Impulse-Response Representation

In addition to difference-equation coefficients, any LTI filter may be represented in the time domain by its response to a specific signal called the impulse. This response is called, naturally enough, the impulse response of the filter. Any LTI filter can be implemented by convolving the input signal with the filter impulse response, as we will see.

Definition. The impulse signal is denoted $\delta (n)$ and defined by

$$\delta(n)\isdef \left\{ {1,\;n=0}\atop{0,\;n\neq 0.} \right.$$

We may also write $\delta = [1,0,0,\ldots]$ .

A plot of $\delta (n)$ is given in Fig.5.2a. In the physical world, an impulse may be approximated by a swift hammer blow (in the mechanical case) or balloon pop (acoustic case). We also have a special notation for the impulse response of a filter:

Definition. The impulse response of a filter is the response of the filter to $\delta (n)$ and is most often denoted $h(n)$ :

$$h(n) \isdef {\cal L}_n\{\delta(\cdot)\}$$

The impulse response $h(n)$ is the response of the filter ${\cal L}$ at time $n$ to a unit impulse occurring at time 0. We will see that $h(n)$ fully describes any LTI filter.^6.3

We normally require that the impulse response decay to zero over time; otherwise, we say the filter is unstable. The next section formalizes this notion as a definition.