Convolution Theorem for the DTFT
The convolution of discrete-time signals and is defined as
(3.22) |
This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length sequences in the context of the DFT [264]. Convolution is cyclic in the time domain for the DFT and FS cases (i.e., whenever the time domain has a finite length), and acyclic for the DTFT and FT cases.3.6
The convolution theorem is then
(3.23) |
That is, convolution in the time domain corresponds to pointwise multiplication in the frequency domain.
Proof: The result follows immediately from interchanging the order
of summations associated with the convolution and DTFT:
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Correlation Theorem for the DTFT
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Shift Theorem for the DTFT