Vritionl Messge-Pssing for Joint Chnnel Estimtion nd Decoding in MIMO-OFDM Gunvor Elisbeth Kirkelund, Crles Nvrro Mnchón,LrsP.B.Christensen, Erwin Riegler nd Bernrd Henri Fleury Deprtment of Electronic Systems, Alborg University, Denmrk Modem Algorithm Design, NOKIA, Denmrk Vienn University of Technology VUT Emil: {gunvor, cnm}@es.u.dk, lrs.christensen@noki.com, erwin.riegler@nt.tuwien.c.t, bfl@es.u.dk Abstrct In this contribution, multi-user receiver for M- QAM MIMO-OFDM operting in time-vrying nd frequencyselective chnnels is derived. The proposed rchitecture jointly performs semi-blind estimtion of the chnnel weights nd noise inverse vrince, seril interference cncelltion nd decoding in n itertive mnner. The scheme relies on vritionl messgepssing pproch, which enbles joint design of ll these functionlities or blocks but the lst one. Decoding is performed using the sum-product lgorithm. This is in contrst to nowdys proposed pproches in which ll these blocks re designed nd optimized individully. Simultion results show tht the proposed receiver outperforms in coded bit-error-rte stte-of-the-rt itertive receiver of sme complexity, in which ll blocks re designed independently. Joint block design nd, s result, the fct tht the uncertinty in the chnnel estimtion is ccounted for in the proposed receiver explin this better performnce. I. INTRODUCTION During recent yers, lgorithms bsed on itertive informtion processing or turbo techniques hve become widespred in wireless receiver design [1] [3]. The success of these lgorithms cn be explined by their remrkble properties: high performnce t trctble complexity nd flexibility in their design. An emblemtic exmple is turbo-codes, which, when ssocited with turbo-decoding, llow for trnsmission close to cpcity t trctble complexity [1]. In this pper, we focus on specific ppliction of itertive informtion processing, nmely to design efficient, fesible lgorithms for chnnel estimtion i.e. estimtion of both the chnnel trnsfer function nd the chnnel inverse noise vrince, interference cncelltion, nd decoding in MIMO- OFDM systems. Some relted work is lredy vilble in the literture. Worth noticing is the itertive lgorithm for detection nd interference cncelltion [4] pplied to multiuser CDMA. This lgorithm is extended for vrious trnsmission Fig. 1. b 1 b K Tx 1 Tx K x 1 x K y 1 y G Chnnel estimtion nd MIMO decoding Bsebnd signl model of the considered MIMO-OFDM system. ˆb 1 ˆb K schemes in [5] [7] to include estimtion of the chnnel response into the itertive process. We coin this receiver the LMMSE-bsed receiver, ccording to the dominnt structure implemented in its constituent blocks. An essentil feture of this receiver is tht its constituent blocks re designed nd optimized individully. These blocks re connected fterwrds to form the itertive structure. In this contribution, we pply vritionl Byesin VB inference [8] nd one of its pplictions, nmely the vritionl Byesin expecttion mximiztion VBEM lgorithm [9] to perform chnnel weight nd noise inverse vrince estimtion s well s seril interference cncelltion in n M-QAM MIMO-OFDM system operting in time-vrint frequencyselective chnnels. Decoding is performed using the sumproduct SP lgorithm [3]. The VBEM lgorithm hs lredy been pplied in [10] for GSM chnnel estimtion nd detection. In [11] it is combined with the sum-product lgorithm for the design of multiuser CDMA receiver. Further relted work is found in [12] [15]. In our pper, we pply the VBEM scheme in [11] to MIMO-OFDM nd reformulte it s vritionl messge-pssing VMP lgorithm on fctor grphs [16]. The proposed VMP receiver nd the LMMSE-bsed receiver from [5] [7] shre similr fetures in their respective structures. Thus, we find it useful to lso include comprison of the two schemes. A crucil difference is tht the estimtion of the noise nd residul interference power in the VMP receiver ccounts for the uncertinty in the chnnel coefficient estimtes, n effect not considered in the LMMSE-bsed receiver. This, combined with the joint design of ll receiver blocks but decoding, yields superior performnce of the VMP receiver, s our simultion results demonstrte. The nottionl convention for the rest of the pper is s follows: the superscripts T nd H denote trnsposition nd Hermitin trnsposition respectively. The symbol denotes proportionlity. The trce opertor is designted s tr. The expecttion opertion with respect to function qx is represented by qx. The newest estimte of the men or covrince of vrible is denoted by ˆ. The opertors dig nd Dig denote the vectorized digonl of mtrix nd the digonlized mtrix of vector respectively. For mtrices A nd B, the Kronecker nd Hdmrd products re represented
by A B nd A B respectively for the Hdmrd product A nd B re ssumed to hve the sme dimension. The identity mtrix of dimension K is designted s I K nd 1 G represents the ll-one mtrix of dimension G G. Weemploy0 K nd [1...1] K to designte respectively the ll-zero column-vector nd the ll-one row-vector of length K. VMP region p p λ λ x k SP region p xk b k p bk K II. SIGNAL MODEL We consider the LTE-like MIMO-OFDM system depicted in Fig. 1 in which we hve K trnsmitters, indexed by k, nd G receivers, indexed by g. Inthekth trnsmitter, denoted by Tx k, the bit-strem b k is encoded, interleved nd modulted into dt symbols, which re then multiplexed with pilot symbols to llow for chnnel estimtion in the receiver. Pilot nd dt symbols re rrnged in n OFDM frme of L OFDM symbols consisting of N subcrriers ech. The OFDM frme of Tx k is represented by x k [x k11...x knl...x knl ] T X k, where l indexes the OFDM symbols nd n indexes the subcrrier number. The set X k of legl M-ry sequences of Tx k is determined by the coding nd modultion scheme nd the multiplexing scheme of dt nd pilot symbols. The OFDM frmes re trnsmitted cross time-vrint frequency-selective chnnel. The smples of the timefrequency response of the sub-chnnel from trnsmit ntenn k to receive ntenn g re conctented in the chnnel weight vector gk [ gk11... gk1l... gknl... gknl ] T. Assuming tht inter-symbol nd inter-subcrrier interferences re negligible, the received signls t ll G ntenn ports re given in vector nottion by y = K A k x k + w 1 k=1 = X + w 2 = Ax + w. 3 The vector y is the conctention of the output vectors of ll receive ntenns, y [y1 T...yg T...yG T ]T with y g [y g11...y gnl...y gnl ] T denoting the output of receive ntenn g. The chnnel mtrix for trnsmitter k is defined s A k Dig k [1...1] T G I N. The noise vector w is white nd circulrly symmetric complex Gussin: w CN0 GNL,σwI 2 GNL, with σw 2 denoting the noise vrince. We define the precision prmeter λ σw 2. The mtrix X is defined s X I G [1...1] K I N Digx nd [ T 11... T gk...t GK ]T. The mtrix A I G [1...1] K I N Dig[1...1] T G I K I N is the MIMO chnnel mtrix. The vector x [x T 1...x T k...xt K ]T contins the conctented OFDM codewords from ll trnsmit ntenns. The receiver outputs n estimte ˆb k of the bit-strem for ny k. III. GRAPHICAL REPRESENTATION In this section, we present grphicl representtion of the signl model introduced in the previous section. This grphicl y p y Fig. 2. Fctor-grph [3] of the signl model in Section II. The prmeter K indictes tht the corresponding block is repeted K times, one for ech trnsmitter. Notice tht the left region is equivlent to the Byesin network representtion from [17]. representtion will be used to derive the messge-pssing lgorithm in Section IV. Let Φ {y,,λ,x 1,...x K, b 1,...b K } 4 denote the set of ll observed nd unobserved vribles in 1. Bsed on the ssumptions mde in Section II, the joint probbility density function pdf of Φ fctorizes s p Φ Φ =p y y,λ,x 1,...x K p p λ λ p xk x k b k p bk b k. 5 The constrints imposed by coding, modultion nd multiplexing of the deterministic pilot symbols re included in the fctor p xk for trnsmitter k. A strightforwrd grphicl representtion of this fctoriztion is the Tnner fctor-grph [3] depicted in Fig. 2. Fctors re represented s squres, vribles s circles. An edge between vrible node nd fctor node indictes tht the vrible is n rgument of the fctor. Bsed on this grphicl representtion of the signl model, we employ itertive lgorithms to estimte the joint pdf p Φ. We split the grph into two regions s depicted in Fig. 2. In the right-hnd region, we pply the SP lgorithm [3] to compute the mrginls p xk nd p bk. In the left-hnd region, we pply the VMP lgorithm [16] to estimte p nd p λ.thevmp lgorithm is used to reformulte the VB inference method proposed in [11] in terms of messges. The motivtion for splitting the Tnner grph in this wy nd pplying two different messge-pssing methods is s follows. The SP lgorithm is well-estblished lgorithm for computing the mrginl probbility mss functions p xk nd p bk in known chnnel conditions. Direct computtion of the chnnel mrginls p nd p λ by mens of the SP lgorithm is, however, computtionlly infesible. In this cse, one hs to rely on techniques for pproximting these mrginls, e.g. prticle filters or the EM lgorithm [18]. Here, we propose nother venue nd compute these mrginls with the VMP lgorithm. We define the set of unknown vribles in the VMP region s Φ VMP {,λ,x 1,...x K } Φ. k
Vritionl Messge-Pssing VMP We consider n rbitrry fctor-grph. The messge from fctor node f to vrible node φ in the set N f of neighbouring nodes of f is m f φ exp ln f mφ f φ N f \φ. 6 The messge from vrible node φ to ny fctor node f in the set N φ of fctor nodes neighbouring φ is m φ f m f φ. 7 f N φ The estimted uxiliry function of φ is b φ m φ f. 8 IV. VARIATIONAL MESSAGE-PASSING In this section, we pply the VMP lgorithm [11], [17] to the left-hnd region in the fctor-grph in Fig. 2, see [16] nd references therein for vritionl inference on fctor-grphs. The messge-pssing rules re summrized in 6-8. Their derivtions re sketched in App. A. The VMP lgorithm pproximtes the joint pdf p ΦVMP,yΦ VMP, y = p y y,λ,x 1,...x K p p λ λ k p x k x k with n uxiliry function b ΦVMP Φ VMP in such wy tht the KL divergence from b ΦVMP Φ VMP to p ΦVMP,yΦ VMP, y is minimized [17]. We constrin the uxiliry function to fctorize ccording to b ΦVMP Φ VMP =b b λ λb x1 x 1...b xk x K. The VMP lgorithm implements sequentil messge updtes to updte the fctors in b ΦVMP Φ VMP. Updting b, b λ λ, nd b xk x k corresponds to estimting the chnnel weights, estimting the precision prmeter, i.e. the chnnel inverse noise vrince, nd interference cncelltion, respectively. A. Estimtion of the Chnnel Weights In this subsection, we derive the messges to nd from the vrible node. These messges re used to updte b by mens of 8. The messge to node from p y is obtined from 6: m py =exp ln p y y,λ,x mλ p y k mx k p y. 16 Solving the expecttion yields m py p CN ˆλ VMP Ĉ py ˆX H y, Ĉp y. 17 Here, p CN µ, C is multivrite complex Gussin pdf with men vector µ nd covrince mtrix C, nd Ĉp y ˆλ VMP ˆXH ˆX + ˆλVMP I G Ĉx 1. The mtrix Ĉx is the block-digonl conctention of the estimtes Ĉx k of the covrince mtrices of x k, k =1...K.BothĈx k nd the estimte ˆλ VMP of the precision prmeter re defined lter in this section. We impose the prior p to belong to the fmily of conjugte pdfs of for p y. This choice gurntees tht the uxiliry pdf b is lso in this fmily. From 16 the conjugte fmily of pdfs of for p y is the Gussin fmily. Thus, from 6 m p = p CN 0 GKNL, C, 18 where C is the prior chnnel covrince mtrix. Inserting 17 nd 18 in 7 yields m py p CN â, Ĉ =b 19 with â = ˆλ VMP Ĉ ˆXH y nd Ĉ =C 1 + Ĉ 1 p y 1.As the Gussin pdf is fully defined by these two moments its nturl sttistics it is enough to pss them to p y. B. Estimtion of the Precision Prmeter In this subsection, we define the messges to nd from vrible node λ. The uxiliry function b λ is then updted by plugging these messges in 8. The messge from p y to λ reds from 6 m py λ =exp ln p y y,λ,x m p y k mx k p y. 20 Evluting the expecttion under the ssumption tht the messges m py nd m xk p y, k =1...K, re Gussin densities [11] yields m py λ p CW1 Ŵ 1,GNL+1. 21 In this expression, p CWF M 1,d is complex Wishrt pdf defined by three prmeters: the dimension F, the degree of freedom d, nd mtrix M of dimension F F [19]. Here, F =1, d = GNL +1, nd M is sclr given s Ŵ try ˆxy ˆxH + ˆXĈ ˆX H + k ÂkĈx k  H k + k 1 G Ĉx k DigdigĈ k. The estimte Ĉ k of the uto-covrince mtrix of k cn be obtined from Ĉ. The estimte Ĉx k of the covrince mtrix of x k is defined lter in this section. We select p λ to be conjugte pdf of λ, which is complex Wishrt pdf of dimension one [20, Sec. IVb]. From 6 m pλ λ = p CW1 M 1 pr,d pr 22 with given prmeters M pr nd d pr. By inserting 21 nd 22 into the messge-pssing rule 7, we obtin the complex Wishrt pdf m λ py p CW1 Ŵ + M pr 1,d pr + GNL = b λ. 23 It is enough to pss the first moment ˆλVMP = d pr + GNLŴ + M pr 1 [20, Eq. 22] of this pdf, since the other messge updtes only depend on this vlue. As we hve no prior informtion on λ, we select p λ to be uniform over the rnge of λ. For this improper prior, we hve M pr =0 nd d pr =0[20]. C. MIMO Decoding To updte b xk, we compute the messges to nd from the vrible node x k. From 6, the messge from node p y to vrible node x k is m py x k = exp ln p y y,λ,x m p y m λ py k =k mx k py. 24
Chnnel Estimtion VMP receiver: LMMSE-bsed receiver: â = C 1 + ˆλ 1 VMP ˆXH ˆX + ˆλVMP I G Ĉx ˆλVMP ˆXH y 9 â = C 1 + H ˆX ˆΛchn ˆX 1 LMMSE ˆXH ˆΛchn LMMSEy 10 MIMO Detection/Interference Cncelltion VMP receiver: ˆx k = ˆλ VMP  H kâk + ˆλ 1ˆλVMPÂH VMP DigdigĈ kg kg k y  k ˆx k g g k =k LMMSE-bsed receiver: ˆx k = C 1 x k + ÂH det 1 k ˆΛ LMMSEÂk ÂH ˆΛdet k LMMSE y  k ˆx k k =k 11 12 Estimtion of the Precision Mtrix VMP receiver: ˆΛ VMP = ˆλ VMP I GNL with y ˆxy ˆxH tr + ˆλ VMP = ˆXĈ ˆX H + 1 k ÂkĈx k ÂH k + k 1G Ĉx k DigdigĈ k 13 GNL LMMSE-bsed receiver: σ 2 is the verge power of the chnnel ˆΛ chn LMMSE = tr y ˆx ˆΛ det LMMSE = tr y ˆx y ˆx H GNL y ˆx H GNL I GNL + σ 2 I G k I GNL + Ĉ xk 1 14 1  k Ĉx k ÂH k =k k 15 Fig. 3. Chnnel estimtion, MIMO detection/interference cncelltion nd the precision mtrix estimtion in the VMP nd LMMSE-bsed receiver. Solving the expecttion, yields m py x k p CN ˆx k, Ĉx k 25 with men vector ˆx k = ˆλ VMP Ĉ xk  H k y k k Âk ˆx k nd covrince mtrix Ĉx k = ˆλ VMP  H k Âk + ˆλ VMP g g digĉ kg kg 1. The estimte Ĉ kg kg of the cross-covrince mtrix of the chnnel vectors kg nd kg cn be obtined from Ĉ. Demodultion nd decoding re performed in the right region of the grph in Fig. 2 using the SP lgorithm. The estimted men of symbol x knl in x k is computed to be ˆx knl = x M xp x knl = x ˆx k, where P x knl = x ˆx k = x k X k,x knl =x m p y x k x k with M denoting the set of constelltion points of the selected M-QAM modultion. For convolutionl codes, these mrginls cn be obtined with the BCJR lgorithm. Likewise, the estimted vrince of x knl is ˆσ x 2 knl = x M x2 P x knl = x ˆx k ˆx 2 knl. Any two distinct symbols re ssumed to be uncorrelted. As result, the estimte of the covrince mtrix of x k fter decoding reds Ĉ xk = Digˆσ x 2 k11,...ˆσ x 2 knl. We pproximte the messge from x k to p y by Gussin pdf. Notice tht the Gussin fmily is the conjugte fmily of x k for p y. With this pproximtion nd from 25 we obtin m xk p y p CN ˆx k, Ĉx k =b xk. 26 We only pss the nturl sttistics ˆx k, Ĉ xk to p y.from8, the messge 26 represents the estimted posterior pdf of x k. V. COMPARISON WITH THE LMMSE-BASED RECEIVER In this section, we compre the VMP receiver derived in the previous section to stte-of-the-rt itertive receiver proposed in [5], further developed for detection in multiuser CDMA [6], nd pplied to MIMO-OFDM systems in [7]. We refer to this receiver s the LMMSE-bsed receiver. Due to lck of spce, the derivtion of the LMMSE-bsed receiver is not included in this work, but the expressions of the different component blocks re summrized in Fig. 3 together with the corresponding expressions obtined for the VMP receiver. The conceptul difference between the two schemes is tht in the LMMSE-bsed receiver the different constituent blocks re designed independently, while in the VMP receiver the blocks corresponding to fctors in the VMP region re designed jointly, by minimizing globl cost function, i.e. KL divergence, in this region. By inspecting the expressions in Fig. 3 we observe tht the LMMSE-bsed receiver nd the VMP receiver shre some structurl properties. For instnce, from 9 nd 10 it is cler tht both lgorithms use n LMMSE-like chnnel estimtor, which minly depends on the chnnel prior covrince, estimtes of the trnsmitted symbols nd n estimte of the precision mtrix, nmely ˆλ VMP I in the VMP receiver nd ˆΛ chn LMMSE in the LMMSE-bsed receiver. Similrly, the detection prt of both receivers consists of interference cncelltion followed by LMMSE filtering of the residul interference. However, we cn highlight two criticl differences between
TABLE I PARAMETER SETTINGS FOR THE SIMULATIONS Cyclic prefix length 4.7 μs Symbol durtion 66.7 μs Subcrrier spcing 15 khz Pilot overhed 4.8% pilots Pilot pttern Regulr spcing/dimond, QPSK Modultion lphbet 16-QAM Number of informtion bits 660 Number of subcrriers N 75 Number of OFDM symbols L 7 Number of trnsmitters K 2 Number of receivers G 2 Chnnel interlever block Convolutionl code 155, 117, 127 8 the two lgorithms: firstly, only one sclr estimte of the precision prmeter is needed in the VMP receiver, while the LMMSE-bsed receiver clcultes two different precision chn mtrices, one for chnnel estimtion ˆΛ LMMSE nd one for det detection ˆΛ LMMSE; secondly, the LMMSE-bsed receiver does not del with the uncertinty in the chnnel weight estimtes nd considers them s the true vlues in the detection prt, while the VMP receiver ccounts for chnnel estimtion errors vi the term Ĉ in 11 nd 13. VI. SIMULATION RESULTS To verify the performnce of the VMP receiver, we perform Monte-Crlo simultions for n LTE-like 2 2 system with the settings reported in Tble I. We consider pilot scheme where ll trnsmitters trnsmit pilots in the sme time-frequency resources. Reliztions of the chnnel time-frequency response re generted using the extended typicl urbn ETU chnnel model from the 3GPP LTE stndrd [21], with Ryleigh-fding chnnel tps, nd ssuming no correltion over trnsmit or receive ntenns. Note tht the chnnel is wide-sense-sttionry nd uncorrelted-scttering WSSUS [22]. We compute the prior covrince mtrix C from the chnnel time-frequency correltion function. We test the OFDM-MIMO system with the two receivers described in Fig. 3. Both receivers use the sme initiliztion, consisting of MMSE pilot-bsed chnnel estimtion nd joint soft-decision mximum likelihood ML detection, followed by soft-in soft-out sequentil decoding. In both receivers n itertion consists of estimtion of the chnnel weights, followed by sequentil detection nd decoding of ll K trnsmitted frmes, nd ending with estimtion of the precision prmeter or mtrices. The bit-error-rte BER performnce of both receivers versus the signl-to-noise rtio E b /N 0 is illustrted in Fig. 4. For the ske of comprison, the initiliztion is lso depicted denoted by the Liner Receiver tg. Both receivers perform 10 itertions. The results show tht both itertive structures significntly improve the performnce of the liner receiver, especilly for E b /N 0 lrger thn 0 db. Moreover, the VMP receiver outperforms the LMMSE-bsed receiver in the considered signl-to-noise rnge. The gin is bout 0.5 db in the opertion rnge of the MIMO-OFDM system. The convergence behviour of both itertive structures is described Bit-error-rte 10 0 10 1 10 2 10 3 Liner receiver LMMSE-bsed receiver VMP receiver 10 4 4 2 0 2 4 6 E b /N 0 [db] Fig. 4. Coded bit-error-rte. 100% Frme-error-rte 10% LMMSE-bsed receiver Eb/N0@2 db VMP receiver Eb/N0@2 db LMMSE-bsed receiver Eb/N0@4 db VMP receiver Eb/N0@4 db LMMSE-bsed receiver Eb/N0@6 db VMP receiver Eb/N0@6 db 1% 0 2 4 6 8 10 Itertions Fig. 5. Frme-error-rte cross itertions. in Fig. 5, which depicts the frme-error-rte versus the number of itertions t the receiver for three different E b /N 0 vlues. Both receivers converge fter pproximtely 5 itertions for ll opertion points. Agin, the VMP receiver outperforms the LMMSE-bsed receiver regrdless of the number of itertions. VII. CONCLUSION We derive novel itertive receiver structure for M-QAM MIMO-OFDM operting in frequency-selective time-vrint chnnels. The scheme performs jointly semi-blind estimtion of the chnnel weights nd of the noise inverse vrince bsed on both dt nd pilot symbols, seril interference cncelltion, nd decoding. The scheme ws lredy proposed for CDMA in [11]. A vritionl messge-pssing VMP interprettion of it is provided here. The VMP receiver is compred with the LMMSE-bsed itertive receiver derived in [5] [7]. Both itertive rchitectures re mde of the sme blocks nd exhibit similr complexity. However, in the VMP receiver ll blocks but decoding re jointly optimized ccording to globl cost function, the KL divergence, while in the LMMSE-bsed receiver ll blocks re designed independently. Furthermore, the VMP frmework yields structure tht tkes into ccount the inccurcy of the chnnel weight estimtes. This inccurcy is neglected in the LMMSE-bsed receiver. In order to ssess the effect of these structurl differences,
we evlute the performnce of both receivers in n LTE-like scenrio. The simultion results show tht the VMP receiver outperforms the LMMSE-bsed receiver with signl-to-noise rtio gin of 0.5 db t relevnt BER vlues. An issue not ddressed in the pper is how to combine efficiently the VMP lgorithm used for chnnel weight nd noise inverse vrince estimtion s well s seril interference cncelltion nd the sum-product lgorithm employed for decoding in the receiver. A solution hs been recently proposed in [23]. VIII. ACKNOWLEDGEMENTS The uthors would like to thnk NOKIA Denmrk for the finncil support which mde this work possible. This work hs lso been supported in prt by the 4GMCT coopertive reserch project funded by Infineon Technologies Denmrk A/S, Agilent Technologies, Alborg University nd the Dnish Ntionl Advnced Technology Foundtion, by the Europen Commission within the FP7-ICT Network of Excellence in Wireless Communictions, NEWCOM++ Contrct No. 216715 nd by WWTF grnt ICT08-44, FWF grnt S10603- N13 within the Ntionl Reserch Network SISE. APPENDIX A THE VARIATIONAL MESSAGE-PASSING ALGORITHM In VB inference [11] we consider s the cost function the KL divergence D KL b Φ f Φ dφ b Φ log b Φ fφ, where f Φ is pdf of set of vribles Φ nd b Φ is n uxiliry function, which pproximtes f Φ. We seek n uxiliry function tht minimizes the cost function. We reformulte the VB inference problem to messgepssing on fctor-grph [16]. We ssume tht f Φ fctorizes ccording to f Φ = f N f, where N f Φ is the set of neighbouring vribles of f. We select n uxiliry function b Φ, which fctorizes ccording to b Φ = φ Φ b φ.asshown in [11], the fctor b φ of b Φ which minimizes D KL b Φ f Φ with ll other fctors b φ, φ Φ\φ fixed is b φ exp ln f bφ φ N f \φ 27 f N φ exp ln f bφ φ N f \φ, 28 f N φ where N φ is the set of neighbouring fctors of φ. With the definitions in 6 nd 7 we cn recst 28 s b φ m f φ = m φ f 29 f N φ for ny f N φ, nd we hve b Φ = b φ m f φ. 30 φ Φ φ Φ f N φ Identity 28 cn be used to design n itertive lgorithm which t ech itertion updtes given fctor b φ of b Φ while keeping the other fctors fixed. The itertive lgorithm converges in the sense of the KL divergence, since D KL b Φ f Φ is minimized t ech itertion. The identities in 29 provide messgepssing interprettion of the updting steps. REFERENCES [1] C. Berrou, A. Glvieux, nd P. Thitimjshim, Ner Shnnon limit error-correcting coding nd decoding: Turbo codes, in Proc. IEEE Int. Conf. Commun. ICC 93, My 1993, pp. 1064 1070. [2] R. Koetter, A. C. Singer, nd M. 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