##

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 .

### Shift Theorem

The *shift theorem* says that a *delay* of samples
in the time domain corresponds to a *multiplication by
* in the frequency domain:

*z*transform .

*Proof: *

where we used the causality assumption for .

### Convolution Theorem

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.*,

*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.

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Z Transform of Convolution

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Existence of the Z Transform