## 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

*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.

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Filter Stability

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Direct-Form-I Implementation