FREQUENCY RESPONSE MASKING BASED DESIGN OF TWO-CHANNEL FIR FILTERBANKS WITH RATIONAL SAMPLING FACTORS

Transcription

1 R. Bregović, Y. C. im an T. Saramäi, Freuency reone maing bae eign of two-channel FIR filterban with rational amling factor, Proc. 5 th Int. orho on Sectral Metho an Multirate Signal Proceing, Riga, atvia, June 25,. 8. FREQUECY RESPOSE MASKI BASED DESI OF TO-CHAE FIR FITERBAKS ITH RATIOA SAMPI FACTORS Robert Bregović, Yong Ching im, 2 an Taio Saramäi Intitute of Signal Proceing, Tamere Univerity of Technology P. O. Bo 553, FI-33 Tamere, Finlan, bregovic@c.tut.fi an t@c.tut.fi 2 Temae aboratorie, anyang Technological Univerity 5 anyang Drive, Singaore , elelimyc@mail.ntu.eu.g ABSTRACT Two author of thi aer have hown that by uing the freuency-reone maing (FRM techniue for eigning finite-imule reone (FIR filter builing a two-channel filterban (FB with rational amling factor, the imlementation comleity can be reuce comare with the ue of irect-form filter for thi uroe. In thi aer, for eigning FB, a moifie verion of the FRM techniue rooe by im an Yang for eigning interolation (ecimation filter i utilize. Uing uch filter enable one to contruct a FB in uch a manner that all filter are evaluate at the FB lowet amling rate, that i, at the ubban amling rate, thereby further ecreaing the FB imlementation comleity. The benefit of the rooe aroach comare to other eiting techniue are illutrate by mean of eamle.. ITRODUCTIO Multirate filterban (FB are very oular in igital ignal roceing ue to their roerty to ivie the inut ignal into two or more ubban ignal, where each ubban ignal contain only a art of the inut ignal. Proceing thee ubban ignal give in mot alication better overall erformance than thoe being achievable when irectly roceing the inut ignal. Moreover, thee ubban ignal are uually ecimate before roceing, thereby reulting in a maller amount of ata to be rocee [] [3]. In mot FB, the ame ecimation factor i ue in all channel. Such FB are nown a uniform FB. However, in many alication thee FB o not rovie the bet achievable erformance. For eamle, in ome eech roceing alication, it i more beneficial to ue a FB with channel banwith following the Bar cale [4]. A goo aroimation to the Bar cale can be Thi wor wa uorte in art by Temae aboratorie@tu, Singaore, an the Acaemy of Finlan, roject o (Finnih centre of Ecellence rogram (2-25. The wor wa erforme while the firt author wa viiting Temae aboratorie@tu, Singaore. achieve uing a FB having ecimation factor 6/, 6 2 / 5, 6 3 / 5 2, 6 4 / 5 3, etc. with the ecimation factor 6/ belonging to the higha channel. In thi FB, each channel ha a ifferent ecimation factor, an correoningly a ifferent channel banwith. Such FB are nown a non-uniform FB. The mot traightforwar aroach for yntheizing the above FB i to ue a tree tructure a the one ue for octave FB [2] but with non-uniform two-channel FB a builing bloc. At the firt level of uch a tructure, the inut ignal i earate into two channel with ecimation factor 6/ an 6/5. At the econ level, the lowa channel (channel with ecimation factor 6/5 i further earate by uing the ame non-uniform two-channel FB. Thi i reeate until the eire number of channel i obtaine. The two-channel non-uniform FB, a the above-mentione one, with the ecimation factor being /, where an are oitive integer, are nown a FB with rational amling factor (the ecimation factor are ratio of two integer or, horter, a rational FB. There are two alternative aroache for eigning two-channel rational FB. In the firt aroach a ecribe in [5] [9] an to be coniere in thi aer, only one et of filter, locate between the u-amler an own-amler, i eigne. In the econ aroach a ecribe in [] [3], the filter are ivie into two art (ee Figure in [3]. The firt et of filter earate the ignal into the lowa an the higha art, wherea the econ et of filter combine with the uan own-amling of the ubban ignal rovie the eire amling factor. In orer to imlify the notation in thi aer, FB eigne with the above two aroache are referre to a Tye an Tye 2 FB, reectively. Moreover, for both tye, an etra highato-lowa tranform {lowa-to-higha tranform} i reuire for the ubban ignal for generating the higha channel in the analyi ban {ynthei ban} in orer to allow the imlementation of all ecimation factor. ithout thee tranform, a bana filter having comle coefficient woul be reuire in the The ecimation factor enote by / mean that the ignal i firt u-amle by a factor of an then own-amle by a factor of.

2 [n] H ( H ( Proceing Unit F ( F ( [n] 2 [n] + y[n] e jn Analyi filterban Synthei filterban e jn Figure. Two-channel filterban with rational amling factor. higha channel for ome value of ecimation factor [], [4]. Due to the multirate oeration inie the FB (u- an own-amling, in orer to ignificantly reuce the alia error in the channel 2, it i imortant that filter builing the FB have narrow tranition ban. However, uing finite-imule reone (FIR filter with narrow tranition ban imlie that the filter orer are very high [5]. Such filter are har to eign an are numerically eenive to imlement. Therefore, for eigning an imlementing thee filter, it i beneficially to ue the freuency reone maing (FRM techniue [6] [2]. In [3] it ha been hown how the imlementation comleity of a Tye 2 two-channel rational FB can be reuce by utilizing the FRM techniue comare to the irect-form imlementation. The uroe of thi aer i to how that a further reuction in the imlementation cot can be achieve by uing the FRM techniue for eigning Tye two-channel rational FB. Thi will be achieve by contructing the FB uing filter that can be evaluate at the lowet oible amling rate, that i, at the ubban amling rate. For thi uroe, the ynthei cheme rooe by im an Yang in [8], [9] for uing a moifie FRM aroach for imlementing efficient interolator an ecimator will be utilize. Ieally, the FB houl not introuce any itortion to the ignal aing through the ban, that i, the outut ignal ha to be a elaye verion of the inut ignal. FB atifying thi roerty are nown a erfectrecontruction (PR FB. However, in mot ractical alication it i enough that the error introuce by the FB are maller than the change in the ignal ue to the ignal roceing algorithm (alication. Such FB that change the ignal (introuce itortion are nown a near PR (PR FB. Moreover, by giving u the PR roerty, other roertie of the FB an the builingbloc filter can be imrove, for eamle, a better channel earation can be achieve for the ame FB elay an imlementation comleity [3]. Therefore, thi aer concentrate on PR FB. The outline of the aer i a follow: Section 2 give the tructure an the baic relation of the two-channel rational FB uner conieration. The FRM techniue i briefly reviewe in Section 3 with the main emhaize lai on the moifie FRM aroach. The eign metho for FB uner conieration i icue in Section 4. Two comarative eign eamle are inclue in Sec- 2 Alia error in the channel are comenate in the ynthei ban auming that the ubban ignal are not moifie. For mot real (ractical ytem, thi aumtion i not vali. tion 5. Finally, ome concluing remar are rawn in Section TO-CHAE FB ITH RATIOA SAMPI FACTORS In thi ection, ome baic relation for two-channel FB with rational amling factor are coniere. The emhaize i lai on Tye rational FB ecribe in [5] [9]. 2. FB Structure The tructure for imlementing a Tye two-channel FB with rational amling factor i hown in Figure. Thi FB conit of an analyi FB an a ynthei FB earate by a roceing unit. In the euel, it i aume that the roceing unit oe not alter the ignal. Due to the moulation of the ignal in the higha channel by e jn in both the analyi ban an the ynthei ban, all filter in the FB are lowa filter. everthele, for convenience, the channel containing the filter tranfer function H ( or F ( {H ( or F (} i referre to a the lowa {higha} channel. The ecimation factor for the lowa an higha channel in Figure are / an /, reectively. The an for =, can be any oitive integer a long a <. Aitionally, for a maimally ecimate FB, the following relation ha to be atifie: = +. ( Thi imlie that = an =. By uing thee ecimation factor, a ban iviion hown in Figure 2 i achieve. ρ + ρ Figure 2. The ban iviion of the analyi ban of Figure. The uroe of the moulation by e jn i twofol. Firt, a alreay mentione, it mae all the filter in the FB lowa filter. Thi fact imlifie the notation when tating the otimization roblem an ecribing the algorithm for olving thi roblem. Secon, by uing thi moulation, all combination for an can be imlemente with filter having real coefficient. For eamle, by omitting thi moulation, a FB with / = 3 / cannot be imlemente without having

3 bana filter with comle coefficient in the higha channel. If filter with real coefficient are ue, then aliaing occurring after the own-amling cannot be cancelle in the ynthei ban [], [4]. 2.2 Baic Relation The relation between the outut an inut euence for the ytem of Figure i ereible in the z-omain a where X ( = X 2( = an Y = X ( + X (, (2a ( 2 z { = = { = = F ( z H ( z F ( e H ( e j / j / z z 2j X ( z X ( z } } (2b (2c = e. (2 The above euation can alo be written in the following form more commonly ue when analyzing FB 3 : l= l Y ( = T ( X ( + T ( X (. (3 l z Here, T ( i the itortion tranfer function etermining the amount of the itortion caue by the overall ytem for the unaliae comonent X( of the inut ignal an the T l ( for l =, 2,, are the alia tranfer function etermining how well the aliae comonent l X ( z of the inut ignal are attenuate. For an PR FB, T ( an the T l ( houl aroimate a ure elay term z D an zero, reectively. In thi cae, the outut ignal i an aroimately elaye verion of the inut ignal, that i, y[n] [n D]. Here, D i the FB elay in amle. In the z-omain, thi correon to Y( z D X(. 2.3 Filter an FB Reuirement In orer to yntheize a FB having goo channel electivity an a low FB itortion a well a mall alia error, the filter builing the FB houl atify the following reuirement [2], [3]: Firt, the value of ( h + f / an ( h + f / have to be integer an ( + ( + h f + D = h f + D, (4 3 Due to the retriction given by Euation ( an the erioicity of the term, Euation (3 can alo be written a Y ( = T ( X ( + l T l ( X ( z In thi cae, l = an for each T l term there eit an l, l {, 2,, } for which T l= T l. Tl T l for l = l T l term for where h, h, f, an f are the orer of the linearhae FIR filter H (, H (, F (, an F (, reectively. In the general cae, in orer to atify the above euation, one of the arameter D or D i zero, wherea the remaining one i a oitive integer. In the ecial cae, when roerly electing the filter orer, the value of thee arameter become eual to zero. Secon, the FB elay a given by h + f D= + D (5 ha to be o. Thir, the aban an toban ege of the filter builing the FB are given by ( = ( ρ (6a ( = ( ρ + (6b for =,. The arameter ρ ha to be electe accoring to the above euation uch that the ege freuencie given by the eign reuirement are achieve. For all filter in the FB, the center of the tranition ban i at /. Thee toban an aban ege freuencie are efine in uch a way that the tranition ban of both filter are eual after maing thee filter to the inut amling rate, that i, the inut ignal ee filter with the ame tranition banwith of 2ρ, but the tranition ban of the iniviual filter are ifferent ue to ifferent amling rate in the two channel. 3. FREQUECY-RESPOSE MASKI (FRM TECHIQUE In thi ection, a hort overview of the FRM techniue i given with the main emhaize lai on filter being aroriate for eigning FB uner conieration. 3. FRM Overview hen uing the FRM techniue, the tranfer function of an FIR filter H( i tyically eree a [6], [7] ( c z H = ( z E ( + ( z E ( (7a g 2 ( = z ( (7b c with g being the orer of ( that ha to be even. Here, ( an c ( are referre to a the moel an the comlementary moel filter, reectively, wherea E ( an E ( are referre to a the maing filter. Due to the uamling by, (z or c (z have narrow tranition ban a well a are imule reone with only every th imule-reone coefficient value being nonzero, wherea E ( an E ( are filter of low orer ue to the wie tranition ban. Therefore, all thee filter can be imlemente very efficiently. hen eigning an FIR filter uing the FRM techniue, variou arameter, incluing an the orer of (, E (, an E (, mut be etermine bae on the given aban an toban ege an, the filter eign ecification (e.g. toban attenuation, aban rile a well a on the reulting imlementation

4 comleity. Moreover, when electing the arameter, the following two conition mut be atifie [6] [9]:. either / nor / i an integer. 2. / = /. 4 In the cae of a conventional FRM FIR filter, it i alway oible to fin the arameter for which the above two conition are atifie. However, when imlementing in the analyi ban of Figure the filter H ( for =, uing an FRM filter receing by an u-amler by the factor of an followe by an own-amler by the factor of, it i imoible to meet imultaneouly the following eire roertie. Firt, the above-mentione two conition houl be atifie when electing for imlementing H (. Secon, in orer to generate a comutationally efficient FB, there i a nee to eloit the fact that after the u-amler only every th amle i nonzero an after own-amler only every th i evaluate. The ame i true for ynthei ban in Figure. Therefore, in orer to achieve the above goal an to minimize the overall imlementation cot, the original FRM techniue ha to be moifie, a will be icue in the net ubection. 3.2 Moifie FRM Techniue By following the icuion in [8], [9], Euation (7a an (7b can be rewritten a: 2 [ E ( E ( ] + z E ( H ( = ( z g z. (8 Auming that the erioic filter (z i erive from a linear-hae half-ban filter, it can be eree a where = z 2 g 2 ( z + ( z g ( z n=, (9a { g( g 2 + 2n ( g 2+ (2n ( g 2 (2n ( z + z }. = (9b Here, g i the orer of ( an (z i a function of z 2 intea of z. An eamle imule reone for ( i hown in Figure 3. By ubtituting Euation (9a into Euation (8 give where 2 ( H = z g E ( + ( z E (, (a E ( = [ E( + E( ] (b 2 E ( z = E ( E (. (c The tructure ecribe by Euation (a (c i hown in Figure 4 an the correoning freuency reone are hown in Figure 5. The role of each filter in the tructure can be een from the ubfigure. The tructure of Figure 3 will be ue in the net ection for gen- 4 i the larget integer le than. erating Tye rational FB. For more etail on the moifie FRM techniue, an interete reaer i referre to [8], [9]. Amlitue n (amle Figure 3. Eamle imule reone of the filter ( for g = 26. u[n] z g/2 (z E ( E ( + v[n] Figure 4. Bloc iagram for a filter eigne by uing the moifie FRM techniue. (a (b (c ( ( j e j j E ( e E ( e ( j e ( ( ( ( 2( ( H ( e j ( 2+ = = Figure 5. Deign of a lowa filter by uing the moifie FRM aroach. (a Moel filter. (b Perioic moel filter. (c Maing filter. ( Overall filter. 3.3 Prooe Deign Structure The FRM techniue briefly ecribe in Section 3. ha been ue in [3] for eigning filter builing Tye 2 rational FB. Thi techniue reuce the imlementation comleity comare to the ue of irect-form ubfilter.

5 In thi aer, the imlementation comleity i further ecreae by uing the moifie FRM techniue icue in Section 3.2 for generating Tye rational FB. In thi cae, the tranfer function of the four filter builing the FB a hown in Figure are contructe accoring to Euation (a a follow: ( h g 2 ( h ( h ( h H ( = z E ( + ( z E ( (a ( f g 2 ( f ( f ( f F ( = z E ( + ( z E ( (b for =,. Each of thee filter i imlemente by uing the tructure of Figure 4. In orer to evaluate all filter in the FB at the ubban amling rate, the following relation between the arameter an for =, mut be atifie [8], [9]: (2l+ = for even (2a 2 = l for o (2b with l being an arbitrary integer. Moreover, for being ( o, ( ( an E ( for {h, h, f, f } are filter with imaginary coefficient. Thi i ifferent from the imlementation in Section 3., where only filter with real coefficient are ue. An efficient imlementation of one channel of the analyi FB i hown in Figure 6. hen imlementing E ( an E (, the fact that after the u-amler only every th amle i nonzero an after own-amler only every th outut i evaluate can be effectively eloite by uing the imlementation rooe in [22] or [23]. In thi cae, the number of multilication for E ( an E ( er inut amle i only ( e +/ an ( e +/, reectively, where e an e are the orer of E ( an E (. 5 (z i imlemente after the ownamler a (z /, that i, at the ubban amling rate. Since (z i accoring to the reviou icuion a function of z 2 [ee Figure 3 an Euation (9a an (9b], thi imlie that 2/ houl be an integer. hen eloiting the coefficient ymmetry of (z, a given by Euation (9b, the number of multilication for (z / er inut amle i ( / ( g +2/4. How to eign the overall FB uing the moifie FMR aroach i icue net. u [n] z g/2 E ( E ( (z / + v [n] Figure 6. Bloc iagram for one channel of the analyi FB eigne by uing the rooe FRM techniue. 4. FB DESI In thi ection, the eign criteria for the rational FB uner conieration are tate, a ynthei cheme i rooe, an the imlementation comleity i evaluate. 4. FB Deign Problem In orer to emhaize the eenence between the unnown an each of the criteria in the FB eign roblem to be tate later, the FRM filter coefficient reuire for generating the filter H ( for =, are note in the vector form a h = { g (h, e (h, e (h }. The vector g (h, e (h, an e (h contain the unnown FRM filter coefficient. Similarly, the unnown for yntheizing the filter F ( for =, are note a f = { g (f, e (f, e (f }. By uing the above notation, thi aer concentrate on the following otimization roblem: iven the aban an toban rile δ an δ, the maimum allowable amlitue an alia itortion δ a an δ, the ecimation factor /, an ρ, the factor efining the aban an the toban ege freuencie, fin the coefficient of all the ubfilter in the four FRM filter to atify the following criteria: ( [, ] j H ( h, e δ for (3a ( [, ] j F ( f, e δ for (3b ( [, ] j H ( h, e δ for (3c ( [, ] j F ( f, e δ for (3 [, ] j T ( h, h, f, f, e δ for (3e T ( h, h, f l, f, e j for δ a [, ] an l =, 2,..., (3f with the toban ege freuencie efine by Euation (6b. Here, H (h,e j an F (f,e j for =, are the freuency reone of the correoning FB filter 6. Furthermore, T (, an the T l (, a efine by Euation (3, are the itortion an alia-error tranfer function. In the above, the goal i firt to meet all the criteria an then to minimize 7 δ, the common toban rile of the four filter ubject to the remaining criteria. Among thee remaining criteria, firt, the maimum value of the filter amlitue reone i limite to be le than or eual to +δ. 8 Secon, the itortion error ha to be le than or eual to δ, wherea the alia error ha to be le than or eual to δ a. Thi give the control over the PR roerty of the FB. 5 Thee relation are true in the cae of a harware imlementation. In the cae of a oftware imlementation, the number of multilier can be higher. 6 Euation (3a (3 are tate for both = an =. 7 The roblem can be alo tate a a leat-uare roblem, that i, the toban energie of the four FRM filter generating the FB are minimize. 8 The minimum value of the filter amlitue reone in the aban i inirectly limite by Euation (3a, (3b, an (3e.

6 4.2 FB Deign Algorithm There are variou way to olve the above eign roblem. One alternative i to ue the following three-te roceure: Ste. Select the filter orer, ( g ( e, an ( e for {h, h, f, f } an value of factor for =, accoring to the icuion in [6] [2]. Ste 2. Dicretize the interval [, ] into oint m [, ] for m =,,, M. A goo choice for M i M = 5 ma{ }. Ste 3. Solve the eign roblem, a efine by Euation (3a (3f, on the above-efine icrete gri of freuencie. Thi can be one by uing tanar non-linear otimization routine, for eamle, the function fminima inclue in the Otimization Toolbo [2] rovie by Mathor, Inc. If all the filter orer are roerly electe, then olving the above roblem reult in the olution with all the eign criteria being atifie. Otherwie, the filter orer have to be increae or ecreae until achieving the eire goal. To atify the given criteria, before tarting the above eign roceure, the orer of the ub-filter of the FRM filter have to be choen accoring to the icuion in [6] [2]. For eign containing FRM filter the eire ecification can be obtaine for variou combination of the filter orer an the value of the Therefore, everal eign have to be carrie out an the combination of thoe arameter i electe that atifie the eign ecification with the lowet imlementation cot or a roer trae-off between the FB elay an the imlementation cot. 4.3 FB Deign an Imlementation Cot Evaluation The eign comleity een mainly on the number of eign arameter. For a FB eigne by the rooe metho, the number of eign arameter i eual to the number of unnown filter coefficient, that i, Y = Y, (4a where ( ( ( g + 2 e + e + Y = + + (4b for {h, h, f, f }. The maller number of arameter for filter ( i ue to the fact that thee filter are erive from half-ban filter [ee Figure 3 an Euation (9a an (9b]. The imlementation cot C, efine a the number of multilication er inut amle for a FB eigne by the rooe metho, i given by C = C, (5a where 9 for = ( ( ( g + 2 e + e + C = + + (5b an for > C ( ( ( g + 2 e + e + = + + (5c 4 with {h, h, f, f }. In the above euation, = for {h, f } an = for {h, f }. 5. EXAMPES Thi ection how, by mean of two eamle, the efficiency of the rooe metho. In both eamle, the rooe eign are comare with the irect-form imlementation an the FRM bae imlementation ecribe in [7] an [3], reectively. All filter in Figure are lowa filter, a ointe out earlier in Section 2. However, in all eamle figure, the filter tranfer function H ( an F ( are hown a their higha counterart in orer to mae them more itinguihable from H ( an F (. Moreover, in all figure, the filter amlitue reone are normalize. hen comaring the rooe eign with the FB eign given in [7] an [3], the notation efine in thee aer are ue. For eamle, H ( an F ( for =, in [3] are filter before an after the amling rate alternation (ee Figure in [3] an not inie the amling rate alternation a in the cae of thi aer. Moreover, to mae a fair comarion, all eign are erforme for the ame ban iviion from the viewoint of the inut ignal, even if ome arameter, uch a ρ, an, are ue or efine ifferently. Eamle. It i eire to eign a two-channel rational FB with = 5, =, = = 6, an ρ =.2 atifying following reuirement: δ = δ =. an δ = δ a =.. Such a two-channel FB can be ue in a tree tructure for builing an M-channel FB aroimating the Bar Scale [4]. In orer to achieve FB with goo roertie, the filter have to be eigne with arameter a given in Table I. Thee arameter have been electe to rovie a roer traeoff between the imlementation comleity an the FB elay. Table I Eamle Reuire filter arameter Deign Prooe [3] [7] ( = 26, = 2 g Filter arameter ( ( = = 65 } for {h, f } e e ( = 8, = 4 g ( ( = = 65 } for {h, f } =, = 8 ( g ( ( = = 55 } for {h, h } e e la = lb = 2, la = lb = 24, h = f = 665, h = f = 33 9 For =, the coefficient ymmetry of filter can be eloite. Since = in thi eamle, a ( = b ( = z 3.

7 The roertie of the FB eigne uing the filter arameter given in Table I are hown in Table II. It i een from Table II that the FB eigne uing the rooe metho can be imlemente more efficiently comare with the other two ynthei techniue uner conieration. hen comare with the irect-form imlementation, the ecreae in the imlementation comleity i ai by a lightly higher FB elay an a higher eign comleity ue to the higher number of unnown. It i intereting to oberve that the Tye 2 FRM-bae FB eign reult in a higher FB elay becaue of the two et of filter that are earately eigne. Thi elay can be ecreae by imultaneouly eigning all filter in the Tye 2 FB. Variou reone for the otimize rooe two-channel rational FB with are hown in Figure 7. Table II Eamle FB roertie: D - elay, Y - number of eign unnown an C - number of multilication er inut amle. Deign D Y C Prooe [3] [7] Amlitue in B Amlitue Amlitue in B Amlitue ormalize freuency (w/(2 (a T (w T l (w ormalize freuency (w/(2 (b ormalize freuency (w/(2 (c Figure 7. Variou reone for the otimize FB in Eamle. (a Analyi filter. (b Synthei filter. (c Amlitue an alia itortion. Eamle 2. It i eire to eign a two-channel rational FB with =, = 3, = = 4, an ρ =.2 atifying following reuirement: δ = δ =., δ =., an δ a =.5. In thi cae, a FB with goo roertie i ( achieve by electing = 4, = 6, an ( ( ( = = 5 for {h, f } an = 8, = 8, e e ( ( an e = e = 7 for {h, f }. Aitionally, a elay of D = 2 ha to be ae in the lowa channel. Figure 8 how variou reone for the otimize overall FB. The eign comleity an the imlementation cot of thi eign a well a a comarion with eign obtainable uing the techniue ecribe in [7] an [3] are hown in Table III. Again, the rooe eign ha a lower imlementation comleity at the eene of a higher eign comleity when comare with the irectform imlementation. g g

8 T (w T l (w Amlitue Amlitue in B Amlitue Amlitue in B ormalize freuency (w/( (a ormalize freuency (w/(2 (b ormalize freuency (w/(2 (c Figure 8. Variou reone for the otimize FB in Eamle 2. (a Analyi filter. (b Synthei filter. (c Amlitue an alia itortion. Table III Eamle 2 FB roertie: D - elay, Y - number of eign unnown an C - number of multilication er inut amle. Deign D Y C Prooe [3] [7] COCUDI REMARKS A een from Table II an Table III, by uing the rooe metho to eign FB, ytem with a lower imlementation comleity can be achieve. Thi i mainly ai by a higher eign comleity ue to the nee of earately eigning all four filter builing the FB. Deign in [7] an [3] ue the relation between the ynthei an analyi filter effectively, which enable one to reuce the number of eign unnown by a factor two. Comarion with eamle reente in [5] [4] ha not been inclue becaue in mot of thoe aer filter with wie tranition ban have been eigne. It i nown that the FRM techniue i efficient only for filter with narrow tranition ban. A mentione in thi aer (ee alo [3], filter with narrow tranition ban are imortant when the alia error are eire to be reuce. Thi aer concentrate on eigning PR FB. It i obviou that by reucing the value of δ a an δ in Euation (3e an (3f ractically PR FB can be achieve with the PR roerty atifie u to the numerical accuracy of the eign rogram in ue. However, enforcing the PR roerty tyically reult in FB having lower channel electivity for the ame elay an imlementation comleity. 7. REFERECES [] P. P. Vaiyanathan, Multirate Sytem an Filter Ban. Englewoo Cliff, J: Prentice-Hall, 993. [2]. J. Fliege, Multirate Digital Signal Proceing, Chiceter: John iley an Son, 994. [3] T. Saramäi an R. Bregović, Multirate Sytem an Filter Ban, Chater 2 in Multirate Sytem: Deign an Alication eite by. Jovanovic-Dolece. Herhey PA: Iea rou Publihing, 22. [4] E. Zwicer an H. Fatl, Pychoacoutic. ew Yor: Sringer, 99. [5] T. atanabe, Y. Shibahara, T. Kia, an. Sugino, Deign of non-uniform FIR filter ban with rational amling factor, in Proc. Aia-Pacific Conf. Circ. Syt., 998, [6] K. ayebi, T.P. Barnwell III, an M.J.T. Smith, onuniform filter ban: Recontruction an eign theory, IEEE Tran. Signal Proce., vol. 4,. 4 27, March 993. [7] R. Bregović an T. Saramäi, Deign of linear hae two-channel FIR filter ban with rational amling factor, in Proc. 3 r Int. Sym. Image an Signal Proceing an Analyi, Rome, Italy, Set. 23, [8] T. Blu, A new eign algorithm for two-ban orthonormal rational filter ban an orthonormal rational wavelet, IEEE Tran. Signal Proceing, vol. 46, , June 998. [9] C. S. in an C. Kyriaai, Freuency reone maing aroach for eigning filter ban with rational amling factor, in Proc. 23 orho on Alication of Signal Proceing to Auio an Acoutic, ew Paltz, Y, Oct. 23, [] B. iu an.t. Bruton, The eign of non-uniform-ban maimally ecimate filter ban, in Proc. IEEE Int. Sym. Circuit Syt., 993, [] S. aa, Deign of non-uniform iviion multirate FIR filter ban, IEEE Tran. Circ. Syt. II, vol. 42, Feb. 995, [2] J.-H. ee an D.-C. Tang, Minima eign of twochannel nonunirom-iviion FIR filter ban, IEE Proc.- Vi. Image Signal Proce., vol. 45, , Aril 998. [3] R. Bregović an T. Saramäi, Deign of two-channel FIR filterban with rational amling factor uing the

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