The 2-D signal representations of variables rather than time and frequency have been proposed based on either Hermitian or unitary operators. As an alternative to the theoretical derivations based on operators, we propose a joint fractional domain signal representation (JFSR) based on an intuitive understanding from a time-frequency distribution constructing a 2-D function which designates the joint time and frequency content of signals. The JFSR of a signal is so designed that its projections on to the defining joint fractional Fourier domains give the modulus square of the fractional Fourier transform of the signal at the corresponding orders. We derive properties of the JFSR including its relations to quadratic time-frequency representations and fractional Fourier transformations. We present a fast algorithm to compute radial slices of the JFSR

Arikan, O.

Durak L.

Song I.

Özdemir, A.K.

Bilkent University Institutional Repository

GENERALIZATION OF TIME-FREQUENCY SIGNAL REPRESENTATIONS TOJOINT FRACTIONAL FOURIER DOMAINSLutfiye Durak∗, Ahmet Kemal Özdemir∗∗, Orhan Arikan†, and Iickho Song∗∗ Department of Electrical EngineeringKorea Advanced Institute of Science and Technology373-1 Guseong Dong, Yuseong Gu, Daejeon 305-701, Korealutfiye@ieee.org, i.song@ieee.org∗∗WesternGeco Ltd. Schlumberger HouseGatwick Airport, West Sussex RH6 0NZ, Englandkozdemir@ieee.org†Department of Electrical EngineeringBilkent University Ankara, 06800, Turkeyoarikan@ee.bilkent.edu.trABSTRACTThe 2-D signal representations of variables rather than time and fre-quency have been proposed based on either Hermitian or unitary op-erators. As an alternative to the theoretical derivations based on op-erators, we propose a joint fractional domain signal representation(JFSR) based on an intuitive understanding from a time-frequencydistribution constructing a 2-D function which designates the jointtime and frequency content of signals. The JFSR of a signal is so de-signed that its projections on to the defining joint fractional Fourierdomains give the modulus square of the fractional Fourier transformof the signal at the corresponding orders. We derive properties ofthe JFSR including its relations to quadratic time-frequency repre-sentations and fractional Fourier transformations. We present a fastalgorithm to compute radial slices of the JFSR.1. INTRODUCTIONTime-frequency representations are frequently utilized to analyzeand process non-stationary signals [1, 2]. One of the major areas ofresearch in time-frequency signal processing is the design of noveltime-frequency representations for different applications. Unfortu-nately, time-frequency distributions do not always convey desirablequalifications in every application. Hence the demand for powerfulsignal representations has led to a substantial amount of researchon the design of 2-D signal representations defined by alternativevariables other than time and frequency. Joint time-scale represen-tations, which have attracted much interest especially in the fieldsof sonar and image processing, constitute one of the earliest exam-ples of this type of representations. Other popular choices of jointvariables include higher derivatives of the instantaneous phase ofsignals for radar and sonar problems [3, 4], dispersive time-shiftsfor wave propagation problems and analogs of quantum mechan-ical quantities such as spin, angular momentum, and radial mo-mentum [5], and scale-hyperbolic time, warped time-frequency andwarped time-scale.These signal representations have been mathematically derivedby using two alternative approaches which are both based on theoperator theory: The variables are associated with either Hermitianoperators as in [2] or unitary operators as in [5]. Recently, ajoint fractional signal representation has been derived by associ-ating Hermitian fractional operators to fractional Fourier transform(FrFT) variables constituting the joint distribution [6]. The frac-tional Fourier domains are the set of all domains interpolating be-tween time and frequency. The fractional Fourier domains corre-sponding to a = 0 and a = 1 are the time and frequency domains,respectively. The FrFT with order a transforms a signal into theath-order fractional Fourier domain and the ath-order FrFT of x(t)is given by [7]xa(t) ≡ {F ax}(t)=∫Ba(t, t ′)x(t ′)dt ′ , −2 < a < 2, (1)whereBa(t, t ′) =e− j(p sgn(a)/4+f /2)|sin f |1/2 ejp (t2 cot f −2tt ′ csc f +t ′2 cot f ) (2)is the transformation kernel, f = a p2 and sgn(·) is the sign function.In this paper, we derive the joint fractional signal representation(JFSR) which designates the energy contents of signals in fractionalFourier domain variables instead of time and frequency. To thisend, rather than using cumbersome mathematical equations basedon operator theory, we extend our intuitive understanding from atime-frequency distribution, i.e., a function which designates jointtime and frequency contents of signals. Then, we derive some im-portant properties of the JFSR including its relation to quadratictime-frequency representations and fractional Fourier transforma-tions, and present a simple formula for its oblique projections. Wealso present a fast algorithm to compute radial slices of the JFSRand numerically computed JFSRs of some synthetic signals.The outline of the paper is as follows. In Section 2, a concisederivation of JFSR is presented as an alternative to the derivationgiven in [6]. Then, in Section 3, properties of the JFSR are exam-ined. After presenting a fast computation algorithm in Section 4,numerically computed JFSRs of some synthetic signals are shownin Section 5. Finally, conclusions are drawn in Section 6.2. DISTRIBUTION OF SIGNAL ENERGY ON JOINTFRACTIONAL FOURIER DOMAINSOne of the primary expectations from a time-frequency distributionDx(t, f ) associated with a signal x(t) is that it accurately representsthe energy distribution of x(t). It is desired that the signal energy ata frequency f for a time instant t is given by Dx(t, f ). Because ofthe uncertainty relationship between time and frequency domains, itis impossible to satisfy this point-wise energy density requirement.Therefore, one is to be usually satisfied with a looser condition onthe marginal densities∫D(t, f )d f = |x(t)|2 (3)∫D(t, f )dt = |X( f )|2 , (4)and integration of the distribution on the whole time-frequencyplane ∫ ∫D(t, f )dt d f = ||x||2. (5)where X( f ) is the Fourier transform of x(t), and || · || denotes the L2norm. One of the prominent energetic distributions which satisfythe desired relations (4) and (5) is the Wigner distribution (WD)which is defined as [2]Wx(t, f ) =∫x(t + t /2)x∗(t− t /2)e− j2p f t dt . (6)The WD of a signal can be roughly interpreted as an energy densityof the signal, since it is real, covariant to time and frequency do-main translations and moreover signal energy in any extended time-frequency region can be determined by integrating Wx(t, f ) over thatregion [2]. Another nice property of the WD is that, its oblique pro-jections give the energy distribution with respect to the correspond-ing fractional Fourier domain [8]. Such properties and its ability toprovide high time-frequency domain signal concentration make theWD attractive compared to other representations.Although the WD and its enhanced versions are useful in time-frequency analysis, in some applications such as signal design andsynthesis, it is more useful to have a time-fractional Fourier do-main representation. In a mathematical framework, by associatingHermitian operators to individual FrFT domains, a joint fractionalrepresentation of signals has been derived in [6].In this section, the joint fractional domain signal representation(JFSR) is constructed using conditions similar to the ones given in(3)-(5). Let the JFSR of a signal be denoted by Eax (u,v) wherea = (a1,a2) denotes the orders of fractional Fourier domains u andv, respectively. It is desired that the marginal densities satisfy∫Eax (u,v)dv = |xa1(u)|2 (7)and ∫Eax (u,v)du = |xa2(v)|2 (8)where |xa1(u)|2 and |xa2(v)|2 are the energy contents of the signalat the ath1 and ath2 fractional Fourier domains, respectively. Similarto (5), the overall integral on the u− v plane is desired to be∫ ∫Eax (u,v)du dv = ||x||2. (9)The conditions stated in (7), (8), and (9) make the JFSR a gen-eralization of the WD, since the JFSR reduces to the WD when(a1,a2) = (0,1).To construct the distribution Eax (u,v) satisfying the conditionson the marginal densities and the total energy, we make use of theprojection property of the WD [8]∫Wx(ucos f − vsin f ,usin f + vcos f )dv = |xa(u)|2 . (10)In Fig. 1, we observe that the value of the time-frequency distribu-tion at a point P contributes to the energy densities of the fractionalFourier domains u and v at points u = u0 and v = v0, respectively.Therefore, the JFSR can simply be formed by redistributing the WDso thatEax (u,v) = C ·Wx(P(t(u,v), f (u,v))) , (11)where the coordinates (u,v) and (t, f ) are related by[cos f 1 sin f 1cos f 2 sin f 2] [tf]=[uv](12)0u v 0P( , )v 00uuvtfFigure 1: The value of the time-frequency distribution at a point Pcontributes to the energy densities of the fractional Fourier domainsu and v at points u = u0 and v = v0, respectively. The JFSR cansimply be formed through redistributing the WD so that, Eax (u,v) =C ·Wx(P(t(u,v), f (u,v))).and f i = aip /2 for i = 1,2. By using the total energy constraint (9),the constant C in (11) is determined asC = |csc(f 12)| (13)where f 12 = f 2− f 1. Thus, Eax (u,v) is explicitly given by|csc(f 12)|∫x(usin f 2− vsin f 1sin f 12+ t /2)x∗(usin f 2− vsin f 1sin f 12− t /2)×e− j2p t−ucos f 2+vcos f 1sin f 12 dt .An equivalent and more compact form of Eax (u,v) can be ob-tained by using the rotation effect of the FrFT on the WDWxa1 (u,v) = Wx(ucos f 12− vsin f 12,usin f 12 + vcosf 12) , (14)which verbally translates that WD of the ath1 order FrFT of a signalx(t) is the same as the WD of the signal x(t) which is rotated byf 1 radians in the clockwise direction in the time–frequency plane.Consequently, Eax (u,v) can be derived in terms of the fractionallyFourier transformed signal xa1(t) asEax (u,v) =∫xa1(u+t sin f 122)x∗a1(u− t sin f 122)× e− j2p (v−ucos f 12)t dt (15)which has the same form as given in [6].The JFSR can be generalized to define a cross-JFSR distributionof signals x(t) and y(t)Eaxy(u,v) = |csc(f 12)| ·Wxy(P(t(u,v), f (u,v))) (16)=∫xa1(u+t sin f 122)y∗a1(u− t sin f 122)× e− j2p (v−ucos f 12)t dt .Defining Eax (u,v) through its relation to the WD provides aneasily interpretable definition of the JFSR of signals. From the def-inition given in (15), it follows that JFSR is a quadratic distributionwhile it is not a time-frequency distribution. Therefore it belongs toa broader class than the familiar Cohen’s class. Thus, its introduc-tion to the non-stationary signal processing will bring new insightsinto the design, filtering, analysis, and synthesis of signals in manyapplications.3. PROPERTIES OF THE JFSRIn this section, we investigate the properties of the JFSR. In theproperties listed below, the joint fractional Fourier domains havethe orders of a = (a1,a2) making angles of (f 1, f 2) with respect tothe time axis where f i = ai p2 .Property 1. The JFSR is a real distributionEax (u,v) = (Eax )∗(u,v) . (17)Property 2. Orthogonal projection of the JFSR of a signal x(t)on to u and v axes give the magnitude square of the FrFTs of thesignal at orders associated with these axes: that is,∫Eax (u,v)dv = |xa1(u)|2 (18)and ∫Eax (u,v)du = |xa2(v)|2 . (19)Property 3. The area under the JFSR of a signal x(t) gives thetotal signal energy∫ ∫Eax (u,v)dudv =∫|x(t)|2dt , (20)which follows from Property 2 and the unitarity of the FrFT.Property 4 The JFSR and WD of a signal x(t) are related asEax (u,v) = |csc f 12|Wxa1(u,v−ucos f 12sin f 12)= |csc f 12|Wx(usin f 2− vsin f 1sin f 12,−ucos f 2 + vcos f 1sin f 12).Property 5 The JFSR and the FrFT of a signal x(t) are relatedasEaxa′ (u,v) = Ea+a′x (u,v) (21)where a+a′ = (a1 +a′,a2 +a′).Proof: By using (21) in Property 4, the JFSR of xa(t) can bewritten asEaxa′ (u,v) = |csc f 12|× Wxa′(usin f 2− vsin f 1sin f 12,−ucos f 2 + vcos f 1sin f 12).Then, by using the rotation property of the WD given in (14), righthand side of this expression is simplified toEaxa′ (u,v) = |csc f 12|Wx(u′,v′)u′ =usin(f ′+ f 2)− vsin(f ′+ f 1)sin f 12,v′ =−ucos(f ′+ f 2)+ vcos(f ′+ f 1)sin f 12which proves the property where f ′ = a′ p2 .Property 6 Any oblique projection of JFSR of a signal x(t) ontoan oblique axis making an angle of f isPf [Eax ](r) = |x(a′)(r/M)|2 , a′ =2f ′p, (22)wheref ′ = arctan2(cos f 1 +cos f +cos f 2 sin f ,sin f 1 cos f + sin f 2 sin f )(23)andM =√1+ sin2f cos f 12 . (24)Proof: By the projection-slice theorem, the oblique projectionof the JFSR of x(t) at an angle f is given byPf [Eax ](r) =∫Fx(z cos f , z sin f )e− j2p z rdz , (25)whereFax (z , h ) =∫ ∫Eax (u,v)ej2p (z u+h v)dudv (26)is the radial slice of the 2–D inverse Fourier transform of the JFSRof x(t). By using (21), the following expression can be obtained forF(a)x (z , h ) in terms of the ambiguity function Ax(.) of x(t)Fax (z , h ) = Ax(z cos f 1 + h cos f 2, z sin f 1 + z sin f 2) . (27)Thus, the radial slice Fax (z cos f , z sin f ) of Fax (z , h ) can be ex-pressed asFax (z cos f , z sin f ) = Ax(z M cos f′, z M sin f ′) , (28)where f ′ and M are as given in (23) and (24), respectively. It isknown that the radial slice of the ambiguity function of a signal x(t)at the angle f has the following relation to the (a′)th FrFT of thesignal x(t)Ax(z cos f ′, z sin f ′) =∫|x(a′)(r)|2e j2p z rdr . (29)Then, the relation in (22) can be obtained by combining (25), (28),and (29).4. FAST COMPUTATION OF THE JFSRIn this section, we provide an efficient computation algorithm ofthe JFSR of a signal on arbitrary radial slices. Throughout the com-putations, we assume that the signal x(t) is scaled to x(t/s) beforesampling, so that its WD is approximately confined into a circle ofradius D x/2. Here, if the time-width and bandwidth of the signalis approximately D t and D f , respectively, then the scaling parame-ter s becomes s =√D f /D t providing a signal which has negligibleenergy outside the interval [−D x/2, D x/2].To compute the radial slice of the JFSR of a signal x(t), we usethe relationEaxa(r cos f ,r sin f ) = |csc f 12|Wx(r cos f ′,r sin f ′) (30)wheref ′ = arctan(cos f cos f 2 + sin f cos f 1,cos f sin f 2− sin f cos f 1)(31)and f 1 and f 2 are the corresponding angles of the fractional Fourierdomains u and v with respect to the time axis. It has been shownin [9] that the radial slice of the WD along the line (r cos f ′,r sin f ′)isWx(r cos f ′,r sin f ′) =∫x(a′−1)(−l2)x∗(a′−1)(l2)e− j2p rl dl .(32)Therefore, (30) and (32) can be used to construct the radial slice ofEaxa(u,v) asEaxa(r cos f ,r sin f ) = |csc f 12|∫x(a′−1)(l2)x∗(a′−1)(−l2)× e− j2p rl dl . (33)Figure 2: The JFSR of x(t) = e−p ((t/3)2+0.3 jt3) at joint fractionalFourier domains with order (a) (a1,a2) = (0,0.25), (b) (a1,a2) =(0,0.5), (c) (a1,a2) = (0,0.75), and (d) (a1,a2) = (0,1). The distri-bution given in (d) is the same as the WD of x(t) since a1 = 0, a2 =1.If the double-sided bandwidth of x(a′−1) is D x and the time-bandwidth product is N, then the integral in (33) can be discretizedbyEaxa(rD x cos f ,rD x sin f ) =|csc f 12|D xN−1åk=−Nq[k]e−j2p rkD x (34)where q[k] = x(a′−1)[k]x∗(a′−1)[−k] and x(a′−1)[k] = x(a′−1)(k/(2D x))is computed using the algorithm given in [10] with O(NlogN) com-putational complexity.As the relationship (33) depends on the FrFTs of the signal x(t),computation of any M uniformly spaced samples on the line seg-ment r∈ [ri,r f ] along the radial slice of Eaxa(rD x cos f ,rD x sin f ) canbe performed through the chirp-z transform algorithm in O((N +M) log(N + M)) computational complexity [11]. In the followingsection, the results of the algorithm are presented for a syntheticsignal on various joint fractional Fourier planes.5. SIMULATIONSIn this section, the JFSR of a quadratic frequency modulated (FM)signal which has a non-convex time-frequency support on the time-frequency plane are evaluated for four different joint fractionalFourier order pairs of a = (0,0.25), (0,0.5), (0,0.75), and (0,1).The JFSRs of a quadratic FM signal x(t) = e−p ((t/3)2+0.3 jt3)which has a non-convex time-frequency support on the time-frequency plane are presented in Fig. 2. The distribution given in(d) with fractional Fourier order pair (0,1) is the same as the WD ofx(t). It is easier to observe the localization of the signal componentthat depends on the order of the joint fractional Fourier domains.Because the uncertainty relation of the fractional Fourier domainshas a tighter lower-bound when compared to the time and frequencydomains [7].6. CONCLUSIONSA joint fractional domain signal representation is developed usingthe energy density interpretation of the WD on the time-frequencyplane. The (u,v) axes defining the joint representation are chosenas the a = (a1,a2)-th order fractional Fourier domains. The dis-tribution is designed so that its projection of the JFSR on to the uand v axes gives the modulus square of the fractional Fourier trans-form of signals at the corresponding orders a1 and a2 as |xa1(t)|2and |xa2(t)|2, respectively. It is shown that the distribution Eax (u,v)depends on the WD through a coordinate transformation. There-fore, the JFSR is a real-valued distribution, too. The overall integralof the JFSR on the (u,v) plane gives the total energy of the sig-nal. In this paper, as part of the novel results, oblique projections ofthe JFSR is also derived and a fast computation algorithm designedfor the computation of arbitrary radial slices of the WD in [9] isextended to the computation of the JFSR. The JFSRs of various sig-nals at various fractionally-ordered domains are presented and thelocalization of the signal components are compared.The JFSR cannot be analyzed in the framework of the famil-iar Cohen’s class. However, its introduction to the non-stationarysignal processing will bring new insights into the design, filtering,analysis, and synthesis of signals.AcknowledgementThis research was supported by Brain Korea 21 Project under a2004 grant from the School of Information Technology, KAIST, forwhich the authors would like to express their thanks.REFERENCES[1] F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear andquadratic time–frequency signal representations,” IEEE Sig-nal Process. Magazine, vol. 9, pp. 21–67, Apr. 1992.[2] L. Cohen, Time–Frequency Analysis, Prentice Hall, Engle-wood Cliffs, 1995.[3] S. Mann and S. Haykin, “The chirplet transform: Physicalconsiderations,” IEEE Trans. Signal Process., vol. 43, pp.2745–2761, Nov. 1995.[4] R. G. Baraniuk and D. L. Jones, “Matrix formulation of thechirplet transform,” IEEE Trans. Signal Process., vol. 44, pp.3129–3135, Dec. 1996.[5] R. G. Baraniuk, “Beyond time-frequency analysis: Energydensity in one and many dimensions,” IEEE Trans. SignalProcess., vol. 46, pp. 2305–2314, Sep. 1998.[6] O. Akay and G. F. Boudreaux-Bartels, “Joint fractional signalrepresentations,” Journal of the Franklin Institute, vol. 337,pp. 365–378, 2000.[7] H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Frac-tional Fourier Transform with Applications in Optics and Sig-nal Processing, John Wiley & Sons, New York, 2000.[8] A. W. Lohmann and B. H. Soffer, “Relationships be-tween the Radon–Wigner and fractional Fourier transforms,”J. Opt. Soc. Am. A, vol. 11, pp. 1798–1801, June 1994.[9] A. K. Özdemir and O. Arıkan, “Efficient computation of theambiguity function and the Wigner distribution on arbitraryline segments,” IEEE Trans. Signal Process., vol. 49, pp. 381–393, Feb. 2001.[10] H. M. Ozaktas, O. Arıkan, M. A. Kutay, and G. Bozdagi,“Digital computation of the fractional Fourier transform,”IEEE Trans. Signal Process., vol. 44, pp. 2141–2150, Sep.1996.[11] L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z–transform algorithm and its applications,” Bell Syst. Tech. J.,vol. 48, pp. 1249–1292, May 1969.

Generalization of time-frequency signal representations to joint fractional Fourier domains

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Generalization of time-frequency signal representations to joint fractional Fourier domains

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