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cicrular_convolution_examples

cicrular_convolution_examples - EE406 Discrete-Time Signal...

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EE406 Discrete-Time Signal Processing Fast Fourier Transform (FFT) Algorithms ZA-041 Page - 8 Linear Convolution vs Circular Convolution in the DFT The linear convolution of two sequences ] [ 1 n x of 1 N -point and ] [ 2 n x of 2 N -point is given by ] [ ] [ ] [ ] [ ] [ ] [ ] [ 2 1 0 1 2 1 2 1 3 1 k n x k x k n x k x n x n x n x N k k = = = = −∞ = ] [ 3 n x is a ( ) 1 2 1 + N N -point sequence. If we choose N =max( 2 1 , N N ) and compute a N - point circular convolution ] [ ] [ 2 1 n x n x , then we obtain a N -point sequence, which is obviously different from ] [ 3 n x . If we choose N = ( ) 1 2 1 + N N and perform a ( ) 1 2 1 + N N -point circular convolution, the result becomes ( ) ] [ ] [ ] [ ] [ ] [ ] [ 2 1 0 1 2 1 4 n R m n x m x n x n x n x N N N m = = = ] [ ] [ ] [ 1 0 2 1 n R rN m n x m x N N m r = = −∞ = ] [ ] [ ] [ 1 0 2 1 n R rN m n x m x N r N m = ∑∑ −∞ = = ] [ ] [ 3 n R rN n x N r = −∞ = Thus the circular convolution is an aliased version of the linear convolution. Since ] [ 3 n x is a N = ( ) 1 2 1 + N N -point sequence, we have 1 0 [n] ] [ 3 4 = N n x n x which means that there is no aliasing in the time domain. If we make both ] [ 1 n x and ] [ 2 n x a N = ( ) 1 2 1 + N N -point sequence by padding an appropriate number of zeros, then the circular convolution is identical to the linear convolution. In order to use the DFT for linear convolution, we must choose N properly. However, when N is large, there is an immense requirement on memory. If N is chosen to be less than the required value, an error will be introduced.
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