the fractional fourier transform and time-frequency representaton

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3084 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. 11, NOVEMBER 1994

The Fractional Fourier Transformand Time-Frequency Representations

Luis B. A l m e i d a , Member, IEEE

Abstract-The fractional Fourier transform (FRFT ), which is ageneralization of the classical Fourier tran sform, was introduceda number of years ago in the mathematics literature but app earsto have remained largely unknown to the signal processingcommunity, to which it may, however, be potentially useful. TheFRFT depends on a parameter cy and can be interpreted as arotation by an angle a in the time-frequency plane. An FRFTwith a =n-/2 corresponds to the classical Fourier transform, andan FRFT with Q =0 correspo nds to the identity operator. On theother h and, the angles of successively performed FRFTs simplyadd up, as do the angles of successive rotations. T he FRF T of asignal can also be interpreted as a decomposition of the signal interms of chirps.In this paper, we briefly introduce the FRFT and a numberof its properties and then present some new results: the in-terpretation as a rotation in the time-frequency plane, and theFRFTs relationships with time-frequency representations suchas the Wigner distribution, the ambiguity function, the short-time Fourier tran sform a nd the spectrogram. These relationshipshave a very simple and natural form and support the FRFTsinterpretation as a rotation operator. Examples of FRFTs ofsome simple signals are given. An example of the application ofthe FRFT is also given.

I . INTRODUCTIONOURIER analysis is one of the most frequently usedF ools in signal processing and is frequently used in manyother scientific disciplines. Besides the Fourier transform itself,time-frequency representations (TFRs) of signals, such as theWigner distribution (WD) [11, [3], the ambiguity function [11,[3] the short-time Fourier transform (STFT) [1]-[3] and thespectrogram [11-[3] are often used, e.g., in speech processing,radar, or quantum physics.In the mathematics literature, a generalization of the Fouriertransform known as the fractional Fourier transform (FRFT),was proposed some years ago [4], [SI. Although potentiallyuseful for signal processing applications, this transform ap-pears to have remained largely unknown to the signal process-ing community. Recently, the FRFT has been independentlyreinvented by a number of researchers, including this author

[6]-[9]. New results, concerning the FRFTs interpretation asa rotation in the time-frequency plane, and its relationshipswith time-frequency transforms, have also been independentlyfound, more or less simultaneously, by this author and bysome of the above referenced ones.Manuscript received December 6 , 1992; revised January i8 , 1994. Theassociate editor coordinating the review of this paper and approving it forpublication was Prof. K. M. Buckley.The author is with INESCAST, Instituto Superior Tecnico, Lisbon, Portugal.IEEE Log Number 9403739.

The purpose of this paper is twofold: First, to brieflyintroduce the fractional Fourier transform and its mainproperties and, second, to present the new results, includingthe FRFTs interpretation as a rotation in the time-frequencyplane and the simple relationships that the FRFT ha swith several time-frequency re resentations that supportthe interpretation as a rotation. We will also see thatcomputing the FRFT of a signal corresponds to expressingit in terms of an orthonormal basis formed by chirps, i.e.,complex exponentials with linearly varying instantaneousfrequencies.This paper is organized as follows. Section I1 presentsthe FRFT, emphasizing its interpretation as a rotation oper-ator, lists some of its properties, and gives some examples.Section I11 discusses the relationships with the Wigner dis-tribution, the ambiguity function, and other quadratic TFRs.Section IV discusses the relationships with the short-timeFourier transform and the spectrogram. Section V brieflypresents an example of an application. Section VI concludesthe paper.Notes on the Formalism: We will represent by j theimaginary unit and by a superscript asterisk * the complexconjugation operation. We will often use square roots ofcomplex numbers, and we w ill reserve the square root symbolJ to denote the square root whose argument lies in theinterval ] - ~ / 2 , ~ / 2 ] .

11. THETRANSFO RM AND ITS PROPERTIESA . Definition

In time-frequency representations, one normally uses aplane with two orthogonal axes corresponding to time andfrequency, respectively (Fig. 1). If we consider a signal x ( t )represented along the time axis and its Fourier transform X ( w )represented along the frequency axis, we can view the Fouriertransform operator, which we shall designate by F, as achange in the representation of the signal corresponding toa counterclockwise axis rotation of ~ / 2ad . This is consistentwith the result of the repeated application of th e F operator,

W e consider the Fourier transform defined as S (UJ )& Tm ( t )x p - J d f d t . In the mathematics literature, the Fourier transform is normallydefined with a plus sign in the exponent. We prefer to use the engineeringconvention here, although this results in a number of sign differences betweenthe expressions of this paper and those of the references (which amounts toa change in the sign of the angle ct in the expressions given ahead for theFRFT).

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Note that this kernel is continuous in a , in the generalizedfunction sense, even at multiples of 7 r , i.e.

Fig. I .angle c1 relative t o the original coordinates ( t . .).Time-frequency plane and a set of coordinates ( U . ) rotated by an

given that FFz(t)=x ( - t ) and that two successive rotationsof 7r/2 of the t axis result in an axis directed along -t .In this context, one may ask which linear operator wouldcorrespond to a rotation by an angle N that is not a multipleof 7 r / 2 , or equivalently, what would be a representation ofthe signal along an axis U (Fig. 1) making an angle N withthe time axis. Let us assume, for the moment, that such anoperator exists, and let us represent it by RO. This operatorshould have the following properties:1) Zero rotation: Ro=I ( 1 )2) Consistency with Fourier transform: R"/2=F (2 )

(3 )4) 27r rotation: R2"=1 (4 )Note that property 4 is a consequence of properties 2 and 3and of the fact that four successive applications of the Fouriertransform correspond to the identity operator:

3) Additivity of rotations: RaRO =R04-d

( 5 )2"=R4("I2)=F4 =1The FRFT, which has these properties, is defined by means ofthe transformation kernel2

K,(t. U )JFl- c ot a - p t c s r a6 ( t - U ) (6)f cy is not a multiple of 7rif N is a multiple of 271.=i( t+U ) if (Y+T is a multiple of% .

The square root factor that precedes the exponential, in this definition,can be written in a number of equivalent forms, some of which appear in thereferences. Note, however, that 141 contained a flaw in the definition of thisfactor, a correction having been made in [SI. Another useful form for thisfac to r i s J 2 ~ ~ ~ " / ( 7 r s i i i c r ) .

lirn K , =K,, for integer no+nri (7 )The kernel has the following properties, which will be usefullater in this paper:

The first three properties are trivial. The derivation of (1 1) israther long and will not be given here. Property (12) can bederived using (S), (9),( 1 I) , and the definition of K , fo r Q =0.This latter property means that the kernel functions K,(t, U ) ,taken as functions o f t with parameter U , form an orthonormalset. Given (S), the same can be said if we take U as a variablean d t as a parameter. We also note that for N =7r/2, the kernelcoincides with the kernel of the Fourier transform.

The fractional Fourier transform of a function 2, with anangle cy, is defined [4], [ 5 ] as the function R"z =X,, givenby (13) and (14), which appear at the bottom of the page.The last equation shows that for angles that are not multiplesof 7 r , the computation of the fractional Fourier transformcorresponds to the following steps:

1) a product by a chirp2 ) a Fourier transform (with its argument scaled by csccu)3) another product by a chirp4) a product by a complex amplitude factor.

Since chirps have constant magnitude, this imm ediately allowsus to make a rather general statement about the existenceof the transform. In fact, if x ( t ) is in L1, L2, or is ageneralized function, its product by the chirp is also in L1, L2,or is a generalized function, respectively. Therefore, in thesesituations, the FRFT of z ( t ) xists in the same conditions inwhich its Fourier transform exists.

It is easy to see that the FRFT, as defined in (13), satisfiesthe properties that were discussed in the introduction. Property1 holds directly by definition. Property 2 is a consequence ofthe fact that for cy = 7r/2, K , coincides with the kernel ofthe Fourier transform. Property 3 is easily derived, using ( 1 1 ).


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IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL. 42, NO . 11 , NOVEMBER 1994

Fractional Founer transform with an gle aX , ( u - r c o s a ) e 2

2I -ma m a - l m m a

In fact, if we defineY(U) =X , ( u )

then

Finally, property 4 is a consequence of properties 2 an d 3 ashad been noted in the introduction. From these properties, wealso easily conclude that the inverse of an FRFT with an a nglea is the FRFT with angle -a:

3=

z ( t )= X,(U)K_,(U,t)dU. (20).II,From (20), we can see that the FRFT consists of expressing~ ( t )n a basis formed by the set of functions K-,(u,t)(with u acting as a parameter for spanning the set of basisfunctions). This basis is orthonormal, according to (12). Thebasis functions are chirps, i.e., complex exponentials withlinear frequency modulation. For different values of U , theyonly differ by a time shift and by a phase factor that dependson U :

K,(t ,u) =e-+ano K,(t - s e c a . 0 ) . (21)An interesting issue is w hether this is the only transform thatsatisfies properties 1-4. It is easy to see that if we wouldmultiply the transform by e4 jna with integer n, he fourproperties would still hold. However, some of the resultspresented below would become more complex, and therefore,the FRFT appears to be the most natural one among these.We do not know if there is still any other transform, besidesthese, obeying the four properties. However, m ost probably, ifsuch a transform exists, its relationships with time-frequencydistributions will not be as simple as those derived below forthe FRFT.

In summary, the FRFT is a linear transform, continuousin the angle a , which satisfies the basic conditions for beinginterpretable as a rotation in the time-frequency plane.B. Further Properties

In Table I, we list a number of additional useful propertiesof the FRFT, which are extensions of the correspondingproperties of the Fourier transform. Their proofs, except forthat of the scaling property (row 8) , can be found in [4] an d[ 5 ] . The proof of the scaling property is given in AppendixA .3 Property 6 of the table is naturally subject to the existenceof z ( t ) / t tself. Property 7 leads to the parity property: If

To our knowledge, this property wa s first presented in [ IO ] and [ I l l . Itsderivation wa s not given there, however, for lack of space, and this is whywe give it in this paper.

-I-ianas e c a e 2 l X a ( z ) e ' ~ ' " a dz if a-x I 2 is no t a multiple of xI 4 I / ( X ( I 8 , d l r if a -TI 1 2 IS a multiple of x , the classical Fourier transform property

where p=a r cm ( c 2 tana)

z ( t ) s even, X,(u) is also even; if z ( t ) s odd, X,(u) salso odd.The scaling property, although somewhat complex, is im-portant since it shows the effect of a change of units (or ascaling) of the independent va riable t. In the classical Fouriertransform, the effect of such a change is only a correspondingchange of units (or scaling) of the frequency variable and ascaling of the amplitude. In the FRF T, the effects are a scaling

of the u variable by sin p / ( c sin a ) , a (complex) amplitudescaling, a product by a chirp, and most important, a changein the angle at which we compute the transform from (Y to,O = arctan(c2 ana) . Th is ang le change can be understoodif we think of contracting the time axis, in the time-frequencyplane, by a factor c. As we know from the classical Fouriertransform, we will then also have to expand the frequency axisby c. With these two operations, the axis along which we w erecomputing the transform, which was originally at an angle Qwith the time axis, will move to a new position at the angle,O given above.The Parseval relation

W 00z ( t ) y * ( t ) d t= X,(u)Y,*(u)du (22)1, .I,L L

is easily derived by expressing x ( t ) n the left-hand side ofthis equation as the inverse transform of X , (U ) and then using(9). A consequence of this equality is the energy-preservingproperty of the FRFTDCI ocIx(t)I2 t = I X , ( U ) ~ ~u. (23)

The Parseval relation and the energy-preserving property canalso be viewed as consequences of the fact that the FRFT isbased on an set of orthonormal basis functions.Due to the energy-preserving property of the Fourier trans-form, the squared magnitude of the Fourier transform of asignal ~ X ( L J ) ~ ~s often called the energy spectrum of thesignal and is interpreted as the distribution of the signal'senergy among the different frequencies ,jut. Although it is less


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Signal

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Fractional Fourier transform with angle a

intuitive, ( 3 8 ) also allows us to call I X , ( U ) ~ * he fractionalenergy spectrum of the signal z ( t ) ,with angle a , an d tointerpret it as the distribution of the signal's energy amongthe different chirps K-,( t , U ) .If z ( t ) s real, X , ( u ) enjoys one further property:

x-, U ) =x; U ) (24)1

which is a consequence of (9). Note, however, that thesymmetry properties of the Fourier transform for even or od dreal sequences do not extend to the FR IT with arbitrary anglea since the FRFT of a real function is not in general Her-mitean. Another important property of the Fourier transformthat does not extend in a simple way to the FRFT is theconvolution theorem.

6 ( r - r ) i"* i ~ m l a - , u r , c a if a is not a multiple of x

C. xamplesTable I1gives the FRFT's of a number of common signals.An expression for the FRFT of a rectangle can also beobtained, but that expression is rather complex and does notappear to be particularly instructive. Instead of presenting

it here, we prefer to give a feeling of the behavior of thetransform by showing the variation of the transform of arectangle [ z ( t )=1 fo r It1


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3088 IEEE TR A N SA CTI O N S O N SI G N A L PROCESSING, VOL. 42, NO. 11, NOVEMBER 1994

alpha =0.01 alpha =0.05

(a) (b)alpha =0.2 alpha =0.4

alpha = Pii4 alpha = Pi/2

(e )Fig. 2.The right-hand side of this equation is the WD of X ,computed with arguments ( U , U ) . The left-hand side is the WDof II computed with arguments ( t .U ) . The equation showsthat the WD of X , coincides with the WD of s f we takeinto account the rotation that corresponds to the fact that weare using different axes on the left- and right-hand sides ofthe equation. This is equivalent to saying that the WD of

FRFT of a rectangle, computed at various angles. Solid line: real part. Dashed line: imaginary part.X , is the WD of z, rotated by an angle -cy, or that it issimply the WD of z expressed in the new set of coordinates(U. ). The fact that the FRFT induces a simple rotation ofthe WD is remarkable and enhances our view of the FRFT asa rotation of the axis along which the signal is represented.On the other hand, this fact also shows that a rotated W D isstill a legal WD. It is well known that the W D has similar


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5Fig. 3 . Variation of magnitude of the FRFI of a rectangle with the angle 0.

Fig. 4. Representation of a swept-frequency system. The input signal isfirst down modulated by an exponential with linearly varying instantaneousfrequency, then passed through a time-invariant filter and finally up-modulatedagain. The constant c controls the sweep speed of the system.

(MSTFT), which consists simply of the normal STFT with asuitable phase modification:

L ( T ) l l i * ( t - ) e - J d T d 7 . (37)From (37), the MSTFT can also be computed as- 1 JtX(t.w)= - e - j T X ( v ) W * ( w- v)eJutdv (38)properties relative to time and frequency. This result shows G

be obtained from their definitions in terms of the WD andfrom the fact that the FFUT induces a rotation in the WD. Forexample, it can be easily shown that the FRFT also induces asimple rotation in the ambiguity function.Iv. RELATIONSHIPSITH TH E SHORT-TIMEFOURIER RANSFORM AND TH E SPECTROGRAM

The short-time Fourier transform (STFT) is another impor-tant time-frequency analysis tool. It is frequently used, e.g.,for speech processing. A related tool is the spectrogram, whichcorresponds to the squared magnitude of the STF T. The STFTof a signal z ( t ) s defined as [l], [3 ]

X ( t ,w )=- ~ ( ~ ) w * ( t) e - J d T .r ( 3 5 )S32

1 utX ( t . w ) = - e 3 r X,(z)W:(u - ) e - J Z t d z . (39)The right-hand side of this equation is the MSTFT of X,computed with window W , and with arguments (..TI). Theleft-hand side is the MSTFT of 5 computed with window wand with arguments ( t . ~ ) .s in the case of the WD, thisequation shows that the MSTFT of X , is the same as theMSTFT of z, again taking into account the rotation, i.e., thatit is simply a rotated version of the MSTFT of z or that it isthe MSTFT of z expressed in the rotated axes ( U , w ) . ~ gain,this enhances our view of the FRFT as a rotation operator.The spectrogram [ l ] , [3 ] is simply the squared magnitudeof the STFT and, therefore, of the MSTFT as well. Theresults we obtained on the MSTFT immediately lead us toconclude that the effect of the fractional Fourier transform on

Jz;; 1,

the spectrogram is identical to the one it has on the MSTFT:the spectrogram of X, computed with window W, is a rotatedversion of the spectrogram of x computed with window IU .

where w ( t ) s a suitably chosen a nalysis win do^.^ The STFTcan also be computed in a simple way from the Fouriertransform of z ( t ) :, r c cX ( t , w )= --&-jut X ( v ) W * ( w- )eJutdv (36)fi

where X an d W are the Fourier transforms of z an d w ,respectively. This equation is similar to (35), except for thepresence of the exponential factor e - J w t .This is an asymmetrybetween time and frequency that we wish to avoid since wewant to deal with rotations in the time-frequency plane. Wewill therefore define a mod$ed short-time Fourier transform

40 fte n, in the literature, the windows that are considered are real, and thecomplex conjugation of the window is not mentioned in the definition. Here,however, we prefer to adopt a more general viewpoint, allowing the windowto be complex.

V . APPLICATIONIn this section, we will briefly present, as an example, theapplication of the FRFT to the study of swept-frequency filters.These filters are used, for example, in frequency analyzers forhigh-frequency signals. Swept-frequency filters are linear time-

varying systems that can be represented in the form shown inFig. 4. They can also be represented by their time-varyingWhen we compute the MSTFT of S,, e have to use the transformedwindow TI, as (39) shows. Note that even for a real-valued w , 147n will

normally be complex. If we use the Gaussian window ~ ( t ) thewindows FRFT is equal to to for any cr . Therefore, in this case, theFRFT induces a simple rotation of the MSTFT without any window change.


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impulse response, h ( t , ~ ) ,hich is the response at time t toan input S ( t - ) . It is easy to see that

(40)where g ( t ) is the impulse response of the shift-invariant filterin Fig. 4.

h ( t . i - ) =e J I ( t L - T L ) ( t - - )

The output of the swept-frequency filter is given byx

g ( t ) =1 ( r ) h ( t ,) d T . (41)Let us compute the FRFT of y(t), choosing the angle a =-arccot c: -,

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO . 1 , NOVEMBER 1994

di-dt (42)( t - j- cot a - ju t r sc a

1-jcotcu O0 .(+- co ta=.J 2T 1,or finally

where G is the (classical) Fourier transform of the impulseresponse g. Therefore, G( u csccr) can be called the transferfunction of the swept-frequency filter in the FRFT domain.The use of the FRFT and of this transfer function allows atreatment of swept-frequency filters that is very similar to theclassical treatment of time-invariant filters with the Fouriertransform.It is also worth mentioning that the FRFT is useful forsolving certain classes of differential equations that appear,

for example, in quantum mechanics and optics, and that mayalso be useful in signal processing applications (see [4], [SI,[lo], [111).

VI . CONCLUSIONWe have presented an extension of the Fourier transform,

which is designated fractional Fourier transform. This lineartransform depends on a parameter a and can be interpretedas a rotation by an angle a in the time-frequency plane.Particularly, the following hold:1) When a = 7r/2, the FRFT coincides with the conven-2) When a =0, the FRFT is the identity operator.3) Two successive FRFTs with angles (IY. an d p, respec-tively, are equivalent to a single FRFT with an angle

On the other hand, the FRIT was shown to induce rotationsin various time-frequency transforms, including the Wignerdistribution and the short-time Fourier transform, further en-hancing its interpretation as a rotation operator.The FRF T was also shown to correspond to a representationof the signal on an orthonormal basis formed by c hirps, whichare essentially shifted versions of one another. Finally, anapplication example was given, showing how the use of theFRFT allows a treatment of sw ept-frequency filters that is verysimilar to the classical treatment of shift-invariant filters withthe Fourier transform.The work presented here opens several areas for furtherresearch. Among these is the study of discrete versions of thistransform. On the other hand, the relationships between theFRFT and time-frequency transforms suggest that it can be auseful tool to further study the properties of these transformsand to develop time-frequency transforms better suited tospecific applications.

tional Fourier transform.

cy +g.

We shall proveof Table I . Let usare related by

APPENDIXthe scaling propertyconsider two signals

Y(t) =

given in row 8x an d y, which

(53)

(54)


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and let us compute the FRFT of y:309

ACKNOWLEDGMENT

(47)where we have used the integration variable t for convenience.Making the change of variable t =ct, we obtain

If we define p = arc cot[(cot a) / c 2]= a rc t an (c2 tarla),we have (50)-(54), which appear at the bottom of theprevious page.APPENDIXB

To obtain the relationship between the FRFT and theMSTFT, we will start from (37), replacing Z ( T ) by the FRFTof X , with angle --cy:

x w*( t - ) e CJ d d r . ( 5 6 )The inner integral is the complex conjugate of the FR FT withangle a of w ( t - r)eJwt,aken as a function of r and withargument z . Using properties 1 an d 2 of Table I, we canconclude that this integral is equal to

W:( - z + cosa +w sin a )(57)

j-in n c o s a + j z ( t sin a--w c o s a ) - j w t Sin aand therefore

03- 1 LiX ( ,w )= e3 X , ( z )W: ( - z + co s +w si n a )&G L ,e j &$ s i n e c o s a + j z ( t s ma - -w c o s a ) - j w t sin a d z .

(58)We now make, on the right-hand side, the change of variables(32) and (33) from ( t , ) to ( U ,U), nd after simplification ofthe exponents, we obtain

The author wishes to acknowledge useful suggestions fromA . F. Santos, J. F. Moura, J. S . Marques, M. D. Ortigueira,and the anonymous reviewers.

REFERENCES[ I ] F. Hlawatsch and G. F. Bourdeaux-Bartels, Linear and quadratic time-frequency signal representations, IEEE Signal Processing Mag., vol. 9,no. 2, pp. 21 47 , Apr. 1992.121 M . R. Portnoff, Time-frequency representation of digital signals andsystems based on short-time fourier analysis, IEEE Trans. Acoust.,Speech, Signal Processing, vol. ASSP-28, no. 1, pp. 55-69, Feb.1980.[3] L. Cohen , Time-frequency distributions-A review, Proc. IEEE, vol.77, no. 7, pp. 941-981, July 1989.[4 ] V. Namias, The fractional order Fourier transform and its applicationto quantum mechanics, J . Insr. Math. Appl. , vol. 25, pp. 241-265,1980.[ 5 ] A. C. McBride and F. H. Kerr, On Namias fractional Fourier trans-forms, /M A J. Appl. Math. , vol. 39, pp. 159-175, 1987.161 L. B. Almeida, An introduction to the angular Fourier transform, inProc. 1993 IEEE Int. Con$ Acoust., Speech, Signal Processing (Min-neapolis, MN), April 1993.171 H. M. Ozatk as, B. Barshan, D. M endlovic, and L. Onural, Convolution,filtering, and multiplexing in fractional Fourier domains and theirrelationship to chirp and wavelet transforms, J. Opt Soc . A mer . A ,

in press.[8 ] A. W. Lohmann, Image rotation, wigner rotation and the fractionalFourier transform, J . Opr. Soc. Amer. A, vol. 10, pp. 2181-2186,1993.[9 ] 0. Seger, Model building and restoration with applications in con-focal microscopy, Ph.D. Dissertation No. 301, Linkoping University,Sweden, 1993.[IO] T . Alieva, V. L6pez, and L. B. Almeida, The ang ular Fourier transformand wave propagation, submitted to Phys. Rev. Lett.[ I I ] T. Alieva, V. L6pez, F. Aguill&L6pez, and L. B . Almeida, The angularFourier transform in optical propagation problems, J. M o de rn O p t , inpress.

Luis B. Almeida (M87) was born in Lisbon, Por-tugal, in 1950. He received the degree in electri-cal engineering from the Instituto Superior Tecnico(IST), Lisbon , in 1972 , and received the Ph.D.degree in signal processing from the UniversidadeTecnica de Lisboa in 1983. He has been with theIST since his graduation and is presently AssociateProfessor of Signal Theory and of Neural Networks.He is head of the Neural Networks and SignalProcessing Group at INESC, which is a researchinstitute associated with Portuguese universities andcommunications companies.Dr. Almeida is vice-president of the European Neural Network Society.In 1985, he received a Sen ior Award from the IE EE for a paper on thenonstationary modeling of voiced speech.

the fractional fourier transform and time-frequency representaton

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