Publication in the conference proceedings of EUSIPCO, Florence, Italy, 200

Andrieu, C

McGeehan, JP

Piechocki, RJ

Sandell, M

English

In this paper we develop an efficient algorithm for inference in Factorial Hidden Markov Models (FHMM), which is particularly suitable for turbo equalisation in Multiple Input-Multiple Output (MIMO) systems. The proposed PDABCJR algorithm can be viewed as a generalisation of the PDA algorithm, which in its basic form handles single latent variables only. Our generalisation replaces each of the single latent variables with a HM

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                          Piechocki, R. J., Andrieu, C., Sandell, M., & McGeehan, J. P. (2006). PDA-BCJR algorithm for factorial hidden Markov models with application toMIMO equalisation. In European Signal Processing Conference (EUSIPCO),Florence, Italy. European Association for Signal Processing (EURASIP).Link to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.htmlTake down policyExplore Bristol Research is a digital archive and the intention is that deposited content should not beremoved. However, if you believe that this version of the work breaches copyright law please contactopen-access@bristol.ac.uk and include the following information in your message:• Your contact details• Bibliographic details for the item, including a URL• An outline of the nature of the complaintOn receipt of your message the Open Access Team will immediately investigate your claim, make aninitial judgement of the validity of the claim and, where appropriate, withdraw the item in questionfrom public view.PDA-BCJR ALGORITHM FOR FACTORIAL HIDDEN MARKOV MODELS WITHAPPLICATION TO MIMO EQUALISATIONRobert J. Piechocki†, Christophe Andrieu∗, Magnus Sandell♠ and Joe McGeehan†♠† University of Bristol, Centre For Communications ResearchWoodland Road, MVB, BS8 1UB, Bristol, UK.E-mail: r.j.piechocki@bristol.ac.uk, Tel. +44 117 9545203, Fax. +44 117 9545206∗ School of Mathematics, University of Bristol, BS8 1TW, UK♠ Toshiba TRL Labs, Bristol, BS1 4ND, UKABSTRACTIn this paper we develop an efficient algorithm for infer-ence in Factorial Hidden Markov Models (FHMM), whichis particularly suitable for turbo equalisation in Multiple In-put - Multiple Output (MIMO) systems. The proposed PDA-BCJR algorithm can be viewed as a generalisation of thePDA algorithm, which in its basic form handles single la-tent variables only. Our generalisation replaces each of thesingle latent variables with a HMM.1. INTRODUCTIONHigh speed wireless communications systems require someform of equalisation to compensate for the dispersive natureof the wireless channel. The subject of (single transmit an-tenna) equalisation has been extensively researched and aplethora of technical papers published. Recently, so calledMIMO wireless systems have gained an enormous interest.Whereas, the sub-optimal equalisation techniques with goodperformance are still sought after, the MIMO sub-optimalequalisation remains a real challenge. Nonetheless, two so-lutions are frequently cited as state of the art: sphere de-coders [1][2] and the probabilistic data association (PDA)[3]. The PDA algorithm was first introduced in communica-tions literature as a sub-optimal multi-user detector (MUD)in CDMA systems. Simultaneously to MIMO, turbo prin-ciple has reached engineering maturity, even thought sometheoretical description remains to be provided. Turbo equal-isation can offer significant gains, especially when equalis-ing ill-posed channels. Turbo equalisation requires all sub-blocks in the system to exchange so-called soft informa-tion i.e. full sets of marginal posterior distributions, ratherthan point estimates (i.e. hard decisions). Both PDA andsphere decoders can operate in the soft mode. As aforemen-tioned, the challenge remains how to efficiently decode wide-band MIMO signals at a reasonable computational cost. TheMIMO equalisation problem can be formulated as inferenceon a Hidden Markov Model, for which an exact and efficientalgorithm is well known: forward-backward (aka BCJR).Unfortunately, the complexity of BCJR increases exponen-tially with the number of transmit antennas and the length ofthe channel measured in symbol intervals. Reduced com-plexity solutions based on heuristics [4] and Monte Carloapproximations [5] have been proposed in the literature toaddress this problem.An interesting line of research was presented in [6],where a more appropriate Factorial Hidden Markov Model(FHMM) was used for MIMO equalisation. The authors useda Structured Variational Inference (SVI) approximation of[7] to compute marginal probabilities required in turbo equal-isation. In this contribution we propose a different type ofapproximation for inference in FHMM based on the proba-bilistic data association (PDA) principle. Simulation resultssuggest improved results as compared to the SVI.2. PROBLEM FORMULATION AND ESTIMATIONTASKSWe consider a MIMO system with N;(1≤ n≤ N) transmitand M;(1≤ m≤M) receive antennas. The system signalsover a wideband channel, modelled as an equivalent multidi-mensional (N×M) FIR filter with L taps:H= hT1,1 · · · hT1,N.........hTM,1 · · · hTM,N (1)Where:hm,n =(h(τ=0)m,n ,h(τ=1)m,n , . . . ,h(τ=L−1)m,n)T(2)We introduce an additional notation: H:,n - is the nth blockcolumn of H i.e:H:,n ≡ hT1,n...hTM,nAt each time instant t, the nth antenna signals a symbolx(t)n , that belongs to a digital modulation with Ω elements:x(t)n ∈ {a1, . . . ,aΩ}. A collection of L consecutive symbolsforms a vector: xtn =(x(t)n ,x(t−1)n , . . . ,x(t−L+1)n)T. StackingN such vectors produces another vector:x(t) =((x(t)1)T,(x(t)2)T, . . . ,(x(t)N)T)T. With this no-tation, the received signal at time t is:y(t) =Hx(t)+n(t) (3)Where: n(t) is a Gaussian noise i.e. n(t) ∼N (0,σ 2n I)We make an assumption that our system operates in aturbo configuration. This is conventional and not detailedhere. It suffices to point out that in a turbo system the MIMOdecoder has to supply a full set of marginal posterior proba-bility mass functions (or an approximation to it):{f(x(1)n |y), f(x(2)n |y), . . . , f(x(T )n |y)}(4)The above problem can be elegantly described in termsof Probabilistic graphical models.2.1 Probabilistic Graphical ModelsProbabilistic graphical models (PGM) have recently emergedas a universal language describing problems in decision mak-ing and/or estimation in the presence of uncertainty. PGMsare used (and were often independently developed) in seem-ingly unrelated disciplines ranging from: Bio-informatics(Bio-statistics), Information retrieval, Speech processing andimage processing Communications to Forensic Science andmany more. It has to be emphasised that PGMs themselvesare nothing more than pictorial representations of families ofprobability density functions, and as such, they do not offerany solutions. However, PGMs may provide an insight intoexisting models, motivation for new models and algorithms,and ease of transferring solutions from one discipline to an-other. There are three types of PGMs [8] that are used mostoften: Directed Acyclic Graphs (DAG), Undirected Graphs(UG) and Factor Graphs (FG). The diversity arises from afact that some problems can be accurately described onlyby one type of graph. Other problems are well describes bymore than one type of graph. Such example is the problemconsidered in this contribution. Reference [6] outlined theproblem using FG, however here we opt for DAG. DAGs areuseful in modelling causal relationships. In communicationsthe causal relationship is appealingly natural: the receivedwaveform arises as a response to the transmitted waveformand realisation of a medium (wireless channel in our case).A constituent model that will build our overall FHMMis a factorial model depicted in figure 1. In DAGs the jointdistribution factors in such a way that each factor (of total D)represents the conditional distribution of a ”child” given its”parents”: f (x1 . . . ,xN) =D∏i=1f (xi |pai )In our case this becomes:f (y,x1, . . . ,xN ,H) = f (y |x1, . . . ,xN ,H) f (H)N∏i=1f (xi)(5)  1x  2x  Nx  y H Figure 1: Directed Acyclic Graph (DAG) of a factorial model(MIMO and MUD/CDMA problems).The task is to calculate a set of marginal posterior proba-bilities for all xi, i ∈ {1,2, . . . ,N}.f (xi |y ) =∫H∑x−if (xi,x−i,H |y )dH=∫H∑x−if (xi |x−i,H,y ) f (x−i |H,y ) f (H |y )dH (6)Where ”−i” stands for ”all except the ith”. This alsoallows to treat the channel as a random variable. As wedo not have access to a particular realisation of the chan-nel H (H is latent), this variable is integrated out. Unfor-tunately, such integral is impossible to calculate in a typi-cal scenario. The first solution (ubiquitous in practice) isto train the channel using known at the receiver data. Thisamounts to approximating the posterior distribution of thechannels with one ”spike” centered perhaps at an ML esti-mate: f (H |y ) = δ(H− HˆML)∫H∑x−if (xi |x−i,H,y ) f (x−i |H,y )δ(H− HˆML)dH=∑x−if(xi∣∣∣x−i,HˆML,y) f (x−i ∣∣∣HˆML,y)(7)A better way to account for the channel uncertainty is viaan EM algorithm as discussed in section 3.2.Even though training simplifies the problem, anotherproblem remains to be tackled. This (computational) prob-lem is brought about by a fact that the joint a posteri-ori distribution is not separable: f (x1,x2, . . . ,xN |H,y ) 6=N∏n=1f (xN |H,y ) since all the variables {x1,x2, . . . ,xN} co-operate in setting the mean of the observation y i.e. E {y}=h(x1,x2, . . . ,xN), which is a root cause of numerical com-plexity for this type of problems.A second component needed for our model is a HMMmodel of figure 2. This model is equivalently described byeq. (8).f(y(1), . . . ,y(T ),s(1), . . . ,s(T ),H)=f (H) T∏t=1f(y(t)∣∣∣s(t) ,H) f (s(t) ∣∣∣s(t−1)) (8)where the states s(t) are defined as s(t) ≡{x(t),x(t−1), . . . ,x(t−L+1)}.    ( )1−ts ( )1−ty ( )ts ( )ty ( )1+ts ( )1+ty H Figure 2: DAG of a Hidden Markov Model.Both single antenna and MIMO equalisation can be rep-resented by a HMM. In a case of known channels (pictorially:if the channel variable H was shaded) the task of marginal-isation is achieved by a forward-backward algorithm (morespecifically a variant known as BCJR).Our overall FHMM model arises by replacing the singlerandom variables {xn} in figure 1 with HMM in figure 2 - asdepicted in figure 3.f(y(1), . . . ,y(T ),s(1)1 , . . . ,s(T )1 , . . . ,s(1)N , . . . ,s(T )N ,H)=f (H) T∏t=1[f(y(t)∣∣∣s(t)1 , . . .s(t)N ,H) N∏n=1f(s(t)n∣∣∣s(t−1)n )](9)In a MIMO system the symbols sent from all N trans-mit antennas are a priori independent (which is a reasonableassumption when interleaves are used). However, the a pos-teriori distribution is again not separable for the very samereason, which again leads to exuberant complexities. ( )1−tNs  ( )1−ty ( )tNs  ( )ty ( )1+tNs  ( )1+ty ( )11−ts ( )ts1  ( )11+ts ( )12−ts ( )ts2  ( )12+ts H Figure 3: DAG of a Factorial Hidden Markov Model.3. THE PDA-BCJR ALGORITHMIn this section we detail the PDA-BCJR algorithm. We alsodescribe modifications to the BCJR algorithm that are re-quired for the use in PDA-BCJR algorithm.In each iteration the algorithm updates a set of marginalposterior distributions{f(x(1)n |y), f(x(2)n |y), . . . , f(x(T )n |y)}via a modified BCJR algorithm. This set pertains to allT ;(1≤ t ≤ T ) symbols sent from the nth transmit antenna. Indoing so, the modified BCJR algorithm exchanges two setsof moments: input (µn,in,Σn,in) and output (µn,out ,Σn,out).The input moments in each iteration are calculated by sum-ming all but the nth output moments, as detailed in table 1.In standard BCJR algorithm as used in MIMO (soft in -soft out) equalisation, the posterior transition probability isfactored:f (s′,s |y)∝ f (s′,y(1:t−1))︸ ︷︷ ︸α(t)f (s ∣∣s′ ) f (y(t) ∣∣s′,s)︸ ︷︷ ︸γ(t)f(y(t+1:T ) |s)︸ ︷︷ ︸β (t)(10)and calculated recursively for numerical efficiency as:α(t) (s) =∑s′α(t−1)(s′)γ(t)(s′,s) (11)β (t−1) (s′)=∑s′β (t) (s)γ(t) (s′,s) (12)the middle factor γ is calculated as:γ(t)(s′,s)∝ exp( −12σ2∣∣∣yt − Hˆxi, j∣∣∣2) N∏n=1f(x(t)n)(13)In the modified BCJR (for the use in PDA-BCJR), themiddle factor (to take into account non zero mean and nondiagonal covariance) is calculated as:γ∗(t)(s′,s)∝ exp(−(z(t)n)H (Σ(t)n,in)−1(z(t)n))f (xn)(14)Where:z(t)n = y(t)− Hˆ(:,n)xn−µ(t)n,inThe α’s and the β ’s are calculated as in the standardBCJR algorithm eq (11) and (12). Additionally, the modi-fied BCJR algorithm calculates ”output moments” (i.e. themean and the covariance of the received signal in an absenceof the noise and other antennae signals) at each time instant:µ(t)n,out = E{Hˆ:,nxn}=S∑j=1K∑k=1Hˆ:,nxn f(s′,s |y) (15)Σ(t)n,out = E{(Hˆ:,nxn−µ(t)n,out)(Hˆ:,nxn−µ(t)n,out)H}=S∑j=1K∑k=1(Hˆ:,nxn−µ(t)n,out)(Hˆ:,nxn−µ(t)n,out)H f (s′,s |y )(16)To initialise the algorithm two options are possible: 1)f (s′,s |y ) = 1SK , which indeed is the only option in prac-tice on the first turbo iteration; 2) f (s′,s |y ) = I (s′,s) f(x(t)n)S ,which becomes an option after the first turbo iteration whenthe ”prior” distributions f(x(t)n)become available. I (s′,s)is an indicator function that is 1 if xn is compatible withstate transition and 0 otherwise. For efficient implementa-tion of the modified BCJR algorithm, one realises that boththe branch metrics Hˆ:,nxn and the posterior probabilitiesf (s′,s |y ) ∝ α(t)β (t)γ(t) are pre-calculated as a part of stan-dard procedures of BCJR algorithm.3.1 Connections with Variational approximationIt is interesting to relate the proposed PDA-BCJR algorithmto the structured variational approximation (SVI) of [7] [6].The SVI approximation associates a fictitious observationwith each Markov Chain, effectively decupling the chains.Our PDA-BCJR algorithm does the same, however it calcu-lates a second moment in addition to the first calculated byVI. To see this and for simplicity of presentation we comeback to the factorial model of figure 1 (relations in FHMMare analogous).In a factorial model a set of variational updates is givenby:fV I (xi = aω |y)appox∝ exp(−1σ2ψThiaω − 12σ 2 |hiaω |2)(17)Table 1: The PDA-BCJR algorithm summary”‘Outer recursions”’for I = 1 : IPDAmax”‘PDA recursions”’for n = 1 : Nµn,in =N∑j=1, j 6=nµ j,out ; Σn,in =N∑j=1, j 6=nΣ j,out[f(x(t),t=1:Tn |y),µn,out ,Σn,out]= BCJR∗(y,Hˆ:,n, f(x(t),t=1:Tn),µn,in,Σn,in)end, endwhere:ψ = y−N∑j=1, j 6=ih j〈x j〉x j |yor equivalently:fV I (xi = aω |y)appox∝ exp( −12σ 2|ψ−hiaω |2)(18)The PDA algorithm updates this distribution via:fPDA (xi = aω |y)appox∝ exp((ψ−hiaω)T M−1 (ψ−hiaω))(19)where:M=σ22I+N∑j=1, j 6=ihTj h jVar{x j} (20)The relation is:MPDA =MV I +N∑j=1, j 6=ihTj h jVar{x j} (21)Hence, variational approximation neglects to update the co-variance when compared to PDA. Here we mean a ”naivevariational approximation” a.k.a mean field approximation.3.2 EM algorithms for Semi-blind equalisationIn the previous section we have presented the PDA-BCJR al-gorithm with an assumption that the channels were known (orestimated). As aforementioned a better approach to accountfor the channel uncertainly is an iterative procedure EM -Expectation Maximisation. The EM algorithm in our casecan be traced directly to the Baum-Welch procedure knownin speech processing problems [9]. We choose set up theproblem so that the hidden (latent) data are the states (setsof transmitted symbols). To provide the algorithm with areasonable initial estimate, a few known symbols are trans-mitted at the beginning of each frame. In standard iterativeprocedure E step is performed first. In our case this is donevia PDA-BCJR algorithm. M step is identical to that of [7]and [6].4. SIMULATIONS RESULTSA MIMO system has been simulated with 3Tx and 3Rx an-tennas. The data bust of 768 bits has been encoded by a sin-gle RCC (7,5) encoder and interleaved prior to transmission.The BPSK signalling has been used and the channel taps aremodelled as hτm,n ∼N(0,L−12)with L= 3 (uniform powerdelay profile).Three turbo MIMO detectors have been investigated. Thedecoders differ in soft in - soft out equaliser block. Thefirst is a full complexity BCJR. This is an optimal decoderfor this case, and it operates over HMM (neglecting FHMMstructure). The two other decoders are reduced complexityversions. The first reduced complexity decoder is based onstructured variational approximation [7] and [6]. The secondis our PDA-BCJR decoder.Figure 4 depicts the BER performance of the developedreduced complexity PDA-BCJR over FHMM, and the per-formance of the optimal BCJR over HMM. After 5 iterationsthe performance is essentially indifferent in both cases. SVI(not depicted) also performs very well in this case.Figure 5 shows performances of semi-blind versions(short training with EM) of the three decoders. M step wasperformed on the outputs of the channel decoder i.e. EMwas incorporated into the ”turbo loop”. First 18 symbolswere known to the receiver to initialise EM. In this casePDA-BCJR clearly outperforms structured variational infer-ence decoder.0 2 4 6 8 10 12 1410−510−410−310−210−1100SNR [dB]BERPDA−BCJR iter #1PDA−BCJR iter #2PDA−BCJR iter #5BCJR over HMM iter #1BCJR over HMM iter #2BCJR over HMM iter #5Figure 4: Performance comparison of PDA-BCJR overFHMM and BCJR over HMM.5. CONCLUSIONSWe have developed a generalisation to a popular PDA algo-rithm. The algorithm iterates on entire hidden Markov mod-els, and as such is suitable for MIMO equalisation problems.2 4 6 8 10 12 14 16 18 20 22 2410-410-310-210-1100SNR [dB]BERPDA-BCJR, Iter #1PDA-BCJR, Iter #6SVI, Iter #1SVI, Iter #6BCJR over HMM, Iter #1BCJR over HMM, Iter #6Figure 5: Performance comparison in a semi-blind setting(all via EM): PDA-BCJR over FHMM, SVI over FHMM andBCJR over HMM.For degenerate Markov chains f(s(t)∣∣∣s(t−1))= f (s(t)) (i.e.no time dispersion), the developed PDA-BCJR becomes astandard PDA [3]. The PDA-BCJR and the FHMM can serveas a framework for the development of further approximatealgorithms by introducing the next level of approximationwithin the BCJR step.AcknowledgmentsThe authors would like to thank C. Vithanage for helpfulcomments and Toshiba TRL Bristol UK for supporting thiswork.REFERENCES[1] B.M. Hochwald and S. ten Brink, ”Achieving near-capacity on a multiple-antenna channel”, IEEE Trans-actions on Communications, vol. 51, No. 5, 2003, pp.389 - 399.[2] M.O. Damen and K. Abed-Meraim and J.C. Belfiore,”Generalised sphere decoder for asymmetrical space-time communication architecture”, IEE Electronics Let-ters, Vol. 36, 2000, pp. 166-167.[3] J. Luo and K. Pattipati and P. Willett and F. Hasegawa,”Near-Optimal Multiuser Detection in SynchronousCDMA using Probabilistic Data Association”, IEEECommunications Letters, Vol. 5, No. 9, 2001, pp. 361-363.[4] P. Thiennviboon and G. Ferrari and K.M Chugg, ”Gener-alized trellis-based reduced-state soft-input/soft-outputalgorithms”, IEEE International Conference on Commu-nications, ICC 2002, Vol. 3, 2002, pp.1667 - 1671.[5] Christophe Andrieu and Robert J. Piechocki and JoeMcGeehan, ”Particle Smoothing Techniques with TurboPrinciple for MIMO Systems”, IEEE Signal ProcessingAdvances in Wireless Communications (SPAWC-2003),2003.[6] M. Nissila and S. Pasupathy,”Reduced-complexity TurboReceivers for Single and Multi-Antenna Systems viaVariational Inference in Factor Graphs”, In Proc. IEEEICC conference, Paris, 2000.[7] Z. Ghahramani and M. Jordan”, ”Factorial HiddenMarkov Models”, Machine Learning, Kluwer AcademicPublishers, Vol. 29, No. 3, 1997, pp. 245-275.[8] Michael I. Jordan, ”Graphical models”, Statistical Sci-ence - Special Issue on Bayesian Statistics, Vol. 19,2004, pp. 140-155.[9] L. Rabiner, ”A Tutorial on Hidden Markov Models andSelected Applications in Speech Recognition”, Proceed-ings of the IEEE, Vol. 77, No.2, 1989, pp. 257-286.

A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition”,

Achieving nearcapacity on a multiple-antenna channel”,

Factorial Hidden Markov Models”,

Generalised sphere decoder for asymmetrical spacetime communication architecture”,

Generalized trellis-based reduced-state soft-input/soft-output algorithms”,

Graphical models”,

Near-Optimal Multiuser Detection in Synchronous CDMA using Probabilistic Data Association”,

Pasupathy,”Reduced-complexity Turbo Receivers for Single and Multi-Antenna Systems via Variational Inference in Factor Graphs”,

PDA-BCJR algorithm for factorial hidden Markov models with application to MIMO equalisation

Andrieu, C.

McGeehan, Joe

Piechocki, Robert

Sandell, Magnus

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

PDA-BCJR Algorithm for Factorial Hidden Markov Models with Application to MIMO Equalisation

https://research-information.bris.ac.uk/files/3005190/Piechocki_EURASIP_Sept2006.pdf

PDA-BCJR algorithm for factorial hidden Markov models with application to MIMO equalisation

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