## Filters and Convolution

A reason for the importance of convolution (defined in §7.2.4) is that every linear time-invariant system8.7can be represented by a convolution. Thus, in the convolution equation

 (8.1)

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

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 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

The complete (continuous) frequency response is defined using the DTFT (see §B.1), i.e.,

where the summation limits are truncated to because is zero for and . Thus, the DTFT can be obtained from the DFT by simply replacing by , which corresponds to infinite zero-padding in the time domain. Recall from §7.2.10 that zero-padding in the time domain gives ideal interpolation of the frequency-domain samples (assuming the original DFT included all nonzero samples of ).

### Amplitude Response

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

From the convolution theorem, we can see that the amplitude response is the gain of the filter at frequency , since

where is the th sample of the DFT of the input signal , and is the DFT of the output signal .

### Phase Response

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

From the convolution theorem, we can see that the phase response is the phase-shift added by the filter to an input sinusoidal component at frequency , since

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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