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’);

Z TRANSFORM

 For discrete time linear time-invariant system with impulse response h[n], the response y[n] of the system to a complex exponential input of the form z^-1 is

Y[n] =H[z].zⁿ

Where H[z] =  zⁿ                  ——-2

For z=e^jw with  real (i.e., with |z| =1), the sum in above equation corresponds to the discrete-time Fourier transform of h[n].

Generally, when |z| is not restricted to unity, then summation in equation 2 is referred as z transform of h[n].

Z transform for the discrete time signal x[n] is determined as

X[z] =  zⁿ          

Where, z= a complex variable.

Program:

clc;

clear all;

close all;

%enter the numerator and denominators coefficients in square brackets

num=input(‘enter numerator co-efficients’);

den=input(‘enter denominator co-efficients’);

% find poles and zeros

poles=roots(den)

zeros=roots(num)

% find transfer function H(s)

h=tf(num,den);

% plot the pole-zero map in s-plane

sgrid;

pzmap(h);

grid on;

title(‘locating poles and zeros on s-plane’);

%plot the pole zero map in z-plane

figure

zplane(poles,zeros);

grid on;

title(‘locating poles and zeros on z-plane’);

Output:

Enter numerator co-efficients [2 -12 4 5]

Enter denominator co-efficients [3 4 5 6]

Poles =

  -1.2653 + 0.0000i

  -0.0340 + 1.2568i

  -0.0340 – 1.2568i

Zeros =

    5.5594

    0.9262

   -0.4855

• GAUSSIAN NOISE:

Gaussian noise will be defined as statistical noise which features a probability density function of the traditional distribution (also referred to as Gaussian distribution). In other words, the valuestha the noise can combat are Gaussian-distributed. It is most ordinarily used as additive noise to yield additive white Gaussian noise (AWGN). Gaussian noise is correctly defined because of the noise with a Gaussian amplitude distribution.

Program:

clc;

clear all;

close all;

%let’s generates the given set of 2000 samples for Gaussian distributed random numbers

x=randn(1,2000);

%to plot the joint distribution for both the sets using dot.

subplot(211)

plot(x,’.’);

title(‘scatter plot of gaussian distributed random numbers’);

ymu=mean(x);

ymsq=sum(x.^2)/length(x);

ysigma=std(x);

yvar=var(x);

yskew=skewness(x);

p=normpdf(x,ymu,ysigma);

subplot(212);

stem(x,p);

title(‘ gaussian distribution’);