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
Thus,
It turns out in general that every linear time-invariant (LTI) system
(filter) is completely described by its impulse response [71].
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.