Nonlinear RLS algorithm for amplitude estimation in class A noise

J.F.Weng and S.H.Leung

Ahstracf: An ;id;iptivc nonliiic;ir recursivc lcest .sr]u;ire (RLS) afyritlrin for ariiplilude cslinxiti~in in cl;iss A noise i s pi-asctiicd. FoI G;iussian input signal aid class A iioisc, i l s iiicaii and iiicxii-sqiiitre k1iavioui.s tiic slutlied. 11 is sliowii that thc lincar. RL.S :md non1inc;ir RLS algotiilitn wiih t h e clippx I'unction i i ic stahlc in llic iiieiiii aiid iiieaii square. For nnii-Gaussiari inpug iunplitudc cstiiiii1tion hi CDMA coininiiiijC:itiiin is prcscntcd. Simulation rcsiilh h o w that the nonlinear. RES can Imvidc good lxl-Yurmuiicc: closc to tlic Crmicr-Rao boiirid and outpcrlbl-in tlic nonlinear LMS and thc cnnvcntional RLS in iinpiilse noisc.

1 introduction

Adaptive: filtering tias hccn wiclcly used in system idcntifica- tion, modcil itig, prediction, ii lid in Icrfctcrm canccllation [I ] . In codc-division intillipfe-scccss (CDMA) communica- tions it idso plays ;in iinportarit rolc iii esliinatiIig (tic sys- tem pwraiiictei-s that arc csscnhl in many adaplivc iiiultiuser dckctovs [2- 41. In pitrticid;irq by consirlering ttie ijliiplittldc cstimalion pi-obIern in synchronous CDMA S ~ S - tem, (he rcccivcd sigi~il y(n) c m be written as 141

Wc consider cqiiipping a rioiilincarity iiito the I< J..S illgo- rithiii for thc purposc of achicvirig wpici coi~vergciicc and robusliicss against impulse noisc. l 'hc rcsultiiig algori~l~i i~ is iclcrred lo as iionliiiear RLS algoritlm. To gain insight into the pcrforinnncc thc iiic;iii and iricnii-sq tirirc Ixhav- iours of tlic nonlinc;ir ItLS in inipiilse iioise wrc anidysed ~ O I - Griiissian iiipiil sigriiil and class A iioisc. Next wc con- sidcr B iiou-Gaussi;iii piublcni in rvliicli liie algorithm is applied LO thc ainplitudc cstiinnlioii iii CDMA system. 'To nic;.isurc the perroriixincc thc Cramcr- I h o bound (CRB) in impiilsc noise is dciived. Simulation results show t1i:~L t t le iiotilinear l iJ,S h a s lhc peilbrtiiancc close to CRB.

2 Nonlinear RLS algorithm

In thc linear I< 1.S algoritlitii, t h ulxlnlc of tlic weight vcc- lor W ( i ) is described by

W(n + 1 ) = W(n) + e ( n ) X : ( n ) (a wlicre thc crror sign;il ~ ( I z ) arid tlic Kalmaii gaiii vcclor h(n) itre given by

The impulsive nature of q(n) cdn bc characterised hg thc class A noisc model with thc p.d.f. defined as [Y]

03 An1 S,!.) = CA ~ f & ) (6)

na=O

wherc A is h e impulsc itidcx,f;,{x) is takcn to be ii normal p.d.f. with xro mean and variance given by

( 7 ) = 0: -tmo, 2

whcre qf and are thc variances of thc puicly Gaussian and impulsive noises, tespcctivcly. The ratio of the powct in the rioininal Gaiissian camponenl to that in thc impul- sive component i s dclined as r‘ and it can be showii that F’

I t is wcll known that the RLS algorithm could provide R fiiskr coiivergence ratc tlran the LMS algorithm in most applications. Howevcr, when thc systcin is contatniiiated by the iinpulsc iioisc, thc performaim c m bc shown dcterio- mtcd. To ovcrcanic thc wcakness, nonlinearity can be iipplied to the error signal. ‘l’hc rcsulting algorithm is refcrreil to as nonlinear RLS, of which the wcight vector is updawcd by

W(-n + 1) = W(~Z) + g{e(n))k(n) (8) whcre g(x) is an odd syniinelric noiilincatity with positive slopc and finite output. The dgorithm is rcferrcd to as non- linear LMS [XI when k(n) is replaced by @(ti) in cqn. 8.

In choosing an appropriate nonlinearity, consider t l c model of the cr~or signal e(.). By defining the weight error voctor V(n) as

= u$IA$.

V(7L) = W(n , ) - w , (9) and tlieii substituting cqns. 9 and 5 inlo eqn. 3ri, the crror signal c(n) can hc written as

C ( 1 B ) = q ( n ) - VT(n)X(n) (10) For the weight update in tlic RLS algorithm it is cnsential to preserve thc misacljustrnent signal, i.e. F’I“’()X(n), espc- ckdlly ilfkr piissing through the iionlineaiity cor compress- ing the large impulse noise. According to this view, thc smooth-limiting nonlinearity of ititercst can be described by the clipper function

Tu IC > To g{x} = 3: 121 5 T, (11) { -?; x < -To

Evidently, whcn To is large cnough, gi.) reduces to I and thus ihc nonlinear RLS simply becomes the linear RLS.

3

We make the following assumptions: {X(n)) are iudepcnci- cnt Gaussian vcctors with Yero mean nnd camlation R, {q(n)rz)l are IlD with thc impulse nnisc niodcl dcfined in cqii. 6 and X(n) atid q(n) arc stalisticidy independent, SO

are X(r!> and &), ‘To facilihte oiir analysis, we subslitute cqns. 9 wnd 3

into cqn 8 which yieids

Analysis of nonlinear RLS algorithm

(12) Vurlhcr, to yield a niathcinaticsllly kactable analysis, we assume that thc input signal vcctor {X(i)) is ergodic, the matrix R an bc approximated by the time average funnula

82

whcre p(n) = 2;~” r r ~ -i is ihe normahation faclor and it equals I I + 1 For w = 1. Accordingly, P(n) c m be approxi- mated to be p-‘(n)R-’ nnd eqn. 12 bccoiiics

whcre

i=fl

The iipdakc c q d o n of eqn. 14 is velid for sufficicntly large iz. Tn the Iirllowing analysis WO apply this cquation to study thc inem itnd mean-square behaviours of thc lionlinear KLS ,

3.1 Mean behaviour Xiking expectitions on bolli sidcs of cqn. 14 yiclds

E{V(rz -1- l)} E { V ( ? $ }

In evaluating h e expcctaiion or thc second term on die righi-himd si& of cqn. 16 we adopi tlie ;ippro:oachehes simi- larly as in [lo]. It is shown in [4] (appcndix 3A) that

.wlicre /I, is given by m

A, := +~(Tt , ,ue , , Jc -“il”’//m! ( I X b ) “=O

v,”,,~ := o;~~ , + t7-[~(~j,jn] ( 1 8 C ) with 42 given by c y . 7 , @I) = E{ V(n)V?’(n)) , and S = (I + Zfi)-]R. The hiiciion 3,(7;,, cr) is ticfincd in cqn. 19 and the rcsull for thc cliplxr ilmctiun in cqn. 11 is cri‘(T,, rr) and thc liiieat funclion is I , [Note I ]

1 xa/2 ,3 CO

+,((T,,o) 1 fJ’{x}-e- d r (.I 9) . -%, &o

where the prime denoks tlic first derivative of thc runclion and in thc sequel. Sirhstihting cqo. 17 into cqn. 16 yiclds

J < { V ( ~ ~ + 1) j E (1 - X,)i~{v(~~)} (20) Using thc ~ / J ] ( T ~ , cr) of clippi. and linear functions, it is stmiglitfoward to show that 0 A , < 1 i s truc for the RLS and iioiiliticar KLS. Hence, the wcight vector con- verges in thc mean.

3.2 Meansquare behaviour Post-multiplying eqn. 14 by its Iransposc md takltig expec- tations c m both sicks, thc correlation inat.rix of the wcight

Note I : ufl.) is the error liiucticm dcfinrrl iis e&) = dU.Jb” exH- r2i21r/f.

1I:k’ Pfor.-Corwiuir.. P’d I<?, A%. 2. Apld 201KJ

error !+I}, clcnolcd by K(Pz), is imirsively cicscl-ibcd hy

n-' X( rr)g { i! ( ra ) } vT (72)

3 1 1 + X''>(?l) R-- lX(n) K(,n+l ) 2 K(,IL) + E

(21.) As in the derivation nf cqn. 17, it can be readily siiown ha^

I dn, + X"'(??>)R .1X(?1) -1 {

n-' x ( I ! . ) g { c (71:) ] v?' ( 1 1 )

w.,, 4- x'!' ( I L ) R- 1 X(.!l,)

rq ,TA) 9 { c: (n ) } X" ' ( ? I ) R- '

(22)

= = E

= - i lnK( ' ra) -

Moreover, il caii hc showti that (see [4], appcntlix 3Ii)

ik3( T,,, a): clipper fmctioii is

tind linear liiuction is 2. Suhslitutiiig cqiis. 22 and 23 inlo cqn. 21 givcs

~ ( 7 1 . + 1) z (1 ~ 5io + &)IC($ + Z 1 ~ - ' -

+ A2tr{K(71.)R}R-i ( 2 8 ) Post-midriplying R on both sides of eqn. 28 :ind dcliniiig An) = tifK(n)R] obtains

;y(rr. + 1) 2 (1 - 2 x 0 + :jTZ2 +Z:<)x(n) 4- iYAl (29) It is shown iu ihc Appcndix Ilia1 0 < 2/1, ~ NA, - A, < 2 is true hor the RLS and nonliiicar RLS. IIcncc, the weight vtxtor converges in thc iiicm-squmc

111 cq11. 1 1 , tllc pilfiIIi1eter 'tiJ is w e d to contd the robuslncss aga?ins1 iriipulsc tioisc. '1'0 obtain a fiist coiivcr- gcncc 121c and a good performance, ?;, can bc sct IO T:,ir,,wheic T:, is a constiin1 and Er: is thc estimatc 01' (4) (ttie varieticc of thc crIcir signill), wliicli cm IE ohtained by

ii," := n.;i: + (:I ~ .).'(.n,) (330) where M is thc weightitig fitclor. I-lcrc wc considcr setting 'I,:) lo the asyiiiplolically optiinum T,,, which iiiaxiiiiiscs thc so-~tllcd nornxllisccl crficacy dcfiiicdhy [ 121

t = [ , ~ . ~ f ~ . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ] '2 // p { x ) j < ( x ) d 3 : - .ocI

(31) whcrcj$x) is Ihc noisc p.d.f. in eqn. 6 but wilti unii vari- mcc. In so doiiig, we expect to imprgvc thc krsymptotic OLII~LII signal-to-noise ratio of the err& signal whcn it is

3.3 Experimental results 'l'lic perform:iuce of the nonlincai- RI ,S (N-RLS) is ewilii- ateri and wmp;ircd with (hc R I 3 in dcaliirg with syslcin idcntificat ioii problcm.

The slriicture of Ihc syslciii idciililicndnn i s shown in [ I ] (b'ig. 16.4, pp. 601). In Ihc cxpwinicrd 1lic impulse re.sponsc or l l ic unknown lincar systerii is set t o W,, [0.2, -0.4, 0.6, -0.8, 1.0, - 0.8, 0.6, -0.4, 0.2 1'': Thc input signnl to thc sys- tem W,, is obtained by p;issing ii 7 m ~ i i c i 1 n w h i t Ciaussim tioisc {K,;} wilh Ihc variancc i$ fhrwgh a hear filter with the imprilsc msponsc,fi = :[I + cos(k(/c -. 2)/4] for k = I, 2, 3 ;ind m * o ohcrwisc. I-Icrc, I I = 3.1 is considercci and the cor-rcspoudirig eigenv:iluc sprcid of thc K tnatrix i s approx- imtely 1 I . Thc signal l o nclisc ratio (SNR) is defitietl by 0 ~ / 2 c r ; f . I n thc cxpcriincnl wc Ict (6' Ix nori~iaIiserI to 1. 'I'he initi;il wluc of Ihc wcighl vcclors in lhc algorithms iirc set cqual to ZCI'O. Tlic Ibrgctting fxtor. 11) is set to 0.995, while the valuc or T, is sct to To,,#,. with TUC, tieing Ihc optiiriiil \liiluc whicli niaximises the efficacy in eqn. 31 and ir,2 is estimatcd according LO cip. 30 with thc initial value 10.0 atitl thc wcighling raclar CI = U.99. The simulation I c S l i l L ? . arc avcr-agcd over 200 l-lltls.

First the tneati convcrgciicc or thc N-RIS, evaluated via eqn. 20, i.il SNR = h.5dB iii thc class A noisc of A = 0.01 :incl 1" 1.0 is plottcd in I';ig. 1 and cuinpitiwl wid1 thc simul;ltinn rcsult. Also, the results of thc RLS arc plotted. In Lhc Figurc thc nicaii of VS(u), which is the fif'lh (dami- ti:inl) clciiicnl or Y(n), is plotted. It ciiii be secn IhaI both tlic N-R1.S aid tlic IZLS have rapid convergcncc in the incaii and they arc sliowii to bc asyiiiplolically uiibiased.

in6 tliIougli tlic nonliiitiii. eIcincn1 [4, 1 I].

H I

(i) Noniincw R LS d g o d l 7 1 ~ The ti Igorilhin is dcscribcd by cqns. 8, 3 tinti 4. (ii) I,kcrrr RLS dgoritlmr: I n 11ic liriuar RLS, tliu updates of error, K~ilmen gain vector, and inverse of the corrclation i11iktrix are the biiiiil: ;IS lhosc in Ihc nonl i~ica~~ K L S , while lhc wcight vcclor i s updatccl by cqn. 2. (ii} Nodinair LMS dgor, i !h: Thc wciglit updiite in the iionlincar LMS is simply to tcplace k(n) in cqn. 8 by pX(n), w l w e ,U is thc stepsize. (iv) M~~,~I.c!/iL‘s:I.Jiltct: By following the theorcm I in [6], the iipdatc equalions in Mnsreliea’s filler for amplitude estima- tion can he ohtaincd straiglit~~rwardly, of which the noisc dcnsity is defiiiccl in cqn. 6. Tlic details of this algorithm c~in bc round in [4] and oini~td Iicrc for ltic siiltc 01’ brcv- ily.

To measure llic pcr-Folm:uice of these algorithms for amplitude est.itn;itiou, thc Craiiicr-Rao bound (CRn) i s ol grewt interest. For tiny unbiased cstilnate of W :: [6i, ?Yr. ___, OAl)”, the CRH is given by [I21

-1 2 Y 10 ’i Y b

-2 10

-3 I n

0 500 1000 1500 2000

; ~ m / t - , y r w twnwpwre e/ /ihcw x(.,T i i ~ d ,%RI..y ( J W c,trrv,v :I

I V

iterations inn Fig. 2

-A- RLS (airiiuliition) .-0- - N-R1.S (thcory) - .X - K1.S (1hctiry)

IWl,S(! U N-RL’3 (siiiiuhtioii)

Sccondly, thc inisadjustmcnt, dcfincd by tr-[KK], is iuves- ligatcd. Both tlieorcticel ~ i i d siinuliition results on thc It[KR] of the N-KLS ai SNR = 65dB in the class A iioisc of A = 0.01 and r’ = 1.0 are plotted in Fig. 2 and coni- pwreed with those of the RLS. In the Figurc thc thcorctical curvcs or the N-RLS and thc RLS arc evaluated via eqn. 29 with the clipper function i l d the lineiir one, respectively. Fig. 2 shows that the N-RI ,S provitlcs kt robust pe~-~orionnancc in idciitifyiiig lhc sys~ciii in the impid- sive channels. In particular, it meiiitains a compar;ible con- vergetice rille to (tie RLS and cxhibils a smallcr misacijustruent than (tic RTS coulitcl*piirt duc to lhc usc or iionliricarity to limit tlic large iinpiilse noise. By c.oinparing ihc Ihcoitclical ctirvcs wilh thc simulation mults, it is of intei-est to note that the analytical ctirves devi;ite froin ttic siinii1:ition ones for smell 11 U tic1 such deviation would becoiric iicgligiblc wi th thc incrcasc of ilcratinns. This plic- nomciion is duc lo thc iisc or the approximatioI1 in eqn. 13, which is generally itiaccurtitc for smnll n but inorc prccisc for large n.

4 estimation

Nonlinear RLS algorithm for amplitude

Now, we would like to apply the nonlinear RLS algorithm For miplitude estitniition i n the CDMA systeiii. For coni- pxison, the RLS, the nodincar LMS, ;md the Mnsrclicz’s filter are also carried oiil. Rclyiug 011 1hc obscr-whn niodcl in cqn. 1, whcrc 11ic impulsc noisc q(1z) i s considcrcd with the p.d.f. in cqn. 6, thc iipclatc cqua~ions Ibr thcsc four iilgorithms arc sumni;iriscd i15 ~ollows:

84

E { ( d , -

j -0

A. - 1

(33)

(34) l o r G:iiissiaii noise ( A = r‘ = a, q,; is constant), g,,,{ x } i s lincar and E { ~ , ( , { x ] ] ~ ) = D , ; ~ . For impulse noise, howevcr, the expectation tias to bc nuriicriailly c v d w k d using eqns. 34 anti G to yicld (hc Claiiicr-Rao bound. The numerical itsul1.s arc providcd in rollowing Scction.

5 Numerical and simulation results

Severd numerical atid siinulation resiilts ;ire prcsciitcd to illiisti-iik lhc pcrlircmancc of 1hc nonlinear RLS (N-RLS), thc R E , 1lic nonlincar LMS (N-L.MS), and lhc Masrelids filter for amplitude estiniation in CDMA systems. To study the Icuning perforiixince of the tilgoritli ins we m s u m that- the data bits {/) , , , (k)} arc nvailable.

[n the e x p i m e n t , Gold sequences with N = 31 are con- sidered its sprcading scc~i~ciiws. Tlw SNli ror tlic kth user i s defined BS N 1 1 . ~ / 2 q ? , The variniice of the iinpulse noise D: is sct to I. ‘The iiic:.iii-square error (MSE) of anipliiude error for user k at time iiistaiit ii is defined as the average ol’ (4 - b$ ovcr 200 runs.

Firsl, lhc cll’ccl or w on lhc pcr~o~~n;mcc 01‘ thc N-RLS with llic clippcr luiiction iri cqn. I1 is studicd. Six values of w = O.%, 0.98, O,W, 0.995, 0.998, 1.0 arc consided. The MSEs of amplitude cr iw for user 1 in a Ih-ee-user system with SNR(1, 2, 3) = 6 .5dU over tlic class A noises of A = 0.01, I” = 1.0, arid A = 0.05, r’ = 0.2 :ire plotted i i i

IRA I ’ , ~ ) , ~ . ~ ~ , ~ , I I , ~ I ~ I , , . , 1’01 J47, Mu 2, A p i l 2Ofll>

, 0 4 L.,,-., ' . . I YLY-LL .

0 500 1000 1500 2000 2600 3000

a

1 0 '

I

6 Conclusions

7 Ackowledgment

l-liis work is supportccl by CERCi undcr grant 9040303.

-4 La-- I A-- - , - ’’ 0 500 l O O O ” ” ~ ~ 0 ~ ‘2000 2500 3000

a

- (:I niiicr R;io low^ hourid ,I /I = 0.01, I* = 1.0 h A = 0.05. r = 0.2

8

1

2

3

4

5

6

7

8

9

I 1 KASSAM, S.A.: ‘Signal dclccliun in iion-(Aius=.i:iti ticlisc’ (Springr-

12 ‘I‘RfFS. I I.L,V.: ‘ I ~ ~ I C C I ~ O I I , cstiinalion, :und timdulahm tlicory, 1’1. I’

13 C;RADSIII’IIYN, 1,s.. and ICYZHIK, 1.M.: ‘Tablc of iiitcgials, serics,

Vcrlag, Ncw York, 1988)

(Wiley, Ncw Yoi’k, 1968)

:ind pr~xlucts’ (Acadcitiic I’itaa, 1980)

9

‘1’0 prow that tlic wight vectcm in the R I 3 aiid nonlitmu- RLS coiivcrgc in the inciiii squnrc w c need tn show that I I ( N } := 2.4, - N A , - A? satisfying 0 < H(N) < 2, where N denotes (hc filter longh. Ry eqns. 18, 23 and 24, li(M) can be expt’cssccd its

Appendix: Proof of stability in the mean square

CO

H ( N ) = I(N) cxp-A)A’n/rn! (35) 7 I l - 0

where

x cxp(-8wr:)d3 (Xi)

11 is noteworthy that 1/11. i/i2 ;mi ~3 in eqn. 36 arc positive fuiiclioiis of p. According 10 rqn. 35, 10 show Ihr: stablility or the algori(hms it is s u k i e n t to prow that 0 2. Deleting lhc two iiegnlivc terms iri cqn. 36 and Itplacing tlic cxponential fiinction and I/), by oiic since oiic i s their iippzcr bound, obtains

II(N)

To show H ( N ) of using (he cliplxr function is positivc, wc substitute lhc fiinctions in Sections 3.1 and 3.2 into eqn. 36. Retainiiig thosc 1cms with U[(.) irnd dclcling all olhcr positive) tcmis,

0 3

whcrc h(P) is a positive fiinclion given by h(p) = crf(Tju) CX~(-/~W,~). For the RLS, il is noted (hat H(N) cquals the iiilcgrd in eqn. 38 with thc error function in h(f l rcplaced by one. It is straigtitforwwcl 10 show thai the derivatiw of !I(/]) i s LL monotonic decreasing hindion for both KLS and nonlincar. RLS. By applying tlic scconri incan value thco- rem in [ 131,

(39) If E < 2!(N .2), thcn r/(Aq is shown positive. Otherwise

Uascd on the rcsults in eqns. 37 iund 40, we sliow h i t the R1,S and noiiline;ir liLS converge i i i the mean square.

8 6