proof of sampling theorem

proof of sampling theorem

0.1

Set-up

Let

w

>

0

be the (two-sided)

bandwidth

. The variable

ξ

below will denote
frequency, and the variable

t

will denote time.
(Both

w

and

ξ

are measured in .)

Consider the

space of functions

:

ℋ

′

=

{

g

∈

𝐋

2

⁢

(

ℝ

)

:

g

⁢

(

ξ

)

=

0

⁢

for almost all

⁢

|

ξ

|

>

w

/

2

}

which is clearly seen to be a complex

Hilbert space

with the usual

inner product

for

𝐋

2

⁢

(

ℝ

)

.

Let

ℱ

denote the

Fourier transform

on

𝐋

2

⁢

(

ℝ

)

,
which is a

unitary

transform by Plancherel’s theorem.
So,

ℋ

=

{

f

∈

𝐋

2

⁢

(

ℝ

)

:

(

ℱ

⁢

f

)

⁢

(

ξ

)

=

0

⁢

for almost all

⁢

|

ξ

|

>

w

/

2

}

=

ℱ

-

1

⁢

ℋ

′

is also a Hilbert space.

0.2

Computation of orthonormal basis

One

orthonormal basis

for

ℋ

′

consists of the usual Fourier

functions

on the

interval

[

-

w

/

2

,

w

/

2

]

, extended to be zero on

ℝ

∖

[

-

w

/

2

,

w

/

2

]

:

ϕ

n

⁢

(

ξ

)

=

{

1

w

⁢

e

-

2

⁢

π

⁢

i

⁢

n

⁢

ξ

/

w

,

|

ξ

|

≤

w

/

2

0

,

|

ξ

|

>

w

/

2

,

n

∈

ℤ

.

Mapping these by

ℱ

-

1

produces an orthonormal basis for

ℋ

:

(

ℱ

-

1

⁢

ϕ

n

)

⁢

(

t

)

=

1

w

⁢

∫

-

w

/

2

w

/

2

e

-

2

⁢

π

⁢

i

⁢

n

⁢

ξ

/

w

⁢

e

2

⁢

π

⁢

i

⁢

ξ

⁢

t

⁢

𝑑

ξ

=

1

w

⁢

∫

-

w

/

2

w

/

2

e

2

⁢

π

⁢

i

⁢

ξ

⁢

(

t

-

n

/

w

)

⁢

𝑑

ξ

=

1

w

⁢

(

w

⁢

sinc

⁡

(

w

⁢

(

t

-

n

/

w

)

)

)

=

w

⁢

sinc

⁡

(

w

⁢

t

-

n

)

,

where we have used the fact that the Fourier transform of

t

↦

w

⁢

sinc

⁡

(

w

⁢

t

)

(normalized

sinc function

)
is the rectangular box function of bandwidth

w

, and vice versa.

0.3

Expansion by orthonormal basis

Given

f

∈

ℋ

, let

g

=

ℱ

⁢

f

∈

ℋ

′

.
We can expand

g

in a

Fourier series

with respect
to the

{

ϕ

n

}

:

g

⁢

(

ξ

)

=

∑

n

∈

ℤ

⟨

g

,

ϕ

n

⟩

⁢

ϕ

n

⁢

(

ξ

)

,

with the infinite sum converging in

𝐋

2

⁢

(

ℝ

)

.
Taking

ℱ

-

1

of both sides,
we obtain:

f

⁢

(

t

)

=

(

ℱ

-

1

⁢

g

)

⁢

(

t

)

=

∑

n

∈

ℤ

⟨

g

,

ϕ

n

⟩

⁢

(

ℱ

-

1

⁢

ϕ

n

)

⁢

(

t

)

=

w

⁢

∑

n

∈

ℤ

⟨

g

,

ϕ

n

⟩

⁢

sinc

⁡

(

w

⁢

t

-

n

)

.

Moreover,

⟨

g

,

ϕ

n

⟩

=

1

w

⁢

∫

-

∞

∞

g

⁢

(

ξ

)

⁢

e

2

⁢

π

⁢

i

⁢

n

⁢

ξ

/

w

⁢

𝑑

ξ

=

1

w

⁢

(

ℱ

-

1

⁢

g

)

⁢

(

n

w

)

=

1

w

⁢

f

⁢

(

n

w

)

.

(Since

g

is also in

𝐋

1

, its inverse Fourier transform

ℱ

-

1

⁢

g

is a

continuous function

.
Provided that we modify

f

on a set of measure zero, we can assume that

f

=

ℱ

-

1

⁢

(

ℱ

⁢

f

)

=

ℱ

-

1

⁢

g

is

continuous

. So it is legal to talk about the pointwise
values

f

⁢

(

n

/

w

)

.)

0.4

Result

Hence, we arrive at the

representation

:

f

⁢

(

t

)

=

∑

n

∈

ℤ

f

⁢

(

n

w

)

⁢

sinc

⁡

(

w

⁢

t

-

n

)

,

thereby reconstructing any

f

∈

ℋ

— a square-integrable band-limited
function —
from its samples at every time period of length

1

/

w

.

0.5

Uniform and absolute convergence of series

The infinite series for

f

converges in

𝐋

2

by construction, but in fact it also converges uniformly and absolutely.
To see this, first note that by the

Cauchy-Schwarz inequality

,

∑

n

∈

ℤ

|

f

⁢

(

n

w

)

|

⁢

|

sinc

⁡

(

w

⁢

t

-

n

)

|

≤

(

∑

n

∈

ℤ

|

f

⁢

(

n

w

)

|

2

)

1

/

2

⁢

(

∑

n

∈

ℤ

sinc

2

⁡

(

w

⁢

t

-

n

)

)

1

/

2

.

The series

∑

n

|

f

⁢

(

n

/

w

)

|

2

converges by Parseval’s theorem
(

w

-

1

/

2

⁢

f

⁢

(

n

/

w

)

are the Fourier coefficients of

g

).
Also, the series

∑

n

sinc

2

⁡

(

w

⁢

t

-

n

)

is uniformly bounded for all

t

∈

ℝ

. To prove this, it suffices to
restrict to

t

bounded

inside

[

0

,

1

/

w

]

as the function

t

↦

∑

n

sinc

2

⁡

(

w

⁢

t

-

n

)

is

1

/

w

-

periodic

; and then
it becomes an easy estimate using the fact that

∑

n

n

-

2

<

∞

.
It follows that the series

∑

n

|

f

⁢

(

n

/

w

)

|

⁢

|

sinc

⁡

(

w

⁢

t

-

n

)

|

is uniformly bounded for all

t

,
and its tail tends to zero uniformly in

t

.

0.6

Illustrations

Figure 1:

Reconstructing a

Bessel function

(of the

first kind

), using a

sinc

series with

n

=

-

10

,

-

9

⁢

…

,

0

,

…

,

10

. Theoretically exact bandwidth is

w

=

1

/

π

≈

0.32

.

Figure 2:

Effect of under-sampling and over-sampling beyond the actual bandwidth of the function

Figure 3:

The basis functions

sinc

⁡

(

t

)

,

1

2

⁢

sinc

⁡

(

t

±

1

)

,

1

4

⁢

sinc

⁡

(

t

±

2

)

•

http://aux.planetmath.org/files/objects/8650/sample.py

Python program to produce the three figures

Title

proof of sampling theorem

Canonical name

ProofOfSamplingTheorem

Date of creation

2013-03-22 16:28:57

Last modified on

2013-03-22 16:28:57

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

13

Author

stevecheng (10074)

Entry type

Proof

Classification

msc 42A38

Classification

msc 94A20

Related topic

PlancherelsTheorem

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