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
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 . 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.