Theory:

Frequency analysis of discrete time signals is usually performed on digital signal processor, which may be general purpose digital computer or specially designed digital hardware. To perform frequency analysis on discrete time signal, we convert the time domain sequence to an equivalent frequency domain representation. We know that such representation is given by The Fourier transform X(e^jw) of the sequence x(n). However, X(e^jw) is a continuous function of frequency and therefore, It is not a computationally convenient representation of the sequence. DFT is a powerful computational tool for performing frequency analysis of discrete time signals. The N-point DFT of discrete time sequence x(n) is denoted by X(k) and is defined as.

x(n)= n)

Steps in performing operations:

• Get the input sequence.
• We can find the DFT for the input sequence using by direct equation of DFT.
• Find the IDFT using the direct equation.
• Let’s plot the DFT and IDFT for the given sequence by using matlab command stem.
• Display the above outputs.

Program:

clc;

close all;

clear all;

xn=input(‘Enter the sequence x(n)’);

ln=length(xn); %find the length of the sequence

xk=zeros(1,ln); %let’s initialize an array of the same size as that of input sequence

ixk=zeros(1,ln); let’s initialize an array of the same size as that of input sequence

%DFT of the sequence

for k=0:ln-1

 for n=0:ln-1

 xk(k+1)=xk(k+1)+(xn(n+1)*exp((i)*2*pi*k*n/ln));

 end

end

%Plotting input sequence

t=0:ln-1;

subplot(221);

stem(t,xn);

ylabel (‘Amplitude’);

xlabel (‘Time Index’);

title(‘Input Sequence’);

magnitude=abs(xk); % In order to find the magnitudes for the individual DFT points

% plot the magnitude response

t=0:ln-1;

subplot(222);

stem(t,magnitude);

ylabel (‘Amplitude’);

xlabel (‘K’);

title(‘Magnitude Response’);

phase=angle(xk); % In order to find the phases of individual DFT points % plot the magnitude

t=0:ln-1;

subplot(223);

stem(t,phase);

ylabel (‘Phase’);

xlabel (‘K’);

title (‘Phase Response’);

%IDFT of the sequence

for n=0:ln-1

 for k=0:ln-1

 ixk(n+1)=ixk(n+1)+(xk(k+1)*exp(i*2*pi*k*n/ln));

 end

end

ixk=ixk./ln;

%code block to plot the input sequence

t=0:ln-1;

subplot(224);

stem(t,ixk);

ylabel (‘Amplitude’);

xlabel (‘Time Index’);

title (‘IDFT sequence’);

Wiener-Khinchin:

The Wiener-Khinchin theorem is a fundamental theorem useful for the stochastic processes. The Wiener-Khinchin theorem provides a simple relationship between time and frequency representations of a fluctuating process. Its great practical importance comes from the fact that the autocorrelation function may be directly determined from the measured power spectral density, and vice versa. The wiener-Khinchin is widely used theorem to states the power spectral density of a wide sense stationary random process of the flourier transform for the corresponding autocorrelation function.

Program:

clc;

clear all;

close all;

t=0:0.1:2*pi;

%input signal

x=sin(2*t);

subplot(3,1,1);

plot(x);

xlabel(‘time’);

ylabel(‘amplitude’);

title(‘input signal’);

%autocorrelation of input signal

xu=xcorr(x,x);

%fft of autocorrelation signal

y=fft(xu);

subplot(3,1,2);

plot(abs(y));

xlabel(‘f’);

ylabel(‘amplitude’);

title(‘fft of autocorrelation of input signal’);

%fourier transform of input signal

y1=fft(x);

%finding the power spectral density

y2=(abs(y1)).^2;

subplot(3,1,3);

plot(y2);

xlabel(‘f’);

ylabel(‘magnitude’);

title(‘PSD of input signal’);

Extraction Of Periodic Signal Using Correlation:

A signal is masked by noise are often detected either by correlation techniques or by filtering. Actually, the two techniques are equivalent. The correlation technique may be a measure of extraction of a given signal within the time domain whereas filtering achieves precisely the same leads to frequency domain.

Program:

clear all;

close all;

clc;

t=0:0.1: pi*4;

%input signal1

s=sin(t);

subplot(7,1,1)

plot(s);

title(‘signal s’);

xlabel(‘t’);

ylabel(‘amplitude’);

%input signal2

c=cos(t);

subplot(7,1,2)

plot(c);

title(‘signal c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%generating noise signal

n = randn([1 126]);

%signal+noise

f=s+n;

subplot(7,1,3);

plot(f);

title(‘signal f=s+n’);

xlabel(‘t’);

ylabel(‘amplitude’);

%cross correlation of signal1&signal2

asc=xcorr(s,c);

subplot(7,1,4)

plot(asc);

title(‘ correlation of s and c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%cross correlation of noise&signal2

anc=xcorr(n,c);

subplot(7,1,5)

plot(anc);

title(‘ correlation of n and c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%cross correlation of f&signal2

cfc=xcorr(f,c);

subplot(7,1,6)

plot(cfc);

title(‘ correlation of f and c’);

xlabel(‘t’);

ylabel(‘amplitude’);

%extracting periodic signal

hh=asc+anc;

subplot(7,1,7)

plot(hh);

title(‘addition of sc+nc’);

xlabel(‘t’);

ylabel(‘amplitude’);

Sampling theorem:

Sampling theorem is used to convert analog signal into discrete time signal.it has applications in digital signal processing and digital communication. Sampling theorem is determined has that if the sampling rate in any pulse modulation system exceeds twice the maximum frequency of the message signal, with this the receiver will reconstructed the signal with minimum distortion.

Whereas,

 f_s is the sampling rate

f_m is the maximum frequency of message signal

Then sampling theorem can be expressed in mathematical form as

F_s≥ 2f_m

Well 2f_m is determined as Nyquist rate.

It the condition is

F_s≤2f_m

It is defined as aliasing effect it occurred when the sampling rate is less than twice the maximum frequency of the message signal.

F_s=sampling rate

Program:

clc;

clear all;

close all;

t=-10:.01:10;

T=4;

fm=1/T;

x=cos(2*pi*fm*t);

subplot(2,2,1);

plot(t,x);

xlabel(‘time’);

ylabel(‘x(t)’);

title(‘continous time signal’);

grid;

n1=-4:1:4;

fs1=1.6*fm;

fs2=2*fm;

fs3=8*fm;

x1=cos(2*pi*fm/fs1*n1);

subplot(2,2,2);

stem(n1,x1);

xlabel(‘time’);

ylabel(‘x(n)’);

title(‘discrete time signal with fs<2fm’);

hold on;

subplot(2,2,2);

plot(n1,x1);

grid;

n2=-5:1:5;

x2=cos(2*pi*fm/fs2*n2);

subplot(2,2,3);

stem(n2,x2);

xlabel(‘time’);

ylabel(‘x(n)’);

title(‘discrete time signal with fs=2fm’);

hold on;

subplot(2,2,3);

plot(n2,x2)

grid;

n3=-20:1:20;

x3=cos(2*pi*fm/fs3*n3);

subplot(2,2,4);

stem(n3,x3);

xlabel(‘time’);

ylabel(‘x(n)’);

title(‘discrete time signal with fs>2fm’)

hold on;

subplot(2,2,4);

plot(n3,x3)

grid;

Noise removal by using cross-correlation/autocorrelation

Detection of the periodic signal conceal by random noise is of great importance .The noise signal encountered in practice may be a signal with random amplitude variations. A signal is uncorrelated with any periodic signal. If s(t) is a periodic signal and n(t) is a noise signal then.

clc;

clear all;

close all;

t=0:0.2:pi*8;

%input signal

s=sin(t);

subplot(6,1,1);

plot(s);

title(‘signal s’);

xlabel(‘t’);

ylabel(‘amplitude’);

%generating noise

n = randn([1 126]);

%noisy signal

f=s+n;

subplot(6,1,2)

plot(f);

title(‘signal f=s+n’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of input signal

as=xcorr(s,s);

subplot(6,1,3);

plot(as);

title(‘auto correlation of s’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of noise signal

an=xcorr(n,n);

subplot(6,1,4)

plot(an);

title(‘auto correlation of n’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of transmitted signal

cff=xcorr(f,f);

subplot(6,1,5)

plot(cff);

title(‘auto correlation of f’);

xlabel(‘t’);

ylabel(‘amplitude’);

%autocorrelation of received signal

hh=as+an;

subplot(6,1,6)

plot(hh);

title(‘addition of as+an’);

xlabel(‘t’);

ylabel(‘amplitude’);