Convolution

The convolution of two signals $x$ and $y$ in $\mathbb{C}^N$ may be denoted `` $x\circledast y$ '' and defined by

$$\zbox {(x\circledast y)_n \isdef \sum_{m=0}^{N-1}x(m) y(n-m)}$$

Note that this is circular convolution (or ``cyclic'' convolution).^7.4 The importance of convolution in linear systems theory is discussed in §8.3.

Cyclic convolution can be expressed in terms of previously defined operators as

$$y(n) \isdef (x\circledast h)_n \isdef \sum_{m=0}^{N-1}x(m)h(n-m) = \left<x,\hbox{\sc Shift}_n(\hbox{\sc Flip}(h))\right>$$   $$\mbox{($h$\ real)}$$

where $x,y\in\mathbb{C}^N$ and $h\in\mathbb{R}^N$ . This expression suggests graphical convolution, discussed below in §7.2.4.