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Author: Sarat-Raj

Circular Convolution using DFT and IDFT method.m

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+
+% V8_3
+%Program to find Circular Convolution of two sequences
+% using DFT and IDFT method
+
+%%% Developed by Dr. M. Venu Gopala Rao, 
+%%% Email: [email protected],  [email protected],
+
+% -----------------------------------------------------------------------
+
+
+clear all; close all; clc;
+
+% x = input('Enter first sequence');
+% h = input('Enter second sequence');
+
+x = [1 2 2]; h = [1 2 3 4];
+
+if (length(h)<length(x));
+    c=length(x)-length(h);
+    h=[h,zeros(1,c)]
+elseif (length(x)<length(h));
+    c=length(h)-length(x);
+    x=[x,zeros(1,c)]
+else
+    disp('lengths are equal');
+end
+
+N = max(length(x),length(h));
+
+n = 0:N-1;
+X = fft(x);
+H = fft(h);
+Y = X.*H;
+
+y = ifft(Y);    
+
+k = 0 : N-1;
+
+% -------------------------------------------------------------------
+figure();
+subplot(3,2,1); stem(0:length(x)-1,x,'r','fill','LineWidth',1.5);
+xlabel('Time index ---->'); ylabel('Magnitude---->');
+title('Sequence x[n]'); axis([-1 4 0  2.2]);
+
+subplot(3,2,3); stem(0:length(h)-1,h,'b','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Sequence h[n]')
+axis([-1 4 0 4.2]);
+
+subplot(3,2,5); stem(n,y,'m','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Circularly convolved sequence y[n](DFT and IDFT)')
+axis([-1 4 0 max(y)+1]);
+
+% figure();
+subplot(3,2,2); stem(0:length(X)-1,abs(X),'m','fill','LineWidth',1.5);
+xlabel('k---->'); ylabel('Magnitude---->');
+title('Spectrum of Sequence x[n], X[k}'); axis([-1 4 -0.2 5.4]);
+
+subplot(3,2,4); stem(0:length(H)-1,abs(H),'k','fill','LineWidth',1.5);
+xlabel('k---->'); ylabel('Magnitude---->');
+title('Spectrum of Sequence h[n], H[k}')
+axis([-1 4 -0.2 11]);
+
+subplot(3,2,6); stem(k,abs(Y),'b','fill','LineWidth',1.5);
+xlabel('k---->'); ylabel('Magnitude---->');
+title('Spectrum of Circularly convoled sequencey[n], Y[k}')
+axis([-1 4 -0.2 51]);
+

Computing Circular Convolution .m

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+
+% V8_2  Computing Circular Convolution
+
+%%% Developed by Dr. M. Venu Gopala Rao, 
+%%% Email: [email protected],  [email protected],
+
+% -----------------------------------------------------------------------
+
+clear all; close all; clc;
+
+
+x = [1 2 2]; h = [1 2 3 4];
+N = max(length(x),length(h));
+
+y = cconv(x,h,N);
+n = 0:N-1;
+
+figure();
+subplot(3,1,1); stem(0:length(x)-1,x,'r','fill','LineWidth',1.5);
+xlabel('Time index ---->'); ylabel('Magnitude---->');
+title('Sequence x[n]'); axis([-1 4 0  2.2]);
+
+subplot(3,1,2); stem(0:length(h)-1,h,'b','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Sequence h[n]')
+axis([-1 4 0 4.2]);
+
+subplot(3,1,3); stem(n,y,'m','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Circularly convolved sequence,y[n]')
+axis([-1 4 0 max(y)+1]);

Computing Circular Correlation.m

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+
+% V8_4  Computing Circular Correlation
+
+%%% Developed by Dr. M. Venu Gopala Rao, 
+%%% Email: [email protected],  [email protected],
+
+% -----------------------------------------------------------------------
+
+clear all; close all; clc;
+
+x = [1 2 2 1]+1i; h = [1 3 4 1]-2*1i;
+N = max(length(x),length(h));
+n = 0:N-1;
+
+y = cconv(x,conj(fliplr(h)),7);	% Compute Correlation using cconv;
+z = xcorr(x,h);				% Compute using xcorr
+
+M = length(x)+length(h)-1;
+m = 0:M-1;
+
+figure();
+subplot(4,1,1); stem(0:length(x)-1,x,'r','fill','LineWidth',1.5);
+xlabel('Time index ---->'); ylabel('Magnitude---->');
+title('Sequence x[n]'); axis([-1 8 0  2.2]);
+
+subplot(4,1,2); stem(0:length(h)-1,h,'b','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Sequence h[n]'); axis([-1 8 0 4.2]);
+
+subplot(4,1,3); stem(y,'m','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Circularly Correlated sequence using cconv.m');axis([-1 8 0 max(abs(y))+3]);
+
+subplot(4,1,4); stem(z,'k','fill','LineWidth',1.5);
+xlabel('Time index---->'); ylabel('Magnitude---->');
+title('Circularly Correlated sequence using xcorr.m')
+axis([-1 8 0 max(abs(z))+3]);
+
+norm(y-z)

Computing circular shift of a sequence.m

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+
+% V8_1  Computing circular shift of a sequence
+
+%%% Developed by Dr. M. Venu Gopala Rao, 
+%%% Email: [email protected],  [email protected],
+
+% -----------------------------------------------------------------------
+clear all; close all; clc;
+
+N = 10;
+n = 0:7; x = 10*(0.8).^n; 
+n = 0:N-1; x = [x, zeros(1,2)];
+y = cirshftt(x,3,N);
+z = cirshftt(x,-4,N);
+
+figure();
+subplot(3,1,1); stem(n,x,'r','fill','LineWidth',1.5);
+xlabel('n---->'); ylabel('Magnitude---->');
+title('Original sequence'); axis([-1 10 -0.2 11]);
+
+subplot(3,1,2); stem(n,y,'b','fill','LineWidth',1.5);
+xlabel('n---->'); ylabel('Magnitude---->');
+title('Circularly shifted sequence,x((n-3) mod N),N=10')
+axis([-1 10 -0.2 11]);
+
+subplot(3,1,3); stem(n,z,'b','fill','LineWidth',1.5);
+xlabel('n---->'); ylabel('Magnitude---->');
+title('Circularly shifted sequence,x((n+4) mod N),N=10')
+axis([-1 10 -0.2 11]);

circonvt.m

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+
+
+function y = circonvt(x1,x2,N)
+
+% N-point circular convolution between x1 and x2: (time-domain)
+
+% -------------------------------------------------------------
+
+% [y] = circonvt(x1,x2,N)
+%  y = output sequence containing the circular convolution
+% x1 = input sequence of length N1 <= N
+% x2 = input sequence of length N2 <= N
+%  N = size of circular buffer
+%  Method: y(n) = sum (x1(m)*x2((n-m) mod N))
+
+% Check for length of x1
+
+if length(x1) > N
+        error('N must be >= the length of x1')
+end
+
+% Check for length of x2
+
+if length(x2) > N
+        error('N must be >= the length of x2')
+end
+
+x1=[x1 zeros(1,N-length(x1))]
+x2=[x2 zeros(1,N-length(x2))]
+
+m = [0:1:N-1];
+x2 = x2(mod(-m,N)+1);
+H = zeros(N,N);
+
+for n = 1:1:N
+	H(n,:) = cirshftt(x2,n-1,N);
+end
+
+y = x1*H';
+

cirshftt.m

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+
+
+function y = cirshftt(x,m,N)
+% Circular shift of m samples wrt size N in sequence x: (time domain)
+% -------------------------------------------------------------------
+% [y] = cirshftt(x,m,N)
+% y = output sequence containing the circular shift
+% x = input sequence of length <= N
+% m = sample shift
+% N = size of circular buffer
+% Method: y(n) = x((n-m) mod N)
+% Check for length of x
+if length(x) > N
+error('N must be >= the length of x')
+end
+x = [x zeros(1,N-length(x))];
+n = [0:1:N-1]; n = mod(n-m,N); y = x(n+1);