Filters and Convolution

A reason for the importance of convolution (defined in §7.2.3) is that every linear time-invariant system^8.2can be represented by a convolution. Thus, in the convolution equation

$$y = h\ast x \protect$$ (8.1)

we may interpret $x$ as the input signal to a filter, $y$ as the output signal, and $h$ as the digital filter, as shown in Fig. 8.12.

Figure: The filter interpretation of convolution.

The impulse or ``unit pulse'' signal is defined by

$$\delta(n) \isdef \left\{\begin{array}{ll} 1, & n=0 \\ [5pt] 0, & n\neq 0. \\ \end{array}\right.$$

For example, for $N=4$, $\delta = [1,0,0,0]$.

The impulse signal is the identity element under convolution, since

$$(x\ast \delta)_n \isdef \sum_{m=0}^{N-1}x(m) \delta(n-m) = x(n).$$

If we set $x=\delta$ in Eq. (8.1) above, we get

$$y = h\ast \delta = h.$$

Thus, $h$, which we introduced as the convolution representation of a filter, has been shown to be more specifically the impulse response of the filter.

It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response. No matter what the LTI system is, we can feed it an impulse, record what comes out, call it $h(n)$, and implement the system by convolving the input signal $x$ with the impulse response $h$. In other words, every LTI system has a convolution representation in terms of its impulse response.