sampling, resulting in iscrete-time, iscrete-frequency functions, before they can be implemente in any igital system an be of practical use. A consier

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1 DISTANCE METRICS FOR DISCRETE TIME-FREQUENCY REPRESENTATIONS James G. Droppo an Les E. Atlas Department of Electrical Engineering University of Washington Seattle, WA fjroppo ABSTRACT This paper presents a new metho for aressing problems relate to the sparsity of the iscrete time-frequency representation. Although each real signal x[n] has only N free parameters, its representation has N 2 elements, along N(N 1)=2 inepenent imensions. This leas to problems in moeling, classication, an recognition tasks. An overview of our iscrete time-frequency representations is presente, together with a brief iscussion of previous work in continuous time-frequency representations. These iscrete time-frequency representations have all of the escriptive power of conventional iscrete spectral features, together with a powerful framework that escribes how the spectrum evolves over time. Unfortunately, using these features irectly in their natural, high-imensional space tens to be unworkable. A geoesic istance measure is introuce, that leverages our knowlege of the set of vali time-frequency representations to reuce the apparent imensionality of the problem. Applications of this geoesic istance are foun in signal classication an nonlinear time-frequency interpolation. 1. INTRODUCTION Stanar techniques such as Gaussian mixture moels, vector quantization, or k-nearest neighbor can break own when applie to iscrete time-frequeny representations. There are three reasons for these ifculties. The rst reason is ata sparsity. For real signals with length N, the time-frequency representation consists of N(N 1)=2 linearly inepenent imensions. To moel this feature accurately using conventional methos, on the orer of N 2 inepenent training examples woul be neee, which is often impractical. The secon reason is that not all congurations in this high-imensional feature space are vali timefrequency representations. In particular, it can be shown that the set of all vali time-frequency representations is a continuous N imensional surface embee in the feature space. As a result, at every point in the feature space, a istribution has zero extent in at least N(N 1)=2 imensions. The thir problem with moeling with traitional This work was supporte by Microsoft Research an the Ofce of Naval Research. techniques is that the mean feature may not lie near any vali representations. That is, for a given set of ata, the mean of the ata oes not correspon with an actual signal. In other wors, the mapping from the representation to the signal is over-etermine. As an example of the problems that may be encountere, consier the case of a set of three imensional ata restricte to lie on the surface of a cone. While it may take many mixtures of full covariance Gaussians to moel the ata accurately, there is a simple unerlying structure that shoul be exploite. Section 2 presents an abbreviate erivation of the iscrete time-frequency representations that are the focus of this paper, together with an overview of previous work on continuous time-frequency representations an a short iscussion of how to rea an interpret these representations. Section 3 escribes what can be expecte of the set of vali time-frequency representations. The imensionality of the representations is explore, together with the gross shape of the set. Section 4 continues the iscussion by introucing a geoesic istance metric that leverages the information containe in Section 3. Three methos are explore to compute an approximate this metric. Applications in signal classication an non-linear interpolation are iscusse in Sections 5 an 6. These Sections illustrate the usefulness of this geoesic metric for classication an interpolation tasks. The metric incorporates topological knowlege of the feature space, an improves performance for classication in aition to having some novel signal morphing, or non-linear signal interpolation, properties. 2. DISCRETE TIME-FREQUENCY REPRESENTATIONS Our time-frequency representations are base on the work of Leon Cohen, who evelope a systematic way of obtaining bilinear, joint bivariate ensities of time an frequency using operator theory[1]. More recently [2], he presente a generalize approach for obtaining joint representations for arbitrary quantities (not just time an frequency) using the characteristic function operator methos of Moyal [3] an Ville [9]. Although Cohen's class of continuous quaratic TFRs are backe by rich theory, they have to unergo iscrete 1

2 sampling, resulting in iscrete-time, iscrete-frequency functions, before they can be implemente in any igital system an be of practical use. A consierable amount of work has gone into the stuy of sampling the continuoustime TFRs an ameliorating the ill effects introuce by the process, e.g. [5]. Our approach to formulating iscrete-time, iscretefrequency Representations [4] avois these sampling issues by proviing a irect theoretical link between the iscrete time sequence x[n] an its iscrete time-frequency representation P [n; k]. The erivation parallels Cohen's metho for continuous time an frequency operators, but substitutes iscrete operators in their place. As in the continuous case, each iscrete time sequence is associate with not one, but a multitue of time-frequency istributions, each one uniquely specie by the root istribution an a kernel function. A similar metho of eriving iscrete time-frequency representations, base on group theory, has been presente by Richman, et al. in [6]. One notable ifference between the two is that whereas Richman's technique incorporates ifferent calculations for even an o length signals, our technique is invariant to signal length Derivation We begin by assuming that we have a iscrete, perioic sequence x[n] with perio N, whose iscrete Fourier transform is X[k]. By eveloping iscrete (matrix) operators corresponing to iscrete time an frequency variables, we can generate a iscrete representation in time an frequency, P [n; k]. This yiels a technique that is istinct from sampling a istribution prouce with the continuous time an frequency operators. There are problems inherent in sampling the continuous istribution that the iscrete theory circumvents, by mapping irectly from a iscrete signal to a iscrete time-frequency representation. The rst step is, of course, to formulate the iscrete counterparts of the time an frequency operators. The time operator in the time omain, L, an the frequency operator in the time omain, K, obey the equations X N 1 x Λ [n]kx[n] = X Λ [k]kx[k]; an (1) k=0 Lx[n] = nx[n]: (2) Since these operators are iscrete an linear, they can be compactly represente by the matrices L an K. Operator theory ictates that we can compute time-frequency representations accoring to the expectation M[ ; ] =hexp(j2ßl ) exp(j2ßk)i : (3) Simplifying Equation 3 above, an absorbing the constants into the operators for convenience, we get M[ ; ] = hexp(j L) exp(jk)i = x Λ [n] exp(j L) exp(jk)x[n ] = x Λ [n] exp(j L)x[n ]: (4) A two imensional iscrete Fourier transform an further simplication yiels the time-frequency representation, P [n; k] =x Λ [n] exp(jnk)x[k]; (5) which is the iscrete version of the well-known Rihaczek TFR [7]. This result is analogous to the continuous-time result which can be erive in a similar fashion. Just as in the continuous case, we can take avantage of the corresponence rule to represent permutations of the operators in Equation 3 as a multiplicative kernel function f[ ; ]. In this way, the complete set of time-frequency representations within this class can be generate in the usual way. Members of this class of iscrete time-frequency representations inclue the iscrete Rihaczek TFR [7], the iscrete Margeneau-Hill, an the spectrogram Interpretation Time-frequency representations promise to capture both static spectral information an evolutionary spectral information in a single feature. Although representations in time an frequency usually make intuitive sense to even the untraine viewer, representations in an, as in Equation 4, ten to be less intuitive. The representation escribe by Equation 4 contains a lot of information about the signal x[n]. If = 0,then M[ ; ] becomes the stationary autocorrelation of the signal x[n], with taking its familiar role as the time lag variable. X N 1 M[0;]= x Λ [n ]x[n] (6) As a result, values along the = 0 axis are always interprete as the autocorrelation of the signal x[n]. Conversely, any stationary signal will concentrate its energy within this region. If = 0, then M[ ; ] becomes the spectrum of the instantaneous energy of the signal x[n]. This leas us to refer to as the moulation frequency variable, because it relates to how quickly the envelope of the signal is changing. M[ ; 0] = jx[n]j exp 2 j2ß n (7) N As a result, values along the = 0 axis show how the moulation envelope" of the signal x[n] ischanging with time. A signal with=20 little correlation between its samples will concentrate its energy in this region. Any point M[ ; ] refers to how quickly a specic correlation coefcient is being moulate. It can be interprete as the output of a non-stationary sinusoial lterbank, where istance from the origin is analogous to banwith, an the angle from the axis represents the linear chirp rate of the lter-bank. To see this, consier a kernel function f[ ; ] = f[ a]f[ b]; where (8) ρ 1 n =0 f[n] = (9) 0 otherwise. 2

3 In the (n k) plane, this kernel becomes a complex exponential,=20 with parameters controlle by a an b. Φ[n; k] = exp ( j2ß(an bk)) (10) a four-imensional space, with extent only in three imensions. This agrees with the preicte value, since the expecte imensionality of the cone is 2(2 1)=2 = 3. This case is examine in etail in Section 4.1. In summary, a stationary process has no extent in, a spectrally flat process has no extent in, an more interesting structures appear when 6= 0 an 6= 0. Time moulations increase a signal's extent in, an colore spectra are responsible for extent in Unleashing Discrete Time-Frequency Representation's Power The traitional isavantage to using these representations as features is the eluge of information. Whereas stationary spectral features prouce vectors with at most a few ozen imensions, a real length N signal will always prouce a representation with N(N 1)=2 linearly inepenent imensions. For a 250 ms signal sample at 16 khz, this amounts to no less than eight million imensions. There have been several attempts in the literature to reuce this information into salient features, incluing taking autocorrelation coefcients of the representation in time or frequency to achieve a time-shift or frequency-shift invariant representation of the ata [10], an ning two imensional moments of the time-frequency representation [8]. Given the apparently immense imensionality of the feature space, one might assume that extracting useful information in an automate an efcient way woul be impossible. This is not the case. The next sections present a new way of looking at these representations. 3. THE SET OF VALID TIME-FREQUENCY REPRESENTATIONS Consier the iscrete-time signal x[n], with length N. The iscrete time-frequency representation M[ ; ] associate with x[n] consists of N 2 complex numbers, an is containe in the set of length N 2 complex vectors. X N 1 M[ ; ] = x Λ [n]x[n ] exp (2ß n=n) : This, however, is somewhat misleaing as to the true size of the feature space. A iscrete Fourier transform from into n yiels, A[n; ] =x Λ [n]x[n]: (11) Clearly, there are N(N 1)=2 unique proucts when 6= 0,an N magnitues when = 0. For complex signals, this yiels a feature space with N 2 linearly inepenent imensions, an for real signals, the feature space has N(N 1)=2 linearly inepenent imensions. Furthermore, the mapping between the signal x[n] an its representation M[n; k] iscontinuous, an therefore the set of representations that correspon to actual signals forms a continuous subspace. For real signals with N = 2, the set of vali timefrequency representations is a 90 egree cone embee in Figure 1: Conical structure of vali TFR For larger values of N, the surface can not be name, but it can be functionally escribe. The set of vali iscrete time-frequency representations always lies on a conical structure, regarless of N. The axis of the cone correspons to the energy of the signal x[n], which is store at the origin of the ( ; ) plane, M[0; 0]. For a xe energy, the vali representations lie on a hyper-sphere with a raius proportional to M[0; 0]. X X L 1 X L 1 X L 1 L 1 =20jA[m; n]j 2 = =20jx[m]x Λ [n]j 2 m=0 m=0 L 1 X = jx[m]j 2 L 1 X jx[n]j 2 m=0 = jm[0; 0]j 2 (12) Although all vali representations lie on this conical structure, not all points on this structure are vali representations. That is, meeting Equation 12 is a necessary, but not sufcient test for vali time-frequency representations. 4. A GEODESIC DISTANCE MEASURE Measuring the istance between New York an San Francisco by rawing a straight line through three imensional 3

4 space woul not seem to be an acceptable measure. Similarly, measuring the istance along a line connecting two vali representations, but lying entirely outsie of the surface of vali representations, may not be the best solution. This is exactly what is going on with a Eucliean istance metric in the time-frequency representation space. The structure of the surface of vali time frequency representations, along with the calculus of variations, allows us to n the shortest line connecting two representations, that remains entirely within the set of representations of true signals. This line is a geoesic, an its length is the geoesic istance between the enpoints. This line can be use to morph one signal into another, along the shortest path in quaratic time-frequency space. That is, given two signals, it species a continuous path along which one signal slowly changes into the other. This iea is explore in Section 6. The line can also be use to n the true" istance between istributions, which woul not necessarily be a straight line in the feature space. Without explicitly solving the geoesic problem, approximate solutions can be foun. Two solutions for the geoesic istance are presente, an algorithmic approximation an a numerical approximation Exact Solution for Length Two Signals This section eals with the case of real signals with N =2. As mentione previously, the set of vali representations for this case is a three imensional cone embee in a four imensional space. An exact solution for the geoesic between two given enpoints can be foun using the calculus of variations. For convenience, let a an b represent the two samples of the signal x[n]. In the ( ; ) plane, this signal's representation only has three non-zero values. M[0; 1] = 2ab M[1; 1] = 0 M[0; 0] = a 2 b 2 M[1; 0] = a 2 b 2 A simple change of variables reveals the traitional formula for a cone, x = M[0; 1] = u sin v =20 y = M[1; 0] = u cos v z = M[0; 0] = u, where u 0 To etermine the istance between two representations, we measure the shortest line along the surface that joins the two points. This is exactly the type of problem that the calculus of variations is meant to solve. The problem is pose as one of minimizing the line integral between the two points. The calculus of variations tells us that the geoesic function v(u) is given by Z p P v = c1 pr(r c 2 1 ) u; where Q = f P = f fu x f fv x fu x 2 f fu y 2 f f fu y f fv y f fu z f fv z fu z 2 an so f fu f fu R = f 2 f 2 f 2 fv x fv y fv z f x(u; v) = sin(v) f y(u; v) = cos(v) f z(u; v) =1 f fu fv fv x(u; v) =ucos(v) y(u; v) = usin(v) z(u; v) =0 fv P =2 Q=0 R=u 2 Combining these equations, the result is v = c1 Z p 2 p(u 2 )(u 2 c 2 1 ) u Z = c1 p 2 p u v = p 2 cos 1 c1 u 1 u 2 c 2 1 c2 The constants c1 an c2 are epenent on the esire enpoints of the geoesic, (u1;v1) an (u2;v2). c1 = ±u cos( v c2 p 2 ) c2 = v p 2 cos 1 c1 u u p c2 = 2 tan 1 u1 cos p v 1 2 u2 cos p v 2 u1 sin p v 1 2 u2 sin p v Algorithmic Approximation To n the shortest istance between two points, lying entirely along vali time-frequency representations, an initial solution is pose, an then iteratively re-estimate. This algorithm prouces both an estimate of the geoesic istance, an evenly space points along the geoesic. The initial estimate is to form a set of points equally space in the signal space, with the proper enpoints. If one enpoint is the signal a[n], an the secon enpoint is b[n], the initial set of points in the signal space woul be, x i[n] =a[n](b[n] a[n]) ; for 0» i<m: M 1 The re-estimation assumes that links between neighboring signals are elastic, an tries to pull each signal towars its neighbors in the time-frequency representation space. First, a irection is compute that will move a signal closer to its neighbors in the time-frequency representation space. If the representation for y[n] moves in this irection, the istance to the representations x[n] an z[n] tens to ecrease. v[n; ] = 1 (x[n]x[]z[n]z[]) y[n]y[] 2 This vector is then mappe to an equivalent irection in signal space. The erivatives of the representation with respect to the original signal samples are y[n]y[] =f[n a]y[]f[ a]y[n]: y[a] i 4

5 Now, this is a scalar-value function of n,, an a. Unwrap it into a matrix, whose columns are a, an whose rows consist of all combinations of n an. Call this matrix V. Its columns represent the irections the representation will move with changes in the signal values. Now, all that remains is mapping the esire vector v, which exists in the representation space, onto the available irections containe in V. This is a simple matrix operation, an consists of the minimum square solution to the linear equation, V = v; = (V H V ) 1 V H v: If the signal y is move in the irection, the istances between the representations for y an z, an y an x, is reuce. If the representations of x, y, an z are sufciently close, the linear istance an the geoesic istances shoul be approximately equal Numerical Approximation The numerical approximation to the geoesic istance consists of rst ening a path in the representation space, an then precisely computing its length. Accoring to Equation 11, the surface of vali timefrequency representations consists of N 2 imensions y ij, where 0» ; jg < N. This surface is parameterize in terms of the inepenent variables x i, where 0» i < N, which are the iscrete signal samples. y ij = x ixj The rst step is to assume the signal values x i are functions of a single inepenent variable, t. As t varies from 0 to 1, a path is trace through both the signal space an the representation space. Let ^u ij be a unit vector in the N 2 time-frequency representation space such that h^u ab ; ^u c i = f[a c]f[b ]: The erivative of the path with respect to the inepenent variable is given by t ^s = X i;j = X i;j = X i;j yij ^uij t t xi(t)xj(t)^uij (x j(t)_x i(t)x i(t)_x j(t) = 20) ^u ij To n the istance along the path, integrate the function f(t), where f(t) = = ^s t ψ X i;j (x j(t)_x i(t)x i(t)_x j(t) = 20) 2! 1 2 In many imensions, it is ifcult to solve this equation exactly for the functions x i(t), 0» i<n, but an approximate solution can be foun by assuming a linear form for the functions. The slope an offset for each function are entirely specie by the initial value, x i(0), an the nal value, x i(1). x i(t) = x i(0)(x i(1) x i(0)) t _x i(t) = x i(1) x i(0) Once the enpoints for the calculation are known, the function f(t) takes the form of a square root of a quaratic function. This can be quickly estimate with great precision by calculating the integral Z 1 = 0 f(t)t: This approximation is equivalent to the starting pointof the iterative solution presente previously. Although this approximation is rather gross, in practice the ifferences between the two solutions o not amount to much. That is, if you raw a straight line between two signals in the signal space, the length of the corresponing line in the time-frequency representation space is close to optimal. 5. CLASSIFICATION IMPROVEMENT Figure 2 shows the performance of three istance metrics for classication in aitive white Gaussian noise. The L2-Signal" metric is equivalent tothe matche lter solution, an serves as an upper boun on performance. The Geoesic Approximation" correspons to a classier using the numerical approximation to the geoesic istance. The L2-Representation" classier uses the istance from the mean in representation space. The geoesic approximation oes much better than computing istances irectly in the representation space. It comes close to being the optimal etector, although it is just an approximation to the true geoesic istance. 6. NON-LINEAR INTERPOLATION, MORPHING The points along a geoesic correspon to signals that are between the enpoints an close to neighboring points in the geoesic. By ening between an close in the iscrete timefrequency representation space, it is reasonable to assume that as one travels along the geoesic, the signal's time an frequency structure are both changing. If the enpoints are chosen to be similar signals that iffer by a time shift, the algorithmic approximation presente in Section 4.2 converges to a sequence of similar signals, with an increasing time shift. If the enpoints are chosen to be impulse response functions for igital lters with two ifferent resonances, the algorithmic approximation converges to a sequence of signals, whose resonance is shifting from being close to the starting signal, to being close to the ening signal. 5

6 100 L2 Signal (o), Geoesic R (x), L2=Rihaczek () Percent Correct Classication RMS Noise Figure 2: Classication in the presence of aitive noise 7. CONCLUSION AND FUTURE WORK This paper presente a new way of computing istances an paths between iscrete time-frequency representations. A geoesic is compute, which incorporates topological knowlege of the feature space to reuce the apparent imensionality of the problem. Using the geoesic istance, a classier was constructe that outperforms classiers that o not incorporate any knowlege of the feature space. Furthermore, if the geoesic in the time-frequency space is use as a non-linear interpolation algorithm, reasonable results are obtaine. The kernel function f[ ; ] is almost completely ignore in this work. It coul be re-introuce before beginning the geoesic calculation. Since it is a multiplicative mask on M[ ; ], the net effect woul be a cost function on the path moving along irections with high f values. One coul use the kernel to impose no cost for changing the moulation of the signal, or vary the cost along to penalize ner moications to the spectrum more or less. REFERENCES [1] Leon Cohen. Generalize phase-space istribution functions. J. Math. Phys., 7: , [2] Leon Cohen. Time-Frequency Analysis. Prentice Hall Signal Processing Series, New Jersey, [3] J. E. Moyal. Quantum mechanics as a statistical theory. In Proc. Camb. Phil. Soc., volume 45, pages , [4] Siva Bala Narayanan, Jack McLaughlin, Les Atlas, an James Droppo. An operator theory approach to iscrete time-frequency istributions. In Proceeings of the IEEE-SP International Symposium on Time- Frequency an Time-Scale Analysis, pages , [5] A. H. Nutall. Alias-free Wigner istribution functions an complex ambiguity functions for iscretetime samples. Technical Report 8533, Naval Unerwater Systems Center, April [6] M. Richman, T. Parks, an R. Shenoy. Features of a iscrete Wigner istribution. In 1996 IEEE Digital Signal Processing Workshop Proceeings, pages , [7] A. W. Rihaczek. Signal energy istribution in time an frequency. IEEE Transactions on Information Theory, 14: , [8] B. Tacer an P. Loughlin. Time-frequency base classication. In Proceeings of the SPIE The International Society for Optical Engineering, volume 2846, pages , Denver, CO, USA, August [9] J. Ville. Theorie et applications e la notion e signal analytique. Cables et Transmission, 2 A:61 74, [10] E. J. Zalubas, J. C. Oneill, W. J. Williams, an A. O Hero III. Shift an scale invariant etection. volume 5, pages