The convolution theorem for z transforms states that for any (real or)
complex causal signals
and
,
convolution in the time domain is multiplication in the
domain, i.e.,
or, using operator notation,
where
Proof:
The convolution theorem provides a major cornerstone of linear systems theory. It implies, for example, that any stable causal LTI filter (recursive or nonrecursive) can be implemented by convolving the input signal with the impulse response of the filter, as shown in the next section.