##

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:

SHIFT

or, using more common notation,
*z*transform .

*Proof:*

### Convolution Theorem

The*convolution theorem for*states that for any (real or) complex causal signals and ,

*z*transforms*convolution in the time domain is multiplication in the domain*,

*i.e.*,

*Proof:*

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Z Transform of Convolution

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Existence of the Z Transform