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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