The Wavelet Transform-Domain LMS Adaptive Filter Algorithm with Variable Step-Size

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1 [ DOI:.68/IJEEE Downloaded from jeee.ust.ac.r at 3:44 IRS on Wednesday ovember 4th 8 he Wavelet ransformdoman LS Adaptve Flter Algorthm wth Varable StepSze. Shams Esfand Abad*(C.A.), H. esgaran* and S.. Khademyan* Abstract: he wavelet transformdoman leastmean square (WDLS) algorthm uses the selforthogonalzng technque to mprove the convergence performance of LS. In WDLS algorthm, the tradeoff between the steadystate error and the convergence rate s obtaned by the fxed stepsze. In ths paper, the WDLS adaptve algorthm wth varable stepsze (VSS) s establshed. he stepsze n each subflter changes accordng to the largest decrease n mean square devaton. he smulaton results show that the proposed VSSWDLS has faster convergence rate and lower msadjustment than ordnary WDLS. Keywords: Adaptve Flter, Wavelet ransform Doman LS (WDLS), Varable StepSze, ean Square Devaton. Introducton Adaptve flters are wdely used n varous applcatons such as system dentfcaton, channel equalzaton, nose cancellaton, actve nose control, and so on [, [. he most popular adaptve flters are the least mean squares (LS) and normalzed LS (LS) algorthms due to ther smplcty. However, these algorthms have slow convergence for colored nput sgnals [3, [4. o solve ths problem, the transform doman adaptve flter (DAF) algorthms have been proposed [5. he DAF algorthms explot the decorrelaton propertes of some wellknown sgnal transforms, such as the dscrete Fourer transform (DF) and the dscrete cosne transform (DC), n order to prewhten the nput data and speed up flter convergence [6, [7, and [8. In the wavelet transform doman least mean square (WDLS) adaptve flterng, the projectons of the nput sgnal onto the orthogonal subspaces are used as nputs to a lnear combner. he weghts of the lnear combner can hence be updated by the LS algorthm whle normalzng the power at each resoluton level to acheve faster and unform convergence of all weghts to the optmal [9, [. Iranan Journal of Electrcal & Electronc Engneerng, 7. Paper frst receved 5 December 6 and accepted July 7. * he Author s wth the Faculty of Electrcal Engneerng, Shahd Rajaee eacher ranng Unversty, P.O.Box: , ehran, Iran. Emals: mshams@srttu.edu ** he Authors are wth the Faculty of Basc Scences, Department of Appled athematcs, Shahd Rajaee eacher ranng Unversty, P.O.Box , ehran, Iran. Emals: hmesgaran@srttu.edu, m_khademyan@srttu.edu Correspondng author:. Shams Esfand Abad In the above mentoned algorthms, the fxed stepsze can change the convergence rate and the steady state mean square error (SE). Wth optmally selectng the stepsze durng the adaptaton, we obtan fast convergence rate and low steady state mean square error at the same tme. In the case of varable stepsze (VSS) methods, varous approaches have been proposed n the lteratures [, [. One of the most mportant strategy n ths ssue was pre Fg.. Structure of the WDLS algorthm. Iranan Journal of Electrcal & Electronc Engneerng, Vol. 3, o. 3, September 7 3

2 [ DOI:.68/IJEEE Downloaded from jeee.ust.ac.r at 3:44 IRS on Wednesday ovember 4th 8 sented n [3. hs approach was successfully extended to the dfferent adaptve flter algorthms n [4. In ths paper, the VSSWDLS s ntroduced. In the proposed VSSWDLS, the stepsze n each subflter changes accordng to the largest decrease n mean square devaton. In comparson wth ordnary WDLS [5, the VSSWDLS has faster convergence speed and lower steadystate SE. he remnder of ths paper s organzed as follows. In Secton, the WDLS algorthm s brefly revewed. he new VSSWDLS s proposed n Secton 3. he computatonal complexty of the VSSWDLS s dscussed n Secton 4. Fnally, before concludng the paper, the usefulness of ths algorthm s demonstrated by presentng several smulaton results. hroughout the paper, represents transpose, takes the squared Eucldean norm, and shows the Expectaton.. he WDLS Adaptve Algorthm Consder a lnear data model for as wth a smoothng factor D g y (n ) h g h (n ) g h (n ) P h (n ) z h (n ) V h (n ) (5) In (5), Ph (n ) s a varable stepsze n subflter. akng the squared Eucldean norm from the both sdes of (5) and then the expectaton leads to (6) E [ g h (n ) ' where (n )z h (n ) () and the error sgnal s obtaned by d (n ) y (n ). he update equaton for each subflter n WDLS s gven by g h (n ) P P h (n )E [ z h (n ) V h (n ) V h (n ) P h (n )E [ e (n ) z h (n ) (7) axmzng Δ wth respect to Ph (n ) leads to the followng optmum stepsze g h (n )z h (n ) V h (n ) $ h P (n ) e (n ) z h (n ) (8) Snce e a (n ) h g (n )z h (n ), for a pror error as, e a (n ) we use the approxmaton g h (n )z h (n ). herefore we have Ph$ (n ) e a (n ) V h (n ) e a (n ) z h (n ) By q h (n ) defnng (3) a z h (n ) VX E [ q h (n ) g h (n )z h (n ) g h (n ) () D. where wt s an unknown dmensonal vector that we expect to estmate, υ(n) s the measurement nose wthvarance συ, and x(n ) [x (n ), x (n ), }, x (n ) denotes an dmensonal nput (regressor) vector. It s assumed that υ(n) s zero mean, whte, Gaussan, and ndependent of x(n). Fg. shows the structure of the WDLS algorthm [9. In ths fgure, the x matrx s an orthogonal matrx that s derved from a unform band flter bank wth flters denoted by h, h,..., h followng the procedure gven n [9. In matrx form, the orthogonal W can be expressed as z(n)=x(n). hs vector can be rep resented as z (n ) [z h (n ), z h (n ),}, z h (n ) where z h (n ) s are output vectors of an band flter bank. By splttng the g(n) nto subflters, each havng coeffcents, g (n ) [gh (n ), gh (n ),}, gh (n ), the output sgnal can be stated as (4) 3. he VSSWDLS Adaptve Algorthm $ By defnng the weght error vector g h (n ) g h g h (n ), $ where g h s the true unknown subflter coeffcents, the weght error vector update equaton for WDLS for each subflter can be represented as ' E [ g h (n ) x (n )wt X (n ) d (n ) V h (n ) DV h (n ) ( D ) z h (n ) z h (n ) (9) z h ( n )e a ( n ) V h ( n ), we obtan that. hen, the optmum stepsze s where V h (n ) can be computed teratvely by gven by 4 Iranan Journal of Electrcal & Electronc Engneerng, Vol. 3, o. 3, September 7

3 [ DOI:.68/IJEEE Ph$ (n ) E [ q h (n ) z h (n ) E [ q h (n ) V X E [ () Applyng the expectaton nto the leads to the E [q h (n ) z h (n ) V h (n ) () We propose to estmate E [q h (n ) by tme averagng as follows: Downloaded from jeee.ust.ac.r at 3:44 IRS on Wednesday ovember 4th 8 ber of subbands, s the number of past values of the th transform coeffcent, and L s number of past squared values of the error. he VSSWDLS algorthm needs +6+5 multplcatons and 4 dvsons. It s nterestng to note that usng Haar wavelet transform (HW) leads to the only 6+5 multplcatons whch s sgnfcantly lower than other VSS transform doman adaptve algorthms. E qˆ h (n ) ( E ) qˆ h (n ) z h (n ) () h V (n ) nstead of E [ q h (n ), where the VSSWDLS for each subflter s establshed as g h (n ) P h (n ) g h (n ) V h (n ),, for n x(n ) [x (n ), x (n ), }, x (n ) z (n ) E. Usng qˆ h (n ) z h (n ) able. he VSSWDLS Adaptve Algorthm. (3) x(n ) z (n ) [zh (n ), zh (n ), }, zh (n ) g (n ) [gh (n ), gh (n ),}, gh (n ) d (n ) gh (n )z h (n ),, for where V h (n ) DV h (n ) ( D ) z h (n ) Ph (n ) qˆ h (n ) V qˆ h (n ) X Vh qˆ h (n ) (4) E qˆ h (n ) ( E ) Ph (n ) he fully update equaton for VSSWDLS can be expressed as qˆ h (n ) qˆ h (n ) V X V h g h (n ) (5) g (n ) C(n )z (n ) z h (n ) V h (n ) g (n ) g h (n ) Ph (n ) z h (n ) V h (n ) end end Where Ch (n ) Ch (n ) Ch (n ) ¹ C(n ) Ph ( n ) (6) and Ch (n ) V ( n ).I s the u matrx. able summarzes the procedure of the VSSWDLS adaptve algorthm. h 5. Smulaton Results We demonstrated the performance of the proposed algorthm by several computer smulatons n a system dentfcaton scenaro. he unknown mpulse response s randomly selected wth 6 taps (=6). he nput sgnal s an AR() sgnal generated by passng a zeromean whte Gaussan nose through a frstorder system H (z ).9 z. An addtve whte Gaussan nose was added to the system output, settng the sgnaltonose rato (SR) to 3 db. he Haar wavelet transform (HW) was used n all smulatons whch leads to the reducton of computatonal complexty due to the elements (+ and ) n HW. he values of α and β were set to.995 and.9 respectvely. 4. Computatonal Complexty able descrbes the computatonal complexty of the proposed VSSWDLS algorthm. he number of multplcatons and dvsons have been calculated for each terms. In the followng, able 3 compares the computatonal complexty of varous VSSDLS algorthms. hese algorthms are from [7, [, [ and [4. In ths able, s the number of flter coeffcents, s the num Iranan Journal of Electrcal & Electronc Engneerng, Vol. 3, o. 3, September 7 5

4 [ DOI:.68/IJEEE able. he Computatonal Complexty of VSSWDLS Algorthm n subband Equaton u z (n ) x(n ) d (n ) g (n )z (n ) V h (n ) DV h (n ) ( D ) z h (n ) qˆ h (n ) z h (n ) V h (n ) P h (n ) qˆ h (n ) u y E qˆ h (n ) ( E ) y Downloaded from jeee.ust.ac.r at 3:44 IRS on Wednesday ovember 4th 8 otal V qˆ h (n ) X Vh g h (n ) g h (n ) P h (n ) z h (n ) V h (n ) otal ultplcatons : 6 5 able 3. he Computatonal Complexty of Varous VSSWLS Algorthm DCLS [7 VSSDLS [ ultplcatons Dvsons ( t +4) 5 L VSSDLS [ 8 8 VSSDLS [4 5 8 VSSWDLS VSSWDLS (HW) able 4. he Parameters In VSSDLS and VSSWDLS Algorthms. DCLS [7 VSSDLS [ 8 u 3 E.9985 J t D.99 E.9 J L μmax 5 u μmn VSSDLS [.98 J 4.7 u 3 VSSDLS [4 D E 3.98 μmax.995 E.5 C.9 J.9 uvx VSSWDLS D.995 E.9 Fgs. 4 show the mean square devaton (SD) learnng curves of proposed VSSWDLS and ordnary WDLS algorthm for dfferent values of. In WDLS, dfferent values for the stepsze have been selected. We observe that VSSWDLS has faster convergence speed and lower steadystate error than ordnary WDLS algorthm for all values of. he comparson of VSSWDLS wth recently and famous VSSDLS algorthms has been presented n Fg. 5 [7, [, [ and [4. he parameters of the smulated algorthms have been chosen accordng to the able 4. hs fgure shows that, the proposed VSSWDLS has better convergence speed an lower steadystate error than other VSSDLS algorthms for all values of. Also, the computatonal complexty of the proposed algorthm s lower than other 6 Fg.. he SD learnng curves of WDLS and VSSWDLS algorthms wth =. Iranan Journal of Electrcal & Electronc Engneerng, Vol. 3, o. 3, September 7

5 [ DOI:.68/IJEEE Downloaded from jeee.ust.ac.r at 3:44 IRS on Wednesday ovember 4th 8 Fg. 3. he SD learnng curves of WDLS and VSSWDLS algorthms wth = 4. Fg. 4. he SD learnng curves of WDLS and VSSWDLS algorthms wth = 8. Fg. 5. he SD learnng curves of varous VSSDLS and VSSWDLS algorthms. Fg. 6. Varaton of the stepsze n each subband durng the adaptaton for VSSWDLS. VSSDLS algorthms due to usng HW. Fg. 6 presents the varaton of the stepsze n each subband for VSSWDLS algorthm durng the adaptaton. As we can see, the stepszes start from large values and end wth low values for =, 4, and 8. Fnally, Fg. 7 shows the number of flter coeffcents versus the flter length for VSSDLS, WDLS, and VSSWDLS algorthms. hs fgure ndcates that the computatonal complexty of VSSWDLS wth HW s sgnfcantly lower than other algorthms. Fg. 7. he number of flter coeffcents versus the flter length () n varous VSSDLS, WDLS, and VSSWDLS algorthms. 6. Concluson In ths paper, the WDLS wth VSS was establshed. he stepsze n each subband changes accordng to the largest decrease n mean square devaton. he smulaton results ndcated that the proposed VSSWDLS had Iranan Journal of Electrcal & Electronc Engneerng, Vol. 3, o. 3, September 7 7

6 [ DOI:.68/IJEEE Downloaded from jeee.ust.ac.r at 3:44 IRS on Wednesday ovember 4th 8 faster convergence speed and lower steadystate error than ordnary WDLS and other VSSDLS algorthms. Also, the computatonal complexty of VSSWDLS wth HW was sgnfcantly lower than other algorthms. of SubbandCoeffcents." Dgtal Sgnal Processng, vol. 69, pp. 945, Oct. 7. References [ B. Wdrow and S. D. Stearns, "Adaptve Sgnal Processng". Englewood Clffs, J: PrentceHall, 985. [ S. Haykn, Adaptve Flter heory. J: PrentceHall, 4th edton,. [3 B. FarhangBoroujeny, Adaptve Flters: heory and Applcatons. Wley, 998. [4 A. H. Sayed, Fundamentals of Adaptve Flterng. Wley, 3. [5 A. H. Sayed, Adaptve Flters, Wley, 8. [6 S. S. arayan, A.. Peterson, and. J. arashma, ransform doman LS algorthm, IEEE rans. Acoust., Speech, Sgnal Processng, vol. ASSP3, pp , 983. [7 D. I. Km and P. D. Wlde, Performance analyss of the DCLS adaptve flterng algorthm, Sgnal Processng, vol. 8, pp , Aug.. [8 S. Zhao, Z. an, S. Khoo, and H. Wu, Stablty and convergence analyss of transformdoman LS adaptve flters wth secondorder autoregressve process, IEEE rans. Sgnal Processng, vol. 57, pp. 9 3, Jan. 9. [9 S. Attallah, he wavelet transformdoman LS algorthm: a more practcal approach, IEEE rans. Crcuts, Syst. II: Analog and Dgtal Sgnal Processng, vol. 47, no. 3, pp. 9 3, ar.. [ S. Attallah, he wavelet transformdoman LS adaptve flter wth partal subbandcoeffcent updatng, IEEE rans. Crcuts, Syst. II: EXPRESS BRIEFS, vol. 53, no., pp. 8, Jan. 6. [ R. Blcu, P. Kuosmnen, and K. Egazaran, A transform doman LS adaptve flter wth varable step sze, IEEE Sgnal Processng Letters, vol., pp. 5 53, Feb.. [ K. ayyas, A transform doman LS algorthm wth an adaptve step sze equaton, n Proc. ISSPI, Rome, Italy, 4, pp [3 H. C. Shn, A. H. Sayed, and W. J. Song, Varable stepsze LS and affne projecton algorthms, IEEE Sgnal Processng Letters, vol., pp. 3 35, Feb. 4. [4 S. Zhao, D. L. Jones, S. Khoo, and Z. an, ew varable stepszes mnmzng meansquare devaton for the LStype algorthms, Crcuts, Systems, and Sgnal Processng, vol. 33, pp. 5 65, 4. [5. Shams Esfand Abad, H. esgaran, and S.. Khademyan. "he Wavelet ransformdoman LS Adaptve Flter Employng Dynamc Selecton. Shams Esfand Abad receved the B.S. degree n Electrcal Engneerng from azandaran Unversty, azandaran, Iran and the.s. degree n Electrcal Engneerng from arbat odarres Unversty, ehran, Iran n and, respectvely, and the Ph.D. degree n Bomedcal Engneerng from arbat odarres Unversty, ehran, Iran n 7. Snce 4 he has been wth the Department of Electrcal Engneerng, Shahd Rajaee Unversty, ehran, Iran, where he s currently an assocate professor. Hs research nterests nclude dgtal flter theory, adaptve dstrbuted networks, and adaptve sgnal processng algorthms. 8 H. esgaran receved the.sc. and Ph.D. degrees n appled mathematcs from Iran Unversty of Scence and echnology, ehran, Iran, n 996 and, respectvely. He s an Assocate Professor wth the Department of athematcs at the Shahd Rajaee eacher ranng Unversty, ehran, Iran. He joned the Department n. Hs research nterests nclude Integral equatons and Wavelet functons. S.. Khademyan receved the.sc. degree n appled mathematcs from Iran Unversty of Scence and echnology, ehran, Iran, n. Currently, he s a Ph.D. canddate at the Department of athematcs, Faculty of Scence, Shahd Rajaee eacher ranng Unversty, ehran, Iran. Hs research nterests nclude dgtal flter theory and adaptve sgnal processng algorthms. Iranan Journal of Electrcal & Electronc Engneerng, Vol. 3, o. 3, September 7