## 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.Theimpulse signalis denoted and defined by

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.Theimpulse responseof a filter is the response of the filter to and is most often denoted :

^{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.

**Next Section:**

Filter Stability

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Direct-Form-I Implementation