## Differentiation Theorem

Let denote a function differentiable for all such that and the Fourier Transforms (FT) of both and exist, where denotes the time derivative of . Then we have

where denotes the Fourier transform of . In operator notation:

Proof: This follows immediately from integration by parts:

since .

The differentiation theorem is implicitly used in §E.6 to show that audio signals are perceptually equivalent to bandlimited signals which are infinitely differentiable for all time.

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