Adaptive Minimum Symbol-Error-Rate CDMA Linear Multiuser Detector for Pulse-Amplitude Modulation

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1 Adative Minimum Symbol-Error-Rate CDMA inear Multiuser Detector for Pulse-Amlitude Modulation AK Samingan, S Chen and Hanzo Deartment of Electronics and Comuter Science University of Southamton, Southamton SO BJ, UK Abstract An adative minimum symbol error rate (M) linear multiuser detector (MUD) is roosed for direct sequence code division multile access (DS-CDMA) systems emloying multilevel ulseamlitude modulation (M-PAM) scheme Based on a kernel density estimation for aroximating the symbol error rate () from training data, a least mean squares (MS) style stochastic gradient algorithm called the least () is develoed for training linear MUDs Comuter simulation is used to investigate the erformance of this MUD I INTRODUCTION DS-CDMA [] constitutes an attractive multiuser scheme that allos users to transmit at the same carrier frequency in an uncoordinated manner Hoever, this creates multiuser interference (MUI) hich, if not controlled, can seriously degrade the quality of recetion A variety of MUDs have been roosed to combat MUI [] [9] Multi bits er symbol modulation schemes can better utilize recise bandidth In this aer, e consider DS-CDMA systems hich emloy M-PAM modulation scheme Within the class of linear MUDs, the MMSE detector [],[9] is oular, as it often erforms adequately and has simle adative imlementation Hoever, it is ell knon that in general the MMSE solution is not the minimum bit error rate (MBER) solution For a binary modulation scheme, it has been demonstrated that a linear detector that minimizes the bit error rate (BER) can outerform the MMSE detector considerably at least in certain situations and adative MS style MBER algorithms can be derived [0] [] The stochastic gradient algorithm of [0] is referred to as the aroximate MBER (AMBER), hile the stochastic gradient algorithm of [],[] is called the least BER (BER) The AMBER algorithm has simler comutational requirement but the BER algorithm is knon to have better erformance in terms of convergence seed and steady state BER misadjustment [],[] In this aer, e extend the BER MUD to DS-CDMA systems ith M-PAM symbol constellation We first derive the of the linear detector for a knon system The M linear detector can be obtained by minimizing the Using a kernel density aroximation of the from VTC 00, Jeju Island, Korea The suort of the Euroean Union and that of the EPSRC, UK is gratefull acknoledged training data, a stochastic gradient algorithm call the is develoed for training linear detector Three versions of the algorithm are resented, each in turn having simler comlexity The simlest version has a comlexity similar to that of the very simle AMBER algorithm The AMBER algorithm can also be extended to the M-PAM case, and e refer to the resulting algorithm as the AM In simulation, e comare the erformance of the linear detector ith that of the AM one s (k) s (k) s (k) N C (z) C (z) C (z) N code filters Σ channel H(z) noise Σ r(k) chi rate samler Fig Discrete-time model of don-link synchronous CDMA II SYSTEM MODE The don-link synchronous DS-CDMA system ith N users and chis er symbol is deicted in Fig, here s i (k) denotes the k-th symbol of user i, hich is assumed to take the value from the symbol set S = fs l =l M ;» l» M g ; () the unit-length signature sequence for user i is μc i = [μc i; μc i; ] T, and the channel imulse resonse (CIR) is H(z) = n h i=0 h i z i : () The received signal samled at chi rate is given by: r(k) =μr(k) +n(k) =P s(k) s(k ) s(k D +) + n(k) ; () here s(k) =[s (k) s N (k)] T is the symbol vector of N users at k, D is the intersymbol interference san hich de-

2 ends on n h and, n(k) =[n (k) n (k)] T is the Gaussian noise samle vector ith E[n(k)n T (k)] = ffn I, and the DN system matrix P has the form P = H μca CA μ μ CA ith the D CIR matrix H given by H = h 0 h h nh h 0 h h nh h 0 h h nh () the user code matrix given by C μ =[μc μc N ] and the diagonal user signal amlitude matrix A = diagfa A N g The linear detector for user i takes the form: () ; y(k) = T r(k) =μy(k) +e(k) () here = [ ] T is the detector eight vector, e(k) is Gaussian ith zero mean and variance T ff n, and μy(k) = T μr(k) Denote the N s = M DN ossible combinations of [s T (k) s T (k ) s T (k D + )] T as s j,» j» N s The noise-free received signal vector μr(k) only takes value from the set R = fr j = Ps j ;» j» N s g, hich can be divided into M subset deending on the value of s i (k): R l = fr (l) j Rjs i (k) =s l g;» l» M: () Similarly, μy(k) can only take value from the set Y = fy j = T Ps j ;» j» N s g, hich can be artitioned into M subset deending on the value of s i (k): et Y l = fy (l) j Yjs i (k) =s l g;» l» M: (8) g T = T P = T [ DN ]=[g g DN ] : (9) Then μy(k) = D N d=0 g j+dn s j (k d) : (0) Among the DN terms in (0), the term g i s i (k) is the desired signal and the rest of the DN terms are residual interferences Thus, the decision thresholds are 0; ±g i ; ; ±(M )g i, and the decision is made according to: ^s i (k) = 8 >< >: s ; y(k)» (s +)g i s l ; (s l )g i <y(k)» (s l +)g i ; for l =; ;M ; s M ; y(k) > (s M )g i : () g s g i s i l g i sm g i(s l-) g i (s l+) Fig Illustration of the symmetric distribution of Y l around g i s l III THE M INEAR DETECTOR It is straightforard to see that Y l+ = Y l +g i, for l = ; ;M We ill assume that Y l+ and Y l are linearly searable This assumtion is necessary for a linear detector to ork We further oint out that the distribution of Y l is symmetric around the symbol oint g i s l, as illustrated in Fig These roerties allo us to consider just one subset Y l in the derivation of the for the linear detector () The conditional robability density function (df) of y(k) given s i (k) =s l is ψ! (y s y (l) j y (y s js l )= ßff ) n T ff n T () here = N s =M is the number of oints in Y l Folloing [], it can be shon that the of the user i linear detector ith eight vector is here P E () = Q(x) = ß Z x Q (f l;j ()) () y dy; () y (l) j g i (s l ) f l;j () = ff ; () n T and =(M )=M The gradient of P E () ith resect to is rp E () = ßff n T ψ (y (l) ψ (y (l) j g i (s l )) ff n T j g i (s l )) T r (l) j + i (s l )!! y : () A steeest-descent or simlified conjugate gradient algorithm [], [] can be used to minimize the ression () to arrive at the M solution It is comutationally advantageous to normalize to a unit-length after each iteration as = = T Comutational requirements can further be simlified by considering the subset Y l ith l =+M=, hich results in s l =0

3 IV ADAPTIVE M INEAR DETECTOR To derive an adative M algorithm, it is more convenient to rite don the df of y(k) licitly M ψ! (y s y (l) j y (y s )= N s ßff ) n T ff l= n T () and ress the alternatively as P E () = N s M l= Q (f l;j ()) : (8) As the df of y(k) is unknon, a kernel density estimate [] can be constructed from data samles Given a block of K training samles fr(k);s i (k)g, a kernel density estimate of the df () is given by ^ y (y s )= K ßρ n T ρ k= n T (9) here the radius arameter ρ n is related to ff n from this estimated df, the estimated is given by here ^P E () = K K K k= ^f k () = y(k) ^g i(s i (k) ) ρ n T (ys y(k)) Q( ^f k ()) (0) ; () ^g i = T ^ i, and ^ i an estimate of i With r ^P E (), the block-data adative steeest-descent or conjugate gradient algorithm can be derived Using a single-samle estimate of y (y s ), a samle-bysamle stochastic gradient adative algorithm can be derived, hich is referred to as the Three versions of the algorithm are resented, each involving a different level of aroximation Version Notice that i is the convolution of H ith the code vector c i = μc i A i Assume that a searate channel estimate ^h(k) = [ ^h 0 (k) ^h (k) ^h nh (k)] T is rovided, hich can then be used to calculate ^g i (k) = T (k)^ i (k) Using a re-scaling after each udate to ensure T (k)(k) =, the stochastic gradient at k is r ^P E ((k);k)= (y(k) ^gi (k)(s i (k) )) ρ n ßρ n f(y(k) ^g i (k) (s i (k) ))(k) r(k) +(s i (k) )^ i (k)g : () The eight udating equation is then given by (k +)=(k) +μ r ^P E ((k);k) here μ is an adative ste size Version Atk, use a moving average udate for ^g i (k) ^g i (k) =( )^g i (k ) + y(k) s i (k) () () here 0 < < is a ste size Using a re-scaling after each udate to ensure T (k)(k) =and further assuming that g i does not deend on leads to a simlified stochastic gradient r ^P E ((k);k)= ßρ n (y(k) ^gi (k)(s i (k) )) ρ n f(y(k) ^g i (k) (s i (k) ))(k) r(k)g : () The eight udating equation has the same form as () Version In this version of the, further simlifications are made Secifically, a one-samle kernel density estimation of ßρ n (ys y(k)) ^ y (y s ;k)= () ρ n is used, and ρ n is assumed to be indeendent of The simlified stochastic gradient at k is thus given by ßρ n r ^P E ((k);k)= (y(k) ^gi (k)(s i (k) )) ρ n r(k) : () The move average () is used to udate ^g i (k) Notice that, for this version, a eight normalization is not necessary The eight udating equation is given by μ ßρ n (k +)=(k) (y(k) ^gi (k)(s i (k) )) ρ n r(k) : (8) For a comarison urose, e also extend the AMBER algorithm [0] to the M-PAM case The resulting AM algorithm has the form (k +)=(k) +μi k (fi )sgn(^e(k))r(k) (9) here the error function ^e(k) is given by ^e(k) =s i (k) y(k) ^g i (k) ; (0)

4 mmse solution mser solution SNR Fig Symbol error rate comarison of the MMSE and M linear detectors for user of Examle SNR =SNR AM version version version MMSE M Training samles Fig earning curves of the four stochastic gradient adative M algorithms for user of Examle, given SNR =SNR =0dB ^g i (k) is udated using the moving average (), and the indication function I k (fi ) is defined by: 8 >< >: y(k) ; ^g <s i(k) i(k) +fi; s i (k) = M +; I k (fi )= y(k) ^g >s i(k) i(k) + fi; s i (k) = M ; 0; otherise : () The adative gain μ and the nonnegative threshold fi are the to algorithm arameters In terms of comutational requirements, the versions and of the are more comlex than the AM, hile the version of the has a similar comlexity to the AM V SIMUATION RESUTS Examle This as a to-user system emloying -PAM modulation ith chis er symbol The to code sequences ere (+; +; ; ) and (+; ; ; +), resectively, and the CIR at chi rate as H(z) =0:8 +0:z : () The to users had equal signal oer, that is, the user signal to noise ratio SNR as equal to SNR of user Fig comares the erformance of the MMSE linear detector ith that of the M one for user, given a range of SNR Convergence erformance of the and AM algorithms ere investigated given SNR = 0 db For the version of the, the receiver as assumed to kno the CIR h, and the to algorithm arameters ere chosen to be μ =0:00 and ρ n =:ff n For the versions and, the same μ =0:00 and ρ n =:ff n ere used ith =0:0 for moving average udating of ^g i (k) Only the versions and emloyed a eight normalization after each udating The AM had a same = 0:0 ith the to algorithm arameters given by μ =0:00 and fi =0: Fig deicts the convergence erformance of these four stochastic gradient adative algorithms, here the results ere averaged on 00 runs It can be seen that for this examle the very simlified version of the algorithm as only slightly inferior to the full algorithm of version, and it had better erformance than the AM algorithm in terms of convergence rate and steady-state misadjustment Examle This as a -user system emloying - PAM modulation ith 8 chis er symbol The four code sequences ere (+; +; +; +; ; ; ; ), (+; ; +; ; ; +; ; +), (+; +; ; ; ; ; +; +) and (+; ; ; +; ; +; +; ), resectively, and the CIR as H(z) =0:9 +0:z : () The users had equal signal oer Fig shos the s of the MMSE and M linear detectors for user Notice that in this case the MMSE detector did not result in linearly searable subsets Y l,» l», and its curve exhibited a oor Hoever, the M detector roduced a linearly searable result and hence had better erformance With SNR = db, learning curves of the and AM algorithms for user are deicted in Fig, here the results ere averaged over 0 runs For the version of the, again the receiver as assumed to kno the CIR h, and the to algorithm arameters ere set to μ = 0:000 and ρ n = ff n For the versions and, the

5 0 0 AM version version version 0 - MMSE 0 - mmse solution mber solution SNR Fig Symbol error rate comarison of the MMSE and M linear detectors for user of Examle SNR i,» i», are identical Fig M SNR earning curves of the four stochastic gradient adative M algorithms for user of Examle, given SNR i =db,» i» same μ = 0:000 and ρ n = ff n together ith = 0:00 ere used Only the versions and emloyed a eight normalization after each udating The AM had =0:00, μ =0:000 and fi =0: It can be seen that the three versions of the algorithm had similar convergence erformance, hich ere better than that of the AM algorithm VI CONCUSIONS A stochastic gradient M adative algorithm has been derived for DS-CDMA systems ith M-PAM modulation scheme The algorithm has been motivated from the kernel density estimation of the as a smooth function of training data, and three versions of this algorithm have been resented The most simlified version of the algorithm is comutationally very simle, and a desired feature of this stochastic gradient algorithm is that the amount of the eight udating is a continuous and decreasing function of a soft distance from the decision boundary Simulation results indicate that this adative linear detector outerforms an existing adative M detector called the AM [] S Miller, An adative direct-sequence code-division multileaccess receiver for multiuser interference rejection, IEEE Trans Communications, Vol, No//,, 99 [] S Moshavi, Multi-user detection for DS-CDMA communications, IEEE Communications Magazine, Vol, No0,, 99 [] HV Poor and S Verdú, Probability of error in MMSE multiuser detection, IEEE Trans Information Theory, Vol, No, 88 8, 99 [8] S Verdú, Multiuser Detection Cambridge University Press, 998 [9] G Woodard and BS Vucetic, Adative detection for DS-CDMA, Proc IEEE, Vol8, No,, 998 [0] CC Yeh, RR oes and JR Barry, Aroximate minimum biterror rate multiuser detection, in Proc Globecom 98 (Sydney, Australia), Nov 998, 90 9 [] S Chen, AK Samingan, B Mulgre and Hanzo, Adative minimum-ber linear multiuser detection, in Proc ICASSP (Salt ake City, Utah, USA), May -, 00 [] S Chen, AK Samingan, B Mulgre and Hanzo, Adative minimum-ber linear multiuser detection for DS-CDMA signals in multiath channels, IEEE Trans Signal Processing, Vol9, No, 0, 00 [] S Chen, B Mulgre, AK Samingan and Hanzo, Stochastic gradient adative minimum symbol-error-rate equalization for ulseamlitude modulation, submitted to IEEE Trans Signal Processing, 00 [] BW Silverman, Density Estimation ondon: Chaman Hall, 99 REFERENCES [] R Prasad, CDMA for Wireless Personal Communications Artech House, Inc, 99 [] S Verdú, Minimum robability of error for asynchronous Gaussian multile-access channels, IEEE Trans Information Theory, VolIT-, No, 8 9, 98 [] Z ie, RT Short and CK Rushforth, A family of subotimum detectors for coherent multiuser communications, IEEE J Selected Areas in Communications, Vol8, No, 8 90, 990 [] U Madho and M Honig, MMSE interference suression for direct-sequence sread-sectrum CDMA, IEEE Trans Communications, Vol, No, 8 88, 99

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