Appendix A: Linear Time-Invariant Filters and Convolution
A reason for the importance of convolution is that every linear time-invariant system8.9 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. 8.11.
The impulse or “unit pulse” signal is defined by
For example, for , . The impulse signal is the identity element under convolution, since
If we set in the filter equation above, we get
Thus, is 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. No matter what the LTI system is, we can give 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.