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
Proof: This follows immediately from integration by parts:
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.
Fourier Series (FS) and Relation to DFT