Iraj Sodagar. “Time-Varying Analysis-Synthesis Filter Banks” 2000 CRC Press LLC. <http://www.engnetbase.com>. Time-VaryingAnalysis-Synthesis FilterBanks IrajSodagar DavidSarnoffResearchCenter 37.1Introduction 37.2AnalysisofTime-VaryingFilterBanks 37.3DirectSwitchingofFilterBanks 37.4Time-VaryingFilterBankDesignTechniques ApproachI:IntermediateAnalysis-Synthesis(IAS) • Approach II:InstantaneousTransformSwitching(ITS) 37.5Conclusion References 37.1 Introduction Time-frequencyrepresentations(TFR)combinethetime-domainandfrequency-domainrepresen- tationsintoasingleframeworktoobtainthenotionoftime-frequency.TFRofferthetimelocalization vs.frequencylocalizationtradeoffbetweentwoextremecasesoftime-domainandfrequency-domain representations.Theshort-timeFouriertransform(STFT)[1,2,3,4,5]andtheGabortransform[6] aretheclassicalexamplesoflineartime-frequencytransformswhichusetime-shiftedandfrequency- shiftedbasisfunctions. Inconventionaltime-frequencytransforms,theunderlyingbasisfunctionsarefixedintimeand defineaspecifictilingofthetime-frequencyplane.Thetermtime-frequencytileofaparticular basisfunctionismeanttodesignatetheregionintheplanethatcontainsmostofthatfunction’s energy.Theshort-timeFouriertransformandthewavelettransformarejusttwoofmanypossible tilingsofthetime-frequencyplane.ThesetwoareillustratedinFig.37.1(a)and(b),respectively.In thesefigures,therectangularrepresentationforatileispurelysymbolic,sincenofunctioncanhave compactsupportinbothtimeandfrequency.Otherarbitrarytilingsofthetime-frequencyplaneare possiblesuchastheexampleshowninFig.37.1(c).Inthediscretedomain,lineartime-frequency transformscanbeimplementedintheformoffilterbankstructures. Itiswellknownthatthetime-frequencyenergydistributionofsignalsoftenchangeswithtime. Thus,inthissense,theconventionallineartime-frequencytransformparadigmisfundamentally mismatchedtomanysignalsofinterest.Amoreflexibleandaccurateapproachisobtainedifthe basisfunctionsofthetransformareallowedtoadapttothesignalproperties.Anexampleofsuch atime-varyingtilingisshowninFigure37.1(d).Inthisscenario,thetime-frequencytilingofthe transformcanbechangedfromgoodfrequencylocalizationtogoodtimelocalizationandviceversa. Time-varyingfilterbanksprovidesuchflexibleandadaptivetime-frequencytilings. c  1999byCRCPressLLC FIGURE 37.1: The time-frequency tiling for differenttime-frequency transforms: (a) The STFT,(b) thewavelettransform,(c)anexampleofgeneraltiling,and(d)anexampleofthetime-varyingtiling. The concept of time varying (or adaptive) filter banks was originally introduced in [7]byNayebi et al. The ideas underlying their method were later developed and extended to a more general case in which it was also shown that the number of frequency bands could also be made adaptive [8, 9, 10, 11]. De Queiroz and Rao [12] reported time-varying extended lapped transforms and Herley et al. [13, 14, 15] introduced another time-domain approach for designing time-varying lossless filter banks. Arrowood and Smith [16] demonstrated a method for switching between filter banks using latticestructures. In[17], the authors presentedyet another formulation for designing time-varying filterbanksusingadifferentfactorizationoftheparaunitarytransform. ChenandVaidyanathan[18] reported a noncausal approach to time-varying filter banks by using time-reversed filters. Phoong and Vaidyanathan [19] studied time-varying paraunitary filter banks using polyphase approach. In [11, 20, 21, 22], the post filtering technique for designing time-varying filter bank was reported. The design of multidimensional time-varying filter bank was addressed in [23, 24]. In this article, we introduce the notion of the time-varying filter banks and briefly discuss some design methods. 37.2 Analysis of Time-Varying Filter Banks Time-varying filter banks are analysis-synthesis systems in which the analysis filters, the synthesis filters,thenumberofbands,thedecimationrates,andthefrequencycoverageofthebandsarechanged (in part or in total) in time, as is shown in Fig. 37.2. By carefully adapting the analysis section to the temporal properties of the input signal, better performance can be achieved in processing the signal. Intheabsenceofprocessingerrors,thereconstructedoutput ˆx(n) shouldcloselyapproximate a delayed version of the original signal x(n). When ˆx(n − ) = x(n) for some integer constant, , then we say that the filter bank is perfectly reconstructing (PR). The intent of the design is to choose the time-varying analysis andsynthesisfiltersalongwith thetime-vary ing down/up samplers so that the system requirements are met subject to the constraint that the analysis-synthesis filter bank be PR at all times. c  1999 by CRC Press LLC FIGURE 37.2: The time-varying filter bank structure with time-vary ing filters and time-dependent down/up samplers. One general method for analysis of time-varying filter banks is the time-domain formulation reported in [10, 22]. In this method, the time-varying impulse response of the entire filter bank is derived in terms of the analysis and synthesis filter coefficients. Figure (37.3) shows the diagram of a time-vary ing filter bank. In this figure, the filter bank is divided into three stages: the analysis filters, the down/up samplers, and the synthesis filters. The signals x(n) and ˆx(n) are the filter bank input and output at time n, respectively. The outputs of the analysis filters are shown by v(n) =[v 0 (n), v 1 (n), ,v M(n)−1 (n)] T ,wherev i (n) is the output of the ith analysis filter at time n. The outputs of the down/up samplers at time n is called w(n) =[w 0 (n), w 1 (n), ,w M(n)−1 (n)] T . FIGURE 37.3: Time-varying filter bank as a cascade of analysis filters, down/up samplers, and synthesis filters. The input/output relation of the analysis filters can be expressed by v(n) = P(n)x N (n) . (37.1) P(n) is an M(n) × N(n) matrix whose mth row is comprised of the coefficients of the mth analysis filter at time n and x N (n) is the input vector of length N (n) at time n: x N (n) =[x(n), x(n − 1), x(n − 2), ,x(n− N(n) + 1)] T . (37.2) The input/output function of down/up samplers can be expressed in the form w(n) = (n)v(n) (37.3) where (n) is a diagonal matrix of size M(n) × M(n).Themth diagonal element of (n), at time n,is1 if the input and output of the mth down/up sampler are identical, otherwise it is zero. c  1999 by CRC Press LLC To write the input/output relationship of the synthesis filters, Q(n) is defined as Q(n) =        g 0 (n, 0)g 0 (n, 1)g 0 (n, 2) g 0 (n, N(n) − 1) g 1 (n, 0)g 1 (n, 1)g 1 (n, 2) g 1 (n, N(n) − 1) g 2 (n, 0)g 2 (n, 1)g 2 (n, 2) g 2 (n, N(n) − 1) . . . . . . . . . . . . . . . g M(n)−1 (n, 0)g M(n)−1 (n, 1)g M(n)−1 (n, 2) g M(n)−1 (n, N(n) − 1)        =  q 0 (n) q 1 (n) q 2 (n) . . . q N(n)−1 (n)  (37.4) where q i (n) =[g 0 (n, i), g 1 (n, i), g 2 (n,i), ,g M(n)−1 (n, i)] T , is a vector of length M(n) and g i (n, j) denotes the j th coefficient of the ith synthesis filter. At time n, the mth synthesis filter is convolved with vector [w m (n), w m (n − 1), ,w m (n − N (n) + 1)] T and all outputs are added together. UsingEq. (37.4), the output of the filter bank at time n can be written as: ˆx(n) = N(n)−1  i=0 q T i (n) w(n − i) . (37.5) If s(n) and ˆw(n) are defined as s(n) =  q T 0 (n), q T 1 (n), q T 2 , ,q T N(n)−1 (n)  T (37.6) ˆw(n) =  w T (n), w T (n − 1), w T (n − 2), ,w T (n − N(n) + 1)  T , (37.7) then Eq. (37.5) can be wr itten in the for m of one inner product, ˆx(n) = s T (n) ˆw(n) (37.8) where s(n) and ˆw(n) are vectors of length N(n)M(n). Using Eqs. (37.1), (37.3), (37.7), and (37.8), the input/output function of the filter bank can be written as: ˆx(n) = s T (n)        (n) P(n) x N (n) (n − 1) P(n − 1) x N (n − 1) (n − 2) P(n − 2) x N (n − 2) . . . (n − N(n) + 1) P(n − N (n) + 1) x N (n − N(n) + 1)        . (37.9) As the last N(n) − 1 elements of vector x N (n − i) are identical to the first N(n) − 1 elements of vector x N (n − i − 1), the latter equation can be expressed by ˆx(n) = s T (n)       (n) P(n)  O O O  (n − 1) P(n − 1)  O O OO  (n − 2) P(n − 2)  O O . . . O O  (n − N (n) + 1) P(n − N(n) + 1)               x(n) x(n − 1) x(n − 2) . . . x(n − 2N (n) + 1)         (37.10) c  1999 by CRC Press LLC where O is the zero column vector with length M(n). Thus, the input/output function of a time- varying filter bank can be expressed in the form of ˆx(n) = z T (n)x I (n) (37.11) where x I (n) =[x(n), x(n − 1), ,x(n− I + 1)] T and I (n) = 2N(n) − 1 and z(n) is the time- varying impulse response vector of the filter bank at time n: z(n) = A(n) s(n). (37.12) The matrix A(n) is the [2N (n) − 1]×[N(n) M(n)] matrix A(n) =            P(n) T (n)   O T O T . . . O T O T   P(n − 1) T (n − 1)   O T . . . O T . . . O T . . . O T   P(n − N (n) + 1) T (n − N (n) + 1)            . (37.13) Forap erfectreconstructionfilterbankwithadelayof,itisnecessaryandsufficientthatallelements but the ( + 1)th in z(n) be equal to zeroat all times. The ( + 1)th entry of z(n) must be equal to one. If the ideal impulse response is b(n), the filter bank is PR if and only if A(n) s(n) = b(n) for all n. (37.14) 37.3 Direct Switching of Filter Banks Changingfromone arbitraryfilterbank toanotherindependently designedfilterbankwithout using any intermediate filters is called direct switching. D irect switching is the simplest switching scheme and does not require additional steps in switching between two filter banks. But such switching will result in a substantial amount of reconstruction distortion during the transition period. This is becauseduringthetransition,noneofthesynthesisfilterssatisfiestheexactreconstructionconditions. Figure(37.4)showsanexampleofadirectswitchingfilterbank. Figure(37.5)showsthetime-varying impulse response of the above system around the transition periods. In this figure, z(n, m) is the response of the system at time n to the unit input at time m.ForaPRsystem,z(n, m) hasaheight of 1 along the diagonal and 0 everywhere else in the (m, n)-plane. As is shown, the time-varying filter bank is PR before and after but not during the transition periods. In this case, each switching operation generates a distortion with an 8-sample duration. One way to reduce the distortion is to switchthesynthesisfilters with anappropriatedelaywithrespecttotheanalysis switchingtime. This delay may reduce the output distortion, but it can not eliminate it. 37.4 Time-Varying Filter Bank Design Techniques The basic time-varying filter bank design methods are summarized in Table 37.1. These techniques can be divided into two major approaches which are briefly described in the following sections. c  1999 by CRC Press LLC FIGURE 37.4: Block diagram of a time-varying analysis/synthesis filter bank that switches between a two- and three-band decomposition. TABLE37.1 Comparison of Time-Varying Filter Bank Different Designing Methods Intermediate Changing freq. Filter bank Computational analysis resolution requirement complexity Arrowood Lattice Smith Yes Indirect structures Low de Queiroz Rao Yes Indirect ELT Low Intermediate Gopinath analysis Burrus Yes Indirect Paraunitary Low synthesis Herley et. al Yes Direct Paraunitary Low (IAS) Chen Noncausal Vaidyanathan Yes Direct synthesis Low Least square General Instantaneous synthesis No Direct (not PR) Low transform Redesigning switching analysis No Direct General High (ITS) Post filtering No Direct General Low 37.4.1 Approach I: Intermediate Analysis-Synthesis (IAS) In the first approach, both analysis and synthesis filters are allowed to change during the transition period to maintain perfect reconstruction. We refer to this approach as the intermediate analysis- synthesis (IAS) approach. In [16], the authors have chosen to start with the lattice implementation of time-invariant two- band filter banks, originally proposed by Vaidyanathan [25] for time-invariant case. Consider the latticestructureshowninFig.37.6. Figure37.6(a)representsalossless two-band analysisfilterbank, consistingofJ + 1 latticestages. ThecorrespondingsynthesisfilterbankisshowninFig.37.6(b). As is shown, for each stage in the analysis filter bank, there exists a corresponding stage in the synthesis filter bank with similar, but inversefunctionality. As long as each twocorrespondinglattice stages in theanalysisandsynthesissectionsarePR,theoverallsystemisPR.Toswitchonefilterbanktoanother, the lattice stages of the analysis section are changed from one set to another. If the corresponding latticestagesofthesynthesis sectionarealsochangedaccordingtothe changesoftheanalysissection, thePRpropertywillholdduringtransition. Duetotheexistenceofdelayelements,anychangeinthe analysis section must be followed with the corresponding change in the synthesis section, but with an appropriate delay. For example, the parameter α j of the analysis and synthesis filter banks can c  1999 by CRC Press LLC FIGURE 37.5: The time-vary ing impulse response for direct switching between the two- and the three-band system. The filter bank is switched from the two-band to the three-band at time n = 0 and switched back at time n = 13. (a) Surface plot, (b) contour plot. be changed instantaneously. But any change in parameter α j−1 in the analysis filter bank must be followed with the similar change in the synthesis filter bank after one sample delay. Because of such delays,switchingbetween twoPRfilter bankscanoccur onlybygoingthrough atransition period in which both analysis and synthesis filter banks are changing in time. In [12, 26], the design of time-varying extended lapped transform (ELT) [27, 28] was reported. The extended lapped transform is a cosine-modulated filter bank with an additional constraint on thefilterlengths. Here,thedesignprocedureisbasedon factorizationofthetime-domain transform matrix into permutation and rotation matrices. As the ELT is paraunitary, the inverse transform can be obtained byreversingtheorder ofthematrix multiplication. Sinceanyorthogonal transform is a succession of plane rotations, any changes in these rotation angles result in changing the filter bank without losing the orthogonality property. The authors derived a general frame work for M- band ELT transforms compared to the two-band case approach in [16]. This method parallels the lattice technique[16]exceptwith the mild modification of imposing the additional ELT constraints. In [17], the authors presented yet another formulation for designing time-varying filter banks. In this paper, a different factorization of the paraunitary transform has been shown which is not based on plane rotations unlike the ones in [12, 26]. Using this factorization, a paraunitary filter bank can be implemented in the form of some cascade structures. Again, to switch one filter bank to c  1999 by CRC Press LLC FIGURE 37.6: The block diagram of a two-band paraunitary filter bank in lattice form: (a) analysis lattice, (b) synthesis lattice. another, the corresponding structures in the analysis and synthesis filter bank are changed similarly but with an appropriate delay. If the orthogonality property in each cascade structure ismaintained, the time-varying filter bank remains PR. This formulation is very similar to the ones in [12, 16, 26], but represent a more general form of factorization. In fact, all above procedures consider similar frameworks of structures that inherently guarantee the exact reconstruction. Herley et al. [ 13, 14, 15, 29] introduced a time-domain method for designing time-varying pa- raunitary filter banks. In this approach, the time-invariant analysis transforms do not overlap. As a simple example, consider the case of switching between two paraunitary time-invariant filter banks. The analysis transform around the transition period can be written as T =             P 1    P T    P 2             . (37.15) The matrices P 1 and P 2 represent paraunitray transforms and therefore are unitary matrices. Their nonzero columns also do not overlap with each other. The matrix P T represents the analysis filter bank during the transition period. In order tofindthisfilterbank,thematrix P T is initially replaced with a zero matrix. Then, the null space of the transform T is found. Any matrix that spans this subspace can be a candidate vector for P T . By choosing enough independent vectors of this null space and applying the Gram-Schimidt proceduretothem, an orthogonal transform can be selected for P T . This method has also been applied to time-varying modulated lapped transforms [24] and two-dimensional time-varying paraunitary filter banks [30]. The basic property of all above procedures is the use of intermediate analysis transforms in the transitionperiod. Thecharacteristicsoftheseanalysistransformsarenoteasytocontrolandtypically the intermediate filters are not well-behaved. c  1999 by CRC Press LLC 37.4.2 Approach II: Instantaneous Transform Switching (ITS) In the second approach, the analysis filters are switched instantaneously and time-varying synthesis filters are used in the transition period. We refer to this approach as the instantaneous transform switching (ITS) approach. In the ITS approach, the analysis filter bank may be switched to another setofanalysisfiltersarbitrarily. Thismeansthatthebasisvectorsandthe tilingofthetime-frequency plane can be changed instantaneously. To achieve PR at each time in the transition period, a new synthesis section is designed to ensure proper reconstruction. Intheleastsquares(LS)method[10],foranygivensetofanalysisfilters,aLSsolutionofEq.(37.14) can be used to obtain the “best” synthesis filters of the corresponding system (in L2 norm): s ( n ) LS =  A ( n ) T A(n)  −1 A ( n ) T b(n) (37.16) The advantageoftheLSapproach isthatthere isnolimitationonthenumberofanalysisfilterbanks that can be used in the system. The disadvantage of the LS method is that it does not achieve PR. However, experiments have shown that the reconstruction is significantly improved in this method compared to direct switching [10]. In the LS solution, b(n) is projected onto the column space of A(n). For PR, the projection error should be zero. Thus, to obtain time-varying PR filter banks, the reconstruction error,||A(n)s(n) − b(n)|| 2 , can be brought to zero with an optimization procedure. The optimization operates on the analysis filtercoefficients and modifiestherangespaceofA(n) until b(n) ∈ range(A(n)). Although the s(n)’s at different states are independent of each other, since the A(n)’s have some common elements, optimization procedures should be applied to all analysis sections at the same time. This method is referred to as “redesigning analysis” [10]. The last ITS method, post filtering, uses conventional filter banks with time-varying coefficients followedbyatime-varying postfilter. The post filterprovides exactreconstructionduring transition periods, while it operates as a constant delay elsewhere. Assume at time n 0 the time-varying filter bank is switched from the first filter bank to the second. If the length of the transition period is L samples,theoutputofthefilterbankintheinterval[n 0 ,n 0 +L− 1] isdistortedb ecauseofswitching. The post filter removes this distortion. The block diag ram of such a system is shown in Fig. (37.7). In this figure, z(n) and y(n) are the analysis/synthesis filter bank and post filter impulse responses, FIGURE 37.7: The block diagram of time-varying filter bank and post filter. respectively. If the delays of the filter bank and the post filter are denoted  and ,respectively,we can write ˆx(n) =  Distorted if n 0 ≤ n<n 0 + L x(n − ) otherwise . (37.17) The desired output of the post filter is ˜x(n) = x(n −  − ) (37.18) c  1999 by CRC Press LLC [...]... 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[4] Portnoff, M.R., Time-frequency representation of digital signals. to the temporal properties of the input signal, better performance can be achieved in processing the signal. Intheabsenceofprocessingerrors,thereconstructedoutput