Convolution Theorem for the DTFT

The convolution of discrete-time signals $ x$ and $ y$ is defined as

$\displaystyle (x \ast y)(n) \isdef \sum_{m=-\infty}^\infty x(m)y(n-m).

This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length $ N$ sequences in the context of the DFT [248]. 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

$\displaystyle \zbox {(x\ast y) \;\longleftrightarrow\;X \cdot Y}

That is, convolution in the time domain corresponds to multiplication in the frequency domain.

Proof: The result follows immediately from interchanging the order of summations associated with the convolution and DTFT:

\hbox{\sc DTFT}_\omega(x\ast y) &\isdef & \sum_{n=-\infty}^{\i...\mbox{(by the shift theorem)}\\
&\isdef & X(\omega)Y(\omega)

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Correlation Theorem for the DTFT
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Shift Theorem for the DTFT