Functions of A Discrete Variable... Sequences
By similar arguments, it can be shown that the discrete convolution of sequences and is given by:
where DTFT represents the discrete-time Fourier transform.
An important special case is the circular convolution of and defined by where is a periodic summation:
It can then be shown that:
where DFT represents the discrete Fourier transform.
The proof follows from DTFT#Periodic_data, which indicates that can be written as:
The product with is thereby reduced to a discrete-frequency function:
- (also using Sampling the DTFT).
The inverse DTFT is:
QED.
Read more about this topic: Convolution Theorem
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