Dual of the Convolution Theorem

The *dual*^{7.19} of the convolution
theorem says that
*multiplication in the time domain is
convolution in the frequency domain*:

**Theorem: **

*Proof: *The steps are the same as in the convolution theorem.

This theorem also bears on the use of *FFT windows*. It implies
that
*windowing in the time domain* corresponds to
*smoothing in the frequency domain*.
That is, the spectrum of
is simply
filtered by
, or,
. This
smoothing reduces *sidelobes* associated with the
*rectangular window*, which is the window one is using implicitly
when a data frame
is considered *time limited* and therefore
eligible for ``windowing'' (and zero-padding). See Chapter 8 and
Book IV [73] for further discussion.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University