# 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:

## 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

*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. 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.Time Duration Frequency Bandwidth c

### 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

*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:

### 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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Fourier Transforms for Continuous/Discrete Time/Frequency