Convergence Analysis of the LCCMA in an Asynchronous DS-CDMA System

Transcription

1 Convergence naysis of the LCCM in an synchronous DS-CDM System James Whitehead and Fambirai aaira Schoo of Eectrica Eectronic and Computer Engineering University of Nata Durban South frica bstract he proof of goba convergence of the ineary constrained constant moduus agorithm (LCCM in the absence of noise for an asynchronous DS-CDM system is given. fter initia convergence by LCCM sitch over to decision directed (DD agorithms is standard practice. measure of the non-idea properties of the channe is used to modify the sitch over to the DD-LMS agorithm to improve its on stabiity and steady-state performance. Simuation resuts are used to demonstrate the increase in performance. I. INRODUCION he performance of persona ireess communications systems using direct sequence code division mutipe access (DS-CDM can be greaty enhanced by using signa processing techniques such as mutiuser detection (MUD []. MUD improves the performance by reducing the amount of mutipe access interference (MI at the receiver. he MUD considered in this paper is categorized as a inear scheme and is appicabe hen short spreading codes are used. In this paper bind indicates that ony the timing and spreading code information of the desired user is non at the receiver. Bind impementations are desirabe because they are spectray more efficient than their trained counter parts (they circumvent the need for piot channes. daptive schemes are desirabe because they offer a oer computationa compexity and have the abiity to trac a time varying channe. he constant moduus agorithm (CM is perhaps the most idey studied and used practica bind equaization agorithm [] [3] due to its computationa simpicity and robustness. he CM has recenty been appied to the bind adaptive MUD probem here the various versions: ineary constrained (LC- CM [] [5] mutiuser (MU- CM [6] cross correation (CC- CM [7] and ineary constrained differentia (LCD- CM [8] have modified the CM cost function to ensure that the desired user is captured. he LCCM scheme is considered in this paper because of its reative simpicity and proven resiience under mismatch conditions []. he upin of a future generation mobie ireess communications system oud typicay mae use of an asynchronous DS-CDM system ith short spreading codes. he his or as partiay supported by cate tech eecoms and eom S as part of the Centres of Exceence Programme. he financia assistance of the Department of Labour (DoL toards this research is hereby acnoedged. Opinions expressed and concusions arrived at are those of the author and are not necessariy to be attributed to the DoL. goba convergence in the mean of the LCCM as proven in [5] but ony for the synchronous case. his paper extends the proof of [5] to the asynchronous case by using a unique boc matrix formuation of the Hessian matrix of the LCCM cost function. he Hessian matrix defines the curvature of the cost surface at every point. Goba convergence is proved by shoing that the Hessian matrix is positive definite. positive definite Hessian matrix indicates that the cost surface is convex hich in turn proves that there is a unique goba minimum to the cost function. If the cost function has a unique minimum then a stochastic gradient agorithm is guaranteed to converge to the goba minimum irrespective of the initiaization of the adaptive fiter. he LCCM is a poerfu agorithm for converging to the MMSE receiver in a competey unnon operating environment. Once the LCCM agorithm has converged to the neighborhood of the MMSE receiver there are hoever better agorithms to sitch over to that i have a oer excess mean square error (EMSE in the tracing and steady-state phases of the adaptive agorithm. hese agorithms convergence properties are not as robust as that of the LCCM hich is hy LCCM is required to get the receiver into the neighborhood of the MMSE receiver. One idey touted agorithm [9] to use is the decision-directed LMS agorithm. he DD-LMS agorithm as originay formuated for the bind equaization of intersymbo-interference (ISI channes. hese happen to be quite different from the mutiuser CDM channe and hence their performance is reduced. his paper suggests a method of quantifying the deviation aay from the idea conditions that the DD-LMS as intended to operate in. his metric is then used to improve the performance of DD-LMS hen appied to the bind MUD appication. Improvements in both stabiity and steady-state EMSE of the modified DD-LMS are demonstrated via simuation. his paper is organized as foos: section II outines the signa mode and LCCM section III proves the goba convergence of the LCCM in an asynchronous DS-CDM system and section IV outines the modifications made to the DD-LMS agorithm. Simuation resuts are given in section V and concuding remars are made in section VI. II. SYSEM MODEL. ransmitter n asynchronous (ideband DS-CDM transmitter mode for the upin of a mobie radio netor is considered. he baseband representation of the th user s transmitted

2 signa is given by i x ( t b ( i s ( t i τ ( here and s denote the ampitude and normaised spreading aveform of the th user respectivey and is the data symbo duration. he reative offset of the th user s asynchronous signa is given byτ hich taes on integer vaues in the range 3. he th user s ith transmitted symbo b ( i taes on the vaues { + } ith equa probabiity. he spreading aveform taes the form N c n s ( t c ( n ψ ( t n t [ ] ( here N is the processing gain and c is the th user s spreading code sequence of ± s ψ ( t is the chip puse shape of duration c N. It is noted that s ( t ony taes on vaues in the interva [ ]. B. Receiver he received signa is passed through a chip-matched fiter and samped at the chip-rate. hese sampes are concatenated into a ength N vector of received sampes. Let s [ ] ( ( τ s s N τ be the don shifted (by amountτ version of discrete signature sequence vector s of ength N. Define ( N c ( c ( N s (3 [ ] [ ] [ τ ] [ τ ] S s s s ( S[ ] s ( N τ s ( N τ here ( N τ (5 s is the up shifted version of s aso padded ith zeros. he first entry in (5 is the zero vector as the receiver is bit synchronized to the desired user ( τ. Using this notation the received vector of sampes from the chip-matched fiter is given by [ ] ( i + [ ] ( i+ + b ( i b ( i b ( i diag ([ ] r S b S b n (6 is a diagona matrix ith the users ampitudes aong the main diagona and n ( i is an additive hite Gaussian noise vector ith covariance matrix σ I N. he vector r ( i is fitered by a finite impuse response (FIR fiter structure hose coefficients form the vector ( i ( i N ( i. he desired user from here on i be user. he output of the fiter hich constitutes the decision statistic is given by ( ( (. y i i r i (7 C. LCCM he non-canonicay constrained LCCM is based on the CM- [] agorithm and attempts to minimize the cost function ( J( ( i E ( i ( i r (8 subject to the inear constraint s. he LCCM uses its estimate of the desired user s signature sequence s to prevent the capture of unanted users. he cost func- tion as it stands is not amenabe to conventiona stochastic gradient techniques due to the constraint but by maing use of the canonica representation of the MMSE fiter [9] it is possibe to spit the fiter coefficients into to orthogona components: a fixed (or non-adaptive part and an adaptive part given by ( i + ( i s (9. he fiter component orthogona to the spreading code is adapted ithout a constraint. he update step of the adaptive agorithm is given by ( ( µ (( ( ( ( i+ i + y i y i r i ( here µ is a sma positive constant (step size. he orthogona projection of the received vector r ( i ( i ( i r s r s ( is used in ( to ensure that remains orthogona to s this in turn ensures that the constraint s is aays met. Reference [5] shoed that the LCCM converges to a scaed version of the MMSE fiter hich is given by opt C s sc s { ( (} [ ] [ ] [ ] [ ] σ C E r i r i S S + S S + I. ( (3 III. CONVERGENCE NLYSIS FOR SYNCHRONOUS DS-CDM he technique used to prove the goba convergence property of the LCCM in an asynchronous DS-CDM system foos the same technique as that presented in []. his paper differs in the definition of the u and u vectors as e as the boc matrix notation used for the first and second derivatives of the LCCM cost function in order to accommodate the asynchronous data transmission of the various users. Define vectors u and u hose eements correspond to the contribution at the output of the fiter of the users current and successive transmitted bits ( [ ] ( [] u S and u S. ( he th eement of u and u are given by respectivey ( [ τ ] ( τ ( u s and u s N. (5 he output of the chip-matched matched fiter may no be ritten as r bu. + bu (6 Expanding the cost function in (8 e have { } ( E ( ( J r r +. (7 Using the reationship in (6 the cost function J ( may be expressed in terms of u and u {( } E r uu + uu (8

3 6 5 φ(u u 3 E u u Fig.. Surface Pot of LCCM cost function ( for a to user asynchronous system. {( r } E ( bubu + ubbu + ubbu 3( uu ( u { } ( uu ( u ( uu ( uu (9 since the inear constraint in (8 is equivaent to u the cost function in (7 can be ritten as u ( u u ( uu ( u + ( uu min φ 3 3 ( u ( uu ( uu ( uu uu ( We need to find the stationary points of the cost function to do this define a ne coumn vector u u ( u here u u u and u u u. Define a ne cost function in terms of u hich is equivaent to ( φ u φ u u ( ( ( he first derivative taes the form (using boc matrix notation u u u here the eements of are given by u (3 u ( ( uu ( uu ( ( u ( u 3 u + u u 3 uu + 3 uu ( u ( 3 + u + 3 u j + 3 u j j j j and simiary for : u ( u u ( 3 + u + 3 u j + 3 u j u j j j for. From ( and (5 ceary if ( stationary point is hen (5 3 then the ony u u (6 hich corresponds to the decorreating detector since the MI has been competey removed. he goba convexity of the cost function can be assured by anaysing the Hessian matrix of φ ( u. Using boc matrix notation the Hessian matrix taes the form φ φ u uu. u φ φ uu u φ Where the eements of the sub matrices are given by 3 ( ( uu 3( uu + u u u u u u u u and simiary for (7 (8 (9

4 3 ( ( uu 3( uu + u u u u (3 u u. (3 u u When 3 and thus u then the main diagona of the Hessian matrix equas 3 ( and off diagona eements equa. Ceary under this condition the Hessian matrix is positive definite the decorreating detector is the ony stationary point hich is aso the goba minimum and thus goba convergence is guaranteed. his is the same condition hich appies to the synchronous system hich as proved in [] he cost function as given in ( is potted in Fig. for a to user asynchronous system. In this case u and u as it is assumed that the spreading code and timing of the desired user is non perfecty. he height of the surface in Fig. is equa to the cost function of the remaining to degrees of freedom u and u. It can be seen that athough the curvature of the cost surface is not constant there does exist a unique goba minimum corresponding to u u the decorreating receiver. (he decorreating receiver competey removes MI. he terms u and u are a measure of the residua MI corresponding to user and hen they equa it indicates that a MI has been removed and thus the receiver is operating as the decorreating receiver. IV. IMPROVING SWICH OVER O DD-LMS. Reationship beteen LCCM and MMSE fiter coefficients It is e non that the optimum tap eight vector of the MMSE fiter is given by ( C s (3 M. Mutipying the optima tap eight vector of the non-canonicay constrained LCCM as given in ( by e get: opt C sc ( s ( s M s M (33 here it is non that the receiver gain is defined as (. θ " M s Substituting the s term of (33 using the receiver gain definition e get M opt θ (3 M θ hich shos that the LCCM and the MMSE fiter coefficients are coinear and are reated via the scaing factorθ. B. DD-LMS he update step of the DD-LMS agorithm is given by ( i+ ( i + µ ( sgn( y y r (35 he DD-LMS agorithm as originay formuated for the bind equaization of ISI channes. he presupposed operating conditions ere high SNR and perfecty bind equaizabiity of the channe. hese assumptions differ from the conditions found hen performing bind adaptive MUD in a DS-CDM communications system. For exampe in a ireess communications channe phenomena such as fading and shadoing can significanty oer the SNR. so there is residua MI at the output of a MMSE MUD and the amount increases as SNR decreases. hus the perfect equaization condition is vioated. It is proposed in this paper that the receiver gain is used to tae into account the deviation of the channe aay from the idea conditions expected by DD-LMS. Under idea conditions the SNR increases to infinity and the receiver approaches the decorreating receiver. When this occurs the residua MI goes to and θ approaches. Under non-idea conditions θ is in the range ( ith corresponding to a SNR of infinity. he optima tap eight vector of the DD-LMS agorithm is the MMSE tap eight vector. he reationship beteen the MMSE and LCCM optima tap eight vector as derived in this paper and given in (3 in terms of the receiver gainθ. he receiver gain is then used to scae the fiter coefficients in a once-off manner at the moment of sitch over to DD-LMS. his has the effect of removing the scaing beteen the optima DD-LMS fiter coefficients (hich are the MMSE fiter coefficients and the optima LCCM fiter coefficients. he scaing coud possiby ead to instabiity at sitch over or convergence toards oca minima (scaed MMSE. C. Estimation of Receiver Gain he receiver gain θ can be estimated by maing use of an approximation of the MMSE given in [9] and the expression for the MMSE achievabe ith a inear transformation ε given in [] as: min ε min sc s. (36 Using the definition of the receiver gain (36 can be ritten as ε min θ. (37 he approximation given in [9] for ε min hods hen the signa vectors are approximatey orthogona and is ε. min + σ Comparing (37 and (38 it is cear that (38 θ. (39 + σ his bound is asymptoticay tight ith SNR (see Fig. and upper bounds the true vaue as MI increases. he LCCM requires the estimation of to compute step (9 of the adaptive agorithm. he added compexity to estimate θ is tantamount to estimating the SNR hich is a common operation in error decoding and maxima ratio combining. It thus constitutes a margina increase in overa receiver compexity.

5 Exact vaue of θ pproximate vaue of θ SNR (db Fig.. Pot of θ and the estimate of θ using (39 as function of SNR. user system ith MI eve of 3dB s is used. V. RESULS the resuts ere generated using ength 3 God codes. he MI ratio is defined as here a the interfering users transmit at the same ampitude. he reative timing offsets of the users ere initiaized in the range [ 3 at the start of a particuar simuation and then fixed for the duration of the simuation. he output SINR at time instant i is defined as ( SINR i ( ( i s ( i + ( i σ s ( ( and as obtained from simuating the received vector of sampes given in (6 and impementing the LCCM as given in (. In Figs. 3-5 the LCCM is seen to converge correcty in an asynchronous DS-CDM system as proved in section III. Fig. 3 iustrates the performance increase obtained hen sitching over to DD-LMS from LCCM. he ensembe average SINR of independent simuation runs of the receiver agorithm is potted as a function of iterations (bits here the sitch over to DD-LMS is made at each time. user system ith MI ratio of 5dB s and SNR of 3 db s is considered. fixed step of is used. he LCCM is initiaized to the singe user matched fiter (MF and thus the output SINR is seen to start at the MF eve and converge toards the optima (MMSE SINR. It is observed that the LCCM requires approximatey bits to converge under this system configuration. It is assumed there is a 5% error in the estimate of the desired user s ampitude. his yieds a arge EMSE at steady-state. he DD-LMS agorithm does not require any ampitude estimation and its steady-state EMSE is significanty oer than the LCCM as it is observed that as soon as the sitch over to DD-LMS is made the output SINR immediatey increases. In Fig. the use of θ to correct the magnitude of the LCCM fiter coefficients at sitch over is seen to mae the DD-LMS mode of operation possibe as the DD-LMS agorithm is no stabe. he SINR of DD-LMS agorithm ithout the use of θ at sitch over is seen to decrease hich indicates that it is unstabe. system ith users 5dB MI ratio 5dB SNR and step size of 5 5 as considered. Fig. 5 demonstrates the performance gain hen θ is used on sitch over in the situation here DD-LMS is stabe ithout the use of θ. he system here θ as used has a higher output SINR as the DD-LMS agorithms converges to the correcty scaed version of the MMSE fiter coefficients. he system considered had users db SNR 5dB MI ratio and a step size of. VI. CONCLUSION In this paper the LCCM as proven to be gobay convergent in an asynchronous DS-CDM system here the receiver has enough degrees of freedom to competey suppress a MI. his paper aso derived the reationship beteen the optima non-canonicay constrained LCCM tap eight vector and the MMSE tap eight vector in terms of the receiver gain. his reationship as expoited at the sitch over to the DD-LMS mode of operation to both enhance the stabiity and increase the output SINR of the DD-LMS mode. simpe method for estimating the receiver gain as aso presented. 5 MMSE SINR Output SINR (db 5 LCCM DD LMS 5 Singe user matched fiter DD LMS sitch over No DD LMS sitch over Iterations x Fig. 3. Improvement in SINR hen sitching over to DD-LMS for a user asynchronous system.

6 Output SINR (db 3 MMSE SINR LCCM DD LMS 5 Singe user matched fiter θ used θ not used Iterations x Fig.. Stabiity due to the use of θ at the point of sitch over to DD-LMS. REFERENCES [] S. Verdú Mutiuser Detection Cambridge University Press 998. [] S. Hayin ed. Unsupervised daptive Fitering Chapter : he Core of FSE-CM Behavior heory Wiey 999. [3] C. R. Johnson Jr. P. Schniter. J. Endres J. D. Behn D. R. Bron R.. Casas Bind Equaization Using the Constant Moduus Criterion: Revie Proc. of IEEE Vo. 86 No. pp Oct [] J. Miguez L. Castedo Lineary Constrained Constant Moduus pproach to Bind daptive Mutiuser Interference Suppression IEEE Commun. Letters Vo. No. 8 pp. 7-9 ug [5] Changjiang Xu Guangzeng Feng yung Sup a modified constrained constant moduus approach to bind adaptive mutiuser detection IEEE rans. Commun. Vo. 9 No. 9 pp Sep.. [6] C. B. Papadias. J. Pauraj Constant Moduus gorithm for Mutiuser Signa Separation in Presence of Deay Spread Using ntenna rrays IEEE Sign. Proc. Lett. Vo. No. 6 pp May 997. [7] P. rasaratnam S. Zhu and. G. Constantinides "Fast Convergent mutiuser constant moduus agorithm for use in mutiuser DS/CDM Environments" IEEE Proc. Int. Conf. coust. Speech Signa Process. (ICSSP Orando Forida pp May. [8].Miyajima Bind daptive Detection Using Differentia CM for CDM Systems rans IEICE. Vo.J83- No. pp Nov.. Output SINR (db MMSE SINR. LCCM DD LMS θ used θ not used.5.5 Iterations x Fig. 5. Increased Performance due to the use of θ at the point of sitch over to DD-LMS. [9] M. Honig U. Madho S. Verdú Bind daptive Mutiuser Detection IEEE rans. On Information heory Vo. No. pp Juy 995. [] C. Xu G. Feng Comments on Lineary Constrained Constant Moduus pproach to Bind daptive Mutiuser Interference Suppression IEEE Commun. Letters Vo. No. 9 pp. 8-8 Sept.. James Whitehead hods a BScEng (Eectronic and MscEng degree from the University of Nata. t present he is studying a PhD at the University of Nata. His research interests incude CDM systems ith emphasis on mutiuser detection and muti-antenna systems. Fambirai aaira hods a PhD degree from Cambridge University Cambridge U.. He is presenty a Professor at and the head of the Schoo of Eectrica and Eectronic Engineering at the University of Nata Durban South frica. His research interests are in digita communication systems and netors ith particuar emphasis on CDM ceuar systems