13

CHAPTER-2

SURVEY OF LITERATURE

2.1 INTRODUCTION

The work in this chapter is an attempt to report the literature on

Nonparametric and parametric power spectrum estimation problems

in the past, present and to the best of knowledge, the available

literature is reported in the following sections.

Power spectrum estimation can be defined as the method of finding

power values of hidden frequency components in the harmonics of a

measured noisy signal, and is a highly recommended problem in

practice. Many applications in engineering and biomedicine ranging

from synthetic aperture radar (SAR) for image analysis, radar for

determining range of a target, sonar for positioning, speech

recognition , heart rate variability (HRV) analysis, time series analysis

in seismology etc., can be recognized as spectrum estimation

problems. Non parametric power spectrum estimation methods do not

assume any rational functional form but allow the form of estimator to

be determined entirely by the data. These methods are based on

discrete Fourier transform of either signal segment or its

autocorrelation function. These methods do not make an assumption

of how the data is being generated. While the parametric methods

14

make use of a specific parametric model (pole-zero or harmonic model)

and also these methods require sufficient amount of apriori

information. Without sufficient apriori information it is very difficult

to estimate the values of a signal using parametric methods. The filter

bank approach is the advanced version of non parametric methods to

smooth the discrete Fourier coefficients.

In many practical solutions, samples of noisy signals are available

and it is required to develop a suitable spectral estimate to find the

hidden frequency contents in the harmonics of the noisy signals.

Consequently, many methods have been proposed and developed

achieving the spectrum estimation. Some of these methods are called

classical methods and others are called modern methods.

2.2 Power spectrum estimation methods during the period

1960-1970:

Dimtri S. Bugnolo [14] has proposed autocorrelation function and

power spectrum density for an electromagnetic wave corrupted with a

non stationary dielectric noise. He also suggested the relationship

between the input power spectrum density and the output power

spectrum density with the help of a transfer function of a linear

network. The output power spectrum density of a measured signal is

evaluated as input power spectrum density of source times the

squared magnitude response of the linear network. The algorithm is

applied to radio link signals and observed the auto correlation and

power spectrum density of the corrupted noise signal.

15

Peter D.welch [15] proposed the estimation of power spectrum

density using the Fast Fourier Transform. It involves partitioning the

whole data record into small partitions, taking the modified

periodograms of these partitions and averaging these modified

periodograms. The use of Fast Fourier transform reduces the number

of computations in the estimation of signals , decreases the storage

space and this spectral estimate is inherently limited in frequency

resolution by the data.

John W. Tukey et.al [16] discussed the influence of Fast Fourier

transform algorithms on the spectral analysis of time series data. In

this algorithm the Fourier transform coefficients are employed to

calculate the average lagged products in a faster rate and also the

inverse fast Fourier transform is applied to a complex Fourier

coefficients sequence. They had also discussed the classical and

modified Fourier periodograms based on the data windowing before

the application of Fourier transform.

Charles M. Rader [17] has proposed the use of high speed

autocorrelation functions in the estimation of power spectrum for the

desired number of autocorrelation lags where a data sequence is

extremely large. The high speed auto correlation functions are based

on the linearity of discrete Fourier transform and their circular

shifting properties. The spectrum estimation is used where the data

sequence is essentially unlimited. On the other hand, the

requirement of good spectral resolution will lead to the necessity to

measure the autocorrelation function for many lags. Therefore the

16

technique should be suited to the computation of the auto correlation

estimate needed in the estimation of power spectrum.

2.3 Power spectrum estimation methods during the period

1970-1980:

Otis L. Frost and Thomas M. Sullivan [18] have proposed high

resolution spectral analysis of data fields in two or more dimensions.

The technique consists of extrapolating the observed data beyond the

observation window by means of autoregressive data generation

model. High resolution spectral analyses are then obtained by the

conventional Discrete Fourier Transforms (DFTs) of the extrapolated

data.

Lawrence R. Rabiner and Joint B. Allen [19] have proposed the

spectrum estimation using the short time Fourier transforms. The

technique removes the effect of windows (biased estimates) by

considering the linear weighted combination of all the biased

estimates. The influence of removing the windows makes the smaller

FFT lengths and thus increasing the speed of the technique without

affecting the accuracy. By using this technique with more number of

samples which leads to a quick solution (optimum solution) with good

least squares approximation. The algorithm uses fixed lengths fast

Fourier transforms which are independent of data samples that are

being analyzed.

2.4 Power spectrum estimation methods during the period

1980-1990:

17

Jae S. Lim and Naveed A. Malik [20] suggested an iterative

algorithm for Maximum Entropy power spectrum estimation. The

technique is based on the computational efficiency of fast Fourier

transform algorithm that is applicable to one as well as two

dimensional signals for estimating the power spectral density. This

algorithm is also useful for the maximum entropy power spectrum

estimation of signals the dimensions of which are higher than two.

Farid U.Dowla and Jae S.Lim [21] have proposed that in

multidimensional power spectrum estimation there exits a

relationship between the Maximum likelihood method (MLM) and the

spectra obtained by the AR signal modeling for non uniformly sampled

data sequences where as Burg has shown a relation ship between the

Maximum Entropy Method and MLM method for one-dimensional

uniformly sampled functions.

Jean Pierre Schott and James H. Mc Clellan [22] described the use

of multidimensional MEM algorithm that approximates the correlation

constraint to the nonuniformly sampled arrays. In this algorithm the

elements of the covariance matrix are formed by taking the difference

in weights of the autocorrelation function estimates and the true

power spectral density values. The algorithm provides a better

resolution than the traditional MEM algorithm.

Yujiro Inouye [23] suggested a maximum entropy spectral

estimation for multichannel time series of degenerate rank. He has

shown that the autoregressive method is equivalent to the maximum

entropy method even in the degenerate rank case. He observed that all

18

the deterministic relationships in any regular random process

matching the data of autocorrelation sequence.

Bruce R. Musicus [24] proposed a maximum likelihood estimate for

uniformly sampled correlation function. The technique measures the

weighted prediction filter coefficients using the Levinson recursions

coefficients. This algorithm provides a tradeoff between the Maximum

entropy method and the maximum likelihood method.

R. Raghuveer and L. Nikias [25]] have suggested a parametric

spectrum estimation method of third order called bispectrum. It can

be defined as the double Fourier transform of its third moment

sequence. The lower order power spectrum does not contain much

information about the random process, hence, we are going for the

higher order spectrum. The higher order spectra contains more

information about the random process, hence, bispectrum is

estimated. The bispectrum provides the information about the

quadratic phase coupling between the harmonic components and non

normal processes. This algorithm proposes a parametric modeling of

non white Gaussian data that fits to proper order of AR model. The AR

filter coefficients are obtained by solving third order recursive

equations. The algorithm provides better resolution and fidelity over

the conventional method of spectral estimates.

Linus M. Blaesser [26] has proposed power spectrum estimation

based on complex Walsh functions that are applied to auto and cross

power spectrum. The Walsh power spectrum is applied to only auto

power spectrum based on real functions for wide sense stationary

19

random process. The Walsh power spectrum can be achieved by

taking the linear transformation on the Fourier spectrum. This

relation enables us to obtain the Fourier estimates from the Walsh

estimates.

Hiroshi Kanai et.al [27] proposed phase matching technique to get

autoregressive power spectrum estimation at low signal to noise ratio.

This algorithm is based on phase matching technique that minimizes

the phase of all zero models to the phase of reconstructed signal from

the spectrum of the observed signal. The AR model coefficients are

estimated from the coefficients of the all zero model. The order for the

AR model should be known apriori. The least squares method is

applied since the phase matching technique satisfies the condition of

least squares. The main advantages of the method are to calculate the

AR parameters at low SNR and to reconstruct the exact phase from

the power spectrum.

Nicholas and Sergios [28] have proposed two fast adaptive least

squares algorithms for power spectral estimation of a time series. This

is achieved by modeling the input signal as an AR signal of order m

and simultaneous minimization of sum of the forward and backward

prediction error energies of mth order prediction. The first algorithm

and the second algorithms require m3 multiplications and additions

while the Burg’s technique requires only m2 multiplications and

additions.

Yung Chi and David Long [29] have presented an analysis for the

noise due to finite word length effects for digital signal power

20

processors using Welch’s power spectrum estimation method to

measure the power of a Gaussian random signal over a frequency

band of interest. The input of the digital signal processor contains a

finite length time interval in which the true Gaussian signal is

corrupted by Gaussian noise. In this algorithm the round off signal to

noise ratio is analytically derived in the measurement of signal power.

Nailong Wu [30] suggested a nonlinear method of power spectrum

estimation by using the uniformly spaced autocorrelation functions.

This can be achieved on imposing an iterative algorithm in Maximum

entropy method and this method does not require imposing the

conditions such as causality, minimum- Phase, etc., on the signal.

We can have reasonably large zero lag autocorrelation functions which

determine the positive background level in the spectrum with the

apriori knowledge of the data sequence.

Moeness G. Amin [31] suggested the use of exact values of

autocorrelation in place of their estimates at one or more lags may

lead to two opposite effects on the variance of the corresponding non

parametric power spectrum estimator. In non parametric spectral

estimation problems, PS estimate is provided via Fourier transform of

the time average estimates of the autocorrelation function. The direct

use of exact values of autocorrelation in place of their estimates does

not necessarily result in an improved spectrum estimator at all

frequencies. This placement can yield two opposite effects on the

estimator’s variance, i.e, increasing the variance within some

frequency bands while reducing the variance in other frequency bands

21

along the Nyquist interval. The location as well as the width of these

different bands is primarily a function of the lag numbers at which the

autocorrelation is known. Therefore a decision whether to use or

discard the exact autocorrelation values in power spectrum estimation

depends on the bands of interest in relation with the lags of known

autocorrelation values.

Michael J. Villalba and K. Walker [32] have proposed an approach

to improve the frequency resolution of the power spectrum estimation

of signals with rational spectrum when the errors are observed in the

autocorrelation calculations. This algorithm is not able to resolve the

two closely spaced poles. The first procedure is employed by applying

the discrete time techniques to estimate the power spectrum of

continuous process. The sampling period of autocorrelation function

is used to separate the closely spaced poles. The second method

employs to resample the autocorrelation sequence to separate the

poles to get a high resolution. They have shown that autocorrelation

poles can be placed in locations which reduce error sensitivity by a

proper choice of either the autocorrelation sampling period or

resampling the interval. The resolution of the spectral estimate is then

improved considerably. These results are applicable to a variety of

situations requiring high resolution spectrum estimation.

Ernest G.Baxa [33] has discussed the application of short time

Fourier analysis to the problem of spectral estimation with the DFT.

Emphasis has been made on the resolution capability associated with

coherent Fourier domain smoothing which is inherent in Short-time

22

unbiased spectrum estimation algorithm. An analysis has been made

on effective spectral window associated with the power spectrum

estimation obtained from short time Fourier transforms. The finite

window length spectral leakage effects on the data sequences can be

reduced by linearly combining the biased estimates.

2.5 Power spectrum estimation methods during the period 1990-

2000:

Cheng Liou and Bruce R. Musicus [34] have presented an

approach for power spectrum estimation modeled by separable cross

entropy .For multidimensional and multichannel models the Gaussian

process is sampled with nonuniform sampling and an appropriate

model is selected in which the frequency samples are treated as

independent samples. For good approximation of data samples two

cross entropy methods are suggested. One is based on the Capon

method and the other is based on the windowing technique.

Chrysostomos and Taikang Ning [35] have suggested the power

spectrum estimation with correlation measurements randomly

displaced from a uniform distribution. Due to randomness, the

resolution capability of the Maximum entropy power spectrum

decreases and its frequency bias increases as location uncertainties

increase. To avoid these effects, three algorithms, namely, the

ensemble averages, minimum variance, and the extended region

approaches have been proposed to generate extendable and uniformly

placed correlation measurements which are more reliable for power

spectrum estimation. To utilize the ensemble average approach,

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information regarding the distribution of location uncertainties of

correlation samples must be available in order to recover the true

power spectrum from the attenuated estimate. The same information

is also required for the extended region approach in order to define the

extended region that encompasses the uniformly spaced correlation

samples. Such information is not required for the minimum variance

approach.

Sergio D.Cabrere and Thomas W.Parks [36] have developed an

iterative procedure for a periodogram spectrum estimate obtained

from samples of signal extrapolation found at one iteration to define

the weight that is used to estimate at the next iteration. The frequency

resolution extrapolation lengths are controlled by the length of a time

domain window used to obtain the smooth spectral estimates between

iterations. This method des not require the apriori knowledge and

provides the comparable resolution to the parametric methods with

more accurate values of the relative strengths of the narrow-band

components. This algorithm is also known as nonparametric

frequency-stationary extension of the data. This algorithm has good

performance on a narrow band portion of the spectrum by the choice

of a relatively long window size.

Aharon Berkovitz and Rusnak [37] have presented the influence of

sampling instabilities on spectral estimation by Fast Fourier

Transform (FFT). Two types of random samples are considered,

independent jitter, and accumulated jitter. For accumulated jitter in

sampling instants, the distortion level is relatively high and the

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resolution level is considerably degraded. The use of FFT with this

kind of instability is limited to a small amount of jitter, low input

frequencies, and short sequences. In the independent jitter case, the

distortion level is relatively low and the resolution capability is

considerably conserved, even for a relatively large amount of jitter and

high input frequencies.

Mohammad A .Maud and Azim I. Bruno [38] have suggested an

approach for parametric spectrum estimation with good frequency

resolution using the rational spectra for the estimation of signals. In

this algorithm the errors in the autocorrelation function can be

precisely reduced by the double autocorrelation method. The solution

for the Yule walker equations can be carried out by forward and

backward linear predictor method of Marple. The algorithm provides

good results even when the poles are closely spaced. The sensitivity to

calculated autocorrelation error is acute without resorting to pole

manipulation.

Langford’B White [39] has presented a method for spectrum

estimation based on minimization of Csiszar’s I-divergence measure.

The blurring effect of the observation window is minimized by the

application of a nonlinear deconvolution procedure which was

originally formulated in connection with positron emission

tomography. In this algorithm the method is applied to the spectral

estimation problem for stationary processes. A reblurring method is

used to regularize the method. The method is iterative in nature

allowing a tradeoff between resolution and error performance to be

25

obtained. This method is implemented using the Fast Fourier

transform.

Jun Yin and Zhaoda Zhu [40] have emphasized the estimation of

power spectrum using the neural–type structured network. Based on

this structured network, a new autoregressive (AR) modeling method

is presented. The algorithm involves solving the Yule-Walker normal

matrix equations for model coefficients using the structured network.

This provides advantages like parallel architecture, suitable for

realization directly by VLSI hardware and no divisions are involved in

all the calculations, so that it still works for unconditioned Yule-

Walker type matrix equations. This algorithm is applied for narrow

band sources and combinations of narrow band and broad band

sources subject to various level to Gaussian white noise.

E. Turkbeyler and A. G. Constantinides [41] have proposed the

usage of higher order statistics in estimating the power spectrum of

signals corrupted with Gaussian noise. The method based on higher

order statistics is developed to obtain noise free power spectrum

estimation when the signal is corrupted by Gaussian additive noise.

The method employs the trispectrum and bispectrum to calculate the

power spectrum and correspondingly the autocorrelations. The

trispectrum is defined by the Fourier transform of fourth order

cumulants (fourth order moment spectra). Non parametric and

parametric methods can be employed to estimate the trispectrum and

bispectrum. The method introduced gives a power spectrum with less

bias, but higher variance than classical estimation methods.

26

Ling Chen et.al [42] presented the comparison between the

Maximum Entropy and minimum relative entropy spectral analysis of

time series data. The maximum entropy spectral analysis is much

safer than minimum relative entropy spectral analysis even though

the latter can offer a better spectral analysis when autocorrelations

are few under certain circumstances. This safety is considered

important in most of the spectral analysis applications. Under those

circumstances which favor the minimum relative entropy spectral

analysis certainly exclude the use of prior spectra which offer

inaccurate or wrong shape information of the true spectrum.

Pierre Moulin [43] has proposed a nonparametric approach based

on logarithmic wavelet transform of unknown power spectrum. This

method provides the ability to capture statistically significant

components of logarithmic power spectrum at different resolution

levels and obtains a nonnegative spectrum estimator. The power

spectrum estimation is a problem of estimating the wavelet

coefficients of a signal interfered by additive white Gaussian noise.

The wavelet coefficients are assumed as independent random

variables. For the distribution of noise coefficients, the threshold

values are based on saddle point approximation. The estimation

techniques studied in this algorithm do not assume a apriori

knowledge about the underlying spectrum, besides the presumption

that the signal contains significant coarse scale coefficients. When a

priori information is available, special techniques may be used to

improve the frequency resolution.

27

Peter T. Gough [44] has proposed a fast spectral estimation

algorithm for spectral estimation based on the Fast Fourier

Transform. The algorithm is recursive and the FFT is used many times

in a systematic way to search for the individual spectral lines. This

algorithm is able to detect multiple sinusoids in additive noise. It is

certainly better than the single phase FFT in separating closely spaced

sinusoids. Since it is based on an iterative application of the FFT, the

spectral estimation algorithm described here is simple to program and

fast to execute.

David .L and John .A [45] have described a quadratic power

spectrum estimation based on implementation of orthogonal

frequency division multiple access windows. The windows are

obtained from frequency shifted version of a single window. The

algorithm is based on the minimum mean squared error criterion.

Quadratic spectral estimators are nonparametric and are quadratic

functions of the data being analyzed. This estimation has good

frequency resolution; its statistical properties are good with respect to

the best spectral estimators.

T. Umemoto et.al [46] suggested an algorithm based on constant

Q-value filter banks with spectral analysis using LMS algorithm. In

spectral analysis of temporarily varying signals, constant Q-value filter

banks using short time spectral analysis method is known to be

effective because the frame length can be changed freely with the

method depending on the frequency. In this method, however, the

parameter to control the stability and convergence factor of system

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was a scalar value, so that the constant Q-value of higher frequency

becomes larger. In this algorithm the time constant of coefficient

adjustment and the resolution in frequency are inverse by

proportional to the parameter of convergence factor. The proposed

constant Q filter bank is superior to the adaptive spectrum analysis

method to analyze the human voices and acoustical waves generated

by musical instruments.

S.V. Narasimhan et.al [47] proposed power spectrum estimation for

complex signals using group delay approach. Even though the basic

periodogram spectral estimate has low bias, good resolution, and good

spectral detectability even at high noise levels, its variance is large.

The averaging of periodograms results in a lower spectral variance

only at the cost of frequency resolution. The group delay (GD), the

negative derivative of phase, provides an improved frequency

resolution over the averaged periodograms method. However, the

smoothed group delay method like periodograms method reduces

variance only at the cost of frequency resolution. The model based

approaches provides both high resolution and low variance, but with a

high signal to noise ratio. The zeros which are close to the unit circle

significantly reduce the variance in the spectrum estimation. The

modification approach removes the zeros close to the unit circle

without disturbing the signal poles and hence reduces the variance

without scarifying the resolution. The algorithm provides a low

variance spectral estimate than that of periodogram.

29

Norikazu. I, Hiroshi. M [48] presented a method to make an

adaptive estimation of non stationary power spectrum. The method

uses model based on time varying coefficient autoregressive (AR)

model in which order of autoregression is also varied with time. Non

stationary power spectrum can be obtained by varying the time

varying coefficients, and abrupt change of the structure of the

spectrum can be estimated by the time varying order. The model is

written in the state space representation with system model that

defines smoothness of time varying parameters AR model. Monte

Carlo filter and genetic algorithm are used for estimation of AR

coefficients and order respectively. From the estimated parameters we

can obtain the time varying power spectrum.

Philippe Ciuciu et.al [49] examined the problem of non parametric

spectral estimation for discrete time compound random process which

is a mixture of narrow band and wide band components. They have

shown that separable spectral estimates based on convex penalized

criteria provide a quite accurate narrow band response and to improve

the quality of wide band response a Markovian penalization is

introduced in the criterion. In this case the closely spaced sinusoids

are not resolved whereas the broadband component is well retrieved.

To avoid such a disadvantage, they have proposed an original model

and an adapted regularization function. Since estimate is obtained via

the minimization of a convex criterion, it is computed by an

optimization procedure. They have used a fast algorithm, whose

convergence to the global optimization of variance values.

30

2.6 Power spectrum estimation methods during the period

2000-2009:

Ch. Rebai et.al [50] presented a non coherent spectral analysis of

ADC using filter bank. The spectral analysis of ADC digital data has

traditionally been done with the Discrete Fourier Transform. This

method imposes restrictions to optimize the coherent sampling. In this

algorithm they have presented a filter bank structure used for

decomposition of signal into its main spectral components. The main

drawback of the spectral analysis is spectral leakage which appears

when the transmitted and received frequencies are not coherent. Non

coherent sampling would make the first and last samples

discontinuities and affects the dynamic specifications. To overcome

this problem, spectral parameters like signal to noise ratio and

harmonic distortion are computed using the windowing method. The

proposed structure based on biquadratic filter has been used for the

spectral analysis of ADC. The estimation of spectral parameters with

digital filtering without coherent sampling are close to the calculated

with coherent sampling by FFT.

Alberto Cristan and Andrew T. Walden [51] have presented a

multitaper power spectrum estimation and thresholding using the

Discrete Wavelet Transforms (DWT).The algorithm is based on the

computation of logarithm power spectrum by applying an orthonormal

transform derived from a wavelet packet tree to the log multitaper

31

spectral ordinates, thresolding the empirical wavelet packet

coefficients, and then inversing the transform. For a small number of

tapers suitable transforms for the logarithm of the multitaper

spectrum estimator are derived using a method matched to a

statistical thresholding property. The partitions thus derived starting

from different stationary time series are all similar and any differences

between the wavelet packet and discrete wavelet transform

approaches are minimal. For large number of tapers, the simple DWT

again emerges as an appropriate method. Hence using the approach

to thresholding and the method of partitioning, they conclude that the

DWT approach is a very adequate wavelet based approach and that

the use of wavelet packets is unnecessary.

Piet M. T. Broersen. et.al [52] presented an application of

autoregressive spectral analysis to missing data problems. The finite

interval likelihood maximization algorithm is numerically stable in

estimating AR models from incomplete data. For a few missing data,

the performance of ML methods is better than that of other known

methods, including all methods that reconstruct the missing data

before the spectral density is estimated. The quality of the estimated

model with a selected model order is good in simulations where the

true process is a low order AR (p), often comparable to Cramer-Rao

lower bound. This algorithm requires no user provided initial solution,

is suited for order selection, can give accurate spectra even if less than

10% of data remains.

32

Jing Deng et.al [53] proposed predictive differential power

spectrum based cepstral coefficients and sub band mel-spectrum

centroid based cepstral coefficients for robust speaker recognition in

stationary noise environments. The proposed algorithms have been

proved effective to enhance the robustness of speech with stationary

noises and so it may also be effective for non stationary noisy speech.

Ivo Batina et.al [54] proposed a method of spectral estimation for

noisy speech signal. The method is based on the combination of an

autoregressive (AR) model with a Kalman filter to get the noise Power

spectral density estimation. The proposed algorithm exhibits good

noise tracking capabilities. The introduction of time varying model

parameters will lead to improve noise PSD estimates at the cost of

higher computational load.

Petre Stoica and Xing Tan [55] have suggested a method for spatial

power estimation that outperforms the beam forming method as well

as the capon method. The proposed method is user parameter free,

unlike the more other spectral estimation methods. In this algorithm

they emphasized a covariance matrix fitting approach to spatial power

estimation. The method uses the Pisarenko frame work for spatial

power spectrum estimation.

John Lataire and Rik Pintelon [56] have presented an estimation

method for disturbing noise of linear continuous-time slowly time

varying dynamic systems. Two methods have been discussed to

reduce the deterministic contributions of the signal. The first consists

of differencing the output signal. The second approximates the signal

33

by a superposition of hyperbolas. The second method is shown to give

significantly lower bias than the first at the price of a more involved

algorithm. Both the estimations of the noise and of the speed variation

only use one well designed experiment and are performed in the

frequency domain, revealing the important benefits of using multistins

as excitation signals.

Zhu Min et.al [57] presented power spectrum estimation based low

temperature weak signal detection in light, sound and laser materials.

The low temperature target radiation energy has smaller absolute

value and the signal is weaker. There are a lot of noises such as

temperature noise, thermal noise and so on. It is very difficult to

detect if the conventional methods are used. As power spectral density

and autocorrelation forms a Fourier transform pair, power spectrum is

used to realize autocorrelation calculation so that low temperature

target is detected. Using this method can get the autocorrelation

function values by calculating the values of power spectrum and the

unknown amplitude of the weak signal can be detected.

Kaushik Mahata and Damian Marelli [58] have proposed

interpolation and spectral analysis of signals from finite number of

samples. When the observed data is of finite length, interpolation and

spectral analysis of band limited signals using the Shanon’s

framework leads to erroneous results causing the spectral leakage

problems. This algorithm deals with this issue from a minimum

variance estimation perspective, and treats a general case where the

signal is not necessarily band limited. In contrast to traditional

34

windowing based methods, the minimum variance estimation leads to

a convolutional transformation of data, which employs a linear

predictor. The performance of the estimator is somewhat sensitive to

the underestimation of ARMA model order, while overestimation of the

order does not cause major issues. For this reason we use a model

order some what higher than that returned by the Akaike’s

information criterion.

Zishu He and Ting Cheng [59] have presented a new spectral

analysis based on multi stage nested wiener filter. The multistage

wiener nested filter performs the wiener filtering with a nested

structure. The MSNWF has the advantage of extracting the signal and

noise subspace with out eigen decomposition and convergence in a

speed much more quickly than the LMS or RLS. The MSNWF

approach is applied successfully in several applications, including

adaptive beam forming in communication, multi-user access

interference (MAI) suppression for asynchronous CDMA, direction of

arrival in radar etc. In this algorithm it can be seen that the spectral

analysis can be performed with different patterns. The multiple signal

classification (MUSIC) method based on signal subspace, the linear

predictive (LP) and smoothed linear prediction (S-LP) method. With

computational advantages, this new algorithm can adopt to both

continuous and discrete spectra. The MUSIC method is good at

distinguishing the frequency components, while the LP method is good

at frequencies estimation and S-LP method is suitable for estimation

35

of continuous spectrum. These different patterns can be shown

synchronously or selected by a priori knowledge.

Lu zhu et.al [60] suggested non coherent spectral analysis of ADC

using resampling methods. The proposed structure using decimation

and interpolation to change the sampling rate by a non integer factor

reduces the spectral leakage and improve estimation accuracy in

frequency analysis of noncoherent sampling. The effectiveness of

resampling methods compared to windowing methods is explained.

Petre Stoica et.al [61] proposed a spectrum estimation method of

iterative adaptive algorithm based least squares method. The method

can be applied to uniform, nonuniform data and as well as for data

with missing samples. MIAA uses the IAA spectrum estimates to

retrieve the missing data, based on a spectral least squares criterion

similar to that used by IAA. The MIAA spectral estimation is much

lower in computational cost.

M.Sreelatha et.al [62] proposed a method of estimating the power

spectral density of a wide sense stationary random signal with

available low resolution samples. A modified maximum entropy

inference engine algorithm for power spectral density estimation of

random signal is explained in this algorithm. The proposed technique

is based on subband multichannel autoregressive spectral estimation

(SMASE). The method filters the input samples by M in length and

decimates M times to yield M subsequences at the output of each

decimator. Theses decimated sequences are expressed using the

multichannel AR modeling. The resulting signals from one subband

36

are then processed using maximum entropy inference engine. This

method makes use of the apriori information provided by the whole

knowledge of autocorrelation function of the filtered signal on one

branch of the filterbank. This prior knowledge allows improving the

spectral estimation performance.

Peter stoica and Jian Wang [63] have presented a non parametric

spectral analysis with missing data using expectation maximization

(EM) algorithm. This algorithm explains the nonparametric complex

spectral estimation for uniform and nonuniform data samples and

also for missing samples in arbitrary pattern, two missing data

amplitude and phase estimation (MAPES) algorithms, namely EM1

and EM2 have been derived by formulating an ML based problem

which is solved iteratively using the two EM algorithms. The two

algorithms have performed quite similarly, but EM2 is

computationally more appealing than EM1.

2.7 Statement of the problem:

Though the non parametric spectral estimation has good dynamic

performance, it has a few 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 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

37

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 and uneven data sequences using

Lomb Transforms and resampling, linear interpolation and cubic

interpolation methods are proposed. The simulation results 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, nonlinear overlapping

38

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.

2.8 The main contributions of the thesis are:

To observe the spectral efficiency, frequency resolution, bias,

variance and other higher order statistical characteristics like

skewness and kurtosis values, the following spectral estimation

techniques are proposed in the next chapter.

Power Spectrum Estimation of stationary and nonstationary

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 and cubicspline interpolation

techniques.

The algorithms and their simulations results are considered in

chapter 4.