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174 Chapter 5 THE DISCRETE FOURIER TRANSFORM
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
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);
In the second approach, the property (5.37) can be used in the frequency
domain. This is explored in Problem P5.20.
□ EXAMPLE 5.12 Given an 11-point sequence x(n) =10(0.8) n , 0 ≤ n ≤ 10, determine and plot
x ((n − 6)) 15
.
Solution
MATLAB script:
>> n = 0:10; x = 10*(0.8) .^ n; y = cirshftt(x,6,15);
>> n = 0:14; x = [x, zeros(1,4)];
>> subplot(2,1,1); stem(n,x); title(’Original sequence’)
>> xlabel(’n’); ylabel(’x(n)’);
>> subplot(2,1,2); stem(n,y);
>> title(’Circularly shifted sequence, N=15’)
>> xlabel(’n’); ylabel(’x((n-6) mod 15)’);
The results are shown in Figure 5.18.
□
6. Circular shift in the frequency domain: This property is a dual
of the preceding property given by
DFT [ W −ln
N x(n)] = X ((k − l)) N
R N (k) (5.38)
7. Circular convolution: A linear convolution between two N-point
sequences will result in a longer sequence. Once again we have to
restrict our interval to 0 ≤ n ≤ N − 1. Therefore instead of linear
shift, we should consider the circular shift. A convolution operation
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