%DFT
close all;
clear all;
N=input('Howmany point DFT do you want?');
x2=input('Enter the sequence=');
n2=length(x2);
c= zeros(N);
x2=[x2 zeros(1,N-n2)];
for k=1:N
for n=1:N
w=exp((-2*pi*i*(k-1)*(n-1))/N);
%prev.step=>evaluating w-matrix
x(n)=w;
end
c(k,:)=x;
end
r=[c]*[x2']
%plotting magnitude and angle
subplot(211)
stem(abs(r));
title('DFT-absolute value');
subplot(212)
stem(angle(r));
title('DFT-angle'); OUTPUT WAVEFORM 
%DFT linearity property
close all;
clear all;
N=input('Howmany point DFT do you want?');
x1=input('Enter the first sequence=');
x2=input('Enter the second sequence=');
a=input('Enter the first constant a=');
b=input('Enter the second constant b=');
n1=length(x1);
n2=length(x2);
c=zeros(N);
x1=[x1 zeros(1,N-n1)];%make both x1 and x2
x2=[x2 zeros(1,N-n2)];%of same dimension
x3=a*x1
x4=b*x2
x5=x3+x4 %a*x1+b*x2
for k=1:N
for n=1:N
w=exp((-2*pi*i*(k-1)*(n-1))/N);
%prev.step=>evaluating w-matrix
x(n)=w;
end
c(k,:)=x; %Every row of w-matrix is inserted &
end %finally c becomes the w matrix
%for n-point DFT
r1=c*x1'; %x1(k)
r2=c*x2'; %x2(k)
R3=a*r1+b*r2 %a*x1(k)+b*x2(k)
R4=c*x5' %DFT of a*x1(n)+b*x2(n)
%plotting magnitude and angle
subplot(211)
stem(abs(R4));
title('DFT of {a*x1(n)+b*x2(n)}');
subplot(212)
stem(angle(R3));
title('DFT of {a*x1(k)+b*x2(k)}');
OUTPUT WAVEFORM
%multiplication property x1(n)*x2(n)=(1/N)*circonv(X1(k)*X2(k))
clear all;
x1=input('enter the sequence=');
x2=input('enter the second seq of same length=');
N9=input ('enter the number of samples=');
x3=(x1).*(x2); %multiplication of 2 signals
x4=fft(x1,N9);%dft of first seq
N1=length(x4);%length of sequence
x5=fft(x2,N9);%dft of secnd sequence
N2=length(x5);%length of second sequence
%finding circonvolution of 2 signals
x=x4;
h=x5;
N1=length(x);
N2=length(h);
N=max(N1,N2);
x=[x zeros(1,N-N1)];
h=[h zeros(1,N-N2)];
for n=0:N-1
y(n+1)=0;
for i=0:N-1
j=mod(n-i,N);
y(n+1)=y(n+1)+x(i+1)*h(j+1);
end
end
y=y/N; %rhs
x6=fft(x3,N); %lhs
n=0:1:N-1;
subplot(121);
stem(n,x6);
title('dft of 2 multiplied signals');
grid on;
subplot(122);
stem(n,y);
title('Nth part of circular convolution of 2 dfts');
grid on; OUTPUT WAVEFORM 
%dft frequecy shift property
close all;
clear all;
N=input('how many point dft do you want?');
x1=input('enter the seq');
n2=length(x1);
c=zeros(N);
x1=[x1 zeros(1,N-n2)];
for k=1:N
for n=1:N
w=exp((-2*pi*i*(k-1)*(n-1))/N);
x(n)=w;
end
c(k,:)=x;
end
disp('dft is ');
r=c*x1';
a1=input('enter the amount of shift in frequency domain');
for n=1:N
w=exp((2*pi*i*(n-1)*(a1))/N);
x2(n)=w;
end
r1=x2.*x1;
subplot(221);
stem(abs(r));
grid on;
title('orginal dft magnitude plot');
subplot(222);
stem(angle(r));
grid on;
title('orginal dft angle');
for k=1:N
for n=1:N
w=exp((-2*pi*i*(k-1)*(n-1))/N);
x(n)=w;
end
c(k,:)=x;
end
disp('dft is');
r2=c*r1';
subplot(223);
stem(abs(r2));
grid on;
title('shifted dft magnitude');
subplot(224);
stem(angle(r2));
grid on;
title('shifed dft angle');OUTPUT WAVEFORM
clear all; x= input ('enter the first sequence'); h= input ('enter the second sequence'); No=input('number of samples?'); N1= length(x); N2= length(h); N=max(N1,N2);%length of sequence x=[x zeros(1,N-N1)]; %modified first sequence h=[h zeros(1,N-N2)]; %modified second sequence for n=0:N-1; y(n+1)=0; for i=0:N-1 j=mod(n-i,N); y(n+1)=y(n+1)+x(i+1)*h(j+1); %shifting and adding end end n=0:1:No-1; x4=fft(x); x5=fft(h); x6=fft(y); %seq 2 subplot(121) stem(n,x6); title('dft of two convoluted signals'); x7=(x4).*(x5);%product of two dfts subplot(122); stem(n,x7); title('product of two dfts'); grid on;
OUTPUT WAVEFORM
%exponential wave
t=0:0.1:10;
s=0.1+1i;%complex number
y=exp(s*t);%expression for complex exponential
stem3(real(y),imag(y),t,'r');
%to show the 3d graph
%'r'indicates the colour-red
view(-39.5,42);%azimuth and elevation
xlabel('real');
ylabel('imag');
zlabel('amplitude');
title('complex exponential');
rotate3d;%to rotate in 3-dimensions
legend('complex exponential');OUTPUT WAVEFORM 
function y=circular_convolution(x,h)
x= input ('enter the first sequence');
h= input ('enter the second sequence');
N1= length(x);
N2= length(h);
N=max(N1,N2);%length of sequence
x=[x zeros(1,N-N1)]; %modified first sequence
h=[h zeros(1,N-N2)]; %modified second sequence
for n=0:N-1;
y(n+1)=0;
for i=0:N-1
j=mod(n-i,N);
y(n+1)=y(n+1)+x(i+1)*h(j+1); %shifting and adding
end
end
%to display and plot circular convolution
n=1:N;
disp('output sequence of circular convolution');
disp(y);%to view output in command window
pause;
stem(n,y);%plotting circular convolution
grid minor;
xlabel('time index');
ylabel('amplitude');
title('circular convolution sequence of x and h');OUTPUT WAVEFORM
clear all;
close all;
x= input ('enter the first sequence');
h= input ('enter the second sequence');
N1= length(x);
N2= length(h);
x1=x;
h1=h;
x=[x zeros(1,N2-1)]; %modified first sequence
h=[h zeros(1,N1-1)]; %modified second sequence
for k=1:N1+N2-1;
s=0;
for j=1:k
p(j)=h(j);
end
u=eye(k);
v=flipud(u);
m=(p)*(v);
for i=1:k
s=s+x(i)*m(i);
end;
d(k)=s;
end
disp(d);
subplot(211);
stem(d);
xlabel('time index');
ylabel('amplitude');
title('linear convolution');
%circular convolution
N=max(N1,N2);
x1=[x1 zeros(1,N-N1)];
h1=[h1 zeros(1,N-N2)];
z=x1';
for i=1:N-1
k=x1(N);
for i=N-1:-1:1;
x1(i+1)=x1(i);
end
x1(1)=k;
z=[z x1'];
end
m=z*h1';
disp(m);
subplot(212);
stem(m);
title('circular convolution');
xlabel('time');
ylabel('amplitude'); OUTPUT WAVEFORM
%chebyshev HPF
%for specification
clear all;
alphap=input('pass band attenuation?=');
alphas=input('stop band attenuation in db?=');
wp=input('pass band frequency in rad?=');
ws=input('stop band frequency in rad?=');
%to find cuttoff freq and order of filter
[n,wn]=cheb1ord(wp/pi,ws/pi,alphap,alphas);
%system function of filter above expression
[b,a]=cheby1(n,alphap,wn,'high');
w=0:.01:pi;
[h,ph]=freqz(b,a,w);
m=abs(h);
an=angle(h);
subplot(221);
%plot the graph
plot(ph/pi,m);
grid;
ylabel('gain in db');
xlabel('normalisd frequency');
subplot(212);
plot(ph/pi,an);
grid;
ylabel('phase in rad');
xlabel('normalised frequency'); OUTPUT WAVEFORM 
%chebyshev LPF
% specification
clear all;
alphap=input('pass band attenuation?=');
alphas=input('stop band attenuation in db?=');
wp=input('pass band frq in rad=');
ws=input('stop band freq in rad=');
% find cut-off freq and order of filter
[n,wn]=cheb1ord(wp/pi,ws/pi,alphap,alphas);
%system function above
[b,a]=cheby1(n,alphap,wn);
w=0:.01:pi;
[h,ph]=freqz(b,a,w);m=20*log(abs(h));
an=angle(h);
subplot(221);
plot(ph/pi,m);
grid;
ylabel('gain in db');
xlabel('nor fre');
subplot(212);
plot(ph/pi,an);
grid;
ylabel('phase in rad');
xlabel('norm freq'); OUTPUT WAVEFORM 
x1=input('Enter first seq='); x2=input('Enter second seq='); n1=length(x1); n2=length(x2); N=max(n1,n2); %padding zeros at the end of smaller sequence if n2>n1 x1=[x1 zeros(1,N-n1)]; x3=x2;x2=x1;x1=x3; elseif (n1>n2) x2=[x2 zeros(1,N-n2)]; end x3=x2; y=[]; %shifting and multiplying for n=1:N y(n)=0; for j=1:N y(n)=y(n)+x1(j)*x2(j); end x=x2(N); for i=N-1:-1:1%loop for shifting and adding zeros x2(i+1)=x2(i); end x2(1)=0; y(n)=y(n)/N end n=1:N; %plotting crosscorrelation output disp('output seq of cross corr='); disp(y); pause; stem(n,y); grid on; grid minor; xlabel('time index n'); ylabel('amplitude y'); title('cross correlation of sequence x1 and x2');
clear all;
x=input('enter the sequence x=');
N=length(x);
y=x;
z=x;
for i=1:N-1
for i=N-1:-1:1
x(i+1)=x(i);
end
x(1)=0;
z=[z; x];
end;
m=[z]*[y'];
m=m/N
stem(m);
title('auto-correlation');
xlabel('time index');
ylabel('amplitude');