This paper considers distributed closed-loop extended orthogonal space-time block coding (EO-STBC) for amplify-forward relaying over time-varying channels. In between periodically injected pilot symbols for training, the smooth variation of the fading channel coefficients is exploited by Kalman tracking. We show in this paper that the joint variation of both relay channels still motivates the use of a higher-order auto-regressive model for the a priori prediction step within a decision-feedback system, compared to a first-order standard Kalman model. Simulations results compare these two case and highlight the benefits of the proposed higher-order Kalman filter, which offer joint decoding and tracking

Alrmah, Mohamed Abubaker

Hussin, Mohammed Nuri

Weiss, Stephan

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryHussin, Mohammed Nuri and Alrmah, Mohamed Abubaker and Weiss, Stephan (2012) Distributedclosed-loop EO-STBC for a time-varying relay channel based on kalman tracking. In: 9thIMA International Conference on Mathematics in Signal Processing, 2012-12-17 - 2012-12-20,Birmingham.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/DISTRIBUTED CLOSED-LOOP EO-STBC FOR A TIME-VARYING RELAY CHANNELBASED ON KALMAN TRACKINGMohamed Nuri Hussin, Mohamed Alrmah and Stephan WeissDepartment of Electronic & Electrical Engineering, University of Strathclyde, Glasgow, Scotland, oUK{mohamed.hussin, mohamed.alrmah, stephan.weiss}@eee.strath.ac.ukABSTRACTThis paper considers distributed closed-loop extended or-thogonal space-time block coding (EO-STBC) for amplify-forward relaying over time-varying channels. In betweenperiodically injected pilot symbols for training, the smoothvariation of the fading channel coefficients is exploited byKalman tracking. We show in this paper that the joint varia-tion of both relay channels still motivates the use of a higher-order auto-regressive model for the a priori prediction stepwithin a decision-feedback system, compared to a first-orderstandard Kalman model. Simulations results compare thesetwo case and highlight the benefits of the proposed higher-order Kalman filter, which offer joint decoding and tracking.1. INTRODUCTIONIn cooperative communications, the range of communica-tions is extended in the absence of a dedicated infrastructurethrough the use of relaying nodes. We here consider the caseof an amplify and forward (AF) scheme, whereby devices inthe relay layer receive from a source and retransmit signalstowards the destination. The relay nodes are assumed to beequipped with single antennas, and although they are notlinked, can be utilised as a virtual MIMO system to achievecombined diversity and array gain.Transmit diversity in a distributed environment has beendiscussed forN = 2 using distributed orthogonal STBC (DO-STBC) [5]. For N = 4, either full diversity or full rate has tobe sacrificed, unless channel state information (CSI) can beexploited akin to closed-loop extended orthogonal STBC [8]in order to achieve maximum diversity and array gain by in-troducing phase rotation to two of the transmit antennas. Theoptimum angles are estimated at the destination, and can befed back to the relay nodes through quantisation and differen-tial encoding [3], exploiting smooth variations of the channelcoefficients in Doppler-fading environments.If the time-varying channel coefficients can be identifiedand tracked, a joint maximum likelihood (JML) decodingapproach proposed in [9] for slow fading channels can beutilised. Based on regular intervals of training, trackingcan be accomplished by a Kalman filter in decision-directedmode [6].In this paper, we formulate the smoothly time-varyingchannel that is formed by the products of the source–relayand relay–destination links. In order to exploit this smoothvariation a higher-order prediction mode akin to [2] is embed-ded in the Kalman filter. Based on a definition of the systemmodel in Sec. 2, the higher-order model Kalman tracker is in-troduced in Sc. 3, with results and conclusions presented inSecs. 4 and 5.In our notation, lower and upper-case bold face variablessuch as, h and H represent vector and matrix quantities re-spectively. For a matrix H, the transpose is denoted by HT,the Hermitian byHH, and the complex conjugate byH∗. Thestatistical expectation operator is given by E{·}.2. SYSTEM MODELWe consider the AF scheme shown in Fig 1, where a sourceS transmits to a destination D via a relay layer consisting ofN = 4 devices Ri, i ∈ (1, N), which only possess singleantennas and are not interconnected. We assume half-duplexmode, where during a first time slot tS transmits to Ri andduring the second time slot, the relay to destination link op-erates in EO-STBC, whereby D provides feedback to relaydevices R1 and R3.S Df2[n]f3[n]f4[n]w2[n]w3[n]f1[n] w1[n]w4[n]R4R3R2R1g1[n]g2[n]g3[n]g4[n]v[n]Fig. 1. System model with source S, relay devices Ri,i = 1 . . . 4 and destination D; channels between nodes arecharacterised by complex fading gains fi[n] and gi[n].2.1. Transmit SignalsIn the first transmission phase, S broadcasts an informationvector sn with variance σ2s , which is received at relay nodesand processed under a constrain of link quality to produce anEO-STBC block. In the second phase, the relay nodes for-ward the signal to a destination node D. As shown in Fig. 1,the channel links fi[n] and gi[n] are spatially independentRayleigh identically distributed wide-sense stationary (WSS)with Doppler spreads Ωf and Ωg and variances σ2f and σ2g ,respectively. The overall transmitted signal isr[n] = b2hTnΛnsn + w[n] (1)w[n] = b1gTnΛnvn + n[n] , (2)where the fixed amplification factor b1 =√p2/ (p1 + 1) isto scale average Ri nodes noise power and b2 = b1√p1 is tomaintain average signal power. Assuming the total averagepower is p, therefore p1 = p2 = p/2 , where p1 is the averagepower of S and p2 is the average power of Ri nodes. Theaggregate source–relay channels are contained in the channelvectorhn =[f1g1 f2g2 f∗3 g3 f∗4 g4]. (3)During even and odd symbol periods, the transmitted vectorwith EO-STBC encoding can be written assn ={12 [s[n], s[n], s[n+ 1], s[n+ 1]] , n even12 [−s∗[n],−s∗[n], s∗[n− 1], s∗[n− 1]] n odd.The beam steering matrix Λn in (1) is diagonal,Λn = diag{ejϑ1[n], 1, ejϑ2[n], 1}, (4)where ϑ1[n]and ϑ2[n] are acting on the channels h1[n] andh3[n] in order to achieve maximum system gain. The op-timum values for beam steering angles will be shown inSec. 2.4.2.2. Relay Node ProcessingThe broadcast signal is received asyi =√p1 fisn + vn . (5)at the relay nodes, where linear processing retransmits thedata based on distributed EO-STBC codewords. For com-plex signal constellations, the processing matrices Ai, Biat each relay node are designed such that A1 = A2 = I2,A3 = A4 = 02 and B1 = B2 = 02,B3 = B4 =[0 −11 0].The forwarded signal therefore isti = b1(Aiyi +Biyi) . (6)The channel gains of all the communications links are as-sumed to be independent Rayleigh fading, giving rise to acovariance matrixRh[τ ] = E{hn−τhHn}= 4σ2fJ0(Ωfτ)σ2gJ0(Ωgτ) I , (7)where J0(·) is the zeroth order Bessel function of firstkind [2].2.3. Received SignalDuring the time intervals n and n+1, the received vector canbe written asrn =[r[n]r∗[n+ 1]]= b2Hnsn +wn , (8)based on the equivalent transmit and noise vectors, sn =[s[n], s[n + 1]]T and wn = [w[n], w∗[n + 1]]T , where thelatter accounts for both noise propagated through Ri and ad-ditive noise at D. The EO-STBC equivalent channel Hn canbe formulated asHn =[h11[n] h12[n]h21[n+ 1] h22[n+ 1]]. , (9)where the components of Hn,h11[n] = ejϑ1[n]f1[n]g1[n] + f2[n]g2[n] (10)h12[n] = ejϑ2[n]f∗3 [n]g3[n] + f∗4 [n]g4[n] (11)h21[n+ 1] = e−jϑ2[n+1]f3[n+ 1]g∗3 [n+ 1]+f4[n+ 1]g∗4 [n+ 1] (12)h22[n+ 1] = −e−jϑ1[n+1]f∗1 [n+ 1]g∗1 [n+ 1]−f∗2 [n+ 1]g∗2 [n+ 1] , (13)are a mixture of dual-hop channel coefficients and rotationsdue to beam steering.2.4. Signal Detection & System GainDetection is performed over two successive symbols periods,over which in standard space-time block coded systems thechannels are assumed to be stationary. This guarantees or-thogonality of the equivalent space-time channel matrix Hn,and enables linear decoding according to sˆn = HˆHn rn. Incases, where the matrix is no longer orthogonal, degradeddetection performance has motivated approaches such as thejoint maximum likelihood method discussed in [9].For deriving the maximum attainable gain, we assumeblock stationarity, i.e. Hˆn = Hn, and h[n] ≈ h[n + 1]. Inthis case, the maximum achievable gain isGn = HHnHn = b22[α+ β 00 α+ β]. (14)withα =4∑m=1|fm[n]gm[n]|2 +4∑m=3|f∗m[n]gm[n]|2 (15)β = +ℜ{ejϑ1[n]f1[n]g1[n]f∗2 [n]g∗2 [n]+ ejϑ2[n]f∗3 [n]g3[n]f4[n]g∗4 [n]} . (16)The maximum factor α+maxϑ1,ϑ2 β = 4+ pi24 can be shownto be attained by setting the beamsteering angles toϑ1[n] = −∠{f1[n]g1[n]f∗2 [n]g∗2 [n]} (17)ϑ2[n] = −∠{f∗3 [n]g3[n]f∗4 [n]g4[n]} . (18)3. KALMAN TRACKINGBased on the model and observation equations (1) and (2), thissection proposes a higher-order auto-regressive (AR) modelfor inclusion in a Kalman tracker.3.1. Channel ModellingAs shown in [2], a narrowband time-varying channel can beapproximated by an M th order AR (AR-M ) prediction modelwith coefficient vector a = [a0 a1 · · · aM−1]T, which lin-early combines a past data vector such thath[n] = aHh[n− 1]h[n− 2]...h[n−M ] = aHhn−1 + e[n] , (19)where e[n] is the prediction error. Minimisation of this er-ror leads to E{hn−1e∗n} = 0, which in our context can beexpressed as E{hn−1h[n]∗} = E{hn−1hHn−1}a. Includingtemporal correlation in (7) leads to a = Rhp with an ap-propriately defined cross-correlation vector p. The identicalprocess noise variance σ2e to the mean squared prediction er-ror isσ2e = E{e[n]e∗[n]} = σ2h −ℜ{aHP}+ aHRha . (20)Thus, the state-space model incorporated in the Kalman filterwith a finite order AR process can be set up for the case of Mashn|n−1hn−1|n−1...hn−M−1|n−M−1 = Ahn−1|n−2hn−2|n−2...hn−M|n−M+w[n]0...0 , (21)with the system matrixA =a1 a2 · · · aM−1 aM1 0 · · · 0 00 1 · · · 0 0...............0 0 · · · 1 0. (22)SNR/[dB] 5 15 25M = 1 0.165 0.134 0.112M = 2 0.016 0.013 0.011M = 3 0.015 0.012 0.010Table 1. Average MSE for use of different AR-M models inthe Kalman tracker..3.2. DD-Based Tracking SchemeA Kalman estimator estimator based on decision-directed(DD) updating akin to [6] is adopted, and extended to thedual-hop relay link and AR-M model. Assuming indepen-dence between the channel gains and observation and processnoises for different links enable the application of the stan-dard Kalman filter approach [10].In the DD approach, a periodic insertion of known sym-bols along with the transmitted data is employed to inhibitKF divergence. Note that the initial channel estimation is theoptimal method only in the linear sense. In updating step, a-piriori estimates are performed during two samples intervalswhich are used to detect coarse symbols. Thereafter, in cor-rection step the predicted state is subsequently refined usingthe current observation such that the coarse symbols makefeedback to produce a posteriori.The modified channel coefficients require to omit thephase rotation during tracking scheme. It can be noted thatthe feedback angles can be absorbed either into channel vec-tor or into transmit vector as in equation 1. Therefore, asimple correction can be used with compensation for phasemodification. Based on this approach, a low complexityDD-based Kalman estimator can be implemented.4. SIMULATIONS AND RESULTSFor the simulations below, it is assumed that (i) the initialstate h0, is previously estimated and known and (ii) the com-pensation for phase modification is based on true phase an-gles. Simulations are performed over an ensemble of 104randomised channel realisations. We consider that the bothdual-hop links are in moderate fading with the normalisedDoppler spread Ωf = Ωg = {0.005pi}. The transmissionis initially interleaved every K = {24} symbol periods byan inserted pilot symbol for channel estimation, incurring 4%loss in bandwidth efficiency.Tab. 1 shows the average MSE per channel of D-EO-STBC with Kalman tracking based on different AR ordersM , with M = 1, 2, 3. It can be seen that M = 2, 3 have asignificant advantage over the standard first order moded withM = 1. In addition, it can be noted that M = 3, althoughit incurs a higher computational complexity than the case of0 5 10 15 20 2510−410−310−210−1SNR/ [dB]BER  AR−1, no trackingAR−1, trackingAR−2, no trackingAR−2, trackingideal CSIFig. 2. BER performance of EO-STBC system with Kalman-based channel tracking based on an AR-M with M = 1, 2compared to perfect CSI.30 40 50 60 70 80 90 100 110 12010−310−210−1pilot symbol insertion, KBER/MSE  MSEBERFig. 3. Average MSE and BER performance at 20dB SNRfor D EO-STBC system with Kalman-based tracking basedon different tracking periods K .M = 2, does not offer a significant improvement over thelatter.Fig. 2 shows the BER performance, whereby AR-M sys-tems of first and second order are compared. In between apilot injection after every K = 24th time slot, the Kalmanfilter is either operated in a prediction mode only (labelled as“no tracking”) or with a correction step based on DD updating(“tracking”). The It can be seen that the AR-2 performanceis very close to the one where the receiver has perfect knowl-edge of the channel state information (CSI).We now study the impact of the pilt insertion period Kon the performance of the AR-2 system. Fig. 3 shows theBER and MSE performances for pilots interleaved after everyK = {24, 48, 72, 96, 120} symbol periods. It can be seen thatK significantly affects the performance. For the moderateDoppler spread selected here, the simulation indicates thanan average BER of 10−3 can be maintained for K = 24.5. CONCLUSIONSWe have considered distributed EO-STBC for a two-hop AFrelay channel, whereby maximum diverity and array gain canbe attained by feedback of appropriate beamsteering anglesto the relay layer. The estimation of these angles is basedon tracking the aggregate channel at the destination. Thisis accomplished by employing a Kalman tracker which in-cludes a higher-order prediction model, that can suitably ex-ploit the smoothly time-varying characteritic of the equivalentchannel due to Rayleigh fading of the source–relay and relay–destination channels.6. REFERENCES[1] J. Boyer D.D. Falconer and H. Yanikomeroglu, “Mul-tihop diversity in wireless relaying channels,” IEEETrans. 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Distributed closed-loop EO-STBC for a time-varying relay channel based on kalman tracking

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Distributed closed-loop EO-STBC for a time-varying relay channel based on kalman tracking

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