# Selected Continuous-Time Fourier Theorems

This appendix presents Fourier theorems which are nice to know, but
which do not, strictly speaking, pertain to the DFT. The
*differentiation theorem* for Fourier Transforms
comes up quite often, and its dual pertains as
well to the DTFT (Appendix B).
The *scaling theorem* provides an important basic
insight into time-frequency duality. Finally, the very fundamental
*uncertainty principle* is related to the scaling theorem.

## 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

*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.

## Scaling Theorem

The *scaling theorem* (or *similarity theorem*) provides
that if you horizontally ``stretch'' a signal by the factor
in the time domain, you ``squeeze'' 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,

*Proof: *
Taking the Fourier transform of the stretched signals gives

The absolute value appears above because, when , , which brings out a minus sign in front of the integral from to .

The scaling theorem is fundamentally restricted to the
continuous-time, continuous-frequency (Fourier transform) case.
The closest we came to the scaling theorem among the DFT
theorems was the stretch theorem (§7.4.10). We found that
``stretching'' a *discrete-time* signal by the integer factor
(filling in between samples with zeros) corresponded to the
spectrum being *repeated* times around the unit circle.
As a result, the ``baseband'' copy of the spectrum ``shrinks'' in
width (relative to ) by the factor . Similarly,
stretching a signal using *interpolation* (instead of zero-fill)
corresponded to the same repeated spectrum with the spectral copies
zeroed out. The spectrum of the interpolated signal can therefore be
seen as having been stretched by the inverse of the time-domain
stretch factor. In summary, the stretch theorem for DFTs can be
viewed as the discrete-time, discrete-frequency counterpart of the
scaling theorem for Fourier Transforms.

## The Uncertainty Principle

The *uncertainty principle* (for Fourier transform pairs) follows
immediately from the scaling theorem. It may be loosely stated as

where is some constant determined by the precise definitions of ``duration'' in the time domain and ``bandwidth'' in the frequency domain.Time Duration Frequency Bandwidth c

If duration and bandwidth are defined as the ``nonzero interval,'' then we obtain , which is not very useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse in §B.2.

### Duration and Bandwidth as Second Moments

More interesting definitions of duration and bandwidth are obtained
for nonzero signals using the normalized *second moments* of the
squared magnitude:

where

By the DTFT power theorem, which is proved in a manner
analogous to the DFT case in §7.4.8, we have
. Note that writing ``
'' and
``
'' is an abuse of notation, but a convenient one.
These duration/bandwidth definitions are routinely used in physics,
*e.g.*, in connection with the *Heisenberg uncertainty principle*.^{C.1}Under these definitions, we have the following theorem
[52, p. 273-274]:

**Theorem: **If
and
as
, then

with equality if and only if

*Gaussian function*(also known as the ``bell curve'' or ``normal curve'') achieves the lower bound on the time-bandwidth product.

*Proof: *Without loss of generality, we may take consider to be real
and normalized to have unit norm (
). From the
Schwarz inequality (see §5.9.3 for the discrete-time case),

The left-hand side can be evaluated using integration by parts:

The second term on the right-hand side of Eq.(C.3) can be evaluated using the power theorem (§7.4.8 proves the discrete-time case) and differentiation theorem (§C.1 above):

If equality holds in the uncertainty relation Eq.(C.2), then Eq.(C.3) implies

### Time-Limited Signals

If for , then

*Proof: *See [52, pp. 274-5].

### Time-Bandwidth Products are Unbounded Above

We have considered two lower bounds for the time-bandwidth product
based on two different definitions of duration in time. In the
opposite direction, there is *no upper bound* on time-bandwidth
product. To see this, imagine filtering an arbitrary signal with an
*allpass filter*.^{C.2} The allpass filter cannot affect
bandwidth
, but the duration can be arbitrarily extended by
successive applications of the allpass filter.

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Sampling Theory

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Fourier Transforms for Continuous/Discrete Time/Frequency