Design of Linear-Phase Two-Channel FIR Filter Banks with Rational Sampling Factors

Transcription

1 R. Bregović and. Saraäi, Design of linear hase two-channel FIR filter bans with rational saling factors, Proc. 3 rd Int. Sy. on Iage and Signal Processing and Analysis, Roe, Italy, Set. 3, Design of inear-phase wo-channel FIR Filter Bans with Rational Saling Factors Robert Bregović and aio Saraäi Institute of Signal Processing aere University of echnology P. O. Box 553, FIN-33 aere, Finland e-ail: bregovic@cs.tut.fi and sara@vi.fi or ts@cs.tut.fi Abstract he design of two-channel filter bans with rational saling factors is considered based on the use of linearhase finite-iulse resonse (FIR filters whose stoband erforances are iniized in the least suares sense. It is shown that a roer selection of the filter orders enables one to synthesize a erfect reconstruction filter ban. he otiization of the overall filter ban is carried out by roerly odifying the algorith of Dutta and Vidyasagar. he odified algoriths enables one to otiize both the erfect-reconstruction and the near erfect-reconstruction filter bans. Several exales are included illustrating the roerties of the resulting otiized filter bans as well as the trade-off between the stoband attenuation and the alitude and the aliasing errors of the otiized filter ban.. Introduction Multirate filter bans have nowadays an iortant role in digital signal rocessing [] [3]. hey have becoe very oular due to their roerty to divide a signal under consideration into two or ore sub-signals, where each of sub-signals contains only a art of the inut signal. hese sub-signals are deciated before actually rocessing the. his is beneficial since a saller aount of data has to be rocessed. Furtherore, in any alications, better overall erforances are achieved coared with the case where the original signals are directly rocessed [] [3]. Usually, the sae deciation factor is used in all the channels. Such filter bans are nown as unifor filter bans. However, there exist any alications where these filter bans do not rovide the best achievable erforance. For exale, when rocessing iages, it is ore beneficial to use the so-called octave filter bans []. hey belong to the grou of non-unifor filter bans. hese filter bans divide the inut signal into non-unifor bands. he deciation factors in these filter bans are his wor was suorted by the Acadey of Finland, roject No (Finnish centre of Excellence rogra (-5. he deciation factor denoted by / eans that the signal is first u-saled by a factor of and then down-saled by a factor of. /, 4/, 8/, 6/, etc. he ost straightforward aroach for synthesizing an octave filter ban is to use one or ore unifor two-channel filter bans as building blocs in an octave tree structure [] [3]. Octave filter bans reresent the silest, although the ost coonly used, non-unifor filter bans. In addition to octave filter bans, where the deciation factors in channels are owers of two, there are alications where non-unifor filter bans with deciation factors being ratios of integer are used. For exale, in soe seech rocessing alications, it is beneficial to have a filter ban with deciation factors being 6/, 6 / 5, /, 6 /, etc. [4] 3. Such a filter ban can also be generated by using an octave-band tree, but the building bloc two-channel filter ban should be a non-unifor one, that is, the deciation factors of the two-channel filter ban have to be 6/5 and 6/. Such non-unifor filter bans with the deciation factors being reresented as /, where and are ositive integers, are nown as rational filter bans (the deciation factors are ratios of two integers or filter bans with rational saling factors 4. here are three ain aroaches to designing an M- channel non-unifor filter ban. In the first aroach, the filters of the filter ban are designed without any redeterined connections with each other. his aroach is coutationally very colex due to a high nuber of unnowns [5]. he second aroach is based on designing a unifor filter ban with M channels, where M is the least coon ultile of all deciation factors. his filter ban can be designed, for exale, as a cosineodulated filter ban, thereby reducing the nuber of unnowns and allowing an efficient ileentation [6], [7]. After that, the individual channels are erged to generate the desired non-unifor filter ban. he roble with this aroach is that the nuber of channels M can be uch higher than the desired nuber of channels M. his fact In an octave filter ban there are always two channels with eual (highest deciation factors. For exale, in a 4-channel filter ban the deciation factors are /, 4/, 8/, and 8/. 3 his selection of deciation factors generates a filter ban that aroxiates the Bar scale. 4 Rational filter bans are a sub-class of non-unifor filter bans. However, in the literature, as well as in this aer, when referring to rational filter bans, soeties, the ore general ter non-unifor filter bans is used.

2 x[n] H (z H (z Processing Unit x [n] F (z x [n] F (z y[n] e jπn Analysis filter ban Synthesis filter ban e jπn Figure. wo-channel filter ban with rational saling factors. can drastically increase the design colexity of the filter ban with M channels. he third aroach is based on designing adeuate two-channel filter bans and on using the in a tree structure to obtain the desired M-channel filter ban [8] [3]. How to design such two-channel filter bans is the ain toic of this aer. he choice of the deciation factors in an M-channel filter ban is restricted by following two constrains. First, in order to achieve a axially deciated filter ban, the su of the recirocal deciation factors in all the channels has to be one. Second, if all the filters in the filter ban are reuired to be real, then soe cobinations of the deciation factors are not allowed. his is due to the fact that alias errors occurring in the analysis filter ban cannot be reoved in the synthesis ban. 5 he second restriction can be always get around by generating the filter ban using only suitable two-channel filter bans in a tree structure. In the ast, various ethods have been roosed for designing two-channel rational filter bans. For orthogonal filter bans 6, an efficient iterative ethod has been roosed by Blue [8]. In the biorthogonal case, two alternative aroaches can be highlighted. In the first aroach introduced in [9] [], all the filters are divided into two arts. he first set of filters searates the signal into the lowass and the highass arts, whereas the second set of filters cobined with down and u salings of the subband signals rovides the desired saling factors. Due to the structure of this syste, only nearly PR (NPR filter bans can be synthesized. In the second aroach described in [], [3], only one set of filters is designed. his aroach is used in this aer and will thus be addressed in ore details in Section. his aroach enables one to synthesize both PR and NPR filter. his is aid by the increased coutational colexity of the filter ban design and ileentation when coared to the first aroach. In both aroaches, extra highass to lowass transfor {lowass to highass transfor} is reuired for generating the highass channels in the analysis ban {synthesis ban} in order to allow ileentations for all deciation factors. ithout these transfors, a colex coefficient assband filter would be needed for soe deciation factors (see Subsection. in the highass channel [5], [9]. 5 In the M-channel case, a set of deciation factors enabling one to generate a valid filter ban is nown as a coatible set [5]. 6 he iulse-resonses of the synthesis filters are tiereversed versions of those of the analysis filters. he urose of this aer is to show that by iosing certain extra constrains in the actual synthesis enables one to generate a PR filter ban using linear-hase FIR filters. For designing these filter bans, a ethod based on the use of the Dutta-Vidiyasagar otiization algorith is roosed. he synthesis of both PR and NPR filter bans is considered.. wo-channel Filter Bans with Rational Saling Factors In this section basic relations for two-channel filter bans with rational saling factors are considered. he ain ehasis is laid on synthesizing these filter bans with the aid of linear-hase FIR filters.. Filter Ban Structure A two-channel filter ban with rational saling factors is shown in Figure. his filter ban consist of an analysis filter ban and a synthesis filter ban searated by a rocessing unit. In the seuel, it is assued that the rocessing unit does not change the signals. Due to the odulation of the signal in the highass channel by e jπn in both the analysis ban and the synthesis ban, all the filters in the overall filter ban are lowass filters. Nevertheless, for convenience, the channel containing the filter transfer function H (z or F (z {H (z or F (z} will be referred to as the lowass {highass} channel. he deciation factors for the lowass and highass channels in Figure are / and /, resectively. he s and s for =, can be any ositive integers as long as <. Additionally, for a axially deciated filter ban, the following relation has to be satisfied: = his ilies that = and =. he urose of the odulation by e jπn is twofold. First, as already entioned, it aes all the filters in the filter ban lowass filters. his fact silifies the notations when stating the otiization roble and describing the algorith for solving this roble. Second, by using this odulation, all cobinations for and can be ileented with filters having real coefficients. For exale, by oitting this odulation, filter ban with / = 3 / can not be ileented without having assband filters with colex coefficients in the highass channel. In the case, if filters with real coefficients are used, then aliasing (

3 occurring after down-salings can not be cancelled in the synthesis ban [5], [9].. Basic Relations he relation between the outut and inut seuences for the syste shown in Figure is exressible in the z- doain as where X ( z = X ( z = and Y z = X ( z X (, (a ( z { = = { = = F ( z H ( z F ( e H ( e jπ / jπ / z z πj X ( z X ( z } } (b (c = e. (d As for a unifor two-channel filter ban or a cosineodulated filter ban, the above euation can be rewritten as 7 l l ( z X ( z l= Y ( z = ( z X ( z. (3 (z is the distortion transfer function deterining the aoun of the distortion caused by the overall syste for the unaliased coonent X(z of the inut signal and the l (z's for l =,,, are the alias transfer functions deterining how well the aliased coonents X ( z of the inut signal are attenuated..3 inear-phase Constraints In order to synthesize a PR filter ban having linear-hase FIR filters, following condition has to be satisfied: ( N h N f = ( N h N f, (4 where N h, N h, N f, and N f are the orders of the filters H (z, H (z, F (z, and F (z, resectively. he filter ban delay is then given by N h N f K=. (5 7 Due to the restriction given by E. ( and the eriodicity of the ter, E (3 can also be written as Y (z = ( z X ( z l l ( z X ( z In this case l l for l = l, l = but for each l, l {,,, } for which l ter there exist an l= l. l l ter for he selection of filter orders so that a roer filter ban delay is achieved is also iortant when using nonlinearhase FIR filters. If the filter orders are not selected aroriately, an extra delay has to be included in one of the channels in order to achieve good filter ban roerties [3]. 3. Design Proble In order to be able to write the relevant euations in a coact for in the otiization roble stateent, the following notation is used: g = h, g = h, g = f, g 3 = f. Here, h, h, f, and f are the vectors containing the coefficients of H (z, H (z, F (z, and F (z, resectively. A set of all the filter coefficients is denoted by g = {g, g, g, g 3 }. Corresondingly, the filter freuency resonses for H (e, H (e, F (e, and F (e are denoted by G (e, G (e, G (e, and G 3 (e, resectively. Siilarly, N h, N h, N f, and N f are denoted by N, N, N, and N 3, resectively. his aer concentrates on the following otiization roble: Given the filter orders N and the stoband edge ( freuencies ω s for =,,, 3, the deciation factor /, the assband rile δ, the distortion error δ d, and the allowable alias error δ a, find g to iniize: where ε ax( ε ( g, (6a ( g = G ( g, e dω for =,,, 3 subject to ω and ax ( (, ω s π ( ωs G (, e δ for =,,, 3 g, (, e δ d ax g, ω [, π] ( ax l g, e δ a for l =,, K,. [, π] ω Here, G N n (, e g [ n] e for =,,, 3 = n=, (6b (6c (6d (6e g (6f is the freuency resonse of filters used for generating the filter ban and (z, and the l (z's, as defined by E. (3, are the distortion and alias-error transfer functions. he filter ban delay K is defined according to E. (5. Due to the linear-hase roerty of the filters aroxiately only half the filter coefficients are unnowns. his drastically silifies the design rocedure. In the above roble, the goal is to iniize the axiu of the stoband energies of the four filters in the filter ban subject to the following constrains. First axiu value of the filter alitude resonses is liited to

4 be less than or eual to δ. Second, the distortion error has to be less than or eual to δ d whereas the alias error has to be less than or eual to δ a. his eans that the PR roerty, that is, y[n] = x[n K] will be aroxiately satisfied rovided that δ a and δ d are sall enough. If δ a and δ d are of order or less, then the filter ban can be considered as a PR one. he following attractive roerty for the filter bans under consideration has been exerientally observed. By sily liiting the axiu alitude values for all four filters to be at ost δ results in filter ban with good erforance. For each filter, this constraint gives autoatically a assband region, where the alitude resonse oscillates aroxiately within ±δ. his assband region is not redeterined, but can be deterined after otiizing the filter ban. Hence, there is no need to control the assband behaviors of the filters in the otiization algorith. 4. Design Algorith In order to conveniently solve the roble stated in the revious section, the interval [, π] is be discretized into oints ω [, π] for =,,,. 8 he resulting discretized roble is to find g to iniize: subject to and G Here, where ε = ax( ε ( g for =,,,3, (7a =,,,3 (, e δ for ( g, (7b =,, K, ( e j ω, δ for =,,, g, (7c d K (, e δ for =,, K g. (7d l a, ( ( s = ax( ω ω, (7e ε ( g = g R g, (7f ( sin[( µ ν ω s ]/( µ ν for µ ν r [ µ, ν ] = ( (7g π ω s for µ = ν for µ =,,, N and ν =,,, N. In order to aly the second algorith of Dutta and Vidyasagar roosed in [5] with roer odifications [4], [6], the following cobined objective function, for the discretized version of functions given in E. (7, is considered: 3 P( g, φ = [ ε ( g φ ] = ε ( g φ > w = g, e δ d > ( g, e δ d w (, l g e δ a = l =, g e δ > l a ( 3 w G (, ( g e δ = = G (, e > g δ In the above suations only the ters not satisfying the given criteria are resent. For the PR case, δ d and δ a are zero. It has been exerientally observed that the selection δ d = δ a = result in a filter ban being ractically a PR ban. w, w and w are weight factors given by the user. he basic idea in introducing an additional araeter φ in the objective function P(g, φ, as given by E. (8, is that the original constrained roble has been converted into a seuence of unconstrained robles. he goal is to gradually find the inial value of φ for which function P(g, φ becoes zero. his inial value is the desired iniu of the stoband energies of the four filters subject to the given constraints. he roosed algorith is carried out in the following stes: Ste. Find initial vectors 9 for g, denoted by g ( et φ ( be the axiu of the stoband energies for this solution. Set β =. and l =. Ste. Set φ (l = ( βφ (l. Find ĝ to iniize P(g, φ (l using g (l as initial vectors. (l ( l Ste 3. If P( ĝ, φ =, then set g = gˆ, l = l, and go to Ste. Otherwise, go to the next ste. Ste 4. If β < 9, then the otiized vectors are g (l. Otherwise, set β = β/ and go to Ste. he above otiization algorith has been described in ore details in [4]. For convenience, only the algorith itself has been reeated here. More details of the use of the original Dutta and Vidyasagar algorith in solving digital signal rocessing robles can be found, for exale, in [6].. (8 8 A good choice for is = ax{n, N, N, N 3 }. 9 he initial filters have to be filters with good freuency selectivity. he stoband freuencies of these filters are deterined by the rational saling factors of the designed filter ban.

5 5. Exales his section shows, by eans of two exales, the efficiency of the roosed ethod. In both exales, as starting oints, filters designed in the least-ean-suare sense have been used. All the filters in Figure are lowass filters, as ointed out earlier in Section. In all exale figures, the filter transfer functions H (z and F (z are shown as their highass counterarts in order to ae the ore distinguishable fro H (z and F (z. Exale. It is desired to design a PR two-channel rational filter ban with the aid of linear-hase FIR filters for N h = N f = 3, N h = N f = 6, δ =.3, and / = 3/ as well as ω s =.475π for H (z and F (z and ω s =.46π for H (z and F (z. For these reuireents, the filter ban delay is eual to K = 3 according to E. (5. Figure shows the overall alitude resonses for the otiized analysis and synthesis filters resulting when alying the roosed otiization schee. Alitude in db Alitude in db Synthesis filters Analysis filters Noralized freuency (w/( Figure. Alitude resonses for the analysis and synthesis filters of Exale. Exale. It is desired to design a NPR two-channel rational filter ban with the aid of linear-hase FIR filters for N h = N f = 34, N h = N f = 5, δ =., δ d = δ a = 3, / = 5/, and ω s =.3π for all four filters. For these reuireents, the filter ban delay is eual to K = 34 according to E. (5. Figure 3 shows the overall alitude resonses for the otiized analysis and synthesis filters together with assband details. Also the alias and alitude distortions of the overall filter ban are included in this figure. Alitude Alitude (ω l (ω Alitude in db Alitude in db Noralized freuency (ω/(π Noralized freuency (w/( (b x 3 x 3 (c Noralized freuency (w/( (a Figure 3. Various resonses for the filter ban of Exale. (a Analysis filters. (b Synthesis filters. (c Alitude and alias distortions. 6. References [] P. P. Vaidyanathan, Multirate Systes and Filter Bans. Englewood Cliffs, NJ: Prentice-Hall, 993. [] N. J. Fliege, Multirate Digital Signal Processing, Chicester: John iley and Sons, 994. [3]. Saraäi and R. Bregović, Multirate Systes and Filter Bans, Chater in Multirate Systes: Design and Alications edited by G. Jovanovic-Dolece. Hershey PA: Idea Grou Publishing,. [4] E. Zwicer and H. Fastl, Psychoacoustics. New Yor: Sringer, 99. [5] J. Kovačević and M. Vettereli, Perfect reconstruction filer bans with rational saling factors, IEEE rans. Signal Process., vol. 4, , June 993.

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