Multiplicity of FRFT

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 1, JANUARY 2000 227

Multiplicity of Fractional Fourier Transforms andTheir Relationships

Gianfranco Cariolaro, Member, IEEE, Tomaso Erseghe, Peter Kraniauskas, and Nicola Laurenti

AbstractThe multiplicity of the fractional Fourier transform(FRT), which is intrinsic in any fractional operator, has beenclaimed by several authors, but never systematically developed.The paper starts with a general FRT definition, based on eigen-functions and eigenvalues of the ordinary Fourier transform,which allows us to generate all possible definitions. The multi-plicity is due to different choices of both the eigenfunction and theeigenvalue classes. A main result, obtained by a generalized formof the sampling theorem, gives explicit relationships between thedifferent FRTs.

Index TermsFourier transform, fractional Fourier transform,sampling theorem.

I. INTRODUCTION

S INCE ITS inception in 1980 [1], [2] and rediscovery in1993 [3][5], the fractional Fourier transform (FRT) has be-come a fundamental tool for optical information processing andhas seen various developments in fiber optics [6], bulk optics[4], [7], [8], and signal processing [9]. As expected from anyfractional operator definition, various forms of the FRT haveemerged from the literature, but apart from some isolated at-tempts, no systematic treatment of the inherent multiplicity canbe found.

The customary FRT of a continuous-time signal has theform

(1)

where is the fraction, and ,, , where de-

notes the complex fourth root , with .In particular, for , the FRT becomes the ordinary Fouriertransform (FT)

and for , the kernel in (1) degenerates into the delta func-tion so that the FRT is the signal itself ,

Manuscript received May 28, 1998; revised June 2, 1999. The associate editorcoordinating the review of this paper and approving it for publication was Dr.Shubha Kadambe.

G. Cariolaro and N. Laurenti are with the Dipartimento di Elettronica ed In-formatica, Universit di Padova, Padova, Italy.

T. Erseghe was with Snell & Wilcox Ltd., Petersfield, Hampshire, U.K. He isnow with the Dipartimento di Elettronica ed Informatica, Universit di Padova,Padova, Italy.

P. Kraniauskas is an Independent Consultant, Southampton, U.K.Publisher Item Identifier S 1053-587X(00)00129-X.

where coincides with . Owing to the presence of chirp com-ponents, we shall identify the form (1) as the chirp FRT (CFRT).

A different form of the FRT was proposed in [10] as

(2)

that is, as a weighted combination of the given signal ,its ordinary FT and their reflected versions and

, where the weights are given by. For this reason, we shall describe

(2) as a weighted FRT (WFRT).1 The WFRT also has the mar-ginal properties of giving the ordinary FT for and thesignal for . This form of FRT has also been considered fordiscrete-time signals in connection with the DFT. For a discretesequence , with DFT , it takes the form [12]

(3)

which has also been discussed in [13, App.], where it was shownthat it cannot yield a discretized version of the CFRT. The possi-bility of two or even more different FRTs is inborn in the natureof fractional operators and has been claimed by several authors[9][11]. A discussion on multiple roots of the FT and identityoperators, mainly from an optical point of view, can be found in[14] and [15].

In a recent paper [16], we determined which signal classesadmit an FRT operator and gave the general FRT definitionfor all of them. Such classes include aperiodic continuous-vari-able signals, periodic discrete-variable signals, and their mul-tidimensional extensions. Following that general definition, inthis paper, we investigate the multiplicity of FRTs of contin-uous variable, giving explicit forms for the whole class of defi-nitions and, moreover, their relationships.

The paper is organized as follows. In Section II, we formu-late a general definition for the whole class of FRTs of con-tinuous variable. The approach is based on the eigenfunctionsof the ordinary FT. The multiplicity of FRTs is a consequenceof the possible choices of eigenfunctions and eigenvalues, andwe show that (1) and (2) correspond to two different choices. InSection III, we study the behavior of the FRT in the domain ofthe fraction for fixed values of frequency . This study clar-ifies several aspects related to bandlimitation and the samplingtheorem in the domain of the fraction. In Section IV, we obtainthe standard forms (1) and (2) of the FRT and their relationships,and in Section V, we introduce several examples of nonstandard

1Very recently, the WFRT was extended to 4L weights [11] (see SectionIV).

1053587X/00$10.00 2000 IEEE

228 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 1, JANUARY 2000

FRTs with the aim of showing how broad the class of FRTs isand how they all ultimately relate to the standard forms. Finally,the dependence on the eigenfunctions is discussed in SectionVI.

II. GENERAL FRT DEFINITION

In this section, we apply the general definition given in [16] tocontinuous-domain signals. Let be an operator generating anFRT, i.e., an operator that for any given fraction mapsa signal , into an FRT , . will be anFRT operator if it

1) is linear;2) verifies the FT condition ;3) has the additive property for every choice

of and .As stated in [16], these constraints assure certain fundamentalproperties for the operator , such as satisfying the marginalconditions , , , where

identity operator;reflection operator;ordinary inverse Fourier transform.

The constraints also imply periodicity in , with period 4. From 1), the FRT operator can be written, with a

suitable kernel , as

(4)

In terms of the kernel, the FT property 2) is expressed as. Moreover, the kernel becomes

when ;when ;

when ;and is periodic in with period 4. Both the CFRT (1) and theWFRT (2) belong to this class as their respective kernels are

(5a)

(5b)

Both satisfy the additive property and the marginal conditions:the former through the structure of the parameters , , and

and the latter through the properties of the weights .We shall see that the whole kernel class can be generated by

the eigenfunctions of the ordinary FT, which we now examinein detail.

A. Basis of Eigenfunctions and Diagonalization of OperatorFollowing the approach of [16], we consider a complete or-

thonormal basis of FT eigenfunctions with. We assume the basis to be ordered

so that we can write the corresponding FT eigenvalues as

(6)

Such a basis permits a signal expansion of the form

(7)

whose coefficients are given by

(8)

Considering that the FT of is , from (7), we getthe expansion

(9)

Inserting (8), we find

(10)

which, compared with (4), yields the expansion for the FT kernel

(11)

The above relationships lead to the decomposition of the FToperator as in

(12)

where

(13)

A signal theory interpretation of (12) is given in Fig. 1. Theoperator yields the coefficients for the signal expansioninto a series of eigenfunctions. As it works on a continuous-argument signal and produces a discrete sequence, it can bethought of as a sampling-like operator. The operator is diag-onal and performs a simple multiplication of the sequenceby the eigenvalues . Finally, converts the sequenceinto the continuous-argument transform and can thereforebe thought of as an interpolation-like operator. It can be easilyseen that the operators and are adjoints of each other(hence justifying our notation). Moreover, since the eigenfunc-tions are orthonormal, they are also the inverse of eachother, i.e., they are unitary operators, in symbols

, with the identity operator. Since all the eigenvalues haveunit amplitude, the operator is unitary as well.

Since is unitary, (12) represents the diagonalization ofthe FT operator. From the theory of linear operators in Hilbert

CARIOLARO et al.: MULTIPLICITY OF FRACTIONAL FOURIER TRANSFORMS AND THEIR RELATIONSHIPS 229

Fig. 1. Decomposition of the FT operator F on .

spaces [17], [18], we know that a unitary operator with only apoint spectrum can be decomposed in the form , whereis determined by the eigenfunctions and by the correspondingeigenvalues. Moreover, the decomposition is unique if, and onlyif, all the eigenvalues are distinct. This is not the case with theoperator , which has only four distinct eigenvalues. The de-composition (12) is thus not unique, and we can find other or-thogonal bases such that .

The best known basis for is given by the HermiteGauss(HG) functions [1], [19]

(14)

where is the th-order Hermitepolynomial. Recently, another basis has been identified [20],whose lowest order functions are

(15)

Different bases can be obtained by linear combinations of agiven basis. Specifically (see Section VI and Appendix E), wehave the following theorem.

Theorem 1: Given an orthonormal basis of FT eigenfunc-tions with associated eigenvalues , everyother orthonormal basis of FT eigenfunctionsis obtained by

(16)

where the coefficients satisfy the four-blockunitary condition

(17)

B. Construction of a General FRTStarting from the decomposition , we can con-

struct an FRT operator by replacing the diagonal operator byits th power, namely

(18)

where is the diagonal operator [see (13b)] : .The operator can be interpreted by the scheme of Fig. 1. Sub-stituting for , the scheme produces the FRT

of the input signal . We can see immediately that the de-fined by (18)

1) is linear;2) verifies the FT condition;3) has the additive property.

In fact, considering that , we have

where . Moreover, we have the following.a) Since is the identity operator, the inverse

FRT operator is .b) Consider that , is unitary; therefore, is

unitary.c) Equation (18) is a diagonalization of the unitary operation

.

d) Since is unitary for every , the Parseval relationholds for every .

e) The basis consists of eigenfunctions of theFRT, with eigenvalues .

To find an FRT kernel, it is thus sufficient to replace the eigen-values in (11) by their th power, to give

(19)

C. FRT Multiplicity, Generating and Perturbing SequencesThe multiplicity of the FRT is twofold: First, a real power of a

complex number is not unique. Considering (6), the possiblevalues of are given by

(20)

where is an arbitrary sequence of integers. Different choicesof lead to different kernels and, hence, to different FRT def-initions. We shall call the function

(21)

the generating sequence (GS) of the FRT. A GS can be ex-pressed in the alternative form2

(22)

where the are also arbitrary integers, and the two forms arerelated by . Table I collects examples of GSs,some of which may be of doubtful interest for applications, butare introduced to illustrate the wide variety of FRTs.

The second multiplicity arises because, as stated byTheorem 1, the basis is not unique. Its effect on the FRTdepends on the fraction and on the GS. This will be discussedin Section VI, where we will take the HG basis defined by(14) for reference and determine any other basis with theinfinite-dimensional matrix of Theorem 1. We willalso call a perturbing sequence (PS), with respect tothe HG basis, for which .

2(n) indicates nmodM , and bn=Mc indicates the integer part of n=M .


TABLE IEXAMPLES OF GENERATING SEQUENCES

FOR THE FRT

In conclusion, an FRT operator is unambiguously deter-mined by choosing a GS and a PS ,which may be indicated by rewriting (18) in the form

(23)

Once the ambiguity of has been resolved by choosing aspecific GS and the basis has been fixed, by choosing a specificPS, the FRT expression becomes unique, and the kernel is deter-mined by means of (19). In Namias [1], the implied choices werethe HG basis and , which lead to the CFRT (see SectionIV). The choice leads to the WFRT (2), which doesnot depend on the basis. We will see in Section IV that the choice

permits generalizing the WFRT to any order .

D. Properties of the General FRTIt is well known that many properties of the FRT are quite

different from those of the ordinary transform [3]. Some of theproperties that hold for every FRT were listed above in 1)3)and a)e). Further properties can be achieved with specific PSsand GSs. If the basis consists of real eigenfunctions (such asHG functions), the kernel satisfies ; fora real signal , this gives , which permitslimiting the study to . If the basis is HG and the GS is

(standard chirp FRT), the FRT is related to the Wignerdistribution and, moreover, turns out to be a canonical operator(see Section IV-D).

E. Summary of FRT DefinitionsWe choose an eigenfunction basis and a GS, , and form the corresponding FRT kernel

(24)

For any given signal decomposed as in (7), we could cal-culate the FRT as

(25)

However, evaluating the FRT according to (25) could be im-practical since every signal would require an evaluation of the

coefficients by (8) and a summation of the series (25). TheFRT can be calculated more directly from (4), provided that thekernel defined by (24) is available in closed form,, asit was for the standard cases (5).

The range of the GS and its cardi-nality play a fundamental role in the classification of pos-sible FRTs. For , we obtain a chirp FRT, for ,we obtain a weighted FRT of order . The kernel (24) can bewritten in terms of the set as

(26)

where the subkernels

(27)are independent of the fraction . As a consequence, the expres-sion of the FRT becomes

(28)

where the terms

(29)

are also independent of .The main problem is to find a closed-form solution of (26),

i.e., a result expressed with a finite number of terms, whereasthe above expression has infinite terms. A numerical evaluationwould be of doubtful interest as it would give no insight. Notethat the problem of finding a closed-form evaluation persists,even when the cardinality is finite. In fact, although (26)has a finite number of terms, it turns out that some subkernels(27) are expressed as infinite series. Further problems are theinterrelationships between FRTs generated by different GSsand different PSs. In the next three sections, we assume thatthe basis is given. The effect of changing the basis is discussedin Section VI.

III. ANALYSIS IN THE DOMAIN OF THE FRACTION

We solve two of the above problems by investigating the be-havior of the kernel in the domain of the fraction forthe different GSs . For this, we write the kernel (24) in theabbreviated form

(30)

where and and are considered fixed.For clarity, we denote the general period of by andits general frequency by , keeping in mind that and

. The periodic signal (30) can also be written as aFourier series [see (26)]

(31)


whose Fourier coefficients are given by

(32)

and when is empty, i.e., for . Note thatthe coefficients are independent of the GS , whereas theFourier coefficients depend on .

Inspection of (31) reveals the role of the cardinality ; itgives the number of degrees of freedom of the dependence onthe fraction , i.e., the number of harmonics that are present inthe signal . In other words, the kernel (26), with respect tothe fraction , is bandlimited for a WFRT and not bandlimitedfor a CFRT.

A. Sampling Theorem

It is well known that a periodic signal containing only con-secutive harmonics can be perfectly recovered from equallyspaced samples within a period . To formulate a more generalversion of this theorem, we need the following preliminaries.

The harmonic support of a periodic signal of the form (31)is defined by . The support can be ob-tained starting from the kernel form (30) of the periodic signaland, more specifically, is given by the image of the GS

. For instance, with , we find.

A unit cell of of size is any subset of with elementssuch that the sets represent a partitionof , i.e.

(33)For example, is a unit cell of size 8, and

are unit cells of size 8for every choice of . A unit cell may consist of noncon-secutive integers, e.g., is again aunit cell of size 8.

Bandlimitation of a periodic signal is stated in terms of itsharmonic support , but the formulation of the sampling the-orem requires the choice of a specific unit cell containing thesupport.

Theorem 2: Let be a periodic signal of period witha limited harmonic support , and let be a unit cell of size

containing , i.e., . Then, the signal can beperfectly recovered from samples , with ,according to

(34)

where the interpolating function is

(35)

The proof is given in Appendix A. In practice, is alwayschosen as the smallest unit cell containing the support

. When consists of consecutive integers, i.e.,, the interpolating

function takes the standard form

(36)

where sinc is the periodicversion of the sinc function.

Three examples of interpolating functions , obtainedwith three different unit cells of size , are illustratedin the complex plane in Fig. 2.

A marginal consequence of the sampling theorem is the pos-sibility of evaluating the nonzero Fourier coefficients from thesamples, namely (see Appendix A)

(37)

which states that the are obtainable from the DFT of thesignal samples.

B. Sampling RelationshipDifferent GSs lead to different kernels. However, the kernels

may have some samples in common.Theorem 3: Let

(38)

Then, and satisfy the sampling conditions, , if and only if

mod (39)The proof of the if part is trivial; for the only if, part seeAppendix B.

The preceding result can be generalized to sequences that areperiodic (mod )

mod (40)For instance, both the sequences andhave this property for any .

Theorem 4: Given two GSs and , if both are periodic(mod ) and the values of the first sequenceare distinct (mod ), then the samples of the can be ob-tained from the samples of as

(41)

where .


Fig. 2. Interpolating functions generated by three different unit cells of size N = 8.

The proof is given in Appendix C. Note that if, as in Theorem 3, then

, and .Hence, Theorem 4 generalizes Theorem 3.

C. Other RelationshipsWe continue the comparison of two kernels and

generated by different sequences and , as in (38). The fol-lowing results are direct consequences of expressions (30) and(31).

Shifting: If , then .Scaling: If (and it is necessary that

in order for to be itself a GS), then .Perturbation of a GS: If the two GSs are equal, ,

except for a point , where , then

(42)

This rule is easily extended to an arbitrary number of perturba-tions. If the perturbation is periodic, that is, for

( and for fixed), then

with

(43)

IV. STANDARD FRT FORMS

In this section, we assume that the basis is given by HG func-tions and investigate the FRTs generated by the sequences

1) 2)which yield the familiar forms of the FRT.

A. Standard Chirp FRTThe FRT generated by has the kernel

(44)

When the are the HG functions (14), the sum of the seriescan be derived in closed form using Mehlers expansion

(45)and yields the expression

(46)where , , and are the parameters of (1).


Fig. 3. CFRT kernel u (a) = (f; t) versus a for f and t fixed (f =1=2, t = 1).

We note from (44) that has an infinite number of har-monics so that it is not bandlimited. Fig. 3 illustrates asa function of for and fixed. Note the turbulent behavioraround the points and , where the kernel is repre-sented by the distributions and . This causesproblems in the numerical evaluation of the CFRT for around

. These can be overcome [13] by decomposing the FRT oforder into the cascade of an ordinary FT and an FRT oforder since the kernel is smooth around odd integers.

As an example of closed-form evaluation, we consider thestandard CFRT of the rectangular signal rect ,which can be expressed in terms of the Fresnel integral

as

forand can be obtained for a general by using symmetriesand periodicity. This transform is illustrated in Fig. 4 for dif-ferent values of the fraction with .3

B. Standard Weighted FRT of OrderFor the GS , the kernel (26) becomes

(47)

where the subkernels are given by (27) as

(48)The sum of this series is only known for (Mehlers ex-pansion) but not for the present case, where . Inspec-tion of (47) shows that has finite harmonic support

3This example was considered in [9] and [11] but without giving aclosed-form expression for S (f).

Fig. 4. Example of CFRT on for several values of a. Complex transformsare plotted with the real part in solid, and the imaginary part in dashed lines.

. We can therefore apply the sam-pling theorem, which assures the reconstruction of from

samples , with , to find

(49)

where the interpolating function is given by (36). We are thusable to capture the full -dependence of the WFRT kernel bymeans of the interpolation formula, provided that we know thesamples . However, the latter require knowledge of thesummation in (48).

The solution lies in using the sampling relationship betweenthe GS and the GS of the standard chirp


Fig. 5. Real parts of u (a) = (f; t) and u (a) = (f; t)versus a for f = 1=2 and t = 1.

FRT. In fact, Theorem 3 holds for , and this allowsus to state that the CFRT kernel and the WFRT kernel take thesame values at . This is illustrated in Fig. 5, where

and are compared as functions of , for, when and are fixed. Applying the sampling relationship

simultaneously solves the two problems of evaluating the WFRTkernel and finding the relationship between the two kinds ofFRTs, with the result

(50)

where is known in closed form, and thesum has a finite number of terms. Hence, (50) yields the WFRTkernel in closed form for any order .

A more explicit result is obtained by considering the marginalvalues of the samples (see Section II), namely

, , etc. In (50), these samples cor-respond to , respectively; hence

(51)In the particular case , the last term disappears, and wefind

(52)which corresponds to a WFRT of order 4.4

Having established the kernel (34) in the form (50), we canobtain the expression of the standard WFRT of order as

4This WFRT is slightly different from the one previously discussed. Thechoice made here gives a closer correspondence with the CFRT.

although a more detailed result can be obtained by using (51).This relationship states that to find the WFRT of a signal ,for any fraction , we must first calculate the CFRT for frac-tions, namely, , , , , and then buildup with the weights . Note that this se-quence contains the terms

for any order . Thus, in the particular case , the chirpedsamples disappear, in accordance with (52). Fig. 6 illustratesthe standard WFRTs of order 4 of the rectangular signal fordifferent fractions . Fig. 7 illustrates the standard WFRTs ofdifferent orders for the fraction .

Remark 1: The standard WFRT was obtained by Shih [10]for and recently extended by Liu et al. [11] to a general-order . These authors, however did not appreciate the roles ofthe sampling theorem and of the sampling relationship.

Remark 2: Incidentally, we note that the sampling relation-ship allows the closed-form evalua-tion of the subkernels (48). In fact, are Fourier coeffi-cients, and by (37), we find

(53)

where is the standard chirp kernel defined by (46),and .

Remark 3: Again, from the sampling relationship, we seethat the WFRT enjoys the same properties as the CFRT and,in particular, the relationship with the Wigner distribution (55)(see Section IV-D) when takes the discrete values .

C. Standard WFRT as an Approximation of the StandardCFRT

The sampling relationship between the CFRT and the WFRTof order states that the WFRT is an approximation ofthe CFRT, which improves as increases (see Fig. 7). Indeed

(54)that is, as increases, the eigenvalues of the WFRT tend tobecome the eigenvalues of the CFRT. Considering the signal

, as expressed in the form (7), its CFRT , and itsth-order WFRT , as expressed as in (25), we find that

the convergence of to is in the norm. In fact

which is assured by the unit amplitude of the eigenvalues andby the orthonormality of the eigenfunctions. If is a signalin , as increases, the above sum converges to zero.


Fig. 6. WFRT of order 4L = 4 of the rectangular signal for several values ofa (real part solid, imaginary part dashed lines).

D. Relationship Between CFRT and Wigner DistributionWe recall that the Wigner distribution operator is defined as

(55)and is not linear. The CFRT is related to the Wigner distributionby the equivalence [4]

(56)where is the image-rotation operator

(57)

Fig. 7. Standard WFRT with weights 4L = 4, 8, 16, 64, and 256, comparedwith the CFRT of the signal s(t) = rect(t), for a fraction a = 0:3 (real partsolid, imaginary part dotted lines).

Such equivalence states that the Wigner distribution of theCFRT equals that of the signal rotated clockwise


by an angle . Since the proof of the above equivalenceis based on the specific structure of the CFRT kernel givenby (46), the conclusion is that (56) holds only whenis interpreted as the CFRT operator. However, we can takeadvantage of the sampling relationship linking the -WFRTto the CFRT, namely, for and(see Section IV-B), to conclude that (56) holds also for the

-WFRT for the fractions . Note that as the numberof weights increases, the identity (56) holds at more andmore values of the fraction or, equivalently, for an increasingnumber of rotated angles of the Wigner plane.

Incidentally, we note a further property that is unique to theCFRT. Again from the kernel structure (46), the CFRT turns outto be a canonical transformation [19], i.e., one with a kernel

proportional to with ,, real. Once again, from the sampling relationship, we can

conclude that for the fractions , the -WFRT is acanonical transformation.

V. NONSTANDARD FORMS OF FRTS

In this section, we introduce new examples of FRTs of bothchirp and weighted forms to show that their expressions can beevaluated as combinations of the standard FRTs of the previoussection. We continue to assume the HG basis.

A. Nonstandard Chirp-FRTs

We consider the specific GS examples

1) shifting2) scaling

3) forfor perturbation

4) forfor periodic perturbation

5) periodicity mod

All these sequences have cardinality so that the cor-responding FRTs are of the chirp type. In the list, we indicatedthe relationship of each sequence with the standard sequence

so that we can use the results of Section IV.For the sequences 1) and 2), we obtain the FRTs

1) 2)

For the sequence 3), we obtain from (42) the kernel relation-ship , where

. This yields the FRT

3)

where is the sixth coefficient of the HermiteGauss expan-sion given by (8).

For the GS 4), from (43), we get

with

(58)

The latter series can be evaluated by a filtering operation onthe function , with the result (see Appendix D)

. The corre-sponding FRT is then given by

4)

For the GS 5), the kernel is given by

The closed-form summation of this uncommon series, whereis the product of HermiteGauss functions,

would appear to be a very hard, even impossible task. However,considering that the GS is periodic mod , thatis (mod ), hence the same as the standard GS

, we can use Theorem 4 to obtain samples of interms of samples of . Specifically

with

The corresponding relationship for the FRT is

5) (59)

We are thus able to evaluate this FRT in closed form for anyrational fraction.

All these examples of nonstandard chirp FRTs, together withthe standard chirp FRT, are illustrated in Fig. 8 for the signal

rect and the fraction .

B. Nonstandard Weighted FRTs

A compression of a GS by a (mod ) operation gives anew GS with finite cardinality, namely

(60)

with and . Hence, startingfrom any nonstandard sequence of the previous subsection,for any order , we can obtain a corresponding nonstandardweighted FRT.


The relationship between a given weighted FRT and the cor-responding chirp FRT is easily expressed by the sampling re-lationship and by the sampling theorem. In fact, (60) satisfiesthe hypothesis of Theorem 3, and hence, for

. Moreover, , andtherefore

This result generalizes (50) to nonstandard FRTs. Now, isthe kernel of a nonstandard chirp form, and is the kernel ofthe corresponding nonstandard weighted form, the interpolatingfunction being the same as in (50). For instance, ifand , we obtain the GS of period 12:

Hence, this WFRT is the weighted combination of the nonstan-dard CFRT given by (59), which in turn is a combination ofsamples of the standard CFRT evaluated for 12 fractions.

Another class of weighted FRTs is obtained by using peri-odic GSs: those with the property

For these sequences, the kernel is given by [see (47) and (48)]

(61)

where are the subkernels of the standard WFRT of ordergiven by (53). Hence

with

(62)

In conclusion, every periodic GS leads to a WFRT that is di-rectly related to the standard CFRT. As an example, we considerthe GS of period

whose kernel can be calculated from (62), with , interms of 16 values of . However, we note that the range

Fig. 8. Nonstandard examples of chirp FRTs 1)5) compared with thestandard chirp 0) for the same value of the fraction a = 0:3 (real part solid,imaginary part dashed lines).

of is the unit cell of size 8 (seeFig. 2). The sampling theorem can thus be applied withto give

where the eight samples can be calculated from (62).Hence, we end up with two different but equivalent expressionsfor the same kernel.


VI. FRT DEPENDENCE ON THE EIGENFUNCTION BASIS

In this section, we assume that the GS is fixed. We haveseen that the possible bases are governed by Theorem 1, whichcan be interpreted in terms of operators as

where1) is determined by the reference basis ;2) is determined by the PS ;3) gives the perturbed basis .

Note that by (17), the operator is unitary. Since is a validbasis, from a given FRT operator , we can obtaina new FRT operator

and this can be done for every PS .However, it is not clear how effectively the perturbation

changes the FRT. We know some general facts, namely, thatthe ordinary FT operator is independent of the basis, whichassures that no changes occur at the fraction or at anyother integer fraction (see marginal conditions). Moreover,by inspecting its structure, we see that the 4-WFRT is alsoindependent of the basis. To gain insight into the problem andto test quantitative effects, we next consider a very simple case.

A. Perturbation of Two EigenfunctionsStarting from an orthonormal basis of eigenfunctions

, we modify two of them, say, and , whichhave the same FT eigenvalue , according to

with , otherwise. Hence, the PS is ,, , , and for the rest. The

new eigenfunctions preserve the eigenvalues, and the conditions

(63)

[see (17)] assure that the modified set is still orthonormal.These conditions can be put into a matrix form as

which states that is a unitary matrix. Since the transposemust also be unitary, it follows that (63) are equivalent to

(64)

The reference FRT and the perturbed FRT are given by [see(25)]

where [see (8)] , , andfor the rest. Hence, the perturbation

is given by

where we used (63) and (64). Moreover, considering the orthog-onality condition, we find

(65)

Note that if , the perturbation has no effect. In anycase, since mod [see (21)], then forevery integer fraction . In general, however, ,although since , this difference is bounded.

The perturbation is illustrated in Fig. 9 for the CFRT of therectangular signal shown in Fig. 4, where , and

.

B. Example of Perturbation of an Arbitrary Number ofEigenfunctions

A curious perturbation sequence is provided by the DFT ma-trix , which is orthonormal for every order .Therefore, we can let

(66)

and , otherwise.By doing so, we obtain perturbed eigenfunctions

(67)


Fig. 9. Effect of perturbation of two eigenfunctions with c = 1 and d = 0on the CFRT of a rectangular signal with fraction a = 1=2 compared with theunperturbed CFRT (dashed).

and as many perturbed coefficients

(68)

whereas, for , , and remain unchanged. Ac-cording to (67), for and fixed, applying the DFT to the se-quence gives the perturbed sequence

. Similarly, in (68), the perturbed coef-ficients are obtained by applying the inverse DFT.

The above procedure can be applied for any order . In thiscase, the GS is , and the perturbation has energy

(69)

with

(70)Note that any other orthonormal matrix can be used to obtainfinite blocks of perturbations. In general, (69) becomes

(71)

with

where the GS is expressed in the form (22). We see that theorthogonality condition assures us that is zero for all integer

.

The above examples show that the perturbation, althoughbounded, can be very significant. More generally, the bounded

effect of the perturbations on the FRT could be inferred fromthe theory of perturbations of unitary operators [18].

Here, we observe that a very interesting derivation of the HGbasis is outlined in [19] (where they are called harmonic oscil-lator wave functions) and in [2], based on the linear combinationof differentiation and multiplication by the argument operators.Unfortunately, such an interpretation is not possible for an arbi-trarily perturbed basis.

VII. CONCLUSIONS

We discussed the multiplicity of the FRT and the relationshipsbetween the different versions. We saw that the class of FRTsis quite vast, and perhaps only a few may be of practical interestfor applications. Until now, interest had been confined mainly tothe standard CFRT, particularly for applications in optics. Thepractical relevance of the WFRT remains to be investigated.

An important classification of FRTs was expressed in termsof the cardinality of the GS or, equivalently, in terms of bandlim-itation of the kernel with respect to the domain of the fraction .If the bandwidth is not finite, the FRT turns out to be of the chirptype, whereas with finite bandwidth, the FRT is of a weightedtype. Moreover, the WFRT may be viewed as a filtered version(in the domain of the fraction) of the CFRT. Our aim was toestablish results in closed form, and this was achieved for thewhole, although vast, class of specific cases considered. Thisinvestigation has also shown that the evaluation of all the spe-cific forms of FRT is ultimately obtained in terms of the standardCFRT.

Although, in this paper, we confined our investigations tocontinuous-domain signals, most considerations also apply toperiodic discrete-domain signals, as suggested in the unified ap-proach of [16]. For periodic signals in a discrete domain, thecardinality of a complete orthonormal set of eigenfunctions isalways finite so that only the WFRT form can be considered.However, it can be shown in an analogous way that WFRTswith a lower number of weights are filtered versions of WFRTsof higher order.

APPENDIX APROOF OF THE SAMPLING THEOREM FOR COMPLEX

PERIODIC SIGNALS

The sampling theorem for periodic signals is well known; his-torically, it was certainly the first to be considered [21], [22], andit can be regarded as a particular case of trigonometric interpola-tion [23, ch. X], [24, ch. VII]. Nevertheless, recent authors [25]state that a proper formulation is not easily found in the litera-ture, even for the particular case of real signals with harmonicsupport of consecutive integers. Therefore, we give the prooffor the general formulation of Theorem 2 that is suitable to ourpurposes.

We write the periodic signal in terms of its Fourier coeffi-cients

(72)


The sampled values are then given by

(73)

where and . On the other hand, theDFT of the samples is given by

(74)

where the sum enclosed in braces is 1 if ,and 0 otherwise. Hence

(75)

Now, from the unit cell definition [see (33)], bandlimitation as-sures that in the periodic repetition (75), the nonzero coefficients

do not overlap and can be recovered as , .Hence, by (74), we can express the in (72) in terms of thesamples as

and (34) follows. The proof of (37) is a consequence of, , and of (74).

APPENDIX BPROOF OF THEOREM 3

For a fixed fraction , we evaluate the energy of the differencein the plane as

where and are expressed as in (38). Considering thatand that the are orthonormal, we find

. Now, thisenergy is zero for if and only if (39) holds. Note that

means that almost everywherein the plane.

APPENDIX CPROOF OF THEOREM 4

With and given by (38), we let .Then, considering that and , forthe samples, we find

(76)

where . Now, if the values forare a permutation of or,

more generally, fill a unit cell of size , the first of (76) canbe put into a standard DFT form byletting with (mod ). Then, the inverse DFTyields

Using the latter in the second part of (76) completes the proof.

APPENDIX DIDENTITIES ON HERMITEGAUSS FUNCTIONS

In this Appendix, we prove the identity

(77)

which may be viewed as a generalization of Mehlers expansion(45). Note that for , this identity gives (53).

For the proof, we use polyphase decomposition, whichis a fundamental result of the theory of multirate systems[26]. Given a -transform , the terms

, are

called the polyphase components of . The result statesthat can be obtained from by the closed-formrelationship

(78)

where . Now, if we let and, we find

and (77) becomes a consequence of (78).

APPENDIX EEIGENFUNCTION BASES OF THE FT

Let be the given basis, which we assume to be or-dered so that , and denote with the eigenspace ofthe eigenvalue ; then,is a complete orthonormal set for . We claim that with

is spanned by the linear combinations

with and (79)

Clearly, . Conversely, if , the expansion withrespect to the basis gives so


that its FT is given by .Since both expansions are unique, it follows that ,which implies that for . Hence, must havethe form (79), with .

Now, if is an arbitrary orthonormal basis, using (79)for each , we find that must have the form (16).Condition (17) is thus a consequence of the orthonormality ofboth and .

ACKNOWLEDGMENT

The authors wish to thank Prof. M. Morandi-Cecchi of theDepartment of Mathematics, University of Padova, for her crit-icism and helpful discussions.

REFERENCES[1] V. Namias, The fractional order Fourier transform and its applications

to quantum mechanics, J. Inst. Math. Appl., vol. 25, pp. 241265, 1980.[2] A. C. McBride and F. H. Kerr, On Namiass fractional Fourier trans-

form, IMA J. Appl. Math., vol. 39, pp. 159175, 1987.[3] L. B. Almedia, An introduction to the angular Fourier transform, in

Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Minneapolis,MN, Apr. 1993.

[4] A. W. Lohmann, Image rotation, Wigner rotation, and the fractionalFourier transform, J. Opt. Soc. Amer. A, vol. 10, no. 10, pp. 21812186,Oct. 1993.

[5] D. Mendlovic and H. M. Ozaktas, Fractional Fourier transforms andtheir optical implementationI, J. Opt. Soc. Amer. A, vol. 10, no. 9,pp. 18751881, Sept. 1993.

[6] H. M. Ozaktas and D. Mendlovic, Fractional Fourier transforms andtheir optical implementationII, J. Opt. Soc. Amer. A, vol. 10, no. 12,pp. 25222531, Dec. 1993.

[7] L. M. Bernardo and O. D. D. Soares, Fractional Fourier transforms andimaging, J. Opt. Soc. Amer. A, vol. 11, no. 10, pp. 26222626, Oct.1994.

[8] H. M. Ozaktas and D. Mendlovic, Fractional Fourier optics, J. Opt.Soc. Amer. A, vol. 12, no. 4, pp. 743751, Apr. 1995.

[9] L. B. Almeida, The fractional Fourier transform and time-fre-quency representations, IEEE Trans. Signal Processing, vol. 42, pp.30843091, Nov. 1994.

[10] C.-C. Shih, Fractionalization of Fourier transform, Opt. Commun., no.118, pp. 495498, Aug. 1, 1995.

[11] S. Liu, J. Zhang, and Y. Zhang, Properties of the fractionalization of aFourier transform, Opt. Commun., no. 133, pp. 5054, Jan. 1, 1997.

[12] B. W. Dickinson and K. Steiglitz, Eigenvectors and functions of thediscrete Fourier transform, IEEE Trans. Acoust, Speech, Signal Pro-cessing, vol. ASSP-30, pp. 2531, 1982.

[13] H. M. Ozaktas, O. Arkan, A. Kutay, and G. Bozdag, Digital computa-tion of the fractional Fourier transform, IEEE Trans. Signal Processing,vol. 44, pp. 21412150, Sept. 1996.

[14] M. E. Marhic, Roots of the identity operator and optics, J. Opt. Soc.Amer. A, vol. 12, no. 7, pp. 14481459, July 1995.

[15] J. Shamir and N. Cohen, Root and power transformations in optics, J.Opt. Soc. Amer. A, vol. 12, no. 11, pp. 24152423, Nov. 1995.

[16] G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, A unifiedframework for the fractional Fourier transform, IEEE Trans. SignalProcessing, vol. 46, pp. 32063219, Dec. 1998.

[17] K. Yoshida, Functional Analysis. Berlin, Germany: Springer-Verlag,1968.

[18] M. Hamermesh, Group Theory and its Application to Physical Prob-lems. Reading, MA: Addison-Wesley, 1962.

[19] K. B. Wolf, Integral Transforms in Science and Engineering. NewYork: Plenum , 1979.

[20] G. Cariolaro, M. Fregolent, and N. Laurenti, A novel set of eigenfunc-tions of the Fourier transform,, in progress.

[21] A. L. Cauchy, Mmoire sur diverses formulaes d analyse, ComptesRendus, vol. 12, pp. 283298, 1841.

[22] C. F. Gauss, Nachlass: Theoria interpolationis methodo nova tractata,in Werke. Gttingen: Kniglichen Gesellschaft der Wissenschaften,1866, pp. 265303.

[23] A. Zygmund, Trigonometric Series II, 2nd ed. New York: CambridgeUniv. Press, 1959.

[24] J. Kuntzmann, Methods Numeriques, InterpolationDerives. Paris,France: Dunod, 1959.

[25] H. Stark, Sampling theorems in polar coordinates, J. Opt. Soc. Amer.,vol. 69, pp. 15191525, Nov. 1979.

[26] P. P. Vaidyanathan, Multirate Systems and Filter Banks. EnglewoodCliffs, NJ: Prentice-Hall, 1993.

Gianfranco Cariolaro (M66) was born in 1936. Hereceived the degree in electrical engineering from theUniversity of Padova, Padova, Italy, in 1960. He re-ceived the Libera Docenza degree in electrical com-munications in 1968 from the same university.

He was appointed Full Professor in 1975, andpresently, he is Professor of electrical commu-nications and signal theory at the University ofPadova. His main research is in the fields of datatransmission, image, digital television, multicarriermodulation systems (OFDM), cellular radios, and

deep space communications. He is the author of several books includingUnified Signal Theory (Torino, Italy: UTET, 1980).

Tomaso Erseghe was born in Valdagno, Italy, in 1972. He received the degreein telecommunication engineering from the University of Padova, Padova, Italy,in 1996, with a degree thesis on the fractional Fourier transform. He is nowpursuing Ph.D. degree at the University of Padova.

He has worked as an R&D Engineer at Snell & Wilcox, Petersfield, U.K., aBritish broadcast equipment manufacturer, in the areas of image restoration andmotion compensation. His research interests include wideband communicationsystems and the fractional Fourier transform.

Peter Kraniauskas was born in Lithuania in 1939.He received the electromechanical engineering de-gree from the University of Buenos Aires, BuenosAires, Argentina, in 1962 and the Ph.D. degree fromthe University of Newcastle upon Tyne, Newcastleupon Tyne, U.K., in 1974.

After 20 years in industry, including research anddevelopment posts with Xerox Research and theRecal Electronics Group, he became an IndependentEngineering Consultant, teaching signal and systemsfundamentals in high-tech industry and research

establishments. He is the author of the textbook Transforms in Signals andSystems (Wokingham, UK: Addison-Wesley, 1992), whose graphical approachhe is currently extending to multidimensional signals.

Nicola Laurenti was born in 1970 in Adria, Italy.He graduated from the University of Padova,Padova, Italy, where he received the laurea in elec-trical engineering in 1995, with a thesis on imagereconstructions from projections. He received thePh.D. degree in electrical and telecommunicationsengineering from the same University in February1999 with a thesis on implementation issues inOFDM systems.

He is currently a Postdoctoral Fellow at the Univer-sity of Padova, where he is doing research on variable

bit-rate transmission systems for multimedia applications. His research interestsinclude the theory of projections, multicarrier modulation systems, multidimen-sional signal theory, and the fractional Fourier transform.

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Multiplicity of FRFT