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
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
It turns out in general that every linear time-invariant (LTI) system (filter) is completely described by its impulse response . 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.
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
Definition: The amplitude response of a filter is defined as the magnitude of the frequency 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  in the music signal processing series mentioned in the preface.