### Differentiation Theorem Dual

**Theorem: **Let
denote a signal with DTFT
, and let

(3.40) |

denote the derivative of with respect to . Then we have

where denotes the DTFT of .

*Proof: *
Using integration by parts, we obtain

An alternate method of proof is given in §B.3.

**Corollary:** Perhaps a cleaner statement is as follows:

This completes our coverage of selected DTFT theorems. The next section adds some especially useful FT theorems having no precise counterpart in the DTFT (discrete-time) case.

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Scaling Theorem

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Downsampling and Aliasing