Approximate Joint MAP Detection of Co-Channel Signals

Transcription

1 Approxmate Jont MAP Detecton of Co-Channel Sgnals Danel J Jaubsn and R Mchael Buehrer Moble and Portable Rado Research Group (MPRG), Wreless@VT, Vrgna Tech, Blacsburg, Vrgna, USA E-mal: {djj,buehrer}@vtedu Abstract We consder jont detecton of co-channel sgnals specfcally, sgnals whch do not possess a natural separablty due to, for example, the multple access technque or the use of multple antennas Iteratve jont detecton and decodng s a well nown approach for utlzng the error correcton code to mprove detecton performance However, the jont maxmum a posteror probablty (MAP) detector may be prohbtvely complex, especally n a multpath channel In ths paper, we present an approxmaton to the jont MAP detector motvated by a factor graph model of the receved sgnal The proposed algorthm s desgned to approxmate the jont MAP detector as closely as possble wthn the computatonal capablty of the recever I INTRODUCTION Detecton of a desred sgnal n the presence of one or more nterferng sgnals s a prevalent problem n dense wreless communcaton systems As a result, desgnng recevers capable of detecton n the presence of nterference has been a very actve area of research wth numerous algorthms proposed n lterature Iteratve multuser detecton problems have been consdered for code dvson multple access (CDMA) [], [], spatal multplexng [] [5], and multuser MIMO [6], [7], among others Recever algorthms generally ft nto one of three categores: lnear flterng, nterference cancellaton, and jont detecton Lnear flterng may be appled n the tme, space, or spacetme dmenson(s) and ncludes technques such as matched flterng, mnmum mean square error (MMSE) equalzaton, and beamformng In systems whch employ spreadng sequences or multple antennas, lnear flterng can be an effectve means of nterference mtgaton, specfcally when the spreadng gan or number of antennas s greater than or equal to the number of sgnals present Interference cancellaton refers to algorthms n whch each user s sgnal s canceled from the receved sgnal after detecton (eg, [8], [9] and the references theren) Lnear flterng combned wth nterference cancellaton may further mprove detecton and has been a very successful approach for spatal multplexng [] Soft cancellaton n conjuncton wth soft decodng of the channel code often referred to as turbo nterference cancellaton has been shown to acheve good results n a CDMA system [] Optmal maxmum a posteror probablty (MAP) detecton s performed by jontly detectng both the desred and cochannel sgnals The detecton stage s separated from decodng and probablstc nformaton s passed between the jont MAP detector and a collecton of sngle user decoders The separaton of detecton and decodng s justfed by message passng algorthms whch operate on a factor graph of the jont probablty densty functon [] Yet, even wth the separaton of detecton and decodng, jont MAP detecton may be prohbtvely complex as a result of hgh-order modulatons, numerous users, or nter-symbol nterference (ISI) A challengng case s detecton n the presence of nonorthogonal, asynchronous nterferng sgnals usng a sngle receve antenna That s, recepton of co-channel sgnals whch do not possess a natural separablty due to a multple access technque (such as CDMA) or multple antennas As a result, lnear flterng and nterference cancellaton are neffectve especally when the sgnal power levels are smlar Jont MAP detecton n such a sgnal model s developed and studed n [] The separablty s acheved due to both frame and symbol tmng offsets and an error correcton code Jont detecton whch accounts for the strongest ISI terms s proposed Thus, the algorthm s exponentally complex n the number of co-channel sgnals For ths reason ts applcaton s lmted to users and BPSK modulaton n [] A large number of users or hgh-order modulatons n addton to ISI due to the asynchronous sgnals maes the optmal jont MAP detector extremely complex Jang and L consder sngle antenna nterference cancellaton n a frequency selectve, multple access channel [] The same channel code, nterleaver, and modulaton s assumed for all co-channel sgnals Sgnal separablty s obtaned through the ndependence of each user s multpath channel Jang and L propose a concurrent MAP (CMAP) algorthm n whch a Gaussan approxmaton s used for co-channel nterference and MAP equalzaton for ISI The CMAP algorthm s compared to jont MAP detecton, the Rae Gaussan method proposed n [4], and soft nterference cancellaton wth MAP equalzaton Whle the CMAP algorthm s the state-of-the-art n addressng the dffcult detecton problem descrbed above, performance s degraded when the Gaussan approxmaton s made for strong co-channel nterference terms In ths paper, we present a new approxmaton to the jont MAP detector whch s motvated by a factor graph model of the receved sgnal The proposed algorthm s desgned to approxmate the jont MAP detector as closely as possble wthn Due to the complexty of the jont MAP detector, ths method s only evaluated for two users wth BPSK modulaton n []

2 the computatonal capablty of the recever The complexty of the algorthm s adjustable and can be set to account for the capabltes of the recever, the desred performance, or the dffculty of the detecton tas The paper s organzed as follows The system model s presented n Secton II followed by development of the MAP detector n Secton III The complexty of the proposed algorthm s compared wth algorthms from the lterature n Secton IV and a detaled descrpton of the proposed algorthm s provded n Secton V The algorthms are compared va smulaton n Secton VI and conclusons are drawn n Secton VII Notaton: Let x denote a column vector x = [x,, x K ] T We use the shorthand x to denote the summaton over the doman of x Smlarly, x denotes the summaton over the doman of the vector x and x\x denotes the summaton wth respect to all varables except x II SYSTEM MODEL In ths wor we consder sngle antenna recepton of U cochannel sgnals (users) Let the nformaton bts, coded bts, and symbols of the uth user be denoted by column vectors b (u), c (u), and x (u), respectvely We defne the collecton of these terms for all users as B = [b,, b (U) ] C = [c,, c (U) ] X = [,, x (U) ] The nth sample of the receved sgnal s gven by U L r n = h (u) x (u) l n l + w n, u= l= where h (u) = [h (u), h(u),, h(u) L ]T denotes the combned effect of the multpath channel and the transmt pulse for the uth user, L s the number of channel taps, and {w n } N are ndependent and dentcally dstrbuted crcularly-symmetrc complex Gaussan random varables wth varance σ The collecton of all receved samples s denoted r = [r,, r N ] In general the transmtted sgnals may be symbol-asynchronous For the sae of notatonal smplcty, the model provded n maes a number of assumptons about the receved sgnal for example, that the channel duraton of the users L s dentcal and that the receved sgnal s sampled at a sngle sample per symbol However, the multuser detecton and equalzaton algorthms presented n ths paper are applcable to the more general cases III MAP DETECTION The goal of the recever s to detect all nformaton bts B gven observaton r Because of the complexty of sequence detecton of B, we desre to perform MAP symbol-by-symbol (n our case, bt-by-bt) detecton The detector for the th bt of user u s gven by ˆb (u) = arg max b (u) B\b (u) p(b r), where the margnal s computed for b (u) Accordng to Bayes rule, s equvalent to ˆb (u) = arg max b (u) f (r, B), () B\b (u) where the term / f (r) s a constant whch has been removed By the Total Probablty Theorem, () can further be expressed as a margnalzaton over the full jont dstrbuton as gven by ˆb (u) = arg max b (u) X,C,B\b (u) f (r, X, C, B) (4) The margnalzaton n (4) cannot be performed drectly, but an teratve mplementaton of the sum-product algorthm s well suted for ths tas A Probablty Dstrbuton Tang nto account condtonal ndependence of the varables, the jont dstrbuton s gven by N f (r, X, C, B) = f (r n,, x (U) ) U p(x (u) c (u) )p(c (u) b (u) )p(b (u) ) (5) u= Factorzatons of the modulaton p(x (u) c (u) ) and code p(c (u) b (u) ) constrants have been explored n the lterature (see, for example, [], [5]) From the lelhood functon for each term r n s dependent on a subset of the symbols We defne, x (u) [n] = [x (u) n L+,, x(u) n ] T to denote the symbols from user u whch have components n the r n sample The dstrbuton s then gven by N f (r, X, C, B) = f (r n [n],, x(u) [n] ) U p(x (u) c (u) )p(c (u) b (u) )p(b (u) ) (6) u= Soft output MAP equalzaton of an ISI channel may be accomplshed va the BCJR algorthm [6] Ths algorthm was extended to the case of jont detecton of a desred and co-channel sgnal n ISI by Moon and Gunther [] The algorthm reles on the ntroducton of state varables m,, m N nto the lelhood functon as follows: N f (r n [n],, x(u) [n] ) = N m n f (r n, m n+ n,, x (U) n, m n )p(m ) (7) where m n = [ ) n L+,, x n,, x(u) n L+,, x(n n ]T At a hgh level, local margnals for the symbols are computed by a forward and bacward pass of the BCJR algorthm

3 n f rn x n n f rn x n m n p ( c ) p ( c b ) p ( b ) n f rn x n n f rn x n p ( x c ) p ( c b ) p ( b ) p ( c ) p ( c b ) p ( b ) n f rn x n m n n f rn x n p ( x c ) p ( c b ) p ( b ) m n x n f r n x n f r n Fg Factor graph of f (r, X, C, B) for U = and L = 4 based on the factorzaton n (6) Fg Factor graph of f (r, X, C, B) for U = and L = 4 based on the state space model factorzaton of (7) substtuted nto (6) (also nown as the forward-bacward algorthm) The forward messages are gven by and α(m + ) = β(m ) = m n f (r n, m n+ n,, x (U) n, m n )p(m ) (8) N n= m n+ f (r n, m n+ n,, x (U) n, m n ) (9) The messages may be defned recursvely as gven by α(m + ) = γ(m +, m )α(m ) () m and β(m ) = γ(m +, m ) β(m + ), () m + where γ(m +, m ) = f (r, m +,, x (U), m ), α(m ) = p(m ) =, and β(m N ) = A margnal for a partcular symbol x (u) s gven by α(m )γ(m, m + ) β(m + ) The jont [] (U ),, x [] \x (u) MAP detector s developed for the case of a receved sgnal wth two samples per symbol n [] B Factor Graph Model The sum-product algorthm performs effcent margnalzaton by explotng the factorzaton of the jont dstrbuton f (r, X, C, B) As an example, consder the case of U = and L = 4 The factor graph of the jont dstrbuton n (6) s gven n Fg Smlarly, the factor graph of the jont dstrbuton wth the ntroducton of the state varables s shown n Fg In Fg, f rn denotes the factor f (r n [n], x [n]) and, n Fg, f rn denotes the factor f (r n, m n+ x n,, x n (U), m n ) We refer to the factor graphs n Fg and as the fully connected graph and the state-space model (SSM) graph, respectvely The generalzaton of the BCJR algorthm to the factor graph of the jont dstrbuton s gven by the sum-product algorthm [5] The factor nodes p(x (u) c (u) )p(c (u) b (u) )p(b (u) ) are further factored when mplementng the sum-product algorthm The factor nodes related to the observatons f rn and the symbol varable nodes mae up the detecton bloc of the factor graph The fully connected graph contans cycles wthn the detecton bloc; the SSM graph elmnates these cycles Cycles have a negatve mpact on the convergence of the sumproduct algorthm In Secton V, we develop an algorthm to reduce the complexty of jont MAP detecton based on the fully connected factor graph of Fg In Secton VI, we quantfy the loss n performance when performng message passng on the fully connected graph versus the SSM graph IV COMPLEXITY For both graphs, the complexty assocated wth each of the detecton factor nodes s O(M U L ) where M s the modulaton order of the symbols (assumed to be the same for each user) The complexty s exponental n the number of users and channel taps and therefore complexty prohbts use of the jont MAP detector n many potental co-channel sgnal scenaros Specfcally when ether M >>, U >>, or L >> and especally when ths s the case for two of these terms As an example, the complexty for QPSK, 4 users, and 4 channel taps (e, M = 4, U = 4, and L = 4) s O( 9 )

4 Because of the problem of complexty wth jont MAP detector, approaches wth lower complexty have been consdered for ths problem Interference Cancellaton: Cancellaton may be performed based on ether hard or soft decsons Detecton s performed startng wth the strongest sgnal and contnung to the weaest Soft cancellaton may be combned wth teratve processng to teratvely mprove the soft estmates Rae Gaussan: Ths method was proposed n [4] for nterleave-dvson multple access In ths method, for the detecton of symbol x (u) all other symbols are modeled as Gaussan random varables Ths ncludes the symbols of all other users and all other symbols of the desred user, e, {x (u ) } u u, The mean and varance of the Gaussan dstrbuton are computed from the extrnsc symbol probabltes obtaned from demodulaton and decodng Concurrent MAP (CMAP): Ths method was proposed n [] to mprove upon the performance of the Rae Gaussan method In ths method, MAP equalzaton of each user s sgnal s performed whle all other user s sgnals are modeled as Gaussan random varables Thus, the complexty of the method s O(U M L ), that s, lnear n the number of users and exponental n the number of channel taps Vsual comparsons of the Rae Gaussan and CMAP algorthms are gven usng factor graphs The factor node f r from the example n Fg s used to represent the approxmatons made by the Rae Gaussan and CMAP algorthms when computng the message m fr n Fgs and 4, respectvely The sngle arrow represent messages contanng dscrete dstrbutons and the double arrow represent messages whch contan a mean and varance based on a Gaussan approxmaton The graphcal models of Fgs and 4 motvate a new approach n whch the dstrbuton of weaer terms n the sgnal component of r n are modeled as Gaussan random varables Sum-product message passng s performed for the stronger terms n r n Ths hybrd approach has a complexty determned by the number of messages wth dscrete dstrbutons and mantans a sngle, connected graph The graphcal model for the hybrd approach s shown n Fg 5 where symbols, x, x, and x are the strongest component n r for users and (e, the power of the channel coeffcent s strongest for these terms) Ths model s motvated by common transmt pulse shapes whch contan the majorty of ther energy wthn the center of the pulse and multpath channels whch often exhbt an exponental decay A detaled descrpton of the algorthm s provded n the followng secton h (u) l V APPROXIMATE MAP DETECTION ALGORITHM Consder a generc nterference model (to represent ntersymbol nterference, co-channel nterference, or both) n whch x n ======= x ======= m fr f r x ======= fr nx Dscrete Dst x Mean and Varance Fg Factor graph motvated representaton of the Rae Gaussan method x x m fr f r x fr Dscrete Dst x Mean and Varance Fg 4 Factor graph motvated representaton of the CMAP method As shown, ths factor s a slce of the overall graph to mplement MAP equalzaton of user whle modelng the nterference from user x ========= x m fr f r x ========= fr Dscrete Dst x Mean and Varance Fg 5 Factor graph motvated representaton of the approxmate MAP method developed n ths wor

5 K sgnal components x, x,, x K are receved wth channel coeffcents h, h,, h K, respectvely The receved sgnal s gven by K y = h x + w = where y represents one sample of a larger sequence of receved samples and the nose w s modeled as a crcularly symmetrc complex Gaussan random varable wth varance σ The factor assocated wth the receved sample y s gven by ( K ) f (y x,, x K ) = CN y; h x, σ where the channel coeffcents and the nose power σ are assumed to be nown The message from factor node f y to varable node x s denoted m fy x Smlarly, the message from varable node x to factor node f y s denoted f y Accordng to the sum-product algorthm, the message m fy x s gven by m fy x (x ) = f (y x,, x K ) f y (x ) x\x (x ) The proposed algorthm modfes the sum-product algorthm computatons as follows: The mean and varance of the nput messages are computed accordng to µ x = x f y (x ) x σ x = x µ x f y x = for all =,, K For computaton of the outgong message m fy x (x ), the remanng varables for are sorted by ther channel coeffcent power h Let the set A ndex the varables assocated wth the strongest channel coeffcents These varables reman a part of the local margnalzaton as gven n The number of varables n the set A wll depend on the acceptable complexty n mplementaton The ndces of the weaer components are ncluded n the set B and the dstrbutons of these varables are approxmated by Gaussan random varables to elmnate the margnalzaton over these varables Let the varables assocated wth sets A and B be gven by x A and x B, respectvely The message s computed wth the followng approxmate sum-product computaton: where m fy x (x ) = x A f (y x, x A ) f y (x ) A f (y x, x A ) = ( CN y; h x + h x + h l µ xl, σ + h l σ x l ) A l B l B () FER Sngle User Jont MAP (SSM) Jont MAP (FG) Approx MAP Concurrent MAP Soft IC SNR (db) Fg 6 FER comparson of the jont MAP state-space model (Jont MAP (SSM)), the jont MAP fully connected factor graph (Jont MAP (FG)), the proposed approxmate MAP algorthm (Approx MAP), CMAP (Concurrent MAP), and soft nterference cancellaton (Soft IC) algorthms wth SIR= db Ths algorthm s appled to the computaton of the sumproduct messages at each of the detecton factors n Fg The complexty of the proposed algorthm for each factor s O(UL M A + ) where A s the number of symbols ncluded n the set A Thus, by choosng the sze of A, the complexty of the algorthm may be adjusted to match the computatonal capablty of the recever and performance requrements VI NUMERICAL RESULTS We frst smulate the performance for a scenaro n lne wth the one consdered n []: two users (U = ) each employng BPSK modulaton (M = ) ISI results from symbol tmng offsets between the users and a transmt pulse wth a duraton of four symbol perods (L = 4) The selecton of these parameters allows us to smulate the jont MAP detector for the purpose of comparson The smulaton parameters are summarzed as follows: Code: /-rate turbo code wth 5 coded bts Modulaton: BPSK Pulse: Square root rased cosne wth L = 4 and roll-off factor 5 A relatve tme delay between the users of T/4 s chosen where T s the symbol perod A relatve phase offset between the channel coeffcents of the users of π/6 s chosen 5 teratons of message passng are performed The FER performance s shown n Fg 6 for the jont MAP (wth the SSM and fully connected factor graphs), the proposed novel approach, CMAP, and soft nterference cancellaton algorthms The approxmate MAP algorthm s mplemented wth A = Thus, the proposed approxmate MAP algorthm and the CMAP algorthm have the same order of complexty (per teraton) The performance of the fully connected factor graph demonstrates a loss of about 5 db compared to the SSM factor graph The proposed approxmate

6 SNR = 6 db FER FER SNR = 6 db SNR = db SNR = db Proposed CMAP 4 4 SIR (db) Proposed CMAP Number of teratons Fg 7 FER of the proposed and CMAP algorthms wth respect to SIR Both sgnals are detected, and the FER of the desred sgnal s shown Ten teratons of the recever are performed Fg 8 FER of the proposed and CMAP algorthms wth respect to the number of teratons performed The SIR s -4 db MAP algorthm s based on the fully connected graph and we observe that t acheves nearly dentcal performance to the recever whch uses exact sum-product computatons At a FER of the proposed approxmate MAP approach and Concurrent MAP approach demonstrate losses of 5 db and 5 db, respectvely, compared to jont MAP detecton based on the SSM We observe that the Soft IC method becomes lmted by nterference as sgnal-to-nose rato (SNR) ncreases We also consder a -user scenaro wth QPSK modulaton wth a 4-tap multpath channel The average power n each multpath component s gven by [644, 7, 87, ] In Fg 7 the FER of the proposed approxmate MAP algorthm and the CMAP algorthm s shown Both SNR and sgnal-to-nterference rato (SIR) are computed wth respect to the nstantaneous power n the multpath channel The most sgnfcant mprovement n FER s acheved by the proposed approxmate MAP algorthm when the sgnals have smlar power levels ( SIR db) and the SNR s hgh In Fg 8, the FER s shown wth respect to the number of teratons where we observe that the proposed algorthm converges - teratons faster than CMAP Thus, for A = L, the proposed algorthm reduces computatonal complexty by 4% due to faster convergence VII CONCLUSION In ths paper, an algorthm s developed whch approxmates jont MAP detecton and equalzaton n co-channel nterference The approxmate MAP algorthm s based on a fully connected factor graph of the jont probablty dstrbuton The algorthm was shown to operate wthn 5 db of the jont MAP state-space model recever where the degradaton n performance was due to the assocated factor graph model Addtonally, the proposed algorthm both mproves performance and reduces complexty when compared to the state-of-the-art REFERENCES [] X Wang and H Poor, Iteratve (turbo) soft nterference cancellaton and decodng for coded CDMA, IEEE Trans Commun, vol 47, pp 46 6, Jul 999 [] J Boutros and G Care, Iteratve multuser jont decodng: unfed framewor and asymptotc analyss, IEEE Trans Inf Theory, vol 48, pp 77 79, Jul [] B Hochwald and S ten Brn, Achevng near-capacty on a multpleantenna channel, IEEE Trans Commun, vol 5, pp 89 99, Mar [4] S Hayn, M Sellathura, Y de Jong, and T Wlln, Turbo-MIMO for wreless communcatons, IEEE Commun Mag, vol 4, pp 48 5, Oct 4 [5] R Vsoz and A Berthet, Iteratve decodng and channel estmaton for space-tme BICM over MIMO bloc fadng multpath AWGN channel, IEEE Trans Commun, vol 5, pp 58 67, Aug [6] J Lee, H Kwon, and I Kang, Interference mtgaton n MIMO nterference channel va successve sngle-user soft decodng, n Proc Inform Theory Appl (ITA) Worshop,, pp 8 85 [7] P Hammarberg, F Ruse, and O Edfors, Iteratve recevers wth channel estmaton for mult-user MIMO-OFDM: complexty and performance, EURASIP J on Wreless Commun and Netw, vol, pp 7, Mar [8] H Arslan and K Molnar, Cochannel nterference suppresson wth successve cancellaton n narrow-band systems, IEEE Commun Lett, vol 5, no, pp 7 9, Feb [9] J Andrews, Interference cancellaton for cellular systems: a contemporary overvew, IEEE Wreless Commun Mag, vol, no, pp 9 9, Apr 5 [] E Bgler, R Calderban et al, MIMO Wreless Communcatons Cambrdge Unversty Press, 7 [] H Wymeersch, Iteratve Recever Desgn Cambrdge Unv Press, 7 [] T Moon and J Gunther, Multple-access va turbo jont equalzaton, IEEE Trans Commun, vol 6, no, pp, Oct [] W Jang and D L, Iteratve sngle-antenna nterference cancellaton: algorthms and results, IEEE Trans Veh Technol, vol 58, no 5, pp 4 4, Jun 9 [4] L Png, L Lu, and W K Leung, A smple approach to near-optmal multuser detecton: nterleave-dvson multple-access, n Proc IEEE Wreless Commun Netw Conf, vol,, pp 9 96 [5] F R Kschschang, B J Frey, and H A Loelger, Factor graphs and the sum-product algorthm, IEEE Trans Inf Theory, vol 47, no, pp , Feb [6] L Bahl, J Coce, F Jelne, and J Ravv, Optmal decodng of lnear codes for mnmzng symbol error rate (corresp), IEEE Trans Inf Theory, vol, no, pp 84 87, Mar 974