uncertainty theorem

uncertainty theorem

The

uncertainty principle

, as first formulated by Heisenberg, states that the

product

of the

standard deviations

of two

conjugated

variables

, cannot be less than some minimum. This statement has been generalized to a precise mathematical

theorem

, in the frame of

wavelet

theory.

1

THE UNCERTAINTY THEOREM

Let

f

⁢

(

t

)

be a

real function

of the

real variable

, satisfying the

L

2

condition (see below), and

F

⁢

(

ω

)

its

Fourier transform

. The standard deviation

Δ

t

and

Δ

ω

of

t

and

ω

respectively, satisfy the following inequality:

Δ

t

⁢

Δ

ω

≥

1

2

For this formula to make sense,

Δ

t

and

Δ

ω

must be precisely defined.

2

THE

𝑳

𝟐

CONDITION

A real function

f

⁢

(

t

)

of the real variable

t

will be said to satisfy the

L

2

condition if

f

⁢

(

t

)

,

t

⁢

f

⁢

(

t

)

and the

derivative

f

′

⁢

(

t

)

are all in

L

2

.

If

F

⁢

(

ω

)

is its Fourier transform,

-

i

⁢

ω

⁢

F

⁢

(

ω

)

is the transform of

f

′

⁢

(

t

)

. All the following

functions

belong to

L

2

:

f

⁢

(

t

)

,

f

′

⁢

(

t

)

,

t

⁢

f

⁢

(

t

)

,

F

⁢

(

ω

)

,

ω

⁢

F

⁢

(

ω

)

The first three functions are just the

definition

and the two last ones result from Parseval’s

identity

, recalled here in its integral form:

∫

-

∞

∞

U

⁢

(

ω

)

¯

⁢

V

⁢

(

ω

)

⁢

𝑑

ω

=

2

⁢

π

⁢

∫

-

∞

∞

u

⁢

(

t

)

¯

⁢

v

⁢

(

t

)

⁢

𝑑

t

U

⁢

(

ω

)

and

V

⁢

(

ω

)

are the Fourier transforms of

u

⁢

(

t

)

and

v

⁢

(

t

)

.

3

DEFINITIONS

f

⁢

(

t

)

and

F

⁢

(

ω

)

being in

L

2

, we may define their finite norms:

∣

∣

f

∣

∣

2

=

∫

-

∞

∞

∣

f

⁢

(

t

)

∣

2

⁢

𝑑

t

∣

∣

F

∣

∣

2

=

∫

-

∞

∞

∣

F

⁢

(

ω

)

∣

2

⁢

𝑑

ω

By Parseval’s identity, they are related:

∣

∣

f

∣

∣

2

=

1

2

⁢

π

⁢

∣

∣

F

∣

∣

2

We are now able to define the probability distributions

T

⁢

(

t

)

and

Ω

⁢

(

ω

)

for the ”random” variables

t

and

ω

:

T

⁢

(

t

)

=

∣

f

⁢

(

t

)

∣

2

∣

∣

f

∣

∣

2

Ω

⁢

(

ω

)

=

∣

F

⁢

(

ω

)

∣

2

∣

∣

F

∣

∣

2

Since the

L

2

integrals

of

T

and

Ω

are 1, they are proper probability distributions. The

mean value

t

0

of

t

is defined the usual way:

t

0

=

∫

-

∞

∞

t

⁢

T

⁢

(

t

)

⁢

𝑑

t

Note that

ω

’s mean value is always 0 because

f

⁢

(

t

)

is a real function.
Finally, we have the standard deviations for the

uncertainty theorem

:

Δ

t

2

=

∫

-

∞

∞

T

⁢

(

t

)

⁢

(

t

-

t

0

)

2

⁢

𝑑

t

Δ

ω

2

=

∫

-

∞

∞

Ω

⁢

(

ω

)

⁢

(

t

-

t

0

)

2

⁢

𝑑

ω

4

PROOF OF THE THEOREM

The heart of the proof is the

Cauchy-Schwarz inequality

in the

L

2

Hilbert space

: the product of the norms of two functions

u

⁢

(

t

)

and

v

⁢

(

t

)

is greater than, or equal to, the norm of their

scalar product

:

∫

-

∞

∞

∣

u

⁢

(

t

)

∣

2

⁢

𝑑

t

⁢

∫

-

∞

∞

∣

v

⁢

(

t

)

∣

2

⁢

𝑑

t

≥

∣

∫

-

∞

∞

u

⁢

(

t

)

¯

⁢

v

⁢

(

t

)

⁢

𝑑

t

∣

2

Equality occurs if, and only if, one of the functions is proportional to the other. For the two functions

u

⁢

(

t

)

=

(

t

-

t

0

)

⁢

f

⁢

(

t

)

and

v

⁢

(

t

)

=

f

′

⁢

(

t

)

, we have therefore:

∫

-

∞

∞

(

t

-

t

0

)

2

⁢

f

⁢

(

t

)

2

⁢

𝑑

t

⁢

∫

-

∞

∞

f

′

⁢

(

t

)

2

⁢

𝑑

t

≥

∣

∫

-

∞

∞

(

t

-

t

0

)

⁢

f

⁢

(

t

)

⁢

f

′

⁢

(

t

)

⁢

𝑑

t

∣

2

The integral at the right hand side can be integrated by parts. Using the definition of

∣

∣

f

∣

∣

:

∫

-

∞

∞

(

t

-

t

0

)

2

⁢

f

⁢

(

t

)

2

⁢

𝑑

t

⁢

∫

-

∞

∞

f

′

⁢

(

t

)

2

⁢

𝑑

t

≥

1

4

⁢

∣

∣

f

∣

∣

4

But

∣

∣

f

∣

∣

and

∣

∣

F

∣

∣

are related by a

2

⁢

π

factor, so:

∫

-

∞

∞

(

t

-

t

0

)

2

⁢

f

⁢

(

t

)

2

⁢

𝑑

t

⁢

∫

-

∞

∞

f

′

⁢

(

t

)

2

⁢

𝑑

t

≥

1

8

⁢

π

⁢

∣

∣

f

∣

∣

2

⁢

∣

∣

F

∣

∣

2

Applying Parseval’s identity to the second integral of the left hand side, we get:

1

2

⁢

π

⁢

∫

-

∞

∞

(

t

-

t

0

)

2

⁢

f

⁢

(

t

)

2

⁢

𝑑

t

⁢

∫

-

∞

∞

ω

2

⁢

F

⁢

(

ω

)

2

⁢

𝑑

ω

≥

1

8

⁢

π

⁢

∣

∣

f

∣

∣

2

⁢

∣

∣

F

∣

∣

2

We have used the fact that the Fourier transform of

f

′

⁢

(

t

)

if

-

i

⁢

ω

⁢

F

⁢

(

ω

)

.

Now, dividing both sides by the norms, and simplifying by the

2

⁢

π

factor, we get exactly the uncertainty theorem:

∫

-

∞

∞

(

t

-

t

0

)

2

⁢

T

⁢

(

t

)

⁢

𝑑

t

⁢

∫

-

∞

∞

ω

2

⁢

F

⁢

(

ω

)

2

⁢

𝑑

ω

≥

1

4

5

THE GAUSSIAN FUNCTION

The Cauchy-Schwarz inequality becomes an equality if, and only if, one of the functions is proportional to the other. In our case, this condition is expressed by

f

′

⁢

(

t

)

=

λ

⁢

(

t

-

t

0

)

⁢

f

⁢

(

t

)

where

λ

is a constant. This differential equation is readily solved:

f

⁢

(

t

)

=

k

⁢

e

λ

⁢

(

t

-

t

0

)

2

.

f

⁢

(

t

)

must be in

L

2

so that

λ

must be

negative

. Defining

λ

=

-

1

2

⁢

σ

2

, we get the

gaussian function

in its traditional form:

f

⁢

(

t

)

=

e

-

(

t

-

t

0

)

2

2

⁢

σ

2

The constant

k

has been omitted because it cancels anyway in the probability distributions. The standard deviations are easily computed from their definitions:

Δ

t

=

σ

2

Δ

ω

=

1

σ

⁢

2

Their product is

1

2

, independent of

σ

. There is no other function with this

property

.

References

1

Roberto Celi

Time-Frequency visualization of helicopter noise

http://celi.umd.edu/Jour/NoisePaperColor.pdf

Despite its frightening title, this paper is mostly theoretical and it is the only place where I saw the uncertainty theorem clearly stated.

2

Robi Polikar

The wavelet tutorial

http://users.rowan.edu/ polikar/wavelets/wttutorial.html

Title

uncertainty theorem

Canonical

name

UncertaintyTheorem

Date of creation

2013-03-22 18:50:37

Last modified on

2013-03-22 18:50:37

Owner

dh2718 (16929)

Last modified by

dh2718 (16929)

Numerical id

14

Author

dh2718 (16929)

Entry type

Theorem

Classification

msc 42C40

Related topic

UncertaintyPrinciple

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