## 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

*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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Introduction to Sampling

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Scaling Theorem