Power Spectrum Estimation Methods Computer Science Essay

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The work in this chapter is an attempt to report the art of literature on Non parametric 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 a spectrum estimation problem. 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 make use of a specific parametric model (pole-zero or harmonic model) and also these methods requires sufficient amount of apriori information. Without sufficient apriori information it is very difficult to estimate the values of 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.

Power spectrum estimation methods during the period 1960-1970

Dimtri S. Bugnolo [1959] described the correlation function and corresponding power spectrum of an electromagnetic wave affected by random dielectric noise (non stationary noise) and is related to the power spectrum of the source by an extension of the notation of the time variable linear networks. In general it will be shown that the power spectrum of the received signal can be regarded as the output of a network characterized by a transfer function. The results are applied to a long line of sight radio link and used to predict the error in the received signal in a mean squared sense.

Peter D.welch [1967] described the use of Fast Fourier transform in power spectrum analysis, which involves partitioning the whole data record into small partitions, taking the modified periodograms of these partitions and averaging these modified periodograms. The advantage of this method is reduction in the number of computations and in required core storage but this spectral estimate is inherently limited in resolution by the data.

John W. Tukey and Michael D. Godfrey [1967] discussed the influence of Fast Fourier transform on the spectrum of time series. They had shown that the computationally fastest way to calculate mean lagged products is to begin by calculating all Fourier coefficients with a fast Fourier transform and then taking the fast Fourier retransform to a sequence made up of the complex Fourier coefficients. They had also discussed the classical and modified Fourier periodograms based on the data windowing before the application Fourier transform.

Charles M. Rader [1970] 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 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 [1979] 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 (DFT's) of the extrapolated data.

Lawrence R. Rabiner and Joint B. Allen [1979] have proposed the short time Fourier transform analysis technique in which the influences of the window(biased estimates) on a spectral estimate can essentially be removed entirely (an unbiased estimator) by linearly combining the biased estimates. As a result section FFT lengths for analysis can be made as small as possible, thereby increasing the speed of the algorithm without limiting the accuracy. The algorithm has the important property that as the number of samples used in the estimate increases; the solution quickly approaches to a least squares (optimum) solution. This algorithm also uses a fixed Fourier transform length independent on the number of samples in the data being analyzed, allowing the estimate to be recursively updated as more data is made available.

2.4 Power spectrum estimation methods during the period 1980-1990

Jae S. Lim and Naveed A. Malik [1981] suggested an iterative algorithm for Maximum Entropy power spectrum estimation. This algorithm is applicable to two dimensional signals as well as one dimensional signal, utilizes the computational efficiency of Fast Fourier Transform (FFT) algorithm and has been empirically observed to solve the power spectrum estimation problem. This algorithm is also useful for the maximum entropy power spectrum estimation of signals whose dimensions are higher than two.

Farid U.Dowla and Jae S.Lim[1984] 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 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 [1984] have proposed a multidimensional MEM algorithm, valid for nonuniform sampled arrays, which satisfies a correlation approximating constraint. In this algorithm, the correlation matching equally constraints of the usual MEM are replaced by a single inequality constraint whose form is based on a measure of the noise in the given autocovaraince function (ACF). The covariance matrix of the correlation estimates is used in a quadratic form tha weights the difference between the given ACF and the one matched by the power spectrum. The maximization of entropy under this inequality constraints leads, ultimately, to a steepest-descent algorithm. The algorithm provides the better resolution than the traditional MEM algorithm.

Yujiro Inouye [1984] had 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 the degenerate rank case. He observed that all the deterministic relationships in any regular random process matching the data of autocorrelation sequence.

Bruce R. Musicus [1985] had derived a fast algorithm for calculating the capon maximum likelihood method (MLM) power spectrum estimate when given the uniformly spaced samples of correlation function. This algorithm computes a weighted correlation of predictor coefficients found by running the Levinson recursions. The Fourier transform of the result gives the MLM spectrum. This approach also suggests a comparison between maximum likelihood method and maximum entropy method.

R. Raghuveer and L. Nikias [1985] have suggested the bispectrum estimation (parametric), which is third order spectrum. The bispectrum of a third order stationary process can be defined as the double Fourier transform of its third moment sequence. Higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of non linearity and deviations from the nonlinearity. Estimation of the bispectrum, which is a third order spectrum, has been applied in various fields to obtain information regarding the quadratic phase coupling among the harmonic components and non Gaussian processes. Existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum which are known to possess certain limitations. This algorithm proposes a parametric approach to bispectrum estimation based on a non Gaussian white noise driven autoregressive (AR) model. The AR parameter estimates are obtained by solving the third order recursive equations which may be Toeplitz in form but not symmetric. This algorithm provides bispectrum estimates that are far superior to the conventional estimates in terms of bispectral fidelity and better resolution.

Linus M. Blaesser [1986] has developed a Walsh power spectrum which is based on a system of complex Walsh functions and thus applies to auto and cross spectra as well. Existing concepts of Walsh power spectra for wide sense stationary stochastic processes are restricted to the case of auto power spectra because they are based on real Walsh functions. It is shown that Walsh power spectrum is related to the Fourier power spectrum by a linear transformation. This fact makes it possible to calculate the Fourier power spectrum estimates from corresponding Walsh power spectrum estimates.

Hiroshi Kanai and Keniti Kido [1987] have proposed accurate autoregressive spectrum estimation at low signal to noise ratio using a phase matching technique. Based on the phase matching technique, it minimizes the difference between the phase of the all zero models and the phase of the maximum phase signal reconstructed from the power spectrum of observed the signal. The parameters of the AR model are obtained from finite length sequence of the estimated all-zero model. The proposed method works only when the order of the AR model is known a priori at present. However, since the phase matching technique satisfies the conditions needed to apply least means square method, the AR parameters are estimated accurately even at low signal to noise ratio. The proposed method allows one to accurately reconstruct the phase from the power spectrum in such cases.

Nicholas and Sergios [1987] 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[1987] have presented an analysis for the noise due to finite word length effects for digital signal power processors using Welch's power spectrum estimation method to measure the power of a Gaussian random signals 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 [1988] 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 in 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 [1988] had 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 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 [1989] have proposed an approach to improve the frequency resolution of the power spectrum estimation of signals with rational spectrum when the effect of errors in the calculated autocorrelation values is considered. Analysis and numerical computations reveal that the sensitivity of the error is acute when the poles are closely spaced, demonstrating one of the difficulties encountered when high resolution spectrum estimates are required. To overcome this problem two modified spectrum estimation procedures are proposed for separating the closely spaced poles. The first applies to situations where discrete time techniques are used to estimate the spectrum of continuous time processes, and involves selecting the autocorrelation function sampling period to separate the poles. The second procedure involves resampling the autocorrelation sequence to artificially separate closely spaced poles. 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 [1989] 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 un biased 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 [1990] have presented an approach for power spectrum estimation based on a separable cross entropy modeling procedure. Starting with a model of multi channel and multidimensional stationary Gaussian random process which is sampled on a nonuniform grid, an approximate model is then fit to this, in which selected frequency samples of the random process are modeled as independent random variables. Two cross entropy criteria are used to select optimal separable approximations. One of the method yields a spectral estimation algorithm which is generalized version of Capon's maximum likelihood method, and the other is similar to classical windowing methods.

Chrysostomos and Taikang Ning [1990] 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 increases. To avoid these effects, three algorithms, namely, the ensemble averages, minimum variance, and the extended region approaches, have been proposed to generate extendible and uniformly placed correlation measurements which are more reliable for power spectrum estimation. To utilize the ensemble average approach, 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 [1991] 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 had good performance on a narrow band portion of the spectrum by the choice of a relatively long window size.

Aharon Berkovitz and Rusnak [1991] have suggested 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 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 [1992] have suggested an approach to improve the frequency resolution of parametric power spectrum estimation for signals with rational power spectra. 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 [1993] 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 obtained. This method is implemented using the Fast Fourier transform.

Jun Yin and Zhaoda Zhu [1993] 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 of Gaussian white noise.

E. Turkbeyler and A. G. Constantinides [1993] 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.

Ling Chen and Su-Shing Chen [1993] have 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 [1994] has proposed a nonparametric approach based on wavelet representation for the logarithm of the unknown power spectrum. This approach offers the ability to capture statistically significantly components of logarithmic power spectrum at different resolution levels and guarantees the nonnegative spectrum estimator. The spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive white Gaussian noise. In this algorithm he has shown that the wavelet coefficients of the additive noise may be treated as independent random variables. The thresholds are using a saddle point approximation to the distribution of the noise coefficients. 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.

Peter T. Gough [1994] has proposed a fast spectral estimation algorithm for spectral estimation based on the Fast Fourier Transform. The proposed algorithm is iterative 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 [1995] have suggested a quadratic power spectrum estimation based on implementation of orthogonal frequency division multiple windows. The windows are constructed from a single window shifted in frequency. The orthogonal constraint ensures that the estimator meets a minimum mean squared error performance criterion. The solutions for the windows are obtained numerically. Quadratic spectral estimators are nonparametric and are quadratic functions of the data being analyzed. This estimation has good frequency resolution, and its statistical properties are comparable to the best multiple window estimators.

Donald B. Percival and Emma .J. McCoy [1998] have suggested a method of spectrum estimation based on wavelet thresholding of multitapers. They explained that all methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor estimate when the true spectrum has a high dynamic range of variations. In addition, because the distribution of the log periodogram is markedly non Gaussian, special wavelet dependent thresholding schemes are needed. These difficulties are bypassed by starting with a multitaper spectrum estimator. The logarithmic of this estimator is close to Gaussian distribution if moderate numbers of tapers are used. In contrast to log periodogram the log periodogram, log multitaper estimates are not approximately pair wise uncorrelated at the Fourier frequencies, but the form of the correlation can be accurately approximated. For scale independent thresholding, the correlation acts in accordance with the wavelet shrinkage paradigm to suppress small scale noise spikes while leaving informative coarse scale coefficients relatively un attenuated. In this algorithm the progression of the variance of wavelet coefficients with scale can be accurately calculated and thus by allowing the use of scale dependent thresholds. For Large number of observation of data samples this estimator converges in some sense to true power spectrum.

T. Umemoto and T. Yoshida [1998] have 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 was a scalar value, so that the constant Q-value of higher frequency become larger. In this algorithm the time constant of coefficient adjustment and the resolution in frequency are inverse 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 wave generated by musical instruments.

S.V. Narasimhan and M. N. S. Swamy [1999] have 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, but 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. Depending on the signal scenario, its variance ranges from 70 to 2% of that periodogram.

Norikazu. I, Hiroshi. M [1999] 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 and Jerome [1999] have 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 a 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

2.5 Power spectrum estimation methods during the period 2000-2010

Ch. Rebai and D. Dallet [2002] have 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. Indeed, 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 closed to the calculated with coherent sampling by FFT.

Alberto Cristan and Andrew T. Walden [2002] have presented a multitaper power spectrum estimation and thresholding using the Discrete Wavelet Transforms (DWT). This algorithm consist of computing the logarithm of multitaper spectrum estimator, applying an orthonormal

transform derived from a wavelet packet tree to the log multitaper 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 properties. 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. 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. and Robert [2004] have 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.

Jing Deng and Wen-Hu Wu [2005] have proposed predictive differential power spectrum based cepstral coefficients and sub band mel-spectrum centroid based cepstral coefficients for robust speaker recognition in statioanary 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.

Jesper Jensen and Ivo Batina [2006] have proposed a method for estimating the non stationary noise power spectral density (PSD) given a noisy speech signal. The method is based on an autoregressive (AR) model of the speech PSD dynamics combined with a Kalman filtering based noise PSD estimation technique. The proposed noise PSD tracking method 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 Jian Ali [2007] have considered the two dimensional nonparametric complex spectral estimation of data matrices with missing samples occurring in arbitrary patterns. Previously, missing amplitude and phase estimation expectation maximization algorithms are developed for the general 1-D missing data problem and shown to have excellent spectral estimation performance. This algorithm is based on MAPES-CM, which solves a maximum likelihood problem iteratively using cyclic maximization(CM). This spectral estimation is suitable for longer data sequences and 2-D applications such as synthetic aperture radar (SAR) imaging.

Petre Stoica and Xing Tan [2008] 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 [2009] 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 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 and Yang Chunling [2009] have 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 be detected 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 is detected.

Kaushik Mahata and Damian Marelli [2009] 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 reslts cuasing 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 a band limited. In contrast to traditional 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 model order some what higher than that returned by the Akaike's information criterion.

Zishu He and Ting Cheng [2009] 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 of continuous spectrum. These different patterns can be shown synchronously or selected by a priori knowledge.

Lu zhu and Chuanyun Wang [2009] have 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 than windowing methods is explained.

Petre Stoica and Yubo Cheng [2009] have proposed a method based on a weighted least squares iterative adaptive approach (IAA). The referred method is missing data IAA(MIAA) and it can be used for uniform and non uniform sampling as well as for arbitrary data missing patterns. 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 computational cost.

M.Sreelatha and T.anilkumar [2009] have 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 sing the multichannel AR modeling. The resulting signals from one subband are then processed using maximum entropy inference engine. This method makes use of the a priori 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 [2009] have presented a non parametric spectral analysis with missing data using expectation maximization (EM) algorithm. This algorithm explains the nonparametric complex spectral estimation for a data sequence with missing samples occurring in arbitrary patterns. 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.

Statement of the problem:

Though the non parametric spectral estimation has good dynamic performance, it has 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 alternative 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 on nonuniform and uneven data sequences using Lomb Transforms and resampling, linear interpolation and cubic interpolation methods. 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 show the 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 show the 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.

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 data sequences using nonlinear overlapping of samples.

Power spectrum estimation of data sequences in wide dynamic range using pre whitening and post coloring technique.

Power spectrum estimation of weakly stationary Gaussian data sequences using averaged weighted least squares algorithm.

Power spectrum estimation of uneven data sequences using resampling methods like spline and cubicspline interpolation techniques.

These proposed estimates are compared to the true values of power spectral density of data sequences and hence observed that these estimates consistent estimates for longer data sequences. An estimate is said to be consistent, if its bias is zero and the variance for large number of observation of data samples approaches to zero, skewness is 3 and the good kurtosis values.