discrete time Fourier transform in relation with continuous time Fourier transform

discrete time Fourier transform in relation with continuous time Fourier transform

Fourier Transforms

of Sampled SignalsFernando Diego Sanz Gamiz

Introduction

Computers

are able to handle only finite number of data. Hence, if
we are to study and treat real world signals (i.e.

functions

ℝ

→

ℂ

) in a computer, a way to characterize signals by a finite number
of data has to be found.

If we sample the values of the signal

x

⁢

(

t

)

at periodic
times

t

=

n

⁢

T

,

n

∈

ℤ

we form the

sequence

x

d

⁢

[

n

]

=

x

⁢

(

n

⁢

T

)

. Does this
sequence contain all the

information

relative to

x

⁢

(

t

)

?

We know from the

sampling theorem

that if the signal

x

⁢

(

t

)

is bandlimited, the sampled sequence allows us to recover the
original

continuous

time signal provided the

sampling

frequency

1

/

T

is at least twice the

maximum

frequency of the signal. However,
real signals are not of finite bandwidth as this would imply the
signal to be of

infinite

time duration. Therefore, a problem arise
of how well can we approximate the original signal

x

⁢

(

t

)

by the
sampled sequence

x

d

⁢

[

n

]

. In fact, we are interested in studying
the spectrum of the original signal based upon the samples

x

d

⁢

[

n

]

.

While the

relation

between

x

d

⁢

[

n

]

and the spectrum of

x

⁢

(

t

)

is widely used in communication and electronic engineering books, it
is difficult to find a

rigorous

proof. We cover here the gap between
engineering daily knowledge and rigorous mathematical proof of the
named relations establishing under what

assumptions

those relations
are valid.

Relation between

Discrete Time Fourier Transform

and Continuous time Fourier transform

Theorem 1

.

Let

x

:

R

→

C

be a

bounded variation

(it can be, in particular,
piecewise smooth)

L

1

function and let

X

⁢

(

ω

)

be its Fourier
transform

1

1

in the present entry we will take the Fourier
transform of

x

⁢

(

t

)

to be

X

⁢

(

ω

)

=

∫

-

∞

+

∞

x

⁢

(

t

)

⁢

e

-

j

⁢

w

⁢

t

⁢

𝑑

ω

. Let

T

∈

R

. If

g

n

⁢

(

ω

)

=

∑

k

=

-

n

+

n

X

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

converges

a.e as

n

→

+

∞

and is there
is

M

such that

|

g

n

⁢

(

ω

)

|

<

M

a.e. then

1

T

⁢

∑

k

=

-

∞

+

∞

X

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

=

∑

k

=

-

∞

+

∞

x

⁢

(

k

⁢

T

+

)

+

x

⁢

(

k

⁢

T

-

)

2

⁢

e

-

j

⁢

ω

⁢

k

⁢

in

⁢

L

2

⁢

(

[

-

π

,

π

]

)

(1)

If additionally

∑

k

=

-

∞

+

∞

X

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

is continuous, the
right hand side of (

1

)

converges uniformly

to the left hand
side in any

closed interval

[

a

,

b

]

⊂

[

-

π

,

π

]

.

Proof.

By

hypothesis

we can form the function

g

⁢

(

ω

)

=

lim

n

→

+

∞

⁡

g

n

⁢

(

ω

)

=

lim

n

→

+

∞

⁡

∑

k

=

-

n

+

n

X

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

This function is obviously periodic of period

2

⁢

π

and bounded,
hence it can be

expanded

in its

Fourier series

which converge in

L

2

⁢

(

[

-

π

,

π

]

)

; the Fourier theory shows that the

convergence

is
uniformly in

[

a

,

b

]

⊂

[

-

π

,

π

]

if

g

⁢

(

ω

)

is
continuous.

g

⁢

(

ω

)

=

∑

k

=

-

∞

+

∞

X

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

=

∑

k

=

-

∞

+

∞

c

k

⁢

e

j

⁢

ω

⁢

k

where the coefficients

c

k

are given by

c

k

=

1

2

⁢

π

⁢

∫

-

π

π

g

⁢

(

ω

)

⁢

e

-

j

⁢

ω

⁢

k

⁢

𝑑

ω

=

1

2

⁢

π

⁢

∫

-

π

π

lim

n

→

+

∞

⁡

g

n

⁢

(

ω

)

⁢

e

-

j

⁢

ω

⁢

k

⁢

d

⁢

ω

As

|

g

n

⁢

(

ω

)

|

<

M

we can appeal the dominated convergence
theorem to write

1

2

⁢

π

⁢

∫

-

π

π

lim

n

→

+

∞

⁡

g

n

⁢

(

ω

)

⁢

e

-

j

⁢

ω

⁢

k

⁢

d

⁢

ω

=

lim

n

→

+

∞

⁡

1

2

⁢

π

⁢

∫

-

π

π

g

n

⁢

(

ω

)

⁢

e

-

j

⁢

ω

⁢

k

⁢

𝑑

ω

=

1

2

⁢

π

⁢

∑

k

=

-

∞

+

∞

∫

-

π

-

2

⁢

k

⁢

π

π

+

2

⁢

k

⁢

π

X

⁢

(

ω

T

)

⁢

e

-

j

⁢

ω

⁢

k

⁢

𝑑

ω

=

T

2

⁢

π

⁢

P

.

V

.

∫

-

∞

+

∞

X

⁢

(

ω

)

⁢

e

j

⁢

ω

⁢

(

-

k

⁢

T

)

⁢

𝑑

ω

Now, as

x

⁢

(

t

)

is of bounded variation, the Jordan theorem on
Fourier transform

inversion

says that

1

2

⁢

π

⁢

P

.

V

.

∫

-

∞

+

∞

X

⁢

(

ω

)

⁢

e

j

⁢

ω

⁢

(

-

k

⁢

T

)

⁢

𝑑

ω

=

x

⁢

(

-

k

⁢

T

+

)

+

x

⁢

(

-

k

⁢

T

-

)

2

and the result follows.

∎

There are, however, very which do not

satisfy

the conditions required in the

theorem

above. Such is the
case of the

pulse

function

r

⁢

e

⁢

c

⁢

t

⁢

(

t

)

=

{

1

if

⁢

|

t

|

<

1

2

,

0

if

⁢

|

t

|

≥

1

2

.

whose Fourier transform is

s

⁢

i

⁢

n

⁢

(

x

/

2

)

x

/

2

-with the

definition

made in
footnote 1-.

s

⁢

i

⁢

n

⁢

c

⁢

(

x

)

behaves as

1

x

, so

∑

k

=

-

∞

+

∞

s

⁢

i

⁢

n

⁢

c

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

will not converge in general

2

2

for

monotone
decreasing

functions ”series behaves as

integrals

”, that is, if

∑

k

=

0

+

∞

f

⁢

(

x

-

k

⋅

M

)

converges or diverges so does

∫

0

+

∞

f

; it will converge for those

T

that make
the series an alternating (

http://planetmath.org/AlternatingSeries

) one, but not for the rest values of

T

.
Therefore we need another result which somehow relates both sides of
eq

1

.

As we have pointed out, the problem with the pulse
function is that its Fourier transform does not decay rapid enough
for the series to converge. So we will try smoothing the signal out
so that its Fourier transform will decay faster and, hopefully, the
series converges. We wish the smoothed version of

x

⁢

(

t

)

to resemble
the original signal, so uniform approximation seems reasonable. But,
as we will see, for an infinite number of samples

{

n

⁢

T

,

n

∈

ℤ

}

each of these might require a different degree of approximation and
it could be impossible to find an uniform approximation for all the
samples. So, we will focus on time limited signal, for which we have
the following result.

Notation

.

𝒟

⁢

(

ℝ

)

will denote the set of test functions on

ℝ

and

𝒮

the set of rapidly

decreasing functions

on

ℝ

-the Schwartz
space-. The symbol

^

above a function will denote its Fourier
transform. We know that

𝒟

⁢

(

ℝ

)

⊂

𝒮

and that the Fourier
transform is an

isomorphism

of

𝒮

onto itself. The

product

of
two functions

X

and

ϕ

at

ω

will be denoted by

X

⁢

ϕ

⁢

(

ω

)

.

Convolution

will be denoted by

∗

.

Theorem 2

.

Let

ϕ

∈

D

⁢

(

R

)

be a test function and

{

ϕ

j

⁢

(

t

)

=

j

⋅

ϕ

⁢

(

j

⋅

t

)

,

j

=

1

,

2

,

…

}

be an approximate identity.
Let

x

⁢

(

t

)

be a time limited bounded variation signal. If

∑

k

=

-

∞

+

∞

X

⁢

ϕ

j

^

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

converges for all

j

=

1

,

2

,

…

to a

continuous function

, then

lim

j

→

∞

⁡

1

T

⁢

∑

k

=

-

∞

+

∞

X

⁢

ϕ

j

^

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

=

∑

k

=

-

∞

+

∞

x

⁢

(

k

⁢

T

+

)

+

x

⁢

(

k

⁢

T

-

)

2

⁢

e

-

j

⁢

ω

⁢

k

(2)

Proof.

Take the signal

(

x

∗

ϕ

j

)

⁢

(

t

)

whose Fourier transform is

X

⁢

ϕ

^

⁢

(

ω

)

. This signal satisfies, by hypothesis, the
conditions of Theorem 1, so we can write

1

T

⁢

∑

k

=

-

∞

+

∞

X

⁢

ϕ

j

^

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

=

∑

k

=

-

∞

+

∞

(

x

∗

ϕ

j

)

⁢

(

k

⁢

T

+

)

+

(

x

∗

ϕ

j

)

⁢

(

k

⁢

T

-

)

2

⁢

e

-

j

⁢

ω

⁢

k

The right hand side is actually a sum of a finite number of terms
since the signal

(

x

∗

ϕ

j

)

⁢

(

t

)

is time limited -being the
convolution of two compact supported functions-. This makes the
Fourier series a continuous function, which, together with the
hypothesis that

1

T

⁢

∑

k

=

-

∞

+

∞

X

⁢

ϕ

j

^

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

is continuous, shows that
equality in the above equation is

pointwise

.

Now let

j

→

∞

and use the fact that

{

ϕ

j

⁢

(

t

)

}

is an approximate identity to obtain equation
(

2

).
∎

The rationale behind choosing test functions in the last
theorem is that, in most
cases,

∑

k

=

-

∞

+

∞

X

⁢

ϕ

j

^

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

will converge even
though

∑

k

=

-

∞

+

∞

X

⁢

(

ω

-

2

⁢

π

⋅

k

T

)

do not. This is because
rapidly decreasing functions decay faster than

1

x

α

for any

α

. So, for example, the Fourier transform of the
pulse function has been tamed enough to make the series converge.

Remark 1

.

When the signal

x

⁢

(

t

)

is continuous, the right hand side of eqs.
(

1

) or (

2

) reads

∑

k

=

-

∞

+

∞

x

d

⁢

[

k

]

⁢

e

-

j

⁢

ω

⁢

k

where

x

d

⁢

[

k

]

=

x

⁢

(

k

⁢

T

)

. This is defined as the

(Discrete Time) Fourier Transform, DTFT

of the sequence

x

d

⁢

[

n

]

=

x

⁢

(

n

⁢

T

)

Title

discrete time Fourier transform in relation with continuous time Fourier transform

Canonical

name

DiscreteTimeFourierTransformInRelationWithContinuousTimeFourierTransform

Date of creation

2013-03-22 17:37:45

Last modified on

2013-03-22 17:37:45

Owner

fernsanz (8869)

Last modified by

fernsanz (8869)

Numerical id

10

Author

fernsanz (8869)

Entry type

Theorem

Classification

msc 94A20

Related topic

SamplingTheorem

Related topic

ApproximatingFourierIntegralsWithDiscreteFourierTransforms

Related topic

Distribution4

Related topic

SpaceOfRapidlyDecreasingFunctions

Defines

DTFT

Defines

Discrete Time Fourier Transform

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