Variable Step LMS Algorithm Using the Accumulated Instantaneous Error Concept

Transcription

1 Proceedgs of the World Cogress o Egeerg 2008 Vol I WCE 2008, July 2-4, 2008, Lodo, U.K. Varable Step LMS Algorthm Usg the Accumulated Istataeous Error Cocept Khaled F. Abusalem, Studet Member, IEEE. Yu Gog, Member, IEEE. Abstract The covergece speed of the stadard Least Mea Square adaptve array may be degraded moble commucato evromets. Dfferet covetoal varable step sze LMS algorthms were proposed to ehace the covergece speed whle matag low steady state error. I ths paper, a ew varable step LMS algorthm usg the accumulated stataeous error cocept s proposed. I the proposed algorthm, the accumulated stataeous error s used to update the step sze parameter of stadard LMS s vared. Smulato results show that the proposed algorthm s smpler ad yelds better performace tha covetoal varable step LMS. Idex Terms Adaptve flters, adaptve array, varable step LMS, movg object trackg. I. INTRODUCTION Adaptve beam former play a mportat rule radar, soar, speech processg ad, more recetly, moble wreless commucatos. It s desred to have a fast coverget adaptve atea wth good trackg capabltes of desred ad terferg sgals. Ths s to mprove the user capacty for the base statos ad the moble hadset wreless commucato system. Potetal performace mprovemets for cludg terferece reducto,of movg sources, through adaptve beam formg motvates the developmet of fast coverget LMS algorthm. As oe may kow, stadard LMS s the most lkely searchg adaptve algorthm due to ts smplcty, stablty ad performace prospertes. As a result may LMS based algorthms have bee developed amg to mprove the covergece characterstcs of the stadard LMS. I the stadard LMS, the step sze s fxed ad the flter weghts are updated accordg to: wˆ ( + ) = wˆ ( ) + μu( )[ d * = wˆ ( ) + μu( ) e ( ) () * ( ) u ( ) wˆ( )] It ca be show that the value of the step sze parameter s fxed ad govered by: Mauscrpt receved March 20, Yu Gog, s wth the Departmet of Electrcal Egeerg, Uversty of Readg, Readg UK (y.gog@readg.ac.uk) Khaled F. Abusalem s wth the Departmet of Electrcal Egeerg, Uversty of Readg, Readg UK (k.f.abusalem@readg.ac.uk). (2) 0 < μ < λmax λ max Where s the maxmum egevalue of the uderlyg correlato matrx R. The stadard LMS was smple, both the umber of calculatos requred for ts update ad ts dervato from the method of steepest descet. Moreover t was robust a umber of applcatos. The adaptve feedback costat µ the LMS cotrols the covergece rate of the flter coeffcets, addto to determato of the fal excess error. Sce the covergece tme s versely proportoal to µ, a large µ for fast covergece trackg applcatos s always selected. owever, large step sze wll result creased msadjust met. ece, a set of LMS based algorthms kow as varable step sze algorthms were proposed to overcome ths problem. I these algorthms the step sze of the LMS s vared usg dfferet approaches. Amog these varable step sze LMS algorthms are the covex combatos of adaptve flters. Recetly,. Sayed ad others CONVE COMBINATIONS OF ADAPTIVE FILTERS, they used two depedet flters wth large ad small step sze. I ther approach they used a mxg parameter to scale the output of both flters order to combe advatages of both flters. The draw back of ths algorthm s that they used two flters workg parallel. Moreover the MSE the swtch over from the hgh step sze to the small step sze MSE o smooth way. I ther approach, the output of the flter s gve by: Where ad are the outputs of two trasversal flters at tme. The dea s that f s assged approprate values at each terato, the the above combato wll extract the best propertes of flters ad. Ths algorthm, however, troduce computatoal complexty as two dfferet flters are used. Moreover, the mxg parameter, amely λ, s a fucto of the mea square error what meas more flters eed to work (3)

2 Proceedgs of the World Cogress o Egeerg 2008 Vol I WCE 2008, July 2-4, 2008, Lodo, U.K. parallel to produce the mea value. I addto, the MSE does ot coverge smoothly to ts fal steady state value. I ths page, a ew algorthm called accumulated stataeous error drve LMS or AIED LMS s troduced whch s utlzg the accumulated stataeous error to cotrol the step sze µ. The way whch µ s chagg depeds oly o the accumulated error. The advatage of usg accumulated stataeous error s that less complexty as oly oe flter s requred. I addto, the trasto from larger step sze to smaller oe s takg place smoothly, as wll be see from the results. The AIED LMs provdes good covergece characterstcs wth less complexty. It out performs stadard LMS as well as covex combatos flters. The fal secto of ths paper show smulato results for applcato of the algorthm beam former. II. TE AIED LMS ALGORITM The step sze flueces two mportat parameters, amely the MMSE (steady state behavor) ad the covergece speed (traset behavor). As we have see the step sze s drectly proportoal to the covergece speed. owever t s versely proportoal to the MMSE. What makes the compromse process dffcult. To overcome ths problem, oe ca start wth large step sze, to ehace the covergece speed, ad gradually ( jumpg steps) reduce t to atta ts mmum value, to acheve desrable MMSE. Oe should ote that the step sze should vary wth the stablty boudares. The way whch the step sze s vared s very mportat. To acheve best performace the step sze should jump to the ext, smaller step, at the rght momet. avg these jumps at the rght momets wll make the MSE coverge smoothly ad fast to the MMSE value. So the key factor for achevg ths behavor s the approach of selectg the break pot for the MSE curve. By the break pot we mea the pot at whch the MSE curve starts covergg to ts steady state, specfcally the ed of the traset porto of the curve as oe ca see from fgure. Smaller step sze wll result a curve havg ts break pot shfted from that of the larger step sze, as oe ca see from the fgure above. So oe ca start wth large step szes ad chage over to the ext, predetermed, step sze at the stat whe the MSE curve start bedg (break pot). Ufortuately dog the trasto maually s ot practcal way of dog the trasto from oe step sze to aother. The MSE gradet s a good measure for the break pot. Actually the gradet wll be close to zero as the curve starts covergg to the steady state values. Moreover the MSE s decreasg fucto of tme ad t fluctuates hghly, due to the stochastc ature of the adaptve flter. Smoother MSE ca be acheved but t eeds esemble averagg ad as a result more complexty as parallel flters are requred. Fg. MSE for 6 Elemets Lear Array for Dfferet Step Sze Values Fg. 2 AIE for 6 Elemets Lear Array for fxed stp sze To overcome ths problem we used the accumulated stataeous error curve to detfy the break pot for smooth trasto. The accumulated stataeous meas square error, abbrevated AISE here after, s always creasg fucto of tme wth mmum fluctuato as oe ca see from fgure 2. It ca bee see easly that the AISE curve behave more tdy way tha the MSE. It evolves smoothly wth small fluctuato to ts steady state value. Oe ca dfferetate betwee two ma segmets of the AISE curve. The frst porto s the curved le where the gradet of the SISE starts from hgh values ad coverge to fxed steady state value. The other segmet of the curve s lear at whch the dervatve s fxed. It s easer to mapulate the AISE tha the stadard MSE. Ths s due to the fact that the AISE has defte break pot whch ca be detfed easly. Actually the break pot s the pot where the curve takes ts lear behavor as reflected from fgure 2.

3 Proceedgs of the World Cogress o Egeerg 2008 Vol I WCE 2008, July 2-4, 2008, Lodo, U.K. Fg. 3 Predcto Error of AISE for 6 Elemets Lear Array To specfy ths break pot we used a predctor. The predctor s a good tool for dfferetatg betwee curved ad lear portos of the same curve. Ths s due to the fact that the predcto error wll almost zero whe the predctor work the lear segmet of the curve. The predcto error for the predctor s show fgure 3. Ths fgure shows that the predcto error for the AISE s eve better to use to detfy the break pot for the trasto from oe step sze to aother smaller oe. Actually the predcto error curve has two ma lear segmets wth clear easy to fd break pot. Ths break pot ca be detfed by troducg a threshold such that whe the predcto error decrease behd ths threshold the algorthm wll automatcally trasfer to the ext smaller step sze. A mathematcal terpretato for the approach wll be preseted the ext few paragraphs. III. MATEMATICAL FORMULATION The optmum flter weght vector w opt, Weer optmum vector, s gve by: wmmse = wopt = Rxx rxr (4) Where R s the put correlato matrx ad r s the cross correlato vector betwee the desred respose ad the flter put sgal. Let the dfferece betwee the flter weght at tme ad the optmal flter weght be gve by: v = w wopt (5) The t ca be show that: v + = ( I 2μR) v (6) Sce R s postve defte, t has postve egvalues. As a result t ca be decomposed to a orthogoal matrx Q ad a egevalue matrx Λ as follows: R = Q ΛQ (7) Where Λ = dagoal λ, λ,..., λ ) (8) ( 0 M λm Where s the mth egevalue of R. Usg equatos above, t ca be show that the above equato ca be wrtte as: ' v = ( I 2μ Λ) v (9) ' v 0 Where s a rotated verso of by Q.A beam former satsfyg ths equato s stable ad coverget provded that the step sze s wth boudares gve below: 0 p μ p (0) λmax λ Where max s the largest egevalue of the correlato matrx R. The accumulated stataeous mea squared error s the summato of dvdual stataeous squared error values for dfferet tme stats. The AISE s gve by: AISM ( ) vk 2 = = ( e( )) () The above fucto s a creasg fucto of. The fucto has a terestg characterstc over the MSE fucto. It ca bee see easly that the AISE curve behave more tdy way tha the MSE. It evolves smoothly wth small fluctuato to ts steady state value. Oe ca dfferetate betwee two ma segmets of the AISE curve. The frst porto s the curved le where the gradet of the SISE starts from hgh values ad coverge to fxed steady state value. The other segmet of the curve s lear at whch the dervatve s fxed. It s easer to mapulate the AISE tha the stadard MSE. Ths s due to the fact that the AISE has defte break pt whch ca be detfed easly. Actually the break pot s the pot where the curve takes ts lear behavor as reflected from fgure 4 above. To specfy ths break pot we used a predctor. The predctor s a good tool for dfferetatg betwee curved ad lear portos of the same curve. Ths s due to the fact that the predcto error wll almost be zero whe the predctor works the lear segmet of the curve. The predcto error for the predctor s show fgure 3. The predctor s a LMS flter ts ow ad has a pre determed order. The order of the predctor should be selected carefully to ft the job perfectly. It s better to keep the order as small as possble to mprove the covergece propretes havg maxmum of three taps. I the proposed STVS-LMS algorthm, at every terato tme the predcto error, of the AISE predctor, s compared to a predetermed threshold. If the error s less tha the threshold, the step sze s chaged to a smaller oe. The process wll proceed utl the mmum step sze s acheved. The proposed STVS-LMS s summarzed below: Proposed STVS_LMS Algorthm a. Italzato

4 Proceedgs of the World Cogress o Egeerg 2008 Vol I WCE 2008, July 2-4, 2008, Lodo, U.K.. Decde order of the beam former M. Decde tal step sze μ for =, the step sze decremet ad boudares. Decde the legth ( M ) of the predctor ad ts p step sze ( μ ) v. Decde N, adopto course terato umbers v. Decde the threshold for the smooth trasto b. For =,2,3,..N. Fd stataeous squared error, e = d p W. Fd accumulated stataeous squared error, AISM ( ) = ( e( )) =. Fd predcto error for the AISM (PEAISM) v. If PEAISM s less tha threshold decremet the step sze v. Fd ew step sze v. v. e W = d + = W W + μ e IV. SIMULATION RESULTS AND DISCUSSION The smulated results for lear array wll be preseted ths secto. The array parameters, ter-elemet separato ad umber of elemets, wll be fxed durg the beam formg adopto course except for complex weghts, ampltude ad phase, of dvdual elemets. The umber of elemets wll be sx ad the ter-elemet separato wll be fxed at dx = 0.5λ, where lambda s the wave legth of the operatg frequecy. The proposed AIED-LMS algorthm s used to drve/steer the lear beam formers. The algorthm parameters wll be determed for each scearo. These parameters clude the step sze decremet, predctor order, predctor step sze, threshold amog others wll be gve as the talzato parameters for the algorthm at the begg of the adopto course. I each scearo the umber of terferers, co-chael terferece, wll be vared. Usually the umber of terferers wll be below that of the atea/beam-former elemets. The drecto of arrval of dfferet terferers ad target wll be vared too. Moreover dfferet combato of target sgal to terferece ad desred sgal to ose ratos are appled to dfferet scearos. The performace of the beam former ad as a result the algorthm drvg t wll be tested for these scearos wth 2 may varables. Specfcally the speed of covergece of the steerg process for the ma beam as well as ulls wll be llustrated by performace measurg dces. These performace dces are maly the mea square error ad the array factor. Accumulated error ad weght s magtude wll be used whe requred. I the smulato, a bary phased shft keyg (BPSK) modulato scheme wth a ut eergy pulse was employed. The chael s assumed to be a AWGN chael. All dsplayed results have bee averaged over two hudred depedet rus. I what follows, the proposed AIED algorthm performace wll be hghlghted vew of the below scearo. Smulato was ru for a sx elemets lear array wth ter-elemets separato of half wave legth. The chael s AWGN wth SNR rato of 0 db ad 0dB SIR. Both the desred ad uwated terferers are bary phase shft keyg (BPSK) modulated wth uty power. The agle of arrval for the desred sgal s θ = 0, whle that of the uwated terferer s located at θ = 35. The stadard LMS was ru for two dfferet values of step sze, μ, whch are.008 ad.00. O the other had a value whch betwee to.00 was selected for the AIED-LMS algorthms. The expermet has bee ru a two hudred depedet tmes. At frst t s mportat to see how the AIED-LMS out performs stadard LMS as well as Covex Combatos ull steerg capabltes. Fgure 4 shows the MSE for AIED-LMS ad that of the stadard LMS for dfferet step sze values. As oe ca see the STVS coverge faster wth mmum steady state error. Actually the proposed algorthms combe covergece speed characterstcs of LMS wth large step sze ad at the same tme steady state characterstcs of the small step sze LMS. Smlarly fgure 5 shows the covex combatos versus the stadard LMS for the same scearo. Comparg both fgures oe ca easly see that AIED-LMS out performs both stadard LMS as well as Covex Combatos. It coverges fast ad smoothly to ts steady state values V. CONCLUSION The LMs s a smple ad robust adaptve algorthm ad has bee used varety of applcatos. Recetly, more advace versos of the LMS have gve sgfcat mprovemets covergece prospertes. owever these algorthms troduced complexty to meet a satsfactory performace. A ew algorthm, the AIED LMS, troduced the cocept of accumulated stataeous error to cotrol the step sze parameter. Based o smulato results for adaptve beam former, t s apparet that the AIED LMS s fast coverget algorthm ad outperforms both stadard as well as covex LMS algorthms. Ths ca be see easly form fgures 4 & 5 Moreover the structure of the AIED s les complex tha that of covex where oly oe flter s requred.

5 Proceedgs of the World Cogress o Egeerg 2008 Vol I WCE 2008, July 2-4, 2008, Lodo, U.K. Fg. 4 AIED versus LMS (µ=.00 ad µ=.008) Fg. 5 Covex combato versus LMS (µ=.00 ad µ=.008) REFRENCES [] S. ayk, Adaptve Flter Theory, Pretce all, fourth edto, 200. [2] C. A. Balas, Atea Theory Aalyss ad Desg, Joh Wley, New York, secod edto, 997. [3] A.. Sayed steady-state Performace of Covex Combato of adaptve Flters, Acoustcs, Speech, ad Sgal Processg, Proceedgs. (ICASSP apos;05). IEEE Iteratoal Coferece o Volume 4, Issue, 8-23 March 2005 Page(s): v/33 - v/36 Vol. 4