### Dual of the Convolution Theorem

The*dual*

^{7.18}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 [70] for further discussion.

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Correlation Theorem

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Convolution Theorem