## Filters and Convolution

A reason for the importance of convolution (defined in §7.2.4) is that*every linear time-invariant system*. Thus, in the convolution equation

^{8.7}can be represented by a convolutionwe 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

*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:

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Correlation Analysis

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Spectrograms