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uncertainty principle

The Uncertainty Principle

Bob Williamson

Australian National University

November18,2000

The uncertainty principle shows that one can not jointly localize a signal in time and frequency arbitrarily well;either one has poor frequency localization or poor time localiza-tion.The degree of localization is measured in the theorem below by the quantitities d and D;these are like the standard deviation of a probability distribution.

The proof we present was originally due to Weyl[H.Weyl,Theory of groups and Quan-tum Mechanics,Dover,NY,(1950);Appendix A]

Theorem1(Uncertainty Principle)Suppose f(t)is afinite energy signal with Fourier

transform F(ω).Let

E:= ∞−∞|f(t)|2dt=12π ∞

−∞

|F(ω)|2dω

d2:=1

E

−∞

t2|f(t)|2dt

D2:=

1

2πE

−∞

ω2|F(ω)|2dω

If |t|f(t)→0as|t|→∞,then

Dd≥1 2

and equality holds only if f(t)has the form

f(t)=Ce−αt2.

In order to prove the theorem we need the following Lemma which is very useful in many other situations.

Lemma2(Cauchy-Schwarz Inequality)For any square integrable functions z(x)and w(x)defined on the interval[a,b],

b

a

z(x)w(x)dx

2≤

b

a

|z(x)|2dx b a|w(x)|2dx(1)

and equality holds if and only if z(x)is proportional to w∗(x)(almost everywhere on [a,b]).

Proof Assume z(x)and w(x)are real(the extension to complex-valued functions is straight-forward).Let

I(y)= b a[z(x)−yw(x)]2dx

= b a z2(x)dx

A −2y b a z(x)w(x)dx

B

+y2 b a w2(x)dx

C

=A−2yB+y2C

1

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