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 and
defined by

We may also write
.

A plot of 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 and is most often denoted :

The impulse response is the response of the filter at
time to a unit impulse occurring at time 0. We will see that
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.