Convolution Theorem for the DTFT
The convolution of discrete-time signals

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(3.22) |
This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length

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(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