## 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 is any nonzero real number (the abscissa stretch factor). A more commonly used notation is the following:

(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.

**Next Section:**

Spectral Interpolation

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Fourier Theorems for the DTFT