Convolution Theorem for the DTFT

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

$$(x \ast y)(n) \isdef \sum_{m=0}^{N-1}x(m)y(n-m).$$

This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length $N$ sequences in the context of the DFT [232]. 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.^3.4.

The convolution theorem is then

$$\zbox {(x\ast y) \leftrightarrow X \cdot Y}$$

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:

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