A Mathematical Journal
Vol. 7, No 2, (201 - 221). August 2005.

An introduction to the Fractional
Fourier Transform and friends

A. Bultheel 1

Department of Computer Science
K.U.Leuven, Belgium

Adhemar.Bultheel@cs.kuleuven.ac.be

H. Mart́ınez
National Experimental Univ. of Guayana,

Port Ordaz, State Boĺıvar, Venezuela
hmartine@uneg.edu.ve

ABSTRACT
In this survey paper we introduce the reader to the notion of the fractional

Fourier transform, which may be considered as a fractional power of the clas-
sical Fourier transform. It has been intensely studied during the last decade,
an attention it may have partially gained because of the vivid interest in time-
frequency analysis methods of signal processing, like wavelets. Like the complex
exponentials are the basic functions in Fourier analysis, the chirps (signals sweep-
ing through all frequencies in a certain interval) are the building blocks in the
fractional Fourier analysis. Part of its roots can be found in optics and mechan-
ics. We give an introduction to the definition, the properties and approaches to
the continuous fractional Fourier transform.

1The work of the first author is partially supported by the Belgian Programme on Interuniversity
Poles of Attraction, initiated by the Belgian Federal Science Policy Office. The scientific responsibility
rests with the authors.



202 A. Bultheel and H. Mart́ınez
7, 2(2005)

RESUMEN

En este art́ıculo de prospección introducimos al lector en la noción de la
transformada de Fourier fraccional, que puede ser considerada como una poten-
cia fraccional de la transformada de Fourier clásica. Ha sido objeto de intensos
estudios durante la última década, que puede deberse parcialmente al interés
respecto de los métodos de análisis tiempo-frecuencia en el proceso de señales,
como es el caso de los wavelets. Tal como las exponenciales complejas son las
funciones básicas del análisis de Fourier, los llamados chirps (señales que barren
todas las frecuencias en un intervalo dado) son los elementos básicos del análisis
de Fourier fraccional. Parte de sus oŕıgenes se pueden encontrar en la óptica y la
mecánica. Damos una introducción a la definición, las propiedades y acercamien-
tos a la transformada de Fourier fraccional.

Key words and phrases: Fourier transform, fractional transforms,
signal processing, chirp, phase space

Math. Subj. Class.: 42A38, 65T20

1 Introduction

The idea of fractional powers of the Fourier operator appears in the mathematical
literature as early as 1929 [32, 8, 11]. It has been rediscovered in quantum mechanics
[19, 16], optics [17, 21, 2] and signal processing [3]. The boom in publications started
in the early years of the 1990’s and it is still going on. A recent state of the art can
be found in [22].

The outline of the paper is as follows. Section 2 gives a motivation for our defini-
tion of the fractional Fourier transform (FrFT) given in the next section. Whereas in
the classical Fourier transform, the harmonics and the delta functions play a promi-
nent role, these are for the FrFT replaced by a more general class of chirp functions
introduced in Section 4. The Wigner distribution is a function that essentially gives
the distribution of the energy of the signal in a time-frequency or phase plane. The
effect of a FrFT can be effectively visualized with the help of this function. This is
described in Section 5. Relations with the windowed or short time Fourier transform,
with wavelets and chirplets can be found in Section 6. The FrFT may be seen as a
special case of a more general linear canonical transform (LCT). Whereas the FrFT
corresponds to a rotation of the Wigner distribution in the time-frequency plane, the
LCT will correspond to any linear transform that can be represented by a unimodular
2 × 2 matrix. This is the subject of Section 7. Thus everything that is explained in
this section will also hold for the fractional Fourier transform. This includes computa-
tional aspects, filtering in the transform domain, generalization to higher dimension,
etc. To define the LCT in higher dimensions, we give a brief introduction to a group
theoretic approach in Section 8. We conclude by a section giving a quick review of
some closely related transforms. Because of page limitations, we shall have to refer



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 203

the reader for all the details to the literature.

2 The classical Fourier Transform

We recall some of the definitions and properties that are related to the classical
continuous Fourier transform (FT) so that we can motivate our definition of the
fractional Fourier transform (FrFT) later.

On an appropriate function space L like e.g., L2(IR), the classical FT operator
F : f → F and its inverse are defined as

F(ξ) =
1√
2π

∫ ∞
−∞

f(x)e−iξxdx, f(x) =
1√
2π

∫ ∞
−∞

F(ξ)eiξxdξ. (1)

In signal processing applications, f is often a time depending signal so that x denotes
time and ξ frequency. Therefore f(x) is a time domain description of the signal and
F(ξ) a frequency domain description.

Furthermore, it is immediately verified that (F2f)(x) = f(−x), (F3f)(ξ) =
F(−ξ), and (F4f)(x) = f(x). This means that for a ∈ Z we may identify Fa with a
rotation in the (x,ξ)-plane over an angle α = aπ/2. The idea of the FrFT is to define
Fa for any a ∈ IR.

It will be useful to introduce some notation. Let Ra denote the rotation matrix

Ra =
[

cos α sin α
−sin α cos α

]
= eJα, J =

[
0 1
−1 0

]

and suppose that (xa,ξa)T = Ra(x,ξ)T , or switching to complex variables z = x−iξ,
then za = eiαz. Note that with this notation ξ = x1, and in general ξa = xa+1.

The notation Ra will also be used as an operator working on a function of two
variables to mean Raf(x,ξ) = f(xa,ξa) and to indicate that Ra(x,ξ) = (xa,ξa).

3 The fractional Fourier transform

In [22] the authors give 6 different possible definitions of the FrFT and others can
be found elsewhere. We prefer to follow an intuitive approach and define it as an
extension of Fa for a ∈ Z to a ∈ IR.

3.1 Eigenfunctions

How to define Fa for a ∈ IR? The key is the eigenvalue decomposition of F. It is
known that F has a complete set of eigenvectors that span L2(IR). Since F4 = I, the
different eigenvalues are {1,−i,−1, i} each with an infinite dimensional eigenspace.
The eigenvectors are thus not unique, but a possible choice of orthonormal eigenfunc-
tions is given by the set of normalized Hermite-Gauss functions:

φn(x) =
21/4√
2nn!

e−x
2/2Hn(x), where Hn(x) = (−i)nex

2Dne−x2, D = −i d
dx
,



204 A. Bultheel and H. Mart́ınez
7, 2(2005)

is an Hermite polynomial of degree n. We have Fφn = λnφn with λn = e−inπ/2. So,
provided we properly define λa for a ∈ IR, we may set Faφn = λanφn, and since {φn}
is a complete set, this defines Fa on L.

If we define the analysis operator Tφ, the synthesis operator T ∗φ and the scaling
operator Sλ as

Tφ : f �→ {cn = 〈f,φn〉2}, Sλ : {cn} �→{λncn}, T ∗φ : {dn} �→
∞∑

n=0

dnφn,

(〈·, ·〉2 is the inner product in L2(IR)) then it is clear that we may write

F = T ∗φ SλTφ and Fa = T ∗φ SaλTφ. (2)

Note that the operator Tφ is unitary on L2(IR) and that T ∗φ is its adjoint.
The formula (2) gives a general procedure to define the fractional power of any

operator that has a complete set of eigenfunctions.
This definition implies that Fa can be written as a operator exponential Fa =

e−iαH = e−iaπH/2 where the Hamiltonian operator H is given by H = 1
2
(D2 +U2−I)

with D = −id/dx and U the shift operator of L2(IR) defined as (Uf)(x) = xf(x) or
U = FDF−1 (see [16, 19, 22]). The form of the operator H can be readily checked
by differentiating the relation

e−iαH
(
e−x

2/2Hn(x)
)

= e−inα
(
e−x

2/2Hn(x)
)

with respect to α, setting α = 0 and then using the differential equation (D +
2iU)DHn = 2nHn satisfied by the Hermite polynomials.

Note that this form identifies Fa as a unitary operator, and hence the Parseval
equality holds in L2(IR).

Several simple properties can now be derived, the most glamorous one being
FaFb = Fa+b, which reflects the group structure of the rotations.

3.2 Integral representation

Any function f ∈ L2(IR) can be expanded as f = ∑n 〈f,φn〉2 φn, so that after
application of Fa we have (Faf)(ξ) = 〈f(x),∑n φn(x)λanφn(ξ)〉2, which identifies Fa
as an integral transform with kernel Ka(ξ,x) =

∑
n φn(x)λ

a
nφn(ξ)/

√
2π. For a = ±1

this reduces to the FT kernel K±1(ξ,x) = e∓ixξ/
√

2π. For a �= ±1, this is not so
simple. Using the eigenvalues and eigenfunctions for the transform Fa, we obtain

Ka(ξ,x) =
∞∑

n=0

e−inaπ/2Hn(ξ)Hn(x)
2nn!
√
π

e−(x
2+ξ2)/2

=
1√

π
√

1 −e−2iα
exp

{
2xξe−iα −e−2iα(ξ2 + x2)

1 −e−2iα
}

exp
{
−ξ

2 + x2

2

}



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 205

where in the last step we used Mehler’s formula ([19, p. 244] or [4, eq. (6.1.13)])

∞∑
n=0

e−inαHn(ξ)Hn(x)
2nn!
√
π

=
exp

{
2xξe−iα−e−2iα(ξ2+x2)

1−e−2iα
}

√
π(1 −e−2iα)

.

To rewrite this expression, we observe that the following identities hold (they are
easily checked)

2xξe−iα

1 −e−2iα = −ixξ csc α

1√
π
√

1 −e−2iα
=
e−

i
2 (

π
2 α̂−α)√

2π|sin α|
e−2iα

1 −e−2iα +
1
2

= − i
2

cot α

where α̂ = sgn (sin α). Obviously, such relations only make sense if sin α �= 0, i.e., if
α �∈ πZ or equivalently a �∈ 2Z. The branch of (sin α)1/2 we are using for sin α < 0 is
the one with 0 < |α| < π. With these expressions, we obtain a more tractable integral
representation of Fa for a �∈ 2Z viz.

fa(ξ) := (Faf)(ξ) =
e−

i
2 (

π
2 α̂−α)e

i
2 ξ

2 cot α√
2π|sin α|

∫ ∞
−∞

exp
{
−i xξ

sinα
+
i

2
x2 cot α

}
f(x)dx,

(3)
where α̂ = sgn (sin α) and 0 < |α| < π.

Previously we defined (Faf)(ξ) = f(ξ), if α = 0, and (Faf)(ξ) = f(−ξ), if
α = ±π. That is consistent with this integral representation because for these special
values, it holds that lim�→0 fa+� = fa. Thus, with this limiting property, we can
assume that the integral representation holds on the whole interval |α| ≤ π. When
|α| > π, the definition is taken modulo 2π and reduced to the interval [−π,π].

Defining the FrFT via this integral transform, we can say that the FrFT exists for
f ∈ L1(IR) (and hence in L2(IR)) or when it is a generalized function. Indeed, in that
case, the integrand in (3) is also in L1(IR) (or L2(IR)) or is a generalized function.
Thus the FrFT exists under exactly the same conditions as under which the FT exists.
Thus we have proved

Theorem 3.1 Assume α = aπ/2 then the FrFT has an integral representation

fa(ξ) := (Faf)(ξ) =
∫ ∞
−∞

Ka(ξ,x)f(x)dx.

The kernel is defined as follows: For a �∈ 2Z, then with α̂ = sgn (sin α),

Ka(ξ,x) = Cα exp
{
−i xξ

sin α
+
i

2
(x2 + ξ2) cot α

}



206 A. Bultheel and H. Mart́ınez
7, 2(2005)

with

Cα =
e−

i
2 (

π
2 α̂−α)√

2π|sin α|
=

√
1 − i cot α

2π
.

For a ∈ 4Z the FrFT becomes the identity, hence K4n(ξ,x) = δ(ξ −x), n ∈ Z and
for a ∈ 2 + 4Z, it is the parity operator: K2+4n(ξ,x) = δ(ξ + x), n ∈ Z.
If we restrict a to the range 0 < |a| < 2, then Fa is a homeomorphism of L2(IR) (with
inverse F−a).
The last statement is proved in [16, p. 162].

It is directly verified that the kernel Ka has the following properties.

Theorem 3.2 IF Ka(x,t) is the kernel of the FrFT as in Theorem 3.1, then

1. Ka(ξ,x) = Ka(x,ξ) (diagonal symmetry)

2. K−a(ξ,x) = Ka(ξ,x) (complex conjugate)

3. Ka(−ξ,x) = Ka(ξ,−x) (point symmetry)
4.
∫ ∞
−∞ Ka(ξ,t)Kb(t,x)dt = Ka+b(ξ,x) (additivity)

5.
∫ ∞
−∞ Ka(t,ξ)Ka(t,x)dt = δ(ξ −x) (orthogonality)

4 The chirp function

A chirp function (or chirp for short) is a signal that contains all frequencies in a
certain interval and sweeps through it while it progresses in time. The interval can
be swept in several ways (linear, quadratic, logarithmic,. . . ), but we shall restrict us
here to the case where the sweep is linear.

The complex exponential eiωt contains just one frequency: ω. This type of func-
tions is essential in Fourier analysis. In fact, they form a basis for the space of
functions treated by the FT. Indeed, the relation f(t) = 1√

2π

∫ ∞
−∞ F(ω)e

iωtdω can be
seen as a decomposition of f into a (continuous) combination of the basis functions
{eω(t) = eiωt}ω∈IR.

On the other hand, if the frequencies of the signal sweeps linearly through the
frequency interval [ω0,ω1] in the time interval [t0, t1], then we should have ω = ω0 +
ω1−ω0
t1−t0 (t − t0). Thus, a chirp will have the form exp{i(χt + γ)t}. The parameter
χ is called the sweep rate. Now consider the FrFT kernel Ka(ξ,x), then, seen as a
function of x and taking ξ as a parameter, this is a chirp with sweep rate 1

2
cot α.

So, by rearranging the kernel (see also Section 7), it can be seen that one way of
describing a FrFT is

1. multiply by a chirp
2. do an ordinary FT
3. do some scaling
4. multiply by a chirp.



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 207

The inverse FrFT can be written as f(x) =
∫ ∞
−∞ fa(ξ)ψξ(x)dξ where ψξ(x) = K−a(ξ,x)

is a chirp parameterized in ξ with sweep rate −1
2

cot α. Thus we see that the role
played by the harmonics in classical FT, is now taken by chirps, and the latter relation
is a decomposition of f(x) into a linear combination of chirps with a fixed sweep rate
determined by α. Note also that in this expansion in chirp series, the basis functions
are orthogonal by property 5 of the previous theorem. However, there is more. The
chirps are in between harmonics and delta functions. Indeed, up to a rotation in the
time-frequency plane, the chirps are delta functions and harmonics. To see this, take
the FrFT of a delta function δ(x−γ). That is (Faδ(·−γ))(ξ) = Ka(ξ,γ), which is a
chirp with sweep rate 1

2
cot α. Thus, given a (linear) chirp with sweep rate 1

2
cot α, we

can transform it by a FrFT F−a into a delta function and hence by taking the FT of
the delta function, we can take the chirp by a FrFT F1−a into an harmonic function.

5 The Wigner distribution and the FrFT

The relation between the multiplication operator U and the complex differentiation
operator D, in the case of the classical Fourier transform is UF = FD, which can be
generalized as follows

Fa
[
U
D
]

=
[
Ua
Da

]
Fa where

[
Ua
Da

]
=
[

cos α sin α
−sin α cos α

][
U
D
]
.

Thus Ua and Da correspond to multiplication and complex differentiation in the vari-
able of the FrFT domain. It are rotations of the usual U and D: (Ua,Da) = Ra(U,D).
This property is intuitively clear: by first applying U or D (i.e., multiplication in the x,
respectively ξ direction) followed by a rotation in the (x,ξ)-plane must be the same as
the rotation followed by the same operations applied to the rotated variables. Because
the rotation is an orthogonal transformation, we also have D2a +U2a = D2 +U2, so that
the Hamiltonian is rotation invariant: Ha = 12 (D2a + U2a −I) = 12 (D2 +U2 −I) = H.

The rotation property of the FrFT that has been mentioned several times now,
can be visualised by the Wigner distribution which is what will be defined next. Let
f be in L2(IR), then its Wigner distribution or Wigner transform Wf is defined as

(Wf)(x,ξ) = 1√
2π

∫ ∞
−∞

f(x + u/2)f(x−u/2)e−iξudu.

Its meaning is roughly speaking one of energy distribution of the signal in the time-
frequency plane. Indeed, setting f1 = Ff, we have∫ ∞

−∞
(Wf)(x,ξ)dξ = |f(x)|2 and

∫ ∞
−∞

(Wf)(x,ξ)dx = |f1(ξ)|2,

so that
1√
2π

∫ ∞
−∞

∫ ∞
−∞

(Wf)(x,ξ)dξdx = ‖f‖2 = ‖f1‖2,

which is the energy of the signal f.
An important property of the FrFT is the following.



208 A. Bultheel and H. Mart́ınez
7, 2(2005)

Theorem 5.1 The Wigner distribution of a signal and its FrFT are related by a
rotation over an angle −α:

(Wfa)(x,ξ) = R−a(Wf)(x,ξ)
where α = aπ/2, fa = Faf. Equivalently

Ra(Wfa)(x,ξ) = (Wfa)(xa,ξa) = (Wf)(x,ξ)
with (xa,ξa) = Ra(x,ξ).
This theorem says that if we have the Wigner distribution of f, then the Wigner
distribution of fa is obtained by rotating it clockwise over an angle α in the (x,ξ)-
plane. The proof is tedious but straightforward. For details see [3, p. 3087]. Looking
at Figure 1, the result is in fact obvious since it just states that before and after a
rotation of the coordinate axes, the Wigner distribution is computed in two different
ways taking the new variables into account, and that should of course give the same
result.

Figure 1: Wigner distribution of a signal f and the Wigner distribution of its FrFT are
related by a rotation.

This implies for example∫ ∞
−∞

(Wfa)(x,ξ)dξ = |fa(x)|2 and
1√
2π

∫ ∞
−∞

∫ ∞
−∞

(Wfa)(x,ξ)dxdξ = ‖f‖2.

The ambiguity function is closely related to the Wigner distribution. Its definition
is

(Af)(x,ξ) = 1√
2π

∫ ∞
−∞

f(u + x/2)f(u−x/2)e−iuξdu.

Thus it is like the Wigner distribution, but now the integral is over the other variable.
The ambiguity function and the Wigner distribution are related by what is essentially
a 2-dimensional Fourier transform. Whereas the Wigner distribution gives an idea
about how the energy of the signal is distributed in the (x,ξ)-plane, the ambiguity
function will have a correlative interpretation. Indeed (Af)(x, 0) is the autocorrelation
function of f and (Af)(0,ξ) is the autocorrelation function of f1 = Ff.



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 209

6 Windowed transform, wavelets and chirplets

The short time Fourier transform or windowed Fourier transform (WFT) is defined
as

(Fwf)(x,ξ) =
1√
2π

∫ ∞
−∞

f(t)w(t−x)e−iξtdt

where w is a window function. It is a local transform in the sense that the window
function more or less selects an interval, centered at x to cut out some filtered infor-
mation of the signal. So it gives information that is local in the time-frequency plane
in the sense that we can find out which frequencies appear in the time intervals that
are parameterized by their centers x.

It can be shown that

(Fwf)(x,ξ) = e−ixξ(Fw1f1)(ξ,−x) = e−ixξ(Fw1f1)(x1,ξ1)
where w1 = Fw and f1 = Ff. Because of the asymmetric factor e−ixξ, it is more
convenient to introduce a modified WFT defined by

(F̃wf)(x,ξ) = eixξ/2(Fwf)(x,ξ).
Then it can be shown that

(F̃wf)(x,ξ) = (F̃wafa)(xa,ξa).
For more information on windows applied in the FrFT domain see [27].

From its definition fa(ξ) =
∫
Ka(ξ,u)f(u)du, we get by setting x = ξ sec α and

g(x) = fa(x/ sec α)

g(x) = C(α)e−i4x
2 sin(2α)

∫ ∞
−∞

exp

[
i

2

(
x−u

tan1/2 α

)2]
f(u)du.

C(α) is a constant that depends on α only. Although, there are some characteristics
of a wavelet transform, this can not exactly be interpreted as a genuine wavelet
transform. We do have a scaling parameter tan1/2 α and a translation by u of the
basic function ψ(t) = eit

2
but since

∫ ∞
−∞ ψ(x)dx �= 0 and it has no compact support,

this is not really a wavelet.
Multiscale chirp functions were introduced in [5, 15]. A. Bultan [6] has developed

a so called chirplet decomposition which is related to wavelet package techniques. It
is especially suited for the decomposition of signals that are chirps, i.e., whose Wigner
distribution corresponds to straight lines in the (x,ξ)-plane.

The idea is that a dictionary of chirplets is obtained by scaling and translating
an atom whose Wigner distribution is that of a Gaussian that has been stretched
and rotated. So, we take a Gaussian g̃(t) = π−1/4e−x

2/2 with Wigner distribution
(Wg̃)(x,ξ) = (2/π)1/2 exp{−(x2 + ξ2)}. Next we stretch it as g(x) = s−1/2g̃(x/s)
giving (Wg)(x,ξ) = (Wg̃)(x/s,sξ). Finally we rotate (Wg) to give (Wc)(x,ξ) with
c = Fag. The chirplet c depends on two parameters s and a and its main support



210 A. Bultheel and H. Mart́ınez
7, 2(2005)

in the (x,ξ)-plane can be thought of as a stretched (by s) and rotated (by a) ellipse
centered at the origin. To cover the whole (x,ξ)-plane, we have to tile it with shifted
versions of this ellipse, i.e., we need the shifted versions (Wg)(x − u,ξ − ν) corre-
sponding to the functions c(x−u)eiνx. With these four-parameters ρ = (s,a,u,ν) we
have a redundant dictionary {cρ}.

The next step is to find a discretization of these 4 parameters such that the
dictionary is complete when restricted to that lattice. It has been shown [31] that
such a system can be found for a = 0 that is indeed complete, and the rotation does
not alter this fact.

If the discrete dictionary is {cn} with cn = cρn , then a chirplet representation
of the signal f has to be found of the form f(x) =

∑
n ancn(x). Such a discrete

dictionary for a signal with N samples has a discrete chirplet dictionary with O(N2)
elements. Therefore a matching pursuit algorithm [14] can be adapted from wavelet
analysis. The main idea is that among all the atoms in the dictionary the one that
matches best the data is retained. This gives a first term in the chirplet expansion.
The approximation residual is then again approximated by the best chirplet from the
dictionary, which gives a second term in the expansion etc. This algorithm has a
complexity of the order O(MN2 log N) to find M terms in the expansion. This is far
too much to be practical. A faster O(MN) algorithm based on local optimization has
been published [10].

This approach somehow neglects the nice logarithmic and dyadic tiling of the plane
that made more classical wavelets so attractive. So this kind of decomposition will
be most appropriate when the signal is a composition of a number of chirplets. Such
signals do exist like the example of a signal emitted by a bat which consists of 3 nearly
parallel chirps in the (x,ξ)-plane. Other examples are found in seismic analysis. For
more details we refer to [6]. An example in acoustic analysis was given in [10].

7 The linear canonical transform

As we have seen, the FrFT is essentially a rotation in the (x,ξ)-plane. So, it can be
characterized by a 2 × 2 rotation matrix which depends on one parameter, namely
the rotation angle. It is a subgroup SO(2) of the group GL(2) of 2 × 2 real in-
vertible matrices. Most of what has been said can be generalized to a more gen-
eral linear transform, which is characterized by a general matrix M in the subgroup
SL(2) = {M ∈ IR2×2 : det(M) = 1}. These generalizations are called linear canonical
transforms (LCT).

7.1 Definition

Consider a 2×2 unimodular matrix (i.e., whose determinant is 1). Such a matrix has
3 free parameters u,v,w which we shall arrange as follows

M =
[
a b
c d

]
=
[

w
v

1
v

−v + uw
v

u
v

]
=
[

u
v

−1
v

v − uw
v

w
v

]−1
=
[

d −b
−c a

]−1
.



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 211

The parameters can be recovered from the matrix by

u =
d

b
=

1
a

(
1
b

+ c
)
, v =

1
b
, w =

a

b
=

1
d

(
1
b

+ c
)

A typical example is the rotation matrix associated with Rα where a = d = cos α
and b = −c = sin α. Let us call this matrix Rα. Although M ∈ IR2×2 is a matrix, we
shall for typographical reasons often write M = (a,b,c,d).

The linear canonical transform FM of a function f is an integral transform with
kernel KM (ξ,x) defined by

KM (ξ,x) =
√

v

2πi
e

i
2 (uξ

2−2vξx+wx2) =
1√
2πib

e
i
2b (dξ

2−2ξx+ax2).

Note that, just like in the case of the FrFT, there is some ambiguity since we have to
choose the branch of the square root in the definition of the kernel.

7.2 Effect on Wigner distribution and ambiguity function

Note that if M is the rotation matrix Rα, then the kernel KM reduces almost to
the FrFT kernel because M = Rα implies u = w = cot α while v = csc α. Hence
FRα = e−iα/2Fa. If f denotes a signal, and fM its linear canonical transform, then
the Wigner transform gives

(WfM )(ax + bξ,cx + dξ) = (Wf)(x,ξ). (4)

The latter equation can be directly obtained from the definition of linear canonical
transform and the definition of Wigner distribution. Thus if RM is the operator
defined by RMf(x) = f(Mx), then W = RMWFM . Note that this generalizes
Theorem 5.1, since (up to a unimodular constant factor which does not influence the
Wigner distribution) FRα and Fa are the same. Similarly for the ambiguity function:
A = RMAFM . The group structure can be used to show that FM is unitary in
L2(IR) and it holds that FAFB = FC if and only if C = AB.

7.3 Special cases

When we restrict ourselves to real matrices M, there are several interesting special
cases, the FrFT being one of them. Others are

• The Fresnel transform: This is defined by

gz(ξ) =
eiπz/l√
ilz

∫ ∞
−∞

ei(π/lz)(ξ−z)
2
f(x)dx.

This corresponds to the choice M = (1,b, 0, 1) with b = zl
2π

, because, with this
M we have gz(ξ) = eiπz/l(FMf)(ξ).



212 A. Bultheel and H. Mart́ınez
7, 2(2005)

• Dilation: The operation f(x) �→ gs(ξ) =
√
sf(sξ), can be also obtained as a

LCT because with M = (1/s, 0, 0,s) we have gs(ξ) =
√

sgn (s)(FMf)(ξ).
• Gauss-Weierstrass transform or chirp convolution: This is obtained by the

choice M = (1,b, 0, 1):

(FMf)(ξ) = 1√
2πib

∫ ∞
−∞

exp{i(x− ξ)2/2b}f(x)dx.

• Chirp (or Gaussian) multiplication: Here we take M = (1, 0,c, 1) and get
(FMf)(ξ) = exp{icξ2/2}f(ξ).

7.4 On the computation of the LCT

To compute the LCT, it is only in exceptional cases that the integral can be evalu-
ated exactly. So in most practical cases, the integral will have to be approximated
numerically. Two forms depending on different factorizations of the M matrix are
interesting for the fast computation or the LCT and thus also for the FrFT.

The first one reflects the decomposition
[
a b
c d

]
=
[

1 0
(d− 1)/b 1

][
1 b
0 1

][
1 0

(a− 1)/b 1
]

(5)

which means (see section 7.3) that the computation can be reduced to a chirp multi-
plication, followed by a chirp convolution, followed by a chirp multiplication. Taking
into account that the convolution can be computed in O(N log N) operations using
the fast Fourier transform (FFT), the resulting algorithm is a fast algorithm.

Another interesting decomposition is given by
[
a b
c d

]
=
[

1 0
db−1 1

][
b 0
0 b−1

][
0 1
−1 0

][
1 0

b−1a 1

]
(6)

and this is to be interpreted as a chirp multiplication, followed by an ordinary Fourier
transform (which can be obtained using FFT), followed by a dilation, followed even-
tually by another chirp multiplication. Again it is clear that this gives a fast way of
computing the FrFT or LCT.

In Figure 2, the effect of the LCT on a unit square is illustrated showing the
different steps when the matrix M, which is for this example M = (2, 0.5, 0, 1, 0.525),
is decomposed as in (5) or as in (6). As we can see the two methods compute quite
different intermediate results. In the example given there, it is clear that the second
decomposition on the right stretches the initial unit square much more and shifts it
over larger distances compared to first decomposition on the left. This is an indication
that more severe numerical rounding errors are to be expected with the second way
of computing than with the first one.

The straightforward implementation of these steps may be a bit naive because for
example in the FrFT case, the kernel may be highly oscillating for certain values of



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 213

0 1 2 3 4 5
0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10
−4

−3

−2

−1

0

1

2

3

4

5

6

7

8

9

10

Figure 2: The effect of a LCT on a square. Left when the matrix M is decomposed as in
(5) and right when it is decomposed as in (6).

a. It is clear that those values should be avoided. Therefore it is best to evaluate
the FrFT only for a in the interval [0.5, 1.5] and to use the relation Fa = FF1−a
for a ∈ [0, 0.5) ∪ (1.5, 2]. A discussion in [20] follows the approach given by the first
decomposition (5).

7.5 Filtering in the LCT domain

One may now set up a program of generalizing all the properties that were given in
the case of the FrFT to the LCT. Usually this does not pose a big problem and the
generalization is smoothly obtained. We just pick one general approach to what could
be called canonical filtering operations. For its definition, we go back to the classical
Fourier transform. If we want to filter a signal, then we have to compute a convolution
of the signal and the filter. However, as we know, the convolution in the x-domain
corresponds to a multiplication in the ξ-domain. Thus the filtering operation is char-
acterized by f∗g = F−1[(Ff)(Fg)]. The natural fractional generalization would then
be to define a fractional convolution f ∗a g = (Fa)−1[(Faf)(Fag)] and the canonical
convolution would be f ∗M g = (FM )−1[(FMf)(FMg)]. Clearly, if M = I or a = 1,
the classical convolution is recovered. This definition has been used in many papers.
See for example [18] and [22, p. 420]. Similar definitions can be given in connection
with correlation instead of convolution operations. The essential difference between
convolution and correlation is a complex conjugate, so that a canonical correlation
can be defined as f �M g = (FM )−1[(FMf)(FMg)∗]. One could generalize even more
and define for example an operation like FM3 [(FM1f)(FM2g)] (see [24]).

If we consider the convolution in the x-domain and the multiplication in the ξ-



214 A. Bultheel and H. Mart́ınez
7, 2(2005)

domain as being dual operations, then we can ask for the notion of dual operations in
the fractional or the canonical situation. A systematic study of dual operations has
been undertaken in [12], but we shall not go into details here.

The windowed Fourier transform can be seen as a special case. Indeed, as we
have seen, applying a window in the x-domain corresponds to applying a transformed
window in the xa-domain. So it may well be that in some fractional domain, it may
be easier to design a window that will separate different components of the signal, or
that can better catch some desired property of the signal because its spread is smaller
in the transform domain [27].

Also the Hilbert transform which is defined as

1
π

∫ ∞
−∞

f(x)
x−x′ dx

′ (7)

(integral in the sense of principal value) corresponds to filtering in the x domain with
a filter g(x) = 1/x.

A somewhat different approach to the definition of a canonical convolution is taken
in [1]. It is based on the fact that a classical convolution f ∗ g = f ∗0 g is an inner
product of f with a time-inverted and shifted version of g:

(f ∗0 g)(x) =
1√
2π

∫ ∞
−∞

f(x′)g(x−x′)dx′ = 〈f(·),g∗(x−·)〉2 .

If we denote a shift in the x-domain as (T0(x′)f)(x) = f(x− x′) and recalling that
time-inversion is obtained by the parity operator F2, it is clear that g∗(x − x′) =
(F2T0(x)g∗)(x′). So f ∗0 g =

〈
f,F2T0(x)g∗

〉
2
. If we now define a canonical shift

as TM (xM ) = (FM )−1T0(xM )FM that is, we transform the signal, shift it in the
transform domain, and then transform back, then another definition of a canonical
convolution could be (f ∗M g)(x) =

〈
f,F2TM (x)g∗

〉
2
. It still has the classical convo-

lution as a special case when M = I, but it is different from the previous definition.

8 Groups and generalization to higher dimensions

There is a nice interpretation of the LCT as a group representation. The purpose
of [28] is to find a unitary operator V on L2(IRn) such that it has the effect that
the Wigner transform of Vf is the Wigner transform of f subject to a general linear
transformation. The n-dimensional Wigner transform is defined as

(Wf)(x, ξ) = 1
(2π)n/2

∫
IRn

f(x + u/2)f(x − u/2)e−iξ·udu.

The dot represents the standard inner product in IRn. Thus we want to find the
unitary operator V = FM on L2(IRn) for which a matrix M ∈ GL(2n) = {M ∈
IR2n×2n : det M �= 0} can be found such that W = RMWV, where as before
(RMf)(x) = f(Mx). GL(2n) is a Lie group and some subgroups are SL(2n) =
{M ∈ GL(2n) : det M = 1}, O(2n) = {M ∈ GL(2n) : MT M = 1} (1 is the identity



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 215

in IRn×n) and SO(2n) = SL(2n) ∩O(2n). A symplectic form on IR2n can be defined
as a Lie bracket: [f, g] = f T Jg with f, g ∈ IR2n and

J =
[

0 1
−1 0

]
.

The symplectic group Sp(n) is the group of real 2n× 2n matrices that leave a sym-
plectic form invariant, i.e., that satisfy MT JM = J. This implies that a symplectic
matrix has determinant ±1. We have in fact Sp(n) ⊂ SL(2n) with equality if n = 1.

The Heisenberg group Hn is identified with IR
n × IRn × IR with group law

(x1, ξ1, t1)(x2, ξ2, t2) = (x1 + x2, ξ1 + ξ2, t1 + t2 + (ξ1 · x2 − x1 · ξ2)/2).
A representation of a topological group G on a Hilbert space H is a mapping μ from G
to the space B(H) of bounded operators on H such that μ(x)μ(y) = μ(xy), μ(e) = I
with e the identity in G and I the identity operator in B(H) and x → μ(x)f is a
continuous mapping for all f ∈ H. The representation μ is unitary if B(H) can be
replaced by U(H), the unitary operators on H. And μ is called irreducible if {0} and
H are the only invariant subspaces of H under the group action μ(x) for all x ∈ G.
It can be shown that a unitary irreducible representation of Hn in the space L2(IR

n)
is the Schrödinger representation defined as

(μ(x, ξ, t)f)(u) = ex·uei(t+x·ξ/2)f(u + ξ).

It takes a couple of lines to show that the relation with the Wigner distribution is
that we can write

(Wf)(x, ξ) = [F〈μ(·, ·, 0)f,f〉2](x, ξ)
where 〈·, ·〉2 is the inner product in L2(IRn) and F the 2n-dimensional FT acting on
the variables indicated by a dot.

Given a unitary V ∈ U(L2(IRn)), another equivalent unitary representation would
be given by ρ(h) = V∗μ(h)V for all h ∈ Hn so that

(WVf)(x, ξ) = [F〈ρ(·, ·, 0)f,f〉2](x, ξ)

=
1

(2π)n/2

∫
IRn

∫
IRn
〈ρ(u, v, 0)f,f〉2 e−iu·xe−iv·ξdu dv.

This implies that if there is a matrix M ∈ IR2n×2n such that μ(g, 0) = ρ(Mg, 0) for
all g ∈ H′n = {g ∈ IR2n : (g, t) ∈ Hn}, then by a change of variables in the last
expression, we get

(WVf)(g) = |det M|(Wf)(Mg).
Now consider the subgroup of U(L2(IRn))

G = {V ∈ U(L2(IRn)) : ∀(g, t) ∈ IR2n+1,∃g′ ∈ IR2n s.t. V∗μ(g, t)V = μ(g′, t)}
The g′ is uniquely defined so that there is a homomorphism ν(V) : IR2n → IR2n given
by ν(V)g = g′. This ν is a continuous mapping from G onto Sp(n) in the subspace



216 A. Bultheel and H. Mart́ınez
7, 2(2005)

topology of G ⊂ U(L2(IRn)) with kernel {cI : |c| = 1}. This means that ν−1 is
only defined up to a unimodular factor. We obtain the metaplectic group which is a
twofold covering of the symplectic group. This shows up in the formulas in the form
of a square root for which the sign has to be chosen.

With these tools, our original problem of finding a unitary V that causes an
arbitrary linear transform of the Wigner distribution, can be solved. It requires some
more lines to show that W = RMWV, if and only if V ∈ G and M = ν(V)−1 ∈ Sp(n).

Several simple examples from the group G can be found.

• Fourier transform
For example, the n-dimensional FT F satisfies all the properties and ν(F) = JT .
• dilation

A second example is the dilation operator: D∗bμ(x, ξ, t)Db = μ(bT x, b−1ξ, t)
with b ∈ GL(n). We have now

ν(Db) =
[

b−1 0
0 bT

]
.

Note that if b is symmetric then (Dbf)(x) = (det b)−1/2f(b−1x).
• chirp multiplication

A third example is a chirp multiplication C∗s μ(x, ξ, t)Cs = μ(x + sξ, ξ, t).

ν(Cs) =
[

1 0
s 1

]
.

With an n-dimensional chirp defined as cs(x) = exp{ i2 xT sx}, and the effect is
that (Csf)(x) = cs(x)f(x). It is clearly no restriction if we assume that s is
symmetric.

In view of the decomposition of the LCT in the scalar case, it is natural to define the
n-dimensional LCT as FM = cCdb−1DbFCb−1a with |c| = 1. It is represented by a
matrix

ν(Cdb−1DbFCb−1a) =
[

a b
c d

]
.

The special case of the separable n-dimensional FrFT corresponds to a = d =
diag(cos α1, . . . , cos αn) and b = −c = diag(sin α1, . . . , sin αn). Then M ∈ Sp(n) ∩
SO(2n), the orthogonal symplectic group. For more details on this approach see e.g.
[9, 28] and for the integral representation [29].

9 Other transforms

Probably motivated by the success of the FrFT and the LCT, quite some effort has
been put in the design of fractional versions of related classical transforms.



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 217

9.1 Radial canonical transforms

It should be clear that for problems with circular symmetry, this symmetry should be
taken into account when defining the transforms. Take for example the 2-dimensional
case. Instead of Cartesian (x,y) coordinates, one should switch to polar coordinates
so that, because of the symmetry, the transform will only depend on the radial dis-
tance. For example, it is well known that the Hankel transform appears naturally
as a radial form of the (2-sided) Laplace transform [34, sec. 8.4]. Giving directly
the n-dimensional formulation, we shall switch from the n-dimensional variables x
and ξ to the scalar variables x = ‖x‖ and ξ = ‖ξ‖, and the n-dimensional LCT will
become canonical Hankel transforms [33, 30]. It is a one-sided integral transform∫ ∞
0
KM (ξ,x)f(x)dx with kernel

KM (ξ,x) = x
n−1 e

− π2 ( n2 +ν)

b
(xξ)1−n/2 exp

{
i

2b
(ax2 + dξ2)

}
Jn/2+ν−1

(
xξ

b

)
,

where Jν is the Bessel function of the first kind of order ν. The fractional Hankel
transform is a special case of the canonical Hankel transform when the matrix M is
replaced by a rotation matrix.

9.2 Fractional Hilbert transform

The definition of the Hilbert transform has been given before in (7). Note that
the convolution defining the transform can be characterized by a multiplication with
−isgn (ξ) in the Fourier domain. Since −isgn (ξ) = e−iπ/2h(ξ) + eiπ/2h(−ξ) with h
the Heaviside step function: h(ξ) = 1 for ξ ≥ 0 and h(ξ) = 0 for ξ < 0, we can now
define a fractional Hilbert transform as

(FM )−1[(e−iφh(ξ) + eiφh(−ξ))(FMf)(ξ)],

with M the rotation matrix M = Ra. For further reading see [35, 13, 23, 7].

9.3 Cosine, sine and Hartley transform

While in the classical Fourier transform, the integral is taken of f(x)e−iξx, one shall
in the cosine, sine, and Hartley transform replace the complex exponential by cos(ξx),
sin(ξx) or cas(ξx) = cos(ξx) + sin(ξx) respectively. Since cos and sin are the real and
imaginary part of the complex exponential, one might think of defining the fractional
cosine and sine transforms by replacing the kernel in the FrFT by its real or imaginary
part. However, this will not lead to index additivity for the transforms. We could
however use the general fractionalization procedure given in (2). We just have to note
that the Hermite-Gauss eigenfunctions are also eigenfunctions of the cosine and sine
transform, except that for the cosine transform, the odd eigenfunctions will correspond
to eigenvalues zero and for the sine transform, the even eigenfunctions will correspond
to eigenvalue zero. This implies that the odd part of f will be killed by the cosine
transform. So, the cosine transform will not be invertible unless we restrict ourselves



218 A. Bultheel and H. Mart́ınez
7, 2(2005)

to the set of even functions. A similar observation holds for the sine transform: it
can only be invertible when we restrict the transform to the set of odd functions.
This motivates the habit to define sine and cosine transforms by one sided integrals
over IR+. See [26]. The bottom line of the whole fractionalization process is that to
obtain the good fractional forms of these operators we essentially have to replace in the
definition of the FrFT the factor eiξx in the kernel of the transform by cos(ξx), sin(ξx)
or cas(ξx) to obtain the kernel for the fractional cosine, sine or Hartley transforms
respectively. In the case of the cosine or sine transform, the restriction to even or odd
functions implies that we need only to transform half of the function, which means
that the integral over IR can be replaced by two times the integral over IR+. Besides
the fractional forms of these operators there are also canonical forms for which we
refer to [26]. Also here simplified forms exist [25, 26].

9.4 Other transforms

The list of transforms that have been fractionalized is too long to be completed here.
Some examples are: Laplace, Mellin, Hadamard, Haar, Gabor, Radon, Derivative,
Integral, Bragman, Barut-Girardello,. . . The fractionalization procedure of (2) can
be used in general. This means the following. If we have a linear operator T in a
complex separable Hilbert space with inner product 〈·, ·〉2 and if there is a complete set
of orthonormal eigenvectors φn with corresponding eigenvalues λn, then any element
in the space can be represented as f =

∑∞
n=0 anφn, an = 〈f,φn〉2, so that (T f) =∑∞

n=0 anλnφn. The fractional transform can the be defined as

(T af)(ξ) =
∞∑

n=0

anλ
a
nφn(ξ) =

∞∑
n=0

λan 〈f,φn〉2 φn(ξ) = 〈f,Ka(ξ, ·)〉2 ,

where

Ka(ξ,x) =
∞∑

n=0

λ
a

nφn(ξ)φn(x).

Of course a careful analysis will require some conditions like for example if it con-
cerns the Hilbert space L2μ(I) of square integrable functions on an interval I with
respect to a measure μ, then we need Ka(ξ, ·) to be in this space, which means that∑∞

n=0 |λn|2a|φn(ξ)|2 < ∞ for all ξ.
In view of the general development for the construction of fractional transforms,

it is clear that the main objective is to find a set on orthonormal eigenfunctions for
the transform that one wants to “fractionalize”. There were several papers that give
eigenvalues and eigenvectors for miscellaneous transforms.

Zayed [36] has given an alternative that uses instead of the kernel Ka(ξ,x) =∑
n λ

a
nφ

∗
n(x)φn(ξ) the kernel

Ka(ξ,x) = lim
|λ|→1−

∑
n

|λ|einαφn(ξ)∗φn(x).



7, 2(2005)
An introduction to the Fractional Fourier Transform and friends 219

Thus λan is replaced by |λ|neinα and the φn can be any (orthonormal) set of basis
functions. In this way he obtains fractional forms of the Mellin, and Hankel trans-
forms, but also of the Riemann-Liouville derivative and integral, and he defines a
fractional transform for the space of functions defined on the interval [−1, 1] based on
Jacobi-functions which play the role of the eigenfunctions.

To the best of our knowledge, a further generalization by taking a biorthogonal
system spanning the Hilbert space, which is very common in wavelet analysis, has
not yet been explored in this context.

Received: June 2003. Revised: March 2004.

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