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OPTIMALITY OF PRINCIPAL COMPONENT

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OPTIMALITY OF PRINCIPAL COMPONENT FILTER BANKS FOR DISCRETE MULTITONECOMMUNICATION SYSTEMSP. P. Vaidyanathan†, Yuan-Pei Lin‡, Sony Akkarakaran†, and See-May Phoong∗†Dept. Electrical Engr., Caltech, Pasadena, CA 91125‡Dept. of Elec. and Control Engr., National Chiao Tung Univ., Hsinchu, Taiwan∗Dept. Electrical Engr., National Taiwan University, Taipei, Taiwan.Abstract.1Discrete multitone modulation is an at-tractive method for communication over a non flatchannel with possibly colored noise. The uniform DFTfilter bank and cosine modulated filter bank have in thepast been used in this system because of low complex-ity. We show in this paper that principal componentfilter banks, which are known to be optimal for datacompression and denoising applications, are also opti-mal for a number of criteria in DMT communication.1. INTRODUCTIONFigure 1 shows a maximally decimated analysis/syn-thesis system traditionally used in subband coding andmore recently in signal denoising (all notations are as in[18]). A dual of this system called the transmultiplexercircuit is shown in Fig. 2. This is commonly used forconversion between time domain and frequency domainmultiplexing. More recently this system has found ap-plication in the digital implementation of multicarriersystems, more popularly known as the DMT (discretemultitone) modulation systems. In this paper we willprimarily be concerned with this application. Eitherof the systems shown in the two figures is said to be abiorthogonal system if the filters are such thatHk(ejω)Fm(ejω)↓M= δ(k − m)This is equivalent to the perfect reconstruction prop-erty, that is, x(n)=x(n) for all n in Fig. 1, andyk(n)=xk(n)for all k,n in Fig. 2. The set of M filters {Fk(z)}is said to be orthonormal if Fk(ejω)F∗m(ejω)|↓M=δ(k − m) (equivalently the polyphase matrix is parau-nitary [18]). In this case biorthogonality or perfect1Work supported in parts by the ONR grant N00014-99-1-1002, USA, by Microsoft Research, Redmond, WA, and by theNSC 88-2218-E-009-016 and 88-2213-E-002-080, Taiwan, ROC.reconstruction is achieved by the choosing Hk(ejω)=F∗k(ejω).The use of filter bank theory in the optimization ofDMT systems has been of some interest in the past[10], [11]. In this paper we will show using the resultsof [3] that the principal component filter bank, which isknown to be optimal for several problems involving thesubband coder, will also be optimal in many respectsfor the DMT communications system.x(n)MMMF (z)0F (z)1F (z)M 1− synthesisfilters expandersMMMH (z)1H (z)M 1− H (z)0analysisfiltersdecimatorsv (n)0v (n)1v (n)M-1x(n)Figure 1. The subband coder system.MMMF (z)0F (z)1F (z)M 1− x (n)0x (n)1x (n)M-1x(n)synthesisfilters expandersH (z)1H (z)M 1− H (z)0analysisfiltersMMMy (n)0y (n)1y (n)M-1decimatorsFigure 2. The digital transmultiplexer.2. PRINCIPAL COMPONENT FILTER BANKSThe optimality of principal component filter banks(PCFBs) in the context of progressive transmission andI-1280-7803-5482-6/99/$10.00 ©2000 IEEEISCAS 2000 - IEEE International Symposium on Circuits and Systems, May 28-31, 2000, Geneva, Switzerlandsubband coding was observed to various degrees by anumber of authors [2,12,15,17, 19]. To define a PCFBfirst consider two sets of M nonnegative numbers {an}and {bn}. We say that {an} majorizes {bn} if, afterreordering such that an≥ an+1and bn≥ bn+1, wehavePn=0an≥Pn=0bnfor 0 ≤ P ≤ M − 1, with equality for P = M − 1.Thus all the partial sums in {an} dominate those in{bn}. Consider a given class C of M-band uniform or-thonormal filter banks. This class can be the class Ctcof transform coders (filter lengths ≤ M ), or the classCidealof ideal filter banks (filters allowed to have in-finite order, like brickwall filters). Or it could be apractically attractive class like the FIR class Cfirwithfilter orders bounded by a fixed integer, or the cosinemodulated class Ccmfb. Given such a class C and an in-put power spectrum Sxx(ejω) we say that a filter bankF in C is a principal component filter bank or PCFBif the set {p2k} of its subband variances (i.e., variancesσ2vkof the signals vk(n) in Fig. 1) majorizes the set{q2k} of subband variances of all other filter banks inthe class. That is, with p2n≥ p2n+1and q2n≥ q2n+1,p20≥ q20,p20+ p21≥ q20+ q21,...and so forth. The equalityM−1k=0p2k=M−1k=0q2kfol-lows automatically from orthonormality.The advantage of PCFBs is that they are optimal forseveral problems. This includes subband coding witharbitrary (not necessarily high) bit rates, the denois-ing problem, and so forth, as elaborated in [3]. Thesearise from the result (proved in [3]) that any concavefunction φ of the subband variance vectorv =[σ2v0σ2v1... σ2vM −1]Tis minimizedbya PCFB when one exists. Using this weshow in this paper that PCFBs also serve as optimal so-lutions to certain problems in communication systemswhich use DMT modulation. It possible that PCFBsdo not exist for certain classes but when they exist,they have the stated optimality. Whenever we say thatthe PCFB is optimal for a problem, the implicit as-sumption is that the class of filter banks searched issuch that a PCFB exists.For the transform coder class Ctc, the M ×M KLT ofthe input serves as the PCFB. For the ideal filter bankclass Cideal, there is a systematic method to constructa PCFB by designing a sequence of compaction filters[19]. For example the filter bank in Fig. 3(b) is aPCFB for the power spectrum in Fig. 3(a).ω02πinput power spectrum(a)(b)H0H1H2H3ω02πPCFB for M=4Figure 3. A power spectrum and its PCFB (M =4).3. THE DMT COMMUNICATION SYSTEMFigure 4 shows the essentials of discrete multitone com-munication. Background material on the DMT systemand more generally on the use of digital filter banks incommunications can be found in [1,4,5,7,8,16].x(n)MMMF (z)0F (z)1F (z)M 1− x (n)0x (n)1x (n)M-1transmittingfilters H (z)1H (z)M 1− H (z)0receivingfiltersMMMy (n)0y (n)1y (n)M-1C(z)channely(n)noise e(n)Figure 4. The discrete multitone communication system.Briefly, here is how the system works: the signals xk(n)are bk-bit symbols obtained from a PAM or QAMconstellation [13]. Together these signals representkbk= b bits, and are obtained from a b-bit blockof a binary data stream [4]. The symbols xk(n) arethen interpolated M-fold by the filters Fk(z). Typi-cally the filters {Fk(ejω)} constitute an


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