ESTIMATION OF PERIODOGRAM

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.