The Uncertainty Principle
The
uncertainty principle (for
Fourier transform pairs) follows
immediately from the scaling theorem. It may be loosely stated as
Time Duration
Frequency Bandwidth
c
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.
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.1Under these definitions, we have the following theorem
[
52, p. 273-274]:
Theorem: If

and

as

, then
 |
(C.2) |
with equality if and only if
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 (see §
5.9.3 for the discrete-time case),
 |
(C.3) |
The left-hand side can be evaluated using integration by parts:
where we used the assumption that

as

.
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):
Substituting these evaluations into Eq.

(
C.3) gives
Taking the square root of both sides gives the uncertainty relation
sought.
If equality holds in the uncertainty relation Eq.

(
C.2), then
Eq.

(
C.3) implies
for some constant

, which implies

for
some constants

and

. Since

by hypothesis, we have

while

remains arbitrary.
If

for

, then
where

is as defined above in Eq.

(
C.1).
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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