### 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