Shift and Convolution Theorems
In this section, we prove the highly useful shift theorem and convolution theorem for unilateral z transforms. We consider the space of infinitely long, causal, complex sequences , , with for .
The shift theorem says that a delay of samples in the time domain corresponds to a multiplication by in the frequency domain:
where we used the causality assumption for .
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.,
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.
Z Transform of Convolution
Existence of the Z Transform