## 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

*input*signal to a filter, as the

*output*signal, and as the

*digital filter*, as shown in Fig.8.12.

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

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

*impulse response*of the filter.

It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response [68]. 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.

### Frequency Response

**Definition: **The *frequency response* of an LTI filter may be defined
as the Fourier transform of its impulse response. In particular, for
finite, discrete-time signals
, the sampled frequency
response may be defined as

*i.e.*,

### Amplitude Response

**Definition: **The *amplitude response* of a filter is defined as
the *magnitude* of the frequency response

*gain*of the filter at frequency , since

### Phase Response

**Definition: **The *phase response* of a filter is defined as
the *phase* of its frequency response:

The topics touched upon in this section are developed more fully in the next book [68] in the music signal processing series mentioned in the preface.

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Correlation Analysis

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Spectrograms