approximating Fourier integrals with discrete Fourier transforms

approximating Fourier integrals with discrete Fourier transforms

Suppose we have a

continuous function

f

:

ℝ

ν

→

ℂ

for which we want to numerically
compute the

continuous

Fourier transform

:

f

^

⁢

(

ξ

)

=

∫

ℝ

ν

f

⁢

(

x

)

⁢

e

-

2

⁢

π

⁢

i

⁢

ξ

⋅

x

⁢

𝑑

x

,

ξ

∈

ℝ

ν

.

Assume that either the function

f

is supported
inside the

compact rectangle

C

=

[

a

1

,

b

1

]

×

⋯

×

[

a

ν

,

b

ν

]

,
or that the

Fourier integral

is well approximated
by truncating the domain to

C

.

We develop an approximation to

f

^

based on the

discrete Fourier transform

(which can be computed efficiently using the

fast Fourier transform

algorithm

), and analyze the error in making this approximation.

Derivation

Performing the linear

substitution

t

j

=

x

-

a

j

b

j

-

a

j

,

j

=

1

,

…

,

ν

,

we find that

∫

C

f

⁢

(

x

)

⁢

e

-

2

⁢

π

⁢

i

⁢

ξ

⋅

x

⁢

𝑑

x

=

Vol

⁡

(

C

)

⁢

e

-

2

⁢

π

⁢

i

⁢

ξ

⋅

a

⁢

∫

0

1

⋯

⁢

∫

0

1

g

⁢

(

t

)

⁢

e

-

2

⁢

π

⁢

i

⁢

(

(

b

1

-

a

1

)

⁢

ξ

1

⁢

t

1

+

⋯

+

(

b

ν

-

a

ν

)

⁢

ξ

ν

⁢

t

ν

)

⁢

𝑑

t

1

⁢

⋯

⁢

𝑑

t

ν

=

Vol

⁡

(

C

)

⁢

e

-

2

⁢

π

⁢

i

⁢

ξ

⋅

a

⁢

g

^

⁢

(

(

b

1

-

a

1

)

⁢

ξ

1

,

…

,

(

b

ν

-

a

ν

)

⁢

ξ

ν

)

,

where

g

⁢

(

t

1

,

…

,

t

ν

)

=

f

⁢

(

a

1

+

(

b

1

-

a

1

)

⁢

t

1

,

…

,

a

ν

+

(

b

ν

-

a

ν

)

⁢

t

ν

)

,

t

j

∈

[

0

,

1

]

,

and

g

^

denotes the Fourier transform of

g

on the unit cube

[

0

,

1

]

ν

.

Let

u

denote the vector with

components

u

⁢

(

k

1

,

…

,

k

ν

)

=

g

⁢

(

k

1

/

N

,

…

,

k

ν

/

N

)

, for

k

j

=

0

,

…

,

N

-

1

.
The discrete Fourier transform
of

u

is

u

^

⁢

(

l

)

=

∑

k

∈

{

0

,

…

,

N

-

1

}

ν

u

⁢

(

k

)

⁢

e

-

2

⁢

π

⁢

i

⁢

k

⋅

l

/

N

,

l

=

{

0

,

…

,

N

-

1

}

ν

,

and approximates

N

ν

⁢

g

^

⁢

(

l

)

since it is a

Riemann sum

(step size

1

/

N

) for the Fourier integral.

Therefore, for

l

j

=

0

,

…

,

N

-

1

,
we have the approximation:

f

^

⁢

(

l

1

b

1

-

a

1

,

…

,

l

ν

b

ν

-

a

ν

)

≈

Vol

⁡

(

C

)

N

ν

⁢

e

-

2

⁢

π

⁢

i

⁢

(

a

1

⁢

l

1

b

1

-

a

1

+

⋯

+

a

ν

⁢

l

ν

b

ν

-

a

ν

)

⁢

u

^

⁢

(

l

)

.

Analysing the error

Assume that

g

can be represented by a pointwise-convergent

1

1

Since the discrete Fourier transform requires sampling of points,

𝐋

2

convergence of the

Fourier series

is not sufficient. Also, the Fourier series
may only be

conditionally convergent

. If there are difficulties in assuring
its convergence, Fejér

summation

can be
used instead in this

analysis

.

Fourier series:

g

⁢

(

t

)

=

∑

k

∈

ℤ

ν

⟨

g

,

E

k

⟩

⁢

E

k

⁢

(

t

)

,

where

E

k

⁢

(

t

)

=

e

2

⁢

π

⁢

i

⁢

k

⋅

t

,

⟨

α

,

β

⟩

=

∫

[

0

,

1

]

ν

α

⁢

(

t

)

⁢

β

⁢

(

t

)

¯

⁢

𝑑

t

.

The Fourier coefficient of

g

is

g

^

⁢

(

l

)

=

⟨

g

,

E

l

⟩

, and in contrast

u

^

⁢

(

l

)

=

N

ν

⁢

⟨

g

,

E

k

⟩

N

, where:

⟨

α

,

β

⟩

N

=

1

N

ν

⁢

∑

k

∈

{

0

,

…

,

N

-

1

}

ν

α

⁢

(

k

/

N

)

⁢

β

⁢

(

k

/

N

)

¯

is a

positive semi-definite inner product

.

We have the exact formula:

f

^

⁢

(

l

1

b

1

-

a

1

,

…

,

l

ν

b

ν

-

a

ν

)

=

Vol

⁡

(

C

)

⁢

e

-

2

⁢

π

⁢

i

⁢

(

a

1

⁢

l

1

b

1

-

a

1

+

⋯

+

a

ν

⁢

l

ν

b

ν

-

a

ν

)

⁢

g

^

⁢

(

l

)

;

and we wish to know what error is introduced
if we were to replace

g

^

⁢

(

l

)

=

⟨

g

,

E

l

⟩

by the discrete version of the

inner product

⟨

g

,

E

l

⟩

N

=

u

^

⁢

(

l

)

/

N

ν

.

Expand

⟨

g

,

E

l

⟩

N

as:

⟨

g

,

E

l

⟩

N

=

⟨

∑

k

∈

ℤ

ν

⟨

g

,

E

k

⟩

⁢

E

k

,

E

l

⟩

N

=

∑

k

∈

ℤ

ν

⟨

g

,

E

k

⟩

⁢

⟨

E

k

,

E

l

⟩

N

.

Since

⟨

E

k

,

E

l

⟩

N

is

1

if

k

j

≡

l

j

(

mod

N

)

for all

j

,
and is

0

otherwise,
the discrete inner product reduces to:

⟨

g

,

E

l

⟩

N

=

∑

k

∈

ℤ

ν

⟨

g

,

E

l

+

k

⁢

N

⟩

=

⟨

g

,

E

l

⟩

+

∑

k

∈

ℤ

ν

∖

{

0

}

⟨

g

,

E

l

+

k

⁢

N

⟩

.

In other words, the approximation entails replacing
the true Fourier coefficient

g

^

⁢

(

l

)

with the same

coefficient

plus other Fourier coefficients

⟨

g

,

E

l

+

k

⁢

N

⟩

corresponding to higher
frequencies, in

multiples

of

N

.
The magnitude of the approximation error
for

f

^

thus depends on how fast the partial sums
of the Fourier coefficients

⟨

g

,

E

k

⟩

decay to zero.

Title

approximating Fourier integrals with discrete Fourier transforms

Canonical name

ApproximatingFourierIntegralsWithDiscreteFourierTransforms

Date of creation

2013-03-22 16:28:23

Last modified on

2013-03-22 16:28:23

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

9

Author

stevecheng (10074)

Entry type

Derivation

Classification

msc 42B10

Classification

msc 65T50

Classification

msc 42A38

Related topic

DiscreteTimeFourierTransformInRelationWithItsContinousTimeFourierTransfrom

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