CHAPTER – 4 PROPOSED ALGORITHMS - …shodhganga.inflibnet.ac.in/bitstream/10603/8536/11/11_chapter 4.pdf · CHAPTER – 4 PROPOSED ALGORITHMS ... Essentially, the Modified Periodogram

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CHAPTER – 4

PROPOSED ALGORITHMS

The work in this chapter is an attempt to propose the techniques

on Non parametric spectrum estimation problems and is reported in

the following sections.

4.1 Need for new algorithms:

Though the non parametric spectral estimation has good dynamic

performance, it has few a drawbacks such as spectral leakage effects

due to windowing, requires long data sequences to obtain the

necessary frequency resolution, assumption of auto correlation

estimate for the lags greater than the length of the sequences to be

zero which limits the quality of the power spectrum and the

assumption of available data are periodic with period N which may not

be realistic. Hence alternatives must be explored to reduce the

spectral leakage effects, to decease the uncertainty in the low

frequency regions, to improve the frequency resolution, to reduce

variance with the increased percentage of overlapping data samples a

consistent spectral estimate with minimum amount of bias and

variance.

The study of spectral leakage effects methods have been discussed

by many authors. In this work, a non parametric power spectrum

estimation method for nonuniform data based on interpolation and

cubicspline interpolation methods is proposed. The simulation results

104

show the reduction in spectral leakage, improved spectral estimation

accuracy and shifting of frequency peaks towards the low frequency

region. The simulation results present a good argument with the

published work.

To reduce the spectral leakage effects and to resolve the spectral

peaks at higher frequencies of non uniform data sequences, a

nonparametric power spectrum estimation method using prewhitening

and post coloring technique is proposed. The combination of

nonparametric with parametric method as preprocessor is proposed in

large active range situations. The simulation results present a good

argument with the published work.

To reduce the variance of a spectral estimate, a non parametric

spectral estimation method based on circular overlapping of samples

is proposed. The existing Welch nonparametric power spectrum

estimation method has increased variance with the increased

percentage of overlapping of samples. Welch estimate uses the linear

overlapping of the samples. Hence the Welch estimate is not a

consistent spectral estimate. To overcome this, circular overlapping of

samples is proposed. The variance of the proposed estimate decreases

with increased percentage of circular overlapping of samples, the

spectral variance is found to be nonmonotonically decreasing

function. The simulation results show the robustness of proposed

estimate with the existing Welch estimate in the published work.

105

The following algorithms are considered to increase the large active

range in spectral estimation, to increase the frequency resolution and

to decrease the variance and these are as follows.

Power Spectrum Estimation of nonuniform data sequences

using nonlinear overlapping of samples.

Power spectrum estimation of nonuniform data sequences in

wide dynamic range using prewhitening and postcoloring

technique.

Power spectrum estimation of nonuniform data sequences using

resampling methods like spline, cubicspline interpolation

techniques and averaged weighted least squares algorithm.

4.2 Power Spectrum Estimation of nonuniform data sequences

using nonlinear overlapping of interpolated samples.

4.2.1 Introduction: The Power spectrum estimation of randomly

spaced samples using nonparametric methods is well known and is

reconsidered in this algorithm. Commonly used nonparametric

methods are Periodogram, Modified Periodogram and Welch methods.

The Periodogram and Modified Periodograms are asymptotically

inconsistent spectral estimators for non uniform samples.

Interpolation techniques (58) like cubicspline and linear interpolations

are employed to convert the nonuniform samples to uniformly

distributed samples and then the estimate is evaluated using the

nonlinear overlapping of samples. Welch method is a consistent

estimate for the random samples and the method is reconsidered for

106

achieving very low variances using different percentage of nonlinear

overlapping of the samples than the existing Bartlett’s method of

estimation. Even though the nonlinear overlapping of samples

produces an amount of discontinuity on the random process; it is

observed that for a Gaussian distributed ergodic wide sense stationary

(WSS) random process the spectrum estimate of power is

asymptotically unbiased. The variance of the proposed estimate

decreases due to nonlinear overlapping of the samples.

4.2.2 Spectral Analysis with Nonlinear Overlaps:

The variance of power spectrum estimation is decreased by

considering the average of the spectrum estimate. In Bartlett’s Method

the entire data length N is divided into K number of segments, with

each segment having the length L=N/K. The Periodograms are

evaluated for each segment and then taking the average of K

periodograms. The average of the resulting estimated power

spectra is taken as the estimated power spectrum. Here the

variance is reduced by a factor K, but a price in spectral resolution is

paid. The Welch Method, [15], eliminates the trade off between

spectral resolution and variance in the Bartlett Method by allowing

the segments to overlap. Furthermore, the truncation window can

also vary. Essentially, the Modified Periodogram Method is applied to

each of the overlapping segments and averaged out. However, the

Welch estimator only uses regular overlap at the signal x (n) with

n=0,1,2, ,…, N-1 is a wide sense stationary Gaussian process. We

divide the signal x(n) in K independent segments such that every

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segment has length L = N/K. Further, we extract different nonlinear

overlapping sub-records. The i-th overlapping sub-record of the signal

x (n) satisfies,

N)1i)(r1(Lnx)n(x i (4.1)

With 1r0 the fraction of overlap, n=0,…,L-1 and where

N)1i)(rI(Ln denotes )1i)(r1(Ln( ) modulo N imposing

circular overlap. The important property of nonlinear overlap is that

every time sample is an equal number of times member of a sub

record, further the different sub records need to overlap an integer

number of time samples. One can show that the two properties are

respectively satisfied if

m11

Lv1r (4.2)

Where m and v are the integer number of time samples also m N and

v N.

The discrete Fourier transform of the i-th sub-record )(nx ias

)1i)(r1(k2j1L

0n

L/kn2jii e.e)n(x)k(X

(4.3)

We made that all initial phases of the sub records same. In the

beginning of the section, we assumed the signal x(n) to be ergodic and

weakly stationary. The Wide sense stationary random process is

generated by passing a White Gaussian process over a low pass filter

whose impulse response is h(n) which is absolutely square summable.

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The block diagram for the proposed spectrum estimate is as shown in

the Figure 4.1.

Figure 4.1.Block diagram for proposed spectral estimate

The periodogram for each segment )(nx i for a time window w (n) with

n=0 , … ,L-1, is given by

221

01

0

2

^)()(

)(

1)( LknL

n

iL

n

i

xx ennxn

kP

(4.3)

Averaging over the different overlapping segments in equation (4.3),

the spectrum estimation with nonlinear overlapping of samples is

given by

r1/k

1i

i

xx^

xx^

)k(Pk

r1)k(P (4.4)

The method of nonlinear overlapping is achieved by overlapping the

last segment of the data with the first segment of the data. A small

amount of discontinuity exits over the statistical properties of the PSE

with nonlinear overlap. The following result can be shown as

kPkHkP ee

2xx for L (4.5)

where )k(H denotes the normalized DFT of the filter h(n) as explained

and the convergence in equation (4.5) is in Mean Square sense,[4.1].

NonuniformGaussiansequence e(n)

Cubicspline

Interpolation

LowpassFilterh(n)

DataFrami

ng

Nonlinearoverlapping

Spectrumestimation

Px (ω)

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Convergence in Mean Square implies that the first and second

moment converges as well. Therefore,

kPEkHkPE ee

2xx for L (4.6)

Where E[.] denotes the Expected value . From the statistical

computations it is shown that 1)]k(P[E ee^

, e(n) is zero mean and unit

variance white noise and )k(Pee^

power spectrum of white

noise.Therefore,

forLkHkPE 2

xx (4.7)

proving that the PSE with nonlinear overlap is asymptotically

unbiased. This is the same result as for the PSE with Welch’s

method, using the setup from Figure 4.1.

4.2.3 The effect of nonlinear overlap on proposed Estimate:

The effect of nonlinear overlapping of samples on the proposed

estimate is as follows

(i) The variance is non-increasing as a function of the fraction of

overlap.(ii) The variance converges to a non-zero lower bound if the

fraction of overlap tends to 1.

The expression for the minimum variance with the overlapping of

samples is observed with respect to all fractions of overlap.

21

1

0

22

222

2^)(

)(21))((

rk

s

xx

n

xx

kksWk

ksWX

kPnk

rkPVar

(4.8)

110

Where

ksW denotes the normalized DFT of w(n) at

kLs2

.In the

classical case, the variance for Bartlett's PSE equals expression (4.8)

for a fraction of overlap 0r .Therefore, it is clear that by using

nonlinear overlaps the variance is reduced to a minimum quantity.

4.2.4 Flow chart for the proposed algorithm:

Figure 4.2: Flow chart of PSE with nonlinear overlapping method.

Design a Chebyshev Type-I/Butterworth filter h (n)

Process White noise e(n) over the filter to get random data x(n)

Segment the random data into number of segments (k) using Hanningwindow

Circular overlap the segments for the desired % of overlapping(r)

Evaluate the PSE of the individual segments and add all segments.

Is Variance < Truevariance

No

Get the consistent estimate with circular overlaps

Yes

Run for number of iterations i =10000

Start

Stop

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4.2.5 Simulation analysis for White noise:

Figure 4.3.1

Figure 4.3.2

Figure 4.3.3

Figure 4.3.1 Nonuniform Gaussian data sequence, Figure4.3.2 Uniform

Gaussian data sequence, Figure 4.3.3 response of Gaussian noise of zero mean

and unit variance.

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Figure 4.4 Average realizations 0f 100 simulations of PSE with circular

overlap with non overlapping and 3 segments (chebyshev type-I filter)

From figure 4.4 the power spectrum estimation of response of

chebyshev type-I filter for nonuniform WSS data sequence

which is divided into 3 segments and using no circular

overlapping of samples reveals that dynamic range of spectral

peak is nearly at 0 db level with variance of 0.332

Figure 4.5 Average realizations 0f 100 simulations Welch estimate

with non overlapping and 3 segments (chebyshev type-I filter) .

From figure 4.5 the spectral analysis of nonuniform WSS data

sequence using the classical Welch estimate which uses 50%

linear overlapping of samples observes that the dynamic range

is not nearly at 0db level and the variance of 0.843

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Figure 4.6 True spectrum estimation of W.S.S Gaussian Ergodic

process.

Figure4.7 Comparison of spectral estimates for 0% overlapping samples.

Figure 4.6 depicts that true power spectral density (PSD) of

nonuniform WSS data sequence with variance of 0.301 and Figure 4.7

shows that the comparison of power spectrum estimate of nonuniform

WSS data sequences using circular overlapping of samples provides

less variance of 0.332 compare to Welch estimate and true estimate,

which provides variances of 0.843 and 0.301 respectively and also the

range of circular estimate is nearly at 0 dB level.

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Figure 4.8 Average realizations 0f 100 simulations of PSE

Figure 4.8 reveals the power spectrum estimation of response

of chebyshev type-I filter for nonuniform WSS data sequence

which is divided into 3 segments and using 50% circular

overlapping of samples reveals that dynamic range of spectral

peak is nearly at 0 db level with variance of 0.325


with 50% overlapping and 3 segments (chebyshev type-I filter) .

Figure 4.9 the spectral analysis of nonuniform WSS data



is not nearly at 0db level and the variance of 0.843

115


process.

Figure 4.10 shows that true power spectral density (PSD) of

nonuniform WSS data sequence with a dynamic range nearly at

0 dB and variance of 0.301

Figure 4.11 Comparison of spectral estimates for 50% overlapping

Figure 4.11 shows that the comparison of power spectrum estimate of

nonuniform WSS data sequences using circular overlapping of

samples provides less variance of 0.325 compare to Welch estimate

and true estimate, which provides variances of 0.843 and 0.301

respectively and also the range of circular estimate is at 0 dB level.

116


Figure 4.12 shows that the power spectrum estimation of

response of chebyshev type-I filter for nonuniform WSS data

sequence which is divided into 3 segments and using 80%

circular overlapping of samples reveals that dynamic range of

spectral peak is nearly at10 dB level with variance of 0.318



Figure 4.9 shows the spectral analysis of nonuniform WSS data



is not nearly at 0db level and with the variance of 0.859

117

Figure 4.14 The true spectrum estimation of W.S.S Gaussian Ergodic

process.

Figure 4.14 it is observed that true power spectral density (PSD)

of nonuniform WSS data sequence having a dynamic range

nearly at 0 dB and a variance of 0.301


Figure 4.15 gives the comparison of power spectrum estimate of

nonuniform WSS data sequences using 80% circular overlapping of

samples provides variance of 0.318 compare to Welch estimate and

true estimate, which provides variances of 0.859 and 0.301

respectively.

118


Figure 4.16 shows that the power spectrum estimation of response of

chebyshev type-I filter for nonuniform WSS data sequence which is

divided into 3 segments and using 100% circular overlapping of

samples reveals that dynamic range of spectral peak is nearly at 0 dB

level with variance of 0.305



Figure 4.17 shows the spectral analysis of nonuniform WSS data

sequence using the classical Welch estimate which uses 100% linear

overlapping of samples observes that the dynamic range is not nearly

at 0db level and with the variance of 0.842

119


process.

Figure 4.18 it is observed that true power spectral density (PSD)

of nonuniform WSS data sequence having a dynamic range

nearly at 0 dB and a variance of 0.301



nonuniform WSS data sequences using 100% circular overlapping of

samples provides variance of 0.305 compare to Welch estimate and

true estimate, which provides variances of 0.842 and 0.301

120

respectively.

Figure 4.20 Average realizations of 100 simulations of PSE


nonuniform WSS data sequences of 4 equal segments and using 60%

circular overlapping of samples provides variance of 0.192 compare to

Welch estimate, which provides variances of 0.756 and 0.301

respectively.



nonuniform WSS data sequences of 4 equal segments and using

70.5% circular overlapping of samples provides variance of 0.186

compare to Welch estimate, which provides variances of 0.716 and

0.301 respectively.

121



nonuniform WSS data sequences of 4 equal segments and using

76.6% circular overlapping of samples provides variance of 0.179

compare to Welch estimate, which provides variances of 0.741 and

0.301 respectively.


overlap with non overlapping and 3 segments (Butterworth filter).

Figure 4.23 the power spectrum estimation of response of Butterworth

filter for nonuniform WSS data sequence which is divided into 3

segments and using no circular overlapping of samples reveals that

minimum uncertainty is observed.

122


with 0% overlapping and 3 segments (Butterworth filter) .



segments and using no linear overlapping of samples reveals that

maximum uncertainty is observed in the above figure 4.24.

Figure 4.25 the true spectrum estimation of W.S.S Gaussian Ergodic

process.

Figure 4.25 provides the power spectrum estimation of response of

Butterworth filter for nonuniform WSS data sequence with the desired

amount of uncertainty is observed.

123


of samples.

Figure 4.26 reveals that PSE with circular overlapping reduces the

maximum uncertainty up to 20% then the PSE (Welch estimate) with

linear overlapping of samples.


overlap with non overlapping and 3 segments (Butterworth filter).



segments and using 50% circular overlapping of samples reveals that


124





segments and using 50% linear overlapping of samples reveals that



process.




125

Figure 4.30 Comparison of spectral estimates with 50% overlapping

of samples.





overlap with 80% overlapping and 3 segments (Butterworth filter).





126








process.




127


of samples.





overlap with 100% overlapping and 3 segments (Butterworth filter).





128








process.




129


of samples.



linear overlapping of samples also the dynamic range using circular

overlapping of the samples observed near the 0 db level.

The Table 4.1 provides the variances for the Power spectrum

estimator using the circular overlapping of the samples with different

segments and different overlapping percentages.

Table 4.1 variance values for different percentages of overlapping

Variance(PXX(ω))/PXX2(ω) r=0 r=0.6 r=0.8 r=1

K=6 0.16 0.11 0.09 0.08

K=5 0.2 0.14 0.12 0.11

K=4 0.25 0.19 0.17 0.15

K=3 0.33 0.23 0.21 0.19

K=2 0.5 0.42 0.35 0.31

130

Figure 4.38: The variance versus the % of overlapping of PSE withnonlinear overlap.

Figure 4.38 reveals that in circular overlapping method the samples

exist for an equal number of times the sub records of data sequence,

by averaging these overlapping of samples of each sub record the

variance of the data sequence is further reduced.

4.26 Results and Discussions:

As the percentage overlapping of the samples increases the

variance is reduced as the case of chebyshev filter from Figure 4.4 to

Figure 4.22 to a minimum quantity and also for the case of

Butterworth filter as shown from Figure 4.23 to Figure 4.38, the bias

also approaches to true value hence the power spectrum estimation

with nonlinear overlapping of samples is said to be a consistent

estimate. Since in the circular overlapping method the samples are

existing for an equal number of times the sub records of data

sequence, by averaging these overlapping of samples of each sub

record the variance of the data sequence is further reduced. The

131

computation time is also reduced by 20% by allowing the samples to

have circular overlapping. Hence the power spectrum estimation with

circular overlapping of the samples is proposed over the linear

overlapping of samples to reduce the variance to smaller nonzero

quantity. The variances for different percentage of overlapping of

samples are tabulated in the Table 4.1 and from Figure 4.38 it can be

shown that the variance is reduced to lowest value for the increases of

percentage overlapping of samples. Hence we can conclude that the

power spectrum estimation with nonlinear overlapping of the samples

is an asymptotically consistent estimate for the nonuniform data

samples.

4.3 Power Spectrum estimation of nonuniform data in wide

dynamic range:

4.3.1 Introduction: Power spectral density (PSD) estimation

techniques are widely used in many applications, such as sonar,

radar, geophysics and biomedicine. Similar to single channel power

spectral density (PSD) estimation, there are basically two broad

categories of MPSD estimators. One is the nonparametric approach,

among which the Fourier-based estimators are the most popular. The

other is the parametric method, which assumes a model for the data.

Spectral estimation then becomes a problem of estimating the

parameters in the assumed model. The most commonly used model is

the autoregressive (AR) model because accurate estimates of the AR

parameters can be found by solving a set of linear equations .Similar

to the single channel case for short data records the Fourier-

132

based methods can suffer from significant bias problems while

AR model-based methods can suffer from inaccuracies in the

model as well as from imprecise model order selection.

Furthermore, some effective AR model-based approaches cannot be

easily extended to the multichannel case. In addition, as pointed out

by Jenkins and Watts, spurious cross-correlation or spurious

cross-spectral content may arise unless a prewhitening filter is

applied before PSD estimation. One such prewhitening filter was

suggested by Thomson for single channel PSD estimation.

The filter system function is given by kp1k z)k(a1)z(A and the

filter parameters a [1], a [2], a [3]…a[p]can be estimated from the data

using any AR-model based method. Denoting the output of this FIR

filter by u[n], a Fourier-based estimator is then used to generate the

PSD estimate )( fPu

. Finally, the PSD estimate of the original data is

found as

2

p1k

ux

fk2jexpka1

)f(P)f(P

(4.8)

where pa,...,2a,1a

are the estimated AR filter parameters. We term

this as the AR Prewhitened (ARPW) spectral estimator. Because of the

inconsistency of the definitions in the literature concerning MPSD

estimation, the following definitions will be made. A complex

multichannel sequence x[n] is defined as the complex 1L vector

x TL21 nX,...,nX,nXn where nX i represents the data

133

observed at the output of the ith channel and L is the number of

channels. For a wide sense stationary (WSS) multichannel random

process, the autocorrelation function (ACF) at lag k is defined as the

LL matrix function

nXKnXEkR HX

`

[k]LLr.[k]2Lr[k]1Lr::::[k]L2r.[k]22r[k]21r

[k]L1r...[k]12r[k]11r

(4.9)

where E[·] is the mathematical expectation, the superscript H denotes

conjugate transpose and krij is the cross-correlation function (CCF)

between nX i and nXj at lag k

nXknXEkr jiij (4.10)

For multichannel data of N samples, the sample vector, which is

1NL , is defined as

TTTTT 1NX,...,1X,...,1X,0XX (4.11)

The NLNL multichannel autocorrelation matrix of order N is

defined as

HXXERX (4.12)

.

0 0 1 ... 11 0 ... 2

.1 2 ... 0

Rx Rx Rx NRx Rx Rx N

Rx N Rx N Rx

it is seen that ,kRkR xxH so XR is hermitian. Because the

multichannel process is assumed to be wide sense stationary, XR is

134

also block Toeplitz. The power spectral density matrix or cross-

spectral matrix is defined as

)f(P...)f(P)f(P...

)f(P...)f(P)f(P)f(P...)f(P)f(P

)f(P

LL2L1L

...L22221

L11211

X

The diagonal elements Pii(f) are the PSDs of the individual channels or

auto-PSDs, while the off-diagonal elements Pil(f) for 1i are the

cross-PSDs between nX i and nX1 , which are defined as

fk2jexpkr)f(Pil il

(4.13)

The magnitude squared coherence (MSC) between channels i and j is

a quantity that indicates whether the spectral amplitude of the

process at a particular frequency in channel i is associated with large

or small spectral amplitude at the same frequency in channel j. It is

defined)t(jjP)f(iiP

)f(ijP)f(ij

22 (4.14)

A classic Fourier-based spectral estimator is the periodogram, which

is given as the LL matrix

)f(X)f(XN1)f(P H

PER (4.15)

where the Fourier transform is the L × 1 vector

).fn2jexp(nx)f(X1N

0n

(4.16)

The p th order AR model is defined as

135

nuinXiAnXp

1i

where pA,...,2A,1A are LL

AR coefficient matrices and nu is the excitation white noise or

kkRu and is the LL excitation noise covariance matrix

with being the discrete delta function.

The ARPW estimator given in (1) is readily extended to the

multichannel case. With the notations defined above, the

multichannel version of (1) is

)f(A)f(P)f(A)f(PH

u1

X

(4.17)

In addition to reducing spurious cross-spectral content, this

Prewhitened spectral estimator also gives an auto PSD spectral

estimate with much less bias than a Fourier based spectral estimator.

This is because the prewhitener reduces the dynamic range of the

PSD. However, this estimator is still inferior to the method proposed in

this paper. Instead of the FIR prewhitening filter, the proposed

estimator uses a prewhitening matrix, which is essentially a time

varying filter that is less susceptible to end effects. The new estimator

for a single channel PSD has been proposed in [4], while in this paper

it is extended to MPSD estimation. Assume the signal

1N,...,1,0n,nwnf2jAceXpnX O (4.18)

where nX is an 1L vector, Ac is an 1L complex amplitude to be

estimated, f0 is a known frequency, and

TTTT 1Nw,...,1w,0ww is a 1NL complex Gaussian noise

136

Whitenoise

e (n)x1(n)

vector with zero mean and known NLNL covariance matrix Rw .

The ML estimate of Ac is XREERoEA 1w

Ho

1

o1

wH

C

where X is the data sample vector and

LNL1Nf2jexpI,...,f2jexpI,IE T0L0LL0 with IL being an

LL identity matrix. The LL covariance matrix of this estimator is

1O

1w

HOCA EREC

(4.19)

Therefore, for a general WSS multichannel random process nX the

MVSE is defined as 1X1H )f(ER)f(Ep)f(PMV

where XR

is the estimated pLpL ACM of X and

.LpL1pf2jexpI,...,f2jexpI,I)f(E TLLL

4.3.2 Proposed System and Algorithm:

The proposed system and algorithm for the power spectrum estimator

is as shown below

)( fPx

Figure 4.39 Proposed system for ARPW spectral estimator

The nonuniform data sequence is generated using the Poisson

distributed sampling instants and also random distributed

sampling instants.

(15)

Estimated model

order ^p (EEF)criteria

Prewhitenmatrix Filter

NonParametricPSD

2)(1zH

AR(P)

)(1

)(zA

zH

137

The data sequence is generated using the given AR(P) process as

the white Gaussian noise with zero mean and unit variance as

the input for the system.

Choose a model order which fits for the AR(P) process using the

exponential embedded family criterion (EEF) for the given length

of the sequence.

Estimate the AR(P) coefficients using the Yule-Walker method

and so the AR model parameters for all the lower order models

are available.

Get residues at the output of a prediction error filter, the power

spectrum of the residues is evaluated using a nonparametric

technique as

The prewhitening technique increases the spectral flatness and

decreases the dynamic range of the residues.

To obtain the power spectrum estimation of data sequence in

wide dynamic range postcoloring technique is applied on the

spectrum of the residues

Segment the data into K equal length blocks, with LK .

Construct the LNLN matrix and 1XR using the estimated

parameters then average all the blocks to get the final estimate.

4.3.3 Simulation Results and Analysis:

To evaluate the effectiveness of the above algorithm, consider an

AR (4) process which has a wide dynamic range and a nonparametric

Welch estimate that suffers from leakage problem. The simulation

results are as shown in the following figures. Using the system

P

1k

*k )kn(xa)n(x)n(e 1NnP

)(eP)z(H)(P 2

X

138

function of the model

4321 z8978.0z7865.2z1012.4z6707.21

1)z(H

and white

Gaussian noise of zero mean and unit variance , input sequence is

generated for different lengths of 64,128,256and 512.The signal

samples are interpolated and then Prewhitened using the system

model to increase the spectral flatness to avoid leakage problem.

Figure 4.40 True estimate of AR(4) process for N=64.

Figure 4.40 provides the true power spectral density for the AR(4) data

sequence of length N=64 (nonuniform data sequence), the two spectral

peaks are clearly resolved and has the wide dynamic range power

levels.

139

Figure 4.41 Welch estimate, the two peaks are not resolve for N=64.

Figure 4.41 provides the Welch power spectral density for the AR(4)

data sequence of length N=64 (nonuniform data sequence), the two

spectral peaks are not clearly resolved and suffers from the spectral

leakage effects also it has no wide dynamic range of power levels.

Figure 4.42 Prewhitening estimate, two peaks are resolved for N=64.

Figure 4.42 provides the Prewhiten/postcolor estimate for the AR(4)


spectral peaks are clearly resolved and has wide dynamic range of

power levels.

140

Figure 4.43 Comparison of three estimates for N=64.




power levels that is almost equivalent to the True spectrum whereas in

Welch estimate the two spectral peaks are not clearly resolved.



sequence of length N=128 (nonuniform data sequence), the two

spectral peaks are clearly resolved and has the large dynamic range

power levels.

141

Figure 4.45 Welch estimate, the two peaks are not resolved for N=128.



spectral peaks are not clearly resolved and suffers from the spectral

leakage effects also it has no wide dynamic range of power levels.





power levels.

142










spectral peaks are clearly resolved and has the large dynamic range

power levels.

143




spectral peaks are just begin to resolve and suffers from the spectral

leakage effects also it has narrow dynamic range of power levels.





power levels.

144










spectral peaks are clearly resolved and has the large dynamic range of

power levels.

145




spectral peaks are resolved but suffers from the spectral leakage

effects also it has narrow dynamic range of power levels.





power levels.

146





power levels that is equivalent to the true spectrum whereas in Welch

estimate the two spectral peaks are resolved with small range of power

levels.

4.34 Conclusions: As we know, the nonparametric methods do not

resolve the two peaks in true spectrum and suffers from leakage at

high frequencies. Hence the combination of nonparametric with

parametric resolves two peaks with ease also follows the true

spectrum. By observing from Figure 4.40 to Figure 4.55, the two

peaks are not resolved in the Welch estimate for lesser amount of data

samples(N=64) whereas in the proposed estimate the two peaks are

resolved. If we increase the number of samples from N=64 to N=256,

the existing Welch estimate does not resolve the two spectral peaks,

on the other hand using the prewhitening and post coloring method

the two spectral peaks are greatly resolved. Therefore the use of

147

parametric method as a preprocessor is highly recommended in the

wide dynamic range of spectral estimations.

4.4 Power spectrum estimation using interpolation techniques:

4.4.1 Introduction: The power spectral analysis of uniformly sampled

data is a well known topic. We are having enough number of regularly

sampled methods for appropriate applications. The survey suggests

that the analysis of nonuniformly sampled data sequences is also to

be considered. The purpose for the analysis of nonuniform data

sequence includes in the applications such as oceanic data,

biomedicine, seismology, astronomy, and particularly in economics

data is nonuniform rather than uniformly sampled data. Hence the

power spectral analysis of nonuniform data sequence is some what

under developed data.

The power spectrum estimation of uneven and nonuniformly

sampled random sequences can be carried out by least squares

periodogram (LSP), and coherent sampling methods. The proposed

algorithm for the estimation of nonuniform random sequences uses

the interpolation methods as resampling methods thus eliminates the

low pass filtering effects and also Lomb transforms as well as weighted

least squares methods are suggested.

The computational methods used in digital computer for the

evaluation of the library functions, such as sin(x), cos (x), requires

polynomial approximations using Taylor series. The data required to

construct a Taylor polynomial is the value of the function f(x) and its

148

derivative. The main disadvantage of this method is to know the

higher order derivatives which are hard to compute.

In statistical signal analysis and scientific analysis arise the

situations where the function y=f(x) is available only for N+1 tabulated

data values, and a technique is needed to approximate the function

f(x) at nontabulated abscissas. If there are a significant amount of

errors in the tabulated values the curve fitting techniques are used.

On the other hand if the points are known to have a high degree of

accuracy, then a polynomial curve p(x) that passes through them is

considered. When the polynomial approximation is considered within

an interval, the approximation p(x) is called an interpolated value. If

the approximation is considered outside the interval, then p(x) is

called an extrapolated value. Polynomials are used to design

algorithms to approximate functions, for numerical differentiation, for

numerical integration and for making computer drawn curves that

must pass through specified points.

Polynomial interpolation for a set N+1 points Nkkkx yx 0, is

generally not quite approximated. A polynomial of degree N can have

N-1 number of maxima and minima, and the graph can wiggle in

order to pass through the points. Another method is to piece together

the graphs of lower degree polynomials )(xS k and interpolate between

the successive nodes ),(,, 11 kkkx yxyx . The set of the function forms a

piecewise polynomial curve. Interpolation is a technique of making a

perfect approximation of given function within the interval of values.

149

Different interpolation techniques involve linear interpolation and

cubic spline interpolation. The linear interpolation involves the linear

relationship of the interval values whereas the cubic interpolation

involves the nonlinear relationship of the interval values. A different

cubic spline technique involves clamped spline, natural spline,

extrapolated spline, parabolically terminated spline and endpoint

curvature adjusted spline. A practical feature of splines is the

minimum of oscillatory behavior that they posses.

Unevenly or nonuniformly sampled data sequences are not

sampled with the Nyquist rates. Hence these data sequences require

resampling methods. Linear interpolation and cubicspline

interpolation techniques are employed as resampling methods to

estimate the power of the nonuniform data sequences.

The least squares spectral analysis is a technique of estimating

the power spectrum of nonuniformly sampled sequences based on the

least squares fit of sinusoids to data sequences. In this method data

sequences are approximated using the weighted sum of sinusoidal

frequency components using a linear regression method or a least

squares fit method. The number of sinusoids that are used for

approximation should be less than or equal to the number of data

samples. A data vector Y can be represented as a weighted sum of

sinusoids as AXY . The elements of the matrix A can be calculated

by evaluating each function at the sampling time instants, with the

weighted vector X . The weighted vector x is chosen such that to

minimize the sum of squared errors in approximating the data vector

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Y and the solution for X is given as a closed form AYAAX1T

where the matrix AAT is a diagonal matrix. Then the power spectrum

estimate using the interpolated least squares spectral analysis can be

given by

XAXA

N1)(P

T

IS .

4.4.2 Algorithm:

The block diagram and algorithm for power spectrum estimation

using proposed techniques is as follows.

Generate nonuniform data sequences of different lengths ‘N’.

Interpolate the data sequences Nk0kkx y,x within the interval

of values N0 x,x .

By applying the Linear as well as Cubic Spline interpolation

techniques within the interval N0 x,x , the data sequence is

approximated as x (n).

Figure4.56 (i) Proposed system model.

AR (P)Filter

PeriodogramAnalysis

Input datax (n)

WhiteNoise w (n)

PeriodogramUsing WLSP

PeriodogramUsing LSP

Y(n)

Non Parametric SpectrumMethods

Pxx(ω)

PWLSP(ω)

PLS ω)

+

151

The power spectrum of the interpolated data sequence x(n) is

calculated employing the nonparametric methods.

XAXA

N1)(P

T

IS

The Lomb periodogram is evaluated for the nonuniform data

sequence

N

1nn

2

2N

1nn

N

1nn

2

2N

1nn

xx))t((sin

))t(sin()tn(x

))t((cos

))t(cos()tn(x

21)(P

where is defined by)2cos(

)t2sin()2tan( N

1nn

N

1nn

The weighted least squares periodogram can be given by

XAXAba

N1)(P

T22

WLSP , where ‘a’ and ‘b’ are data

dependent weights and ‘A’ is the data dependent matrix.

4.4.3 Simulation Analysis:

The simulations results for the nonuniform data sequences of

white Gaussian noise, real sinusoids in Gaussian noise and narrow

band sinusoidal components in broad band noise are illustrated as

follows. The Interpolation methods are employed as resampling

methods to convert the nonuniformly sampled sequences into

uniformly sampled sequences. The proposed interpolation techniques

employed are linear and cubic spline interpolation techniques. The

simulation results are observed for both the interpolation techniques

152

and are compared to the least squares methods and Lomb transform

methods.

Figure 4.56 Real sinusoids in white Gaussian noise for N=30.

Figure 4.56 provides the real sinusoidal components in nonuniform

data sequence of white Gaussian noise of length 30. The figure shows

the amplitude versus time index of Gaussian noise.

Figure 4.57 Interpolated random sequence using linear interpolation.

Figure 4.57 provides the linear interpolated data sequence of white

Gaussian noise of length N=30 using linear interpolation technique

which converts the nonuniform data sequence to uniformly sampled

sequence with sampling period of 1 sec.

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Figure 4.58 Periodogram analysis for the random sequence of N=30.

Figure 4.58 provides the nonparametric power spectrum estimation

using the periodogram technique having the spectral component at

0.5 normalized frequency with power levels of 680 dBs and no other

components are exists below 0.2 normalized frequencies.

Figure 4.59 Spectral analysis of random sequence using linear interpolation

Figure 4.59 provides spectrum analysis using proposed linear

interpolation technique shows that there spectral components existing

even below 0.2 normalized frequency and are at 0.08, 0.1, and 0.15

frequencies respectively with good power levels above 800dbs which

were not visible in the classical nonparametric periodogram

estimation.

154

Figure 4.60 Power Spectrum analysis using Lomb Transforms.

Figure 4.60 provides the spectral analysis with using the Lomb

transforms which shows that the existing spectral component at 0.5

normalized frequency with 710 dbs of power levels that is higher

compared to the periodogram estimate.

Figure 4.61 Narrowband components in broadband noise for N=30.

Figure 4.61 provides the narrow band components in nonuniform data

sequence of broadband noise of length 30. The figure shows the

amplitude versus time index of narrow components in broadband

noise.

155

Figure 4.62 Interpolated random sequences using linear interpolation.

Figure 4.62 provides the linear interpolated data sequence of narrow

components in broadband noise of length N=30 using linear

interpolation technique which converts the nonuniform data sequence

to uniformly sampled sequence with sampling period of 1 sec.

Figure 4.63 Spectral analysis of random sequence using linear interpolation

method.

Figure 4.63 provides spectrum analysis using proposed linear

interpolation technique shows that there spectral components existing

even below 0.2 normalized frequency and are at 0.08, 0.1, and 0.15

frequencies respectively with good power levels above 1000dbs which

were not visible in the classical nonparametric periodogram

estimation.

156

Figure 4.64 Power Spectrum analysis using Lomb Transforms.

Figure 4.64 provides the spectral analysis with using the Lomb

transforms of narrow band components in broadband noise of length

30 which shows that the existing spectral component at 0.5

normalized frequency with 710 dbs of power levels that is higher

compared to the periodogram estimate.

Figure 4.65 Nonuniformly sampled white Gaussian Noise AWN(0,1) for N=30.

Figure 4.65 provides the nonuniformly sampled white Gaussian noise

of length N=30 .The figure shows the amplitude versus time index of

white Gaussian noise.

157

Figure 4.66 Spectrum analysis using the periodogram.



0.5 rad/sample normalized frequency with power levels of 0.09 dBs

and the other components are exists with power levels of 0.052 dBs

and 0.06 dBs.

Figure 4.67 Spectral analysis using Least Squares Periodogram.

Figure 4.67 provides power spectrum estimation using the Least

squares periodogram technique having the spectral component at 0.5

normalized frequency with power levels of 0.18 dBs and the other

components are exists with power levels above 0.1 dBs.

158

Figure 4.68 Comparison of spectral estimates for LSP and Periodogram.

Figure 4.68 provides the comparison of two spectral estimates using

Least squares periodogram and periodogram techniques. In LSP

technique the power levels are increased than the power levels of

periodogram by more than 100% so that one can realize the noise

spectral components in the signal content.

Figure 4.69 Nonuniformly sampled real sinusoids in AWN(0,1) for N=30.

Figure 4.69 provides the nonuniformly sampled real sinusoids in

white Gaussian noise of length N=30 .The figure shows the amplitude

versus time index of white Gaussian noise.

159

Figure 4.70 Spectrum analysis using the periodogram.



0.5 rad/sample normalized frequency with power levels of 6.5 dBs and

the other components exists with power levels of 3 dBs and 1 dBs.




normalized frequency with power levels of 13 dBs and the other

components are exists with power levels above 2.2dBs and 6 dBs.

160







161

Figure 4.73 Nonuniformly sampled narrowband components in wideband noise

Figure 4.73 provides the nonuniformly sampled narrowband

components in wideband noise of length N=30 .The figure shows the

amplitude versus time index of white Gaussian noise.

Figure 4.74 Spectral analysis using Periodogram for N=30.



0.5 rad/sample normalized frequency with power levels of 6.5 dBs and

the other components exists with power levels of 3 dBs and 2 dBs

respectively.

162




normalized frequency with power levels of 13 dBs and the other

components are exists with power levels of 6dBs and 4 dBs.







163

Figure 4.77 periodogram analysis for the data sequence of length N=256

)1.0cos(4)952.0cos(4)04.0cos(2)( 321 nnnny

Figure 4.77 provides power spectrum estimation of the above data

sequence y(n) using the periodogram technique having the spectral

component at 0.04 , 0.952 and 0.1 normalized frequencies. It observes

that two spectral peaks at 0.952 and 0.1 normalized frequencies are

not properly resolved.

Figure 4.78Least squares periodogram analysis for the data sequence of length

N=256 )1.0cos(4)952.0cos(4)04.0cos(2)( 321 nnnny


sequence y (n) using the Least squares periodogram technique having

the spectral component at 0.04, 0.952 and 0.1 normalized

frequencies. It observes that two spectral peaks at 0.952 and 0.1

normalized frequencies are clearly resolved.

164

Figure 4.79 Weighted Least Squares Periodogram analysis for the data

sequence )1.0cos(4)952.0cos(4)04.0cos(2)( 321 nnnny


sequence y(n) using the weighted Least squares periodogram

technique having the spectral component at 0.04 , 0.952 and 0.1

normalized frequencies. It observes that two spectral peaks at 0.952

and 0.1 normalized frequencies are clearly resolved with wide dynamic

range of 20 to 40 dBs.

Figure 4.80 Comparison of Periodogram, LSP and WLSP.


sequence y(n) using the periodogram , LSP and WLSP techniques. It

observes that the Weighted Least Squares periodogram technique

provides the efficient spectral estimation components in wide dynamic

range with less variance compared to other methods.

165

Figure 4.81 Nonuniform data sequence.

Figure 4.81 provides the discrete non uniform data sequence with

amplitude versus time index plot. It observes that all the samples are

randomly spaced within the time interval 0 to 1 second.

Figure 4.82 Linear interpolations of nonuniform data

Figure 4.82 provides the linear interpolation of discrete non uniform

data sequence with amplitude versus time index plot. It observes that

all the samples are uniformly spaced after applying the interpolation

technique within the time interval 0 to 1 second.

166

Figure 4.83 Cubic spline interpolation of nonuniform data.

Figure 4.83 provides the cubic spline interpolation of discrete non

uniform data sequence with amplitude versus time index plot. It

observes that all the samples are uniformly spaced after applying the

interpolation technique within the time interval 0 to 1 second.

Figure 4.84 Spectral analysis using cubic spline interpolation.

Figure 4.84 provides the cubic spline interpolation of discrete non

uniform data sequence with amplitude versus time index plot. It

observes that all the samples are uniformly spaced after applying the

interpolation technique within the time interval 0 to 1 second.

167

Figure 4.85 Comparison of Spline and Linear spectral analysis.

Figure 4.80 provides power spectrum estimation of the data sequence

y(n) using cubic spline , Interpolation techniques. It observes that the

cubic spline interpolation technique provides the efficient spectral

estimation components in wide dynamic range.

4.4.4 Results and Discussions: As observed from the above figure it

is clear that by the application of interpolation technique the

frequency components of the data sequence are shifted from high

frequency region to low frequency region and thus avoid the effect of

low pass filtering, also the range of the power values increases when

compared to the periodogram plot. The use of Lomb transform on the

spectral analysis reduces the variance in the spectral estimate and

also the power values are increased. We can now conclude that the

use of interpolation techniques like linear and cubic spline methods

shift the frequency components from higher to lower frequency

increases the dynamic range of spectral estimates and decreases the

168

variance of the spectral estimate. But the use of Lomb transforms

increases the power values than the periodogram and also the

variance of the spectral estimate decreases. Hence the superiority of

Lomb transforms over the interpolation techniques are studied in this

method.

The periodogram does not resolve the two spectral peaks of the

taken data sequence of length N=256.Hence the least squares

periodogram based on the least squares approximation of data

sequence using sinusoidal functions is used. The least squares

periodogram technique merely resolves the two spectral peaks with

reduced variance. The weights of the approximated data sequences

are calculated using the Yule walker’s method of solving the set of

normal equations. These weights are not appropriate if we consider

the narrowband components in wideband noise. Hence weighted least

squares method of periodogram analysis is adapted. The appropriate

weighted coefficients are estimated using the linear prediction

technique where these weights are multiplied with the power values to

get the weighted estimates. The resulting weighted least squares

method of spectral analysis resolve the two near by spectral peaks

with ease and also the variance of the spectral estimate is further

reduced compared to the least squares periodogram method. Hence

for the narrowband components in wideband noise the weighted least

squares periodogram method is proposed. It uses the weights which

are data dependent.

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