#### Proof of Aliasing Theorem

To show:

or

From the DFT case [264], we know this is true when and are each complex sequences of length , in which case and are length . Thus,

(3.38) |

where we have chosen to keep frequency samples in terms of the original frequency axis prior to downsampling,

*i.e.*, for both and . This choice allows us to easily take the limit as by simply replacing by :

(3.39) |

Replacing by and converting to -transform notation instead of Fourier transform notation , with , yields the final result.

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