Filters and Convolution

A reason for the importance of convolution (defined in
§7.2.4) is that *every linear time-invariant
system ^{8.3}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

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

It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response [66]. 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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