### 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:

Next Section:
Correlation Theorem for the DTFT
Previous Section:
Shift Theorem for the DTFT