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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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o c .... "'0 C .... > • > ,.. • ~ o .... ~ I
m )( . ,.. A , en A ~
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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
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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.
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I .37
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0 •• ' UJ e;
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0.4' 0 Q.
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X'O-I FRACT. OF SAf1PlING FRED.
.. 83 P ISI4~NKO (PHD, .. thod
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0.57
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• ~ o ~
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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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AP • Nal..., .1...,.ld
MtAX • ~
SA ... FIt • • 1.000
NR.I T\D.J • 1.00
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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
-2.08 ~ Q.. ,
-3.08 f:l ~ -4.07 ..... -' ~
~ ,
-~.07
-6.07
-7.06
-8.06
E.C.. Fr('" o.~
XIO' YUlE-WALKER m~thod ~ 0.51~1------------------~==~~~------------_, Q) 'll -o.~ EsC.. Frl, ,. O.7.JIJ
" -0.61
-1.17 ~ Q.. ,
-1.14
@ ,...., -2.30 ..... -' -2.86. ~ Q:
~ -3.42.
-3.98. , -4.'"
o C -4 ." c -4
> · > r
I
~ o -4
~ , •
-9.05 1 usee ,e" '''''''fUC';'''';; "i'eu .. ,,,''os e eei ee he ',"'h' UfO" ,sc,,1 0.08 0.57 1.06 1.55 2.05 2.51 3.03 3.52 1.02 1.515.00
XIO- 1
-5. , OJ .. ,,,,,,,,,,,, .eo" ))..4",,,,,,,,,,00'" .,""""',. ''''''''''.".~u .,""'" ., ~ 0.04 0.54 1.03 1.53 2.02 2.~2 3.02 3." 4.01 4.50 '.00 ~
~
~
" ~ Q.. , Cl
~ ..... -' ~ ~ ~ ,
FRACT. OF SANPLING FREO.
Xl02 8LACKNAN- TUKEY mt;tthod 0.09·~. ___________________________________________ , ~
-.01 E.C.. Frl I ,. o.~ ~
-. " " -.22. ~
Q..
-.32 ,
-.43. Cl
~ ..... -'
-.74
~ ~ C!: ,
-.84
-.9",. . ,. " , , h hi e ,,,,,U"'f e i. eel e. U I""'''' est
0.0" 0.'" 1.03 1.'3 2.02 2.'2 3.02 3." f.O' f.,O 5.00 XIO-'
FRACT. OF SANPLING FREO.
XIO-I FRACT. OF SAffPL ING FREO.
XJ.O;" fJUf?G Algor; thm mt-thod i
-O.Of Est. Fro(',. O.~ -0.47.
-0.89
-1.32
-1.74
-2.17.
-2.60.
-3.88) " U'''''''''f U e s,,, h' .S. u. , """""" , c. eo" .... l..t. eo •• f 0.08 0.57 1.06 1.5' 2.05 2.,f 3.03 3.'2 f.02 f." '.00
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.. , ~ ,.." .... ..... ~ ~ ~ ,
\.N I
\..Q '" ~ \",
~ Q.. , t:l ~ .... ..... ~ ~ ~ ,
XIOI 8~G Algorjthm m~thod 0.79-
-0.081 A E.'. Frl"· D.ISfl
-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
Antl.1 TtfJS • 1.00 I .00
Flt!0!. I 0.1!OO 0.'700
.... n.Phe... 0.00 0.00
!JNRs ( dB,. 20.000 20.000
STAtCJ.(~y. • 0.1000000000000000
RE50... linn 10.0'S
o c ~ 'lJ c -4
> . > r-
, ~ o -4
!1 , I
m JC o ,.. w , en .. XIO' BURG Algorithm m~thod X'O ' 0.80 '" 0.82, BlRG Algor i thllt .. thod I ~
-0.08 E.,. Frl ". D.ISfl ~ -0.08, Ed. Frl,,. O. I,.,
-0.96 \",
-0.99,
-I .8~ ~ -1.89, Q..
E.t. Frl~" 0.17"
-2.13 ,
-2.80
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
-8.221 ..... ", .. "",,., us,,, 'I' • e $ .,ee,,,,,,,,. e "US" ." c,eu"" "",.,.. os.c;. .. "" •• , 0.08 0.~7 1.06 ,.~, 2.0' 2.'i 3.03 3.'2 i.02 i." '.00
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
j
Q) ~ -0.00 E.,. Frl 1 ,. tI.lSO:l
'" -o.~
~ Q. -, ..... ,
-2.' 2
~ -2.91 .... ~
~
~ ,
'" ~
'" ~ Q. , Cl
~ .... ~
~ ~ ~ ,
-J.~
- ... 19
-".97
-~.~6
-6. 24 1 rn .... n .. n. n pun .; """"'cus""", """"i so, ... ;;-""".J'III("; .. ec, "I 0.04 0.54 1.03 1.53 2.02 2.52 3.02 3.51 1.01 1.50 5.00
XIO-I FRACT. OF SANPLING FREO.
XJ r, 8U1?G Algor i th", ",t- thod
I -0.01 E.,. ~rl I,. II. '56' -0.13. E.,. ~rll" 11.'71'
-0.2~
-0.36
-0.48.
-0.60
-0.72.
-0.83
-O.~
-, .Oi'y:;;;;, ,e u ' .. "'" "eu, U us,,, """'I' e. h, uVil't',,,.J1II(:f. $ e .1 0.08 0.~7 1.06 I." 2.0' 2.,i 3.03 3.'2 i.02 i." '.00
XIO-I
FRACT. OF SANPLING FREO.
... MDI.., .1,..,.,ld
MM. • 14
SA" .FIt • I .000
ANIlITtm • 1.00 '.00
FRE05. • O.I!CJO 0.1'700
I"n........ 0.00 0.00
_. ( dB ,. 30.000 30.000
SUND.~ •• 0.031577778G015838
II:SG... Linn 'O.OIS
No. of coeffs. = 6
FIG. ( 3.6 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerenl PSDE melhods *
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
-1.78.
-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.
I1L ff sppc t,,.tJ
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.
IAfIt .FI. • , .000
Mft. 1 TdJS • , .00
FWf05 • • 0.2!OO
1".\.,.,... 0.00
,.. C dB,. 10.000
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 .. .. ~
"'" ~ " ~ Q. , ~ "'" ..... -.J
~ ~ ~ ,
w I "'" ......
~ llo
" ~ Q. , Cl
~ ..... -.J
~
~ ,
XI02 0.12
-0.01
-O.I~
-0.29
-0."2
-O.~
-0.70
-0.8"
-0.97
-I .11
-1.25 w - _ - ~ - - __ ... - - - -
0.00 0.50
X,O' 0.66.
-0.07.
-0.79.
- I .~I
-2.2f
-2.96
-3.68
-4.41
-,." -6.'" . - .
0.00 o.~
PeN sppctro X'O' "'" 0.7~
Ed. t:rl,,. ".~ I ~ "0.071 " -0.90
~ -1.12. Q. , -2.'4
~ -3.37. ..... ..., -4.19.
~ -'.01 ~
;t -'.84 , -6.66
ffUSlC ~thod
I E.'. f:#o(' Ie ". r.ItItJ
o c ~
" C -4
~ . ~ , ~ o -4
~ , •
1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 XIO- I
-1.4':}.iiO ii.so' ,a.DO I'. SiJ'f.Oii'i.;;o 'r.ooU:Uj.';o" 4':00 :".fNJ ~I.DO I ~ XIO-I
FI1ACT. OF SAffPL IN(; FIlEO.
ffEl1 SpPCtro
E_t. Fro(' Ie ".~
FI1ACT. OF SAf1PI...lNG FlIED.
"'" XIO' EVon spPCtra 0.84? ______________________ ~ ______________ ~
~ -0.08
" -, .01 Cl V) -1.94 Q. ,
-2.86 Cl ~ -3.79. .... ..., ~ ~ ~ ,
-4.12.
-'.64
-6.~1.
-7.~
E.t. Frl, Ie 0."""
-8.42t::=:::::, """ .. " .. , , ~, , e~ 1.00 ,.~ 2.00 2.~0 3.00 3.~0 f.OO .,.~ ~.OO 0.00 O.~O 1.00 ,.~O 2.00 2.~ 3.00 3.~ 4.00 4.~ ~.OO
XIO- 1 XIO-I FI1ACT. OF SAffPL IN(; F11ED. FI1Acr. OF SAf1PI...lNG F11ED.
-, eft .. ...,
! z .. .. • ~ ~ I
~
-fIJ
o .. fIJ CD
FIG. ( 3.9 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
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
.102 ~ 0.09 PERIODOGRAN m~thod
j
Q) "0 - .01
'" ~ Q.. , ~ ..... ..,J
~ ~ ~ ,
- .11
-.2' -.J' -."1 -.51
-.61
-.71
-.92
Ed. Frl,,. II.If.f
E.,. Frll,. II.lOJI
XI&?, ________________ ~==~~A~t~gor=_~j~t~ ___ ~~t_hod __ _, ~ 0.12,_
~ -0.01
'" -0.14
-0.27. ~ Q.. ,
-0.40 (l ~ -0.'3. ..... ..,J
~ ~ ~ ,
-0." -0.78.
-0.91
~.,. A-f I,. II. ,,.,
I.,. I'W~" II.IIIJ'
@ ~
~ -t
> ~
• ~ o -4
~ , - .92 1.,. , US" ,,,. C 'US" se." hce ., $Ch"'1""'''''' US"" Ii "",,: ..... I
0.08 0.57 1.06 1.55 2.05 2.51 3.03 3.S2 1.02 1.51 5.00 XIO-I
-I. '~:o'ii''O':S7''';':'ot;· ;'.~, ;:'0, i.~ti.0j"'j:~2--" •• 0;_.~" ~I.oo I E XIO-'
~
~
'" 5l Q.. , fi1 '" ..... ..,J
~
~ ,
FRACT. OF SAf1PLING FREQ. FRACT. OF SAI'fPI..ING FIlEQ.
X'O' YUlE-IiALKER ~I.hod 0.62Ti------------------------------------,
XJO;, BLACKNAN- TlJI(EY trtttl.hod . I ~
-O.~
-0.61
- 1.11
-, .73,
-2.29,
-2.~
-3.4'
-3.97,
-4.'3,
E.,. Frl I,. 11.10"
I.'. Frll,. O.If.f
~ -0.06
'" -0.73
5l Q.. -, .44 , -2.12,
Cl ~ -2.BI ..... ...,J
~
~ ,
-3.~,
-f.'9.
-f.B7.
-'.!76. 1.03 I.~3 2.02 2.~2 3.02 3.~I f.Ol f.5O ~.OO
XIO-I FRACT. OF SAffPL IN(; FIlEO.
I.,. fIIrf,,. II.Mn'
I.,. fIIrf ~,. fl. I»!
1.03 I.~3 2.02 2.~2 3.02 3.~I 4.01 f.~ ~.OO XIO-'
FRACT. OF SAffPt..ING FREQ.
FIG. ( 3.10 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerent PSDE methods -
4rI , m .A
" ~ Z
..., , i , ~
-~ -~ .. ~
W I
..... Q)
Xl0' ~ 0.63 NFN s~tra
j
~ -0.06 E.,. Frll)- D.iI'InO
'" ~ t\. ,
-0.76 E.,. Frll" 0.''''''
-, ."6 -2. '6
~ ~ -2.86 ~ .... ~ ~ ~ ,
~
~
'" ~ t\. , Cl
~ ~ .... ~ ~ ~ ,
-3.~
-".25 -".~
-6.35 1 e e 'cuu a '" us Ii " 0 "" •• S hau use ... .. .... ,,"""'''' ~$, • ....c:;;,,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 F~ACT. OF SAffPLlNG FRED.
X'O' f1l.N sppctra 0.48? ______________________ ~ ______________ ~
j -O.~ E.,. Fro('" D.'5Ot1 -0.'7.
E.,. Frll,. D.1t1«J
-, .'0 -, .63
-2.16
-2.68 .
-3.21
-3.74
-4f .191 • e e e; , u' •• 'I"" , eo .. " D,' , e.,. :=:" .. , J s I 0.00 o.~o 1.00 I.~ 2.00 2.~ 3.00 3.~0 f.OO f.~O '.00
XIO- 1
F~ACT. OF SANPLING F~ED.
"". MDI..., .lftUMld
NnA. . 14
IAfIt .At • • 1 .GOO
.... ITtm • '.00 '.00
FltEO!. • O.I!GO 0.2OOCt
... Il....... 0.00 0.00
,.. Cell,. 3D.GOO 3D.GOO
ITA..,.~Y •• 0.031111771801~
IfIOL.Lln,t 10.0'"
S,tU.."'BJ Ccwerl~ ... " ••
FIG. ( 3.11 ) POWER SPECTRAL DENSITY ESTIMATE ~ For diFFerenl PSDE melhods -
o c ~
~ -4
:. • :. r
, ~ o -4
~ , , m J( . ~
crt , en ~ ..,
~ z -.... S , ~
-'" w w .. ~
~
~ " f;l Q.. , ~ "" .... ...J
~ eli:
~ ,
W ~
I ~ ~ \0
" f;l Q.. , Cl
~ .... ...., ~
~ ,
.10' 0."7
-0.05
-O.~
-, .08
-1.60
-2.12
-2.;3
-3.15
-3.67
-4.19
-1.70 .- .. ---- ... -.---
0.00 0.50
XIO' 0.64
-0.06
-0.77.
- 1.47.
-2.18.
-2.88
-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
-1.81
-2.S7.
-3.54
-4.fO.
-5.2&
-S.I3.
-7.851... ., h' • ec e 0;' e. 5U u .. _ • " ,;;;:: e., 0.00 O.~ 1.00 I.!JO 2.00 2.!JO 3.00 3.~ ".00 ... ~ ~.DO
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
" -D.';' ~ «l -I.~ , -I." 51 -2.ff
"" .... ..,J -3.0f ~
~ -3.';4
-f.24 , -f.Bf
E.l. ~,,. e.,7,. E.I. IIWI" e.' ...
-,. 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
-,." -6.45
'.t. 1Irl,,. 0.'''' ht. flWl" 0.'7' ••. I •. J
-1I·2fl1 e " U • 0 n_,. , U ~« ,I O.Of O.,f 1.03 1.'3 2.07 2.'2 3.02 3." f.Ol f.50 '.00
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 -~
~
~ " ~ 11.. ,
~ ~
-.J
~
~ ,
~ w 1 ~ I\J ~
" ~ 11.. , Cl
~ ~
-.J
~ ~ ~ ,
.10' 0.64
-0.06
-0.77
-1.47
-2.18
-2.88
-3.~9
-4.29
-~.OO
-~.70
-6.11 .- --- --- ... ---_. 0.00 0.50
x,o' 0.",
-o.~
-0.S6
-, .:16
-1.86
-2."1
-3.01,
-3.67,
-'f.118
-,. "B, . . 0.00 O.~
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
~Inm I 1.00 1.00
FWEO!. I O.I!OO 0.'100
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>
~
~
~
" ~ Il. , ~ "" .... ...,J
~ ~ ~ ,
~
w I ~ t\)
t\)
" ~ Il. , Cl
~ .... -.J
~ § ,
XIO' 0.11
-0.01
-0.49
-0.94
-I .39
-1.83
-2.28
-2.73
-J.I7
-3.62
-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. ,,,., ~
" E.t. Frll" ".,f", ~ Il. , 51 "" .... -.J
~ ~ ~ ,
'.00 ,.~ 2.00 2.'0 3.00 3.!f0 f.OO f.!fO '.00 XIO-'
FllACT. OF SAIfPlING FI?EO.
"". 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
IlESOl.lIn" 'O.OlS
51 ...... ATBJ CO¥W,~. "'",.
X,O' 0.80. EVON SPKf.ro
-0.08. E.,. If/Irot,,. ". ,,,,,,
-0.g7. E.,. If/Irot ~ Ie ". , no -'.85 -2.74
-3.62.
-4."
-'.39 -6.28.
-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
~ Q. -0.99 ,
-I .~7
~ -1.91
"" .... -oJ
~ ~ ~ ,
~
~ "-
~ Q. , Cl
~ .... -oJ
~ § ,
-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
-,. '0
-, .62
-,. " -'.67.
-3. ,g
-".,,, -if .171 , e , e " " Ie e , eo.. H ... n~ w'p. ~,c::. os. ,I
0.08 0." 1.06 I." '.0' ,.,,, 3.03 3." ".02 "." '.00 X,O-I
FRACT. OF SAf1PL ING FRED.
... MDI .... 1...,..,141
..... t 14
IMP .FIt. t t .000
.... ITWI • 1.00 1.00
f:ltf0!. • o .• ~ 0.'''' '"Il.Phew. 0.00 0.00
_. ( ttl,. '.000 '.000
IUMJ.DEY •• o .sn4' ",.,.,.:147 IIfIOl. lI"" to.O'"
FIG. ( 3.16 ) POWER SPECTRAL DENSITY ESTIMATES * For diFFerenl PSDE melhods *
@ -4
~ --4
:. • > r
~ o -4
~ , m IC . " ~ 8t .. ..,
~ -~ • ~ • ~
-.. -.. ..., -8
w I
IV ~
.'0' '" 0.51 YUlE-tiALKER ~thod i
~ -0.05 E.t. Fri,,. D. "13 ........... 1...,.ld
" ~ Q.. , ~ '" ...... ...J
~ ~
~ ,
'" ~ '" ~ Q.. , Cl
~ ...... ...J
~ § ,
-o.s. E.t. Fril" ".'~ ... AX • I.
-, .17 IMP .At. ., .000
-1.73
-2.29 ~ITWI • '.00 '.00
-2.95 RtEG!. • O.'!GO 0.,7GO
-3.41 ' .. Il.Phe... 0.00 0.00
-3.97 _. ( dB" '0.000 '0.000
-4.~3 ,tAJCJ.OEY •• 0."'7777810'11371
-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.
-0.06-1 I E.t. Fri'" D. 'S3
-0.141 IV\ E.t. Frill- D. ,1'§
-1.42
-2.10
-2.18
-3.46
-4. ,..
-4.82
-S. I81 • SO • U" , 0 " '" , • ~$~ I 0.08 0.~7 '.06 I.~' 2.0~ 2.~f 3.03 3.~2 f.02 4.~I ~.OO
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
-, .33
-1.97
-2.61
-3.~
-3.98
-1.~2
-S.16
-S.90 . ~ .. 0.00 O.SO
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'
" -0. '3. Cl ~ -0.26 ,
-0.38. Q ~ -O.!JO .... ..,J
~ ~ :;(! ,
-0.52 .
-0."'.
-0.99
E.l. Frf,,. fI.'m"!J
'.t. ,,"1" tJ.'''' ~ ~
~ .... ~ • ~ r-
• ~ o .... ~ ,
-1. 19 1. ea. e.,,,,,,,, IU e •• au" sa", "" n ... n n. n eei' u. , s" .. n ,csl 0.08 0.57 1.06 1.55 2.05 2.51 3.03 3.52 1.02 1.515.00
Xl0- 1 -,. '2J.oii' Oi:~i";~iit; "i'::;~ ;:oj i:5fiAj.03 i 3.52 "j:om, loo I ~
X'O-I F~ACT. OF SANPL ING FREO.
XJ.o;, BLACKffAN- TUKEY Iftt'thod • ~
~ -O.~
" -0.5' Cl ~ -1.15 Q. ,
-, .72 Cl ~ -2.28 .... ..,J
~ ~ ~ ,
-2.113 .
-3.39.
-3.95.
-'f."
E.,. 1%-('" tJ. , •• E.'. 1%-(1" II. ,f.f
FRACT. OF SANPLING FRED.
~ x'o' YUlE-WALKER .. "hod 0.'9. •
~ -0.05
" -0.70. Cl ~ -,.~ ,
-2.00.
~ -2.5'f .... ..,J
~
~ ,
-3.29 .
-3.93.
-'f.~.
-'.23.
'.t. I'rf'" tJ.'" '.1. ""'1" tJ.''''
-'.871....:: e, • " : hi $ a 'U'" e "CUD e eu en. " us e "ee, '''''' I -'.05t:: eo" $ 0 u. sa cu •• ,,, '" e c, hiD Ii ;: .. :.X'l O.Oi 0.'. '.03 I.~3 2.02 2.~2 3.02 3.~' i.O' i.~O ~.OO
XIO- 1 O.Oi O.~. 1.03 I.~3 2.02 2.~2 3.02 3.~I •• 01 i.5O ~.OO
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
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FIG. ( 3.22 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
~ ~ -4
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• ~ o ~
~ • m JC • " (II , GI .A ...,
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FIG. ( 3.23 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerent PSOE methods ~
~ ... ~ ..... > • > r
• ~ o ..... ~ , I
m • • " ~ 4ft .. " ! z .. -" I
5 • ~
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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)=
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
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)=
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
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
" ~ '" ~ Q.. , ~ ~ .... .... ~ ~ <: ,
V'1 I .......
V'1
" ~
'" ~ Q.. , Cl
~ .... .... ~ ~ <= ,
XI02 NLN spect.ra " 0.10
-0.01 E.'. F,.,,)· 0.1700 ~ -0.12 '" E.,. Fr(1). O.I~
~ -0.23 Q.. !iNR , til). if 0 • 0000 ,
-O.~
-0."6 fa "" ....
-0.57 .... ~ -0.68 ~ <: -0.79 ,
-0.90
-1 .01 1",,, euuufll(";" :ow "es S Ii """"'''''''''';:;''''''''''''''''''1'''''''''''''''''''' """"I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
Xl0- 1
FRACT. OF SANPLING FREO. xlo2 0.18 N£N spect.ra
" -0.02 E.,. F,.O). O.'~ ~ -0.22 '" E.'. FrlV. O. '700 ~ -0.42 Q..
!iNR , til ). if 0 • 0000 , -0.62
-0.82 Cl
~ .... -1.03 ....
~ -1.23 ~ -1.43
~ , -1.63
-1.83~uuuLuuuuul U UU'~Ui~~ 0.00 O.~o 1.00 I.~O 2.00 2.~0 3.00 3.~0 i.OO i.~O ~.OO
X10- 1
FRACT. OF SANPL ING FREO.
Xlo2 EVON spec tra 0.301
-0.03 E.'. Frl')· 0.1100
-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 .. ~
FIG. ( 5.2 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
~
~
" ~ 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-'
FRACT. OF SANPL ING FREO.
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
FRACT. OF SANPLING FREO.
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
XIO-I FRACT. OF SANPLING FREO.
t\J til ...,
~ z
t\J en I
::I > ~ I
CD o
t\J .. w
til
FIG. ( 5.3 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
"'" ~ '" 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.
FIG. ( 5.4 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
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
FRACT. OF SANPLING FREO.
FRACT. OF SANPL ING FRED.
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
XIO-I FRACT. OF SANPLING FREO.
FIG. ( 5.5 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
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
XIO-I FRACT. OF SANPL ING FREO.
,,,,. 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
XIO-I FRACT. OF SANPLING FREO.
FIG. ( 5.6 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
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.
FIG. ( 5.7 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
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-'
FRACT. OF SANPL ING FREO.
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
FRACT. OF SANPL ING FREO.
~~~~. 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
FRACT. OF SANPLING FREO.
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
FRACT. OF SANPL ING FREO.
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
XIO-I FRACT. OF SANPLING FREO.
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
XIO-I FRACT. OF SANPLING FREO.
'" 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
-0.81
-0.97
-'.'3. -, .29
X'O-' FRAC T. OF SAI1PL I NG FREO.
l10diFied peN spE'ctra
E.c.. Fro( 1 ,. D.llIDO
E.C.. Frt2'. D.'~
Fro.s.".- D.D3DO !IF
~
-'.44~~~~~ __ ~ __ ~~~~~~~~~ __ ~ __ ~~~ 0.00 O.SO '.00 ,.~ 2.00 2.S0 3.00 3.S0 4.00 4.S0 S.OO
X'O-' FRACT. OF SAI1PLING FREO.
w , IV til
'" ~ z
IV CD I
:::r > ;0 I
CD o
A
o CD .. ~
FIG. ( 5.10 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
\J'I I
N \J'I
Xl02 .... __________________ ~Nl~N~S~P~~~c~t~r~a __________ _, "" 0.10.
~ -0.01 Est. F,,( I)· 0.1600
" -0.12
-0.23 Est. Fri2>- O.I:SOO
~ Q.
F",~,. O.OlOOSF ~ -0.31
~ -0.16 ~
.....J -0.57 ~
~ -0.68
-0.79 ~
-0.90
Xl02 EVON sp~ctra "" 0.32 i
~
'" ~ Q.
~
~ '" ~ -J
~ ~ <: ~
-0.03. Est. Frl 1 ). 0.1600
-0.39. Est. Frl2J. O.I:SOO
-0.74 Fr.~ •• O.OlOOSF
-1.10
-1.4'
-1.BI
-2.17
-2.'2.
-2.88
o c ~ ." C -4
> . > r
I t
" r o -4 ...., ::J , I
-1 .01 liS""".,,,,,,,; SI 5~",,,,,,,s ''''', .. ,,' "US "'''''" .. " .. ""eel , .. ,,"''''''';:;w 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
-3.231.", .. ,,,,,,;; ..... , ""SS3h"c~ D s as, "''''''''''''''''1'''''''''' ':::::""'''''011 ~ 0.00 0.25 0.50 0.75 1.00 1.25 I.SO 1.75 2.00 2.25 2.50 ~
"" ~
" ~ Q.
~
C)
~ ~
.....J
~ ~ ~ ,
Xl0- 1
FRACT. OF SANPLING FREO.
xl02 NEN sp~ctra 0.I7Ti----------------------~--------------~
-0.02. Est. FriO. O.I:SOO
-0.21 Est.. F,.( 2). 0.1600
-0.39. F,..~ •• O.OlOOSF
-0.'8
-0.77.
-0.96
- I. I'
- 1.34
- 1.'2
"" ~
" ~ ~
C)
~ ~
.....J
~ ~ ct ~
-, .7/1 .. ;;, .,,,,;;;;;,, e he ci'''';;;''''''' s "e , .,""""""" ..... " u .:;"',,,,,,,1 0.00 0.2~ O.~O 0.7~ 1.00 I.2~ I.~O 1.7~ 2.00 2.2~ 2.'0
XIO-I
FRACT. OF SANPLING FREO.
XIO-I FRACT. OF SANPLING FREQ.
Xd 0:., Nod j F j ffJd peN SPffJC tra • J
-0.01 Ed. Frl I)· 0.1600
-0.17 E.t. Frl2 ,. 0.1500
-0.33 F,..~ •• O.OlOOSF
-0.49.
-0.65
-0.81
-0.97
-1.12.
-1.28.
-,. "41 ,es ..... ,,, ...... ,,,ec,,r;;;;; .. ,ce,eo,eec .,,"''''''''''; """,,'0 see al 0 .... ",'-:::::::..1 0.00 0.2~ O.~O 0.7~ 1.00 1.2' 1.50 1.7' 2.00 2.2' 2.50
XIO-I FRACT. OF SANPL ING FREO.
FIG. ( 5.11 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
w , ...., UI ...,
~ Z
...., en I :I > ;0 I
CD o
w o UI
.A
V1 I
N (J'\
"" ~
" ~ It , ~ "" ..... ..,J
~
~ ,
"" ~
" ~ It , Cl
~ ..... ..,J
~ ~ <: ,
XI02 .... __________________ ~N=l~N~S~P~~~c~t_r~o __________ _. 0.10 , "" ~ -0.01 E.'. Frl I,. O.I!fr.J
-0.13 " ~.s.p •• o.~
-0.2" ~ -0.3~
, -0."7 51
'" ..... -0.~9 ...,J
-0.70
-0.91
~ ~
~ , -0.93
-1 .011 h'ue "i"'" "a,"~"""U e I""'''''' 'UG""',"'" • $,CU''''''''''''''''''''';;;;;.t 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50
Xl0- 1
FRACT. OF SANPLING FREO.
XI(J2 NEN sp~ctro 0.'6~1 ______________________________ ~ ___________________ ~
-0.02 E.,. Frll,. O.I~
-0.19. E.'. Frl Z). O.I!f!fO
-0.37 F,.. s... O.OO5OSF
-O.,f
-0.72
-0.89
-, .07
- 1 .2f
-1.42
-, . "91 ,af .. ;;;;;.." , e us"-;::: "Ii hU • U; c • ,u" U ... ,0 cue" ell' ;;:;:;.,.. ... , .. t
" ~
" ~ , Cl
~ ..... ..,J
~ ~ ~ ,
0.00 0.2~ 0.50 0.7~ 1.00 '.2~ 1.50 I.7~ 2.00 2.2~ 2.50 XIO-I
FRACT. OF SANPLING FREO.
XI(J2 EVON sp~ctro 0.34)
-0.03. Esl. Frl". O.I5:JO
-0.41 E.,. FrlZJ. O.I~
-0.79 F,.. s. .• O.OO5OSF
-1.16
-1.'4
-1.92.
-2.29.
-2.67.
-3.0'.
o C -4 ." c ~
> > r
• • ;yJ o -4
~ , •
-3. f3t ".,,'" """,,""UOUChu,,;;;;:::: """'h' , "h'I"'" h"""""":::;;:"" hess ,I ~ 0.00 0.25 O.SO 0.75 1.00 1.2' 1.50 1.7' 2.00 2.25 2.50 ~
XIO-I
FRACT. OF SANPLING FREO.
XJ a:., Nod j F' i Etd PCN SPEtC tro • J
-0.01 E.,. Frl I)· O.I5:JO
-0.' 7. E.,. FrlZJ. O.I~
-0.33. F,. • s.p.. O. (J()!f(JSF
-0.49.
-0.65
-0.81
-0.97.
-'.'2. -1.28
-I.ffl """" ''', .. ,,'''';;;:;: 'h"", "" h',,,,,,,,,,,,,,, .... """"IU'; "c,,'~ 0.00 O. ~ 0.50 O. 7~ I. 00 I. ~ I.:SO ,. ~ 2.00 2.25 2. ~
XIO-I
FRAC T. OF SANPL I NG FREO.
w , '" UI
" ~ Z .. ~ • :3 > ,., I
CD o
w o " .. "
FIG. ( 5. 12 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
'" ~ "-
~ a.. , ~ "" ..... -J
~ ~ ~ ,
\J'1 I " N
-...j
~ "-
~ a.. , Cl
~ ..... -J
~
~ ,
X'O' NEN spf'ct,ra 0.90
-0.091 I E.'. Frl I,. O. '010
-1 .08 j 1\ ~.s.p •• O.DD'fO SF
-2.07
-3.05
-1.01
-5.03
-6.02
-7.01
-7.99
-8 .981=3.::, d' CUi U hUOU' u.;e , ,,,;;;- u;eo: u. u" : fa" seo:us, s" as"" ii' ; i i li""~ is'' .. ," C i SOU: S II'"
0.00 0.20 0.10 0.60 0.80 1.00 1.20 1.10 1.60 1.80 2.00 Xl0- 1
FRACT. OF SANPLING FREO. xl02 NodjFj~d peN sp~ctra
" o. I 'f -0.01 E.,. Frl"· O. '0'f0 ~
"--0. 17 E.t. Frll'. O.IDDO ~ -0.33 a..
~.s.p .• O.DD'fO SF , -0.49
-0.65 Cl
~ ..... ...,J -0.80 ~
-0.96 ~ ~
- 1.12 , -1.28.
-,. 44fl .. ,os,';;;;;::;u;ue"euu ee, ."us sue '"'''' • 0;'"'''''''' "ce,eu,,,,,,,,,,,.,. "",;;::;;5",1 0.00 0.20 0."0 0.60 0.80 1.00 1.20 1.'10 1.60 1.80 2.00
XIO-I FRACT. OF SANPLING FREO.
~~~~. NoI.., slnutlOld
..,AX • ~
SA" .FIt. 1 .000
AlA. I TOOl • 1.00 I .00
FREOS. • 0.1000 0.1010
SNt. ( dB)t 10.000 10.000
STANO.DEY •• 0.31 82217B8O I 6831!
AESOL.ll"IT 10.0400
EXACT CovoriOftC. ftGlrl.
Xl02 EVON sp~ctra 0.3'
-0 0 03
1 II E.,. ~(I" O.'DOD
-0.'f2 E.t. ~u,. D.ltHO
-0.80. Fro.s." •• O.DD'fO !IF
-1.19.
-1.57.
-I." -2.3'1
-2.72.
-3.11
-3. 491 .. """" US" i U "U, ,:=: SCi ""su, D,e 0 hi GS'I"" """""""'" .. .:::::-: U hi , e it" JiEeso .1 0.00 0.20 0.'f0 0.60 0.80 1.00 1.20 1."0 1.60 1.80 2.00
XIO-I FRACT. OF SANPLING FREO.
FIG. ( 5.13 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o c ~
" C -4
> > ,.. I • " ,..
o -4
~ , I
m )( . ,.. W , ..., CJI , o ..., 'V
~ .. ..., CJI I
c.... i I
~
..., .. CJI
"
'" ~
'"' fX Il.. ,
~ ..... ..,J
~ ~ <: \
\J1 I
N (l)
'" ~ '"' fX Il.. , Cl
~ ~
..,J
~ Q::
~ ,
Xl02 NEN sp~ctra 0.10
-.0'1 E.,. Frl I ,. 0.1010 ~~~,. Nal~ .I""sold
-. " j I FI- .s.,J •• O.OO2OSF N"IAX • ~ -.22
SAnP.FR. 1.000 -.33
-.13 ANtlI TlI)S • I .00 t .00
-.54 FREOS. 10.1000 0.1020
-.61 SMI. ( dB" 10.000 10.000
-.75 STAtCJ .DEv. • 0.3182277880168379
-.85 RESOL.lIHIT 10.0400
-.96 O· ~ 00" o'~ 20" o"~ 10" o"j;o··o"j)o···"~ 00·· ',"~ 20" ·i"~ 10'· 'i"~60" 1'~80· '2".00 EXACT Covarlonc. Ho~rl.
x,o-, FRACT. OF SANPLING FREO.
XloL O. I tf NodiFi~d peN sp~ctra
'" -0.01 ~
XloL EVON sp~ct.ra 0.3tf
l -0.03, E.,. ~(I" 0.1020
-0.17 '"' -O.tfl
-0.33 fX Il.. -0.79,
E.,. ~(2'. 0.1000
-0.49 ,
-I. 17 Fro.S..- O.OO205F
-0.65 Cl
~ -1.55 .....
-0.80 ..,J -1.93
-0.96 ~ ~
-1.12 ~
-2.30
-2.68, , -1.28 -3.06
- I • tf3, 0'.00 0'.20 0.40 0.60 0'.80 1.00 1.20 1.40 1.60 1.80 i.oo -3.441 e U 01'1"'"'''::;;;''''''''' .,,""'uu, .. ,,""" Di' ,,"h' 01""""'::;;:""0"1"",,, .. 1
0.00 0.20 O. 40 0.60 0.80 I. 00 1.20 I. 40 1.60 I. 80 2.00 XIO- 1 XIO-I
FRACT. OF SANPLING FREO. FRACT. OF SANPLING FREO.
FIG. ( 5. 14 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
c c .... " C -4
:> . :> r
, • ;rJ
C -4
~ , , m )(
• " w , ...., UI 'V
~ z
...., en , ::I > ~ I
CD o
w w
CD
V1 I
N \,(J
XI02 NodiFi~d PCN sp~ctra ~ O.14_j--------------------------------__ ~ ______ __, ~ -0.01
" ~ ~ ,
-0.17
-0.33
-0.19
~ -0.65 ..... ..... ~
~ ,
-0.91
-0.97
-, • I 3
-, .29
E.'. Fro( I,. 0.1500
~.s." •• 0.0010 SF
Xl02 NodiFi~d pcn spectra ~ 0.11Ti--------------------~~~ ________ ~ ______ 1
~ -0.01
" -0.17
~ ~ -0.33 ,
-0.19
~ -0.65 ..... ..... ~
~ ,
-0.9'
-0.97
-I .13
-, .29
E.'. Frl, Ie D. I~
~.s.p.- O.OOOOISF o C -4
~ ~
> . > r-
I
~ o ~
~ I
-1 .41 $..;., u "'1' •• ;1 •• ::;:.' ue",e;s" ••••• ::::::'::!!';"":":,;, a De, e eCCles deSdS: s,uu, su •• , """00"01 :' 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.S0 4.00 1.50 5.00 ~
-, .1"1 ,,,,,au"""';:,,,,,,,,,,, 'SO"""" u ';;=;"""""'1""'"'''''''''''''''' ""0,,,. " ... 1 0.00 0.50 1.00 1.50 2.00 2.S0 3.00 3.50 1.00 1.505.00
~
~
" ~ ~ ,
~ ..... ..... ~ ~ ~ ,
XIO-I FRACT. OF SANPLING FREO.
Xl02 EVON sp~ctra 0.3~Ti------------------------~------------__
-0.03 E.,. Frl I Ie D.I500
-0.41 Fro.s.".- 0.0010 SF
-0.79
-1.17
-I .~4
-1.92
-2.30
-2.67
1 ~:::~ .......... L.~ ...... ............. on ...................... .1
~
~
" ~ ~ , Cl
~ ..... ..... ~ ~ ~ ,
0.00 o.~o 1.00 I.~ 2.00 2.~0 3.00 3.~0 4.00 4.'0 ~.OO XIO-I
FRACT. OF SANPL ING FREO.
X'O-' FRACT. OF SANPLING FREO.
XI02 EVON sp~ctra 0.27~ ______________________ ~ ____________ ~
I -0.03, E.,. Frl' Ie D.I5OO
-0.32 Fro • ..".- 0.0000'.
-0.62,
-0.91
-1.21
-1.'0
-1.80
-2.09.
~: : :1.."." .... n::. L.~ ........ j ..... h ............................. ' .... ,' .... 1 0.00 O.~ 1.00 I.~ 2.00 2.'0 3.00 3.~0 4.00 4.~ '.00
XIO-I FRACT. OF SANPLING FREO.
FIG. ( 5. 15 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
'r: N tJI ...,
~ Z .. N 01 • :::I > ;1Q I
CD o
tJI
w CD .. tJI tJI
'"' ~ "" ~ t\. \
~ '" .... ....J
~
~ \
\J'I I
VJ 0
'"' ~ "" ~ t\. \
Cl UJ
'" .... ....J
~ ~ ~ ,
XI02 NLN spe-ct,ra xl02 EVON spe-ct,ra 0.10 '"' 0.30.
-0.01 E.C.. F,.( ". 0.1700 ~ -0.031 I I E.C.. Frl I,. 0.1700
-0.12 "" -0.36 E.C.. Frl1'. O.I~
~ E.C.. ~(2'. O.I!JOO
-0.23 -0.69 t\. ., .PhG •• • 0.00 o.g .e.PhG .. • 0.00 o.v ,
-0.35 -1.02
-0.4S fr1 '"
-1.3' .... -0.57 ....J -1.68
~
~ -0.S8 0::
-200'1 ~ -0.79 -2.3'1 ,
-0.90 -2.67
-1 .011Ud'''''''''';:;::';'" '''''Iu.ee, ,.""DI;S; .~"uest'd$":eieu,,,,,.ueii''':C'''''i''UUu,,,,1 -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 I.SO 2.00 2.50 3.00 3.50 '1.00 '1.50 5.00
Xl0-l XIO-I FRACT. OF SANPLING FREO. FRAC T. OF SANPL I NG FREO.
xl02 NEN spe-ct,ra Xl02 NodiFie-d peN spect,ra 0.18 '"' O. ",
-0.02 E.C.. Frl 1 ,. 0.1500 ~ -0.01 E.f.. Fr(',. 0.1100
-0.22 "" -0.17 E.C.. Frl1'. 0.1700
~ E.C.. Fr(2). O.'!fOO
-0.'12 -0.33 t\. ., .PhG •• • 0.00 o.g ., .PhG •• • 0.00 o.v
-0.62 ,
-0.'19 Cl
-0.82 ~ -0065
1 JUl ....
-, .03 ....J -0.81 ~ -0.97 -1.23 ~
-1.'13 ~ - 1.13 ,
-1.63 -1.29
-1.83 -1.'1'1 0.00 0.'0 1.00 1.'0 2.00 2.'0 3.00 3.'0 '1.00 1.'0 '.00 0.00 0.'0 1.00 1.'0 2.00 2,'0 3.00 3.'0 '1.00 '1.'0 '.00
XIO-I XIO-I FRACT. OF SANPLING FREO. FRACT. OF SANPL ING FREO.
FIG. ( 5. 16 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c: ~ 'lJ c: -4
> . > r
I
-d r 0 -4 ...., :::J , I
m )(
• ,.. ~ , ...., CJI ~
~ z .. ...., en
I ::J > ;10 I
U) 0
CJI . . CJI CD .. w en
V'1 I
'vJ ~
X 1 02 NL N SPPC t,ra '"' 0.10,
~ -0.01 E.t. Frl 1). 0.1700
"" -0.12 E.t. Frl ~). O.I!JOO
-0.23 ~ Q.
111-•• Phe... 30.00 o.g -0.35 ~
~ ""
-0.1S ..... -J -0.57
-0.S8
-0.79
~ ~
~
Xl02 EVON sppct,ra '"' 0.30~, __________________ ~~~~~ __________ ~
Q) 'tj -0.03. Ed. Frll,. 0.1700
"" -0.36
-0.69. E.t. Frl~)- 0.1!JOO
~ Q.
111-1 .Phe .. · 30.00 o.g -1.02.
, ~ ""
-1.3' ..... -J -1.68
" ~ 0:::
~ -2.01
-2.3i
o c ~
~ C -4
> · > r
• ~ o ~ , ,
-0.90
-1 .01 l:ttcs ,.,,....::; ••• ,eUlUc"", eo use c,u;;., ••• ;;;"',.:",::: .5:', C'i,,' .,I;i";"', ,e"iueal
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 XIO-I
-3 o ooJoo "0': ~"o so~:s.;" j: O'ij " "j:"Sij"too".'':;O J. 00 I E XIO-I
-2.67 ~ , •
'"' ~ "" ~ Q.
\
Cl
~ ..... -J
~ ~ ~ ,
FRACT. OF SANPLING FRED.
Xl02 NEN sppctra 0.18
1 -0.02 E.t. Frl". O.I!JOO
-0.22. E.t. FrlV- 0.1700
-0.'12 111-•• Phe... 3D. 00 o.v
-0.62.
-0.82
- 1.03
- 1.23
-1.'13
-1.63
" ~
"" ~ Q. , Cl
~ ..... -J
~ ~ ~ ,
-1.83~uuu.u4o.uuuu,u.u,.uu,u~.,~ 0.00 o.~o 1.00 I.~O 2.00 2.~0 3.00 3.~0 i.OO i.~O ~.OO
XIO- I
FRACT. OF SANPLING FRED.
FRACT. OF SANPLING FRED.
XJf, NodiFittd peN spttctra
I -0.01 E.t. ~(1). 0.1700
-0.17. E.t. ~(~)- O.1!JOO
-0.33.
-0.49. 111-1 • Phe... 30.00 o.g
-0.65
-0.81
-0.97.
-1.13.
- 1.29.
-1.4" 1 'se" au .-:::: ... ,,,.... .,,,,,us,,,,, ''''';::;... 'h",e '" h., """" hi 10',. """.1 0.00 O.~O 1.00 I.~O 2.00 2.~ 3.00 3.'0 '1.00 '1.'0 '.00
XIO-I FRACT. OF SANPL ING FRED.
FIG. ( 5. 17 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
.. , ..., UI
'" ~ Z
..., <n • ::::r > ;1U
• CD o
<n o .. ~
V'1 I
\J.J N
XIOI '" 0 • 65 ffL ff SpE'C t.ra
i
~ -0.06
" ~ It ,
~ ....
-0.78
-I .19
-2.20
-2.91
-3.62 -.J ~
~ -1.3"
~ -~.O~ , -~.76
E.'. Frl 1 ). D.I:J!SO
E.'. Frl ~). O.I7!JO
N. fOl
Xl02 EVOff spE'ct.ra '" O.'OI~' __________________ ~~~ __ ~ ______________ 1
~ -0.01
" ~ It , fn "" .... -.J
~ Q::
~ ,
E.,. Frl 1 ,. 0.1500
-0.12. E.,. Frl~" 0.1Il00
-0.2'1 ". fOl
-0.3'
-0. '17.
-0.'8
-0.70
-0.81
o c: ~ "V C ..... > . > r
• ." r o ~
~ , , -I .041"""" Ii IU' U, "'" "ue S S Ii.,i CU'US S "i' """'I""'"'''' he s "SU" 01 "" .. .-=: ... leu itUh U .1 ~
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 '1.00 i.SO '.00 ~ -6. 47 1 IICcOUC.,,, .. ,,, •• ,,,,",,,,, "'"'''U''''''''''''''''''''''''' us""""a"'''i''''UUi~,1
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
'" ~
" ~ It , Cl
~ .... -.J
~ ~ ~ ,
Xl0- 1
FRACT. OF SAffPL ING FRE(}.
XIO' ffEff spt:tct.ra 0.87TI----------------------------~ ______________ ~
-0.09 E.,. Frl 1 ). O.I!!JO(J
-1.0' E.'. Frl ~). O.'BOO
-2.01 N- fOl
-2.97.
-3.93.
-i .89.
-'.8' -6.82
-B.14t " c ".,'"'' I'c,:," 0' '"''''1''''' 'hi'"'' II "."ii' h'" ".:::,; US 01,;::"..1
'" ~ " ~ Q.. , Cl
~ .... -.J
~ ~ ~ ,
0.00 O.~O 1.00 I.~ 2.00 2.~0 3.00 3.~0 i.OO i.~O ~.OO XIO-I
FRACT. OF SANPL ING FRE(}.
XIO-I FRAC T. OF SAf1PL I NG FREQ.
XJ~;9 ffodiFit:td peN spt:tct.ra
J -0.09. E.,. Frot 1 ,. 0.1500
-1.07. E.,. Fr-U,. 0.1Il00
-2.0' ". fOl
-3.03.
-'1.01
-i .99.
-'.97.
-6.9i
-7.92.
-B·90101"""'I~"'" .. " .. ,'''''''"., """",;:,....",;se,,,,,,,,, .. ,,,,,,,::o ""',. hU ... ,
0.00 o.~o 1.00 I.~O 2.00 2.~0 3.00 3.~0 i.OO 'I.~O ~.OO XIO-I
FRACT. OF SANPL ING FRE(}.
FIG. ( 5. 18 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
CD , .. o ..,
~ z
~ , ::I > ;10 , CD o
.. Q) .. N o
V1 I
VJ VJ
Xl0' ""' 0 .69 NL N Spt!'c t, 1"0
i
~ -0.07
'" ~ Q , ~ .... -J
~ ~ ~ ,
-0.81
-1 .56
-2.JI
-3.06
-3.BO
-1.~~
-~.30
-6.01
E.t. Frl ". D.lSOD
N. 402
Xl02 EVON spt!'ctra " 0.10,
~ -0.01
'" ~ Q , ~ ,.." .... -J
~ ~ <:: ,
-0.12.
-0.23.
-0.34
-0.46
-0.'7.
-0.6B.
-0.79.
-0.90
E.t. Fri,,. O.I~
E.t. Fri2'. 0.1700
". 402
o C -4
" C -4
> . > r-
, .,J ro -4
~ , , -, .01 1 .. """.,,,. i'''','U: o,eo'i'''''''''':'' "".,ce,,,,,,,,,,,,,,, .. ,,,,,, "~u,,,,::ss"Dul ~
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 ~ -6.79, .. ", .,,'"'''''''''''''''''''''''''''' """'" """ .. ,""""'; """",,,,,,,.uu,;;;;bihul
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 ~.50 5.00
" ~ '" ~ Q , Cl
~ .... -J
~ ~ ~ ,
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."
" ~ '" ~ Q , Cl
~ .... -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
FRACT. OF SANPL ING FREO.
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
XIO-I FRACT. OF SANPLING FREO.
FIG. ( 5.19 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
CD , ~ o N
" ~ Z .. N -.J , ::J > ~ I
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
" ~ Q.. , ~ '" ..... -J
~ ~ <: ,
'" ~
" ~ Q.. , Cl
~ ...... -J
~ Q::
~ ,
-0.75 ". 101
- I .11
-2. I 2
-2.81
-3.49
-1. I 8
-1.97
-5.55
-6. 21 1 .. , ................... ,... .................. ""liSUO"US"",,, cS,,'DDcCO;""""::;';;'''';:'' u~1
" ~ Q.. , ~ "" ..... ..J
~ ~ <: ,
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 X10- 1
FRACT. OF SANPLING FRED.
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
FRACT. OF SANPL ING FRED.
-. " E.t. PrU" 0.I1f1O
-.21 ". 20'
-.31
-.'fl
-.~I
-.61
- .71
-.81
o c .... ~ C .... > > r
I
~ r o -4
~ , I
- . 91 in; .. U .. , hi ;;'so, "US u , ... ,e "", s """ e "".,,,. O"SO I,,,,, ,,,.,, e .. """.::::::: .. , •• ,:e 2 e e ., ~ 0.00 O.SO 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 ~
XIO- I
FRACT. OF SANPL ING FREO.
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.
Q) , ~ " ~ Z
II.)
" I :r > '" I CD o
~
.. tJI
"
FIG. ( 5.20 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
"'" ~
"" ~ It , ~ '" ..... .... ~ 0::
~ ,
IJ'1 I \.N IJ'1
"'" ~ "" ~ It , Cl
~ ..... .... ~ ~ ~ ,
)(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
FRACT. OF SANPLING FRED.
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
-8. '31 •• n. c,eo """,,,,,,,,,,, hue u,s sen '." ." u" SUO; su .... , .. = .. "u55sueul 0.00 o.~o 1.00 I.!W 2.00 2.~ 3.00 3.~0 f.OO f.~O ~.OO
XIO-I FRACT. OF SAffPL /NG FRED.
FIG. ( 5.21 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o C -4
~ -4
> · > r
, ~ o -4
~ , •
m )(
• ..... CJ) , o ~
~ z
'" .... • :I > ,., • U)
o
A ..' '" .. UI N
\J'I I
'vol 0--
XIOI "" 0 .69 NEN Spl"CtrQ
j
~ -0.07 E.,. Frl,,. D.' 7rIO
" ~ 'l ,
~ .... " ~ ~ ~ ,
"" ~ " ~ 'l , t:l ~ .... " ~ ~ ~ ,
-0.92 E.,. Frll). D.'!HID
-, .59 ". .,
-2.33
-3.09
-3.91
-4.59
-5.~
-S.IO
-S .9S I ......... ,,, .... , ........ , .. """., .. ,, .... ,.. "".,""" .. ~oo., 00.-.:..1 0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.50 1.00 1.505.00
XIO- I
FRACT. OF SANPL ING FREO.
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
" ~
" ~ 'l , t:l
~ .... " ~ ~ ~ ,
-., .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
-4 • .,."
-5 • ."
-6.65
-'1.59
-8.'21." " .. " .. """"" ••. ue, ,,,.,us,,,,,,,, .. ,,,. S h',"'" a , " ""'u.::: .. , 50 .1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 ".50 ~.OO
XIO-I FRACT. OF SAf1Pt.ING FREO.
FIG. ( 5.22 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o c ~
" C ~
> • > r
, ~ o ~
~ , , m )(
• ,.. CD , CD ..,
~ z
N -.I I
::I > ,., , CD o
~
en -~
'" ~ "-
~ Q. , ~ "" .... ..,J
~ ~ ~ ,
\J'1 I
\.Jo.I -.J
'" ~ "-
~ Q. , C)
~ .... ..,J
~ ~ ~ ,
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.'~
lie .. ,
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.
NodiPiE"d PeN spE"ctra '"
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
FRACT. OF SAffPL ING FREO.
,,,,. 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
-7.361 •• 'h, • see ., •• e eo,,,,,,, '."". ua en"" .,,' '''''''c .. ; .:::: iSh,,1 0.00 O.~ '.00 I.~ 2.00 2.~ 3.00 3.~ •• 00 •• 50 ~.OO
XIO-' FRACT. OF SANPL ING FREO.
FIG. ( 5.24 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o c ~
" C ~
> > r
, ~ o ~
~ , , m )(
• ,.., CD , .... U' ...,
~ 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
~
~ ,
-3.1 J
-3.71
-1.~
-1.97
E.'. Frl,,. O.,:no E.,. Frfl,. O.,~
". " o c ~ 'V C ~
> • > r-
I
~ o ~
~ , •
-5 .580~OO"~sO" 1':00 1'.50"2':oo"i'~ ':too 'j':;;"::'jio 1':;;";;'.001 ~ XIO-I
-5.7e l a ",,,,"-'=::: e , CUt. os " • "",,~$ e" """., US"" "eo eSiS ,; " ",,,I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
XIO-I
'" ~
" ~ I:l , ~ ...... ...,J
~ ~ ~ ,
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
-3.91
-f.68
-6.991 • 'f , 0, se." , ." hi ,co.;;;::. hh'
0.00 o.~ '.00 ,.~ 2.00 2.::J0 3.00 3.~0 f.OO f.~O ~.OO X'O-,
FRACT. OF SANPL ING FREO.
'" ~ " ~ I:l , Cl ~ ...... ...,J
~ ~ ~ ,
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~
'" ~
"" ~ It , ~ "'f ...... ~
~ ~ ~ ,
\J1 I ~ ~
'" ~
"" ~ Q.. , C)
~ ...... ~
~ ~ ~ ,
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.
FIG. ( 5.27 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
0 c -4
" c -4
> . > r
• ~ 0 -4
~ , , m X • " crt , ..., til ...,
~ z .. ..., ..... • :I > lU , CD 0
-0 -til .. til UI
'" ~ " ~ a.. , ~ ...., ..... -.J
~ ~
~ ,
\J1 I
.J> N
'" ~ " t:) Vl a.. , t:)
~ --.J
~ ~ <: ,
.101 ffLf1 speoctra XIo' EVOff sp~c tra 0.11 '" 0.67.
-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.,
-1 ... t ,
-2.29
-t .97 ~ ""
-3.03 .....
-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
-2.21 t:)
~ -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.
FIG. ( 5.28 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c -. ~ -. > • > r
, ~ 0 -. ~ , , m )(
• " CJ) , '" CJI OJ
~ z .. CJI I
~ ,., , CD 0
I~
'" ~
'" ~ Q.. , ~ "" ..... ...J
~
~ ,
V'I I
.f;:-\.N
'" ~
'" ~ Q.. , t:l ~ ..... --.I
~
~ ,
)('0' NLN spE'ctro x,o' EVON spt-ctro 0.37 '" 0.69.
-0.01 E.'. Frl"· D. ',;,0 ~ -0.071 1\ A E.l. Frl I,. D. ,,,,,,
-0.15 '" -0.82. SIll • 'D.DOD t.,
~ V E.,. ~~,. D. ,..""
-0.86 -1.58 Q.. -. ID.tItItI t., -, .77
, -2.33
-, .68 ~ ""
-3.08 .....
-2.09 --.I -3.84
~ -2.50 ~ -4.'9
-2.92 ~ -,.~ , -l.ll -6.10
-3.74 t $ sec,,,,,,,,, ... "eo , " """e """.,$"""" •• """"'e;::....oc~ ... "",,,,, .. 1 -6.BS 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 0.00 0.50 '.00 1.50 2.00 2.50 3.00 3.SO 4.00 f.50 5.00
)(10-' XIO-I Ff?ACT. OF SANPL ING FREO • Ff?ACT. OF SANPLING Ff?EQ.
X,O' XIO' NEN spE'ctro nodiFi~d Pen s~trQ 0.f8 '" 0.51
-0.05 E.,. FrO" D. f~ ~ -0.06 E.,. Frl I ,. D. ,.",
-0.57 "" -0.68 E.'. Frl2'. D.ffDO 5l E.,. ~~,. D.'<ftI(J
-'.10 -1.31 Q.. SIll • 'D. DOD t., ".. ID.tItItI t.,
-, .62 ,
-1.94 t:l
-2. I' ~ -2.56
/ \ ..... -2.68 --.I -3. '9]
~ -3.82 -3.20 ~ -3.73
~ -f."4 , -f.~ -'.07.
-4.78 -'.69. 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 1.00 1.50 2.00 2.50 3.00 3.SO f.OO f.SO '.00
XIO-' X'O-I Ff?ACT. OF SANPL ING FI?EO. Ff?ACT. OF SAf1PL ING Ff/EO.
FIG. ( 5.29 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c -f ." c -4
> . > r-
, ~ 0 -f
':i , I
m x • " en , ~ ..,
~ z
-II.) ..., , ::J > '" I CD 0
0 . . ~ .. -...,
"" ~ " ~ Q.. , Cl
~ .... ......
~ r)!
~ ,
V'1 I .t:-.t:-
"" ~
" fX Q.. , Cl
~ .... ...... ~ ~ ~ ,
.,0' ffEN sppct.ra 0.~5
-0.0~1 /\ E.t. Fri' ,. D.'~
-0.5~ I I \ E.t. Fril,. D.'"''''
-1 .03 1 I \ SI*I • 1.000 (dl"
-1 .53
-2.02
-2.52
-3.01
-J .51
-~.OO
-".50 , ..... ___ ._ ... __ ... _ ..... ___ ._. ___ , ________ , _________ • _________ • ______ ow. ... -w'" -w
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 ".00 ".50 5.00
XIO' 0."
-0.06
-0.66
-1.21
-1.88.
-2.48.
-3.09
-3.10
-4.31
-4.91
-'.'2 0.00 0.'0
XIO- I
F1?ACT. OF SANPLING F1?EO.
NodjFjpd PeN sppctra ""
E.,. Fri' l- D. 'IIDD ~ "
E.t. Frill- D.'~ fX Q..
SI*I • 1.000 (dl" , Cl
~ .... ...... ~ ~ ~ ,
1.00 I.~ 2.00 2.'0 3.00 3.'0 f.OO f.,O '.00 XIO- 1
F1?ACT. OF SANPL 1M3 F1?EO.
"", Ma ...... lftUMld
MtAX • ~
IMP .At • 1 .000
ANtl.1 TOOS • 1.00 , .00
FllfDl. • O.I!CJO 0.1100
1 ... 1. ...... ' 0.00 0.00
!INt. ( dB" '.000 '.000
STAfC).DEY. • 0.~"87l18:J01734'
A!1Ol. .llnl' '0 .CHOO
S ...... ATEO C ...... I....: .... " ••
X'O' 0.61, EVDN spfitctra
-0.01. E.t. Fro(,,. D. ,.",
-0.90 E.t. Fro(~,. D.'''''
-1.'4 _. ..tJtJtJ (., -2.28
-3.01
-3.1'
-f.f9,
-'.23
-'.96 -6. 701. "CC;;' sa us .... ". h,""""', •• """"h' U """'e,,.~. ,is''' .1
0.00 0.'0 1.00 I.~ 2.00 2.~ 3.00 3.~ 4.00 f.,O '.00 XIO-I
F1?ACT. OF SANPL ING F1?EO.
FIG. ( 5.30 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o c ~
" C ~
> • ~
• • ;JI o .... ~ , , m )(
• ,.. en , '" UI .....
~ Z .. '" .., , ~ ,g , (0 o
-o w .., .. W c.J
\J1 I ~ \J1
X10' ~ 0.12 ~N sp~ctra , ~ -0.01 E.' 0 Frl',. D.llIDO
" ~ Q. , ~ ~ .... ~
~
~ ,
" ~ " ~ Q. , Cl
~ .... ~
~
~ ,
-0.!5' E.'. Frll" D.I~
-0.98 51*1. f • ODD t diU -, ."4
-, .9' -2.37
-2.84
-3.3'
-3.77
-1.24 1 '''''''i' e ",se,. ''''''''', .. "'',,.,'' s ., ""'" )t. .. ..c;- II " """"" I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00
X'O-1 FRACT. OF SAI1PL ING FREO.
Xd ~;9 Nod iF i pd peN sppc t.ra
I -0.06 E.t 0 Frl 1 ,. D.llIOO
-0.71 E.to Frll" D.'~
-1.36 51*1. f • ODD t diU
-2.00
-2.6'
-3.30
-3.95
-f.60
-'.2' -".9()1;t, ue~. e '''''"., " .. ,',. , ,::"""""""eu,, h' eo., hu 0 Of""'" .,
" ~
" ~ Q.. , Cl
~ .... ~
~ ~ ~ ,
0.00 o.~ 1.00 I.~ 2.00 2.~ 3000 3.~ •• 00 •• '0 '.00 XIO-I
FRACT. OF SANPL ING FREO.
,,,,. Hal ... 1,.,.,ld
.... AX • n
SA" .At. 1 t.OOO
ANtLI nm • t .00 t .00
FREO!I. • O.t!OO 0.'800
1ft." .Phew' 0000 0.00
!HI. ( dB" ".000 ".000 STAND.~ •• 0.83095735~'B'6Oti
RESOL.ll"IT 10.0400
SIIU.A'EO Coverlenc. ttI"l.
XIO' EVON sppct.ra 0.69)
-0.07. E., 0 Frl,,. 0.'''' -0.83 E.,. Frl ~,. 0.' 7.1D
-I.~. fNII. f.DDD t.,
-2.34
-3.10
-3.86.
-f.61
-,.37r---6.13.
-6.981. e ea,,,,, s, ." sue. e Sf C "".,." uu, • '''".-:::..., 0"'::: u ,e •• ' 0.00 Oo~ , 000 I.~ 2.00 2.~ 3.00 3.~ f.OO i.~ ~.OO
XIO-I FRACT. OF SAI1PLING Ff?EO.
FIG. ( ).31 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o c ~ ""0 C ..... > · > r
I
~ o ..... ~ , I
m K • " aJ , ~ ...,
~ Z
~ • ~ ,., • ~
o .. 0') .. ..
VI I
,p. Q'\
"" ~
" ~ It. , ~ '" ..... ~
~ ~ ~ ,
"" ~ " ~ It. , t:l ~ ..... ~
~ ~ ~ ,
X'O' f1EN spEtctra 0.39
-0.041 f\ E.t. Fril,- D.'~ "". NoI.., .lrtuMld
-0.47 t --./ \ SMII • 1. (J(J(J it., ..,AX • ~
-0.90 SA" .At. • I .000
-1 .33
-1.76 N9l.1 TOO! • I .00 I .00
-2.19 FRfD5. • O.'!OO O.'BOO
-2.62 1"1" .Phe... 0.00 0.00
-3.~ ,.. ( dB" 2 .000 2 .000
-3.18 BUfC).IJEY. • O.794328n818:J:J9G8
-3.91 0".00 -0".50,".00--1 ".50- 2".00 i. 50i.00 i.50 1".00 • ... 50 5".00 RESOl.llniT .0.0400
XIO- 1
FRACT. OF SANPLING FREO. StrU .. ATEO Coverl..:_ "'''I.
NodiPiEtd PeN SpEtctra X'D' "" 0.67,
XIO' 0.'9
EVON spEtctrtJ
-O.OG E.t. Fril)- D.'~ ~ -0.07, I.t. Fri,,. 0.'""
" -O.B' Ed. Fril,- D.'3DD
~ It. -, ."
-0.71
-, .36 I.t. Frill- D.IJIIO
SMII • 1.(J(J(J f.' , -2.29, -2.01
!NIl. 1.DtJt) f.' -2.66
t:l ~ -3.03, ..... ~ -3.77,
~ -f." ~ ~ ,
-3.3'
-3.96
-f.61
-'.26 -'.99, -'.91 ~iS ~ e " s" , .. ,," us eo.;:;"""o;l'''''", e e ""'" "., ... ,so .. 1 -S. 731 """'U • e " , u eo cu, GU"""." sec "" "u""'".~ ""u,ee,. st
0.00 O.~ 1.00 1.50 2.00 2.'0 3.00 3.'0 i.OO i.'O '.00 0.00 O.~ 1.00 1.50 2.00 2.50 3.00 3.50 f.OO ... ~ '.00 XIO- I X,O-I
FRACT. OF SANPL ING FREO. FRACT. OF SAI1PI..ING FREO.
FIG. ( 5.32 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
o c ~ ~ C -4
> . > r
• ~ o -4
~ , I
m )(
• " 0) , '" UI ...,
r! z .. ~ I
::I > lU I
CD o
o ~ ~
w ...,
'" ~ "-
5l Q. , &1 "" ....... ..,J
~ ~ ~ ,
\JI I
.p. '-oj
'" ~ "-Cl Vl Q. , Cl
~ ....... ..,J
~ ~ ct ,
X10' NEN sppct,ra 0.58
-0.061 r\ E.'. Fri' ,. D.IStItI
-0.70 1 I \ SNIt. ".fX1O t diU
-1 .31
-1.98
-2.62
-3.26
-3.90
-1.~1
-~.18
-5.82 ... _-._.- .. ----- ---.-- ----- .. - - -. - - . .. 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
Xl0- 1
F1?ACT. OF SANPL ING F1?EO.
x'o' NodiPipd PCN sppctra 0.61 '"
-0.06 E.,. FriO. D.lf~ ~ -0.73 "-
E.'. Frll" D.I~ ~ -I. fO Q. !I#I. ". fX10 t"U ,
-2.08
-2.75 ~ ,.... .....
-3.f2 ..,J
-4.09 ~ ~
-f.76 ct ,
-'.f3.
-6. , '1 is see""::;:". II h" , us .. o, ",~"",e sn :sc; e seu:s;""" 'h, .. ;;.1 0.00 O.~O 1.00 I.~ 2.00 2.~0 3.00 3.~0 f.OO f.~ ~.OO
XIO- 1
F1?ACT. OF SANPLING F1?EO.
"". NoI.., .If'lUHld
""AX • n
SA" .FR. , .000
ArW'l.1 TOOS 1 , .00 , .00
FREo!. 1 D.I!GO 0.'700
I .. '" • .."... 0.00 0.00
!INR. ( dB ,. ".000 ".000
STAND.DEY •• 0.000017787794' 1tII
RESOl.ll"IT 10.0400
S UQ .. A TEO Covw IIII'C .... ..,. ,.
X'O' 0.74 Evon sppct,ra
-0.07. E.,. Frl,,. D. 'f'" -0.88
E.t. Frll'" D.'~ -1.69 ... ".tIDtI t,., -2.50.
-3.31
-f. 12.
-4.93.
-'.7f
-7·.161;;, ""., • """". "".,,, •• u sa """., e. .... se"""""., ••• ".;::;U"':o' 0.00 O.~O 1.00 I.~ 2.00 2.~ 3.00 3.~ f.OO f.~ ~.OO
XIO-I F1?ACT. OF SAf1PL ING F1?EO.
FIG. ( 5.33 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
o c .... ~ c .... :. · :. r
• ~ o .... ~ • m )( . " Ol , N (II ..,
~ z -N
" • :I ~ • CD o
o (II (II .. ~
"" ~ '" ~ Q. , ~ ...... .... ~ ~ ~ ,
VI I ~ CD
"" ~ '" ~ Q. , Cl
~ ...... .... ~ ~ ~ ,
XIOI 0.58
-O.OS
-0.70
- I .31
- I .99
-2.62
-3.2S
-3.90
-1.~1
-~. 18
-~.82 0".00··0.50
X'O' 0.6'
-0.06
-0.73.
-'."0
-2.08.
-2.75
-3."2 .
-".09
-".76 -, ..... -6. "
0".00 0·.'0
NEN sppctro
E.t. Fri I'· D. 'SOD
SNR. 5D.DOtJ ttll,
1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO- 1
FRACT. OF SANPL ING FREO.
NbdiP;pd PCN sppctro ""
E.t. Fri',. D.'f5D ~
'" E.t. FreIJe D. '750 ~ Q.
SNR. 5D.DOtJ t., , Cl
~ ...... .... ~ ~ ~ ,
'.00 ,.~ 2.00 2.'0 3.00 3.~ ".00 ... ~ '.00 x'O-,
FRACT. OF SANPL ING FI?EO.
,,,,. MDI.., .lflUMld
NrlAX t~
SA" .FIt • I .000
AN\.lnm t 1.00 '.00
FltE05. • O.I!OQ 0.1700
'"1" ........ 0.00 0.00
_. ( dB,. !O.OOO !O.OOO
SlAHD.OEY •• 0.0031821718801814
RESOl.L'"IT '0.0400
IUO.AlEO C"""'I~ .........
x,o' 0.7"
EVON sppct.ra
-0.07. E.t. Frl I Ie tI. If'"
-0.88. E.t. Pr-Ule tI.I1S
-, .69. !NIl. 5D.DDD t.,
-2.~
-3.3'
- ... '2 .
-4.93
-'.7" -6." -7·35'1 " u. ,,,sue ~. , DiU" • "su,ee u .,ue, ""',h' 0 esc; ". " •• h;;;;u35Ss,,1
0.00 0.'0 '.00 ,.~ 2.00 2.~ 3.00 3.~ ".00 ".'0 '.00 X'O-,
FRAC T. OF SAI1PL I NG FREQ.
FIG. ( 5.34 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o c .... ." C .... > • > r-
I
~ o .... ~ , , m M • " 0) , ~ ..,
~ z
'" "'-J I
:::I > :.J , ~
-o UI W .. N m
V'I I
.po \"Q
'" ~I .~I2 Nod; F' ; pd PCN sPPc t.ra j
~ -0.06
" -0.71 Cl V) -1.12 It ,
-2.10 Cl ~ -2.78 .... .... ~
~ ,
-3.16
-1.13
-1.BI
-~.19
E.l. FrO" II. I~
E.,. Frll)e 1I.'f~
!NIl. '''.IIO(J t.,
XIO I Nod oro ° '" 0.S3 Jrl~ Pen SDf!"ct.ra
~ -0.06
" -0.76
~ It -1.15 , -2.1~
~ -2.84 .... .... ~ Q::
~ ,
-3.54
-1.23
-1.93
-~.62
j i
hi. '*'f'" 1I.'1OD
.. ,. ~I" D.lf.
... ,.tIOtI t.' ~ .... ~ -4
:. . :. r
, ~ o -4
l , -6 • 17 1 """~"",,., 'n" Sf ''is "U ."" .. , ~""" e '''''' "e "''''''''1 CO"" .. ,
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 X10- 1
-6. 3 'O~oo" O~sO' 1'.00 ill '. 50"2':00" i~"jU~ 00" j':;;" .. ':oo"!t;;o";'.00 , ~
.'0-1
'" ~
" ~ It , Cl
~ .... .... ~ ~ ~ ,
FI?ACT. OF SANPL IN(; FI?EO.
X'D' EVON sppcLra O.1'T ______________________ ~ ____________ ~
I -0.07. E.l. F,U,. II.I~
-0.90 E.t. Frll)e D. If",
-1.72 !Mt. I".DDtJ t.,
-2.'f
-3.36
-f.19
-'.01 _,.831 .,. -6.&6
-7. fBl 0, », • , • :" us .. eo i'"",," "'IIICiiV".. • " ... 1
'" ~
" ~ It , Cl
~ .... .... ~ ~ ~ ,
0.00 0.'0 '.00 '.'0 3.00 3.'0 3.00 3.'0 f.OO f.~ '.00 x,o-, FIlACT. OF SAffPL IN(; FI?EO.
FI?ACT. OF SAffPL ING F~O.
X'D' EVON sPE-cLra 0.16,T ______________________ ~ ____________ ~
j -0.08. Ed. h(,,. D.I1OD
-0.9' E.,. '*'fIle D. '" -, .1' SNt. '.IIO(J r.,
-2.~.
-3.fl
-f.~
-'.08. -'.92.
-6.1'
-7.~91 eD "" 0$;: he ~"H"~" i; h' e , sec e 'u", .. ",,,... e"'-:::, SO cu, •• 1 0.00 O.~ '.00 ,.~ 2.00 2.~ 3.00 3.~ f.OO f.~ '.00
XIO-' FI?ACT. OF SAI'fPL ING FI?EO.
FIG. ( 5.35 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods -
crt , "'" VI
"
~ -~ , ~ lD , ~
o .. ~ UI ~
'" ~
" ~ Q.. ,
~ ...... --oJ
~
~ ,
\J'I I
\J'I a
'" ~ " ~ Q.. , Cl
~ ...... --oJ
~ ~ ~ ,
XIOI 0.63
-O.OS
-0.75
-, ..... -2. I 3
-2.82
-J.51
-".20 -".89 -5.57
-6.26 .---------,,-----
0.00 0.50
XIO' 0.75.
-0.07.
-0.90
-1.72
-2.'f
-3.36
-4.18
-'.01
-6.6'
-7.47 .. ---- -"
0.00 O.~
NbdjFj~d PeN s~ctrQ '"
E.,. ~,,. 11.1700 ~
" SMtt. f.ooo f", ~ Q.. , ~ ...... --oJ
~ ~ <= ,
1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO- I
FRACT. OF SAffPLlNG FRED.
EVON spEtctrQ '"
E.&. FrO" D.'7OO ~ " _. f.ooo f", ~ Q.. , Cl
~ ...... --oJ
~ ~ ~ ,
XIOI 0.57
-O.OS
-0.68
-, .JI
-, .9" -2.56
-3.19
-J.81
-".44
-5.07
-5.69 .. -------- .. -----
0.00 0.50
XIOI 0.6',
-0.06.
-0.18.
-1.49.
-2.21
-2.92.
-3.64
-f.".
-'.18
Nod iF' j ~ PeN SptaC lrQ
E.,. ~,,. II. ,.", ... lI.tIt1tI (.,
g -4 "V C -4
> • > r
I
~ o -4
':l , I
m x
t.OO t.SO 2.00 2.50 3.00 3.50 4.00 1.50 5.00 , • XIO-I ;
FRACT. OF SAf1PLING FRED.
EVOff sp~c trQ
E.,. ~,,. D.''''' ... lI.tIt1tI f.'
, ..., tJt
" ~ z .. ..., ..... I
~ • ~
o ~ .. o ...,
-6. f91; • e ceo,ou " us on "e .. " • ..... "se" .n a '" ''':;.. ,ee • .,.c:::;"" '"'''' .1 1.00 I.~ 2.00 2.~ 3.00 3.~ 'f.00 'f.~ '.00 0.00 O.~ 1.00 I.~ 2.00 2.:lD 3.00 3.50 f.OO f.:lD '.00
XIO-I XIO-I FRACT. OF SAffPLING FREO. FRACT. OF SAffPLlNG FRED.
FIG. ( 5.36 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
V'l I
V'l ~
X10' "" 0 .12 I1LN SpEtctra
i
~ -0.01 E.,. Fro('" O. ,:JOt)
" -0.51
-0.97 I.,. Frl1'. 0.1OOtJ
~ Q..
Fro.s." •• 0.05D0 SF -1 ."3
,
~ -1.90 ..... ~ -2.36 ~
~ -2.92
-3.29 , -3.75
XIO' "" 0 • 7 I EVON SPEtC tra I
en " -0.07. E.e.. Frl,,. 0 .... '"
" -O.~
-1.63. I.e.. Prl Z,. O.ZfI'O
~ Q..
-2.40. , Fro.,.".. 0.05D0 SF
fi1 ""
-3.18 ..... ~ -3.96.
~ ~
~ -4.74
-5.51 , -6.29.
o C -4
~ -4
> . > r
I
~ o -f
~ , I
-7 • 07J. 00 ''0.50 "j':Oij"j':':jiJ '2.00' !DiO 3.00 ''i:iO''f':m:,=:t¥' 51.00 I S ,
-1.2' 1 he up ...... est """., •• ce, "I' ee, US;o""""I'"'' en, """"'''''.'''''''.~, 0.00 O.SO 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00
X10- 1
"" ~
'" ~ Q.. , Cl
~ ..... ~
~ ~ ~ ,
FRACT. OF SANPLING FRED.
XIO' NEN spEtctra 0.'3~ ______________________ ~ ______________ ~
1 -0.05 I.,. Fro('" O. , .. ", -0.64 I.e.. Frl1,. 0.1t15O
-, .22. Fr- .!I#p.. ". D500 SF -, .8'
-2.39
-2.98
-3.56
-4. I' -4.73
-'.31 1"0 "".,. u '" "US e " cu. , '"''''''''''01''''''''' he s,u e.;;;;' 5 "5Osl
"" ~ " ~ Q.. , Cl
~ ..... ~
~ ~ ~ ,
0.00 0.50 1.00 1.50 2.00 2.'0 3.00 3.50 f.OO f.50 '.00 XIO-I
FRACT. OF SANPL ING FRED.
FRACT. OF SANPL ING FIlED.
XJ.O~ ffodiFiEtd peN s~t,.tJ I
-O.OG. I.e.. Prl,,. O. , .. ",
-0.69. I.e.. Prll,. 0.""
-1.33. Fro."".. o. 05D0 SF
-1.96
-2.59
-3.23.
-3.86
-4.50
-5.13.
-'.111""iUhsUOi~""'"''''''';''''''''''' e " .' ";a..,,,,uehS5S' su "".,ee' '"'''''''''' I 0.00 0.50 1.00 1.50 2.00 2.'0 3.00 3.50 4.00 4.'0 '.00
XIO-I FRACT. OF SAI1Pt.ING FRED.
FIG. ( 5.37 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
..., CJI ..,
~ z ..., ..... I
::I > :IU , CD o
-~ ..., w
'" ~
" ~ Q.. , ~ "" .... --.I
~ ~ ~ ,
\J1 I
\J1 N
'" ~ " ~ Q.. , C)
~ .... .... ~ ~
~ ,
XIO' f1I..N spEtct.ro X'O' EVON spt-ctrtJ 0."2 '" 0.67.
-0.04 E.t. Frl"· D. ,s~ ~ -0.071 1\ A E.t. Fro( I,. D.'fOD
-o.~ " -0.81 Fr.s.,J.- D.D3tJO SF ~ V ~.t. Prl~" D.'''''
-0.95 -1.5' Q.. ~."" •• D.D"'" • ,
- I .4 I -2.29
-I .87 @ ~
-3.04 ....
-2.32 --.I -3.78.
~ -.. ,zv -2.78 ~ -3.24 ~ -'.26 , -3.69 -6.01
-4.15 t """'" • ",,,.,ee, c •. H
,;;;;;""1 -6.751 " cae; "~,,I " " , Ii """"'''' '"",ee, """ up eo" u, """""""'1""""" see: US," su. sa • 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 ".00 ".50 5.00 0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.50 4.00 4.50 '.00
XIO- 1 XIO-I FI?ACT. OF SANPLING FI?EO. FI?ACT. OF SANPL ING FIlED.
X,O' X'O' /1EN sppct.rtJ Nodjrjpd PCN sppctrtJ 0.49 '" 0.5'
-0.0' E.t. Frl'" D. I~ ~ -0.06 ~.t. Frf I,. D. If""
-0.'9 " -0.66 E.t. h-( 7'. D.lfDD ~ E.t. Prl~'" D.'''''
-'.13 -1.27 Q.. Fr.5.p.- D.D3tJO SF p,. .J.p •• D.D" •
-, .67 ,
-1.88 C)
-2.21 ~ -2.49. ....
/ \ -2.7' .... -3.,01
-3.29 ~ -3.71 ~ ~ -4.32. ,
-4.37 -4.93.
-4.91 -'.!H 0.00 O.~ 1.00 ,.~ 2.00 2.~ 3.00 3.~ 'f.00 'f.~ '.00 0.00 o.~ 1.00 I.~ 2.00 2.~ 3.00 3.~ 'f.00 4.~ '.00
XIO-' XIO-I FI?ACT. OF SAf1PL ING FI?EO. FI?ACT. OF SANPL ING F1?EO.
FIG. ( 5.38 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c: ~ "1J c ~
:> • :> r-
• • ~ 0 ~
~ , , m JC . " -.J , '" (II ..,
~ z .. '" .... I
::J :> ~ , CD 0
I~
"" ~ " ~ Il , ~ .... ...J
~
~ ,
\J1 I
\J1 \..oJ
"" ~ " 5l Il , Cl
~ .... ...J
~
~ ,
X'o' NLN sppctro x,o' EVON sp~ct"'a 0.50 "" 0.7".
-0.05 E.l. Fro( I'· 11.'500 ~ -0.071 (\/\ E.t. Frl,,. II. H",
-0.60 " -0.88 Fro.s.".- D.D~ SF ~ '.t. Frl~" II. I~
-1 .15 -, .69 Il ~ .,.". - tI.IIIDD ., ,
-1.70 -2.~
-2.2" @ ,.... -3.31 .... -2.79 ...J -".'2.
~ -3.3" ~ -f.93
-3.99 <= -~.7" , -....... -6.~'
-1.99l . e eo",,,,, e , CD , • h " •• he, " h' s, e ce, cu,. D" Ii" S ,::;:;;;.. .1 -7.36 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.~ ".00 ".50 '.00
XIO- 1 XIO-I FRACT. OF SANPL ING FREO. FRACT. OF SAI1Pl..ING FREO.
X'O' XIO' NEN sppctra NodjFi~d PeN 5p~ct,..O 0.'" "" 0.61
-0.06 E.l. Frl,,. D.'SOD ~ -0.06 E.t. Frl,,. 11.1.'"
-0.70 " -0.73 Fro.s.".- D.D~ SF 5l E.t. 'WI,. II.I~
-1.33 -I.'" Il
-1.97 ,
-2.09 F,. • .". - 1I.1I1t1t1 .,
-2.61 Cl
~ -2.7' .... -3."21 I \ -3.2f -I
~ -f.IOJ / \ -3.88 ~ ~ -f.77, ,
-~.16 -~.""
-'.79 -6. " 0.00 0.'0 1.00 I.!W 2.00 2.'0 3.00 3.SO ".00 i.SO '.00 0.00 0.'0 1.00 I.!W 2.00 2.SO 3.00 3.!W i.OO i.SO '.00
XIO- 1 XIO-I FRACT. OF SANPLING FREO. FRACT. OF SAI1PLING FREO.
FIG _ ( 5.39 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c: ~
" c: -f
:. . :. r
• ~ 0 -f
~ , , m )(
• " .... , ~ ...,
~ z
-IV .... , :I > lU
• CD 0
1=
I~
'" ~
" ~ It , ~ .... ...J
~ ~ ~ ,
VI I VI .p..
'" ~
" ~ It , Cl
~ .... ...J
~ ~ ~ ,
XIOI 0.S2
-O.OS
-0.74
-I .42
-2.09
-2.77
- J .1'.5
-4. I 2
-'.5.18
-6.IS . ---.-----0.00 0.50
XIOI 0.61
-0.06
-0."4
-1.41
-2.08
-2."6
-3.43
-4.11
-4.18.
-'.4'
-6. 13. 0·.00 o".~
NEN sppctra
E.,. Frl',. 0.'500 Fro. s.".. D. tKHO SF
1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO-I
Ff?ACT. OF SANPL ING FI?EO.
NodiPipd peN sppctra '"
E.,. Frl,,. O.'foo ~
" E.,. Frll" '.,no ~ It
Fr-.s..- D.oof' SF , Cl
~ .... ...J
~ ~ ~ ,
1.00 1.'0 2.00 2.'0 3.00 3.'0 ".00 ".'0 '.00 XIO-I
Ff?ACT. OF SANPL ING Ff?EO.
"". NoI.." .lftUMld
III'tAX .~
SA" .At. 1 .000
NWtl.1 nm • I .00 I .00
FAEG!. • O.'!GO O.I!40
'nil.""'.' 0.00 0.00
!INt. ( ell,. 10.000 40.000
STAND.DEY •• 0.01 OOOOOOOOOOOOOO
RESDL.llniT '0.0400
S IIU .. ATEO CO¥et' I enc .... ..,. ..
X'O ' 0 • .,4 EVON spKtra
-0.07. E.t. Frl,,. D.'''''
-0.89. E.t. Frl~" D. ,.",
-1.71 Fr-.s.. - D.lItHO !IF
-2.'3
-3.34
-4.16
-4.98.
-6.61
-7. f3t sue" ", .... ,,'" .us. sue • ,." c , .... n~ , = eo. g" cui 0.00 0.'0 1.00 I.~ 2.00 2.~ 3.00 3.~ 4.00 ".'0 '.00
XIO-I FI?ACT. OF SAffPL ING FREO.
FIG. ( 5.40 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
o C -4 ." c .... > • > r-
, ~ o .... ~ , I
m )( . '" N , N (JI ...,
~ z
CIt , ~ , ~
-N . . ~
.A
\J1 I
\J1 \J1
X'O' Nod oF' 0 PC ~ 0.64 ,,~d N sp~ct..ra Q) "tl -0.06
" -0.77
~ Q. -1.17 , -2.18
C) lO -2 88 "'" . ..... -.J -3.58
~
~ -1.29
-4.99 , -~.70
j
E.,. Fr'f'" tI. 'S",
E.e.. Fr< '" tI. 'ftJD
Fr • ..". - D.t1t15D SF
X'O' ffod°F'°~ PeN ~ 0.65 "~ 5~lra
~ -0.07
" -0.79
~ Q. -1.50 ,
-2.22 Cl ~ -2.94 ..... -.J
~ ~ <= ,
-3.66
-1.38
-~.IO
-~.82
• I.'. Fr<,,. tI. ,.",
,.,. 1W1" tI.''''
",. ..... 0.'" " o c ~
~ ~
> • > r
• ~ o ~
~ • -6. ~o 1"""",,;:: .. ""., .. " .. , """""'''''''''';::.. """ '''''a ,.,,""", •• ,,"'" , ..... 1
0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.50 1.00 1.505.00 XIO- I
-6 • 5~ ~OO ~ 50 i': 00000":50 ;'.00 7.'50"3'. 00" ]':;; ';':0:',' ~~ r" 5'.00 /s , .." Ut
~
~ " ~ Q. , C)
~ ..... -.J
~ ~ ~ ,
FI?ACT. OF SAffPL ING FI?EQ.
X'O ' EVON sp~ctrtJ 0.77~ ______________________________ ~ __________________ ~
I -0.08. I.,. Fr('" tI. 'SOD -0.93 I.e.. ~,,. tI.'ftJD
-1.78 Fr • ..".- D.~.
-2.63
-3.48
-4.34
-!!J.I9
-6.0f
-6.89
-7.1f1 " ., •• ee, 0 eelS .... u.,. "ue. c .... " '"'' Ie VJltuc,ucfllC'V" i """,I
~
~ " ~ Q. , t:)
~ ..... -.J
~ ~ ;z: ,
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.'0 f.OO f.5O '.00 XIO- I
FI?ACT. OF SANPL ING FI?EO.
FI?ACT. OF SAI1PlING FIlED.
XIO' EVON sPKlra 0.78 I
-0.08, I.e.. Frf,,. o. ,1tID
-0.93, E.t. Frf,,. 0.'""
-1.78,
-2.6f ",. ....... D.""'" •
-3.f9.
-4.3f
-!!J.I9.
-6.90.
-7. 7~ I " SI .. " ",. e he , nn_._UH SO e e heel • e u..,... ; fIIIIIIC: • " • u ... ,
0.00 0.50 1.00 1.50 2.00 2.~ 3.00 3.~ 4.00 f.~ ~.OO XIO-I
FI?ACT. OF SANPL ING FIlEO.
FIG. ( 5.41 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
..,
~ Z -.." ...., • ::J >'It
• ~ .. -...., .. W N
V1 I
V1 CJ'.
XIO' ~ 0.66 NbdiF;~d PCN sp~ctra i
~ -0.07
" -0.79
~ Il. -1.51 , -2.23
~ -2 ~ ....., . .... -' ~ ~ ~ ,
-3.67
-4.40
-~. I 2
-~.81
E.'. Fr('" tI.'5OO
E.,. h-(l" O.,~ Fr . ..".- 0.0t1tJ1 SF
-6.56 1""""';:;,, """"""" ;hee,,,,,, h .... ;:;a..eu:cesu, e OJ ''''''''''.. cu. "'01' h.1
~ XOI~~6 NodiF;f!>d PeN s~tra i
~ -0.07
" -0.79
5l Il. -1.51 , -2.23
Cl ~ -2.96 .... -' ~
~ ,
-3.69
-4.40
-~.I 3
-~.8~
E.,. Fr('" O. 'SlID .. ,. ~1" tI.,~
Fr • ..". - o.tIDtItI SF
-6.57 1: ",.:;:; "hue" D ~. " .;s .. e • ;;..'"'''''' S " hi hue cal """'"'' .. I 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 1.00 4.50 ~.OO
~
~ " ~ Il. , Cl
~ .... -' ~
~ ,
X10- 1
FRACT. OF SAffPL ING FRED.
x'O' EVON sp~ctra 0.77~ ________________________________________ ~ __________________ __
I -0.08. E.,. h-('" O.'SlID -0.93
E.,. Frill- O.,~
-, .18. Fr.,..- 0.0t1tJ1 SF -2.63
-3.49
-4.3f
-5.19.
-6.0f
-6.90
~
~
" ~ Il. , Cl
~ .... -' ~ ~ <! ,
-7.1"1. ,e s e sa e. ,0 us U' ,ue, "su,ec,.":::"'oc"fIIIIC!:: .. ,e" .. "I 0.00 O.~ '.00 ,.~ 2.00 2.~ 3.00 3.'0 f.OO f.~ '.00
XIO-' FRACT. OF SAf1PL ING FRED.
X10- 1
FRACT. OF SAIfPLING FRED.
X'D' EVON sp~tra 0.77?1----------------------------~ ________________ ~
-0.08 E.,. h-(,,. 0.'" -0.93. E.,. ~l" 0.''''' -I.7B. Fr.,..- o.tIDtItI • -2.63
-3.f9.
-f.3f
-5.19.
-6.0f
-6.90
-7. "I hoe • 0, "'U. , • en ., •• s .. ,' e ... , .,,~ •• e"'::, .. , 'he, 0.00 O.~ '.00 ,.~ 2.00 2.~ 3.00 3.~ f.OO f.~ '.00
XIO-' FRACT. OF SAI1PLING FRED.
FIG. ( 5.42 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods -
a c .... ~ .... > . > .-• ~ a .... ~ , m JC • " ...., , N Ut
" ~ z
-!::I , ~
~ • ~
CD
... CD
"" ~
" ~ Q.. , ~ "" ...... ...J
~ ~ :<!: ,
\J1 I
\J1 -.I
"" ~ " ~ Q.. , Cl
~ ...... ...J
~ ~ <: ,
.to' f1I..N s~ctrQ XIO' EVon spf!tctra 0.12 "" 0.67.
-0.01 E.t. Frl',. D. '650 ~ -0.071 f\ 1\ E.t. FrO" D. 'fDO
-O.~ " -0.81 ..... I'rte ••• D.DO o.v ~ V E.t. fIlr( ~,. D.''''
-0.95 -I." Q.. .... "-... D.DO a.v -, .1' , -2.29
-, .87 ~ ""
-3.04 ......
-2.32 ...J -3.78.
-2.78 ~ -f.52 ~ -3.24 :<!: , -3.69 -6.01
-4·'~1 cae"uoue , """",ee, h' , DU" GU". ""." e. us, e. .,." C "i '::::-:=:"'1 -6.7S 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 0.00 O.SO 1.00 I.SO 2.00 2.SO 3.00 3.SO '1.00 4.SO S.OO
.10- 1 XIO-I FRACT. OF SANPLING FRED. FRACT. OF SANPLING FRED.
XIO' NEN sp~ctrQ X'D' NodjFi~d PeN sp~trQ 0.'19 "" 0.55
-0.05 Ed. Frl',. D.'goo ~ -0.06 E.t. Fri,,. D.'fDO
-0.59 "- -0.66 E.t. Frl7 ,. D. 'fDO
~ E.t. Fri~" D.'''' -, • 13 -1.27 Q.. .... "'-... D.DO DwJ lIP, ."-... D.DtI a.v -1.67
, -1.88
~ -20 49) -2.21
J \ ~ ...... -2.~ ...J -3.10
~ -3.11 -3.29 ~ -3.83
~ -f.32. , -f.37 -f.93.
-4.91 -5.54 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 i.SO 5.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 i.OO i.50 5.00
XIO- I XIO-I FRACT. OF SANPLING FRED. FRACT. OF SAffPL ING FRED.
FIG. ( 5.43 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c -4
~ -4
> . > r-
I
~ 0 -4
~ , , m J(
• " CD , N til
" ~ z
-N ..., , :::I > ~ , CD 0
--bt .. ~ c.J
'" ~
" ~ Q. ,
~ .... -.J
~ ~ ~ ,
\J'1 I
\J'1 CD
'" ~ " 5l Q. , Cl
~ .... -.J
~ ~ ~ ,
X101 f1l..N sppctro x,o' EVON sppctro 0.38 '" o.&~.
-0.04 Ed. Frl"· D.lf!JO ~ -0.071 1\ " E.t. f:r('" D.'7.IIJ
-O."S " -0.78. E.C.. Frll,. D.'~ ~ V E.'. f:r( I" 0.'''''
-0.97 -, .50 Q. ., .PP.... 3D.00 o.v • , .~... "'.00 o.v , -, .29 -2.22
-, .71 @ '"
-2.94 ....
-2.13 -.J -3.6&
-2.55 ~ -4.38 ~ <:: -2.97 -!Y.IO ,
-3.38 -~.82
-J.90t .......... ".".,. III ... " 11.1 ..... 11111111 .. 111111'''''''"'' '''''1 '11I11"I~ -6.S4 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 '.00 ,.so 2.00 2.50 3.00 3.50 4.00 ".SO S.OO
XIO- I XIO-I
FRACT. OF SANPL ING FREO. FRACT. OF SAf1PLING FRED.
XIO' X'O ' NEN sppct,ro NodiF;pd peN sp~ctra 0."" '" 0.~2
-o.o~ E.C.. Frl I ,. D.I3!JO ~ -o.o~ E.,. f:r(,,. 0.''''' -o.~., " -0.&3
E.C.. Frll'. O.'~ 5l E.C.. f:r( I,. 0.''''' -1.09 -1.20 Q. ., .PP.... 3D.DO o.v ., .~... "'.00 o.v , -, .&1 -1.7"
-2.13 Cl
~ -2.35 ....
I \ -2.6~ -.J -2.92J ~ -3.SO -3.18 ~ ~ -4.07. ,
-4.22 -".6~
-" . .,,, -~.22
0.00 O.~O 1.00 I.SO 2.00 2.~0 3.00 3.~0 ".00 ".~o ~.OO 0.00 o.so '-.00 '-.SO i.oo i.so 3.00 :i.so "-.00 "-.SO ~·.iJO X10- 1 XIO-I
FRACT. OF SANPLING FRED. FRACT. OF SANPLING FREO.
FIG. ( 5.44 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c .... ." c -t
:. • :. ,.. , ,J ,.. 0 -t
~ , , m )C
• " CD , '" Ut 'V
~ z .. '" .... , ::J :. ;0 , CD 0
I~
"" ~
'" ~ Q. , ~ ..... ..... ~ a= ~ ,
V'I I
V'I \D
"" ~
'" ~ Q. , Cl
~ ..... ..... ~ ~ ~ ,
XIOI I1t.. N SPPc tro X'O' EVON sPPc tro 0.37 "" 0.66.
-0.04 E.t. Fri, ,. D. ,.~ ~ -0.011 1\ " E.,. Fr-f, I- D.'33D
-0.45 '" -0.19. E.t. FrlV. O.,~
~ V E.'. FriZI- D.''''' -0.96 -1.'2 Q.
., .,..... 4'.DO o.v ... ,.~ ... 4'.011 o.v , -I .27 -2.24
-1.S8 fi1 '"
-2.91 .....
-2.09 ..... -3.10
~ -2.51 ~ -"."2
~ -2.92 -!J.I!J , -l.ll -!J.81
-l.741 ee '" : , ""CO, ;'"'' hU """", """CUI' 0 "au ;s,ee, ,ce, """",':::::::"1 -6.60 0.00 0.50 1.00 1.50 2.00 2.50 3.00 l.SO 1.00 1.50 5.00 0.00 0.50 1.00 I.SO 2.00 2.'0 3.00 3.50 ".00 'f.50 !J.OO
X10- 1 XIO-I FRACT. OF SANPLING FREO. FRAC T. OF SAI1PlI NG FREQ.
XIO' XIO' NodjFj~d PeN sppct,ro N£N sp~ctro o. "., "" 0.5"
-O.O!J E.t. Fri,,. 0.'350 ~ -0.0' E.,. Fri'I- D.''''' -0." '" -0.6"
E.t. Fri Z ,. D.''''' ~ E.t. FriZJe D.'"" -1.09 -1.23 Q. ., .,.". ... 4'.DO o.v ... ,.,.. ... .'.011 o.v - 1.61
, -1.82
-2.13 ~ '"
-2.'" .....
/ \ -2.6' ..... -3.11°1 ~ -3.11 ~ -3.'9
-3.69 ~ -".18. ,
-".21 -" .". -4.13 -!J.36
0.00 0.'0 1.00 1.50 2.00 2.'0 3.00 3.'0 'f.00 4.'0 '.00 0.00 0.'0 1.00 1.50 2.00 2.50 3.00 3.50 ".00 4.50 '.00 XIO- I XIO-I
FRACT. OF SANPL ING FREO. FRACT. OF SANPllNG FREO.
FIG. ( 5.45 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c -4
" c -4
> • > r-
• ~ 0 -t
~ • • m X • " CD , ...., UI
" ~ z .. ...., ...., , ::I > ;Q , U) 0
-UI ..... .. 0
~
~ " ~ Q.. , ~ ....... ...,J
~
~ ,
\J1 I 0'\ 0
~
~ " ~ Q.. , Cl
~ ....... ...,J
~ ~ ~ ,
X'o' f1l.N spfftctra X'O' EVON spf-ctra 0.37 ~ 0.66.
-0.04 E.t. Fri',. D.'f~ ~ -0.011 1\ A E.t. Frf,,. D.'~ -0.45 " -0.19.
E.t. Fril'. D.'~ ~ V E.t. Frl~" D.''''' -0.96 -1.'2 Q.. ., .Pt.... (;(1.00 019 • , .",.... 110.00 D.9 , -1 .27 -2.24
-1 .69 ~ "of
-2.96 .......
-2.10 ...,J -3.69
-2.51 ~ -..... , ~
~ -2.92 -'.14 , -3.34 -'.86 -].75l eo c ... "" eo, u'cus,,'" 'sc,;'" 'I'" a eo., .. """,; u ussu, De • Ii hi Oi~ -6.'9
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 1.505.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.'0 4.00 4.50 S.OO XIO-I XIO-I
FRACT. OF SANPL ING FREO. FRAC T. OF SAI1PL I NG FREQ.
XIO' NEN sp~ctra X'O ' NodjFj~d PeN sp~ctra
0.'" ~ 0.S3
-0.0' E.t. FriO. D.'~ ~ -0.0' Ed. Frl,,. D.'''''
-0." " -0.64 E.t. Fril'. D.'~ ~ E .... Frf~" D.'""
-1.09 -1.23 Q.. ., .Pt.... II(J,OO o.v .. , .Pt.... IIO,DD o.v - 1.61
, -1.82
-2.12 Cl
~ -2.fO .......
/ \ -2.6f ...,J -2.991 ~ -3.58 -3.16 ~ :<t -f. 11. ,
-f.20 -f.1'
-4.12 -'.34 0.00 0.'0 1.00 I.SO 2.00 2.'0 3.00 3.'0 f.OO f.,O '.00 0.00 0.:50 1.00 I.SO 2.00 2.SO 3.00 3.!JO 4.00 ".'0 '.00
XIO- 1 XIO-I
FRACT. OF SANPLING FREO. FRACT. OF SANPL ING FREO.
FIG. ( 5.46 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods -
0 c ... " c -4
> . > r
• ~ 0 -4
~ , •
m )(
• '" CD , ..... UI ...,
~ z .. ..... " • ~ XJ I
~
I~
\J'I I
Q'\ -»
X'O' ~ 0.36 f1l..N sppctro j
~ -0.04
" ~ Q.. , ~ '" .... -.J
~ ~ ~ ,
-0 .....
-0.9"
-, .2" -, .64
-2.04
-2.4"
-2.94
-3.24
E.t. Frl"· 0.'400
E.t. Fril'. O.'IJOO
... • Phe •• • go.OO o.v
XIO' ~ 0.6~ EVON sppctro •
~ -0.06
" ~ Q.. , ~ '" .... -.J
~ ~ <:: ,
-0. '1'1,
-1.49,
-2.20
-2.91
-3.62,
-4.33
-'.'1'
E.t. Frf,,. 0.'300
,.t. Frfl" 0.''''' .... ",. ... •• 00 o.v
o C -f ." C -4
:. . :. r-
• ;9 o -f
~ , , -J .641,,1 e US" .n" 'I"'" ,ee, e" .• u ;,eesu, seu h'h" e $ e.,. he US" .. ;:;;:,...f
0.00 O.SO 1.00 1.50 2.00 2.50 3.00 3.50 4.00 1.505.00 X10- 1
-6. i6loo u'ii.5ii";'~oo";'.Sij i:tiij''2:5ii too ''j~;ii'':'':;';'~'.OO , E X'O-'
~
~ " ~ Q.. , Cl
~ .... -.J
~ ~ <:: ,
FRACT. OF SAffPL ING FREO.
XIO' ffEff sppctro 0.46
1 -0.05 E.t. fIrll,. 0.1300
-O.'~ E.t. FriV. o •• goo
-I .O~ .... ",.... go. 00 o.v
-I .~
-2.06
-2.~'1
-3.07
-i.08
~
~
" ~ Q.. , Cl
~ .... -.J
~ ~ <:: ,
- 4. -'Bl .. o 0" ,ee, 'fse, e su ." ,0 c. e; hOC se." see .. " "CU. , "ue,,~ 0.00 O.~O 1.00 I.~ 2.00 2.~0 3.00 3.~ ".00 ... ~ ~.OO
XIO- I
FRACT. OF SAffPL ING FREO.
FRACT. OF SA11PL ING FRED.
KJ.o;2 ffodiF'jpd PCN spt!'ct,ro
1 -o.o~ '.t. Fril,. 0.1300
-0.63, '.t. Frfl,. O. I"", -1.20,
-1.'18, .. , .",.... • .• o.v
-2.35
-2.92,
-3.'0
-4.0'1,
-".6~
-'.22""" . ~ ,n '''''' eo.,,,. us. U,S D e""'~'''Si''''5 CUus a .,,: US" e Sf " .. ,,,.1 0.00 O.~ 1.00 I.~ 2.00 2.~ 3.00 3.~ ".00 i.~ ~.OO
XIO-I
FRACT. OF SAf1PL.ING FRED.
FIG. ( 5.47 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
GD , '" Ut
" ~ Z
N .... • ;!
:lID , CD o
N
o -.a.
\J'I I
a-. N
X'O' '" 0 • 39 f1L N sPPc t,ra
j
~ -0.04 E.t. Frl I,. II. ,,~
" -0.47 • , .Prte •• • '20.00 0.0 ~
Q.. -0.99 , -1.32
~ '"
-1 .75 ...... -.I -2.19 ~
~ -2.60
-3.03 , -3.46
XIO' '" 0.65 EVON sptac tra
i
E.t. ~I" D.'~ en 'tJ -0.07.
" -0.78 .
-I.SO '.t. Prl7,. 0.'''''' ~
Q..
-2.22. , .... ~ .... 110.00 o.g
@
'" -2.93.
...... -.I -3.65
-".37
-5.08.
~
~ , -5.80
o c ~
" C ~
> . > r
I
~ o ~
;t , I
-6.521 D" """"".," us .. ,e s c su. 0" eo CDC e eo , eo., hSue sue ",::::;::;;"":1 ~ 0.00 0.50 1.00 ,.SO 2.00 2.50 3.00 3.50 ".00 ".50 5.00 ~
-3.89 1 .h" CUi " ""'I' "h" Ieee ,ee", eeoc ""eo, • ,us, "" hi """'" SO" ,,,, .. ii,::="I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
'" ~ " 5l Q.. , Cl
~ ...... -.I
~ ~ <: ,
X10-1 FRACT. OF SANPL ING FRED.
X'O ' NEN sppct.ra 0."6T----------------------~~~----------~ I
-0.05 E.t. Frlt)- D.'~
-0.56 Est. Frll" II.'~
- 1.07. ., .""" •• - '20.00 o.v
-1.'8.
-2.09.
-2.60
-3. "
-".Ii
-If .6'"1 .... ", " "e 0 $ , ." poe •• II , e, hU .. o,uu"uo .. ""uo".;t4
'" ~
" ~ , Cl
~ ...... --J
~ ~ ~ ,
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 i.OO i.50 5.00 XIO-I
FRACT. OF SANPL ING FRED.
XIO-I FRACT. OF SANPLING FREO.
Xd.O;3 NodiFipd peN s/1E'ct.ra
I -0.05 '.t. ~ I,. D.''''' -0.6"
'st. Prl7,. D.'''''' -1.23.
-1.82. .... ~ .... 'IO.DD o.g
-2."0
-2.99.
-3.58.
-".16
-".7' -~.3""'" seu":: e U" us. 0 •• 0, ;:.. eo,oue, 152 , ",Ci'se SUi' """I
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 ".00 ".50 5.00 XIO-I
FRACT. OF SANPL ING FRED.
FIG. ( 5.48 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
..... , ~ ...,
~ z .. N ..... I :I > ,g I
CD o
.. UI .. .. w
"" ~ "-
~ Q. , ~ ~ .... ...,J
~
~ ,
V'1 I
0'\ VJ
"" ~ "-
~ Q. , Cl
~ .... -' ~ Q-:
~ ,
XIOI /1lN splI'ctra X'D' Evon sPlI'ctra 0.50 "" 0.73,
-0.05 E.,. Fri,,. II.'~ ~ -0.071 f\/\ I.,. ~, Ie II.''''' -0.60 "- -0.88, ., .,.,. ... ,~.oo 0.0
~ I.,. ~ ~ Ie II. I .. ",
- •• 15 -1.68 Q. .. , .",. ... ,so .• a.v
- •• 70 ,
-2.49
-2.~ @ ..... -3.29 ....
-2.80 -' -4.10
~ -4.90 -3.35 ~
-3.90 <'! -5.71 , - ..... 5 -6.51
-5.00 to » e .. c"s " , e US" leu. " .. ,; • "SO, su. ",.,,""'" e "".'''''.,,~ -7.32 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.SO 5.00 0.00 O.SO '.00 '.SO 2.00 2.SO 3.00 3.50 4.00 f.SO 5.00
.'0-1 XIO-I
FRACT. OF SAI1PLING FRED. FRACT. OF SAffPLING FRED. X,O' XIO' NodiF;~d PeN s~tra NEN splI'ctra 0.'7 "" 0.60
-0.06 E.,. Fril,. 11.'500 ~ -0.06 I.,. ~, Ie II. I.", -0.68 "- -0.72
E.'. ~I" 11.1Il00 ~ I.,. fIifI'f ~ Ie O. , .. ",
-, .31 -1.39 Q. ., .,.,.... ,~.oo o.v .., .",.... ,so .• a.v -, .93
, -2.0'
fi1 -2.11 -2.~
'" I \ I ....
-3.18 -' -3.37] ~
-f.Of -3.8' ~ -f.ff
~ -f.10 , -5.36,
-'.S'9t.I .. "OIIII, ",.". 111,,,.1, '1" .... I;;; , •• ~ -6.02, 0.00 O.!}O '.00 '.!W 2.00 2.!W 3.00 3.!}0 f.OO 'f.!j(J !}.OO 0.00 0.!j(J 1.00 '.!j(J 2.00 2.!j(J 3.00 3.!jO f.OO f.!}O 5.00
XIO- 1 XIO-I
FRACT. OF SAf1PL 1116 FRED. FRACT. OF SAI1PLING F~D.
FIG. ( 5.49 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c -t
~ -t
> · > r
• ~ 0 -4
!l • t
m " • " CD , ~ " ~ z
-~ • ::I > lU
• ~
Ig .. ... "
"" ~
" ~ It , ~ ..... ...J
~
~ ,
\J1 I 0\ po
"" ~ " ~ It , Cl
~ ..... ...J
~ ~
~ ,
X102 I1t.. n SPPC tro x,(J2 Evon sppctro 0.11 "" 0.30
-0.0' E.,. Frl I,. D.3OOD ~ -0.031 I I E.,. Frl'" 0.3t1DD
-0.14 " -0.36 E.,. Frll" 0.1700 ~ I.,. Ilrll" O. '100
-0.26 -0.69 It E.,. Frl'" 0.1500 Ed. Frl,,. O. '!DO
-0.39 ,
-1.02
-O.~I ~ '"
-I.~
..... -0.64 ...J -1.68
-0.76 ~
-2.0/1 ~ ~
-0.89 ~ -2.3f , -, .0' -2.67
-1.131 ",..c:::::: OUt. "c , " US" Ie" "'''' Ie " ;:::... .. ,,"""'" ';c .. 0 h,1 -3.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.505.00 0.00 O.SO '.00 1.50 2.00 2.50 3.00 3.50 4.00 f.!50 '.00
XIO-' XIO-I FRACT. OF SAI1PLING FREO. FRACT. OF SAf1PLING FREQ.
Xl02 11£11 sppct,..o xl02 f1odiFi,.d pcn s~ct,..o 0.23 "" O. If
-0.02 E.'. Frl'" D.3OOD ~ -0.01 E.,. Frl I,. 0.3t1DD
-0.28 " -0.17 E.C.. Frll" 0.'700 ~
I.,. Frll,. 0.'100
-O.,f It -0.33 E.,. Frl,,. 0.1500 Ed. Frl,,. 0.1500
-0.79 ,
-0.49 Cl
-1.05 ~ -0.65
1 JUt II .....
- 1.31 ...J -O.BI ~ -0.97 -,.~ ~
-, .82 ~ -1.13 ,
-2.08 -1.28
-2.33 -1.4f 0.00 O.~ 1.00 '.~o 2.00 2.~0 3.00 3.~ f.OO i.'O ~.OO 0.00 O.~O '.00 I.~ 2.00 2.~ 3.00 3.~ f.OO f.~ '.00
XIO- I X,O-I FRACT. OF SAI1PL ING FREO. FRACT. OF SAI1PLING FREQ.
FIG. ( 5.50 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
0 c -4
" c -4
> . > r
• ~ 0 .... ~ , •
m N . " CD , '" tA
" ~ z .. tA • > ~ , ~
'" -.. w .. -..
'" ~ " ~ Cl , ~ '" ... ...J
~ ~ ~ ,
IJ1 I
'" IJ1
'" ~
" t:) Vl Il... , t:) to
"'" ... ...J
~
~ ,
Xl02 0.11
-0.01
-0.' ..
-0.26
-0.39
-0.51
-0.64
-0.76
-0.88
-I .01
-I .13 . --------.-----0.00 O.SO
Xl02 0.20
-0.02
-0.2tf
-O."~
-0.67.
-0.88
- 1.10
-1.32
-, .~3
-1.7'
-1.97 0.00 O-.~
f1l If SPPC t.ra '"
E.l. FrU" D.2'OOO ~
" E.l. Frll" D.'1OO ~ Cl
E.l. Frl:J,. D. ,,,,,, , fi1 '" ... ...J
~
~ ,
1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 Xl0- 1
FRACT. OF SAffPL ING FRED.
NEN sppctra
E.l. Frl',. D.1OOO
E.l. Frl,,. D. , 100
E.t. Fril" D.''''''
'" ~ , ) )
'-l
~ ... ...J
~ ~ ~ ,
1.00 I.SO 2.00 2.~0 3.00 3.~0 tf.OO tf.,O ~.OO XIO- I
FRACT. OF SANPL ING FRED.
Xl02 0.36. EVON sppctra
-0.0" Ed. Fri'" II.~
-0 ..... E.t. Fri, Ie II. , 100
-0.83. E.l. Frl'" II.'"
-1.23
-1.63
-2.03.
-2."3.
-2.83.
o c ~
~ ~
> . > ,...
• ~ o -4
-3.23.
'-4~
-3.63L, .. ,L .. " .. "~"im' moo ......... , .. " ..... .1 , w 0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.~ ".00 ".SO ~.OO ~
XIO-I
~ I
FRACT. OF SAffPLING FREQ.
XJ ~ Nodi F i E'd peN SPE'C tra
J -0.01 Ed. Frl,,. II.~ -0.17.
E.t. Frl~" 11.'100 -0.33. E.,. Frl,,. D.'''''' -0."9.
-0.". -0.81
-0.97.
-1.13.
-1.28
-, • 441"", s~ " ,h" ... , he hi'"~ ... ". e ,;::;.., 'sc", '" S is •• en, "" .1 0.00 O.~ 1.00 I.~ 2.00 2.~ 3.00 3.SO ".00 ... ~ '.00
XIO-' FRAC T. OF SAIfPl 1 NG FRED.
." , ~ " ~ z .. til • j I
~
-I\,) .. .... .. ~
FIG. ( 5.51 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods ~
'" ~
" Cl V) Q.. , ~ ..... ...,J
~ ~ ~ ,
\J"I I
C]'\ C]'\
'" ~
" ~ Q.. , Cl
~ ..... ...,J
~ ~ ~ ,
XI02 ffLl1 sp~ctro x,02 EVOI1 s~tro 0.11 '" 0.34.
-0.01 E.t. Frl'}· D. 'BOO ~ -0.031 . II E.t. Fri,,. 0.'1f1II
-0.' .. " -0.41 E.t. Frll" D.'~
~ E.t. Frl ~ Ie II.'''''
-0.2S -0.19 Ed. Frf,,. II.''''''
-0.39 ,
-1.11
-0.51 fi1 '" -I." .....
-0.63 -J -1.93.
-0.76 ~
-2.3/1 V' ~
~ -0.98 -2.69 , -, .01 -3.01
-, .131 "",;:;;;: .. sec .,e ""cu,'". """sa, "su" " ;;;;:;..'''''''''''''''' """",,,,,,,,,,,,.1 -3.4' 0.00 O.SO 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 0.00 O.SO 1.00 /.50 2.00 2.50 3.00 3.50 4.00 4.~ '.00
XIO-I X/O- I
FRACT. OF SAI1PL ING FRED. FRACT. OF SAI1PLING FRED. Xl02 Xl02 /1E11 SPE-C tro NodiFiE-d PeN sp.ctro 0.17 '" O. ,..
~ -0.011 -0.02 I II E.t. Frl,,. II. 'Il00 E.t. Frl,,. 0.'''''
-0.20 " ~.'71 ~'\ E.t. Frll" D.'~ ~ Ed. Frllle 11.'7011
-0.39 Q.. -0.33 E.t. ~tl" 11.'100 E.t. Frl,,. II.''''''
-0.!J8 ,
-0.49 Cl
-0.16 ~ ~.651 I \ ..... -0.9' ..... -0.81
~ - / .13 ~ -0.91
-1.32 ~ -1.13. ,
-,." -1.28.
-1.69 -1.44 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 f.OO f.,O 5.00 0.00 0.'0 /.00 /.50 2.00 2.50 3.00 3.50 4.00 f.50 '.00
XIO- I XIO-I FRACT. OF SANPLING FREO. FRACT. OF SAIfPLING FREO.
FIG. ( 5.52 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods -
0 c ~
~ ~
> . :-r
• ~ 0 .... ~ I
m M • " CD , N til ..,
~ z
til I :-~ I
~
-N -.. co .. til
\.J1 I 0'\ -.J
X102 /1Lff sppctra "" 0.11]
~ -0.01
" -0.' 4
~ Q.. -0.26 , -0.39
tl ~ -O.~' ~
..,J
~ -0.64
-0.76
E.'. Frl'" tJ.'5O(J
x I 02 .... ________ --...:E::...VO:.::::'N..:.....:S:!;p::t-C:..=-::t:.:..r..:a:...-___ -, "" 0.36 i
~ -0.0"
" ~ Q.. , ~ '" ~ -J
~ ~
E.t. Fr( r,. tI.'''''' -0."3.
E.t. Fr( ~,. 0.1100
-0.83. E.t. Ifrl,,. O. ,.",
-1.22.
-1.62.
-2.02
-2.'"
o c .... ~ -4
> . > r
, ~ ~ , -0.88
-, .01
-t ., 31 us-=::;-, " "ee,,,," eo,. e , .. us, ,;,,;::;;.., "5 "0,05' S ._nn e; ",," al
~ ,
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 X'O-'
~::;1 .... ~, .. ~... ... .. " .... " ............ .I I :' 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 ".00 ".50 ~.OO ~
XIO-I
o .... ~ , I
F~ACT. OF SAf1PL ING F~EO.
"" Xd.tf,. NEff sppct,ra
Q) 'lj -0.02.
" ~ Q.. , ~ ~
-J
~ ~ ~ ,
-0.18
-O.~
-O.~I
-0.68.
-0.84
- 1.01
-1.1'1
- 1.34
E.,. Frl'" tJ.ISD(J
E.,. FrlI,. tJ. 1100
E.,. Frl'" tJ.'500
"" ~
"" ~ Q.. , ~ '" ~ -J
~ ~ ~ ,
-, ." 1:;;" ,,:::, 0 e. 'us, "e lSUS S • ,,7')t., .,.;:"""" $'''''' c; "I 0.00 o.~o 1.00 1.50 2.00 2.~0 3.00 3.~0 4.00 4.'0 ~.OO
XIO- I
F~ACT. OF SAf1PL ING FREO.
FRAC T. OF SAf1PL I NG FRED.
XI~ NodiF;pd peN sppct,ra 0.1 '~i------------------~----~
-0.01 E.t. Ifrl r,. O. '700
-0.17. E.'. Fr( ~,. O. ,.",
-0.33 Ed. Frl,,. O.I!tItI
-0."9.
-0.6~
-0.81
-0.9'1
- 1.13
-1.28
-, • ff 1"""::::;"""",, • e us,; "Ii '"'' su, ;;:;"'eue S su'l " '" uu: • c .1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.~0 ".00 ... ~ ~.OO
XIO-I FRACT. OF SAf1PLING FREO.
FIG. ( 5.53 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
CD , N Ut
" ~ z .. C,/t
I
~ 'IJ I
~
N .. C,/t .. tJl
"" ~ " ~ Q. , t:l
~ ..... -J
~ Q:
~ ,
\J'I I (j\
co
"" ~ " ~ Q. , Cl
~ ..... -J
~ ~ ~ ,
Xl0l f1L N sPt"c t,,.o X'O' EVDf1 sp~ct,"(J 0.12 "" 0.67.
-0.01 E.,. Frl' ,. D. '630 ~ -0.071 1\ It E.'. Frf,,. D.'<fDD
-O.~ " -0.81
5l V '.t. I'W~" 0.''''' -0.95 -I." Q.
-, .1' , -2.29
-, .87 m ""
-3.04 .....
-2.32 -J -3.78.
-2.78 ~
-4.'2 O!
~ -3.21 -'.26 , -3.69 -6.01
-4. '51 ... ,"'" ''''''''''', ... ,,''''''''''',. """., US" e ;Oil e $ .,"h''''', as'uo i~1 -6.7S 0.00 0.50 '.00 '.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00 0.00 O.so 1.00 I.SO 2.00 2.50 3.00 3.SO 4.00 4.SO '.00
XIO- 1 XIO-I FRACT. OF SAf1PL ING FREO. FRACT. OF SANPL ING FI?EQ.
XIO' XIO' NEN Spftct,,.o NodjFi~d PeN 5~ct"a 0.49 "" o.ss
-0.0' E.r.. Frl". D.'g(J() ~ -0.06 Ed. A--t,,. O.'fO(J
-0.'9 " -0.66 E.,. Frlt,. D. 'fDO ~
E.,. A--t ~,. O.,§tID
- 1.13 -1.27 Q.
-, .67 ,
-1.88 Cl
-2.21 ~ -2.49] / \ ..... -2.7' -J -3.10
~ -3.29 ~ -3.71
C! -4.32 , -4.37 -4.93.
-4.91 -'.'4 0.00 o.SO 1.00 I.SO 2.00 2.'0 3.00 3.'0 4.00 4.SO '.00 0.00 o.so 1.00 I.SO 2.00 2.SO 3.00 3.SO 4.00 4.SO '.00
XIO- 1 XIO-I FRACT. OF SAf1PL ING FREO. FRACT. OF SANPL ING FREO.
FIG. ( 5.54 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
0 c -4
"2 -4
::. . ::. r
I
~ 0 -4
~ , I
m )( . " CD , '" (Jt ...,
~ z
UI I
~ lU I
CD 0
w . . 0 .. CD
~
~ " Cl
'" Q.. , @ ..... ~ ...., ~ ~
~ ,
\J'I , 0'\ '-0
~
~ " ~ Q.. , 0
~ ~ ...., ~
~ ,
X101 11l..f1 s~tro 0.40 ~
-0.041 I.t. Frl, Ie O.f"'" ~ -0.48 "
I I.t. Frll" 0.'100 -0.93 51
Q..
-I .37 ,
- I .81 fi} ..... ....
-2.25 ....,
-2.70
-3.11
~ ~ ~ ,
-J.~9
x'o' EVON SPf*ctrG 0.66
-0.071 I.t. FrI',. O.f"
-0.791 I.t. ~1" 0.''''
-1.5'
-2.23
-2.96
-'.68 -f.fO
-5. I'.
-5.11'
c c ... ~ ... ~ · ~ r
• ~ o ... ~ •
-4.03 K. c. ceca .~n.n. • e eo suus; • ~ •• _ ,ue. ''''''''V' e.".,,,,,, .1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 ~.OO
XIO-I -6.'~, ii.sijhi':OO -i-3iU:iiiJ 2.50'"i:00 hi:", ":O:;~:¥j'J.DD I [
, FRAC T. OF SAffPL I NG FREO.
x,o' nff1 s~trG 0·"1
-0.05
-0.6'.
-, .2' -'.79.
-2.37
-2.~
-3.5'
-f. "
-f.69
I.t. Frl' Ie O •• ~
I.t. ~,,. O.'fOII
I.t. Frll1e o. ,..,
~
~
" ~ Q.. , 0 ~ .... ...., ~
~ ,
-, .211(: " • Oie U au. sa ..... ,.. , • , •• us. • .... "hi 0.00 o.!)O '.00 ,.!)O 2.00 2.!)o 3.00 3.!)o f.OO f.!)O '.00
X'O-, FRACT. OF SAI1PL ING FREO.
FRAC T. OF SAI1PL I NG FRED. x,o' 0.56. nod;F;~ PeN s~trG
-O.OG I.t. Fro(,,. o •• SGrJ
-0.67. I.t. Fro( 1 Ie o. ,.., -'.28
-1.90.
-2.5'
-3.12 .
-3.7f
-f.~
-f.97
-'·5B1ees $ b u, • ..c:: II " •• ee. eo, ice" }J!I.. a i:Cst 50"'::: Us. i hU ... 1 0.00 O.~ '.00 ,.~ 2.00 2.~ 3.00 3.~ f.OO f.~ '.00
X'O-' FRACT. OF SAI1PL ING FRED.
FIG. ( 5.55 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods -
~ ....
~ -UI • j • 8
w o ..... u CD
V'I I
--.-J 0
'(10' '" 0.35 /1LN spE'ctra
i
~ -0.04
" -0.42
~ Q.. -0.81 , -1 .20
~ ~ -1.58 .... -J
~ ~ ~ ,
-1.97
-2.36
-2.74
-3.13
E.'. Frl' Ie 0.:13110
'.t. Frll" 0.1530
X'D ' '" 0.62.
~ -0.061 E.'. h-f' Ie o.%Jt1tI
" -D. 75
~ I E.'. h-f '" O. 'tiIIl Q.. -I.f3. ,
-2. "
~ ""
-2.79. ... -J -3.fB.
~ -f. 16. ~ <!: -f.Bf ,
-'.'3
EVon sp~ctr(J
o C -4
~ -4
> • > r-
• ~ o .... ~ •
-6.21J~'ii:;o "i':'OO"i':;:O ''i:'iiO' 1.';o"'3.00''3:;:O''4':0;;''f'.;;;';.00 I E XIO-I
- J. 52 K "".,... us,,,,,,,. 'C""'iD",'" "".,ue, """,us .. ,,,.,.,,,,, au e "su,; " e ".1 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
XIO-I
XIOI
'" 0.f9
~ -0.05
" -0.58
5l Q.. -i.I2 , -1.6' Cl
~ -2.19 .... -J -2.72 ~ -3.25 §
-3.19 ,
FRACT. OF SAIfPLING FREQ. FRACT. OF SAI1PLING FREQ.
XJ.O;, "odjFi~d PC" s~t.rG I
-0.05
"Eff sp~ctra ~
,.,. ~I" O."'l} ~ E.,. h'f' Ie O.%It1tI
" -0.6'
-I.~ E.,. h-fl" .I~ 5l E.'. ~3" .Ife. Q..
E.t. h-f I,. II. I SOD
, - 1.86.
Cl
~ -2.f6. ... -J -3.06.
-3.66 ~ ~ ~ -f.26 ,
-f.86.
-5 •• 5'1"", ... ,...::, .,""". • • """ • "~,,..c:. 'U""'" "e. ,; .. ,":::;.., SI cl 1.00 1.:50 2.00 2.503.00 3.50 f.OO '1.:50 5.00 0.00 0.:50 1.00 1.:50 2.00 2.:50 3.00 3.50 f.OO ".:50 5.00
XIO- I XIO-I FRACT. OF SAffPL ING FREQ. FRACT. OF SAI'1PL.ING FREQ.
FIG. ( 5.56 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSOE melhods *
ce ~ " ~ z --o • S • ~
u N
-o
\J1 I
-..J -->
X101 ~ 0.52 NEN s~tra
I
~ -0.05 Est. Frl',. tI.3OOO
'" ~ Q.. , @ "" .... -J
~ ~ ~ ,
~
~ '" ~ Q.. , Cl
~ ~
-J
~ ~ ~ ,
-0.62 E.'. Frl, Ie tI.' 100 - I .19
-I.n
-2.31
-2.91
-3.18
-1.05
-1.62
-5. 19 1 """"'''' ""'I""" ,es, "CU'I""""" """"U'''''''''''' ... "., .. ". ''',hS s~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 SAffPL ING FRED.
XJ~;9 NodiFi,-d PeN sp,-ctra
I -O.Ofi E.t. Frl'" tI.3tJ50
-0.71 E.t. Frl~" tI."~
-, .37
-2.02
-2.67.
-3.32
-3.98
-".63
-'.28
-'.94 1 .. ,s: sue; us""::=-s; " .. cu" "".,,,. eo.,,,. "".,""" Of hue;;;' ,ssSS'US ". I
~
~ '" 51 Q.. , Cl
~ ~
-J
~ ~ ~ ,
0.00 o.~ '.00 ,.~ 2.00 2.~ 3.00 3.~0 'f.00 'f.~ ~.OO XIO- 1
FRACT. OF SAffPLING FRED.
",,. NoI .... 1,..,...ld
-.AX t n
IMP .FII. • 1.000
.... ITUJS • 1.00 1.00 2.00
FIlE05. • O.I!GO 0.'100 0.""'"
.,'n .. ,.,.... 0.00 0.00 0.00
!Nt. ( dB" 10.000 40.000 .... 021
'lANI.DEY •• 0.0100000000000000
.sa. .lInlt '0 .O~OO
tltllLATEO CowvleflC. "-v ..
X'O' EVON sp,.ctra 0.671 I ~.07 Es'. Frl' Ie tI.3IItItI
-0.90 E.,. Frl~" tI.'"
-I.'of
-2.27.
-3.01
-3.7'f
-"."8. -'.21
-,.~
-S.SBJ • "" 0' ,; us • cue, e................ , .' .. , ""'f • ....... cu, ••• :;;....1 0.00 O.~ 1.00 I.~ 2.00 2.~ 3.00 3.~ f.OO 'f.~ ~.OO
XIO-I FRACT. OF SAI1PLING FRED.
FIG. ( 5.57 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods -
o c ~
;! ~
> . > r-
• ~ o .... ~ • m )C . " .., , ..., UI
" ~ Z
UI • > ." XJ I
~
w
w .. o
'"
\J'1 I
-......J N
X'02 f1lN spE'ct,rQ ~ o. '0, ~ -0.0'
" -0.12 Cl V\ -0.23 Q.
~ -o.~
~ -0.46 .... .... ~ Q:
~
-0.57
-0.68
-0.79
E.,. Frl,,. 11.'500
E.,. Frll" O. , 1fID
AIIp,U"Mp.tI Ie .",
x,02 EVON SI)E'C t,rQ ~ 0.30, Ql lj -0.0:1.
" 51 Q.. , ~ .... .... ~ ~ ~
-0.36
-0.69
-1.02.
-I.~
-1.68.
-2.01
-2.3f
E.,. Frl, Ie II. ,7fID
I.,. FrlZ,. 0.'''''' AIIpf t I ",."U , Ie • SO
"
o c .... ;! .... ~ • ~ r
, ~ o ....
-, .0' 1 "see"":;;, '''"." so ;;::r.. s es,o sse $ eo e. suseuCi"'" """ I 0.00 0.50 '.00 '.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
X'O-' -.:J.oc,J, ..... L ... ~ moo ..... " .... ". .." ... .1 , w
0.00 O.SO 1.00 I.SO 2.00 2.50 :1.00 3.50 f.OO f.~ '.DO ~ XIO-I
-2.61. ~ ~ , ,
, -0.90
FRACT. OF SANPL ING FRED. FRACT. OF SAffPLlNG FRED.
~ Xl02 lfocliFif!d PCIf slWCtra O.If·~ __________________________________ ~ ________ ~ ______ --,
I ~ XJ.tfs N£N s~trQ ,
-0.01 Q) lj -0.02. I.,. Frl' Ie II. ,,,,,, ~ ,.,. I'rf'" 0.'1fIO
" ~ Q. , ~ .... .... ~ § ,
-0.11. -0.22. E.'. Fro(' Ie II.' 7fID " I.,. Frl Z" 0.''''''
-0.f2. 5l Q. -0.33.
-0.62
-0.92
-1.03
-1.23
-I.f3
-1.63
AIIp' U "..". ( ,,. .,., , Cl
~ .... .... ~ ~ ~
~
-I.B:1~ •• Ls. ...... ,. u,~.,.~ 0.00 O.~ 1.00 I.~ 2.00 2.~ 3.00 3.~0 f.OO f.~ ~.OO
XIO-I
FRACT. OF SANPL ING FRED.
AllpU Z,, __ t • Ie .", -0.f9
-0.6'
-0.91
-0.91
-1.13.
-1.29.
-, • 441,,,,, ... ,.-;;"" e.. US'C' see .;;;... hWUue e S eni en i." au u ..... 1 0.00 O.~ 1.00 ,.~ 2.00 2.~ 3.00 3.~ f.OO f.5O '.00
XIO-I FRACT. OF SAffPL ING FRED.
FIG. ( 5.58 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
U) , ~ ~ o
" JU
2 .. U) , ~ ~ , ~
..., A o .. g
\J1 I -.J \.JJ
XI02 .. ____________________ Nl~~N_=S~p~~~t~r~a __________ __, "" 0.10 J
~ -0.01
" -0. I 2
5l Q.. -0.23 ,
-o.~
~ -0."& ..... -' ~ ~ ~
-0.'7
-0.&8
-0.79
I.'. Fr('" •. ,~ I.,. ~,,. '.,1DtI
...", U ".", (f" • 15
1(,02 EV(Jl1 5ptactra "" 0.30, Q) 't) -0.03
" ~ , ~ "" .... -' ~ ~ ~
-0.36
-0.&9
-, .02.
-'.35 -, .68.
-2.01
-2.3f
I.'. Fr('" '.,7fI/D
,.&. ~I" •• ,,,,,,
MpU ~"AIIpC (I" .15
o C -4
~ -4
~ • ~ r
, ~ o -1 '4
-3.001",," · '" · L .... ~. · """ """"'" """""" ... .1 f W 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 f.OO f.5O '.00 ~ X'O-,
-2.&1 ,
~ , , ,
-0.90
-1 .01 1 'ee",..;:::;" .... , ;un s, "",;;:"ceucuei ae e ,ee" e U , .. ,," he .. I 0.00 O.SO 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
"10- 1
FRACT. OF SAI1PlING FRED.
"" "" XdfB ffEN spflctra i
~ -0.02 .. ,. h(,,. '.'''''' ~
" -0.22 " -0.f2.
I.,. Fr(',. •. , 7fI/D 5l Q..
5l Q.. ,
-0.62 _.u".",t,,. .15 ,
-0.B2 ~ ..... Cl ~ ....
-, .03
-, .23
-1.-43
-' ~ ~ ~
-' ~
~ , , -, .63
-I.B3~. Lx ...... uuo~.,~ 0.00 o.~ 1.00 I.~ 2.00 2.~ 3.00 3.~ ".00 ... ~ '.00
1('0-' FRACT. OF SAI1PlING FRED.
FRACT. OF SAf1PLING FlIED.
1(,02 NodiFi.d PeN s~trG O./fl I
-0.0' I.'. ",.",. •. ,1DtI -0. '1. I.,. I'W ~,. '.'!DO -0.33.
AIIpC U ,,_It ,,. .J! -0.f9.
-0.6'
-O.B'
-0.97
-'.'3. -1.29.
-, • ff 1 us,....:::..." 51 cue. 4ee c ;:,... "USus ssc,,'" e. cus u. U us, .1 0.00 0.'0 1.00 I.~ 2.00 2.~ 3.00 3.50 'f.00 'f.5O '.00
XIO-I FRACT. OF SAffPL ING FRED.
FIG. ( 5.59 ) POWER )I( For
SPECTRAL diFFerenl
DENSITY ESTIMATE PSDE melhods *
CD , ~ • o ..,
i cp ~ ~ , ~
N -.. (JI
en
'" ~
" 5l Q.. , ~ '" .... ...,J
~ ~ ~ ,
\j1
I --.J ~ '"
~
" ~ Q.. , Cl
~ .... ...,J
~ ~ ~ ,
XI02 /1LN Sptltct.ra xl02 EVON sptact.ra 0.10 '" 0.30.
-0.01 E.,. Fro( I,. II. '500 ~ -0.03-:1 I I E.'. Frf'" 11.'7011
-0.12 " -0.36 AIIp' t ~ "AIIp, t I,. .10 5l 6.,. IfIr( ~,. o. ,,,,,, -0.23 -0.69 Q..
AllpU 1,,1WIpf tI,. .10 -O.~
, -1.02
-0.1& fi1 I\i
-I.~ ....
-0.~7 ...,J -1.69.
-0.&8 ~
-2.011 ~ ~
-0.79 ~ -2.3" , -0.90 -2.67
-1 .01 1 '" s "...G: " e , " " .. "c. e ~"s,,"" 5" "" ••• ,""" "'''' , .... I -3.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.505.00 0.00 O.SO 1.00 1.50 2.00 2.50 3.00 3.~ ".00 ".50 ~.OO
XIO-I XIO-I FRACT. OF SAf1PL ING FRED. FRACT. OF SANPL ING FRED.
Xl02 x,02 N£N spt!>Ctra nodiFi~d Pen sppctra 0.18 '" -::::1 I
~ -0.02 •• ,. ~,,. e. ,!DO
I I 6.,. I'rf'" e. , 1011 -0.22 " E.,. ~,,. e.,l011 -0.1"
~ •• ,. I'WI" tI.'''''' -D."2 Q.. -0.33.
AIIp' t I "...". ( ,,. • '" ,
/\ /\ A.pf U,,1WIpf tI,. .'tI
-0.62 -D.49j ~ -0.6~ -0.92 I\i ....
-1.03 ...,J -0.111
-, .23 ~
-0.971 / \ ~
~ -, ."3 -1.13 , -, .63 -1.29
-, .93 -I."f 0.00 o.~ 1.00 ,.~ 2.00 2.~ 3.00 3.~ f.OO f.~ ~.OO 0-.00 o.~ '.00 I.~ 2.00 2.~ 3.00 3.~ f.OO f.~ ~.OO
X'O-I XIO-I FRACT. OF SA/1PL ING FREO. FRAC T. OF SAI1PL I NG FRED.
FIG. ( 5.60 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
a c ... ~ ... ,. . ,. r
• ~ 0 -t
~ , , m Ie • " ~ fi\I
~ .-0 . '" ~ , ~ ~ , ~
N -... ..., -tr
\J'I I
-.J \J'I
J( 1 07 f1l.. 11 SfJE'C t,ro
'" 0.10.
~ -0.01
~ -0. I 2
~ Q.. -0.23 , -o.~
Cl ~ -0.46 ~
..,J
~
~
-0.~7
-0.68
E.,. Frl,,. tI.,~
MipU ~ "AlllpU ,,. .",
x,02 E VDI1 5~C t,ro '" 0.30, Q) 'tJ -0.03. ~
~ Q... , ~ '" .... ..., ~ ~ ~
-0.36
-0.69
-1.02.
-I.~
-1.68.
-2.01
-2.34
E.t. Frf'" tI.,7rID E.t. ~~,. tI.'" AIIpC U "IfIIpU ,,. .CJ!!I
o c ~
~ -4
~ • ~ r-
• ~
, -0.79 , -0.90
-, .01 1 s e""u~e •. n n. e" , • eo::;::'" cu, sus e ,ee'i""'" ,ee, ., .. ,," .. I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 ~.SO 5.00
)(10- 1
~:::t. .... ~ .... ~ " ................. " ............. .1 , :' 0.00 0.50 1.00 I.SO 2.00 2.50 3.00 3.50 4.00 4.SO '.00 ~
XIO-I
" o -4
~ , •
FRACT. OF SAf1PI..lNG FREO.
~ '" ~.~ I1En sppct.ra f
~ Q) 'lj -0.02. .. ,. h'(,,. tI.,!DtJ ~
~ Q.. , Cl
~ .... ..,J
~ ~ ~ ,
-0.22. E.'. Frl,,. tI. ,7rID -0. "2.
-0.62 MipU ~ "Mtpl CI,. .",
-0.83
-1.03
-1.23
-1.43
-1.63
~
~ , Cl
~ .... ..., ~ ~ ~ ,
-'.B3~. Lx , .~u'u~ 0.00 O.!SO '.00 '.!SO 2.00 2.SO 3.00 3.SO ".00 1.!SO '.00
XIO-I
FIlAC T. OF SANPL ING FREO.
FRACT. OF SAI1PL ING FREQ.
XI~ nodiFi~d Pen s~tra o. I I -0.01 E.t. ~,,. tI.,"'" -0. '7.
E.t. ~~,. tI.,,. -0.33.
"",U"IfIIpU,,. .O! -0.49.
-0.6'
-0.81
-0.97.
-1.13.
- 1.2'9
-, • ff 1, $ o..c::: Of • a eu.. e. is ;;:,.., use "" U • sou " ........ 1 0.00 O.SO 1.00 '.SO 2.00 2.SO 3.00 3.50 4.00 4.50 '.00
XIO-I FRACT. OF SAf'fPLING FRED.
FIG. ( 5.61 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
." , ~ .. o ...,
i ... , ~ -4 , ~
~ .. CD .. ~
V"1 I
-....J 0"
X'o' "" 0.43 NLN Spftctra j
Q) 1> -0.04 E.t. FrltJ· D."'"
" -0.51 MpH Z~"""( I , •• ~ 5l
Q. -0.98 , -I ."'5
t:l ~ -1.92 ...... ...J -2."'0 ~
~ -2.87
-J.J1
Xl02 EVON spftctra "" 0.30. Q) "tJ -0.03.
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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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EIGEN-VALUES No. •
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
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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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FIG. (h.4) THE EFFECT OF OBSERVATION LENGTH ON EIGEN-VALUES EIGEN-VALUES No. 11,12,
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FIG. ( 6.~) THE EFFECT OF OBSERVATION LENGTH ON EIGEN-VALUES
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1, 2, 3,
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-. -'- -,-, .. -.-. ~
........ '.000
~'nm • '.00
FWIVI. • 0."""
h.n .• "-_' 0.00
.... LI"" 10.0400
IlfU.AIH ee..r. "'''1.
9 18 27 36 15 Sf 63 72 81
Signal / Nois~ Ratio (dB)
4, 5, 6, 7, 8, 9,10
FIG. ( 0.7) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES
() I ~
ru
X102 "'" O. 3'!J EVON SPPC t.rd
j
~ -0.03
" -0."2 Cl \rt -0.80 Q. ,
-, .19 Cl ~ -1.~ .... ..., ~ ~ ~ ,
-I .~
-2.33
-2.71
-3.09
E.t. Frl,,. ".~ 5MI • -ftl.ootJ f.'
)(10 1 "'" 0 .58 EVCJf1 s~tra
i
~ -0.06
" -0.70 Cl \Il _ I .33 Q.. ,
-1.97 t:l 1O -2.SI P-., .... ..., ~ ~ ~ ,
-3.~
-3.99
-4.~3
-'.16
E.t. Frl'" II.""
".. -'.tItItI f.' o C -4
~ -4
> • > r-
I
~ o -4
~ I
-3.471 "" il" 0, non .. nn L,~ ° I e lil""i "0,"" "" I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 1.00 1.50 5.00
XIO-I -5.~t.OO"O'.SO "':OO~:50~I~OO 2'~50"j·~oo"jl.50"1'.00"1'~;o 5'.00 I ~
X10- 1
FRACT. OF SAffPllNG FRED.
x,o? Nodirj~ PeN sppctro "'" O.'f~i __________________________________________________ ~ ________ --'
~ -0.0'
" -0. '1.
~ Q. -0.33 , -0.f9.
~ -0.65 .... ..., ~ ~ ct
-0.8'
-0.91.
-,. '3
'.t. Fri'" tI.~ 5Mf • -ftl.ootJ f.'
,
~:::L.. 0" 0 Lo~ .. 0 0 0 no 0 ... 1
FRACT. OF SAffPllNG FREO.
"'" XJ.o;3. ffodiFi~ PeN SPf!'Ctro i
Q) 't) -0.05.
" -0.63.
~ Q.. -, .21 , Cl
-'.19. ~ -2.37. .... ..., ~ ~ ~ ,
-2.~
-3.53.
-4.10
-f.68.
'.t. Fri'" II.'"
... -'.tItItI f.,
-5.251. he "e • s euucsee ,,~ e e, eo ~ e, sees US u" .~ " .. I 0.00 O.~ '.00 , .50 2.00 2.~ 3.00 3.~ 4.00 f.~ '.00
X'O-I 0.00 o.~ 1.00 I.~ 2.00 2.~ 3.00 3.~ 4.00 f.~ !f.00
X,O-I FflACT. OF SAffPL ING FRED. FRACT. OF SAffPllNG FREO.
FIG. ( 6.8 ) POWER SPECTRAL DENSITY ESTIMATE * For diFFerenl PSDE melhods *
f\J , ~ ...,
~ Z
W I
:J > ~ I
~
~ UI .. .. N w
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
~ ~ 0::
~I ' I
\
~
:1 ~
·1 ""C
~I Q h. :::, C)
X'O' 2.50
2.~
2.00
,.~
r.~
,.~
~ 1.00 --oJ
~ , O.~ ~ l!) ...... O.!JO llJ
O.~
0.00 0 9
... ..., ........ --... ... -. ~
~.Rt. '.000
~ITtm 1.00'.00
~. • O.I~ 0.1700
1"1\. ....... t 0.00 0.00
wn. .U"I' '0.0400
1ltU." ~o C.,.,.... ...lr I_
X'O' 2.50
2.~
2.00
,.~
, .50
I.~
llJ :::, '.00 --oJ
~ I O.~
~ l!) .... O.!JO llJ
O.~
.. ........... ,-.--., .. ..... • 2'S
....... • 1.000
Nft.lfdII • '.00 '.00
t=IIOI. , 0.'''' 0.1700
I""' ...... I 0.00 0.00
.... .llnlf 10.0400
IUC' c:e..r.~ _tr I.
18 ;;7 36 15 5f 63 7;; 81 D.001 ;:;;, , , , , , , , , ,
90 0 9 18 27 36 f' ,f 63 72 8' 90
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)
~I
~
J ~I ::) ~
() , I I , ~ C\f UJ ......
C) -.J Q \ I
3 ~I ......
51
!J.oo
4.~
4.00
3050n,, ) If (S,;nJ e~~e~ ~{) Itt • MDI-" .1...,..,ld
MWr • 25 3.00
~.At. ".000 2.~ NIIt. , 1UJI • , .00 , .00
lIJ ~ 2.00 RImS. • O.I!GO 0.1700
..... ~ Iftl\....... O.GO O.GO , ,.~ ~ ..... lI"" '0.0400
l!)
\"- (rto~se 9~lA \f~~) ~ , .001 ).'0
1lru.,,~0 Covr ........ 1.
o.~) , r 711r'eshofd stJR. 6 ci-B "'-"-
o.oo! I . . , . i • i i i . , 0 9 '8 27 36 fS Sf 63 72 81 90
Signal / Nois~ Ratio ( dB )
FIG. (6.10) THE EFFECT OF Signal/Noise Ratio ON EIGEN-VALUES EIGEN-VALUES No. : 10,11,
1
""I •• ~
:':1 ~ -I
I C) Ol ~I
~ I I ~
·0
~I ~
(J I I I ,
.... ~ (J)
~I
~ ~ . -.::
~I "-
51
~~~-~~--- - . - - - ---~~----
4.00
3.60
3.20
2.90 " • MIIt.ev •• ...,.,Id
MM. • 2!
2.40 INtt.FIt. ., .000
2.00 NWtllrtm • '.00 1.00
~ '.60 RIEGI. 1 o.la 0.1100
-J
~ .ftn....... 0.00 0.00
I
~ If .... lInn 10.0400
(!) IltU.A.., eo..r. ,.." •• ..... llJ
0.00 0 9 18 27 36 15 51 63 72 81 90
Signal / Noist:' Ratio ( dB )
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
~I c-.... -I
• C) 0) , I
~ ~
~ .. ~ Q::
0 , I I , ~
~ '-l
:1 ~
·1 ~
......
X'O' 2.!lD
2.25
2.00
,.~
, .25
~ '.00 -.I
~ I O.~
~ ~ "-, O.!lO lJ.J
o.~
0.00. 0 9
.. · ... ··v -.~ ... ~. 2'5
~.Rt. '.000
NWIllftm • '.00 '.00
~. • o. ,. 0.2000
I"", ........ 0.01 0.00
IIAO... .U"" '0.0400
II ..... "~O co...-. "-va.
18 21 36 .5 5. 63 72 81 90
Signal / Nojs~ Ratio (dB)
X'O' 2.SO
2.~
2.00
,. r.J
, .50
,.~
~ '.00 -.I
~ I 0.'" ~ ..... O.!lO lJ.J
O.~
EIGEN-VALUES No.
EIGEN-VALUES No. •
1, 2, 3,
11,12,
-. -'- .,-, .. .... • 2S
...... ". • '.000
.... lnIJI • 1.01 '.01
,..,.. • 0.'" 0.2000
I .. n....... 0.00 O.OD
lItGl.ll"" 10.0400
f1UIC' c:...1~ "-lrl_
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
..
~ ~
~I I , ~
:1 ~ . ~
...... :::l Q k
::::> t)
~ ....J
s: I
~ l!) ..... ltJ
fO
3S
37
28
;f
20
IE;
,;
B
f
oJ;w...;: =;;1] :;;;;; 2;;: IX Ii I :'1 1 I:: ii
0.00 0.35 O. 70 '. OS '. 'f0 '. 75 :I.' 0 :I. 'fS :1.80 3.' 5 3.50 XIO-I
" I ..... ev .1.....,ld
.... • J!
....... • '.000 MIll I fUJI • • .00 , .00
RIfOI. • O.ID
",n........ 0.00 0.00 ... ••• 000 10.000
., ....... I O.02lalll'IIO'''''
1InUl.1I"" 10.0400
IlfU.A1!O C ...... "-\PI.
FrEJQUEJncy SEJporotion ( Froct. oF Sompl.Fr )
EIGEN-VALUES No. •
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
... ~ •• (l) ..... •• ~ -C) 0). I
)..
~ I ~
-. ~ Q:
(J , , I ru 0
-.J ~ . ~ . ~
~ -.J
~ I
~ ..... llJ
8.00
7.20
6.fO
5.60
f.SO
f.OO
3.20 -2.fO
1.60
O.SOlr-------______ _
0.001 , • , , , , , • , • o • 00 0.20 O. fO 0.60 0.80 1.00 1.20 '. fO 1.60 1.80 2.00
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 )
FIG. (6.14) THE EFFECT OF FREQUENCY SEPARAT. ON EIGEN-VALUES
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 )
FIG. (6.15) THE EFFECT OF FREQUENCY SEPARAT. ON EIGEN-VALUES
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_ •
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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