'''''''UUCCSO''''"eu,eu "i"'" "'I" a .1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 i.OO i.50 5.00



JV , ..., (.II

" ~ z

~ I

:::I > ;10 I

CD o

JV o •

W en UI (.II

'" ~

'" ~ Q. , Cl

~ ..... ..,J

~

~ ,

\J1 I

N 0

'" ~ '" 5l Q. , Cl

~ ..... ..,J

~ ~ ~ ,

Xl0' NEN sppct,ra 0.60

-0.061 M EsC.. FriO- 0.'700

-0.72 i E.C.. Fril J. D.''''''

-1.39 J SNR (dB). '.DODD

-2.05

-2.72

-3.39

-4.04

-4.71

-5.37

-6.03 0.00 -(j"~SO--'-'~OO --,·~SO-:i.OO i~so 3'~OO 3".SO 1.00 1.50S".00

Xl0-l FRACT. OF SANPLING FREO.

Xl02 NodiP;pd peN sppct,ra O. 14 '"

-0.01 Esc.. FriO- D.'700 ~ -0.17 '" E.C.. FriV. D.'''''' 5l -0.33 Q.

SNR (dB ,. , • DODD , -0.49

-0.65 Cl

~ ..... -0.81 ..,J

-0.97 ~ ~

- I. 13 ~ ,

- 1.29

-, • 4f4f lasue "".....:=::; .. ," "f"""'"'''' Ii''::;''' .. "",,,,,, E t;e """"ue',,'" ,,,,,,,,,,1 0.00 o.~o 1.00 I.~O 2.00 2.~0 3.00 3.~0 i.OO i.~O ~.OO


,,,,. NoI .... '",,"Id

..,AX t ~

SA" .FA. 1 .000

AN'L I TUOS t 1.00 1 .00

FREO!I. t O.I!OO 0.1700

SMt. C dB,. 1.000 1.000

SlANO .DEY. I 0.3518133728111901

AESOL.llnlT '0.0100

EXACT Covorlonc. nolrl.

Xl02 0.30. EVON sppct,ra

-0.03. Est. FreO· D.'700

-0.36 E.t. FreIJ. D.''''''

-0.69. SNR (dB)· '.DODD

-1.02.

-1.35

-1.68.

-2.01

-2.34

-2.67

-3. ooL ....... ,L,~.~ ........................ .................... .I 0.00 0.50 1.00 1.50 2.00 2.~0 3.00 3.50 4.00 i.~O ~.OO



o C -f "V C -f

> • > r

I

~ o -f

~ , I

m )(

• " IV , IV UI ...,

~ z

..., en I

::I > ~ I

CD o

...,

... (oJ

...

"" ~

'" ~ Q , ~ .... ...J

~ ~ <: ,

\J'1 I

N -l>

"" ~

'" ~ Q , Cl

~ .... ...J

~ ~ <: ,

)('0' N£N sppctra 0.58

-0.061 f'1 E.C.. FrO)· D. 15~

-0.69 t E.C.. Fr( I)- D. 16~

-1 .32 ~ SNR (dB)· 11.0000

-1.96

-2.59

-3.22

-3.86

-1.19

-5.13

-5.76 • ____ • ___ ,._ •• _. __ •• - •• ------.-------_ ... -----_.--.--.------.-.---- --O' ._-. __ -.-._-- -_ow

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 )(10- 1

FRACT. OF SANPLING FREO. x,02 0.'4

NodiF;pd PCN sppctra ""

-0.01 E.C.. Frl I)- 0.1700 ~ -0.17 '" E.C.. Fr(I)- 0.1500

~ -0.33 Q SNR (dB)· 11.0000 ,

-0.49

-0.65 Cl

~ .... -0.81 ...,J

-0.97 ~ ~

-I. 13 <: , -1.29

-, • 'I if 1 '" e ,...:::, 'p" e, " "ue .;;... .. , "I"" "su, " .... ,; a eo" """",,1 0.00 O.SO '.00 '.SO 2.00 2.S0 3.00 3.S0 ".00 ".SO 5.00

XIO-' FRACT. OF SANPL ING FREO.

nnl NoI .... lftUSOld

N1AX t ~

SA".FIt. 1 .000

AIW'l.I n.os a 1.00 1 .00

FREas. • O.I!OO 0.1700

!Nt. ( dB)I 8.000 8.000

ST AIC) .DeY. I O.398107'8"~~

AESOl.llnlT 10.0400

EXACT Covarlone. na~rl.

x,02 0.30 EVON sppctra

-0.03. E.C.. Frl,,. D.I1OD

-0.36 E.t. Fr-t 2" 0.1500

-0.69. SNR (dB). II. tJtJtJtJ

-1.02.

-1.35

-1.68

-2.01

-2.34 ~

-2.67.

-3.001. ....... . L ..... ~ .. ".. ..""" .......... " .... " .. " ........ I 0.00 O.SO 1.00 I.!SO 2.00 2.!SO 3.00 3.!SO <f.00 ".50 5.00

XIO-I FRAC T. OF SANPL / NG FREO.


o c ~ ." C -I

> > r-

, • ~ o -f

~ , I

m )(

• ,.. '" , '" CJI

'" ~ Z .. N en I

:::J > 010 I

CO o

'" .. ..., .. N en

""' ~

" ~ n..

-~ .... -J

~ ~

~

-

V1 I

N N

""' ~ " Cl Vl n.. , Cl

~ .... -J

~ ~ <: ,

XIOI 0.15

-0.04

-0.54

-, .03

-1 .52

-2.0'

-2.50

-2.99

-3.18

-3.97

-1.16 ........ 0.00 0.50

Xl02 O. If

-0.01

-0.17

-0.33.

-0. f9.

-0.65

-0.81

-0.97.

- I. 13

-1.29

NEN spE'ctra

E.t. Frl ". 0.1500

911 (.,. D.DOOD

'.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 X'O-'


NodiFiE-d peN spE-ctra ""' E.t. Fr("· 0.1 '1OD ~

" E.t. Frl1'. D.I~ ~ n..

911 (dB'· D.DOOD , Cl

~ .... -J

~ ~ <: ,

-, • "'.,. , e c su"":=::, Ii' e ."ii' e II' hi u;;;.., 'hUSi """", ceo" Di' 0 u; "", .. I 0.00 0.'0 1.00 1.50 2.00 2.'0 3.00 3.'0 f.OO i.'O 5.00

XIO- I


~~~~. Nal~ .lftUtIOld

N1AX I~

SA" .FR • I .000

NA.I TU)S 1 1.00 I .00

FREas. 1 O.I~ 0.1700

!Nt. ( dB)I 0.000 0.000

SU"'.DEV. 1 , .000000000000000

AESOL.ll"IT 10.0400

EXACT CO¥Orlonc. na~rl.

Xl02 0.30. EVON spE'ctra

-0.03. E.t. Frl,,. 0.1700

-0.36 E.t. t:rlz,. O.I~

-0.69 SNR (dB'. O.DOOD

-1.02.

-1.35

-1.68

-2.01

-3.ooL ...... ",L .. ~ ~ ..... " ""', " .. " ............ , .... .I -2.3f

-2.67

0.00 0.50 1.00 1.'0 2.00 2.'0 3.00 3.50 f.OO i.'O '.00 XIO-I


FIG. ( 5.B ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *

o c: ~ ~ C -4

> > r

• • ~ o ~

~ , •

m )(

• " IV , IV CJI ...,

~ Z

IV en • :I > ,., • CD o

IV .. CJI .. ~

~

~

" ~ Q.. , Cl

~ ..... -J

~ ~ ~ ,

V1 I

N VJ

~

~ " Cl V) Q.. , ()

~ ..... -J

~ ~ ~ ,

Xl02 0.11

-0.01

-0.' 7

-0.33

-0.19

-0.65

-0.91

-0.97

-1 .13

-1 .29

-1 .11 0.00 0.50

Xl02 0.30

-0.03

-0.36

-0.69

-1.02

-1.35

-1.68

-2.01

NodiFitad PCN Sptact.ra ~

E.,. FrO" 0.1700 ~

" E.,. Fr(1)- O.I!JOt) ~ Q..

SNIt • -50.000 (ell, , Cl

~ ..... -J

~ ~ ~ ,

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO-I


EVON SPEtC t.ra ~

E.,. Frl". 0.1700 ~ " E.,. Fril'- O.I!JOD ~ Q..

SNR • -50.000 (ell, , ()

~ ..... -J

~ ~ ~

-2.3f

-2.67

,

-3."',1 ...... , .. L,~ ~ ........ , ....... , ...... , ................ .1 0.00 o.~o 1.00 I.~O 2.00 2.~0 3.00 3.~0 f.oo f.~O ~.OO


Xl02 0.11

-0.0'1 I I -0.17

J n n -0.33

I II II -0.19

-0.65

-0.91

-0.97

-I .13

-I .29

NbdiFi~ PCN sptact.ra

E.'. FrO,. 0.1700

E.'. Ilrll" O.I:JDQ

!iIfII • -•• DOD (ell'

o C -t "V c -t

> . > r

I

~ o .... ':l , ,

-1 . 14 1 ""cGsu,"";;::'"",,, .",,,.,uc,,,, ... :o':::;"''''''';::;''5''',,,, "ses,ou, .. , .. ",,;.; .... 1 ~ 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 •

XIO-I " '" , FRACT. OF SANPL ING FREQ. Xl02 0.30 EVON sp~ct,rtJ

-0.03. E.,. Fri,,. 0.1700

-0.36 E.,. Fril,. O.I:JDQ

-0.69. SNII • - •• DOD (ell,

-1.02.

-1.35.

- 1.68.

-2.01

-2.3f ~

-2.67.

-3. ",,l .......... L .. .. ~ ............... no.... .. .. "no .............. .1 0.00 0.'0 1.00 1.50 2.00 2.'0 3.00 3.'0 f.OO f.,O '.00


'" UI

" ~ z

'" en I

:::I > ;IU I

~

UI . . ~

'" c.J

FIG. ( 5. C) ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *

" ~ '" ~ It ,

~ ..... ...,J

~ ~ <: ,

\f1 I

N ~ "

~

'" Cl V) It , Cl

~ .... ...,J

~

~ ,

)(102 0.10

-0.01

-0.12

-0.23

-0.)45

-0.16

-0.~7

-0.68

-0.79

-0.90

-1 .02 .--.-.----.----.

0.00 0.50

Xl02 0.19.

-0.02.

-0.23

-0.44

-0.65

-0.86

-, .06

-, .27.

- 1.48

f1L11 spE'ctra "

E.C.. Frl 1 ,. D.lBOD ~ '" E.C.. Frl2,. D.I~ ~ It

Fr.s.".- D.D3DO SF , ~ "" .... ...,J

~ ~ ~ ,

1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 XIO-l

FRACT. OF SAI1PLING FREO.

11£11 spE'ctra

E.C.. Frl". D.I~

E.C.. Frl2,- D.lBOD

Fr.s.".- D.D3DO SF

" ~

'" ~ It , Cl

~ ..... ...,J

~ ~ <: ,

'.00 '.'0 2.00 2.S0 3.00 3.'0 4.00 4.'0 S.OO XIO-'

FRACT. OF SAI1PLING FREO.

x,02 0.29. EVOI1 spE'ctra

-0.03. E.C.. Fro('" D.IBOD

-0." E.c.. Frl2'. O.I~

-0.67 Fr._. - D.DJDD $F

-1.00.

-1.32

-1.64

-1.96

o C -4 "'0 C -4

> . > r

I

~ r o -2.28

-2.61

-2 ,931 """"""~":,~,, .. , .......... , .. "" .. " .. , " .. """" .I ! 0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.50 4.00 4.50 S.OO ~

x,02 0.14

-0.01

-0. '7.

-0.33.

-0.49.

-0.6S

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X'O-' FRAC T. OF SAI1PL I NG FREO.

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Xl0-l FRACT. OF SANPLING FREO.

XIO ' N£N spt!'ct,ra 0.B4~ ____________________ ~ ______________ ~

I -0.08 E.C.. Frl 1 ,. O.I!JOO

- 1.01 EsC.. Frl2 ,. 0.1700

- 1.93. ". 402

-2.86

-3.7B.

-4.70

-'.63.

-6."

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~ .... -J

~ ~ ~ ,

-B. 401." •• ". """., ... ""',, I""" 001"'" "1""""'1 ".'::-:0000,00' ~I 0.00 O.~O 1.00 I.~O 2.00 2.~0 3.00 3.~0 i.OO i.~O ~.OO

XIO- 1


XIO-I FRACT. OF SANPLING FRED.

XJ.O:7 NodiFi~d peN sp~ct,ra I

-0.09. E.t. Frl". 0.1700

- 1.04 Est. ~t2'. O.I~

-2.00. ". 402

-2.96

-3.91

-4.B7

-'.B3

-6.7B.

-7.74

-8. 701IChouun;.c::: .... ,' hU ... , .. ,,, 1"1"""",;:;:'0:"'"""",,,; """",es, e ",. """.1 0.00 O.~O 1.00 I.~ 2.00 2.~ 3.00 3.'0 4.00 i.~O '.00



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CD o

N -g o

IJ1 I

'vJ ~

XIO' '" 0.62 f1LN sp~ctra

i

Xl02 EVON sp~ctra '" O.OS~i __________________ -=~ __ ~ ______________ -' ~ -O.OS E.t. Frl I)· 0.1600 ~ -.01 Ed. Frll)- 0.'300

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~ ~ <: ,

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" ~ Q.. , Cl

~ ...... -J

~ Q::

~ ,

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- I .11

-2. I 2

-2.81

-3.49

-1. I 8

-1.97

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" ~ Q.. , ~ "" ..... ..J

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0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 X10- 1


XIO' NEN sp~ct.ra 0.74~1 ______________________ ~ ______________ _

-0.07 E.t. Frl'). 0.1700

-0.8S, E.t. Frll)- O.I~

-'.71 N. 101

-2.~3,

-3.34

-'f.16

-'f .S8.

-~.80

-6.61

"" ~

" ~ Q.. , Cl

~ ..... ..J

~ ~ ~ ,

-7.431 ...... , .. '" .. , ........... """"""" .. "" ".,"'" ... ,,,'::::::::::,;;,,,,,,:J 0.00 o.~o 1.00 I.~O 2.00 2.~0 3.00 3.~0 i.OO i.~O ~.OO

XIO- I


-. " E.t. PrU" 0.I1f1O

-.21 ". 20'

-.31

-.'fl

-.~I

-.61

- .71

-.81

o c .... ~ C .... > > r

I

~ r o -4

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XIO- I


Xd~;6 NodjFj~d peN spE'ct.ra )

-0.08. E.t. Prt I,. O.,~

-O.SI E.t. Ff-oU" 0.'70()

-1.75 ". 20'

-2.'S.

-3.43,

-'f .27,

-S. "

-~."

-6.78,

-7 .62 .... 10 3"ei~ou""''''''U'''iO"e''D sa lUi ... ,;;tu'hosI"'" eu'fC""'''';lsus.e; .. ,,,,,,,;,ul 0.00 O.~O 1.00 I.~O 2.00 2.~0 3.00 3.~0 4.00 4.~0 ~.OO

XIO-I FRACT. OF SANPLING FRED.

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)(10' NEN sptact.ro 0.65

-0.071 M E.t. FrO ,. D.I7OO

-0.791 I E.C.. ~1'. D.'!HID

-1.50 I \ N- 10'

-2.22

-2.91

-J.66

-1.38

-5. '0

-5.92

-6.51 0·'<X;"()"'SO'I"~OO".50 2".00 i.50 J".OO J'.50 1'.00 • ... 50 5-.00

)(10- 1


XIO' 0.68

NodiFipd PCN sptact.ro "'"

-0.07 Ed. FrO" 0.'750 ~ -0.82 ""

E.C.. Frl1" D. 14~ ~ -I .~8 Q...

". '0' -2.33

, -3.08

Cl

~ ..... -3.8f .... -f.'9

~ ~

-'.3f ~ ,

-6.10

-S.~1f1$ $ e~ .. e 0, '" , u eo OUI '0' ,,:::.., 5 h: IS i" "us.. 'CO" ,

0.00 O.~O 1.00 I.~O Z.OO z.~O 3.00 3.~ f.OO f.~O ~.OO XIO- I

FI?ACT. OF SANPL ING FRED.

",,. NoI.." .lftUSOld

fl'tAJ( • 101

SA" .At. I .000

N'Fl.1TOO5 • 1.00 I .00

FMo!. I O.I!OO 0.1700

Inll..Phe .. I 0.00 0.00

SNI. ( dB" 40.000 10.000

STAND.~Y •• 0.0100000000000000

IlESOl.ll"' , '0.0099

SIrQ..ATEO Coverlenc: .... ,,1_

X'O' 0.81 EVON sppctro

-0.08. Ed. ~,,. 0.'''''

-0.98. ~.c.. Frll" 0.''''''

-1.87. ". '0' -2.76.

-3.66

-f."

-,.f,

-6.3f

-1.2f

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XIO-I FRACT. OF SAffPL /NG FRED.


o C -4

~ -4

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~ , •

m )(

• ..... CJ) , o ~

~ z

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o

A ..' '" .. UI N

\J'I I

'vol 0--

XIOI "" 0 .69 NEN Spl"CtrQ

j

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"" ~ " ~ 'l , t:l ~ .... " ~ ~ ~ ,

-0.92 E.,. Frll). D.'!HID

-, .59 ". .,

-2.33

-3.09

-3.91

-4.59

-5.~

-S.IO

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XIO- I


XJ~;, Nodif'ifld peN spflct.ro

I -0.07 E.,. Frl'" tI.'!HID

-0.'" E.t. Frll)11 tI.'700

-, .62, ". .,

-2.40

-3.IB,

-3.9'

-4. '13,

-'.50 -6.28

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-., .OGI"", eeL, .. "OJ'" Ii. ,., ,~"" puc ",ee,." .. ee, "" "'"' Ii' I 0.00 0.50 '.00 ,.~o 2.00 2.'0 3.00 3.'0 ".00 ".50 '.00

XIO-' FRACT. OF SANPLING FREO.

"". Nol .. elttuMld

MIA. • "

IAJIt .At • • I .000

N9lITdJS • 1.00 1.00

FREo!. • O.I!OO 0.1700

1"ll."'-.' 0.00 0.00

!INIe ( cfJ,' 40.000 10.000

'TAMJ.DEY •• 0.01 OOOOOOOOOOOOOO

AESOL.llniT '0.012'

SI .... ATEO C.",.,.lerte. ftllrl.

X'O' EVON spftCt,ro O·~I

-0.09, E.,. Frf, Ie tI. '!DO

-1.02, E.,. Frf ~,. tI.''''''' -1.96

". ., -2.90,

-3.84

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-5 • ."

-6.65

-'1.59

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XIO-I FRACT. OF SAf1Pt.ING FREO.


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X'O' 0.54

-0.05

-O.6~

- 1 .24

-1.83

-2.12

-3.02

-3.61

-1.20

-1.79

-5.39 0.00 0.50

XIO' 0.'9.

-0.06

-0.70

-1.35

-1.99.

-2.6f

-3.28

-3.93

-f.,7

-'.21

-'.8G . - . 0.00 0.'0

f1EN spE"ct.ra

Ed. Frl"· D.I.!JO

E.'. Fr-<1'. D.'~

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1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO-I

FI?ACT. OF SANPLING FI?EO.

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E.'. Frl"· D.'.oo ~ "-

E.'. Fr-<1,. D.'I#OO ~ Q.

lie f' , ~ ~ .... ..,J

~ ~ <: ,

1.00 1.'0 2.00 2.'0 3.00 3.'0 f.OO f.,O '.00 XIO-I

FI?ACT. OF SANPL ING FI?EO.

nn. Nol ... lftUtIOld

..,AX . ... SA" .FIt. 1 .000

NPL I TUDS 8 1 .00 1 .00

FRE05. t O.I!OO 0.1700

I" I" .Phe.. t 0.00 0.00

9NR. ( dB" 10.000 10.000

B"'''' .DEY. • 0.01 OOOOOOOOOOOOOO

~5OL.ll"I' 80.0241

SUU.ATEO CO¥erI~ ,,- .... ht

XIO' 0.71 EVON spE"ctra

-0.07. E.t. Frl,,. D.'''''

-o.~ E.t. trrl1,. D.'fOD

-1.63 ". .. ,

-2.41

-3.19.

-3.97.

-f.7'

-'.'3 -6.31

-7.081 " "hi .n" .""co " " . p9n. eo • .ee,,, """ sue su.", :;:;::'50,,1 0.00 0.'0 1.00 1.50 2.00 2.50 3.00 3.'0 f.OO f.,O '.00

XIO-I FRACT. OF SAI1Pt..ING FI1EO.

FIG. ( 5.23 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *

o c ~

" C ~

> . ~

I

~ o ~

':l , •

m JC • " CD , .A

.., ~ Z

N ...., I :I > '" • ~

.A

...., .. S

"" ~ " ~ Q..

~

~ ..... -J

~

~ ~

\J'l I

I..J.j

CD

"" ~

" ~ Q..

~

Cl llJ

'" ..... -J

~ ~ :ct

-

X101 0.59

-0.06

-0.70

-1 .33

-1 .97

-2.6'

-3.2"

-3.98

-1.~2

-~. '6

-5.79 .. - -.-- .. -----0.00 O.SO

X'O' 0.61

-0.06

-0.73

-'.41

-2.08

-2.75

-3.42

-4.10

-4.77.

-5.44

-6. " . . 0.00 O.~O

ffEff sppct.ra

E.,. F~t' ,. D.'SOD ". ",

'.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 Xl0- 1

FRACT. OF SAffPL ING FREO.

NodiFipd peN sppct.ra ""

E.,. F~tl'. D.lf50 ~

" E.,. Frl v- D.'750 ~ Q..

". ", ~

Cl

~ ..... -J

~ ~ ~

~

1.00 ,.~ 2.00 2.~0 3.00 3.~ 4.00 <f.~ 5.00 XIO-I


,,,,. Nal ..... Inuwld

..,AX In

IMP.FIt. , .000

~ITlm I '.00 '.00

FREO!I. I O.'!OO 0.'700

1"1l.""'.' 0.00 0.00

!INt. ( cit,. 40.000 40.000

STANIUJEY. I 0.0' oooooooooooooo

RESOL.llniT 10.0400

SIIU.ATEO CowerIMe. "'., ..

X'O' 0.74. EVON spftCtra

-0.07. E.,. Frl I,. D. 1 f",

-O.BB E.,. p".t~,. D. I~

-1.69

". " -2.50

-3.31

- •• 12.

- •• 93.

-'.74

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XIO-' FRACT. OF SANPL ING FREO.


o c ~

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~ z

.... ..... I :J > :IU , CD o

.. .... w

VI I

\.N '-D

~ Xo'~~ I10djFjtad PeN sptactra i

Q) "tJ -0.06

" -0.G9

~ I:l -1.33 , -I .97

Cl ~ -2.&0 ...... ...,J

~

~ ,

-3.24

-3.98

-1.51

-5.15

E.,. Fri,,. 0.'Il00

E.,. Fril" 0.'400

". ,S

~ XO'~~G f1odiFi~ PeN 5~tra •

~ -O.OS

" -0.67

~ I:l -1.28 , -1.90

~ -2.51 ...... ...,J

~

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-3.1 J

-3.71

-1.~

-1.97

E.'. Frl,,. O.,:no E.,. Frfl,. O.,~

". " o c ~ 'V C ~

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I

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XIO-I

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FRACT. OF SAf1PLING FREQ.

x,o' EVON sp~ctra 0. 70

1 -0.07 E.,. Frl'" 0.'''''

-0.8f E.l. Frll" O. '400

-, .6' ". 'S

-2.38

-3. '5

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0.00 o.~ '.00 ,.~ 2.00 2.::J0 3.00 3.~0 f.OO f.~O ~.OO X'O-,


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FRACT. OF SA11PLIAG FREQ.

JtlO' EVON spt'ctra 0.68. •

-0.07, E.I. Frf,,. II. ,3!tI

-0.81 E.l. Frfl" II.'''''

-, .56

". " -2.30

-3.05

-3.19

-f.53,

-6.02,

-5. 711 a, ', .. " ... e U e e eo .,. c. , sue us.,s ., "..~ :1 0.00 o.~ '.00 ,.~ 2.00 2.~ 3.00 3.~ f.OO f.~ ::J.OO

X'O-' FRACT. OF SANPL ING FREO.

FIG. ( 5.25 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods -

.. , CJt

" ~ Z .. N

" I

~ ~ , ~

-~ N .. ....

~

~

" Cl (/l Q.. , Cl

~ .... .... ~ ~ ~ ,

V1 I

.t::=-O

~

~

" Cl (/l Q.. , Cl

~ .... .... ~ ~ <: ,

X,o' NodiFipd peN spectra X10' l10diFipcJ Pen sprctra 0.34 ~ 0.22

-0.03 E.'. Fr('" 11.'300 ~ -0.021 f\ 1\ E.,. FrO" II.''''' -0.4' " -0.26

E.,. Frll" I.''''' ~ V E.,. ~1" 11.'100 -0.78 -0.51 Q..

". • ". 1 -I .15

, -O.~

-I .52 fi1 ~

-0.99 ....

-I .89 .... -I .23 ~

-2.27 ~ -1.48

-2.S4 <: -1.72 , -3.01 -1.96

-3.381 " ,,:t;. .ee , e. , e ,eo", ".,eu~", see, sI" cuu so us,,, 'I'" I -2.20 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 1.SO 5.00' 0.00 0.50 '.00 '.50 2.00 2.50 3.00 3.50 1.00 4.50 5.00

XIO- I XIO-I FRACT. OF SAI1PL ING FREQ • FRACT. OF SAI1PL ING FREQ.

X'O' X'O' EVON sppctra EVON s~t.ra 0.'2 ~ 0.30

-O.~ E.,. Frl I,. I.''''' ~ -0.03 E.,. Frl I,. II.'''''

-0.63 " -0.36 E.,. Frll" I.'M(J ~ E.'. Frll" 11.1100

-, .2' -0.70 Q.. ". • ". 1

-, .78 ,

-, .03 Cl

-2.36 ~ -'.36l/ ~ I .... -2.9f .... -, .69

~ -2.03 -3.51 ~ -f.09

~ -2.36 , -".67 -2.69.

-5.25 -3.02. 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 f.OO f.50 5.00 0.00 0.50 '.00 '.50 2.00 2.50 3.00 3.50 ".00 ".50 5.00

XIO-I XIO-' FRACT. OF SAf1PL ING FREO. FRACT. OF SA/fPL ING FREQ.

FIG. ( 5.26 ) POWER SPECTRAL DENSITY ESTIMATE ~ For diFFerenl PSDE melhods *

0 c -4

~ ..... > . > r

• ~ 0 ..... ~ , , m JC • ..... CD , w v

~ z

-N ..... , ~ :IU I

~

l~

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"" ~ Q.. , C)

~ ...... ~

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x'o' NlN sppctra x,o' EVON sp~ctra 0.11 '" 0.67.

-0.01 E.,. Frl"· D. 'fi~ ~ -0.071 1\ 1\ E.t. Frl, Ie D.'fOCI

-0.50 "" -0.81 5JI6l • ".tJOO (dB' ~ V '.t. FrlZIe O.'IDO

-0.95 -1.55 It !WI. ".tJOO t.,

-1 .11 ,

-2.29

-1 .87 &1 "'f

-3.03 ......

-2.32 ~

-2.78 ~

-f." ~ -3.23 ~ -,.~ , -3.69 -'.99 -".151 uou " cue , " eo I,n" " " c; """ 0, ees ai " , c,ue' ,,::-:::"'1 -6.73

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 4.50 5.00 0.00 0.50 '.00 ,.~ 2.00 2.50 3.00 3.'0 f.OO f.SO '.00 X10- 1 X'O-'

FRACT. OF SANPL ING FREO. FRACT. OF SANPLING FRED.

X'O' X'D' NEN spf?ctra NodiFif?d PeN spf?ctra 0.f9 " 0."

-0.0' E.,. Frl',. D.'§DO ~ -0.06 '.t. Frl, Ie 0. 'fDtJ

-0.59 "" -0.66 E.,. Frll,. D. 'fDtJ

~ '.t. Frll" 0. '§DO

-,., 3 -, .27 Q.. 5JI6l • §!.tJ(JtJ tdB' !WI. 65.tJOO t.,

-, .67 ,

-, .88 C)

-2.2' ~ -2.4

91 / \ ......

-2.7' ~ -3.09 ~

-3.30 -3.70 ~ -3.8f

~ -f.31 , -f.38 -4.92.

-f.92 -'.'2. 0.00 0.50 '.00 '.'0 2.00 2.50 3.00 3.'0 f.OO f.,O '.00 0.00 0.'0 '.00 '.50 2.00 2.50 3.00 3.50 f.OO f.50 '.00

X'O-' X'O-' FRACT. OF SANPL ING FREO. FRACT. OF SANPL ING FREO.


0 c -4

" c -4

> . > r

• ~ 0 -4

~ , , m X • " crt , ..., til ...,

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-0.01 E.'. Frl',. 11.',;,0 ~ -0.071 f\ " E.l. Fro('" tI.'<fOII

-0.50 " -O.SI SI#t. so.""" t,., 5l V ,.,. IW Z,. tI. ,f/DO

-o.~ -1.55 a.. fNII. SO.tK1D t.,

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-2.29

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-2.32 -.J -3.77.

~ -f.51 -2.79 ~ -3.21 ~ -,.~ , -3.69 -6.00

-~ . '5 the. "0; co " .. "eo, , " " "'" "ue i s e "" .. " s '" a .,,". e "" ::::::=::,...t -6.74 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 0.00 0.50 1.00 I.SO 2.00 2.SO 3.00 3.50 4.00 4.50 5.00

.'0-' XIO-I Ff?ACT. OF SAffPL lNG FREO • FRACT. OF SAffPL ING FRED.

XIO' XIO' ffodjFj~d PeN sp~ctra ffEff splIct.ra 0.49 '" 0.'5

-0.0' E.,. Frl',. II. '!DO ~ -0.06 E.l. Fro(,,. tI.,ftI(J

-0.'9 " -0.66 E.,. FrlI" 11.14110 5l E.t. IWZ,. tI.'''''

- 1.13 -, .27 a.. SI#I. !D.""" t", !NIl. SO.IIDO t.,

- 1.67 ,

-1.88

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~ -2.49. .....

J \ -2.'15 -.J -3.,Oj -3.29

~ -3.70 ~ -3.S"

<: -f.31 , -f.38 -f.92

-f.92 -'.53. 0.00 0.'0 1.00 1.50 2.00 2.'0 3.00 3.'0 ".00 ".SO '.00 0.00 0.'0 1.00 I.SO 2.00 2.50 3.00 3.SO 4.00 4.50 '.00

XIO-' X'O-I FRACT. OF SANPLING FREO. FRACT. OF SAffPLING FREO.


0 c -. ~ -. > • > r

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X10-1 FRACT. OF SANPL ING FRED.

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FRACT. OF SAI1PLING FRED. FRACT. OF SAffPLING FRED. X,O' XIO' NodiF;~d PeN s~tra NEN splI'ctra 0.'7 "" 0.60

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XIO-' XIO-I FRACT. OF SAI1PLING FREO. FRACT. OF SAf1PLING FREQ.

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FRACT. OF SAI1PL ING FRED. FRACT. OF SAI1PLING FRED. Xl02 Xl02 /1E11 SPE-C tro NodiFiE-d PeN sp.ctro 0.17 '" O. ,..

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F~ACT. OF SAf1PL ING F~EO.

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FRACT. OF SAIfPLING FREQ. FRACT. OF SAI1PLING FREQ.

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Chapter Six

THE PREDICTION OF THE EVDT PERFORMANCE

FROM THE BEHAVIOUR OF THE EIGEN VALUES

OF THE COVARIANCE MATRIX

CHAPTER SIX

THE PREDICTION OF THE EVDT PERFORMANCE FROM

THE BEHAVIOUR OF THE EIGEN VALUES OF THE COVARIANCE MATRIX

6.1. INTRODUCTION

Eigen vector decomposition techniques gave excellent

performance in terms of detection and frequency resolution

of the two closely separated signals in white Gaussian noise

which formed the basis of investigation. The key to this

excellent performance was the estimated number of signals,

which depends to a large extent on the multiplicity of the

smallest eigen value of the random process covariance

matrix, which is not always obvious as was demonstrated in

chapter 4.

In this chapter, the effect of the different parameters

mentioned in chapter five on the behaviour of the eigen

values will be studied from which we can predict the way in

which these parameters will affect the detection and

resolution of the EVDT approaches. For the purpose of

comprehension, this chapter will deal with the exact as well

as the estimated covariance matrix.

6.2. TEST PROCEDURE : Fig(6.1) shows the flow chart of the Fortran 77

program written to test the effect of these parameters on

the behaviour of the signal and the noise eigen values of

the covariance matrix. The parameter under study is changed

in steps so that the random process data samples or their

6-1

THE EFFECT OF "HICH PARAMETER

DO YOU "ANT ?

1. SHR Changes

3.0bservat. Lenqth Chanqes

4.Relative Phase Chanqu

READ IAHS

SET SHR:0 and READ P(Max)

SET P(Min):0

READ P (MiX.)

READ P(Min) a P(Max)

P(Min) : Initial value for parato\.ttr under stud\J.

P(Max) I naxiMUM value tor the above parAMettr.

HOTE I Resul~s can be .ither plotted or wrltt.n on screen or as HC

READ S IG. PARAM. a GEN. DAtA

INCREMENT PARAMETER

EIGEH STRUCTURE DECOMPOS.

~. 1

PLOT or ~RITE

RESULTS

YES

Fig.(6.~) FLO~ CHART FOR PROGRAM RAGAD ~HICH SIMULATES THE EFFECTS OF THE DIFFERENT PARAMETERS ON THE EIGEN VALUES BEHAV10UR

6-2

covariance matrix is generated, which is then decomposed

into its eigen values and the associated eigen vectors using

the standard methods. Plots for these eigen values are

obtained at the end of the process. The other parameters are

kept constant at values well above their critical values in

order not to affect the behaviour of the eigen values.

6.3. THE EFFECT OF OBSERVATION LENGTH VARIATIONS:

As was mentioned in chapter five section (5.2.2.1),

the detection and resolution capabilities of the eigen

vector decomposition approaches are highly affected by data

length variations, especially when the data length is short.

In this section the behaviour of the signal and noise eigen

values with the data length variations are studied for the

cases of single and multiple sinusoids in white Gaussian noise.

6.3.1. SINGLE SINUSOID CASE:

In chapter three section (3.1.1) the detection

capabilities of the different power spectral density

estimators were studied, from which we saw that nearly all

the estimators were capable of detecting easily the single

sinusoid in white noise though the data length was short.

However, the eigen vector decomposition technique was always

the best technique in detecting signals corrupted by noise,

the effect of this corruption can be severe when the data

length is very short.

Fig(6.2) shows the eigen values of a random process

consisting of a single sinusoid of unit ampl i tude and a

normalized frequency of 0.25£5 in white Gaussian noise

having a signal-to-noise ratio of 10dB, -this is the same

test example used earlier in chapter three-. From this

6-3

figure we can see that the signal eigen value level, i. e

eigen value No. 121 was very high compared with the noise

eigen values levels. Hence we can expect that the wavenumber

and the frequency estimations will be perfect even at short

data length. Fig(6.3) shows the EVM and HPCH estimates for

the PSD of the above random process for sample length of 4

and 11 points from which we can see that these two

algorithms were able to detect the signal (with some bias)

at a data length of as short as four samples , so in this

case we can consider that the data length variations has no

effect on the signal and noise eigen values of the random

process which in turn means that it will not affect the

detection capability of the eigen vector decomposition technique.

6.3.2. MULTIPLE SINUSOIDS CASE:

Fig(6.4) and Fig(6.5) represent the behaviour of the

signal and noise eigen values of the random process used in

testing the resolution capabilities of the different

algorithms in chapter three. These figures show that the

signal eigen values have a damped oscillation around a

constant value as the data length increases and they reached

their steady state values as the data length became high

where we can expect the resolution to be perfect. This

oscillation was caused by many factors such as the

correlation between the two sinusoids, which decreases as

the frequency separation increases causing less oscillations

to the signal eigen values and hence improving the

resolution, -see Fig(6.6)-1 and the fact that the covariance

matrix estimated from a short data record is too far from

the true one, and that is why these eigen values became

nearly constant at long data records.

6-4

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X'O' x,01 Obs~rvatjon L~n9th ( Samplps ) Obs~rvatjon L~n9th ( Samplps )

EIGEN-VALUES No. •


1, 2, 3,

11,12,

4, 5, 6, 7, 8, 9,10

FIG. (6.2) THE EFFECT OF OBSERVATION LENGTH ON EIGEN-VALUES

0\ I

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FIG. ( 6.3 ) POWER SPECTRAL DENSITY ESTIMATE • For diFFerenl PSDE melhods -

-, • .....

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Fig(6.S) shows the signal eigen values behaviour as the

data length varies from 69 to 1000 samples, from which we

can say that these eigen values remain nearly constant at

their higher values and the noise eigen values remain

constant at their lower values which indicates that the

detection and resolution capabilities of the eigen vector

decomposition approaches are very slightly affected by the

data length variations when it is long and these

capabilities tend to be perfect (ideal) when the observation

length approaches infinity where on the other hand we can

expect no estimation bias will exist.

It is obvious that these sets of eigen values curves are

function of frequency separation between the two signals,

- i. e we can get another set of curves as the frequency

separation changes-. Fig(6.6) shows the eigen values as a

function of observation length for two different sets of

signal frequencies, (0. lSf s' O. 2f s) and (0. lSf s' O. 35f s)

from which we can see that the difference between the two

signal eigen values levels became less as the frequency

separation increased allowing the second (lower) signal

eigen value to be increased and to move away from the noise

eigen values levels which in turn means an improvement in

the detection and resolution capabilities of the eigen

vector decomposition technique.

6.4. THE EFFECT OF SNR VARIATIONS:

6.4.1. SINGLE SINUSOID CASE:

The signal and noise eigen values of the single

sinusoidal signal in white Gaussian noise random process

mentioned in section (6.3.1) for both the true and estimated

covariance matrix cases are shown in Fig( 6. 7 ). In both of

these cases the signal eigen value level was well above the

noise eigen values levels even at low values of SNR which

6-7

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Obsprvation Lpngth ( Samplps ) XloZ

Obsprvation Lpngth ( Samplps )

FIG. (h.4) THE EFFECT OF OBSERVATION LENGTH ON EIGEN-VALUES EIGEN-VALUES No. 11,12,

--~--~

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EIGEN-VALUES No_: 1, 2, 3, 4, 5, 6, 7, 8, 9,10 EIGEN-VALUES No. : 11,12,

FIG. ( 6.~) THE EFFECT OF OBSERVATION LENGTH ON EIGEN-VALUES

(J I .... o

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x,tiI x, til ObsEY"'vation LEJngth ( SamplEJS ) ObsEJrvation LfJngth ( Samplps )

EIGEN-VALUES No. • 1, 2, 3, 4, 5, 6, 7, 8, 9,10 •

EIGEN-VALUES No. • 11,12, . THE EFFECT OF OBSERVATION LENGTH ON EIGEN-VALUES

(() C)

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4, 5, 6, 7, 8, 9,10

FIG. ( 0.7) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES

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f\J , ~ ...,

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indicates that the eigen vector decomposition technique will

be able to detect this signal however small the SNR value

is. Fig(6.8) shows the power spectral density estimates of

this random process using the eigen vector method (EVH) and

the new proposed method (HPCH) at low values of SNR, for the

case of true and estimated covariance matrix -40dB and -5dB

respectively, from which we can say that these two methods

can easily estimate the exact signal frequency (i.e without

any estimation bias) and they can estimate it with small

amount of bias when the estimated covariance matrix is

employed.

It is obvious that due to the influence of the noise upon

the signal, the steady state level of the signal eigen value

for the estimated covariance matrix is lower than that for

the true covariance matrix, whereas the steady state levels

of the noise eigen values for the case of estimated

covariance matrix are higher than those obtained for the

true covariance matrix case.

6.4.2. MULTIPLE SINUSOIDS CASE:

The signal-to-noise ratio of the two equipower

signals was varied from OdB to 87dB. The signal and noise

eigen values as a function of this range of SNR variations

are depicted in Fig(6.9) for the case of estimated and true

covariance matrices. The figure shows that all the eigen

values of the random process were gradually decreasing as

the SNR increasing and after a certain limit of SNR values

the eigen values remain constant. This SNR value (l imi t )

when calculated from the graphs, -see Fig(6.10)-, appeared

to be equal the threshold SNR value below which the noise

eigen values levels were comparatively high when compared

with the second (low) signal eigen value and that is why

6-13

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Signal / Nois~ Ratio (dB) Signal / Nojs~ Ratio (de)

EIGEN-VALUES No. : 1, 2, 3, 4, 5, 6, 7, 8, 9,10

EIGEN-VALUES No. : 11,12,

FIG. (h.9 ) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES

{

." C) .. .. :':1 f"') -I C) (J)

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Signal / Nois~ Ratio ( dB )

FIG. (6.10) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES EIGEN-VALUES No. : 10,11,

1

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FIG. (6.11) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES EIGEN-VALUES No. : 1, 2, 3, 4, 5, 6, 7, 8, 9,10

C) ·0

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EIGEN-VALUES No.


1, 2, 3,

11,12,

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9 18 21 36 f, Sf 63 72 81 90

Signal / Nojs~ Ratio (dB)

4, 5, 6, 7, 8, 9,10

FIG. (6.12) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES

there was no resolution achieved when the SNR value

below the threshold value of this technique, Fig. (6.11)-.

was

-see

Fig(6.9) also shows the signal eigen values of the same

random process for the case of true covariance matrix. Since

the noise eigen values remained constant what ever low the

SNR value was, then this means that this technique is

capable of resolving the two closely separated signals at

the worst situation, where lower SNR values are assigned.

Again these sets of curves are function of the frequency

separation between the two signals to be resolved, so

another sets of eigen values curves can be obtained when

other values are assigned to the frequency separation, see

Fig(6.12) for the case of O.OSf frequency separation. s

6.5. THE EFFECT OF FREQUENCY SEPARATION VARIATIONS:

The plots for the signal and noise eigen values for

the different values of frequency separation between the two

signals are presented in Fig(6.13) and Fig(6.15) for the

estimated and true covariance matrices cases respectively.

Fig(6.13) shows the first signal eigen value decreases

sharply and the second signal eigen value increases until

they reach the same level and then continue to be nearly

constant at this level. The value of frequency separation at

which they first met is called the discrimination frequency

separation value at and above which the two signals have no

effect on each other and the estimates become bias free.

Fig(6.15) shows these two eigen values for the case of the

true covariance matrix from which it can be seen that the

second signal eigen value curve is exactly the reciprocal of

the first signal eigen value curve and the decoupling or

6-18

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:1 ~ . ~

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FrEJQUEJncy SEJporotion ( Froct. oF Sompl.Fr )


EIGEN-VALUES No. • •

1, 2, 3,

11,12,

4, 5, 6, 7, 8, 9,10

FIG. (6.13) THE EFFECT OF FREQUENCY SEPARAT. ON EIGEN-VALUES

I I

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XIO-2

___un ___ ,

It! •• 'ev el...-,Id

MIA. • 2!

.... FIt. ".000

NWIl.I nm • I .00 , .00

RIEGI. • O.'!GO

Inll....... 0.00 0.00 ... • •• 000 •• 000

.'MD."Y .• 0.031121771801..,.

~.LI"I' to.04OO

IIIU.AT!G tower. ...tr I.

Fr~qu~ncy S~porotjon ( Froct. of Sompl.Fr )


EIGEN-VALUES No. 10,11,

C) C) -.

~ Ol

~I Q ~ , I

C) C") -. ~I ;:j ~

(J ' I I ru

, ~

f'., '-C) ....J

~

...... :1 ~

-I ~

'-::::> Q '--::::> C)

X'O' 2.~

2.~

2.00

1.75

, • !SO

,.~

UJ ::::l '.00 ...... ~ ~ 0.1'

(!) ..... O.!SO llJ

0.7$

r I r~· /-. . ' 1" i/ J.,Cd.A,l()1.- froio. Ie

Z 0.082 £ 0.35 0.70 1.05 1.'10 1.75 2.10 2.'15 2.80 3.15 3.!KJ

XIO-I

Itt • IID.-V .,...,.Id ~. • JS

tANt.... • 1.000

Nft..naI I 1.00 '.00

RIEGI. • o.la

Itt 1 t. ."-.. 0.00 0.00

... • •• 000 •• 000

"MU.DI!Y •• 0.OJ'e22"&IOI~

.... _lll'" '0.0400

hAC' Cower I....:. ... lr I.

Fr~qu~ncy S~paratjon ( Fract. of Sampl.Fr )


EIGEN-VALUES No. • 12,11,

"IIIIII!Iiiii •

I

..... ~ -. (Xl '\'--. --0 0) , )...

~ , ~

-. ~ Q::

() , I ,

ru (\f ru t-O -J Q , , ...... -J ~ -~ . ~

'-~ Q t--~ 0

--- -- ,

XIO-2 7.00

&.30

,.so I

f.901 I

f.20

3.~

lIJ ~ 2.110 -.I

~ , 2.10

~ t!) ....... I.fO lIJ

0.70

D.OO~ , • • • , • • • • • 0.00 0.20 0 _ fO 0.60 0.80 1.00 1.20 I. fO 1.60 '.80 2.00

XIO-2

........ .., ........ d

.... • !O

....... • '.000

..... ITUII • '.00 '.00

RIAl. • 0.'. Inll...... O.GO 0.00

... • JO.oao JO.oao

.'MG ...... O.OJ'~777IIDI""

.... .L Iftl' 10.0200

fUC' eo..rlenn ... Lrl.

Fr~qu~ncy S~oroLjon ( FracL. of Sompl.Fr )

FIG. (6.16) THE EFFECT OF FREQUENCY SEPARAT _ ON EIGEN-VALUES

EIGEN-VALUES No. 10,11, --

decorrelation time is clear to be equal to 1/0.08f . s

Another point which can be pointed out from these two

figures, is that when the two signals had nearly the same

frequency, (0. 15fs )' -i.e for approximately zero frequency

separation value-I and for the case of true covariance

matrix, the second signal eigen value had a very small value

which was equal to the noise eigen values (0. 001dB), -see

Fig(6. 16 )-. So, in this case, we had only one very large

signal eigen value giving rise to one signal to be detected

and that confirms what has been said in chapter five,

whereas for the estimated covariance matrix case this signal

eigen value had a significantly high level (4. 65dB) when

compared with the noise eigen values at zero separation

(1.04dB) and hence it is expected that two signals will be

obtained from the estimates of these approaches which again

confirm what has been obtained in chapter five.

6.6. THE EFFECT OF THE RELATIVE PHASE VARIATIONS:

The initial phase of the second signal was changed in

steps from (0 degrees) to (360 degrees) and the signal and

noise eigen values were plotted for the case of estimated

covariance matrix only, since as was mentioned in chapter

five, when the true covariance matrix is used the initial

phases and so the relative phase between the two signals

have no effects on the estimates.

Fig( 6. 17) shows the two signal eigen values behaviour,

from which the corresponding eigen value reached its minimum

level at an initial phase of 90 degrees in which case the

two signals were orthogonal. Hence they had no effect upon

each other and we can expect that they have the best

resolution. As the relative phase increased, this eigen

value reached its maximum level at relative phase of 270

degrees and that explain why the worst resolution have been

6-23

obtained, -see chapter five section (5.2.2.4)-.

When the first signal was given an initial phase of 60

degrees and 180 degrees the sets of eigen values curves were

shifted by those amounts as shown in Fig(6.18) and Fig(6.19)

from which we can decide whether or not the first signal has

been assigned an initial phase shift. The amount of phase

shift assigned to the first signal can also be calculated

from these plots by plotting two horizontal lines, one

through the mid-level point of its eigen value curve, and

another line at the value on the same eigen value curve

corresponding to 180 degrees phase shift. Now, if the two

lines coincide, then there must be an initial phase of 0 or

180 degrees being assigned to the first signal, depending

upon wheather the eigen value curve is a negative going or

positive going sine shape as shown in Fig(6.17) and

Fig( 6. 19 ). On the other hand, if the two lines do not

coincide, see Fig(6.18), then we read the phase angle

corresponding to the intersection between the first line and

the eigen value curve, ~ ,and the phase shift of Intersec.

the first signal is calculated as follows;

~ = 180-~ 1 intersec.

(6.6.1)

- 180-224.8 - 64.8 degrees

where ~ 1 is the initial phase of the first signal

with respect to the observation window.

Other sets of frequencies were used to study the effect

of the relative phase variations on the behaviour of the

eigen values. Fig(6.20) shows that the swing of the eigen

values was almost always less when the frequency separation

6-24

was increased l -i.e the amplitude of the swing in the eigen

value curve is inversely proportional to the frequency

separation between the two signals-I and that again confirms

the effect of the correlation between the two signals which

decreases as the frequency separation increases.

6-25

b·~6 ,

cl 0)

~I

~ I I ~

e.

~I ~ Q:: I

t\J I I

...,J , ....

:1 ~ ., ~

~I

(tl ..... :::>: C)

!JO

.. , "0

;r.J

30

2'

~ 20 -J

~ I " ~ ..... 10 ltJ ,

0 0

M'~. ,,~~ s-

t'\o...x. \t~~ -= ~S" ~ .. \&. p~w:t\f"-tu..e ::a 20.

" OJ

ColWo.l~c\ ~ ~~ ~ 6 4·~ !:::!.. Go

36 72 108 Iff 180 216 252 288 32f 360

5~cond 5 j 9no l Pho Sift 5h j Ft (Olftgr"lftlfts)

EIGEN-VALUES No.

EIGEN-VALUES No.

• •

•

1, 2, 3,

11,12,

... -'ev ....... d

.... • 2'!1

....... • '.010 ~.nm • I.GO I.GO

AlGI. • O.I!IOO 0.1700

Iftll.Phe... IO.GO

... • •• GOO •• GOO

.' .... IIIY •• a.OJ'aJJ';WIOI"~

..... u .. " to.CHGO

II .... AfR r:...r. "'~I.

4, 5, 6, 7, 8, 9,10

FIG. (A.18) THE EFFECT OF 2nd SIGNAL PHASE SHIFT ON EIGEN-VALUES

O)i ~ •• r-.... C) .. t.o -C) 0) , ~

~ , ('0

-. ~ Q::

~I , ,

~ ~I ~ . ~

t-::J Q t-::J CJ

~ ....J

~ I

~ l!> ...... UJ

::-:=:====:::===:::~::::::=:=~~===:::===:::=========~~ -- --

~

.. ~

"0

~

30

~

" '0

,

M~~. volu.( ::: S".5"

M~· \ttt1..~ ~ 34·sMtcl. \t~~ ~~\.vJ: -CoJ.Mcd-~& P k~e ~

:lo (')

\~O

0 1 , , , , I , , • , , o 36 72 '08 ''If 180 216 252 288 32'1 360

S~cond Signal Phas~ ShiFt (D~gr~~s)

'" • _I.., .1,.,.,111

MM. t 2S

~.... • 1.000

NWIllfdJI • 1.00 1.00

RIfOI. • 0.'" 0.'100 I"ft. ...... t '''.00 ... • 30.000 30.000

lUND •• '. t

lHIll.lInlf to.0400

SlI'U.A," tow.r. n." I.

FIG. (6.19) THE EFFECT OF 2nd SIGNAL PHASE SHIFT ON EIGEN-VALUES

EIGEN-VALUES No. 11,12,

d Ol

~I Q.. ~

~ -. ~.

0\ n:: I

r-0 , I

\0 , ....

:1 ~ ., ~

~I

X,o' :I.~

~.~

~.OO

, .75

,.~

ll.I :::) '.00 -.J

~ , 0.'"

~ (,!) to 0.50

o.~

0.00 0 36

... . --. . ..., .t ___ ...

~. 2S

~.Rt. 1.000

~Inm I 1.00 '.00

~. t 0.'* 0.2000

Inll."'-_ I 0.00 ... t •• GOO 3O.GOO

., MID .IIEY • t O.OJIUlnI

..... lI"" 10.0400

X,o' I.~

1.35

'.20 1.05

- • -lev -,-, ..

.... I 2'5

........ • 1.000

..... 'nm • '.00 '.GO ..... • 0.''''' 0.""

.ftn ....... I 0.00 ... • JD._ JO._ ., ...... Y •• o .OJIUJ77'I

IUD. .LI"" to.OtGO

D.90'L_~ _____ ~-----""""""'.r--""'"

D.'"

~ 0.60 ..,J

~ I 0 • .,'

~ (!) ..... D.30 UJ

o. "

0.001 • , , , , , ii' • •

72 108 Iff 180 216 :l52 :l88 32f 360 0 36 72 '08 Iff '80 :116 ~2 2fJfI 32'1 360

Sttcond Signal Phastt ShiFt (Dttgrtttts)

EIGEN-VALUES No_ •


1, 2, 3,

11,12,

4,

Sttcond Signa I Pha Stt Sh i Ft (Ottgr

5, 6, 7, 8, 9,10

FIG. (6.20) THE EFFECT OF 2nd SIGNAL PHASE SHIFT ON EIGEN-VALUES

Chapter Seven

CONCLUSIONS

AND SUGGESTIONS FOR FURTHER WORK

CHAPTER SEVEN

CONCLUSIONS

AND

SUGGESTION FOR FURTHER WORK

7.1. CONCLUSIONS:

Most of the power spectral density estimates

approaches have been examined using computer simulated data

during this study. Emphasis has been given to so-called high

resolution methods among which the Eigen Vector

Decomposition Technique has superior resolution capabilites.

A new eigen vector decomposition method has been proposed

whose detection and resolution capabilities have been tested

for the different circumstances and it proved to have

superior performance. Another area where a new method has

been suggested as well is the Partitioning of the random

process covariance matrix into Signal Subspace and Noise

Subspace by separating the signal eigen values from the

noise eigen values.

As a result of the intensive computer simulation

performed during this study, the following conclusions can

be drawn :

1. All the PSD estimators are capable of correctly

estimating the frequency of a single signal in white

Gaussian noise without any bias and are capable of

detecting and resolving multiple signal frequencies

with different amounts of bias when a sufficiently long

data record is employed.

7-1

2. Conventional PSDE methods became computationally

efficient with FFT, but they suffer from ambiguities

due to the side lobe leakage and their poor resolution capabilities.

3. Parametric PSDE methods have higher detection abilities

with better side lobe supression and they possess

higher resolution capabilities with significantly small

estimation biases when compared with conventional approaches.

4. Non parametric approaches to PSDE, and Eigen Vector

Decomposition Technique in particular, possess the

highest detectability and resolution capability.

5. In spite of the resolution capabilities of the

Modelling approaches and Burg algorithm, they suffer

from some practical difficulties such as the order of

the filter is not known a priori. Another major problem

associated with AR modelling is that it exhibits

spontaneous Line Spl i tting which is more I ikely to

occur when the SNR is high, the number of coefficients

is a large percentage of the data samples, the data

length includes some odd number of quarter cycles and

the initial phase is an odd multiple of p/4.

6. The Maximum Likelihood Method (MLM) provides direct

power estimation but it is unable to resolve two

closely separated signals, whereas Maximum Entropy

Method (MEM) gives no indication about the actual power

of the signal but on the other hand it possesses higher

resolution ability than MLM.

7 . Methods with best performance are often based on the

idea of decomposition of the covariance matrix into its

7-2

eigen values and their associated eigen vectors.

8. Intensive study was performed upon MLM, MEM, EVM, and

MPCM which proves that EVDT has superior performance

due to the lowest SNR threshold value it possesses

(>-90dB), the shortest data length -6 or even 4 data

samples on condition that the number of signals is

known a priori-, with which it can still give a

resonable estimate and the minumum frequency separation

(O.002fs ) between the two signals to be resolved when

compared with other approaches to PSDE.

9. Eigen vector Decomposition Approaches pioneered by

Pisarenko suffer from a number of disadvantages such as

the number of computations required by the full eigen

vector analysis and the evaluation of spectrum using

the noise eigen vectors, -except MPCM which uses the

signal eigen vectors-, In addition it shares with

Pisarenko Harmonic Decomposition (PHD) the practical

difficul ty related to the actual number of signals,

upon which the detection and resolution capabilitites

are highly dependent, and which is not normally known a

priori. This number, if overestimated, will give

spurious frequencies -except in the case of EVM-, and

if underestimated it will give highly smoothed spectra.

10. The new proposed method for separating the signal eigen

values from the noise eigen values proves its

effectivness especially when the true covariance matrix

is used.

11. EVDT is capable of resolving the two closely separated

signals with no limits to their relative amplitudes

ratio when the true covariance matrix is known. on the

other hand it is highly affected by the lower ratios

7-3

when the estimated covariance matrix is used.

12. Again when the true covariance matrix is used, the

detection and resolution capabilities of Eigen Vector

decomposition Technique (EVDT) are not affected by the

presence of a third strong nearby signal whereas MLM

and MEM capabilities are highly affected. On the other

hand when the estimated covariance matrix is used, MEM

is the less affected estimator.

13. The new proposed method -Modified Principal Components

Method (MPCM)- has the best resolution capability among

the EVDT approaches especially at low SNR values and

short data records.

14. Intensive study was conducted upon the different

parameters affecting the behaviour of the eigen values

of the covariance matrix from which the following

observations can be drawn :

14.1. It is possible to predict the performance of the

EVDT from the behaviour of the eigen values of

the covariance matrix.

14.2.

14.3.

Data length has a very slight effect on the

detection and resolution capabilities of the

EVDT and this effect vanishes as the signals

become sufficiently separated in frequency.

The threshold SNR value for the EVDT can be

calculated from the Eigen Values vs SNR curves

which show that above this SNR value the SNR

variation has no effect on the detection and

resolution capabilities of the EVDT.

7-4

14.4.

14.5.

Eigen Values vs. Frequency Separation curves

show exactly how the detection and resolution

capabilities of the EVDT improve as the two

signals become more and more apart from each

other. EVDT reaches its superiority in resolving

the two signals when they are separated by more

than Df=1/Td, where T d is the decorrelation time

which can be calculated from these curves.

Finally the initial phase given to one signal

can be calculated from the curves of the Eigen

Val ues vs. the Second Signal's Initial Phase

Variation. These curves show that the amplitude

of the swing associated with the second signal

curve is a function of the frequency separation

between the two signals.

7.2. SUGGESTION FOR FURTHER WORK:

Since it is the first time that the Space Domain

Signal Processing Approaches are being used in the

Time Domain Processing, it is of importance to :

A. Implement these algorithms to provide efficient

tools in time domain signal processing in terms of

computations and implementations especially as we

are now facing a revolution in VLSI design and

manufacturing and these algorithms have already

been implemented in the space domain.

B. Extend the study of the behaviour of the eigen

values of the covariance matrix from which we can

have more possible prediction and further

understanding of the EVDT performance and

limitations.

7-5

APPENDICES

APPENDIX ONE

REPRESENTATION OF THE COVARIANCE MATRIX IN TERMS OF ITS EIGEN DATA

Any matrix R can be analysed into its eigen data, and

computations involving matrices may be largely simplified by

using the two sets of parameters known as eigen values and

eigen vectors of the matrix.

Let R be the covariance matrix with dimensions HxH, and V

an eigen vector corresponding to the eigen value A, i.e.

for V ~ 0

RV = A.V (Al. 1)

For a typical HxM matrix R, there will be H such vectors.

Equation (Al.l) can be rewritten as follows:

(R-A.I)V = 0 (Al.2)

where I is the Identity matrix. Equation (Al.2) has a non

zero solution in vector V if and only if the characteristic

equation is satisfied;

i.e det(R-A.I) = 0 (Al.3)

The polynomial equation generated from this

characteristic equation written as :

f(A.) - det(R-A.I) (Al.4)

Al-l

has H roots (A.I s) and hence there are H eigen vectors

corresponding to these eigen values all satisfying

Equ(Al.l)1 or in other words,

RV. - A..V. ~ ~ ~

(Al.5)

since the covariance matrix of a stationary time series

is almost always positive definite [23}, then its eigen

values are both real and positive.

Let V be an HxH matrix formed from the H eigen vectors of

R as follows :

(Al.6)

and A be an HxH diagonal matrix formed by the H eigen values

of R as follows :

(Al.7)

Now rewrite the set of equations (AI. 4) in a single

matrix form as follows, [4B} and [53} :

RU - VA (Al.B)

and when the eigen values of R are distinct, the

corresponding eigen vectors will be orthogonal and hence

matrix VI -formed from these eigen vectors- is non singular

[ 23}.

Multiply both sides of Equ(Al.B) by U-1

, we get:

U- 1RU - A (Al.9)

Al-2

Now, if U is a unitary similarity matrix, then:

}( Al. 10)

for i=k i.e (Al. 11)

other vise

and so equation (Al.9) can be rewritten as :

(Al.12)

Now, since R is hermetian and A is diagonal matrix, then

Equ(Al.12) can be rewritten as :

OR H

R - L i=l

H A.V.V. ~ ~ ~

Al-3

(Al.13)

(Al.14)

APPENDIX TWO

DERIVATION OF THE OPTIMUM WEIGHT FOR CAPON FILTER

We have the average power output P given by;

p - ~RW (A2. 1)

and we wish to minimize this power subject to the constraint;

(A2.2)

Using Lagrange's method, we can perform this minimization

subject to the above constraint by defining a cost function

as follows :

(A2.3)

where A is an arbitrary constant. The minimization can be

achieved by differentiating the cost function with respect

to Wand equating the derivative to zero, but before that

let us assume, for generality, that Wand C are complex

vectors.

i.e W - W+ ·W

} r J j (A2.4)

and C - C+ ·C r J j

which implies two constraints as follows • •

Re[WTC] - 1 } (A2.5) Im[WTC] - 0

A2-1

and

(A2.6)

substituting this result in equation (A2.3) gives:

(A2.7)

The derivatives of P with respect to Ii and Ii are given r j

by, [48];

8P 2RW ---- -8W r r

8P 2RW ---- -8W J j

So 8H(W )

-----~-- = 2RW - A(C -jc ) 8W r r j r

Equating the derivative to zero will give :

whre

and

* RW = (3C r

f3= ")../2,

8H(W ) _____ 1 __ 8W

j

Or

is

-

a constant.

2RW + A(C + ·C ) - 0 J r j j

RW = -(3(C + 'C ) J J J r

(A2.8)

(A2.9)

(A2.10)

Multiply both sides of Equ(A2.10) by the operator j will

give :

A2-2

jRW = f3(C - jC ) = f3C* J r J

Combining equation (A2.9) and (A2.11) yields;

or

and

* R(W + jW ) = 2{3C r J

W o A = constant - -------1 * R C

(A2.11)

(A2.12)

(A2.13)

Multiply numerator and denominator of equation (A2.13) by

CT gives :

(A2.14)

substituting this value of A in equation (A2.12) gives us

the expression for the optimum weight as follows :

- 1 * R C TIl - ---------

o CTR-1C* (A2.15)

A2-3

APPENDIX THREE I

1. LIST (3A.1) DATA RECORD OF ONE OF THE EXAMPLES USED IN TESTING THE DIFFERENT PSO ESTIMATION APPROACHES

NMAX : 25 AMPLITUDS : 1.00 FREQS. : 0.2500 Init.Phase : 0.00 SNRs ( dB ): 10.000

0.3162277660168379 STAND. DEV . :

THE RANOOM PROCESS DATA SAMPLES :

Y( 0)Y( 1)Y ( 2)Y( 3)Y( 4)Y ( 5)Y( 6)Y( 7)Y( 8)Y( 9)Y( 10)Y( 11)Y ( 12)Y( 13)Y( 14)Y ( 15)-= Y( 16)Y( 17)Y( 18)Y( 19)Y ( 20)Y( 21)Y( 22)Y ( 23)Y( 24)-

0.11695939011609300+01 0.40883060962759580+00

-0.50034814137549390+00 -0.83972401547697030-02

0.14796233552576660+01 -0.13630862285486910+00 -0.12435749406673540+01

0.74144220108680440-01 0.10646211030968030+01

-0.45869413202394970+00 -0.98102902592295880+00

0.4891349243872809D+00 0.18698613223663300+01 0.19172647412666170+00

-0.11863772843188990+01 -0.50192154991385870-01

0.14414023315106200+01 -0.27232837508983960+00 -0.87105885969379300+00

0.10424647742874960+00 0.10282422188596980+01

-0.36665420084601050+00 -0.66296695765149220+00

0.19829341526256390+00 0.89579720112463920+00

"'(3-1

0.68405022969687590+00 0.1120480466941184D+Ol 0.13186811679340900+00

-0.8477784492382966D+00 0.8753906370500357D-Ol 0.11101395470235190+01 0.48364301752567300+00

-0.89094893286010060+00 -0.3016755283664857D+00

0.1009639590877976D+01 0.5057804559784573D+00

-0.1397039850903806D+01 -0.3861102031574317D+00

0.1058091967040075D+Ol -0.4895825204484141D+00 -0.8397691370132612D+00

0.1129023447591659D+00 0.1043549397963129D+Ol

-0.4927149606409684D+00 -0.1029957891621009D+Ol -0.4230550455120892D+00

0.9550033681737977D+00 -0.2861423056190439D+00 -0.1115733180063233D+Ol

0.4403240312434403D+00

2. LIST (A3.2) DATA RECORD OF ONE OF THE EXAMPLES USED IN TESTING THE DIFFERENT PSO ESTIMATION APPROACHES.

NMAX : 64 AMPLITUOS : 1.00 1.00

0.1500 0.1700 FREQS. : Init.Phase : 0.00 0.00 SNRs ( dB ): 30.000 30.000

0.0316227766016838 STAND. DEV . :

THE RANDOM PROCESS DATA SAMPLES :

Y( 0)Y( 1)Y ( 2)Y( 3)Y( 4)Y ( 5)Y( 6)Y( 7)Y( 8)Y( 9)Y( 10)Y( 11)Y ( 12)Y( 13)Y( 14)Y( 15)Y( 16)Y( 17)Y( 18)Y( 19)Y( 20)Y( 21)= Y( 22)= Y( 23)Y ( 24)Y( 25)Y( 26)Y ( 27) .. Y( 28)Y( 29)Y( 30)Y ( 31)Y ( 32)Y ( 33)Y ( 34) .. Y ( 35)Y( 36)Y ( 37)Y ( 38)Y( 39)Y( 40)Y( 41)Y( 42)Y( 43)-

0.20169593901160930+01 0.11104219444630160+01

-0.79487870357107140+00 -0.19499230005668160+01 -0.11868337811718650+01

0.57415466323191980+00 0.17767742717649000+01 0.13265952102058650+01

-0.32194526782997010+00 -0.16159420177177740+01 -0.13071196142775890+01

0.14567576973435620+00 0.13645864983546270+01 0.12189188387874460+01 0.6141013783893589D-01

-0.95607605298665690+00 -0.95225785530533810+00 -0.20777557684917260+00

0.6336538575646932D+00 0.72354289339112550+00 0.19380687808748480+00

-0.35370730827032590+00 -0.33810410578779470+00 -0.86899169891544360-01

0.56869432090719500-01 0.68404935546908480-01

-0.55241610472931370-01 0.11991562241531660+00 0.38702980119326690+00 0.32579544075554600+00

-0.17996957505808200+00 -0.66475441429084300+00 -0.60985393676185520+00

0.15037657840208530+00 0.99736273661938970+00 0.10016340091446680+01

-0.11975363951378470+00 -0.12383582415879420+01 -0.12717902918529340+01 -0.14571837194530440+00

0.1325041211431293D+01 0.15813620259783060+01 0.33275992564866450+00

-0.1368454023224522D+01

.-:3· ,

0.48688459168275140-02 0.16398629409213180+01 0.17575896027623660+01 0.21406486118076930+00

-0.15153275039578020+01 -0.17841671007734520+01 -0.48464694173407720+00

0.1259429011374204D+01 0.17455223657901930+01 0.57738035189176670+00

-0.98222108335509010+00 -0.15231907641338530+01 -0.6723307245186435D+00

0.66354457747172970+00 0.1311926618118088D+01 0.7262144521861759D+00

-0.47063551877488800+00 -0.94482768747610830+00 -0.59314744881948230+00

0.16882009637257360+00 0.60266790186402620+00 0.4141531487162449D+00

-0.1594102677560550D-01 -0.21113714141810770+00 -0.71184252850886460-01 -0.7309643150080884D-02 -0.11448524834536940+00 -0.22119108977502900+00 -0.1459772938039198D-01

0.38349926435481320+00 0.6163854711596512D+00 0.17700336785326500+00

-0.55111291652914510+00 -0.9101769234062238D+00 -0.37575722717636900+00

0.7366112594188643D+00 0.1264744693875607D+01 0.6181013097480524D+00

-0.6690850013025907D+00 -0.1568904111950104D+01 -0.9698313663087803D+00

0.6005573046462739D+00 0.1765873459981662D+01 0.1261971955938591D+Ol

Y ( 44)Y( 45)Y( 46)Y( 47)Y( 48)Y( 49)Y( 50)Y ( 51)Y( 52)Y( 53)Y ( 54)Y ( 55)Y( 56)Y( 57)Y ( 58)Y( 59)Y ( 60)Y ( 61)Y ( 62)Y ( 63)-

-0.18041269704737550+01 -0.63008858194273080+00

0.12302985690438250+01 0.19204688171309850+01 0.83326806943688330+00

-0.10255089441646580+01 -0.19850931691668470+01 -0.10874479235496670+01

0.87454551724698230+00 0.18942063571758590+01 0.13475018290001170+01

-0.55348326064370600+00 -0.18312662688752730+01 -0.13365518778097090+01

0.41250647768454050+00 0.15722426834320010+01 0.12946606473993860+01

-0.13402525834059000+00 -0.12467373135600130+01 -0.12088518594417510+01

.- 3 . .3

-0.4299200430336004D+00 -0.1838808402227356D+01 -0.1511253238302535D+01

0.19621362530223280+00 0.18176761141942830+01 0.1689347069969490D+01

-0.2483149482723509D-01 -0.1686464246947114D+01 -0.1814609169021447D+01 -0.29092166388350360+00

0.14396791111860810+01 0.1814421222204194D+01 0.4633049700921898D+00

-0.12863189733525270+01 -0.1696169806285736D+01 -0.6356809988250748D+00

0.9756929399937595D+00 0.15607791113801520+01 0.72988683146701070+00

-0.6535904428276002D+00

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