**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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