Periodogram and Welch Method Revisited

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  • Topic: Digital signal processing, Signal processing, Discrete Fourier transform
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SPECTRAL ANALYSIS OF NONPARAMETRIC METHODS & IMPROVED WELCH METHOD VIA CIRCULAR OVERLAP Svs Gopala Krishna, S.Mujassim, Prof.D.Ramachandra Reddy School of Electronics Engineering, VIT University, Vellore- 632014, Tamilnadu, India. gopal.svsemails@gmail.com; mujassim@rediffmail.com

Abstract The objective of this paper is to overcome the drawback in the Welch method. The major problem with the Welch method is that the variance is not monotonically decreasing with respect to the amount of fraction of overlap. It is difficult to choose the fraction of overlap which gives optimum variance value as its dependent on the type of the window used. By replacing the regular overlap mechanism of the Welch method with a circular overlap we can achieve the monotonic decrease in variance as the fraction of overlap increases. This paper is analyzed in two fold manner. The first part gives preface to all non-parametric methods and comparison of all the methods in terms of two statistical parameters viz. variance and resolution. The second part of this literature deals with the modification of the standard Welch method to get a monotonic decrease in variance. Index Terms— Non-Parametric, Periodogram, Variance, Resolution, Windowing, Power Spectrum, Bartlett method, Welch method. I. INTRODUCTION SPECTRUM ESTIMATION is an important application of the Digital Signal Processing. Spectral analysis is used in many applications such as to get the target location and its velocity information in the radar applications [9]. In general many practical applications such as Ocean noise, Wind speed give a time series data [10]. This data can be analyzed using spectral analysis. Spectrum estimation is a problem that involves estimating the power spectrum of the signal from a finite number of noisy measurements of the signal. The techniques adopted for the analysis of power spectrum estimation (PSE) are classified into two classes: 1.Parametric and 2.Non-parametric methods. In our literature, we study the methods to carry out the spectrum estimation by the classical methods, viz. non-parametric estimation. The non-parametric estimation of power spectrum is carried out by periodogram method, modified periodogram method, Bartlett method, Welch method and Blackman-tukey method. All these methods can be attained by slightly modifying the properties of the classical periodogram. II. NON PARAMETRIC ESTIMATION Non- Parametric Spectrum estimation does not assume any data generating process or model [11]. The basic idea behind non-parametric spectrum estimation is to estimate the autocorrelation of the finite data and then apply Discrete Fourier Transform (DFT) to this data. This makes the estimation of the power spectrum of the finite data random process. Spectrum estimation will be true if the estimated autocorrelation is unbiased. But in practical applications we take the biased autocorrelation estimate of finite length signal of length N. The periodogram introduced by the Schuster is defined as [11]:  (

where

Y{

{{ is the autocorrelation function given as

J { {



J{ { {J{

("

#

{J - { {J{ {J{ { {

{J{

The signal x(n) is truncated to a finite amount of data by using a Window function wR(n)

Thus the resulted signal, xN(n) is thus truncated to a finite number of values. The autocorrelation of the truncated signal xN(n) is given as [10]:

J { {

(

(

{J - {

{ {

{. {
$

The Fourier transform of this biased autocorrelation gives the estimated power spectrum which is nothing but the periodogram defined as follows: 

The variance of the classical periodogram of an ergodic weakly stationary signal [12] x(n) for n=0:N-1 is asymptotically proportional to the square of the true power of the signal. The major drawback with the periodogram is that it’s a not a consistent estimate of the power spectrum i.e., the variance does not approach to zero for a very large amount of signal information. The classical periodogram...
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