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Signal Processing

Signal Processing 91 (2011) 571–581

0165-16

doi:10.1

� Cor

fax: +9

E-m

(A. Ser

journal homepage: www.elsevier.com/locate/sigpro

The discrete fractional Fourier transform based on the DFT matrix

Ahmet Serbes �, Lutfiye Durak-Ata

Yildiz Technical University, Department of Electronics and Communications Engineering, Yildiz, Besiktas, 34349 Istanbul, Turkey

a r t i c l e i n f o

Article history:

Received 4 November 2009

Received in revised form

4 May 2010

Accepted 5 May 2010Available online 16 May 2010

Keywords:

Discrete fractional fourier transform

DFT matrix

Hermite–Gauss functions

Eigentransform matrices

Rotation property

84/$ - see front matter & 2010 Elsevier B.V. A

016/j.sigpro.2010.05.007

responding author. Tel.: +90 212 383 24 98;

0 212 383 24 86.

ail addresses: ahmet.serbes@gmail.com, aserb

bes), lutfiye@ieee.org (L. Durak-Ata).

a b s t r a c t

We introduce a new discrete fractional Fourier transform (DFrFT) based on only the DFT

matrix and its powers. Eigenvectors of the DFT matrix are obtained in a simple-yet-

elegant and straightforward manner. We show that this DFrFT definition based on the

eigentransforms of the DFT matrix mimics the properties of continuous fractional

Fourier transform (FrFT) by approximating the samples of the continuous FrFT. By

appropriately combining existing commuting matrices we obtain a new commuting

matrix which performs better. We show the validity of the proposed algorithms by

computer simulations comparing DFrFT points and continuous FrFT samples for various

signals.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

The fractional Fourier transform (FrFT) has been popularin signal processing [1], time–frequency analysis [2,3],filter design [4], signal compression [5], parameter estima-tion [6,7], and pattern recognition [8]. As it is a powerfultool in signal processing, a discrete definition of FrFT whichinherits the properties of its continuous counterpart is ofmuch interest in recent years. Santhanam et al. [9] defineda discrete fractional Fourier transform (DFrFT) by usingTaylor series expansion of the DFT matrix followed by theapplication of Cayley–Hamilton theorem and definedDFrFT as a weighted sum of its powers. Howeverthe definition fails to approximate the samples of thecontinuous FrFT, because the proposed algorithm has onlyfour distinct eigenvalues for any transform order whereasthe FrFT has more than four distinct eigenvalues fornon-integer orders.

Except for the work of Santhanam et al., the early workon DFrFT can mainly be split into two major groups. The firstapproach is based on the S matrix introduced by Dickinson

ll rights reserved.

es@yildiz.edu.tr

et al. [10], which is an almost-tridiagonal matrix commutingwith the DFT matrix. As the S matrix commutes with theDFT matrix, they share at least one set of eigenvectors incommon. Candan et al. used the second-order differenceequation by approximating the second-order differentialequation in which the homogenous solution set isthe Hermite–Gauss functions [11]. Recently, Candan useshigher-order approximations to discrete derivative operatorto approximate Hermite–Gauss functions better [12].Instead, in [13] we suggest using infinite-order approxima-tion to second derivative operator and find excellentHermite–Gauss-like eigenvectors.

Pei et al. used the S matrix as a basis for obtaining anew basis, which is closer to the samples of Hermite–Gauss functions by employing Gram–Schmidt algorithm(GSA) and orthogonal Procrustes algorithm (OPA) [14].Specifically, eigenvectors of the S matrix is projected ontothe samples of the continuous Hermite–Gauss functionsand GSA is employed to get eigenvectors, where theperformance criteria is chosen as the norm of errorbetween the discrete Hermite–Gauss-like vectors andthe samples of the continuous Hermite–Gauss functions.OPA has been suggested as another technique, which isreasoned to minimize the Frobenius norm between theeigenvectors of the S matrix and the samples of theHermite–Gauss functions. However, Hanna et al. show in

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581572

[18] that any matrix other than S could also be used as aninitial matrix in this OPA and GSA methods withoutaffecting the result.

The second approach is based on the tridiagonal Tmatrix defined by Grunbaum [15], which is later onrefined by [16]. The T matrix is employed in combinationwith S matrix to furnish the basis of eigenvectors for theDFrFT matrix in [17] as S+kT, where k is an integer. Apartfrom these, Hanna et al. [18] have used the spectraltheorem to decompose the ordinary-DFT matrix andobtained four projection matrices Pi for i=1,2,3,4. Theindependent columns of Pi are used as a basis for theDFrFT and this technique is called as P method. However,when used alone, the P method produces inaccurateresults and the norm of error vector between the samplesof continuous Hermite–Gauss functions and the orthogo-nal eigenvectors obtained by the P method are very high.To overcome inaccurate results, authors have employedOPA and sequential-OPA to minimize the error.

In this work, we focus on a different approach to deriveeigenvectors of the DFT matrix in order to build the DFrFTmatrix that mimics the properties of its continuouscounterpart. A straightforward and refined derivation ofthe eigenvectors of the DFT matrix is proposed withoutusing any commuting matrices, but using only thefactored form of a basic property of the DFT matrixstating that four consecutive Fourier transforms is theidentity transform, i.e.,

W4N ¼ IN ð1Þ

where WN and IN are the centered DFT (CDFT) and theidentity matrices of order N, respectively. We utilizefactorization of the CDFT matrix to obtain new matrices,whose columns are eigenvectors of the CDFT matrix.We thereafter employ GSA to find the orthonormaleigenvectors obtained by factorization of the CDFT matrix.By employing a DFT-shift matrix K, eigenvectors ofthe ordinary-DFT matrix are obtained by using theCDFT matrix. We have also used the S and T matricestogether with our method later on to boost theperformance.

In [14], Pei et al. compute the samples of the Hermite–Gauss functions and employ GSA to orthogonalize thesesamples by projecting on the S matrix, which is equivalentto orthogonalizing the samples of the Hermite–Gaussfunctions without using the S matrix [18]. Hanna et al.claim to find the eigenvectors of the DFT matrix using asimilar method to that of our work. However, they use theordinary-DFT matrix as a basis to their work, but weemploy the CDFT matrix. Additionally, they do notorthogonalize the non-orthogonal eigenvectors and theyseem not to get eigenvectors similar to the samples ofHermite–Gauss functions. Our work is different fromthese works that we first find the non-orthogonaleigenvectors of the CDFT matrix without utilizing any Smatrix or samples of the Hermite–Gauss functions andthen we orthogonalize the obtained eigenvectors byemploying the GSA. We do not use any samples ofcontinuous Hermite–Gauss functions or project ourproposed eigenvectors onto other matrices.

This paper is organized as follows. Section 2 givesintroductory information on the FrFT and DFrFT. Sections 3and 4 describe how to find the eigenvectors of thecentered-DFrFT, the process to refine the eigenvectors toobtain a Hermite–Gaussian-like discrete eigenvector and toobtain the ordinary-DFrFT matrix. Section 5 compares theproperties of the proposed method with the continuousFrFT. The performance of the proposed method is boostedin Section 6 and simulation results are given in Section 7.The paper concludes in Section 8.

2. Preliminaries

In this section, we define the continuous FrFT, givesome of its properties, and present the underlyingprincipals of the DFrFT.

2.1. Continuous FrFT

The a th-order continuous FrFT is defined as a linearunitary operator acting on an integrable function f(u),

faðuÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�jcota

p Z 1�1

f ðuÞexpðjpðcotau02�2cscauu0

þcotau2ÞÞdu0 ð2Þ

where a¼ ap=2 is the transformation angle. The contin-uous FrFT kernel turns into the ordinary continuousFourier transform (FT) kernel and inverse FT kernel fora=1 and 3, respectively. For a=4, the kernel is the identityoperator and for a=2 it approaches to dðuþu0Þ in the limit.The FT has four distinct eigenvalues l 2 f1,�j,�1,jg, sincefour consecutive FT is the identity transform. Hermite–Gauss functions are eigenfunctions of the FT satisfying

F fcnðuÞg ¼ e�jnp=2cnðuÞ ð3Þ

where cnðuÞ is the n th-order Hermite–Gauss functionassociated with the eigenvalue e�jnp=2. The continuousFrFT satisfies the index additivity property [1]. From asignal processing prospect, one of the most importantproperties of the FrFT is its rotation property in time–frequency axis. FrFT rotates the time–frequency (orspace–frequency) axis with an angle proportional to thetransformation order a in counter-clockwise direction. Ananalogous statement of this property is that the FrFTrotates the signal in time–frequency axis of Wignerdistribution with angle a in the clock-wise direction [1].

2.2. Centered-DFT and discrete FrFT

The N-point CDFT matrix WN is defined as a unitarymatrix whose elements are

ðWNÞn,m ¼1ffiffiffiffiNp exp �j

2pNðn�cÞðm�cÞ

� �,

n,m¼ 0,1, . . . ,N�1; c¼N�1

2ð4Þ

Let IN be the identity matrix of order N, the CDFT matrixsatisfies (1) and since four consecutive DFT operationscorrespond to the identity transform, the CDFT matrix hasfour distinct eigenvalues l 2 f1,�j,�1,jg as well. Usingthe eigenvalue decomposition, WN can be written in

Table 1Multiplicities of the eigenvalues of the N �N CDFT matrix.

N l

1 �j �1 j

4m m m m m

4m+1 m+1 m m m

4m+2 m+1 m+1 m m

4m+3 m+1 m+1 m+1 m

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581 573

the form of

WN ¼UNKNUTN ð5Þ

where ð�ÞT is the transpose operator, U is the real Hermite–Gauss-like eigenvectors of the CDFT matrix and K is thediagonal matrix containing the eigenvalues. As the con-tinuous-time FrFT is a generalization of the FT with anorder parameter a where a 2 ½0, 4], we generalize theeigendecomposition-type FT expression into DFrFT bytaking the ath fractional power of WN matrix.

3. Eigenanalysis of the discrete fractional Fouriertransform

The DFT maps the signal from [0, N�1] discrete inputspace to [0, 2p] discrete frequency space, whereas theCDFT maps [�(N�1)/2, (N�1)/2] to ½�p, p�. This allowsCDFT to define even and odd-functions such as Hermite–Gauss functions, that is contrary to the ordinary DFTdefinition. In order to approximate the samples of thecontinuous FrFT and to imply the rotation property,Hermite–Gauss-like eigenvectors of the DFT matrix haveto be obtained.

In this work, we use the CDFT matrix instead of theordinary DFT matrix because the Hermite–Gauss-likevectors are eigenvectors of only the CDFT matrix, not theDFT matrix, i.e., when a discrete Hermite–Gauss-likeeigenvector is transformed using the ordinary DFT matrix,the output is the shifted version of the Hermite–Gaussvector multiplied by a complex sinusoidal.

Proposition 1. The Hermite–Gauss-like eigenvectors of the

CDFT matrix are

V1 ¼1

2ðRfWNgþðRfWNgÞ

2Þ ð6aÞ

V2 ¼�1

2ðRfWNg�ðRfWNgÞ

2Þ ð6bÞ

V3 ¼1

2ðIfWNgþðIfWNgÞ

2Þ ð6cÞ

V4 ¼�1

2ðIfWNg�ðIfWNgÞ

2Þ ð6dÞ

where columns of V1, V2, V3, and V4 are eigenvectors

associated with the eigenvalues 1, �1, j, and �j, respectively.R and I denote the real and imaginary parts and WN is the

CDFT matrix as defined in (4).

Proof. Re-arranging (1) as WN4�IN = 0 and rewriting it in

its factored form as

ðWN�INÞðWNþINÞðWN�jINÞðWNþ jINÞ ¼ 0N ð7Þ

leads us to a very important result when rewritten in fourdifferent eigenequation forms for the DFT as stated earlierby Bose [19]. As for the eigenvectors associated with l¼ 1,we obtain the eigenvalue problem

ðWN�INÞðW3NþW2

NþWNþINÞ ¼ 0N ð8Þ

Hence, the eigenvector matrix whose columns include theeigenvectors is expressed by

V1 ¼ ðW3NþW2

NþWNþINÞ ð9aÞ

which satisfies WNV1=V1 considering WN4 = IN. We note

that some, but not all columns of V1 are linearlyindependent. The other matrices whose columns includeeigenvectors of the CDFT matrix can be obtained in thesame manner

V2 ¼ ðW3N�W2

NþWN�INÞ ð9bÞ

V3 ¼ ðW3Nþ jW2

N�WN�jINÞ ð9cÞ

V4 ¼ ðW3N�jW2

N�WNþ jINÞ ð9dÞ

Obtained matrices are not full-rank and columns of thesematrices are not linearly independent either, but theireigenvalues are of modulus 4, since V1, V2, V3, and V4 keepthe eigenvectors with the associated eigenvalues only. Letus consider V1=WN

3 +WN2 +WN+IN which can be expressed

more explicitly as

V1 ¼XN�1

n ¼ 0

ðe3ðjnp=2Þ þe2ðjnp=2Þ þejnp=2þ1Þvn ¼XN�1

n ¼ 0

4v4n

ð10aÞ

since WN ¼PN�1

n ¼ 0 ejnp=2vn, where vn represents discreteHermite Gaussian eigenvectors. Similarly using (9b)–(9d)V2, V3 and V4 are expressed as

V2 ¼XN�1

n ¼ 0

�4v4nþ2 ð10bÞ

V3 ¼XN�1

n ¼ 0

ð4jÞv4nþ3 ð10cÞ

V4 ¼XN�1

n ¼ 0

ð�4jÞv4nþ1 ð10dÞ

Consequently, columns of V1 keep only Hermite–Gauss-like eigenvectors of order 4n and analogously V2, V3 andV4 hold only Hermite-Gaussians of order 4n+2, 4n+3, and4n+1, respectively. Number of independent columns (orrank) corresponds to the multiplicities of the eigenvalues,shown in Table 1 [20]. Note that V1 and V2 are pure realand V3 and V4 are pure imaginary, since WN

3 +WN and WN2

are pure real and WN3�WN is pure imaginary

W3NþWN ¼ 2RfWNg ð11aÞ

W3N�WN ¼�2jIfWNg ð11bÞ

W2NþIN ¼ 2ðRfWNgÞ

2ð11cÞ

W2N�IN ¼�2ðIfWNgÞ

2ð11dÞ

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581574

Substituting appropriate terms in (9) and (11), we obtain

V1 ¼ 2ðRfWNgþðRfWNgÞ2Þ ð12aÞ

V2 ¼ 2ðRfWNg�ðRfWNgÞ2Þ ð12bÞ

V3 ¼�2jðIfWNgþðIfWNgÞ2Þ ð12cÞ

V4 ¼�2jðIfWNg�ðIfWNgÞ2Þ: ð12dÞ

Hermite–Gauss-like eigenvectors have to be real andunitary. In order to obtain real and unitary eigenvectors,we divide V1, V2, V3, and V4 by 4, �4, 4j and �4j,respectively, since these eigenvectors have eigenvalues ofmodulus 4. The proof is complete. &

Notice that all four Vi are commuting with the CDFTmatrix and therefore have the eigenvectors divided intofour multiplicities. However, since columns of thesematrices are not linearly independent, they cannot be leftin this form. The acquired eigenvectors that are columnsof Vi, for i=1–4 are employed in the calculation of theDFrFT matrix, which is discussed in the following section.

4. The DFrFT matrix

The FrFT operator of order a can be defined as the a th-power of the ordinary DFT operator WN. Hence, the DFrFTmatrix can be expressed by means of its eigenvectordecomposition,

WaN ¼UNKa

NUTN ð13Þ

where KaN is explicitly

KaN ¼ diagðe�j0,e�jp=2a, . . . ,e�jp=2aðN�2Þ,e�jp=2aðN�1ÞÞ ð14Þ

for the centered-DFrFT [20].As the columns of the matrices stated in (6) are not

linearly independent (or full rank) an easy and quick wayto obtain linearly independent and orthonormal columnsof them is to

1.

Compute the reduced echelon form of the matrices. 2. Pick only the columns associated with the pivots. 3. Employ an orthogonalization algorithm, e.g., Gram–

Schmidt orthogonalization (GSO) algorithm.

We employ Gauss–Jordan elimination method to find thepivots and choose the linearly independent columns, there-fore throw away some columns for each Vi, for i = 1–4. Then,we employ the celebrated modified GSO (m-GSO) algorithmto obtain the normalized linearly independent orthonormaleigenvectors [21].

Recall that the reason of employing CDFT matrix is that,discrete Hermite–Gauss-like vectors are eigenvectors of theCDFT matrix, but the eigenvectors of DFT matrix are phase-shifted versions of the discrete Hermite–Gauss vectors.Applying the same method directly to the DFT matrix willproduce incorrect results. Hanna et al. have derived similarequations to (9), however they have used the conventionalDFT matrix and the spectral theorem and they have notobtained the linearly independent columns by utilizing anorthogonalization algorithm in [18]. The eigenvectors that

they have come up are completely different and conse-quently incorrect results are obtained.

4.1. Modified Gram–Schmidt algorithm

Despite the fact that the matrices stated in (6) areorthogonal to each other, columns of each individualmatrix are neither linearly independent nor orthonormal.The Hermite–Gauss-like orthogonal CDFT eigenvectorscan be obtained by employing the m-GSO. Let

�

y: non-orthonormal columns of Vi after Gauss–Jordanelimination. (There are ki columns for each Vi.) � v: orthonormal columns obtained by m-GSO. � ki: number of pivots in Vi,

then for each Vi the algorithm is summarized in Algorithm 1.The m-GSO algorithm orthogonalizes any set of vectors byusing projections and subtractions. As a first step, a vector istaken and the projections of the remaining vectors aresubtracted from the vector, following a normalization step.The process is completed when this operation is carried outfor all vectors.

Algorithm 1. The modified Gram–Schmidt algorithm.

for n=1 to ki do

for m=1 to n do

yn ¼ yn�/ym ,ynS/ym ,ymS ym

end for

vn ¼yn

JynJ

end for

4.2. The centered-DFrFT matrix

Let V i, i=1,2,3,4 be the new linearly independentorthonormal eigenvector set after the m-GSO algorithm.The size of the V i matrix including the new eigenvectors isN � ki, where ki is the rank of Vi, which is already at handafter Gauss–Jordan elimination process. ki can also becalculated by referring to the multiplicities of eigenvalues,which is shown in Table 1.

Obtaining the ordinary CDFT matrix: The ordinary CDFTmatrix can be obtained by calculating the weighted sumof four eigenvalue decomposition of V i associated withthe eigenvalues l¼ f1,�1,j,�jg

WN ¼V1Ik1V

T

1þV2ð�Ik2ÞV

T

2þV3ðjIk3ÞV

T

3þV4ð�jIk4ÞV

T

4

ð15Þ

where Ikiis the ki � ki identity matrix.

Sorting the eigenvectors: Sorting or indexing the eigen-vectors is straightforward in this method. As we employthe CDFT matrix itself to obtain the eigenvectors, we comeup with the sorted eigenvectors. There are n zero-crossingsin a Hermite–Gauss function of order n. Since the zero-crossings in the CDFT matrix is sorted in descending order,i.e., there are no zero-crossings in the middle and there areN�1 zero crossings in the first column, the eigenvectorsautomatically appear in sorted order after the Gauss–Jordan elimination. That is, columns of all four V i contain

1 This operation is also valid as a function named fftshiftð. . .Þ in

MATLABs.

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581 575

the highest order of discrete Hermite–Gauss-like eigen-vector in its first column and the lowest order of discreteHermite–Gauss-like eigenvector in its last column. Forexample, V1 contain the zeroth-order Hermite–Gaussvector in its last column and 4(k1�1)-th order ofHermite–Gauss vector in its first column. The othercolumns in between are sorted in order of

V1 ¼

j j j j

h4k1�4 . . . h8 h4 h0

j j j j

264

375 ð16aÞ

In a similar manner,

V2 ¼

j j j j

h4k2�2 . . . h10 h6 h2

j j j j

264

375 ð16bÞ

V3 ¼

j j j j

h4k3�1 . . . h11 h7 h3

j j j j

264

375 ð16cÞ

V4 ¼

j j j j

h4k4�3 . . . h9 h5 h1

j j j j

264

375 ð16dÞ

are formed. Hence the associated eigenvalues lIkiin (15)

can be written explicitly for all l

Ik1¼ diagðe�j2pðk1�1Þ, . . . ,e�j2p,e�j0Þ

�Ik2¼ diagðe�jðpþ2pðk2�1ÞÞ, . . . ,e�j3p,e�jpÞ

jIk3¼ diagðe�jð3p=2þ2pðk3�1ÞÞ, . . . ,e�j7p=2,e�j3p=2Þ

�jIk4¼ diagðe�jðp=2þ2pðk4�1ÞÞ, . . . ,e�j5p=2,e�jp=2Þ ð17Þ

Obtaining the centered-DFrFT matrix: The centered-DFrFTmatrix can be obtained by combining (13), (9) and (17) as

WaN ¼V1K

a

k1V

T

1þV2Ka

k2V

T

2þV3Ka

k3V

T

3þV4Ka

k4V

T

4 ð18Þ

where

Ka

k1¼ diag e�j2pðk1�1Þa, . . . ,e�j2pa,e�j0

� �ð19aÞ

Ka

k2¼ diag e�jðpþðk2�1Þ2pÞa, . . . ,e�j3pa,e�jpa

� �ð19bÞ

Ka

k3¼ diag e�j 3p=2þðk3�1Þ2pð Þa, . . . ,e�j7p=2a,e�j3p=2a

� �ð19cÞ

Ka

k4¼ diag e�jðp=2þðk4�1Þ2pÞa, . . . ,e�j5p=2a,e�jp=2a

� �ð19dÞ

As V i are orthonormal to each other, (18) implies that theindex additivity rule is supported:

Wa1

N Wa2

N ¼ ðV1Ka1

k1V

T

1þV2Ka1

k2V

T

2þV3Ka1

k3V

T

3

þV4Ka1

k4V

T

4ÞðV1Ka2

k1V

T

1þV2Ka2

k2V

T

2þV3Ka2

k3V

T

3

þV4Ka2

k4V

T

4Þ ¼ ðV1Ka1þa2

k1V

T

1þV2Ka1þa2

k2V

T

2

þV3Ka1þa2

k3V

T

3þV4Ka1þa2

k4V

T

4Þ ¼Wa1þa2

N ð20Þ

In Section 5, the proposed method is tested whether theproposed centered-DFrFT approximates the samples of itscontinuous counterpart.

4.3. Obtaining the DFrFT matrix

There is a close relationship between the CDFT andordinary-DFT matrices; they can be transformed to eachother in some circumstances. The only difference betweenthe CDFT and the ordinary-DFT is the offset in the locationof the signal. The transform domain signal by theordinary-DFT is the shifted version of the CDFT-domainsignal. When CDFT of an odd-length signal is taken andthe left and right halves of the signal is swapped, theordinary-DFT-domain of the signal is obtained. We callthis operation the DFT-shift.1

Let WN and FN be the N � N CDFT and ordinary-DFT matrices, respectively. We analyze the relationshipbetween them for N odd and even separately. For odd N, FN

is related with WN through the permutation matrix KN as

FN ¼KNWNK�1N ð21Þ

where the DFT-shift permutation matrix KN can bewritten by

KN ¼0 Ik

IN�k 0

" #ð22Þ

where Ik and IN�k are k� k and ðN�kÞ � ðN�kÞ identitymatrices and k=(N+1)/2.

In this case, the permutation matrix is equal to itsinverse, i.e., KN=KN

�1. Eq. (21) is reduced to FN = KNWNKN.Therefore, FN and WN are similar matrices. Hence, theireigenvalues are the same and their eigenvectors arerelated through KN matrix. KN swaps left and right halvesof the signal.

The discrete Hermite–Gauss vectors are not eigenvec-tors of the ordinary-DFT matrix, since they are eigenvec-tors of the CDFT matrix. Let hn be the n th-order discreteHermite–Gauss-like vector, the ordinary-DFT transformsatisfies

FNKNhn ¼ lnKNhn ð23Þ

as the eigenvectors of the ordinary-DFT transform iscircularly shifted versions of the Hermite-Gaussians. Here,ln is the eigenvalue associated with the eigenvector hn.

For N even, we have to define WN in a different way asc¼N=2þ1 in (4), to satisfy (23). KN in (22) is modified to

KN ¼0 IN=2

IN=2 0

" #ð24Þ

where IN/2 is an N=2� N=2 identity matrix. When theCDFT matrix is defined accordingly, multiplicities of theeigenvalues change. However, new multiplicities ofeigenvalues are exactly the same as the ordinary-DFTmatrix and computations have to be done by taking theminto consideration (see Table 2). A comparison of Tables 1and 2 shows that the multiplicities of eigenvalues remain

Table 2Multiplicities of the eigenvalues of the N �N ordinary DFT matrix.

N l

1 �j �1 j

4m m+1 m m m�1

4m+1 m+1 m m m

4m+2 m+1 m m+1 m

4m+3 m+1 m+1 m+1 m

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581576

the same for odd N, while it is different for even N. Eq. (19)is modified into Ka

N ¼ diagðe�j0,e�jðp=2Þa, . . . ,e�jðp=2ÞaðN�2Þ,e�jðp=2ÞaNÞ by making a by-pass in the last eigenvalue.Hence, for N=4m+2 we only change (19b) to

Ka

k2¼ diagðe�jðpþk22pÞa,e�jðpþðk2�2Þ2pÞa, . . . ,e�j3pa,e�jpaÞ

ð25aÞ

and for N=4m it is sufficient that only (19c) is changed to

Ka

k4¼ diagðe�jðp=2þk42pÞa,e�jðp=2þðk4�2Þ2pÞa, . . . ,e�j5p=2a,e�jp=2aÞ

ð25bÞ

Eq. (21) also holds for even N. To sum up, the ordinary-DFrFT with order a can be expressed by

FaN ¼KNWa

NK�1N ð26Þ

The acquired eigenvectors V i, for i = 1–4 are employedin the calculation of the DFrFT matrix, which is discussedin the following section.

5. Numerical verification of the proposed V method

The proposed V method satisfies unitarity and angleadditivity properties. In parallel to the continuous FrFT[1], it reduces to the ordinary DFT when a = 1 and itapproximates to the samples of the continuous FrFT forfractional values of a. To numerically verify, we havetransformed a discrete square-wave signal x½n� of lengthN=73

x½n� ¼1 if �6rnr6

0 otherwise

�ð27Þ

Fig. 1 shows DFrFRT of the rectangular function of varioustransform orders and Fig. 2 shows the samples of thecontinuous FrFT of x[n] for the same transform orders.As it is clear from the figures, the proposed methodapproximates the samples of its continuous counterpart.Table 3 shows the normalized error between the samplesof the signal using continuous FrFT and the proposedDFrFT in which the error is defined as

Error¼Jxa�xaJ

JxJð28Þ

where J � J is the norm operator, xa and xa are the samplesof continuous FrFT and the proposed DFrFT of the signalx of order a. Table 3 also gives a comparison of thenormalized errors when the S, ðSþ15TÞ as in [17], andnew ðSþ30T�7VT Þ methods are utilized. A detailedpresentation and discussion of ðSþ30T�7VT Þ method isgiven in Section 6. The derivations of both S and Tmatrices are given in Appendices A and B, respectively.

6. Boosting the performance

Performance of the proposed algorithm can be im-proved by using it in combination with S and T matrices.Pei et al. [17] uses the S method together with theGrunbaum’s T matrix as a linear combination, in which ithas been observed that Hermite–Gauss-like eigenvectorsof (S+15T) are more accurate than both S and T alone.

In this section we suggest determining eigenvectors oflinear combinations of S, T, VT matrices to improve theperformance by defining a new commuting matrix

VT ¼VNKTVT

N ð29Þ

where VN is the N � N sorted eigenvectors in theascending order of zero-crossings obtained by the pro-posed method. KT contains sorted eigenvalues of theT matrix in its diagonal, and ð�ÞT is the transpose operator.

Proposition. VT commutes with the DFT matrix, i.e.,VTWN=WNVT.

Proof. As the columns of VN are eigenvectors of the DFTmatrix, then the DFT matrix can be written as

WN ¼VNKWVTN ð30Þ

where KW is the diagonal matrix containing the eigenva-lues of the DFT matrix. VTWN can be expressed as

VTWN ¼VNKTVTNVNKWVT

N ¼ VNKTKWVTN ¼ VNKWKTVT

N

¼ VNKWVTNVNKTVT

N ¼WNVT ð31Þ

which concludes the proof. &

T matrix is tridiagonal and has discrete Hermite–Gauss-like eigenvectors similar to the proposed eigenvectors.Since we interchange eigenvectors of the T matrix withour proposed eigenvectors, the output VT will be similarto the T matrix. The VT is found to be nearly tridiagonalwith large-magnitude values on three of its diagonals andsmall values on the others.

We propose that eigenvectors of linear combinationsof S, T, and VT matrices as k1S+k2T+k3VT produce moreaccurate eigenvectors. Performance criteria is chosen asnorms of error between the samples of continuousHermite–Gauss functions and the discrete Hermite–Gauss-like eigenvectors.

We have done computer experiments based on geneticalgorithms and pattern search and found out that whenk1 = 1, k2 = 30, and k3 = �7 total norm of error isapproximately minimum. Computation of the eigenvec-tors of the proposed S+30T�7VT method can be donesimilar to [17].

Pei et al. showed that using the linear combination of Sand T as S+15T produces better results than the S or Talone. We also show that using S+30T�7VT produces amore accurate approximation to the continuous FrFT.Using (28) as a measure of error, Table 3 showscomparison of the error for a square function. The resultsshow that S+30T�7VT approximate the continuous FrFTbetter than S, V , or S+15T. Comparative presentation ofperformances of the proposed methods are shown in thefollowing section in detail.

Fig. 1. DFrFT of a discrete rectangular function by the proposed method at various transform orders. Solid: real part, dashed: imaginary part.

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581 577

Fig. 2. Samples of the continuous FrFT of the rectangular function at various transform orders. Solid: real part, dashed: imaginary part.

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581578

Table 3Normalized error between the continuous FrFT and the DFrFTs for the

signal and orders given in Fig. 1.

a Error

V S S+15T S+15T�7VT

0.05 0.0829 0.0612 0.0181 0.0160

0.10 0.1481 0.1155 0.0282 0.0193

0.15 0.1866 0.1524 0.0535 0.0412

0.25 0.2017 0.1538 0.0686 0.0482

0.50 0.3156 0.2038 0.0764 0.0541

0.75 0.2048 0.1567 0.0625 0.0437

1.00 0.0000 0.0000 0.0000 0.0000

Fig. 3. Error norms of the discrete Hermite–Gauss like eigenvectors for

the proposed method (V method) and the S matrix approach for N=33.

Fig. 4. Error norms of the discrete Hermite–Gauss like eigenvectors for

the proposed method (V method) and the S matrix approach for N=65.

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581 579

7. Simulations

We have proposed two techniques for DFrFT.

Fig. 5. Error norms of the discrete Hermite–Gauss like eigenvectors for

the S+15T�7VT method and S+15T matrix approach for N=33.

1. First method is the V method, where V is the m-GSA

output of (6) as discussed in Section 3.

2.

Fig. 6. Error norms of the discrete Hermite–Gauss like eigenvectors for

the S+15T�7VT method and S+15T matrix approach for N=65.

The second method depends on S+30T�7VT matrix, aspresented in Section 6.

Fig. 3 shows the performance of the proposed method inwhich the norm of error vectors between the samples ofcontinuous Hermite–Gauss functions and the ortho-normal eigenvectors are chosen as the performancecriteria. The performance of the proposed V method iscompared to the performance of the S method [11] forN=33. Even though the S method performs better forsmall orders, the V method is better for middle-orders.Comparison of the S method and V method for N=65 isshown in Fig. 4.

Norms of error vectors between the samples of thecontinuous Hermite–Gauss functions and the eigenvec-tors of the S+15T method [17] and the eigenvectors ofthe proposed (S+30T�7VT) method for N=33 arecompared in Fig. 5. It is clear that the proposed(S+30T�7VT) method outperforms compared to theS+15T method.

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581580

Comparison of the S+15T and (S+30T�7VT) ap-proaches for N=65 is shown in Fig. 6. The latter approachwith VT outperforms the former in terms of the errornorm.

8. Conclusions

In this work, we introduce a new way of obtainingan approximation to DFrFT matrix based on the idea thatfour consecutive Fourier transforms of a signal returnsthe signal itself. We employ a simple and elegantprocedure to establish the eigenvectors of CDFTmatrix by using only the CDFT matrix itself and itspowers. After obtaining the new set of eigenvectors ofthe CDFT matrix, we present a straightforward way totransform the centered-DFrFT matrix into the ordinaryDFrFT matrix. The simulation results show that theproposed method outperforms the S method for someorders.

To obtain a better set of Hermite–Gauss-like discreteeigenvectors, we generate a new commuting matrix VT

and appropriately combine it with the other commutingmatrices S and T. Computer experiments show that thenew commuting matrix (S+30T�7VT) generates closersamples to the samples of Hermite–Gauss functions thandeveloped before.

Acknowledgment

The authors would like to thank Dr. Cagatay Candan foruseful discussions on the S method.

Appendix A. The S matrix

The S-matrix approach is a discrete solution to thesecond-order Hermite–Gauss-generating differentialequation [11]

d2

dt2hðtÞ�4pt2hðtÞ ¼ lhðtÞ ðA:1Þ

where h(t) is a continuous Hermite–Gauss function. UsingEuler’s formula for the differentiation

d2

dt2hðtÞ �

hðnþdÞ�2hðnÞþhðn�dÞd2

ðA:2Þ

T¼

1 0:5 0 . . . 0

0:5 cospN

� �� �2 cospN

cos2pN

2cosðp=NÞ. . . 0

0cos

pN

cos2pN

2cosðp=NÞcos

2pN

� �� �2

. . . 0

^ ^ ^ & ^

0:5 0 0 . . .cosðN�2Þp

Nc

2cosðp

2666666666666666664

where d¼ 1=ffiffiffiffiNp

. In order to obtain a linear transforma-tion matrix, the Taylor expansion

cosy¼ 1�y2

2þOðy4

Þ ðA:3Þ

is used as y2� 2ð1�cosyÞ. After appropriate substitutions

using t¼ nd

hðnþ1Þ�2hðnÞþhðn�1Þþ2 cos2pn

N

� ��1

� �hðnÞ ¼ lhðnÞ

ðA:4Þ

is obtained. Consequently the S matrix can be expressedas

S¼

�2 1 0 . . . 0 1

1 2cos2pN

� ��4 1 . . . 0 0

0 1 2cos2pN

2

� ��4 . . . 0 0

^ ^ ^ & ^ ^

1 0 0 . . . 1 2cos2pNðN�1Þ

� ��4

26666666666664

37777777777775

ðA:5Þ

The S matrix commutes with the ordinary DFT matrixand therefore they share at least one set of commoneigenvectors. To obtain the set of eigenvectors commutingwith the centered case we have to transform it by usingthe T-matrix given in Section 4.3. Consider a centereddiscrete Hermite–Gauss-like vector,

SKhðnÞ ¼KhðnÞ ðA:6Þ

where K is defined in (22). To obtain the centered Smatrix, both sides are multiplied with K�1,

ðK�1SKÞhðnÞ ¼ hðnÞ ðA:7Þ

Consequently, the centered version of the S matrix can beobtained by K�1SK.

Appendix B. The T matrix

The T matrix is defined as in (B.1). The derivation ofthis matrix is inspired from the work of Grunbaum [15]and is explained in [17] in detail. The S matrix choosesd¼ 1=

ffiffiffiffiNp

as a sampling rate. However in the derivation ofT matrix a smaller sampling rate d¼ 1=3

ffiffiffiffiNp

is chosen as asampling rate to approximate the second-order differen-tial equation in (A.1) better. After appropriate calculationsand approximations

0:5

0

0

^

osðN�1Þp

N=NÞ

cosðN�1Þp

N

� �� �2

3777777777777777775

ðB:1Þ

A. Serbes, L. Durak-Ata / Signal Processing 91 (2011) 571–581 581

cosðn�1ÞpN cosnp

N

cosðp=NÞhðn�1Þþ 2cos

npN�4

� �h ihðnÞ

þ

cosnpN

cosðnþ1ÞpN

cosðp=NÞhðnþ1Þ ¼ ð2l�4ÞhðnÞ ðB:2Þ

is obtained which produces the T matrix. The centeredversion of the T matrix can be obtained similar to (A.7)as K�1TK.

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