Study on the Use of Error Term in Parallel-form Narrowband Feedback Active Noise Control Systems

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1 Study o the Use of Error erm i Parallel-form Narrobad Feedback Active Noise Cotrol Systems iau He, Woo-Seg Ga ad Yog-Kim Chog Digital Sigal Processig Lab, School of Electrical ad Electroic Egieerig, Nayag echological Uiversity, Sigapore. he7@e.tu.edu.sg, esga@tu.edu.sg, eykchog@tu.edu.sg Abstract Parallel-form arrobad feedback active oise cotrol (FBANC) system has bee sho to perform better tha covetioal iteral model cotrol (IMC) based FBANC system i cacellig multi-toal oise. A previous paper illustrated a ovel approach i estimatig the frequecies of the multi-toe oise, ad usig a iteral toal geerator cum frequecy groupig uit to icrease its frequecy separatio i each chael of the parallel-form FBANC system based o a full-bad error. his paper ivestigates hether it is ecessary to use a arrobad error to further improve o the performace of the parallel-form arrobad FBANC through theoretical aalysis. Computer simulatios are also preseted to validate its performace. I. INRODUCION Active oise cotrol (ANC) techiques have bee commoly used i may applicatios that require the reductio of acoustic oise []-[]. It is based o a simple physical priciple of destructive iterferece by itroducig a secodary ati-oise source to cacel the primary oise source. he ati-oise source must be approximately the same amplitude ad 8 o out of phase ith the primary oise source at the zoe of cacellatio. ypically, a adaptive feedforard active oise cotrol (FFANC) system detects (through a referece microphoe) the primary oise source ad adaptively creates a ati-oise at the zoe aroud the error microphoe through a secodary source. Hoever, i some istaces, referece microphoe may ot be desired due to the feedback of the secodary source to the referece microphoe or physically ot feasible to be set up ear the oise source. I these applicatios, o-acoustic sesors, such as tachometers, are used to estimate the speed (rpm) of the oisy machie ad iterally geerates the primary referece sigal ad its harmoics (correspodig to the rpm) to drive the adaptive filter of the FFANC. here is yet aother approach utilizig a iteral model to sythesize the referece sigal ithout the eed of a referece microphoe. his approach is ko as the iteral model cotrol (IMC) feedback active oise cotrol (FBANC) [3]-[5]. he FBANC, hoever, is subected to a accurate estimatio of the secodary path, S(z) to sythesize the referece sigal. It has bee foud i recet studies [5] that the accuracy of the secodary path estimatio of the FBANC system is much more critical compared to the FFANC system. A e FBANC system based o IMC as recetly proposed i [4], hich uses the idea of toal sigal geeratio, as deployed i the direct/parallel FFANC proposed i [6]. he direct/parallel FFANC system is used to cacel multiple oisy toes at the fudametal frequecy ad its harmoic frequecies, hich are commoly produced by may rotatig machies. he trick i the direct/parallel FFANC system lies i the partitioig of the iterally geerated harmoic oise sources to differet chaels to icrease the frequecy separatio amog the geerated harmoics, before performig parallel adaptive filterig across these chaels. I a similar maer, e have proposed a e parallel-form arrobad FBANC i [7] that allos oisy toal harmoics to be separated ad adapted separately to improve its covergece ad oise reductio. his e FBANC system is based o the assumptio that the umber of toes i the primary oise is ko ad the frequecies of the multi-toe oise are estimated i real time by usig ifiite-impulserespose (IIR) adaptive otch filter (ANF) [8]. Based o the estimated frequecies, the referece sigals are iterally geerated ad grouped to differet chaels so that the frequecy separatio i each chael is icreased, ad thus, the covergece rate ad oise reductio are improved. Furthermore, the proposed e feedback ANC system is foud to be less sesitive to impulsive oise. Hoever, questio o hether the updatig error eeds to be separated to facilitate parallel adaptive filter update remais uasered. his paper attempts to aser this questio ad performs a theoretical aalysis to ivestigate hether there is ay differece i usig a sigle full-bad error or multiple arrobad error updates. he rest of the paper is structured as follos. Sectio II gives the to variat structures of the parallel-form arrobad FBANC based o commo-error ad multipleerror updates. Our theoretical aalysis o the use of differet error terms is discussed i Sectio III, hich is folloed by the computer simulatios i Sectio IV. Fially, e coclude this paper i Sectio V. II. PARALLEL-FORM NARROWBAND FEEDBACK ACIVE NOISE SYSEM Fig. illustrates the block diagram of the covetioal parallel-form arrobad FBANC, here the same full-bad error is used i filtered-x least-mea-square (FxLMS) APSIPA APSIPA 4

2 x x x ˆi ω, W ( z) W ( z) W N( z) ( z) d ˆ( ) yf y ( ) S( z) ya e ( ) d ( ) x ( ) f x ( ) f xf Fig.. A covetioal parallel-form arrobad FBANC system, full-bad error used i FxLMS (adapted from [7]). x x x ˆ ω W ( z) W ( z) W N( z) ( z) d ˆ( ) yf y S( z) ya e ( ) d ( ) e ( ) x ( ) f x ( ) f xf Fig.. A complete parallel-form arrobad FBANC system, arrobad error used i FxLMS (adapted from [7]). R ( z) R ( z) R ( z) algorithm to update the eights of all the chaels [7]. For compariso, the complete parallel-form arrobad FBANC system is illustrated i Fig., here the error term is obtaied from IIR ANF R (z) [9]. Clearly, the oly differece betee the to FBANC systems i Figs. ad is the use of differet error terms. It is assumed that the primary oise d() cosists of domiat arrobad compoets ith frequecy f, =,,...,. he primary path, secodary path, ad secodary path model are expressed as P z, S z, ad Sˆ z, respectively. Each chael of the ( ) ( ) ( ) adaptive filter W ( ) z cosists of to eights, ad all the chaels are coected i parallel. he associated siusoidal referece sigal x ( ) ca be geerated (sychroized by a o-acoustic sesor or obtaied by estimatig the frequecies usig a adaptive otch filter [7], [8]). I the geeratio of the referece sigal, it has bee foud that the amplitude of the siusoids affects the covergece rate [6]. I order to further improve the covergece rate, the amplitudes of all the referece sigals are set as the iverse of the magitude respose of the secodary path at the respective frequecies. Hece, the referece sigal i the th chael x () is expressed as x = g ( f ) cos( π f ), =,,...,, () here g ( f ) =. he filtered sigal x f is Ŝ ( ) f f = f

3 computed as I x ( ) ˆ f = si x ( i), =,,...,, () i= here s ˆi is the coefficiet of Sˆ ( z). Usig FxLMS algorithm, the eight vector is updated as ( + ) = + μ e x, =,,...,, (3) here e() is the error sigal, = [,, ], x f = [ xf xf ( )], deotes traspose operator, ad μ is the step size of chael. he selectio of error term i (3) has bee studied recetly. I [7], the full-bad error that is commo to all adaptive filters has bee used ad validated to perform better tha the covetioal iteral model cotrol based FBANC system. Hoever, accordig to the aalysis i [9], for a complete parallel-form arrobad FBANC system, the arrobad error ca be used to update the idividual adaptive filters. Nevertheless, the ecessity of usig arrobad errors as compared to the full-bad error is ot fully uderstood. I this paper, e exted the aalysis i [9], ad study the differece i the use of arrobad errors ad full-bad error i the FBANC systems [7]. III. HEOREICAL ANALYSIS ON HE USE OF ERROR ERM Withi the scope of the arrobad FBANC, it is assumed that the primary oise d(), as ell as the cacellig oise ya ( ), cosist of arrobad (or toal) compoets. hus, e ca express them as, ( ) ( ) d = d, =,,...,, = ( ) a ( ) y = y, =,,...,, a = here ( ) ad ( ) d ya have the same frequecy (4) f as the referece sigal x ( ). hus, the error sigal, ca also be expressed as the sum of the arrobad errors, i.e., e = e, =,,...,, (5) = here e = d ya, =,,...,. (6) herefore, e ca rerite eight updatig equatio expressed i (3) as ( + ) = + μ ei xf i= (7) = + μe xf + μ ei x f i=, i = + μ e x + μ β x, ( ) ( ) ( ) ( ) ( ) m f f here β = ei, =,,...,. (8) i=, i β ca be cosidered as a disturbace to the th chael FxLMS. Substitute (6) ito (7) ad take the expectatio i both sides, e have ( + ) = I μ R + μ P + μ D, (9) here I is the idetity matrix, ad = E, R = E xif x f, () P = E dm x f,ad D = E β x f. Accordig to the aalysis i [9], he D, it ill act as a Clearly, the error term disturbace, hich ill cause misaligmet of the adaptive eights, ad reduce the step size boud as compared to the case he the arrobad error is used. Hoever, the sigificace of D is ot ell examied. Accordig to (8), β is the sum of the arrobad error excludig the th subbad, ad by substitutig (6) ito (8), e have β = di ui, =,,...,, () i=, i hus, e ca express D D as = E d u x i i f i=, i ( ) x ( ) ( ) x ( ) = Ed Eu i f i f i=, i i=, i () It is ell ko that the expectatio of the cross-correlatio of to toes havig differet frequecies is equal to, i.e., E cos( π f i ) cos( π f) =, fi f. (3) Based o (3), e ca deduce that D =. his fidig idicates that the adaptive filter eights i the steady state ill ot differ betee the to cases: oe usig arrobad error e ( ), ad the other usig full-bad error e( ). Without cosiderig other effects, the above aalysis idicates that there ill ot be much differece o the performace betee the to FBANC systems (Figs. ad ), i terms of step size boud, ad covergece rate. I additio, the arrobad filters are ot ideal due to certai badidth of the badpass filter [9]. It has bee foud i [] that he the badidth is small, the group delay i the arrobad filters becomes a maor factor that degrades the covergece of FBANC. O the other had, if e icrease the badidth, the maximum stable step size decreases. I other ords, he the frequecy separatio of the primary oise is smaller, the maximum stable step size i arrobad error method is smaller. I order to make its steady-state performace be comparable to the full-bad error method, a smaller step size is required,

4 , Full-bad error, μ =. Narrobad error, μ =., ,.5, , 3, , 4, Fig. 3 Covergece of adaptive filter eights i the to methods: full-bad error, ad arrobad error i parallel-form arrobad FBANC. IV. SIMULAION RESULS o validate the aalysis i Sectio III, computer simulatios are coducted. he secodary path estimatio is assumed to be perfect, i.e., S( z) = S ˆ ( z). I our simulatio, e cosider 4 toal ( = 4) sigal as primary oise. heir frequecies are 8, 6, 4, ad 3 Hz, ith samplig frequecy of f s = Hz. For the arrobad filters, e adopted the same filters described i [9] ith p m =.99. After 6th iteratios (i.e., 3 secods), e compute the folloig estimates of () as: R = 6 P = 6 D = f f f = = 6 = 6 = 6 6 xf xf ( ) xf = = x x ( ) x ( ) 6 = 6 = ( ) ( ) dm xf ( ) ( ) dm xf βm βm x f x ( ) f (4)

5 - - Full-bad error, μ =. - Full-bad error, μ =. Narrobad error, μ =. - Narrobad error, μ = Fig. 4 Learig curves of parallel-form arrobad FBANC for the to methods: full-bad error, ad arrobad error. Frequecies of the toes are: 8, 6, 4, ad 3 Hz Fig. 6 Learig curves of parallel-form arrobad FBANC for the to methods: full-bad error, ad arrobad error. Frequecies of the toes are: 4, 8,, ad 6 Hz. - 6 Full-bad error, μ =. - Narrobad error, μ =. 4 Full-bad error, μ =. Narrobad error, μ = Fig. 5 Learig curves of parallel-form arrobad FBANC for the to methods: full-bad error, ad arrobad error. Frequecies of the toes are:,, 3, ad 4 Hz x 4 Fig. 7 Learig curves of parallel-form arrobad FBANC for the to methods: full-bad error, ad arrobad error. Frequecies of the toes are varyig from: 5,, 5, ad Hz to,, 3, ad 4 Hz. hus, e ca calculate these terms as R =,, P =.5 D = R =,, P =.4 D = R3 =, 3 =, 3 =.9.4 P. D R4 =, 4 =, 4 =.9.36 P.8 D.5 (5) It ca be observed from (5) that as compared to R, ad P, D is actually isigificat ad ca hece be igored i (9). Next, e sho the covergece of the eights i these to methods i Fig. 3. Usig the same step size of., the fullbad error method coverges approximately at the same rate as the arrobad error method. here is o misaligmet of the fial eights i the full-bad error method as compared to the arrobad method, ad hece o differece i oise reductio. As a result, the fial MSE of these to methods are very similar, as sho i Fig. 4. Hoever, the covergece of the full-bad error method is faster tha that of the arrobad error method. his is due to the group delay icurred by the arrobad filters i the arrobad error method, as poited out i []. Furthermore, e have tested the primary oise ith other frequecies i simulatios ad preseted the results of

6 learig curves. I Fig. 5, the frequecies for the four toes are set at,, 3, ad 4 Hz ith other settigs remai the same. I Fig. 6, the frequecies are decreased to 4, 8,, ad 6 Hz ith p m =.995 to esure the covergece. I Fig. 7, the frequecies of the toes are set to 5,, 5 ad Hz for the first 4 secods, ad icrease liearly i the folloig secods all the ay to,, 3, ad 4 Hz, respectively. It is obvious that the MSE performace of the full-bad error structure is much better tha the arrobad error structure i the parallel-form arrobad FBANC. V. CONCLUSIONS I this paper, e studied the use of the error term i parallel-form arrobad FBANC systems. Both full-bad error, hich is commo i all adaptive filters; ad arrobad error, hich is obtaied usig IIR ANF, are cosidered. Our theoretical aalysis suggested that the use of full-bad error performs equally ell compared to the arrobad error. his is because i each chael, the disturbace error term is orthogoal to the filtered referece sigal. hus, o misaligmet i the eight vector is icurred ad the maximum step-size boud remais the same i both cases. Hoever, besides the additioal computatioal cost, the use of the arrobad error iheretly itroduces group delay i the secodary path ad slos do the covergece of the FBANC system. I coclusio, the full-bad error is recommeded i the adaptatio of the parallel-form arrobad FBANC system. ACKNOWLEDGMEN We ould like to ackoledge some of the prior ork carried out by ogei Wag. REFERENCES [] S. M. Kuo ad D. R. Morga, Active Noise Cotrol Systems Algorithms ad DSP Implemetatios, Ne York: Wiley, 996. [] Y. Kaikaa, W. S. Ga ad S. M. Kuo, Recet advaces o active oise cotrol: ope issues ad iovative applicatios, APSIPA ras. Sig. If. Process., vol., pp., Aug.. [3]. W. Wag, W. S. Ga ad S. M. Kuo, Effects of frequecy separatio i feedback active oise cotrol systems, Proc. Iter-Noise, pp , Sep.. [4]. Wag, W.-S. Ga, Stochastic aalysis of FXLMS-based iteral model cotrol feedback active oise cotrol systems, Sigal Processig, 4. [5] V. L. Wag, W. S. Ga, A. W. H. Khog ad S. M. Kuo, Covergece Aalysis of Narrobad Feedback ANC System With Imperfect Secodary Path Estimatio, IEEE ras. Audio, Speech, Lag. Process., vol., o., pp. 43-4, Nov. 3. [6] S. M. Kuo ad A. B. Puvvala, Effects of frequecy separatio i periodic active oise cotrol systems, IEEE ras. Audio, Speech, ad Laguage Processig, vol. 4, o. 5, pp , 6. [7]. W. Wag, W. S. Ga, ad S. M. Kuo, Ne feedback active oise cotrol system ith improved performace, Proc. IEEE ICASSP, Florece, Italy, 4, pp [8] L. a ad. iag, Novel adaptive IIR otch filter for frequecy estimatio ad trackig, IEEE Sigal Process. Mag., vol. 6, o. 6, pp , Nov. 9. [9] C.-Y. Chag ad S. M. Kuo, Complete parallel arrobad active oise cotrol systems, IEEE ras. Audio, Speech, Lag. Sigal Process., vol., o. 9, pp , Sep. 3. []. Cheer ad S.. Elloitt, Commets o complete parallel arrobad active oise cotrol systems, IEEE/ACM ras. Audio, Speech, Lag. Sigal Process., vol., o. 5, pp , May 4.