ISSN: 777 ISO 9: Certified Volume, Iue, April Deign of Coine Modulated Filter Bank uing Computationally Efficient Multiplierle FIR Filter Jyotna Ogale, Alok Jain Abtract Thi reearch work preent a computationally efficient and multiplierle deign approach for coine modulated filter bank The prototype filter i deigned uing rounding and harpening technique to provide multiplierle filter The rounding factor, requiring minimum to maximum number of multiplier, are ued to how the performance of the deigned filter bank with the fixed harpening polynomial Linear iterative optimization algorithm i ued for minimization of ditortion function Several example are given to illutrate the performance of the propoed method Index Term Coine modulated filter bank, rounding factor, harpening polynomial I INTRODUCTION Coine modulated filter bank (CMFB) find wide application in different area of digital ignal proceing uch a equalization of wirele communication channel, ub band coding, pectral analyi, adaptive ignal proceing, denoiing, feature detection and extraction Several deign technique have been developed for thee filter bank in the pat [,, ] In CMFB analyi and ynthei filter are coine modulated verion of low pa prototype filter Thu, the deign of whole filter bank reduce to that of the prototype filter and the cot of overall filter bank i almot equal to the cot of one filter with modulation overhead Alo, during the deign phae, only prototype filter coefficient are required to be optimized in cae of near perfect recontruction (NPR) In NPR, aliaing i cancelled approximately and ditortion i a delay only approximately The condition of NPR in term of frequency repone of the linear phae prototype filter i a follow [,, ] j P e for / M () j j M j( k / M ) T, e where T e P e () k Both condition are to be atified a nearly a poible The accuracy of the firt approximation give a meaure of the aliaing error, while the accuracy of the econd approximation give a meaure of the ditortion error Eq () can be atified approximately by appropriately chooing filter coefficient through computer aided optimization technique Traditional deign approache involve nonlinear and linear optimization [,,] In thi work, a linear optimization of the ingle parameter namely the cutoff frequency i varied to minimize the objective function Prototype filter i deigned uing Kaier Window and then computationally efficient technique baed on rounding and harpening i applied [,,, 7, 9, and ] Prototype Filter Deign The impule repone coefficient of a caual N th order linear phae FIR filter h(n) uing window technique i given by Eq (): h( n) w( n) h ( n) () i Where, h (n) i the impule repone of the ideal low pa filter and i expreed a: incnn h i ( n) () nn Where, c i cutoff frequency of the ideal low pa filter and wn ( ) i the window function The prototype filter i deigned uing widely ued Kaier window function The filter order N and tranition width ω i etimated a A 79 N () () for a given value of pa band ( ) and top band ( ), the required number of adder and multiplier are equal to (N) and (N+)/, repectively In cae of FIR filter the filter order i inverely proportional to the tranition bandwidth Thu, for narrow tranition bandwidth, filter order become very high, and hence number of adder and multiplier are alo high Therefore, reearcher are putting effort on developing computationally efficient deign technique In thi propoed work, computationally efficient and multiplierle deign technique of FIR filter i ued to deign the prototype low pa filter a uggeted by Mitra et al [] II ROUNDING AND SHARPENING TECHNIQUE The rounding and harpening technique i applied on window baed FIR filter to atify the given pecification The technique i briefly decribed in thi ection ROUNDING The impule repone rounding i given by h(n) = α g i (n) = α round (h(n)/α) (7) p 7
ISSN: 777 ISO 9: Certified where, h(n) i an impule repone of the FIR filter which atifie the given pecification, round() mean the rounding operation, g i (n) i the new impule repone derived by rounding all the coefficient of h(n) to the nearet integer The rounded impule repone g i (n) i caled by a factor α which determine the preciion of the approximation of g(n) to h(n) The rounding contant i choen in the form of α = N where N i an integer The proce of rounding introduce ome null coefficient in the rounded impule repone The number of nonzero integer coefficient correpond to the number of the um and the number of integer multiplication correpond to the number of a different poitive integer coefficient Computational complexity i expreed in term of number of integer multiplication, which itelf depend on rounding contant Computational reduction = (Multiplier FIR Multiplier Rounding/Sharpening ) / (Multiplier FIR ) Coefficient aving = Number of null coefficient / Total coefficient SHARPENING TECHNIQUE The harpening technique i ued to improve the gain repone characteritic of a linear phae FIR filter [] In thi technique polynomial relationhip of the form H = f(h) i ued, where H and H are the amplitude of the overall and prototype filter, repectively Thi relationhip i known a amplitude change function (ACF) The expreion of ACF for given value of m and l i given a [] m m l ( l )! ( ) l (, )( ) H H H H C l H () x l!! x where (l+, ) i the binomial coefficient The amplitude function hould be horizontal near pa band and top band, ie, ha a derivative of zero at thee point, denoted a m and l, repectively ACF polynomial for m =,, and l =,, are hown in Table Table m l ACFS H H H H H H H H H H +H H H +H H H +H H 7 H H H +H H +H H +H H +H H 7 +7H H +H III COSINE MODULATED FILTER BANK Coine modulation i the cot effective technique for Mband filter bank [, 7, ] a hown in Fig Let H (z) denote the tranfer function of the prototype filter i given by N n H( z) h( n) z, with h(n)=h(n n) (9) n Then the analyi and ynthei filter are given by Volume, Iue, April Fig MChannel Uniform Filter Bank N r ( n) h( n) co r n ( ) M N n h n r n r ( ) ( ) co ( ) M for k M, n N H k F k () Where Hk ( n) and Fk ( n ), repreent the impule repone of the kth channel analyi and ynthei filter, repectively IV OPTIMIZATION TECHNIQUE In thi work objective function given by Eq () i ued to minimize the ditortion function The initial window coefficient are calculated prior to calling the optimization routine Intead of optimizing all the ubfilter and harpening polynomial individually, only the cutoff frequency (ω c) of the rounded filter at each iteration i to be adjuted The tep ize denoted by tep in flowchart, calculate the impule repone coefficient of the new filter and compute the error, (denoted error) uing Eq () a the cot function Whenever the error increae over that of the previou error (ierror), tep i halved and the earch direction labeled dir i changed When difference between previou iteration and current iteration become within a pecified tolerance (tol), the optimization proce halted The flow chart of the optimization algorithm i given below j j( / M ) max h ( e ) h( e for / M () 7
Aliaing error Amplitude ditortion ISSN: 777 ISO 9: Certified Volume, Iue, April Specify top band attenuation (A ), number of band (M) Initialize: pa band (ω p), top band freq (ω ), ierror, tol, tep, dir, and flag Initialize: Rounding contant (α), degree of harpening polynomial (m, l) Conventional Rounded & Scaled Calculate cutoff frequency (ω c) Determine the Filter order (N) and impule repone coefficient of the filter uingwindow technique Obtain rounded and caled rounded impule repone coefficient of the filter (a) Obtain all the ub filter, their freq repone, harpened freq repone I pecification atified? Ye No Ye ( error ) Determine recontruction error tol error I Or error = ierror No (b) Flag = Step=tep/ dir=dir ω c = ω c + dirtep, and determine recontruction error at new cutoff frequency 999 I error < ierror Diplay optimized error value Stop Diplay optimized error value No 99 997 x (c) top V CASE STUDY Example An eightband CMFB i deigned The deign pecification of the filter are: top band attenuation A = db, ω = π, N = The rounding contant α i varied from to 7, accordingly harpening polynomial (m, l) have been elected Figure (ad) how the magnitude repone of optimized prototype filter, optimized rounded and harpened filter, analyi filter bank, the amplitude ditortion function and the aliaing error At α =, m =, l =, the obtained value of E pp = 9x, aliaing error E aa =9x are minimum with 9 % computational reduction (d) Fig Magnitude repone of (a) Optimized prototype filter, rounded and harpened filter for α =, m = and l = (b) Eight band filter bank (c) amplitude ditortion (d) aliaing error Example A band CMFB i deigned with prototype filter top band attenuation A = db, = 9π, N = 9 The rounding contant i varied and accordingly harpening Polynomial ( ml, ) have been elected Fig (ac) how the magnitude repone of optimized filter and filter bank along 7
Amplitude ditortion Amplitude ditortion ISSN: 777 ISO 9: Certified with amplitude ditortion The pa band pecification in the rounding and harpening deign doe not exceed the original et of pecification At =, m = and l = the E = 9x, E a = x are obtained pp Volume, Iue, April Conventional Rounded & Sharpened 7 7 77 Conventional Rounded & Scaled (a) (a) 7 9 (b) (b) 999 999 99 99 997 (c) Fig Magnitude repone of (a) Optimized prototype filter, optimized rounded and harpened filter for α =, m = and l = (b) band filter bank (c) amplitude ditortion Example In thi cae, the filter bank of channel ha been deigned The prototype filter ha the ame pecification, a given in []The top band attenuation around db, = 7π, N = The ampling frequency i π Fig (a) how the overlapped optimized prototype filter of conventional and propoed method (b) band filter bank and Fig (c) how zoom plot of amplitude ditortion At E = x, E a = 7x =, m = and l = the pp 7 are obtained (c) Fig Magnitude repone of (a) optimized prototype & Rounded and harpened filter for =, m =, l =, (b) band filter bank filter (c) amplitude ditortion Performance comparion of the propoed method with the Previouly reported work i given in Table All the reported value in the propoed method how better performance than the previouly reported work with ignificant reduction in computational complexity at very low value of rounding contant and low degree of harpening polynomial Table M A Report ed work Kha et al[] propo ed Kha et al[] Propo ed 9 9 N M ulti pli er Epp x x 9 x 9 x E a 7 x x x 7 x 7
%Computational reduction %Computational aving Lin et al[] Propo ed ISSN: 777 ISO 9: Certified 9 Volume, Iue, April 97 x 7 x x 7 7 x 7 Experimental reult have hown that, ue of rounding and harpening over conventional method yield filter bank with the mallet peak to peak recontruction error and computational complexity a well a achieving a decreae in amplitude error over conventional and Kha et al [] of, repectively 7% and % in eightband FB, 79% and % repectively in cae of ixteenband FB with almot ame aliaing error, when compared with conventional method and a percentage decreae of and repectively in eightband and ixteenband FB,when compared with Kha et al[] Plot of rounding contant α veru coefficient aving i hown in Fig (a) The coefficient aving i expreed in term of number of null coefficient which varie with α For higher value of it number of null coefficient are more due to which coefficient aving i more and vice vera 7 band band Rounding contant Fig (a) number of multiplier) i maximum and decreae with increaing α Table how the computational complexity comparion of the propoed work with repect to conventional and polyphae method Approach M N Arithmetic element Adder Multiplier Method of [] Conventional Propoed Method of [ ] Conventional Propoed Method of [ ] Conventional Propoed 9 9 9 9 7 Computational reduction 9%,9 % 7%,9% %,7% VI DISCUSSION In the cae tudy three example are taken to how the performance of the deigned CMFB The complexity of rounded filter depend on choice of rounding contant At higher value of rounding contant null coefficient are more, number of adder & multiplier are le, ditortion in gain repone & aliaing error i more, which lead to high degree of harpening polynomial, le computational complexity and high computational aving At the lower value of α complexity i more and ditortion in gain repone i le In the field of filter bank deign where ditortion parameter play very important role, compromie with thee can not be agreed upon at the cot of computational reduction hence without compromiing with thee parameter, ignificant reduction in computation i achieved Thi approach can provide % computational reduction at the cot of other performance parameter Unlike to method [7] there i no retriction in the pecification of the filter 9 7 band band Rounding contant Fig (b) Fig (A) Rounding Contant v % Computational Saving Plot (B) Rounded Contant v % Computational Reduction Computational reduction plot Fig (b) how that at lower value of α, computational complexity (expreed in term of VII CONCLUSION In thi propoed work computationally efficient deign for M band NPR CMFB have been preented Different rounding factor with fixed harpening polynomial are ued to improve the magnitude characteritic of the parent filter Thi approach yield linear phae FIR filter that can meet the given pecification with a reduced number of multiplier The technique can be equally applied to narrowband and wideband filter deign and at the lower and higher top band attenuation level However, the method exhibit more efficient reult if the order of the initial filter i le than In that cae the making method [] can be ued to obtain le order filter and to apply the propoed procedure for the model and making filter Since the deign i in a ingle tage, without introducing additional delay, it ignificantly 7
ISSN: 777 ISO 9: Certified reduce the computational complexity of the prototype filter The obtained value of performance parameter are maller than the latet reported work with ignificant reduction in computation The imulation reult how that the rounding and harpening can provide the filter bank with higher performance a compared to conventional method Computation time i almot ame a that of conventional method which make thi method practicable for application where deign mut be carried out in real or quaireal time Volume, Iue, April econdorder cone programming approach, IEEE Tran Circuit Sytem, vol, no, pp March [] Z Zhang, Deign of coine modulated filter bank uing iterative Lagrange multiplier method, in ProcIEEE IntSympMirowave, Antenna, Propagation EMC technologie for Wirele Communication;, vol, pp, 7 [] TQNguyen, Nearperfectrecontructio peudoqmf bank, IEEE Tran Signal Proce, vol, no9, pp7, Jan99 AUTHOR S PROFILE REFERENCES [] Gordana Jovanovich Dolecek and Sanjit K Mitra, Computationally Efficient Multiplierfree Fir Filter Deign, Computation y Sitema, vol, no, pp 7, ISSN, 7 [] G JovanovicDolecek, M MAlvarez and M Martinez, One imple method for the deign of multiplier le FIR filter, Journal of Applied Reearch and Technology, vol No, pp, Augut [] M Bhattacharya and T, Saramaki, Some obervation leading to multiplier le implementation of linear phae filter, proc ICASSP, pp7, [] H H Kha, H D Tuan and T Q Nguyen, Efficient deign of coinemodulated filter bank via convex optimization, IEEE Tran on Signal Proceing, vol 7, no, pp 997, March 9 [] Y C Lim, Frequencyrepone making approach for the ynthei of harp linear phae digital filter, IEEE Tran Circuit and Sytem, CAS, pp 7, April 9 [] P P Vaidyanathan, Multirate ytem and filter bank, Englewood Cliff, NJ: PrenticeHall, 99 [7] S K Mitra, Digital ignal proceing: A computer Baed Approach New York, NY: McGraw Hill, 99 [] C D Creuere and SK Mitra, A imple method for deigning highquality prototype filter for Mband peudo QMF bank, IEEE Tran on Signal Proceing, vol, no, pp 7, April 99 [9] J F Kaier and R W Hamming, Sharpening the repone of a ymmetric no recurive filter by multiple ue of the ame filter, IEEE Tran, AcoutSpeech, Signal Proceing, vol, pp77, February 997 [] W S Lu, TSaramaki and RBregovic, Deign of practically perfectrecontruction coinemodulated filter bank: A econdorder cone programming approach, IEEE Tran Circuit Sytem, vol, no, pp March [] Y P Lin and P P Vaidyanathan, A Kaier window approach for the deign of prototype filter of coinemodulated filter bank, IEEE Signal Proceing Letter, vol, no, pp, June99 [] A Jain, R Saxena and S C Saxena, An improved and implified deign of coine modulated peudoqmf filter bank, Digital Signal Proceing, vol, no, pp, May, [] W S Lu, TSaramaki and RBregovic, Deign of practically perfectrecontruction coinemodulated filter bank: A Alok Jain wa born in Vidiha, India during 9 He received hi BE (Electronic & Intrumentation) degree from Samrat Ahok Technological Intitute, Vidiha, and MTech (Computer Science and Technology) from IIT Roorkee (ertwhile Univerity of Roorkee), in 9 and 99, repectively He obtained hi PhD degree from Thapar Univerity (ertwhile Thapar Intitute of Engineering and Technology), Patiala, India, in He i preently erving a a Profeor and Head in the department of Electronic & Intrumentation Engineering, Samrat Ahok Technological Intitute, Vidiha, India He ha publihed paper in Journal and Conference Proceeding of International repute He authored two text book related with power electronic and monograph on multirate ytem, filter bank and Tran multiplexer He cochair the eion in Int Conf SCI held at Orlando, USA He reviewed the paper and book of Int Journal and Publiher Dr Jain i a life member of IE (I), IETE, ISTE, BMESI, and Intrument Society of India and wa the member of IET, UK for more than year Hi current reearch interet include digital ignal proceing, multirate ignal proceing, filter bank, and their application in biomedical ignal proceing, image proceing, and power electronic Jyotna ogale wa born in Jabalpur, India during 97 She received her BE (Electronic & Telecommunication) degree, ME (Communication ytem) degree from Government Engineering College, Jabalpu, in 99 and 99, repectively and puruing PhD degree from RGTU (Bhopal)She i preently erving a a Aociate Profeor in the department of Electronic & Communication Engineering, Samrat Ahok Technological Intitute, Vidiha, India She ha publihed paper in Journal and Conference Proceeding of International repute Her current reearch interet include digital ignal proceing, multirate ignal proceing, filter bank, and their application in biomedical ignal proceing, image proceing 77