A new non-linear system for estimating and suppressing narrowband interference in PN spread spectrum modulation

A NEW NON-LINEAR SYSTEM FOR ESTIMATING AND SUPPRESSING NARROWBAND INTERFERENCE IN PN SPREAD SPECTRUM MODULATION A. I. Pk ez-Nei a, J. Roca, Ad A. Lagunas' Uni e si a Poli ecnica de Ca alunya. Dep . Signal Theo y and Comm. C1 Jo di Gi ona, 1-3. Campus No d UPC. Edi icio D-5 08034 Ba celona- SPAIN ABSTRACT This wo k de elops a no el dynamic uzzy logic sys em ha , based on a uzzy basis unc ion expansion, success ully sol es he non-linea p oblem o na owband in e e ence p edic ion and ejec ion in DS-SS. A uzzy basis unc ion ep esen a ion p o ides a na u al amewo k o combining bo h nume ical and linguis ic in o ma ion in a uni o m ashion. The esul is a low complexi y non-linea adap i e line enhance , which o e s a as e con e gence a e and an o e all be e pe o mance o e o he well-known non-linea line enhance s. 1. INTRODUCTION Sp ead Spec um (SS) communica ions o e a p omising solu ion o an o e c owded equency spec um amid g owing demand o mobile and pe sonal communica ion se ices. The p oposed applica ions o comme cial use o sp ead-spec um in ol e he o e laying o sp ead spec um signals on exis ing na owband (NB) use s, hus, implying s ong in e e ence o he SS sys em. While SS has inhe en noise supp ession capabili y, sys em pe o mance can be u he enhanced a he decision de ice i an in e e ence ejec ion il e o line enhance is used be o e desp eading [ 11. In he case whe e a single an enna is used and he s a is ics o he in e e en signals a e unknown, he ejec ion il e is usually a ans e sal adap i e il e (adap i e line enhance o ALE) and elies on bo h, he pseudo-whi e p ope ies o he SS signal and he p edic abili y o he na owband in e e ence, bo h p esen in he ecei ed signal z( ). The ecei ed signal z( ) consis s o 3 addi i e componene s: he SS ansmi ed signal s( ), he wide-band noise n( ), and he na ow-band (NB) in e e ence i( ) z( ) = s( ) + n( ) + i( ) (1) s( ) = A c( ) d( ) cos(w, ) (2) The signal s( ) is a modula ed wideband signal gi en by whe e A is a cons an ampli ude, W, is he ca ie equency, d( ) is he in o ma ion, a bina y da a sequence aking on he equip obable alues o 1 each o which las s o T seconds, and c( ) is he sp eading sequence, usually a pseudo andom noise (PN) code o chip sequence, which also akes he alues o l bu which las s o Tc seconds, whe e T,.c<T In ecep ion, o eco e he in o ma ion d( ), z( ) is chip-ma ched and sampled a he chip a e o he PN sequence. We hus ha e zk = sk +nk + ik (3) whe e (s }, (n } and (i } a e he disc e e- ime sequences om (s( )), (n( )) and (i( )] espec i ely. (sk}, (n } and (ik} a e assumed o be mu ually independen . We ha e assumed ha n( ) is bandlimi ed and becomes whi e a e sampling. Fo he in e e ence, we ha e conside ed ha i s bandwid h is small compa ed wi h Inc. Finally, since he PN sequence is andom, we can assume (sk}, o be a sequence o i.i.d. andom a iables aking alues o k 1 wi h equal p obabili y. In equa ion (3), (s ), and (n } a e wideband signals and a e poo ly co ela ed when sampled a he chipping a e. The e o e when he ALE ies o es ima e he nex sample o he signal, i would succeed only in es ima ing he high y co ela ed in e e ence and consequen ly manages o supp ess i . No e, howe e , ha he sequence (sk} is highly non-Gaussian. Thus, he op imum il e o p edic ing a na ow-band p ocess in he p esence o such a sequence will, in gene al, be nonlinea . Only i he SS signal lies below he noise loo , hen he Gaussian assump ion o sl;+nk is mo e easonable and a linea il e achie es good esul s [l]. In [2], Mas eliez de eloped an App oxima e Condi ional Mean o ACM il e , wi h a s uc u e simila o ha o he Kalman il e , in o de o es ima e he s a e o a linea sys em wi h non- Gaussian obse a ion noise. Vijayan and Poo [3] employed his algo i hm o sol e he NB in e e ence supp ession p oblem in he DS-SS. When he AR pa ame e s a e unknown, Vijayan and Poo also de eloped an adap i e nonlinea LMS algo i hm based on he ACM il e ing algo i hm. This algo i hm was la e modi ied by Rush and Poo [4] and Wu [5]. All hese non-linea algo i hms depa om he s a e-space ep esen a ion o he sys em i, = (Pi,-, + w, (44 Z, = hi, + , (4.b) whe e he in e e ence has been modeled as a Gaussian AR p ocess o o de P. The s a e ec o is ik=[ik. i .,,.. i -p+,] and is gene a ed by he Gaussian p ocess wk=[wk 0 ... o , and he ma ix @, o med by he AR pa ame e s. The obse a ion ec o is h=[l 0 ... 01 and he measu emen noise is he non-Gaussian sequence k= sk+nk. This wo k has been suppo ed by Na ional Resea ch Plan o Spain CICYT unde g an numbe TIC-96-0500-C10-01 3261 0-7803-4428-6/98 $10.00 0 1998 IEEE Neu al Ne wo k and Radial Basis Func ion sys ems ha e also been s udied as adap i e line enhance s [6] and [7]. Howe e , he gene al p oblem o non-linea p edic o s is ha hey equi e g ea e complexi y han i s linea coun e pa , which is ypically encompassed in a single DSP chip. Addi ionally, o non-linea il e ing, hough he na u e o he e o su ace is no known, i is highly likely ha he e a e mul iple local minima. The e o e, a g adien sea ch echnique canno be gua an eed o con e ge o he globally op imum pa ame e es ima es. In his wo k we depa om he Kalman and ACM il e s a e space o mula ion in (4) and design an adap i e uzzy p edic o which ou pe o ms he esul s o he ecen wo ks o [3] and [SI and speeds up he con e gence o he adap a ion p ocess. Also he p oposed echnique p esen s an a ac i e pa allel algo i hmic s uc u e, whe e, a each ins an o ime, jus a ixed numbe o p oduc s and sums and one di ision a e needed o p oduce he ou pu . 2. ADAPTIVE FUZZY LINE ENHANCER The uzzy ejec ion il e o design depa s om he s a e space ep esen a ion o mula ed in (4). Taking ad an age o he ela ionship equa ed in (4.b), he uzzy sys em p edic s he in e e ence sample ik om he obse a ion zk. In con as o o he non-linea in e e ence cancelle s, he p oposed sys em does no equi e he ma hema ical model o he in e e ence in (4.a): no AR pa ame e s ha e o be nei he known no es ima ed. The uzzy sys em o design jus elies on he slow a ying na u e o he NB in e e ence in o de o model i s beha io by means o linguis ic IF-THEN ules, which eplaces equa ion (4.a). The in e e ence ange is quan ized in egions o izy se s and, om a e e ence poin , he e olu ion o he in e e ence among his egions is ollowed by means o linguis ic IF-THEN ules o he ype: "IF ik;l is in he egion o posi i e high alues and ikm2 is in he egion o posi i e high alues and ik-, is in he egion o posi i e high alues THEN ik is in he egion o posi i e high alues". These IF-THEN ules a e he co e o he uzzy sys em o design. Addi ionally, he s a is ical knowledge o he measu emen noise ( k] and model noise (Wk} can be easily in oduced in he design o he p oposed sys em uzijica ion s age. A uzzy sys em is a unc ional ne wo k (Fig. 1) ep esen ed as se ies expansions o uzzy basis unc ions g, (x) M Y = m) = g, (XI ei (6) 1'1 whe e 6 E R a e cons an s. Using he S one-Weie s ass heo em, linea combina ion o uzzy basis unc ions p o e o be capable o uni o mly app oxima e any eal con inuous unc ion on a compac se o a bi a y accu acy [8]. In ou case y=Tk and he inpu x is he measu emen zk and wo ee- o wa ded in e e ence es ima ed alues: x = [z, jk-3 ik-2]. The mos impo an ad an age o using uzzy basis unc ions, a he han polynomials, adial basis unc ions, neu al ne wo ks, e c., is ha a linguis ic IF-THEN ule is na u ally ela ed o a uzzy basis unc ion (FBF). In o he wo ds, he FBF p o ide a gene al amewo k o ansla e abs ac concep s in o compu able en i ies. B (.) = m: I Rule base: ma ix R 11 P e Y I I Figu e 1. Adap i e uzzy line enhance . In ig. 1 we dis inguish 4 main pa s: he izzpe maps he c isp inpu s x o uzzy se s de ined on he inpu space; he se o s a emen s comp ise he uzzy ule base, which is a i al pa o a Fuzzy Logic Sys em, he iiy in e ence engine combines he s a emen s in he ule base acco ding o app oxima e easoning heo y o p oduce a mapping om uzzy se s in he inpu space X (i.e. Ai(.) in ig. 1) o uzzy se s in he ou pu space Y (i.e. Bi(.) in ig.2). Finally, he de izi e maps he agg ega ed ou pu uzzy se s o he single c isp poin in he ou pu space, which in ou sys em is he in e e ence es ima e o ik o be used by he communica ion ecei e . Nex , he design o he p oposed uzzy logic sys em (FLS) is desc ibed. 2.1 The ma hema ical amewo k o heo y o uzzy se s p o ides a na u al basis o uzzy logic, which is a gene aliza ion o bina y logic. In o he wo ds, he logical in e encing using uzzy se s is known as uzzy logic [8-111. In uzzy se heo y he e is no sha p bounda y be ween hose objec s ha belong o he class and hose do no . In addi ion, an elemen may also be a membe o mo e han one se . Membe ship unc ion in a uzzy se is a ma e o deg ee. A uzzy se F in a uni e se o discou se, U, is cha ac e ized by a membe ship unc ion pF , which akes alues in he in e al [O,l]; ha is, pF : U 3 [0,1]. Thus, a uzzy se F consis s o a gene ic elemen U and i s g ade o membe ship unc ion; ha is, F =((u,p,(u))IuE U>. A uzzy a iable is cha ac e ized by a e m se o se o uzzy se s (i.e. o linguis ic o uzzy alues) o U. In his wo k A(.) and x will be used o he inpu e m se and he inpu a iable, espec i ely. Also B(.) and y will be used o he ou pu e m se and he ou pu a iable, espec i ely. Due o noise, he measu ed inpu s a e ague and, he e o e, he sys em classi ies o quan ize hem in o e lapping egions o uzzy se s Ai(.), o whom he inpu s belong wi h some membe ship deg ee (e.g. A3(.) s ands o he uzzy alue: " posi i e high alue"). Thus, hese se s con o m in a na u al si ua ion when desc ibing he possible alues o he 3 measu emen s in x. The Fou S ages o he FLS 3262 Figu e 2. Fuzzy e m se o he a iable “ il e inpu ” Figu e 2 plo s Gaussian membe ship unc ions, howe e , he e a e di e en me hods o de e mine a uzzy membe ship unc ion [IO]. I is wo h no ing ha a membe ship unc ion may be subjec i e, bu no a bi a y. Since ou p oblem employs s a is ical inpu s, he design based on hei p obabili y densi y unc ions shall be app op ia e. In his way, we ela e he uzzy membe ship unc ions o physical p ope ies o he sys em. F om (4.b) we know he condi ional p obabili y densi y ( .d.p) unc ion o ik-1. p(ik-l /z~-~). The e o e, i we ela e he uzzy se s AI wi h his .d.p., we can say ha whene e he inpu alls inside hese uzzy se s, his inpu will be ela ed o some deg ee wi h ik- I. This dynamic designed uzzy se s (dynamic because hei posi ion depend on he alue o &.I) ac as a e e ence o loca ing alues ik+ ik.3 in a slow a ying na owband. Addi ionally, o ob ain he uzzy se o zk we no e ha P(Z, /ik-l) can be assumed o be Gaussian by a easoning equi alen o he one used by Mas eliez in [2] o de elop he ACM il e . Thus, he uzzi ica ion o zk is done by means o he same uzzy se e m A(.) as he one depic ed in ig.2. Finally, he ou pu uzzy se s B(.), which quan ized in a uzzy way he possible alues o he es ima ed in e e ence alues k , ha e been designed as M Gaussian unc ions no malized o 1 and o equal a iance. M is he numbe o IF-THEN ules. Thei means a e ini ially he same as hose in ig.2, howe e , hey can be modi ied by a LMS ype algo i hm as we commen la e in his sec ion 2. We no e wo gene al design conside a ions: 1) because o he ela ionship be ween .d.p and membe ship unc ions, he mo e noise p esen , he wide he uzzy se s ha e o be; 2) o sa e compu a ion he inpu uzzy se s o ig. 2 and he ou pu uzzy se s can be designed as iangles wi h he same wid h as he Gaussian noise a iance. Once he inpu uzzy se s a e designed, he uui ie maps a c isp measu emen o alue in o a uzzy se . The mos widely used uzzi ie is he single on uzzi ie : he c isp poin xi is mapped in o a uzzy se F wi h suppo x whe e pF(x)=6(x-x,~. We no e howe e ha in cases when he signal- o-noise a io (SNR) is low o he e is high inpu unce ain y, non-single on uzzy se s [8] a e mo e use ul as he simula ions in his pape show. The uzzy ule base consis s o a se o linguis ic ules in he o m o “IF a se o condi ions a e sa is ied, THEN a se o conseqiiences a e in e ed”. Suppose we ha e a ule base consis ing o M uzzy i - hen ules R, (m=1 ... M) R, : IF iL-, is A,, and lk-? is A,, and whe e {0,+1,+2,+3}. The p edic o o in e e ence ik cons uc ed based on he M ules. Each ule R, can be iewed as a uzzy implica ion which is a uzzy se R,(.) in XxY wi h is A,, THEN k is B, is PRm(X?y)=PAm,(X)*PAw (‘)*PAd (’)*PB,(Y)’ whe e he mos commonly used ope a ions o “*” a e “p oduc ” and ‘“in’’ [8]. In his wo k we ha e used he “p oduc ” ope a ion. The uzzy ules can be sis ema ically de i ed by conside ing all he possible combina ions among he 7 membe ship unc ions (73=343). Howe e , in his wo k, his ule explo ion is d ama ically educed o 72 ules by a oiding hose i ele an ules o slow a ying in e e ences such as: R, : IF -, is A-, and -, is A, and z, is A-, THEN is B.! The uw in e ence engine o uzzy associa i e memo ies is decision making logic which employs uzzy ules om he uzzy ule base o de e mine a mapping om he uzzy se s in he inpu space X o he uzzy se s in he ou pu s space Y. Le F be a uzzy se in X; hen each R, de e mines a uzzy se FOR, in Y based on he sup-s a com osi ion [81: single on uzzi ica ion pF (x) = 6(x - x, ) and esul s in pFoRm (y) = SUPE,y bF (XI *pRm (x, YIP 1’ he case o p,G.Rm (y) = PA, (k-3)*/’!AM (L-2)*pAk (2k)*/’!Bm(y) = wm(x)*p8m ( ) whe e w, is he i ing s eng h o weigh o he m h ule. In summa y, all he M ules o he FAM’s a e ac i a ed pa allely and imply a ixed numbe o sums and mul iplica ions. A ins an o ime “k”, he esul o he in e ence o each ule can be exp essed as a ma ix mul iplica ion [ 111 ( ig. 1). Finally, he indi idual s a emen solu ions a e agg ega ed o p o ide he o e all solu ion A e he uzzy in e ence, he de uui ie pe o ms a mapping om he uzzy se s in Y o c isp poin s in Y. The ollowing cen oid o cen e o mass de uzzi ie [8] is he mos commonly used me hod. I uses all and only he in o ma ion in he ou pu se B in i s domain “y” in a Bayesian sense (see eq. (8)). m=l whe e 7, = cen oid {B, (y)} and 8, = F,,, No e ha we ha e inally come o he unc ional exp ession (6). which depends linea ly on he ou pu pa ame e 8,. The e o e, we p opose o use an LMS (leas mean squa e) ype algo i hm in o de o adjus 0, and e ine he uzzy sys em esul . This LMS is modi ied o inco po a e he app oxima e condi ional mean non linea i y exac ly in he same way as done in [3]. Tha is he adap i e algo i hm is applied o each uzzy sys em ou pu ik in o de o minimize ( zk - i”, - sign ( zk - i; )II* . II 3. SIMULATIONS In his sec ion, we epo on simula ions ca ied ou o e alua e he pe o mance o he p oposed algo i hms. We ollow he commonly used SNR imp o emen , de ined in [3-51. The SNR a he inpu was a ied by changing he powe o he in e e ing examples s udied in [3] and [SI. Ou pe o mance measu e is he 3263 signal. The a iance o he backg ound he mal noise was kep cons an a a * = 0.01. The SS p ocessing gain is 10. AH esul s we e ob ained based on 10 ials and, o each ial, 3000 da a poin s we e compu ed. Table I summa izes he esul s o he 3 se s o simula ions. I can be seen ha adap i e non linea il e ing uzzy echniques o e conside able imp o emen o e con en ional linea il e s (i.e. TS-LMS: Two Sided Leas Mean Squa e il e ) and he non linea algo i hm designed in [5]. We no e ha , i o simpli y compu a ion iangula membe ship unc ions a e used ins ead o Gaussian ones, he esul s jus deg ade in 1 dB. Also, in he case o AR in e e ence no di e ence exis s i he 72 ules a e educed o 32. In he case o single one sinusoidal in e e ence, o high SNR a ios he pe o mance is no so good as in [5]. This ac is due o he quickly speed o change o he alue o he in e e ing signal. I we wish be e esul s, we ha e o assign mo e membe ship unc ions o he inpu s o co e he a ia ions o he in e e ing signal. O he poin o ema k is ha he LMS adap a ion is e en no needed in he case o he AR in e e ence. In any case, he LMS con e ges in ew samples ( ig. 3) due o he good ule ini ializa ion and he good p ope ies o he FBF. Finally, we ha e also ca ied ou a s udy o noisy scena ios. As no hing is said in [3-5] o his case, we compa e ou algo i hm wi h he linea ALE o 4 aps epo ed in [l]. Figu e 4 shows he pe o mance o he p oposed algo i hm when no LMS adap a ion is ca ied ou . The bes esul s a e o he non-single on uzzy il e , which is mo e sui able o noisy scena ios 181. Logically, he pe o mance o he linea il e is imp o ed o low noise. Fo high noise, he designed sys em imp o es he linea p edic o LP and ob ains BER o he same o de o magni ude han he LP wi h ma ched il e . We no e ha in he linea simula ions AR pa ame e s a e conside ed known, while in he uzzy sys em no in e e ence knowledge is assumed. 4. CONCLUSIONS In his wo k we ha e add essed he p oblem o in e e ence ejec ion in SS sys ems. We p esen a low compu a ional algo i hm ha imp o es he pe o mance ob ained wi h ecen non-linea algo i hms. Jus he slow a ying na u e o he NB in e e ence is assumed and used o he ule ini ializa ion, which helps o a oid local minima and o speed up con e gence, 0.1. 0.s.J 0.3 O. 6 0.Z 0.16 0.7 0.0s 5. REFERENCES J.Ke chum, J.P oakis, “Adap i e algo i hms o es ima ing and supp essing na owband in e e ence in PN SS sys ems,” IEEE T ans. Comm. Vol. 30, May 1982. C.J.Mas eliez, “App oxima e non-Gaussian il e ing wi h linea s a e and obse a ion ela ions,” IEEE T ans. Au . Con ., Feb. 1975. 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[9] Zadeh, L.A., “Ou lined o a new app oach o analysis o complex sys ems and decision p ocesses,” IEEE T ans. 0 Sys ems, Man and Cybe ne ics, ol. 3, no. 1, Janua y 1973. [lo] Nakajima S. and Hamada H. “Au oma ic gene a ion o syn hesis uni s based on con ex o ien ed clus e ing”. P oceed. ICASSP. New Yo k, Ap il 1988, pages 659-662. [ 1 I] B.Kosko, Fuzzy Enginee ing, P en ice-Hall, 1997. -A- Single on Fil e Figu e 4.BER a e desp eade o 100 ones. The signal- o-in e e ence a io pe chip is -20 dB. AR in e e ence I TS-LMS/DR2D [SI I 26.9/37 I 22.3132.6 I 17.6/28.1 1 13/23.4 I I Fuzzy I 35.7 I 31.6 I 26.8 I 21.9 I Sinusoidal in e e ence (1 one), no =O. 15 -1 Sinusoidal in e e ence (100 ones in Table 1. SNR imp o emen (dB) o single on uzzi ica ion. 3264