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Unformatted text preview: Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 7 (week 3): Power Spectrum Estimation October 6, 2010 1 Introduction In the first and second weeks of this experiment, we introduced methods of statistically characterizing random processes. The sample autocorrelation and cross correlation are ex- amples of time domain characterizations of random signals. In many applications, it can also be useful to get the frequency domain characteristics of a random process. Examples include detection of sinusoidal signals in noise, speech recognition and coding, and range estimation in radar systems. In this week, we will introduce methods to estimate the power spectrum of a random signal given a finite number of observations. We will examine the effectiveness of the periodogram for spectrum estimation, and introduce the spectrogram to characterize a nonstationary random processes. Questions or comments concerning this laboratory should be directed to Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907; (765) 494- 0340; firstname.lastname@example.org Purdue University: ECE438 - Digital Signal Processing with Applications 2 2 Power Spectrum Estimation In this section, you will estimate the power spectrum of a stationary discrete random process. The power spectrum is defined as S xx ( ) = lim N E 1 N vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle N- 1 summationdisplay n =0 x ( n ) e- jn vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 2 (1) There are 4 steps for calculating a power spectrum: 1. Select a window of length N and generate a finite sequence x (0) ,x (1) ,...,x ( N- 1). 2. Calculate the DTFT of the windowed sequence x ( n ), ( n = 0 , 1 ,...,N- 1), square the magnitude of the DTFT and divide it by the length of the sequence. 3. Take the expectation with respect to x . 4. Let the length of the window go to infinity. In real applications, we can only approximate the power spectrum. Two methods are introduced in this section. They are the periodogram and the averaged periodogram ....
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