Filters and Convolution

A reason for the importance of convolution (defined in
§7.2.4) is that *every linear time-invariant
system ^{8.7}can be represented by a convolution*. Thus, in the
convolution equation

we may interpret as the

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

For example, for sequences of length , .

The impulse signal is the *identity element* under convolution,
since

If we set in Eq. (8.1) above, we get

Thus, , which we introduced as the convolution representation of a filter, has been shown to be more specifically the

It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response [71]. No matter what the LTI system is, we can feed it an impulse, record what comes out, call it , and implement the system by convolving the input signal with the impulse response . In other words, every LTI system has a convolution representation in terms of its impulse response.

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