## 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.

**Next Section:**

Introduction to Sampling

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Scaling Theorem