Continuous-Time Fourier Theorems
Selected Fourier theorems for the continuous-time case are stated and proved in Appendix B. However, two are sufficiently important that we state them here.
Scaling Theorem
The scaling theorem (or similarity theorem) says that if
you horizontally ``stretch'' a signal by the factor
in the
time domain, you ``squeeze'' and amplify its Fourier transform by the
same factor in the frequency domain. This is an important general
Fourier duality relationship:
Theorem: For all continuous-time functions
possessing a Fourier
transform,

where

and

![]() |
(3.41) |
Proof: See §B.4.
The scaling theorem is fundamentally restricted to the
continuous-time, continuous-frequency (Fourier transform) case. The
closest we come to the scaling theorem among the DTFT theorems
(§2.3) is the stretch (repeat) theorem
(page ). For this and other continuous-time Fourier
theorems, see Appendix B.
Spectral Roll-Off
Definition: A function
is said to be of order
if
there exist
and some positive constant
such
that
for all
.
Theorem: (Riemann Lemma):
If the derivatives up to order
of a function
exist and are
of bounded variation, then its Fourier Transform
is
asymptotically of order
, i.e.,
![]() |
(3.42) |
Proof: See §B.18.
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Spectral Interpolation
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Fourier Theorems for the DTFT