IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000 1329
The Discrete Fractional Fourier Transform
Çag
˜
atay Candan, Student Member, IEEE, M. Alper Kutay, Member, IEEE, and Haldun M. Ozaktas
Abstract—We propose and consolidate a definition of the
discrete fractional Fourier transform that generalizes the discrete
Fourier transform (DFT) in the same sense that the continuous
fractional Fourier transform generalizes the continuous ordinary
Fourier transform. This definition is based on a particular set of
eigenvectors of the DFT matrix, which constitutes the discrete
counterpart of the set of Hermite–Gaussian functions. The defini-
tion is exactly unitary, index additive, and reduces to the DFT for
unit order. The fact that this definition satisfies all the desirable
properties expected of the discrete fractional Fourier transform
supports our confidence that it will be accepted as the definitive
definition of this transform.
Index Terms—Chirplets, discrete Wigner distributions, Her-
mite–Gaussian functions, time–frequency analysis.
I. INTRODUCTION
I
N RECENT years, the fractional Fourier transform (FRT)
has attracted a considerable amount of attention, resulting in
many applications in the areas of optics and signal processing.
However, a satisfactory definition of the discrete FRT that is
fully consistent with the continuous transform has been lacking.
In this paper, our aim is to propose (following Pei et al. [1],
[2]) and consolidate a definition that has the same relation with
the discrete Fourier transform (DFT) as the continuous FRT
has with the ordinary continuous Fourier transform. This def-
inition has the following properties, which may be posed as
requirements to be satisfied by a legitimate discrete-input/dis-
crete-output FRT:
1) unitarity;
2) index additivity;
3) reduction to the DFT when the order is equal to unity;
4) approximation of the continuous FRT.
The first two are essential properties of the continuous trans-
form, which we desire to be satisfied exactly by the discrete
transform. The third is necessary for the discrete fractional
Fourier transform to be a consistent generalization of the
ordinary DFT. The last, of course, is the major motivation for
defining the discrete transform in the first place. Beyond these,
it would be desirable for the discrete transform to satisfy as
many operational properties of the continuous transform as
possible.
Manuscript received December 1, 1998; revised October 4, 1999. The asso-
ciate editor coordinating thereviewof this paper and approving it for publicaiton
was Dr. Phillip A. Regalia.
Ç. Candan was with the Department of Electrical Engineering, Bilkent Uni-
versity, Ankara, Turkey. He is now with the School of Electrical and Computer
Engineering, Georgia Institute of Technology, Atlanta, GA 30032 USA.
M. A. Kutay was with the Department of Electrical Engineering, Bilkent Uni-
versity, Ankara, Turkey. He is now with the Department of Electrical and Com-
puter Engineering, Drexel University, Philadelphia, PA 19104 USA.
H. M. Ozaktas is with the Department of Electrical Engineering, Bilkent Uni-
versity, Ankara, Turkey.
Publisher Item Identifier S 1053-587X(00)03334-1.
A comprehensive introduction to the FRT and historical ref-
erences may be found in [5]. The transform has become pop-
ular in the optics and signal processing communities following
the works of Ozaktas and Mendlovic [6]–[8], Lohmann [9] and
Almeida [12]. Some of the applications explored include op-
timal filtering in fractional Fourier domains [13]–[16], cost-ef-
ficient linear system synthesis and filtering [17]–[21], time-fre-
quency analysis [11], [12], [22], [23], and Fourier optics and op-
tical information processing [24]–[26]. Additional recent pub-
lications include [27]–[33]. Further references may be found in
[5].
Up to now, the fractional Fourier transform has been digi-
tally computed using a variety of approaches. However, these
approaches are often far from exhibiting the internal consistency
and analytical elegance we take for granted with the ordinary
DFT. It is the purpose of this paper to offer and consolidate such
a definition of the discrete fractional Fourier transform.
A fast
algorithm for digitally computing the con-
tinuous fractional Fourier transform integral has been given in
[34]. This method maps the
samples of the original function
to the
samples of the transform. Whereas this mapping is
very satisfactory in terms of accuracy, the
matrix un-
derlying this mapping is not exactly unitary and does not exactly
satisfy the index additivityproperty. This makes it unsuitable for
a self-consistent a priori definition of the discrete transform.
Several publications proposing a definition for the discrete
FRT have appeared, but none of these papers satisfy all the
requirements. Most of these provide a satisfactory approxi-
mation to the continuous transform; however, [34] and [35]
do not satisfy requirements 1 and 2, and [36] does not sat-
isfy requirement 3. Reference [37] satisfies requirement 2 for
certain discrete orders, and it is not clear to us whether it
satisfies requirement 4. The definition in [38] and [39] cor-
responds to a completely distinct definition of the fractional
Fourier transform [40].
The definition proposed in this paper was first suggested
by Pei and Yeh [1], [2]. They suggest defining the discrete
FRT in terms of a particular set of eigenvectors (previously
discussed in [38]), which they claim to be the discrete
analogs of the Hermite–Gaussian functions (which are well
known as the eigenfunctions of the continuous transform).
They also justify their claims by numerical observations and
simulations. In the present paper, we provide an analytical
development of Pei's claims with the aim of consolidating
the definition of the discrete FRT.
In Section II, the definition and some properties of the
continuous FRT are presented. In Section III, the definition
of the discrete fractional Fourier transform is given. Certain
extensions are provided in Section IV. The paper concludes
with numerical comparisons and future research directions.
1053–587X/00$10.00 © 2000 IEEE
1330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
II. PRELIMINARIES
A. Continuous Fractional Fourier Transform
The
th-order continuous FRT can be defined for
through its integral kernel
(1)
where
and sgn
. The kernel is defined sepa-
rately for
and as
and . The definition can easily
be extended outside the interval
by noting that
for any integer . The kernel is known
to have the following spectral expansion [3]:
(2)
where
denotes the th Hermite-Gaussian function, and
denotes the variable in the th-order fractional Fourier domain.
Here,
is the th power of the eigenvalue
of the ordinary Fourier transform. When ,
the FRT reduces to the ordinary Fourier transform
, where denotes the frequency-domain vari-
able. As
approaches zero or integer multiples of , the kernel
approaches
and , respectively [4]. The most
important properties of the FRT are
1) unitarity:
, where denotes
Hermitian conjugation;
2) index additivity:
;
3) reduction to the ordinary Fourier transform when
.
Another important property not discussed here is the relation-
ship of the fractional Fourier transform to time–frequency rep-
resentations such as the Wigner distribution [8], [10], [11], [12].
We will define the discrete FRT through a discrete analog
of (2). Therefore, we will first discuss the Hermite–Gaussian
functions in some detail.
B. The Hermite–Gaussian Functions
The
th-order Hermite–Gaussian function is defined as
(3)
where
is the th Hermite polynomial having real zeros.
The Hermite–Gaussians form a complete and orthonormal set
in
[42]. The Hermite–Gaussian functions are well known to
be the eigenfunctions of the Fourier transform operator, as will
also be seen below.
We begin with the defining differential equation of the Her-
mite–Gaussians:
(4)
It can be shown that the Hermite–Gaussian functions are the
unique finite energy eigensolutions of (4) (see [43, p. 337]). We
can express the left-hand side of (4) in abstract operator notation
as
(5)
where
and denote the differentiation and the or-
dinary Fourier transform operations, respectively. The operator
can also be recognized as the Hamiltonian as-
sociated with the quantum-mechanical harmonic oscillator [44].
Here, we will denote this operator by
and thus write (5) as
(6)
A theorem of commuting operators will be used to show that the
Hermite–Gaussian functions, which are eigenfunctions of
, are
also eigenfunctions of
(see [45, p. 52]).
Theorem 1: If two operators
and commute, i.e.,
, there exists a common eigenvector set between and .
The commutation of
and can be shown as
(7)
In passing from the third term to the fourth, we used
. This, in turn, follows from the
fact that
(where ). Thus,
this proves that the Hermite–Gaussian functions, which are the
unique finite energy eigenfunctions of
, are also eigenfunc-
tions of
.
III. D
ISCRETE FRACTIONAL FOURIER TRANSFORM
We will first show that the first three requirements are au-
tomatically satisfied when the fractional transform is defined
through a spectral expansion analogous to (2). Assuming
to be an arbitrary orthonormal eigenvector set of the
DFT matrix and to be the associated eigenvalues, the dis-
crete analog of (2) is
(8)
which constitutes a definition of the discrete fractional Fourier
transform matrix
. This transform matrix is unitary since the
eigenvalues
of the DFT matrix have unit
magnitude [38], [41]. Reduction to the DFT when
follows
from the fact that when
, (8) reduces to the spectral expan-
sion of the ordinary DFT matrix. Index additivity can likewise
be easily demonstrated by multiplying the matrices
and
and using the orthonormality of the [48]. Additionally, it
is easy to show that any definition satisfying these three require-
ments can always be expressed in the spectral expansion form.
Before we continue, we note that there are two ambiguities
that must be resolved in (8). The first concerns the eigenstruc-
ture of the DFT. Since the DFT matrix has only four distinct
eigenvalues
[41], the
eigenvalues are in general degenerate so that the eigenvector set
CANDAN et al.: DISCRETE FRACTIONAL FOURIER TRANSFORM 1331
is not unique. For this reason, it is necessary to specify a partic-
ular eigenvector set to be used in (8). In the continuous case, this
ambiguity is resolved by choosing the Hermite–Gaussian func-
tions as the eigenfunctions or, equivalently, by choosing that
eigenfunction set of the Fourier transform that are also eigen-
functions of
. In other words, we choose the common eigen-
function set of the commuting operators
and . Since our aim
is to obtain a definition of the discrete transform that is com-
pletely analogous to the continuous transform, we will resolve
this ambiguity in the same manner by choosing the common
eigenvector set of the DFT matrix and the discrete matrix analog
of
, which we define to be the discrete counterparts of the Her-
mite–Gaussian functions.
The second ambiguity arises in taking the fractional power of
the eigenvalues since the fractional power operation is not single
valued. This ambiguity will again be resolved by analogy with
the continuous case by taking
. Distinct
definitions based on other choices are discussed in [40]. The
particular choice we are concentrating onis the one that has been
most studied and has overwhelmingly found the largest number
of applications.
Denoting the discrete Hermite–Gaussians as
, the defi-
nition of the discrete fractional Fourier transform becomes
(9)
Now, we must explicitly define the discrete counterparts of
the Hermite–Gaussian functions.
A. Discrete Hermite–Gaussians
We will define the discrete Hermite–Gaussians as solutions
of a difference equation that is analogous to the defining differ-
ential (4) of the continuous Hermite–Gaussian functions. First,
we define the second difference operator
(10)
which serves as an approximation to
. This can
also be seen by examining
(11)
where we have expressed the shift operator in hyperdifferential
form as
[46].
Now, we consider the finite difference analog of
appearing in (6), which is .
(12)
where we used the fact that
, which is
nothing but a statement of the shift property of the ordinary
Fourier transform.
Finally, we replace
in (6) with to obtain an approxi-
mation of
, which we refer to as
(13)
We see that the analogous finite difference operator
is an
approximation of . If we explicitly write the difference
equation
, we obtain
(14)
We will now switch to discrete variables by letting
[46]
and obtain the second-order difference equation analogous to
the defining differential equation of Hermite–Gaussians as
(15)
where
, and . We immediately note
that the coefficients of the above equation are periodic with
,
implying the existence of periodic solutions with the same pe-
riod [47]. When (15) is written explicitly by concentrating on a
single period, say,
, we obtain (16), shown at
the bottom of the next page, where the rows of (16) follow from
the replacement of
in (15) and the utilization
of the periodicity relation
. This completes the
derivation of the discrete analog of
, which we will refer to as
the
matrix.
We will show below that
commutes with the DFT matrix
and that the common eigenvector set of
, as well as the DFT
matrix, is unique and orthogonal.
This unique orthogonal eigenvector set, which we call
,
will be taken as the discrete counterpart of the continuous Her-
mite–Gaussians to be used in the defining (9).
We now demonstrate that the
and the DFT matrices com-
mute.
Theorem 2: The matrix
and the DFT matrix commute.
Proof:
can be written as , where is the
circulant matrix corresponding to the system whose impulse re-
sponse is
, and is the
diagonal matrix defined as
. It can also be seen
that
since is an even function.
Then,
.
In the next subsection, we will show that the common
eigenvector set is unique, and in the following subsection,
we will discuss the issue of ordering (or indexing) of the
eigenvectors in one-to-one correspondence with the continuous
Hermite–Gaussian functions.
1332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000
B. Uniqueness of the Common Eigenvector Set of and DFT
The eigenvectors of the DFT matrix are either even or odd
vectors [41]. It follows from this result that the common eigen-
vector set of
and the DFT matrix, which is known to exist
since they commute, should also consist of even or odd vectors.
For completeness, we present a short proof of this important re-
sult, which will be utilized in developments below (see also [41]
for an alternative proof).
Theorem 3: Eigenvectors of the DFT matrix are either even
or odd sequences.
Proof: Letting
denote the DFT matrix, we know that
and , where is the coordinate inversion matrix,
and
is the identity matrix. Since , the eigenvalues
of can only be . Now, let be an eigenvector
of
satisfying ; then, ; however, since
is equivalent to ,
and that completes the proof.
It is known that when is not a multiple of 4, all of the eigen-
values of
are distinct [38]. Since is a real symmetric matrix,
it followsthat all of its eigenvectorsare orthogonal to each other,
and thus, the set of eigenvectors of
is orthogonal and unique
(within multiplicative constants). Since we have already shown
that
has a common eigenvector set with the DFT matrix, this
unique set of eigenvectors of
must also be a set of eigenvec-
tors of the DFT matrix. The normalized version of this set of
eigenvectors will be defined as the discrete version of the Her-
mite–Gaussian functions.
When
is a multiple of 4, the matrix still has distinct
eigenvalues, with the exception of one eigenvalue, which has
the value of zero with degeneracy two. The eigenvectors cor-
responding to all eigenvalues except this one are orthogonal
to each other. The two eigenvectors corresponding to the zero
eigenvalue can be chosen to be orthogonal, again because
is a
real symmetric matrix. There are many ways of choosing these
two eigenvectors such that they are orthogonal; however, there
is only one way to choose them such that one is an even vector
and one is an odd vector. Since we are seeking the common set
of eigenvectors between
and the DFT matrix and since we
know that all eigenvectors of the DFT matrix are either even
or odd vectors, we have no choice but to choose the even and
odd eigenvectors corresponding to the zero eigenvalue; other
choices could not be an eigenvector of the DFT matrix. This re-
quirement resolves the ambiguity associated with choosing the
eigenvectors corresponding to the zero eigenvalue when
is a
multiple of 4 and again uniquely determines the common set of
eigenvectors of
and the DFT matrix.
We have seen that the common eigenvector set has to be
formed by even and odd vectors. Therefore, we will restrict the
search for the common set on the even and the odd spaces. To do
that, we will introduce a matrix
that decomposes an arbitrary
vector
into its even and odd components. The matrix, as
defined below, maps the even part of the
-dimensional vector
to the first components and the odd part to the
remaining components.
1
For example, the matrix of dimen-
sion 5 is
(17)
The first three components of
(18)
represent the components of the even part of
. Similarly,
the remaining two components represents the odd part of
.
We remember that the arguments are interpreted modulo
,as
in the study of the ordinary DFT. In addition, note that the
matrix is both symmetric and unitary, that is, .
If we consider the similarity transformation
,weex-
pect the resultant matrix to be in the block diagonal form, that is
(19)
Otherwise, the
matrix cannot have all even/odd eigenvector
set. It is clearly seen that eigenvectors of
can be de-
termined separately from
and matrices, and the corre-
sponding eigenvectors of
are simply the even/oddextension of
1
b
x
c
is the greatest integer less than or equal to the argument.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(16)
CANDAN et al.: DISCRETE FRACTIONAL FOURIER TRANSFORM 1333
the eigenvectors of . Therefore, the problem of finding
the common eigenvector set is reduced to finding the eigenvec-
tors of the
and matrices.
The
matrix has a tridiagonal structure, except the two
entries at the upper-right and lower-left corners. After the sim-
ilarity transformation,
becomes exactly tri-diagonal,
meaning that the submatrices
and are also tridiagonal.
From [45], we know that tridiagonal matrices have distinct
eigenvalues, and this result implies the uniqueness of the
eigenvectors of the
and matrices and completes the
proof of the uniqueness of the common eigenvector set of
and the DFT matrix.
Note that in [38], the matrix
is employed as a vehicle
to obtain an orthogonal eigenvector set for the DFT matrix. The
authors of [38] conjecture that the eigenvalues of
are
distinct whenever the dimension of
is not a multiple of
4. In the present paper, we show that the eigenvalue degeneracy
of
is at most two, and even though there exist many ways of
choosing these two orthogonal eigenvectors, there is only one
way to choose them such that one is an even vector and other is
an odd vector. This argument uniquely determines the common
set of eigenvectors of
and the DFT matrix, whatever the di-
mension of
.
C. Ordering the Eigenvectors of
In the previous sections, we have shown the existence and
uniqueness of the common eigenvector set of
and the DFT
matrix. In this section, we will determine which eigenvector of
this set corresponds to which Hermite–Gaussian function. We
will order (index)the eigenvectorset in a manner consistent with
the ordering of the continuous Hermite–Gaussians. Our method
of ordering will be based on the zero-crossings of the discrete
Hermite–Gaussians, in analogy with the zeros of the continuous
Hermite–Gaussians.
We will first clarify what we mean by the zero crossing of
a discrete vector. The vector
has a zero crossing at if
. In counting the number of zero crossings of
the periodic sequence
with period , we count the number
of zeros in the period
, where we also include
the zero crossing at the endpoints of the period such as when
[48]. With this convention,
it can be seen that the shifted periodic sequences have the same
number of zero crossings, regardless of the shift. Therefore, the
number of zero crossings becomes a property of the periodic
sequences and not just a property of a particular period.
Now, we need to show that all eigenvectors have a distinct
number of zero crossings and need to establish a convenient
method for counting the zero crossings. The exhaustivecounting
of the zero crossings can be numerically problematic due to the
difficulty of determining the sign of a component which is of
small magnitude.
To find the eigenvector
with zero crossings, we will
combine two results from [45]. As discussed before, the
common eigenvectors of
and the DFT matrix can be derived
from the eigenvectors of the tridiagonal
and matrices.
An explicit expression for the eigenvectors of tridiagonal
matrices is given in [45, p. 316]. Combining this expression
with the Sturm sequence theorem [45, p. 300], we can show
that the eigenvector of the
or matrix with the highest
eigenvalue has no zero crossings, the eigenvector with the
second highest eigenvalue has one zero crossing, and so on
[48]. Therefore, we can show that the
and matrices
have eigenvectors whose number of zero crossings range from
0to
and from 0 to , respectively [48].
Since the evenand odd eigenvectors of
are derived from the
zero-padded eigenvectors of
and , we can show that after
zero padding and transformation through
, which is nothing
but even/odd extension of the vector, the eigenvector of
with
zero crossings yields the even eigenvector of with
zero crossings, and the eigenvector of with
zero crossings yields the odd eigenvector of with
zero crossings. This eigenvalue-based
procedure enables us to accurately determine the number of zero
crossings of an eigenvector without employing any means of
counting. [48].
An even eigenvector of
can be formed as
.
.
.
, where is the eigenvector of with
zero-crossings . It can be seen at this point
that the vector
.
.
.
(20)
has
zero crossings, when the above convention of zero
crossing counting is exercised.
zero crossings
.
.
.
zero crossings
(21)
where
. The same result can also be shown for
.
Similarly, odd eigenvectors of
are derived from the
eigenvectors of
by zero padding and transformation:
. It can be further shown that the odd
eigenvector
derived from the eigenvector of with
zero crossings is an eigenvector of with zero crossings
[48].
This procedure not only enables us to accurately determine
the number of zero crossings but also demonstrates that each of
the eigenvectors of
has a different number of zero crossings
so that each vector can be assigned an index equal to its number
of zero crossing. The index number
spans different ranges,
depending on the parity of
, that is, for
odd
and for even .
A numerical comparison of the continuous and discrete Her-
mite–Gaussians is presented in Fig. 1. A comparison of the
known properties of discrete and continuous Hermite–Gaus-
sians is presented in Table I. These completely analogous prop-
erties strengthen our belief that discrete time counterparts of the
other properties of continuous time Hermite–Gaussians can be
obtained.
In conclusion, there is a well-defined procedure for finding
and ordering the common eigenvector set of the matrix
and
the DFT matrix, such that the
th member of this eigenvector
