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
![$ N$](http://www.dsprelated.com/josimages_new/sasp2/img61.png)
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:
![\begin{eqnarray*}
\hbox{\sc DTFT}_\omega(x\ast y) &\isdef & \sum_{n=-\infty}^{\infty}(x\ast y)_n e^{-j\omega n} \\
&\isdef & \sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}x(m) y(n-m) e^{-j\omega n} \\
&=& \sum_{m=-\infty}^{\infty}x(m) \sum_{n=-\infty}^{\infty}\underbrace{y(n-m) e^{-j\omega n}}_{e^{-j\omega m}Y(k)} \\
&=& \left(\sum_{m=-\infty}^{\infty}x(m) e^{-j\omega m}\right)Y(\omega)\quad\mbox{(by the shift theorem)}\\
&\isdef & X(\omega)Y(\omega)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img168.png)
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Correlation Theorem for the DTFT
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Shift Theorem for the DTFT