ESTIMATION OF PERIODOGRAM
AIM
To estimate the power spectral density of a given signal using periodogram
in MATLAB.
THEORY
The power spectral density (PSD) of a WSS process is the Fourier transform of the autocorrelation sequence. Periodogram is a non-parametric method to estimate PSD
() = (k)
For an autocorrelation ergodic process and an unlimited amount of data, the autocorrelation sequence may be detemined by using the time average
(k) = (n+k)x*(n)
If x(n) is only measured over a finite interval, say n=1,2,…N-1, then the autocorrelation sequence must be estimated using with a finite sum
(r) = () (n+k)x*(n)
In order to ensure that the value of x(n) that is fully outside the interval [0,N-1] are excluded and written as follows
(k) = () (n+k)x*(n) k=0,1,2….,N-1.
Taking the discrete Fourier transform of rx^(k) leads to an estimation of the power spectrum known as the periodogram.
() = (k)
The periodogram
() = ()() = ()
Where XN(ejw) is the discrete time Fourirer transform of the N-point data sequence XN(n)
() = (n) =
ALGORITHM
STEP 1: Compute the value of x.
STEP 2: Perform periodogram function for x signal.
STEP 3: Using pwelch function, smoothen the output of periodogram signal.
STEP 4: Plot the graph for input and output signal
PROGRAM
##########################################################
clc;
clear all;
close all;
fs=1000;
t=0.1:1/fs:0.3;
x=cos(2*pi*t*200)+0.1*randn(size(t));
figure(1);
plot(x);
title('input signal');
xlabel('time');
ylabel('amplitude');
figure(2);
periodogram(x,[],'one sided',512,fs);
figure(3);
pwelch(x,30,10,[],fs,'one sided');
#############################################################
RESULT
Thus the MATLAB program to estimate the power spectral density of given signal using periodogram is executed and output is plotted.
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