## The Uncertainty Principle

The*uncertainty principle*(for Fourier transform pairs) follows immediately from the scaling theorem (§B.4). 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 useful. This conclusion follows immediately from the definition of the Fourier transform and its inverse (§2.2).

### Duration and Bandwidth as Second Moments

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

where

*e.g.*, in connection with the

*Heisenberg uncertainty principle*[59].Under these definitions, we have the following theorem [202, p. 273-274]:

**Theorem:**If as , then

with equality if and only if

(B.63) |

That is, only the

*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 [264],

^{B.2}

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

(B.65) |

where we used the assumption that as . The second term on the right-hand side of (B.65) can be evaluated using the power theorem and differentiation theorem (§B.2):

(B.66) |

Substituting these evaluations into (B.65) gives

(B.67) |

Taking the square root of both sides gives the uncertainty relation sought. If equality holds in the uncertainty relation (B.63), then (B.65) implies

(B.68) |

for some constant , implying for some constants and .

### Time-Limited Signals

If for , then(B.69) |

where is as defined above in (B.62).

*Proof:*See [202, pp. 274-5].

### Time-Bandwidth Products 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*.

^{B.3}The allpass filter cannot affect bandwidth , but the duration can be arbitrarily extended by successive applications of the allpass filter.

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Relation of Smoothness to Roll-Off Rate

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Sampling Theory