Dual of the Convolution Theorem

The dual^7.13 of the convolution theorem says that multiplication in the time domain is convolution in the frequency domain:

Theorem:

$$\zbox {x\cdot y \;\longleftrightarrow\;\frac{1}{N} X\circledast Y}$$

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 $w\cdot x$ is simply $X$ filtered by $W$, or, $W\circledast X$. This smoothing reduces sidelobes associated with the rectangular window (which is the window one gets implicitly when no window is explicitly used). See Chapter 8 for further discussion.