discrete Fourier transform

discrete Fourier transform

The

discrete Fourier transform

(DFT)

is an

invertible

transform widely employed in signal processing and

analysis

. It can be computed using stable efficient algorithms known as

Fast Fourier Transform

(FFT) algorithms.

1

Definition

Given a

sequence

{

f

n

}

n

=

0

N

-

1

of

N

complex numbers

, the DFT is defined as a sequence

{

F

k

}

k

=

0

N

-

1

of

N

complex numbers according to the

equation

F

k

=

1

N

(

1

-

p

)

/

2

⁢

∑

n

=

0

N

-

1

f

n

⁢

W

N

-

k

,

n

k

=

0

,

1

,

2

,

…

,

N

-

1

where

W

N

k

,

n

=

e

2

⁢

π

⁢

i

⁢

q

⁢

k

⁢

n

/

N

are the basis functions,

p

is an arbitrary real number and

q

is either

1

or

-

1

.

Similarly, we can reconstruct

{

f

n

}

n

=

0

N

-

1

from

{

F

k

}

k

=

0

N

-

1

via the

inverse discrete Fourier transform (IDFT)

:

f

n

=

1

N

(

1

+

p

)

/

2

⁢

∑

k

=

0

N

-

1

F

k

⁢

W

N

k

,

n

n

=

0

,

1

,

2

,

…

,

N

-

1

2

Explanation

Generically, if we have some vector

𝐯

=

(

v

1

,

v

2

,

…

,

v

n

)

that we wish to project on to a set of unit basis vectors

𝐁

^

1

,

𝐁

^

2

,

…

,

𝐁

^

m

, we find that the

k

th

component

of the

projection

𝐯

′

is given by

v

k

′

=

⟨

𝐯

,

𝐁

^

k

⟩

where

⟨

𝐮

,

𝐯

⟩

is an

inner product

. Using the standard

dot product

of

complex vectors

, we have

v

k

′

=

∑

j

=

0

m

v

j

⁢

B

¯

k

,

j

(where

z

¯

represents

the

complex conjugate

of

z

).

We may use a procedure to project a

function

onto basis functions. Here, the components of our vectors are the functions sampled at certain time intervals.

The idea of the

Fourier transform

is to project our discrete function

{

f

n

}

onto a basis made up of sinusoidal functions of varying frequency.

We know from Euler’s

relation

that

e

i

⁢

x

=

cos

⁡

x

+

i

⁢

sin

⁡

x

so we can

construct

a sinusoidal function of any frequency

k

:

W

N

k

,

n

=

e

2

⁢

π

⁢

i

⁢

q

⁢

k

⁢

n

/

N

=

cos

⁡

(

2

⁢

π

⁢

i

⁢

q

⁢

k

⁢

n

/

N

)

+

i

⁢

sin

⁡

(

2

⁢

π

⁢

i

⁢

q

⁢

k

⁢

n

/

N

)

By Nyquist’s theorem, any frequency components of

{

f

n

}

above the

Nyquist frequency

N

2

will be aliased into lower frequencies, so we will only concern ourselves with the frequency components

-

N

2

≤

k

≤

N

2

.

Now we substitute these results into the standard formula for projection to reveal that

F

k

=

∑

n

=

0

N

-

1

f

n

⁢

W

N

-

k

,

n

k

=

0

,

1

,

2

,

…

,

N

-

1

3

Group theoretic interpretation and generalization

This procedure may be interpreted in terms of

group theory

as follows: The basis functions

χ

k

⁢

(

j

)

⁢

e

-

2

⁢

π

⁢

ν

k

⁢

i

⁢

j

/

f

are the

characters

of the

irreducible representations

of the group

ℤ

M

. The

orthogonality relation

⟨

χ

m

,

χ

n

⟩

=

δ

m

⁢

n

which was used to obtain the transform as a projection is the orthogonality relation for characters.

Once we view the discrete Fourier transform in this fashion, it is not hard to generalize it by replacing

ℤ

M

with any

discrete group

G

. As it turns out, there are two versions of the generalization which turn out to coincide exactly in the case of

commutative groups

(such as

ℤ

M

).

First generalization: Let

f

:

G

→

ℂ

be

conjugation

invariant

, which means that, for every

y

∈

G

, it is the case that

f

⁢

(

y

⁢

x

⁢

y

-

1

)

=

f

⁢

(

x

)

. Then one has

f

⁢

(

x

)

=

∑

k

1

d

k

⁢

⟨

f

,

χ

k

⟩

⁢

χ

k

⁢

(

x

)

where the index

k

runs over all irreducible representations and

d

k

is the

dimension

of the

representation

k

. The inner product on vectors is the usual one:

⟨

𝐮

,

𝐯

⟩

=

∑

g

∈

G

𝐮

⁢

(

g

)

⁢

𝐯

¯

⁢

(

g

)

The justification for this comes from the fact that the set of irreducible characters spans the space of conjugation invariant functions and the orthogonality of these characters.

Second generalization: If

f

:

G

→

ℂ

is any function on a group (not necessarily conjugation invariant), then we have the transform

f

⁢

(

x

)

=

∑

k

∑

i

,

j

=

1

d

k

⟨

f

,

ρ

m

⁢

n

k

⟩

⁢

ρ

m

⁢

n

k

⁢

(

x

)

where

k

and

d

k

are as before and

ρ

m

⁢

n

k

are the

matrix elements

of the representation

k

. The justification for this comes from the fact that the set of matrix elements of representations spans the

space of functions

on the group and the orthogonality relation for matrix elements.

Remarks:

DFT is also generalized by

two-dimensional Fourier transforms

(2D-FT),

Fourier-Stieltjes transforms

, and via

groupoid representation

to anharmonic analysis that can also be numerically computed by utilizing FFT, but tend to
be very much slower that the one-dimensional FFT because many series of sequential one-dimensional FFTs and also additional matrix

transformations

are

involved in

2D-FT, or 2D-FFT.

References:

Paul Bourke,

http://astronomy.swin.edu.au/ pbourke/analysis/dft/

”Discrete Fourier Transform”

Title

discrete Fourier transform

Canonical name

DiscreteFourierTransform

Date of creation

2013-03-22 12:39:45

Last modified on

2013-03-22 12:39:45

Owner

stitch (17269)

Last modified by

stitch (17269)

Numerical id

25

Author

stitch (17269)

Entry type

Definition

Classification

msc 65T50

Synonym

DFT

Related topic

DiscreteSineTransform

Related topic

DiscreteCosineTransform

Related topic

FourierTransform

Related topic

LaplaceTransform

Related topic

TwoDimensionalFourierTransforms

Related topic

FourierStieltjesTransform

Defines

inverse discrete Fourier transform

Defines

IDFT

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