A Probabilistic Data Association Based MIMO Detector Using Joint Detection of Consecutive Symbol Vectors

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1 A Probabilistic Data Association Base IO Detector Using Joint Detection of Consecutive Symbol Vectors Shaoshi Yangieun Lv ember IEEE Xiang Yun Xinghui Su an Jinhuan Xia School of Information an Communication Engineering Beiing University of Posts an elecommunications Beiing China s: {banpoheitao Abstract A new probabilistic ata association (PDA approach is propose for symbol etection in spatial multiplexing multiple-input multiple-output (IO systems By esigning a oint etection (JD structure for consecutive symbol vectors in the same transmit burst more a priori information is exploite when upating the estimate posterior marginal probabilities for each symbol per iteration herefore the propose PDA etector (enote as PDA-JD etector outperforms the conventional PDA etectors in the context of correlate input bit streams oreover the conventional PDA etectors are shown to be a special case of the PDA-JD etector Simulations an analyses are given to emonstrate the effectiveness of the new metho Inex erms Gaussian approximation IO etection maximum lielihoo (L etection probabilistic ata association (PDA I INRODUCION he probabilistic ata association (PDA algorithm is a reuce complexity approximation to the a posteriori probability (APP etector an has attracte great interests for suboptimal etection of CDA [] an multiple-input multiple-output (IO systems []-[4] It is pointe out in [5] that if the IO etector is confronte by symbols that are correlate through some form of channel coe which is usually the practical case it is suboptimal for the signal etector an channel ecoer to operate separately an only on iniviual symbol vectors o optimally etect the symbols the etector shoul mae ecisions ointly on all the symbol vectors using nowlege of the correlations across symbol vectors introuce by the channel coe an the channel coe shoul ecoe using lielihoo information on all the symbol vectors obtaine from the signal etector Although this is computationally prohibitive it is still valuable to mae a traeoff between the enhance performance an complexity from the integral system point of view he seminal iterative receiver that combines IO etector an a soft-input soft-output channel ecoer to accomplish oint etection an ecoing [6] is investigate from this motivation In practice a more simplifie receiver where IO etector an soft channel ecoer are concatenate in a serial style without feebac [7] is often his research has been supporte by Program for New Century Excellent alents in University (NCE uner Grant NCE-6-9 aopte Since its output is probabilistic information PDA algorithm is appropriate to play the role of soft IO etector in both the aforementione receivers In this paper we propose a new PDA base IO etector that achieves oint etection (JD of consecutive symbol vectors in the same transmit burst instea of the conventional strategy etecting the symbol vector in a burst one by one he propose PDA-JD etector exploits statistical correlations between consecutive symbol vectors an leas to superior performance to conventional PDA IO etectors in the context of correlate input ata symbols he improve quality of soft reliability information provie for the concatenate channel ecoer by PDA-JD etector can not only help reuce the complexity of both the iterative receiver an the serial receiver but also abate error propagation in the latter scheme II BASIC SYSE ODEL Consier a spatial multiplexing IO system with transmit antennas an NR receive antennas (for generality NR N is not assume here he receive baseban signal at each instant of time is given by y=x+ n ( where is the N R N channel matrix x is the length N vector of transmit symbols taen from a moulation constellation A = { a a a an n is a length N R zero mean complex circular symmetric Gaussian noise vector with covariance matrix σ I where I is an NR NR ientity matrix At the receiver when the PDA-base algorithms are consiere the tas of symbol etection is to estimate the posterior marginal symbol probabilities Px ( = am y = P m( x y for = N m = without maing an exhaustive search in the space of all possible symbol combinations Define N /8/$ 8 IEEE 46 ICCS 8

2 x = a P ( x y ( m m * = m m m { V x ( a x ( a x P ( x y ( = m m m { V x ( a x ( a x P ( x y (4 Base on ( the non-ecorrelate system moel for PDA etectors is formulate as y =h x + h x + n h x + v (5 where h enotes the th column of an x the th element of x he interference an noise term v in (5 is approximate as a multivariate colore Gaussian istribute ranom variable with mean μ ( v = xh v = hh + σ I an pseuo covariance V( V{ x covariance V( V( x v = hh For more information about pseuo-covariance see [8] When NR N the complexity of PDA etector can be reuce by aopting the ecorrelate system moel y = x+ n = x e + x e + n x e + N (6 where y = ( y n is a colore Gaussian noise with zero mean an covariance σ ( an e is a column vector with in the th position an elsewhere he interference an noise term N is still approximate as a colore Gaussian vector with mean μ ( N = xe covariance ( { x covariance V( N V( x V N = V e e + σ ( an pseuo = ee For simplicity we will focus on the ecorrelate system moel in the following III PDA-JD IO DEECOR A On JD of Consecutive Symbol Vectors Note that there are generally hunres of symbol vectors in a single transmission burst of IO systems [9] hese symbol vectors are all receive in a time interval as a bloc Suppose the number of symbol vectors in a burst is enote as L then corresponing to each transmitte symbol vector the receive ( ( ( ( ( ( ( ( signals are y = x +n y = x +n ( l ( l ( l ( l y = x +n l = L respectively Conventionally the receive symbol vectors are etecte on an isolate vector-by-vector basis even in coe system which is suboptimal however while soft information an correlations across symbol vectors introuce by some form of channel coe are taen into consieration Our iea is to ointly etect an appropriate number of consecutive symbol vectors o our best nowlege such a etection strategy in IO systems has not been reporte before Assume the number of symbol vectors etecte simultaneously is enote as which satisfies { L N = L with N being a positive integer hen the basic system moel ( is reesigne as r = Gs+ u (7 ( ( ( ( where r = [( y ( y ] s= [( x ( x ] ( ( ( u = [( n ( n ] G = an u is ( still a zero mean complex circular symmetric Gaussian noise vector with covariance matrix σ I Cov( u = E( uu = in which σ is the σ I ( power of the white noise n corresponing to the th symbol vector = Similarly to (6 (7 can be reformulate as the ecorrelate moel r = s+ u = s e + s e + u = s e + U (8 where r = ( G G G r (( ( ( ( ( ( ( ( ( ( ( = y y u is Gaussian noise with zero mean an covariance σ I Cov( u = ( G G G G( G G σ I ( ( σ ( = ( ( σ ( = N Since the elements of ( are ii ( complex Gaussian the ran of is almost always N ie ( has full column ran So the ermitian matrix ( ( is almost efinitely invertible Notice that the above erivation oes not mae the assumption that is quasi-static which means the erivation maes sense for the fast variant channel as well where changes ranomly from one symbol vector to another When quasi-static channel is taen into account we have ( ( ( = = = = hen r an Cov( u rewritten respectively as (( ( ( ( ( y y are 47

3 σ an ( σ ( Furthermore if the Gaussian noise n ( is consiere ii between ifferent Cov( u can be simplifie as ( σ σ ( From (8 we can clearly see that when any symbol s is etecte the imension of the corresponing interference is N instea of N in the conventional PDA etectors An the conventional PDA etectors are a special case of the PDA-JD etector when = B PDA Algorithm Derivation Base on JD For any element s we associate a probability vector P ( whose m th element P m( s r is the current estimate of the posterior probability that s = a m Following the PDA principle ( Gaussian approximation in [4] the pseuo noise U is approximate by a complex Gaussian istribution with matche mean s e covariance matrix ω e e + Cov( u ξ e e where an pseuo-covariance matrix m m s = a P ( s r (9 * ω = ( am s ( am s P m( s r ( ξ = am s am s P m s ( ( ( r ( ere P m( s r is initialize as a uniform istribution an will be replace with an upate value where the pseuo-covariance typically oes not vanish Let w = r s e s e an R( w R( w φ m( s exp I( w I( w ( ( + ( ( + ( I R ( R( Cov( u I( Cov( u = + Q ( Cov( ( Cov( I u R u where R ( an I ( are the real an imaginary part of a complex variable respectively an Q = ( ω + R( ξ e e + ( ω R( ξ e e + N + N + N + N + I( ξ e e + I( ξ e e Since it is assume that the transmitte symbols have equal a priori probabilities the estimate posterior marginal symbol probability is given as p m( r s P( s = am φm( s P m( s r = (4 p ( r s P( s = a φ ( s m m m o sum up the algorithm procees as follows Determine the optimal etection sequence accoring to the orering criterion propose for the ecision feebac N = etector in [] an enote the sequence as { Initialization: compute r set P m( s r as a uniform istribution for = N ie P m( s r = ; set the iteration counter z = Initialize = P ( n compute 4 Base on the current values of { n n n P ( s r via (9 ~ (4 an set the results equal to the m n corresponing elements of P ( n 5 If < N let = + an go to step 4 Otherwise go to step 6 6 If P ( has converge or the iteration counter has come to a certain given number go to step 7 Otherwise let z = z + an return to step 7 For = N mae a ecision s ˆ for s via s a l = arg max P ( s r ˆ = l { m m= = { sˆ = N hen sˆ is obtaine 8 Finally we have ( x = { sˆ = ( N + ( N + N = From the above algorithm proceure we can get the perspective that the essence of the PDA-JD etector is it exploits more a priori information when estimating the posterior marginal probabilities of symbols being etecte One nees to compute at least N ( probabilities in each iteration Because we are intereste in the funamental property of the PDA-JD etector the refinements such as complexity reuction techniques propose in [] an conversion of the complex-value signal moel into an equivalent real-value signal moel for QA constellation in [] are not incorporate into the above basic algorithm proceure though they both mae sense in this PDA-JD etector 48

4 C Complexity Analysis aing QA moulation as an example the complexity of the propose PDA-JD etector using the matrix spee-up tactic (the Sherman orrison Woobury formula of [] is approximately O(( N = O(8 N real operations for upating the inverse of the complete-covariance matrix per symbol vector per iteration his is actually the worst case complexity since the size of the problem may be lowere by the use of successive cancellation [] What is more the best recor of matrix exponent for the complexity of matrix inverse is 76 up to now [] an many mathematicians are trying to prove the conecture that the value can be reuce to By comparison the corresponing complexity is approximately ON ( [] an O(( N = O(8 N [][4] real operations when BPSK an QA moulation is use respectively It is clear that N plays a ominant role in the complexity of all PDA base IO etectors especially for the system with large N herefore the PDA-JD etector enoys almost the same orer polynomial complexity of N as the conventional PDA etectors o when is properly etermine by the constraint length of specific channel coe scheme an simulations to be an afforable constant coefficient For the overall complexity of the PDA-JD etector many techniques can be use to further reuce it For example when the system moel is ecouple into real an imaginary parts lie [] the number of probabilities that are evaluate per transmit symbol per iteration is reuce to at least ( an + / for square QA an rectangle QA respectively While complex system moel is applie both the values are IV SIULAION RESULS AND DISCUSSIONS A Simulation Results We emonstrate the power of the PDA-JD etector using computer simulations for a V-BLAS system which utilizes N = NR = antennas an QPSK moulation scheme he entries of the IO channel are chosen as ii zero mean stanar complex normal ranom variables which remain constant for the uration of the burst with length L = 64 but change ranomly from one burst to another A rate / convolutional coe with polynomials (77 is applie to introuce correlation across symbols In orer to emonstrate the error probability improvement brought by PDA-JD etector which is far less obvious when entangle with channel ecoer we focus on the part from the input to the output of soft IO etector as shown in [6]an [7] an evaluate the bit error performance at the output of PDA-JD etector irectly his is a generally accepte manner in research of IO etectors he performances of the PDA-JD etector the conventional PDA etector propose in [4] an the L etector are compare in Fig Both the PDA etectors o not use the optimal etection sequence an refinements we have mentione in the above section his is one in orer to provie the same comparison conitions as in [4] It is easy to fin the avantage of the PDA-JD etector over the conventional PDA etector When = 8 an = the BER performance of the propose PDA-JD etector is nearly B an more than B better than that of the conventional PDA etector respectively at BER= Compare with ZF-OSIC [9] the PDA-JD etector substantially improves the BER performance his can be seen in Fig For ZF-OSIC etector when the number of transmit antennas is equal to that of receive antennas the performance of the larger one is much worse than that of the smaller one except in very high SNR regimes But if the PDA-JD etector is invoe the performance gap between the larger one an the smaller one shrins Furthermore the cross-point in ZF-OSIC curve is 8 B while in PDA-JD curve it ramatically ecreases to B ence we can conclue that the PDA-JD etector prefers larger IO systems in terms of BER performance which is a natural result of the fact that it exploits more a priori information when etecting each symbol of the symbol sequence Fig shows the complexity of the PDA-JD an the conventional PDA etectors (the special case of = in the PDA-JD etector It is evient that in both etectors N imposes ominant influence on the complexity his observation is consistent with the theoretical result O(8 N an O(8 N aing the constraint length 7 of (77 into account the more appropriate value of is which gives more favorable traeoff between the BER performance an the computational complexity than other values of B Some Discussions igh-quality probabilistic ata offere by soft IO etector lie PDA-JD etector is capable of giving more reliable a priori information to its following channel ecoer in both receivers in [6]an [7] an thus helps reuce the total complexity accumulate through roun by roun iterations for the whole receiver Aitionally for the straightforwar serial receiver the soft error ecisions mae by the soft IO etector flow into the after channel ecoer an inuce serious error propagation owever by using the PDA-JD etector this problem can be mitigate In [] it is shown that real-value signal moel base PDA etector gives an improve error probability an reuce complexity over the irect PDA approach base on complex-value signal moel he former processes the components of complex signal sequentially an the latter processes both the real an imaginary part in parallel In fact the sequential style correspons to Gauss-Seiel iteration while the parallel is Jacobi iteration Gauss-Seiel is generally lot better than Jacobi in terms of spee of convergence herefore when the real value signal moel is employe in the PDA-JD etector the real-value version of PDA-JD etector has a reuce complexity an improve error 49

5 - Conventional PDA PDA-JD = PDA-JD =8 - - BER SNR in B Fig Comparison of bit error rates of the PDA-JD the conventional PDA an the L etectors for V-BLAS with QPSK an N = N = R BER N = PDA-JD = N =4 PDA-JD = N =8 PDA-JD = N =8 ZF-OSIC N =4 ZF-OSIC N = ZF-OSIC SNR in B Fig Comparison of bit error rates of the PDA-JD an ZF-OSIC etectors for V-BLAS with QPSK an equal number of antennas at transceiver performance than the complex-value PDA-JD etector which is valiate by our numerous simulation results (not shown for page limitation On the other han both the PDA-JD etector an the GPDA etector in [] procees in a parallel style to expan the imension of symbol vector an the former operates on consecutive symbol vectors while the latter only focuses on the interior of one symbol vector PDA base algorithms fall into the category of belief-propagation base iterative methos an most of the algorithms in this category have not yet been well unerstoo he performance analysis an convergence property of PDA base algorithms are still open problems an will be aresse in our future research Number of FLOPS 5 x N = N = 4 N = 8 V CONCLUSION We propose a novel PDA etector using oint etection of consecutive symbol vectors in the same transmit burst for spatial multiplexing IO systems Our analyses an computer simulations show that the PDA-JD etector significantly outperforms the conventional PDA etector especially in high SNR regimes in the context of correlate input bit streams an its worst-case complexity is almost the same orer polynomial of N as that of the conventional PDA etectors In aition the conventional PDA etectors are in fact a special case of the PDA-JD etector Simulation results also show that the PDA-JD etector prefers IO systems with large number of transmit antennas in terms of BER performance REFERENCES [] J Luo K R Pattipati P K Willett an F asegawa Near optimal multiuser etection in synchronous CDA using probabilistic ata association IEEE Commun Lett Vol 5 No 9 pp 6-6 [] D Pham K R Pattipati P K Willet an J Luo A generalize probabilistic ata association etector for multiple antenna systems IEEE Commun Lett Vol 8 No 4 pp [] S Liu an Z ian Near-optimum soft ecision equalization for frequency selective IO channels IEEE rans Signal Process Vol 5 No pp [4] Y Jia C Vithanage C Anrieu an RJ Piechoci Probabilistic ata association for symbol etection in IO systems Electron Lett Vol 4 No pp 8-4 5th Jan Number of simultaneously etecte symbol vectors ( Fig Comparison of complexity of the PDA-JD an the conventional PDA etectors (the special case of = in the PDA-JD etector [5] B ochwal an S ten Brin Achieving near-capacity on a multiple-antenna channel IEEE rans On Communications Vol 5 No pp ar [6] Balur Steingrimsson Zhi-Quan Luo an Kon ax Wong Soft Quasi-aximum-Lielihoo Detection for ultiple-antenna Wireless Channels IEEE rans On Signal Process Vol 5 No pp 7-79 Nov [7] J Frice Sanell J ietzner an P oeher Impact of the Gaussian Approximation on the Performance of the probabilistic ata association IO ecoer EURASIP Journal on Wireless Communications an Networing Vol5 No5 pp Oct 5 [8] Frey D Neeser an James L assey Proper complex ranom process with applications to information theory IEEE rans Inf heory Vol 9 No 4 pp 9- Jul 99 [9] P W Wolniansy G J Foschini G D Golen an R A Valenzuela V-BLAS: an architecture for realizing very high ata rates over the rich-scattering wireless channel Proc ISSSE-98 pp 95- Italy Sept 998 [] K Varanasi Decision feebac multiuser etection: a systematic approach IEEE rans Inf heory Vol 45 No pp 9 4 Jan 999 [] Don Coppersmith Shmuel Winogra atrix multiplication via arithmetic progressions Proc of the nineteenth annual AC conference on heory of computing pp