Iterative Equalization with Decision Feedback based on Expectation Propagation

TO APPEAR ON IEEE J OURNAL ON TRANSACTIONS ON COMMUNICA TIONS - MA Y 2018 1 Iterat i ve Equalization with Dec ision Feedback based on Expectation Propagation Serdar S ¸ ahin, Antonio Maria Cipriano, Cha rly Poulliat and Marie-Laure Bouch e ret Abstract —This paper inv estigates the design and analysis of minimum mean square error (MMS E) turbo decision feedback equalization (DFE), with expectation p ropagation (EP), for single carrier modulations. Classical non iterative DF E structures hav e substantial advantag es at h igh data rates, ev en compare d to tu rbo linear equalizers - interference cancellers (LE-IC), hence making turbo DFE-IC schemes an attracti ve solution. In this paper , we derive an it erativ e DF E-IC, capitalizing on the use of soft feedback based on expectation p ropagation, along with the use of p rior i nform ation for improv ed fi ltering and interference can- cellation. T h is DFE-IC significantly outperforms exact turbo L E - IC, especially at high spectral effici en cy , and also exhibits various advantages and performance impro vements ove r existing variants of DFE-IC. The proposed scheme can also be self-iterated, as done in the recent trend on EP -based equ alizers, and it is shown to be an attractive alternativ e to linear self-iterated receive rs. For time-vary ing (TV) filter equalizers, an efficient matrix in version scheme is also proposed, considerably reducing the computational complexity relati ve to existing method s. Usin g fin ite-length and asymptotic analysis on a sever ely selective channel, the proposed DFE-IC is shown to achi ev e h igher rates than known alternatives, with better waterfall thresholds and faster conv erg ence, whi l e keeping a similar compu t ational complexity . I . I N T RO D U C T I O N C OMMUNICA TION systems operating on wide- band channels suffer from inter-symbol interfere nce (ISI), which can b e mitigated with an appr o priate transceiv er de- sign. In par ticular, for wireless systems where the throu ghput requirem ents increase at each n ew gen eration, more effecti ve receivers are n eeded in order to maintain r obust data links. W ith the discovery of turb o-cod es, iterati ve processing principles were extended to joint detection and decod ing tech- niques via soft-input soft-outp ut (SISO) receivers which use prior information provided by the channel decod er , to further reduce detection errors. Although early turbo equalization technique s, such as m aximum a posterior i ( M AP) detector using BCJR estima tio n [1]– [ 3], can op erate near the channel capacity with pr operly desig n ed coding sche m es, th e ir o p er- ational comp lexity significantly in creases for large chann el delay spre ad or with high modulation orders. Conseq uently , finite im p ulse r esponse ( FIR) filter-based turbo equalizer s with Manuscript recei v ed November 16, 2017; rev ised April 05, 2018 and accep ted on May 24, 2018. S. S ¸ ahin is with both T hales Communication s & Security and IRIT/INPT - ENSEEIHT (e-mail: serdar . sahin@tha lesgroup.com) A. M. Cipriano is with Thales, 4 A v . des Louvresses, 92230, Genne villi ers, France (e-mail: antonio.cipri ano@tha lesgroup.com) C. Poul liat and M. -L. B oucheret are with IRIT / INP T oulouse - ENSEEIHT , 2 Rue Charles Camichel, 31000, T oul ouse, Franc e (e-mail s: { charly .pouillat , marie-laure.bouc heret } @enseeiht.fr) lowered c omputatio nal comp lexity have been pr oposed. These structures can be categorized into th ree group s with regards to its filter updates d e pending on prior inf ormation . Other kinds of adaptive FIR recei vers ar e out o f this paper’ s scope. Time- in variant (TI) structu res update their filters only on ce at each packet reception, using the av ailable channel state. Iteration- variant (IV) equalizers are u pdated at each turb o iteration by additionally using the overall prior infor m ation. T ime-varying (TV) stru ctures u pdate their filters a t ea ch symbo l, using both symbol-wise prior info rmation and channel states, making them particularly suitable f o r doubly selectiv e chann els, where the impulse response varies over time. The first FIR turbo structure, p roposed b y Laot e t al. [4] , uses a tim e - in variant interfere nce canceller [5], and an app li- cation to IV filtering appear e d in [6], [7 ]. Further extension to TV eq ualization is provided in [8 ] and a fo rmal f ramework presented in [ 9] derive th ese rec e ivers fro m the MAP criterio n. An alternative approach f ormalized by T ¨ uchler et. al [10] consists in designin g a T V adaptive LE, by using statistics condition ed on prio r information , while solv in g the MMSE criterion. T his structure ha s b een ap plied to hig h-ord er mod u- lations, time-varying ch annels an d to IV , TI , f requen cy domain structures for lower co mplexity , and also to multi- u ser detec- tion for multiple input-mu ltiple outpu t systems [11]–[14]. Equiv alence of these appr oaches was shown in [15], m ak- ing the T V MMSE LE -IC the mo st widespread refer ence. Although tu r bo LE- IC br ings significan t imp rovements over classical filtering, it falls far beh in d classical DFE [16], [ 17] at hig h spectra l ef ficiency oper ating points. Oppositely , at lower rates, turbo LE-IC is near cap acity-achieving while DFE perfor ms poo rly 1 . This pape r addr e sses the design of iterative time-doma in TV DFE-IC equalizer s, i. e. FIR receivers wh e re pr ior in formatio n and a symbol- wise decision f eedback is respectively u sed on anti-causal and cau sal symb ols, to improve equalization. These receivers are of interest for application s wh ere doub ly -selective channels are in volved, such as HF comm unication s [18]. A. Rela ted W ork There exist several prior works on DFE- IC. Propo sals mainly differ with th e nature of decision feed back, an d with the filter up dating method . Besides, recent com p lex r e ceiv ers use DFEs as con stitue n t elements for concaten a ted eq ualizers. Hence, for clarity , we prop ose to classify related works in three sub-categories. 1 These facts are also shown in subsection V -C, in Fig. 7. c  IEEE. Personal use of this materia l is permitted. Howe v er , permission to reprint/re publish this material for advert ising or promotional purposes or for creati ng new collecti ve wor ks for resale or redistrib ution to s ervers or lists, or to reuse any copyrighte d component of this work in other works must be obtaine d from the IEEE. Final version: https://ieee xplo re.ieee.org/document/8 371591/ 2 T O APPEAR ON IE E E JOURNAL ON TRANSACTIONS ON COMMUNICA TIONS - M A Y 2018 1) Ite rative Hard DFE-I C: Among hard feedb a ck stru c - tures, DFE-IC in [ 1 9] is a c lassical DFE that uses prior informa tio n f or IC o n anti-cau sal symbols. This stru cture is known for its err or propagation issues which makes its TV form even less efficient than TI LE, and its extrinsic infor- mation tran sfer (E XIT) analy sis yields contradic to ry results [19, Fig. 14] . In [ 20], the previous struc tu re is enh anced with a powerful soft demapper that uses the d istribution of residual ISI sequence s for symbo l detectio n . This mod ified structure outperf o rms turbo LE-IC, but this residual ISI dis- tribution is very difficult to deriv e even in the simple BPSK case. A mor e practical solution, prop osed in [21], consists in approx imating the residu al ISI at the DFE -IC ou tput to an additive white Gau ssian noise (A WGN), which simplifies the demapp e r . While th is solution ch allenges TV LE- IC on BPSK, its extension to multilev el m odulation s h as not been explored so far . T o the authors’ knowledge, this is the only DFE-IC outperf orming exact TV LE-IC in the referen ce scenario of Proakis-C channel with BPSK sy mbols. DFE-I Cs in [20], [21] were later used as con stituent elem ents fo r more advanced receivers such as bi-directio nal DFE, or structu r es obtained by parallel concatenation of FIRs [22]. 2) Ite rative Soft DFE-IC: Literatur e on turbo soft DFE - IC is more d iv erse; altho ugh fee dback is mo stly based o n the posterior distribution, th ere is no commo n stra tegy fo r eval- uating its variance [ 23]–[25]. Such iterati ve structure is fir st presented in [26], wher e various TI DFE with sof t feed back are ev aluated with a p erfect dec ision hy pothesis, within a sub - optimal receiver using h a r d decoding . In p articular, it is seen that soft feedback mitigates to some extent err o r prop agation, despite igno ring decision err ors in filter compu tation. An o ther notable structure is the IV sof t interferen ce canc e ller in [23]; using both prior and p osterior L L Rs f or filtering and for interferen ce ca n cellation with BPSK, this scheme significantly outperf orms IV LE-I C, b ut it requires stochastic methods for estimating the correlatio n prop erties of posterio r LLRs. Sev eral other I V sof t feedback structures exist [2 5], [27], with alterna ti ve h euristics for feedb ack q uality assessment. Structural co mparison of IV schemes u sing p osterior f eedback is given in [24], extend ing [2 3] and [27 ] to higher o rder modulatio ns, but requ iring new heuristics w ith LE-I C pre- equalization for filter c omputatio n. These ap proach es h av e drawbacks due to their limitations in usable co nstellations [ 23], [25], [2 7], or due to the sub-optima lity of heuristics used in filter computatio n [2 3], [24], [27]. Ind eed, I V stru ctures n eed static statistics of its soft feed b ack for computin g its filter s, which requires approxim a tions. T ime-varying soft posterio r feed back stru ctures do n ot h av e such issues; they can upda te their filters after each sym bol is detected, as it had been done for MIMO receivers in [2 9]. In equalization , the structure closest to [29] is a block-f e e dback turbo DFE in [ 28], which u p dates its filters every P symbols. A classification of the references a bove is given in T able I. 3) Re ceivers based o n Exp ectation Pr opagation: There is a recent renewal of interest in iter ativ e e qualization, broug ht by the use of an app roximate statistical infere nce meth od, namely expectation prop agation (EP) [30]. This tech nique ca n be used as a message passing algo rithm, wh ich extends the loo py belief propag ation (BP) b y using exchan ge o f expectation s. When EP is used with prob ability d ensity fun c tio ns (PDF) b e longing to the exponential family , it is possible to compu te an extrinsic message passed from the d emapper to the equalizer . This parad igm has alrea dy b een used in chan nel decodin g [31], and in receiver d esign with MIMO receivers [32], bloc k linear equ a lizers [33] and Kalm an smo others [34], [35]. In particular, a concom itan t work has rec e ntly extended these schemes to FIR with a self-iterated LE-IC [36]. In [ 37], EP was applied on multiv ariate wh ite Gaussian distributions to derive a low-complexity self-itera ted fr e quency do m ain eq ualizer . T he receivers above use EP in a parallel in terferen c e c ancellation scheduling throug h self-itera tio ns, i.e. the whole data block is detected, a n d then detection process is r epeated using EP feedback from the demapp er . These structu res are n o t d ecision feedback stru ctures as in [16], wh ich a r e n a tural successi ve interferen ce cancellers. Hence, in this p aper, we p ropose to derive a DFE- IC EP exploiting the successive interfe r ence cancellation schedu le of DFE-IC to operate on an EP-based soft feedb ack. Moreover, we comb ine this serial d etection framework with an outer loo p , as in p rior work on EP , to o btain a self- iterated DFE-IC EP . A low complexity matrix in version strategy f or TV FIR stru c- tures is also derived, significan tly reduc in g the computatio nal complexity difference between DFE - IC and LE-IC. B. Contributions and P aper Ou tline The main contributions of this paper are as follows: • A novel tim e - varying DFE-I C algorith m , using EP to update its filters, an d to cancel residual ISI, is propo sed. It outperf orms other co nstituent FIR receivers k nown to the authors, while provid ing an overall efficient complexity- perfor mance trade-off. • DFE-IC EP is extended to a self-iterated structure, and compare d to prior work on self-itera ted EP rece ivers. • W ell-known h ard [ 19], [21] or su b -optimal [2 4], [ 28] DFE-IC pro posals are extended to TV structures with soft posterior feedback , b y using MMSE Bayesian estimators. • Analytical and asympto tic a n alysis of DFE-IC is carried out on a h ighly selective de terministic chan nel. Perf o r- mance and comp utational complexity comparison be- tween LE- I C and d ifferent DFE-I C structur es is provided . • A new recursive m atrix in version strategy for TV equal- izers is exp osed. Compar ed to the iterative algo r ithm in [10], it brings betwee n 30% (fo r lo ng data blo cks) and 75% (for shorter blocks) com plexity redu ction fo r L E-IC. The rem a inder of this paper is o rganized as follows. The considered BICM co mmunica tio n scheme and the generic FIR receiver m o del are described in section II. Section III pr oposes a factor g raph model for the system an d applies the expectation propag ation fra m ew ork to der iv e the pr o posed eq ualizer in subsection II I-D. A novel matrix inv ersion strategy is detailed in section IV for red ucing TV equ alization complexity . Section V extends prior work on DFE-IC to the state-of-the- art an d compare s with the pr oposed DFE- IC EP . In sectio n VI, DFE- IC EP is self-iterated, and co mpared with several existing self- iterated EP receivers. S ¸ AH ˙ IN et al. - ITER A TIVE EQU ALIZA TION WITH DECISION FEE DBA CK BASED ON EXPECT A TION PROP AGA TION 3 T ABLE I C L A S S I FI C AT I O N O F C O N S T I T U E N T F I R T U R B O E Q UA L I Z E R S V S . T H E U S A G E O F P R I O R I N F O R M AT I O N . Linear Structure Decision Feedback Structures Update T ype TI IV TV Dec. T ype TI IV TV Referen ces [4], [19] [6], [7], [9], [10], [12], [14], [15] [8]–[11], [13], [19] Hard [20], [21 ] [20], [21] [19]–[21] Soft APP [26] [23], [24], [27], [28 ] Proposed Soft EP Proposed C. Nota tions Bold lowercase letters are used for vectors: let u be a N × 1 vector , then u n , n = 0 , . . . , N − 1 are its entries. Capital b old letters deno te matrices: for a g i ven N × M m atrix A , [ A ] n, : and [ A ] : ,m respectively denote its n th row and m th column, and a n,m = [ A ] n,m is the entry ( n, m ) . I N is the N × N iden tity matr ix , 0 N ,M and 1 N ,M are respectively all zeros an d all o nes N × M matrices. e n is the N × 1 ind icator who se on ly n on-zer o en try is e n = 1 . Operator Diag ( u ) d enotes the diag onal ma trix whose diag o nal is defined by u . R , C , and F k are respectively the real field, the complex field and a G a lois field o f order k . Let x and y be two ran d om variables, then µ x = E [ x ] is the expected value, σ 2 x = V ar [ x ] is the variance and σ x,y = Cov [ x, y ] is the covariance. The probab ility of x tak ing a value α is P [ x = α ] , and proba b ility d ensity fun ctions (PDF) are denoted as p ( · ) . If x and y are random vector s, then we defin e vectors µ µ µ x = E [ x ] and σ σ σ 2 x = V ar [ x ] , the covariance matrix Σ x , y = Cov [ x , y ] an d we note Σ x = Cov [ x , x ] . C N ( µ x , σ 2 x ) denotes th e circu larly- symmetric complex Gaussian distribution of mean µ x and variance σ 2 x , and B ( p ) denotes the Bernoulli d istribution with a success probab ility of 0 ≤ p ≤ 1 . I I . S Y S T E M M O D E L A. T r ansmission Over a Multipath Channel W e conside r a single carrier transmission using a bit- interleaved coded mod ulation (BICM) scheme. Let b ∈ F K b 2 be a binary infor mation packet of length K b bits. A channel encoder maps b into a co dew ord c ∈ F K c 2 , with a co de rate R c = K b /K c , which is th en inte r leav ed to g iv e a d ata blo ck d ∈ F K c 2 . A memory less mapping ϕ associates d to the symb ol block of length K , den oted x ∈ X K , where the constellation X ⊂ C has M elemen ts. The q -word associated to a symbo l is deno ted d k = [ d ] qk : q ( k +1) − 1 , and ϕ − 1 j ( x k ) and d k,j denote the value of the j th bit labellin g the k th symbol x k , i.e. d kq + j . W e assume the constellation has zero m ean, and has an av erage symbol power of σ 2 x , with equipr obable symbo ls. For th e sake of clarity , only the single user , single in p ut- single outpu t T -spaced (sym bol spa ced) equalization problem is co nsidered. Th e ch a nnel is mo delled at the base-ban d as an equivalent L -tap linear time-varying filter h [ k ] = [ h k,L − 1 , h k,L − 2 . . . h k, 0 ] , k b eing the time index, and where pulse shaping and transceiver filters are accou nted for . The signal going throug h the channel is then affected by therma l n o ise w k at the receiver side, an d assuming a perfect ch annel state info r mation, ideal time and frequ ency synchro n ization and the ab sence of inter-block interfer ence (IBI), the base-band received samp les ar e given by : y k = P L − 1 l =0 h k,l x k − l + w k , (1) where k = 0 , 1 , . . . , K + L − 2 , an d x k , k < 0 an d k > K are set to 0 . These assumptio n s ca n be satisfactorily app roached in pra ctice with the use o f a unique- word signalling scheme, among o ther op tions, to jointly enab le chan nel estimation and the IBI removal. The n oise is m o delled as w k ∼ C N (0 , σ 2 w ) , i.e. its real and imaginary parts are real ind ependen t zero mean Gau ssian rando m processes with σ 2 w / 2 variance each. The transmission can be rewritten as: y = Hx + w , (2) with y = [ y 0 , . . . , y K + L − 2 ] T , w = [ w 0 , . . . , w K + L − 2 ] T , x = [ x − L +1 , . . . , x K + L − 2 ] T and H is the ( K + L − 1) × ( K + 2 L − 2) m a trix wh o se k th row is [ 0 1 ,k − 1 , h [ k ] , 0 1 ,K + L − 1 − k ] , k = 1 , . . . , K + L − 1 . B. On MMSE FIR Equa lization FIR structure s can be modelled by windowed pro cesses; applying a sliding window [ − N p , N d ] on the observation vec- tor y , we define y k = [ y k − N p , . . . , y k + N d ] T . N p and N d are respectively the n umber of pre-c u rsor and po st-cursor samples, and we den o te N , N p + N d + 1 , and N ′ p , N p + L − 1 to simplify notation s. Then , using th e same window on w , an d [ − N ′ p , N d ] on x , the channel model becomes y k = H k x k + w k , (3) with H k = [ H ] k − N p : k + N d , k − N ′ p : k + N d , for k = 0 , . . . , K − 1 . Below , a generic structure of an unbiased MMSE FIR receiver is given for comparing different stru c tu res and their dynamics in the remainder of the paper . Prior estimates on x with means ¯ x fir k , [ ¯ x fir k − N ′ p , . . . , ¯ x fir k + N d ] and variances ¯ v fir k , [ ¯ v fir k − N ′ p , . . . , ¯ v fir k + N d ] are used for interfer e n ce cancel- lation. Then den oting its output estima te o n x k as x e k , and the variance of the residual interfer ence and noise as v e k , with x e k = f fir k H y k + g fir k v e k = 1 /ξ fir k − ¯ v fir k ,      f fir k , Σ fir k − 1 h k /ξ fir k , g fir k , ¯ x fir k − f fir k H H k ¯ x fir k , ξ fir k , h H k Σ fir k − 1 h k , (4) where Σ fir k , k w σ 2 w I N + H k ¯ V fir k H H k , ¯ V fir k , diag ( ¯ v fir k ) , h k , H k e k and k w = 1 / 2 , wh en sig nals with one r e al degree of freedom are used ( e.g. X is BPSK), and otherwise k w = 1 [17]. A proof of these relationships is in Ap pendix A. Note that ¯ x fir k and ¯ v fir k completely character ize such re- ceiv ers. When ¯ x fir k ′ and ¯ v fir k ′ are indepen d ent of x e k , v e k , ∀ k ′ , k , we ca ll this receiver a LE-I C, an d when ¯ x fir k ′ and ¯ v fir k ′ are depend ent on x e k , v e k , ∀ k ′ < k , we ref er to it as a DFE- IC. I I I . R E C E I V E R D E S I G N W I T H E X P E C TA T I O N P R O PAG A T I O N This section focu ses o n the design o f a FIR rece iver that approx imates the p osterior probab ility distribution o n x k using an EP-based message passing on th e system factor gr aph. 4 T O APPEAR ON IE E E JOURNAL ON TRANSACTIONS ON COMMUNICA TIONS - M A Y 2018 A. F a ctor Graph Model for FIR Receivers The o ptimal joint MAP r e c eiv er satisfies the MAP criterio n ˆ b = max b p ( b | y ) , where, assumin g i.i.d. informa tio n bits, the posterior PDF can be factorized as f ollows p ( b | y ) = p ( b , d , x | y ) ∝ p ( y | x ) | {z } channel p ( x | d ) | {z } mapping p ( d | b ) | {z } encoding . (5) This density can be further factorized by u sing: - the memo ryless mapping: p ( x | d ) = Q K − 1 k =0 p ( x k | d k ) , - the indepen d ence assumption in BICM encoding: p ( d | b ) = Q K − 1 k =0 Q q − 1 j =0 p ( d k,j ) , where p ( d k,j ) , p ( d k,j | b ) is a probab ility mass fun ction (PMF) w h ich is seen as a Berno ulli-distributed p r ior constraint provided b y the decoder, fro m the r eceiv er’ s po int of v iew . The “ch annel” factor in (5) creates constrain ts between th e whole blo ck of r eceived baseb a n d samples an d the transmitted symbols, howe ver to der iv e a r educed com plexity FIR receiver which estimates x k and d k , the win dowed model in (3) is needed. The FIR approxim a tion posterio r is p  ¯ d k , x k | y k  ∝ Q k + N d k ′ = k − N ′ p p ( y k | x k ) p ( x k ′ | d k ′ ) Q q − 1 j =0 p ( d k ′ ,j ) , (6) where ¯ d k = d k − N p − L +1: k + N d . Note that working with p  ¯ d k , x k | y  ≈ p  ¯ d k , x k | y k  is not the only option for estimating x k . Ind eed x k can b e estimated through inferenc e on x k ′ , w ith k ′ = k − N d , . . . , k + N ′ p , but by selecting x k , this option is ind irectly translated to the cho ice of window parameters, which is a common aspect of FIR equalizers. A message-passing based decoding algorithm it eratively estimates the variable nodes (VN) x k and d k,j by using constraints imp osed by factor nodes (FN). Factor nodes are non pro per PDFs for reso lv ing transmission steps. The dec o der FN models BICM encoding constraints with f DEC ( d k,j ) , p ( d k,j ) , (7) and the demapper FN incorpor ates m apping constraints with f DEM ( x k , d k ) , p ( x k | d k ) = Q q − 1 j =0 δ ( d k,j − ϕ − 1 j ( x k )) , (8) where δ is th e Dirac delta functio n. The multipath channel constraints are modelled within the equalization factor n ode f EQU ( x k ) , p ( y k | x k ) ∝ e − y H k y k /σ 2 w +2 R ( y H k H k x k ) /σ 2 w , (9) where the de penden ce on y k is o mitted, as obser vations are fixed during th e message-p a ssing pro cedure. Using the se notations, the posterior (6) g iv es the factor gra ph shown in Fig. 1. B. Exp ectation Pr o pagation Message P assing F ramework EP-based m essage passing algorithm is an extension of loopy belief propag ation, where VNs are assumed to lie in the exponen tial distribution family [ 38]. Co nsequently , the ex- changed messages are depicted b y tractable distributions, and they allow iterative co mputation of a fully-factor ized ap prox i- mation fo r cumber some po sterior PDFs such as p ( ¯ d k , x k | y k ) . y k − N p y k − N p +1 . . . y k . . . y k + N d − 2 y k + N d x k − N ′ p . . . x k . . . x k + N d . . . d k, 0 . . . d k,q − 1 . . . . . . f EQU ( x k ) f DEM ( x k , d k ) f DEC ( d k ) Fig. 1. Fac tor graph for the posterior PDF (6) on x k and d k . Updates at a FN F co nnected to variable nodes v ar e as fo llows. Message s exchanged betwe e n a VN v i , the i th compon ent of v , and factor nod e F a r e m v → F ( v i ) , Q G 6 = F m G → v ( v i ) , (10) m F → v ( v i ) , proj Q v i [ q F ( v i )] /m v → F ( v i ) , (11) where pro j Q v i is the Kullback -Leibler projection tow ards the probab ility distribution Q v i of VN v i . The posterio r q F ( v i ) is an approx imation of the marginal of the true posterior p ( v ) on v i , obtaine d by combin in g th e true factor o n FN F with messages from the neighbour ing VNs q F ( v i ) , R v \ i f F ( v ) Q v j m v → F ( v j ) dv \ i , (12) where v \ i are VNs without v i [38]. The projectio n opera tio n for expon e ntial families is equiv alent to moment ma tching , which simplifies the computation o f messages [30], [38]. In this paper symbol VNs are assumed to lie in the family of mu ltiv ariate circularly symmetric Gaussians with diagon al covariance matrices, making the ap proxim ate d istributions fully factorized to indep endent Gau ssians. Hence, a message on x k will be defined by a mean and a variance. The VNs d k,j are co nsidered to follow Bern oulli distributions (wh ich is included in the exponential family), and their messages ca n b e described by bit log-likelihood r atios (LLR). This forma lism is very generic and allows th e der ivation of many receiver structur e s. It has b een used to derive a MIMO detec to r in [ 32], and a Kalman smooth er in [34]. Howe ver EP receivers can also be d eriv ed withou t a message passing fo rmalism, as r ecently shown for the b lock [33] or FI R [36] equ a lizers. T o the auth ors’ knowledge, message-passing formalism was not previously u sed for FIR de sign, an d it is fa voured in th is paper because of the av ailable scheduling options it allows to clearly identify . C. Derivation of Exchanged Messages This section d etails the EP-based message pa ssing algo- rithm’ s applica tio n to the conside r ed factor g raph. First, ex- changed messages are defined, and th en their characterizing parameters ar e explicitly comp uted. See Fig . 2 for a conven- tional view of the receiver with these q uantities. The messages arriving on the VN x k are Gaussians with m EQU → x ( x k ) ∝ C N ( x e k , v e k ) , (13) m DEM → x ( x k ) ∝ C N  x d k , v d k  , (14) S ¸ AH ˙ IN et al. - ITER A TIVE EQU ALIZA TION WITH DECISION FEE DBA CK BASED ON EXPECT A TION PROP AGA TION 5 EQU Node DEM Node DEC Node y ( x e , v e ) L e ( d ) Π Π Π − 1 ˆ b ( x d , v d ) L a ( d ) Π Π Π SISO Equal- izer Soft Mapper / Demapper SISO Decoder Fig. 2. Factor nodes shown as an iterati v e BICM recei v er . whereas messages arriving on the VN d k,j are Bernoullis m DEC → d ( d k,j ) ∝ B ( p a d ) , m DEM → d ( d k,j ) ∝ B ( p e d ) . (15) During the message passing proced ure, the char acteristic pa- rameters of th ese distributions are upd ated f ollowing a selected schedule. For Bernoulli distributions, it is rath er prefe r able to work with bit LL Rs, rather than the su c cess probability p d : L ( d j ) , ln P [ d j = 0 ] P [ d j = 1 ] = ln 1 − p d p d . (16) W e use L a ( · ) , L e ( · ) an d L ( · ) opera to rs to d enote respec ti vely a prio ri, extrinsic and a poster iori LLRs. When applied to d k,j , this vocabulary rep r esents the receiver’ s perspec tive, i.e. L a ( d k,j ) , L e ( d k,j ) respec tively cha racterize m DEC → d ( d k,j ) and m DEM → d ( d k,j ) . Finally , conside r ing the factor grap h shown on Fig. 1, all variable nodes are on ly conn ected to a pair o f distinct facto r nodes. Conseq uently , using eq. (10), m v → F ( v i ) = m G → v ( v i ) , for all VN v i , and FN F , G , F 6 = G they are conne cted to. 1) Messages fr om DEC to DEM: I n this pap er , we assum e DEC is a SISO decoder providing prior inform ation L a ( d ) to DEM, whenever it receives extrinsic inform a tion L e ( d ) by DEM. The d emapper uses these pr io r LLRs, along with the DEM FN (8) to compute a prior PMF o n x k = α , ∀ α ∈ X with P k ( α ) ∝ Q q − 1 j =0 e − ϕ − 1 j ( α ) L a ( d k,j ) . (17) This is a c a tegorical PMF correspon d ing to the margina l of f DEM ( x k , d k ) m d → DEC ( d k ) on x k [32], used h ereafter to compute approx imate marginals q DEM ( x k ) and q DEM ( d k,j ) . 2) Messages fr om DEM to EQU: The d emapper com putes an approxim ate po ster io r on the VN x k using eq. (12) with q DEM ( x k ) = P d k f DEM ( x k , d k ) m x → DEM ( x k ) Q q − 1 j =0 m d → DEM ( d k,j ) . (18) This is a posterior categor ical PMF o n the e lements x k of X , which ca n be compu ted using eqs. (13) an d (1 7), which will be denoted as D k ( α ) ∝ exp  − k w | α − x e k | 2 /v e k  P k ( α ) , ∀ α ∈ X . (19) For computing messages towards EQU, the posterior PMF is projected into C N thro ugh mo ment matching. The mean and the variance of D k are µ d k , E D k [ x k ] = P α ∈X α D k ( α ) , γ d k , V ar D k [ x k ] = P α ∈X | α | 2 D k ( α ) − | µ d k | 2 . (20) When m x → DEM ( x k ) ∝ 1 , i.e. when th ere is no info rmation from the EQU n o de (equiv alent to x e k = 0 an d v e k = + ∞ ), D k = P k , and we denote the prior mean and v ariances as x p k , E P k [ x k ] , v p k , V ar P k [ x k ] . (21) Note that these values are used as soft f eedback in conv en- tional turbo equalization. Then in or d er to calculate m DEM → x ( x k ) as in ( 1 1), a Gaussian di vision [30] is implemente d x ∗ k = µ d k v e k − x e k γ d k v e k − γ d k , and , v ∗ k = v e k γ d k v e k − γ d k . (2 2 ) This is the major novelty in using EP: the computa tio n of an extrinsic f eedback f rom the dem a pper to the equalizer . Attempting th is with categorica l distributions, as in BP , would completely rem ove m x → DEM ( x k ) , and th e extrin sic “feed - back” to E QU would simply be the prior PMF P k [32], which would yield a recei ver equ ivalent to LE- IC [ 19]. EP message passing algorithm consists in minim iz in g g lobal div ergence th rough iterative minimization of simpler local div ergences. Thus, it might lock o n undesirable fixed po ints, and a damping heuristic, as recommend ed in [3 8, eq. (1 7)], is used to improve accuracy v d ( next ) k = h (1 − β ) /v ∗ k + β / ¯ v d ( prev ) k i − 1 , x d ( next ) k = v d ( next ) k " (1 − β ) x ∗ k v ∗ k + β x d ( prev ) k v d ( prev ) k # , (23) where 0 ≤ β ≤ 1 con figures the dampin g, a nd its effectiv eness has been verified in [36]. 3) Messages fr om EQU to DEM: T h e equ alizer co mputes an approxim ate posterior on the VN x k using eq. (12) with q EQU ( x k ) = R x \ k k f EQU ( x k ) Q k + N d k ′ = k − N ′ p m x → EQU ( x k ′ ) dx \ k k . (24) The integran d of th e equation ab ove is a multiv ariate Gau ssian distribution C N ( µ µ µ e , Γ e ) , hence, using eq. (9), its covariance and mean satisfy Γ e k = ( V d k − 1 + σ − 2 w H H k H k ) − 1 , µ µ µ e k = Γ e ( V d k − 1 x d k + σ − 2 w H H k y k ) , (25) where V d k = diag ( v d k ) , with v d k = [ v d k − N ′ p , . . . , v d k + N d ] , and x d k = [ x d k − N ′ p , . . . , x d k + N d ] . Using some matrix algeb r a, and W oodbury’ s identity o n Γ e , the mean µ e k and th e variance γ e k of the marginalized PDF q EQU ( x k ) are gi ven by γ e k = e H k Γ e k e k = v d k (1 − v d k h H k Σ d k − 1 h k ) , µ e k = e H k µ µ µ e k = x d k + v d k h H k Σ d k − 1 ( y k − H k x d k ) , (26) with Σ d k = k w σ 2 w I N + H k V d k H H k . Message to the dema pper is then extracted with the Gaussian d ensity division in eq . (11) v e k = γ e k v d k v d k − γ e k , an d , x e k = v d k µ e k − γ e k x d k v d k − γ e k . (27) Dev eloping these y ields a FIR expression as in (4) with ¯ x ep k , [ x d k − N ′ p , . . . , x d k + N d ] and ¯ v ep k , [ v d k − N ′ p , . . . , v d k + N d ] for IC. 6 T O APPEAR ON IE E E JOURNAL ON TRANSACTIONS ON COMMUNICA TIONS - M A Y 2018 Algorithm 1 Proposed Self-Iterated DFE- IC E P receiv er . Input y , H , σ 2 w 1: Initialize deco d er with L (0) a ( d k ) = 0 , ∀ k . 2: for τ = 0 to T do 3: ∀ k = 0 , . . . , K − 1 , use L ( τ ) a ( d ) to co mpute P ( τ ) k with (17), and set ( x d (0) k , v d (0) k ) ← ( x p k , v p k ) using (21). 4: for s = 0 to S τ do 5: f or k = 0 to K − 1 do 6: Equalize using (27) and get ( x e ( s ) k , v e ( s ) k ) . 7: Use (1 9)-(20) to u pdate D ( s +1) k , and gen erate EP feedback ( x d ( s +1) k , v d ( s +1) k ) with (22)-(23). 8: If v d ( s +1) k ≤ 0 , then ( x d ( s +1) k , v d ( s +1) k ) ← ( µ d k , γ d k ) and store k in the set I ( s ) err . 9: end for 10: ∀ k ∈ I ( s ) err , ( x d ( s +1) k , v d ( s +1) k ) ← ( x d ( s ) k , v d ( s ) k ) . 11: end f or 12: Compute L ( τ ) e ( d k ) using D ( τ , S τ ) k with (29), ∀ k , and provide them to the decoder, to obtain L ( τ +1) a ( d k ) , ∀ k . 13: end for 4) Messages fr om DEM to DEC: The demapper co mputes an approxim ate po ster io r on the VN d k,j using eq. (12) with q DEM ( d k ) = P x k ∈X f DEM ( x k , d k ) m x → DEM ( x k ) Q q − 1 j =0 m d → DEM ( d k,j ) . (28) As b it LLRs are used to represent m e ssages to DE C, this distribution is marginalized on d k, 0 , . . . , d k,q − 1 [32], an d the division in eq. (11) is directly carried out with LLRs L e ( d k,j ) = ln X α ∈X 0 j D k ( α ) − ln X α ∈X 1 j D k ( α ) − L a ( d k,j ) , (2 9) with X p j = { α ∈ X : ϕ − 1 j ( x ) = p } where p ∈ F 2 . D. Pr oposed Self-Iterated D FE-IC EP R eceiver A factor graph (sec. II I -A) and messages exchan ged over it ( sec. III-C) are n ecessary to der iv e a receiver algorithm, but may be insufficient when considerin g a grap h with cycles. Indeed , specifying a scheduling for the upd ate of VNs and FNs is also required . In this paper, a serial scheduling across variable no des x k is considered . In deta il, whe n E QU u pdates a VN x k , facto r node DEM is immed iately activated in o rder to p rovide its own extrinsic estimation of x k , jo in tly using prior inform ation from the d ecoder and th e eq u alizer’ s extrinsic o utput. This re su lts in a DFE-I C structur e, using a novel k ind of soft feed back, unlike any h ard or sof t APP feedb ack p reviously used in the literature [19]– [21], [23]–[ 27]. Moreover, when detection across the whole block is co mpleted, this serial schedu ling can be repeated by keeping th e previously updated DEM messages, yielding a self-iterated DFE- IC E P structure. T o clarify the dy namics of th e prop osed receiver , τ = 0 , . . . , T den o tes turb o iterations (T I), i. e . exchanges betwe e n the DEM and DEC factor nodes. E ach TI consists of s = 0 , . . . , S τ self-iterations (SI ) (may vary with τ ), i.e. exchanges between EQU and DEM factor node s, which seq uentially y k f k x e k v e k Soft Demapper L e ( d k ) ¯ x a k g a k g c k ¯ x c k x d k − 1 v d k − 1 Gaussian Div ision Soft Mapper L a ( d k ) σ 2 w H k y k + N d + − − x p k + N d v p k + N d + x p k {P k } µ d k − 1 γ d k − 1 Fig. 3. TV DFE-IC EP (dashed) / AP P (no dashed) structure . updates the whole bloc k x . In the following, EQU ↔ DE M messages derived previously are app ended a super scr ipt ( s ) , but τ is omitted for read ability . The propo sed sched uling, given in Algo rithm 1, gener ates an EP FIR receiver which uses the fo llowing means and variances for interfer e n ce can cellation ¯ x dfe-ep k ( s ) , [ x d ( s +1) k − N ′ p , . . . , x d ( s +1) k − 1 , x d ( s ) k , . . . , x d ( s ) k + N d ] T , ¯ v dfe-ep k ( s ) , [ v d ( s +1) k − N ′ p , . . . , v d ( s +1) k − 1 , v d ( s ) k , . . . , v d ( s ) k + N d ] T , (30) for k = 0 , . . . , K − 1 . This lay out shows that this structu re indeed follows a time-varying DFE-IC ev olution, with anti- causal symbols using demapp er’ s ou tput from the p revious self-iteration, and causal symbo ls using cu r rent EP f eedback from the d emapper . The extrinsic f e edback from the demap per is obtain e d by using jointly the prio r inform ation from the previous T I, an d the past e q ualizer ou tputs of the cu rrent and previous self iterations (see (19)-(23)). The Algorithm 1 also incorpo rates a mechanism to deal with EP- b ased feedback ’ s infamous negative variances [32], [3 3], with the set I ( s ) err which stores their in dexes. Th ese values are replac e d with APP-b ased variances in th e curren t SI, and then replaced again with their previous values f o r the n ext SI. Although equ ation (4) is usefu l for FI R an a lysis, c a usal and anti-causal f eedback of DFE-I C should be separated in practice. Using E c = [ I N ′ p , 0 N ′ p ,N d +1 ] , E a = [ 0 N d +1 ,N ′ p , I N d +1 ] , (31) we define H c k = H k E c T and H a k = H k E a T , to respectively operate on ¯ x c ( s ) k = E c ¯ x dfe-ep ( s ) k , and ¯ x a ( s ) k = E a ¯ x dfe-ep ( s ) k , as a generalized interference can cellation scheme. The SI DFE-IC EP of (30), is re written as: x e ( s ) k = ¯ x a ( s ) k + f ( s ) k H y k − g c ( s ) k H ¯ x c ( s ) k − g a ( s ) k H ¯ x a ( s ) k , v e ( s ) k = 1 /ξ dfe-ep ( s ) k − ¯ v a ( s ) k , (32) with f ( s ) k = Σ dfe-ep ( s ) k − 1 h k /ξ dfe-ep ( s ) k , g c ( s ) k = H c k H f k , and g a ( s ) k = H a k H f ( s ) k . When S τ = 0 , the propo sed r e c eiv er is a strict TV DFE-IC EP , with · d ( s +1) k = · d k and · d ( s ) k = · p k , this case is shown on Fig. 3 with the dashed module. In conc lu sion, we have a pplied message passing framework of EP fo r equ alization, using slidin g window o bservations. This results in a novel me ssage compu tation given by (22)-(23) and (27), unlike bloc k wise m essages in [32], [33]. M o reover , by usin g an hybrid serial/par a llel schedule, o ur structure S ¸ AH ˙ IN et al. - ITER A TIVE EQU ALIZA TION WITH DECISION FEE DBA CK BASED ON EXPECT A TION PROP AGA TION 7 Algorithm 2 Cholesky upd ate algorithm fo r LE- I C. Input L k − 1 , σ 2 w , ¯ v k + N d , H k − 1 , H k , ¯ V k − 1 Output L k 1: { Ad d a r ow and a column } 2: [ h 1 k , h 2 k ] ← [0 , e H k + N d H k ] 3: w ← H k − 1 ¯ V k − 1 h 1 k 4: l 12 ← L − 1 k − 1 w 5: l 22 ← q h 1 H k ¯ V k − 1 h 1 k + ¯ v k + N d | h 2 k | 2 − l 12 H l 12 + σ 2 w 6: { Bu ild augmented ma trix and r emove r ow & column } 7:  × 0 1 ,N l 21 L 22  ←  L k − 1 0 N , 1 l 12 H l 22  8: { Ra nk-1 update L k L H k = L 22 L 22 H + l 21 l 21 H } 9: for l = 1 to N do 10: r ← q [ L 22 ] 2 l,l + | [ l 21 ] l | 2 , c ← [ L 22 ] l,l r , s ← [ l 21 ] ∗ l r 11: [ L 22 ] l : N ,l ← c [ L 22 ] l : N ,l + s [ l 21 ] l : N 12: [ l 21 ] l : N ← c [ l 21 ] l : N − s ∗ [ L 22 ] l : N ,l 13: end for 14: L k ← L 22 operates as a self- iter ated DFE-IC, u nlike the self-iterated LE- IC schem e co n curren tly developed in [36]. I n th e fo llowing, a matrix inversion strategy is in troduc e d, that red uces the computatio nal comp lexity difference between DFE-IC and LE- IC. I V . M AT R I X I N V E R S I O N F O R T I M E - V A RY I N G S L I D I N G W I N D O W T U R B O E Q UA L I Z E R S A. Sh ortcomings of Ex isting Appr oaches T ime-varying FIR as in (4) have excessive computationa l costs d ue to symbol-wise filter u pdates, requirin g recursive matrix in version metho ds. T h is section overviews the prob lem of computing f k = Σ − 1 k h k , for k = 0 , . . . , K − 1 efficiently . In [10], T ¨ uch ler et al. prop o se fo r LE-I C, a recursive matrix in version algor ith m, based on co mmon submatrices between successiv e inverses. T he proced u re requires comp uting an initial in verse (Gauss-Jord an inversion) with a c o mplexity order 2 of 4 N 3 / 3 , but f urther re c ursions’ complexity is 2 N 2 . Practical imp le m entations avoid inv ersion by so lv ing th e system Σ k f k = h k for f k with triangular factorizations [3 9], using forward/backward substitutions. T his approach is even more advantageous in eq ualization where the system is sparse. In th is paper, we pro pose a novel recursive in version strategy for LE-I C and DFE-IC, based o n an initial Ch o lesky d ecom- position, and followed by sparse r a n k-1 u pdates/downdates o f the factor s fo r following inversions. Unlike [ 39], our alg orithm is ab le to d eal with channel matrices evolving in time, makin g it more e fficient f o r turbo TV FIR. For LE-I C the complexity order is of N 2 , hence roughly 50% less complex than [10]. B. Cholesky F ac tor Upda te for MMSE LE-I C W e con sider a LE-IC with p r iors variances ¯ v k , let L k − 1 be the lower triangula r Cholesky dec omposition o f th e covariance 2 ”Order” means asymptotic expa nsion as N → + ∞ , assuming N ∝ 3 L , i.e. sliding window operatin g on 4 L symbols. Algorithm 3 Cholesky update algorithm for DFE-IC. Input ˜ L k , ¯ v a k − 1 , ¯ v c k − 1 , [ H k ] : , − 1 Output L k 1: w ← q | ¯ v a k − 1 − ¯ v c k − 1 | [ H k ] : , − 1 2: for l = N p to N do 3: if ¯ v c k − 1 < ¯ v a k − 1 then 4: { Rank-1 do wndate L k L H k = ˜ L k ˜ L H k − ww H } 5: r ← q [ ˜ L k ] 2 l,l − | [ w ] l | 2 , c ← [ ˜ L k ] l,l r , s ← [ w ] ∗ l r 6: [ ˜ L k ] l : N ,l ← c [ ˜ L k ] l : N ,l − s [ w ] l : N 7: else if ¯ v c k − 1 > ¯ v a k − 1 then 8: { Rank-1 up date L k L H k = ˜ L k ˜ L H k + ww H } 9: r ← q [ ˜ L k ] 2 l,l + | [ w ] l | 2 , c ← [ ˜ L k ] l,l r , s ← [ w ] ∗ l r 10: [ ˜ L k ] l : N ,l ← c [ ˜ L k ] l : N ,l + s [ w ] l : N 11: end if 12: [ w ] l : N ← c [ w ] l : N − s ∗ [ ˜ L k ] l : N ,l 13: end for 14: L k ← ˜ L k matrix Σ k − 1 , i.e. L k − 1 L H k − 1 = Σ k − 1 . The r e sulting updated Cholesky d ecompo sition is a rank- 1 upd ate [ 40] of L 22 , defined within algorithm 2 . These steps, followed b y for ward/backward substitutio n s f k = L − H k L − 1 k h k , allow low co mplexity filter comp utation. C. Cholesky F actor Update for MMSE DFE-IC In the case of DFE -IC, the diago nal o f the covariance matrix ¯ V tdfe is compo sed of two in depend ently sliding parts: on e for causal sym b ols ¯ v c k , between symbols k − N ′ p and k − 1 , the oth er for a nti-causal ¯ v a k , b etween symbols k and k + N d . T he LE - IC upd ate p rocedu r e above h a n dles th e addition of ¯ v a k + N d and the rem oval of ¯ v c k − N ′ p − 1 , but the change in ( k − 1) th symbol remains to be updated. Algorithm 3 gives a such u pdate pr ocedur e for DFE-IC, by app lying either a rank- 1 update or do wndate on ˜ L k , the Cholesky factor who h as already been updated by algorithm 2, depending on th e sign of ¯ v c k − 1 − ¯ v a k − 1 . Such updates are carried out using G ivens p lane rotations [40]. D. Compu tational Comp lexity Ana lysis The comp utational complexity of the prop osed algorithm is ev aluated with the number of req u ired multiply and accu- mulate units, estimated b y th e number of rea l add itions a nd multiplications, amoun ting to half a floating point oper ation ( 0 . 5 FLOPs) each. FLOP count ratios between different FIR implem entations are plotted in Fig. 4, depending on the channel sprea d , with a b lock length K = 2 048 and a FIR wind ow giv en by N = 3 L + 2 , N d = 2 L . The blue dashed cu rves show the FLOP count ratio of a L E-IC using ou r strategy relative to using th e algorithm in [1 0], fo r different co n stellation or ders. Up to 50% sa ving is observed as ch annel spread increases. DFE-IC FLOP count is compar ed to L E-IC, b oth u sing th e propo sed inv ersion strategies, with red solid lines. T h is ratio is high for a low nu mber chann el taps, but decre a ses to 7% as L 8 T O APPEAR ON IE E E JOURNAL ON TRANSACTIONS ON COMMUNICA TIONS - M A Y 2018 5 10 15 20 25 −100 −75 −50 −25 0 25 50 75 100 125 150 Chan nel Spread L (Number of Paths) Chan ge in C omputati onal Complex it y (F LO Ps) (%) M=2 M=4 M=8 M=16 M=32 M=64 Prop. DF E-IC vs. Prop. LE -IC Prop. LE-IC vs. LE-IC in [10] MAP vs. Prop. LE-IC Fig. 4. Complexi ty comparison of LE-IC and DFE -IC with proposed matrix in v ersion algorithm. increases, m ore or less qu ickly depe n ding on the modu lation order M . Finally , MAP detector is seen to be a n in teresting alternative to FIR r eceivers for BPSK/QPSK signalling , in channels with very sh o rt channel spreads. V . C O M PA R I S O N W I T H T H E P R I O R W O R K O N T I M E - V A RY I N G D F E - I C S T R U C T U R E S In this section , th e DFE-IC based on EP feedb a ck, pr oposed in section III- D, in its can onical form without self -iterations ( S τ = 0 ) and witho ut dampin g is comp ared to alternative state-of-the- art TV DFE-IC structur es. First, to p rovide a fair perform ance co mparison with alter- natives , existing subo ptimal DFE - IC sch emes [19], [21], [24] are extend ed to time-varying structures using soft posterio r feedback . Next, analy tical an d asympto tic ana lysis, and Monte Carlo simulation s sho w the superio r ity o f DFE-IC based on EP relati ve to LE-IC, classical DFE and concu rrent DFE-IC structures. A. On th e TV DF E-IC based on Bayesian e stimators References on tim e-varying DFE-IC with sof t feedb ack a r e limited. Hence, here existing method s are gener alized and improved befor e co mparison , thanks to o ur framework, in order to provide a fair comparison. Un til EP , soft posterio r feedback was the only imp erfect feedback with a re a sonable complexity in the literature, app licable to any constellation. Nev ertheless, it is not possible to d erive a structur e using such feedback with in the co n ventional BP formalism, but her e its usage is justified with Bayesian inference. One can consider the equ alization prob lem within a Bayesian fram ew ork, wher e a p articular realization of a ran- dom data sym bol is estimated. For instance , the con ven- tional MMSE linear turbo receiver [ 10] is also the MAP estimator, if priors are for ced to lie in the family of Gaus- sian distributions [9]. Hence this equ alizer is the unb iased Bayesian estimator E L k [ x k | y k , H k ] , whe re th e joint pr ior distribution L k ( x k ) ∝ Q k + N d l = k − N ′ p , C N ( x p l , v p l ) is used. How- ev er , in Bayesian estimation theory , the mean square erro r can be fur ther reduce d , using a sequentia l MMSE estimator , which imp roves its po sterior with previously estimated d ata (Sect. 12 .6 in [41]). Follo wing this idea, we p ropose the im- proved estimator E A k [ x k | y k , H k ] , ba sed on the joint posterio r A k ( x k ) ∝ Q k + N d l = k C N ( x p l , v p l ) Q k − 1 l = k − N ′ p C N ( µ d l , γ d l ) , wher e µ d l and γ d l are given by (20). In the following, we deriv e a posterior feedback b ased DFE-IC using this estimator fo r IC, with model (4). 1) Ex act TV DFE-IC with A PP F eedback: Th is equalizer is a genera lization of in v ariant schemes in [2 3], [24] to TV structures. It is derived by u sing the jo int posterio r A k ( x k ) with the mode l ( 4), derived in the Appen dix A . The r esulting APP FIR structure, is fully defined b y ¯ x app k = [ µ d k − N ′ p , . . . , µ d k − 1 , v p k , . . . , x p k + N d ] T , ¯ v app k = [ γ d k − N ′ p , . . . , γ d k − 1 , v p k , . . . , v p k + N d ] T . (33) This stru cture will be ref e rred as DF E-IC A PP in the remainder of this paper, and illustrated in Fig. 3 without the dashed module. 2) TV DFE-IC with P erfect APP F eedb ack: Here we pro- pose to g eneralize [19], [26] to APP f e edback, with perfect decision hyp o thesis. This im poses decision covariances to 0, focusing the MMSE filter design to o nly mitigate an ti-causal symbol interfe r ence. Howe ver , its u se of ha r d feedback , i.e. arg max α D k ( α ) , was sh own to be seriously pro n e to err o r propag ation [19]. While [26] showed im p rovements with soft posterior feedb ack on non-tu rbo, in variant structure s, here , we extend this case to time- varying tur bo structures. This case named DFE- IC P APP , differs from the DFE - IC APP with the v ariance estimates: ¯ x papp k = ¯ x app k , ¯ v papp k = [ 0 T N ′ p , v p k , . . . , v p k + N d ] T . (34) 3) Hyb rid TV DFE-IC with APP F eedba ck: Th is stru cture is an extension of the TV stru cture from [21] to APP feed back. In [ 21], the D FE - IC with per fect h ard decision s fro m [1 9] is improved b y adding an estimate of the decision error to the eq ualizer output variance v e k . This quan tity is given by V ar D k [ g c k H ([ x − µ µ µ d ] k − N ′ p : k − 1 )] , u sing (1 9). Mor eover , this structure checks wheth er this variance causes sign ch anges in extrinsic LLRs, and sets amb iguou s LL Rs to zero. This receiver is extend ed to use APP soft feedba c k, instead of hard decisions, and d enoted DFE-IC HAPP . B. An alytic Comparison o f DFE-IC vs. LE-IC This p aragrap h semi- a n alytically assesses the beh aviour of a DFE-IC relative to a L E-IC to u nderlin e the inter est in join tly using decision feedback and prior informatio n for IC. In fact, LE-IC oper ating with p riors ( ¯ x k , ¯ v k ) p r ovides a lower bou nd for the achiev able infor mation rate of a DFE-IC structure using the same prio r informatio n fo r its anti-cau sal symbols ( ¯ x a k , ¯ v a k ) = ( ¯ x k , ¯ v k ) , alon gside d e cision feedback estimates ( ¯ x c k , ¯ v c k ) (see (32)). By exploitin g the structur a l S ¸ AH ˙ IN et al. - ITER A TIVE EQU ALIZA TION WITH DECISION FEE DBA CK BASED ON EXPECT A TION PROP AGA TION 9 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 Deci si on F eedbac k R eli abi li t y ¯ v c Ouput S NR rati o of DFE-IC / LE-IC: G (dB) ¯ v a = 1 ¯ v a = 0 . 5 ¯ v a = 0 . 2 σ − 2 w = 0 d B σ − 2 w = 5 d B σ − 2 w = − 5 dB Fig. 5. Post-equal izati on SNR ratio G depending on channel SNR σ − 2 w , prior reliab ility ¯ v a and “decision” reliabil ity ¯ v c . similarities betwee n DFE-IC and LE- IC, the causal feedb ack’ s impact is reflected on a ratio of post-equalization SNR 3 G = SNR dfe out SNR le out = σ 2 x E [ v e ( dfe ) k ] E [ v e ( le ) k ] σ 2 x = ξ dfe ξ le 1 − ¯ v ξ le 1 − ¯ v ξ dfe (35) where ¯ v = E [ ¯ v k ] and ξ XX = E [ ξ XX k ] , where XX is “le” or “dfe”. This gain is greater than u n ity iff ξ dfe ≥ ξ le , or equiv alently iff E [ ¯ V le k − ¯ V dfe k ] is po siti ve semi-defin ite. Henc e having ¯ v > ¯ v c , ¯ v c = E [ ¯ v c k ] for D FE - IC is requ ired for achieving improvements. Based on em pirical and experimen tal evidence not pr esented here, the conjecture P [ ¯ v c k > ¯ v k ] < 0 . 5 has been verified over a wide ran ge of input SNRs, and for random con stellations, fo r ¯ v c k = v d k (DFE-IC EP) an d for ¯ v c k = γ d k (DFE-IC APP). This ensures ¯ v > ¯ v c and thus, L E-IC output SNR is a lower b ound on DFE-IC EP/APP , as p ossible detection degradations are small. G is p lotted in Fig. 5, with N = 1 7 , N d = 10 and σ 2 x = 1 for the static Pro akis-C ch a nnel, h = [1 , 2 , 3 , 2 , 1] / √ 19 ; when decisions are m ore reliable than priors, G in c reases, o therwise DFE-IC b rings small improvements. Whe n ¯ v a → 1 , there is no pr ior inform a tio n, and d ecisions brin g a significant g ain. Oppositely , when ¯ v a → 0 , prior informatio n is already close to the ideal, and DFE-IC cannot improve further . This indicates boosted perform a nce at initial turbo -iterations. C. Asymp totic Analysis a nd P erformance Pr ediction T o assess the full poten tial of DFE-IC, asym ptotic an alysis is used to evaluate its ach iev able rates. Extrin sic infor mation transfer (EXI T) an alysis [42] of a SISO modu le is used as a tool fo r characte r izing its asym ptotic limits, b y track ing extrin- sic mutu al informatio n (MI) exchanges be tween the iterative compon ents. Essentially , a SISO receiver ca n be characterized by a simp le transf e r function I E = T R ( I A , H , σ 2 w ) , wh ere I A and I E are the MI b etween coded bits and respectively its 3 SNR XX out = σ 2 x / E [ v e ( XX ) k ] is the post-equa lizat ion SNR, where XX is “dfe” or “le”, (s ee (4) for v e k ). Superscript “le” refers to the use of ( ¯ x k , ¯ v k ) for IC, and “dfe” refers to the use of ( ¯ x a k , ¯ v a k ) and ( ¯ x a c , ¯ v a c ) for IC, as in (32). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 I A, Rece iv er − I E, Decoder I A, D ecoder − I E, Receiv e r RSC [7 , 5] 8 Decoder MFB MAP LE-IC [10] DF E-IC APP DF E-IC EP DF E-IC P APP DF E-IC HAPP Fig. 6. EXIT curve s and av erage MI traject ories of FIR equalizers with BPSK in Proakis C channel at E b / N 0 = 7 dB. 0 2 4 6 8 10 12 14 16 18 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 E b / N 0 (dB) Ac hiev able I nformati on Rate (bits /s/Hz ) T urbo R ate U.B . Non-Iter ati v e Rate MAP LE-IC [10] DF E-IC APP DF E-IC EP Class ical DF E [16] Fig. 7. Achie v able spectral ef ficien cy on determinist ic Proakis C channel with BPSK. input prior LLRs and ou tput extrinsic LLRs, and σ 2 w and H show its dep e ndence on the chan nel an d the re ceiv ed SNR. In Fig. 6, transfer curves T R are p lotted in solid lines for considered receivers alon g with th e reverse transfe r T − 1 D of the BCJR decod e r of a recur si ve systema tic conv olutional (RSC) code. DFE-I C APP y ields a high er I R than L E -IC fo r all I A , u n surprising ly given the po sterior feedb a ck, an d there is little difference with DFE-IC EP , which has slig h tly lower rates at low prior infor mation. In par ticular, the improvement at I A = 0 lets us con jecture a lower waterfall thr eshold in BPSK, and the hig her slope of the T R curve at low I A hints an imp roved conv ergence speed ac ross turb o iterations. EXIT curves provide a fairly accura te waterfall th reshold estima tio n and can be u sed for code design [43]. Another use of EXIT analy sis is perfor mance p rediction, howe ver this in volves strong assumptions on prio r inp uts that often cann o t b e met for FIR turbo eq ualizers in pr actice. Hence, EXIT cu rves on ly provide an uppe r-bound on in for- mation rate for receivers other than MAP . In this respec t, it 10 TO APPEAR ON IEEE J OURNAL ON TRANSACTIONS ON COM M UNICA TIONS - MA Y 2018 2 4 6 8 10 12 14 16 18 20 10 −5 10 −4 10 −3 10 −2 10 −1 E b / N 0 (dB) Bi t E rror Rate (BER) 0 1 2 3 4 5 6 7 8 9 10 6 8 10 12 14 16 18 20 Num b er of Tur bo Iterati ons E b / N 0 (dB) requi red f or BLER=10 − 2 No Iter 1 Iter 10 Iter MFB MAP LE-IC [10] DF E-IC APP DF E-IC EP DF E-IC P APP DF E-IC HAPP Fig. 8. BER and con ver gence performance of the proposed DFE-IC in Proakis C channel with BPSK constell ation. 5 7.5 10 12.5 15 17.5 20 22.5 25 10 −4 10 −3 10 −2 10 −1 E b / N 0 (dB) Bi t E rror Rate (BER) 10 12.5 15 17.5 20 22.5 25 27.5 30 10 −4 10 −3 10 −2 10 −1 E b / N 0 (dB) Bi t E rror Rate (BER) MF MAP LE-IC DF E-IC APP DF E-IC EP No Iter 1 Iter 10 Iter 8-PSK 16-Q AM Fig. 9. BER performance of the proposed DFE-IC in Proakis-C with 8-PSK and 16-QAM constella tions. is then inter esting to comp are transfer curves, with actual MI trajectories (in dashed lines in Fig. 6). It had been no ted in [19], that trajec tories o f DFE-IC with hard, “perf ect” decision assumption do not fo llow EX I T curves; this issue remains with DFE-I C P APP , althou gh less se verely , indicating that the “perfe c t decisions” assumption causes a severe inform ation loss. Other FIRs’ trajectories overall follow receiv er and deco der cu rves and re a ch MFB, but after a few iteration s, they no longer make contac t with transfer cur ves, losing convergence spe e d. Th is is a co mmon disadvantage o f FIR eq ualizers, attributed to short cycles caused by neigh bourin g symb ol corr elations, as shown in Fig. 16 in [19]. Howe ver note that amon g DFE-IC r eceiver , EP fee d back yields trajector ies th a t rem a ins c lo sest to E XI T curves, mak ing it easier to predict. The achiev able spectral efficiency for a given re c e iv er can be measu red with th e help of th e area theorem fo r EXIT charts [ 4 4]. In Fig. 7, achiev able rates for BPSK con stellatio n are p lotted. Note that fo r MAP receivers, this rate is an accurate appro ximation of the channel sym metric informa tion rate (SIR) [4 5]. As n on-itera tive FIR do no t depe nd on prior inputs, th e ir achievable r a te s are also accu rately co mputed . For turbo FIR, upp er b ounds are ob tained by combinin g results of area theorem with the chan nel SIR. Tightness of this bound depend on the closeness o f tru e MI trajectories to EXIT charts in Fig. 6, so APP feedba c k’ s asym ptotic perf ormance is likely to be overestimated c o mpared to EP feedback. D. F inite-Length Comparison with Existing Schemes Monte Carlo integration remains the most reliab le a n alysis approa c h joint detectio n of BPSK symbols is considered with parameters in section V -B, and K u = 20 4 8 , cod ed with a terminated [7 , 5] 8 RSC code. Bit error rate (BER) of various receivers are plotted in Fig. 8. For th e reported iteratio ns, the DFE-IC APP outperf orms other APP feedb ack D FE structures, and th eir convergence speeds are compared on the rig ht side of the figur e, at a block er ror rate ( BLE R) o f 10 − 2 . EP-based feedback p rovides fur ther imp rovement o f the thr eshold b y 0 .5 dB relative to APP , and it is shown to r each MFB limit with in 7 iterations, earlier than DFE-IC APP . Assessing DFE-IC performa nce at lo w spectral ef ficiency condition s, as above, is of interest, to remedy the poor be- haviour of cla ssical DFE at those ope rating p oints (see Fig. 7). Indeed , turbo processing helps DFE structure s to outperf o rm LE at all ra tes. A high er spectral efficiency case is plotted on th e left side o f the Fig. 9, with 8 -PSK con stellation in S ¸ AH ˙ IN et al. - ITER A TIVE EQU ALIZA TION WITH DECISION FEE DBA CK BASED ON EXPECT A TION PROP AGA TION 11 5 10 15 20 25 30 35 10 2 10 3 10 4 10 5 10 6 E b / N 0 (dB) requi re d f or BLER=10 − 2 Complex it y (FLO Ps p er equali zed sy m bol) MAP LE-IC DF E-IC APP DF E-IC EP BPSK 8-PSK 16-Q AM Matched Fi l ter Bound Fig. 10. Performance complexi ty trade-of f in Proakis C. the same co nfiguratio n; DFE-IC APP is shown to improve LE-IC waterfall thresho ld by 2 dB, DFE-IC EP asym ptotically provides an additional 1 . 2 dB. On the rig ht side of the Fig. 9, 16-QAM is consider ed; showing that DFE-IC EP provides further perf ormanc e enhance m ents for o n e or more iteratio ns. Finally , the coded p erform ance of DFE-IC is b alanced with complexity conside r ations. In Fig. 10, the receiver co m puta- tional com plexity (FLO Ps per symb ol) requir e d to deco de with a BLER of 10 − 2 is plotted as a functio n of E b / N 0 . These values are c omputed , assum ing the u se of the p roposed matr ix in version algo rithm in section IV, and by accounting for the equalization , the demapping and the d e coding costs. A cu rve represents the evolution o f BLER a nd the co mputation al costs of a receiv er accr oss turbo iterations. DFE-IC provide s a better trade-o ff th an LE-I C; at any given complexity , it is mo re efficient, especia lly at initial iteration s, and the asympto tic E b / N 0 gap between LE-IC and DFE- IC increa ses with the modu lation order M . T he use of EP feedback is m ore advantageou s at higher iterations, for hig her order constellations, while APP is mor e efficient fo r n on- iterativ e recei vers. In conc lu sion, DFE-IC outperfo rms LE-I C in various as- pects: it c o n verges faster towards MFB, has a lower decodin g threshold than LE-IC, especially at h igher spectral efficiencies. Among DFE - IC with APP feed back, exact der iv ation DFE-IC APP is sup erior accor d ing to bo th finite- length and asymptotic analysis. Alth ough EXIT cha r ts show little difference b e tween DFE-IC EP and APP , in pr actical simulations EP fe e d back tends to outp erform APP . T h is is justified by the tig h tness of EP MI trajectories to EXIT curves; APP is overestimated. Although it DFE-IC EP appears to be able to reach channel SIR at low to me d ium spectral efficiencies, th ere is still a g ap to MAP perfor mance. In the f ollowing, the use of self- iterations will b e a ssessed to further improve perfo rmances. V I . C O M PA R I S O N W I T H T H E P R I O R W O R K O N S E L F - I T E R AT E D E P S T RU C T U R E S Some re cent EP-b ased receivers [32], [3 3], [35]– [37] have observed remarka b le perfor mance improvements in repeating the d etection p rocess in a par allel sche d ule thro ugh self- iterations. As the demap ping process is compu tationally less intensive than ch annel dec o ding, suc h stru ctures are of p rac- tical interest. In this section , the ben efits in using a self- iterated DFE-IC EP compared to structures in prior work is in vestigated. Indepe n dently of our work, an EP-based FIR structure is derived in the conco mitant work [36]. Unlike the message passing fo rmalism used in section III, structure in [3 6] is obtained by appr oximating a self-itera ted block r e ceiv er , de- riv ed by EP-based appro ximation of the posterior PDF (5) . The resulting FI R structure uses a parallel schedule, and correspo n ds to a LE-I C within each SI . Using our form alism, it is equiv alent to up dating, all VNs x k with messages from EQU sequ entially , and only then acti vating DEM to update posterior approx imations. Th is process is then iterated with DEM sending back a n extrinsic message to EQU, and finally DEM compu tes messages to wards DE C. In the f ollowing, the structure den o ted as “EP- F” in [36], is refered as a self-iter ated LE-IC (SI LE-IC), with f o llowing me a n and variances used for IC ¯ x le-ep k ( s ) = [ x d ( s ) k − N ′ p , . . . , x d ( s ) k + N d ] T , ¯ v le-ep k ( s ) = [ v d ( s ) k − N ′ p , . . . , v d ( s ) k + N d ] T . (36) If the com putation s o f messages on EQU is carried out only once ( S τ = 0 ), this r eceiver y ields the same result as th e conv entional turbo LE-I C [1 0]. A. Asympto tic Comparison First, we look into the achiev able rates of SI LE-IC and DFE-IC EP to identify op e rating points where self-iter ations have an advantage. W e consider 8- PSK sign alling on the Pro akis-C channel, an d use the area theo rem to obtain an upper b ound on asymptotic achiev able rates (i.e. τ → ∞ ), plotted on the le f t side of Fig. 11. Inf ormation rates of th e optim a l MAP d e tector, LE-IC and DFE-IC EP witho ut SI, and SI LE-IC and SI DFE-I C are co nsidered. For self-iterated receivers, a static da m ping with β = 0 . 6 is used. Nu merical re su lts show that SI is not required for LE-IC up to 0.7 5 b its/s/Hz (i.e. using a cod e rate less than 1/4 ) , as LE-I C is close to the SIR, whereas DFE- IC EP continues to follow MAP ra te s up to 1 bit/s/Hz (up to a code r a te of 1/3 ). On the other h and, when u sing 5 self- iterations, DFE-I C EP follows MAP r a te s within 0 . 5 dB up to 2.25 bits/s/Hz, while L E-IC fo llows up to 1.85 bits/s/Hz. I t is also interesting to note that DFE- IC EP with 2 SI outper forms LE-IC with 5 SI, at all rates, indicating at faster conver gence of DFE-IC EP tow ards asymptotic lim its. At th e r ight side of Fig. 11, non -turbo iterative achiev able rates of these receivers, an d those o f the classical DFE [16], are compare d . Th ese rates are accu rate, and not an u pper boun d, unlike asympto tic rates, and no te that MAP d etector is a mere 12 TO APPEAR ON IEEE J OURNAL ON TRANSACTIONS ON COM M UNICA TIONS - MA Y 2018 0 2.5 5 7.5 10 12.5 15 17.5 20 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 E b / N 0 (dB) Ac hiev able I nformati on Rate (bits /s/Hz) 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 E b / N 0 (dB) Ac hiev able I nformati on Rate (bits /s/Hz) 1 Self. I ter. 2 Self. I ter. 5 Self. I ter. MAP Detec tor LE-IC DF E-IC EP SI LE-IC SI DF E-IC EP Class ical - DF E Asy mptot i c: τ → + ∞ Non I terat iv e: τ = 0 Fig. 11. Achie va ble Rates of Self-itera ted L E-IC and DFE-IC in Proakis-C with 8-PSK constellati on. maximum likeliho od (ML) detec tor in this case. Althou g h self-iterations significantly improve LE-IC pe rforma n ce, at rates ab ove 2.7 5 bits/s/Hz, classical DFE still o utperfo rms these rece iv ers. DFE-IC E P o n the other hand o utperfo rms alternative FIRs at any giv en self iteration. Note th at the gap to capac ity still remains significan t for non tu rbo itera ti ve rates, and to some extent, for asymptotic rates. Hence with the objec ti ve of deriving capacity achieving practical receivers in mind , future work should exp lore the usage of the prop osed DFE-IC EP as a co n stituent elemen t for bidirection al DFE [20] or for con catenated FIR [ 22] re c e iv ers. B. F inite-Length Comparison In this section, numerical finite-length results complete the previous an alysis. In addition to r eceivers above, th e self- iterated block linear r eceiv er ( SI BLE-IC), deno ted n uBEP in [36], is consider ed. W ithout self-iteratio n s, this r eceiver is equiv alent to turbo b lock L E -IC [46], an d it ou tperfo r ms the self-iterated b lock r e ceiv er and Kalman smooth er in [ 33], [35]. SI BLE-IC provides a lower bo u nd to th e BER perfor m ance of SI LE-IC. A low density parity chec k (LDPC) coded 16-QAM trans- missions over the Proakis C chan nel, with rate 1/2 and 3/4 encodin g of K b = 204 8 bits (Fig. 12). Th e pr oposed SI DFE - IC EP u ses r espectively β = min(0 . 5 , 1 − e τ / 2 . 5 / 10) an d β = min(0 . 1 , 1 − e τ / 1 . 5 / 10) fo r dampin g , in th ese two cases, whereas the op timized dampin g repo rted in [36] is kept for SI BLE-IC an d SI LE-IC. Th e LDPC codes a re obtaine d by path edge growth metho d, a n d a BP decoder up to a 100 iter ations is used. The low rate case, with (3,6) regular LDPC, shows that wh ile all self-iterated r e ceiv ers reach the same asympto tic perfor mance as S τ increases, DFE-IC conv erges m uch faster at intermediar y iterations. On th e other hand, a t the high rate configur ation, with (3,12) r egular LDPC, DFE-IC is strictly superior to LE-I C, even withou t self-iterations. Asymp totically ev en the exact SI BLE-IC is 3. 8 dB beh ind the pr oposed SI DFE-IC. These nu merical perfor mance resu lts are comp leted with computatio nal co mplexity co nsideration s in Fig. 13, where decodin g threshold for BLER = 10 − 2 is ev aluated for τ = 0 , . . . , 5 , for each receiver . In the me dium rate (2 b its/s/Hz: 16 - QAM with rate 1 /2 co de) case the thr e e con - sidered recei vers co n verges to th e same asymp totic limit near 17 dB, but DFE-I C offers lower complexity at interm ediary iterations. At 3 bits/s/Hz config uration (16 -QAM with rate 3/4 code), with 5 TI and 3 SI, DFE-I C r equires 3 dB less energy , and 3 times less compu tational resources than BLE-IC. W ith τ = s = 0 , LE-IC is unable to decod e, BLE- I C decod es around 39 dB, and DFE-IC decodes w ith 13 dB less en ergy . These nume rical resu lts confirms co nclusions drawn by th e asymptotic analysis; the p ropo sed SI DFE-I C is of a significant interest fo r hig h da ta rate applications where linear stru ctures are less efficient. Using the efficient imp lementation meth od of section IV, DFE-I C ou tperfor m s prior work in term s of b oth complexity and per forman ce. V I I . C O N C L U S I O N This pap er in vestigates on the use of decision feed back with tur bo equalizatio n , for imp r oving the limitation s of lin ear equalizers for high data rate applications. T urbo DFE structures in th e literature consist in eithe r using har d feedback with symb ol-wise adaptive filters, o r soft posterior feed back with symbol-wise inv ariant filters. Th e former p erform po o rly at low spectral efficiency , and require complex mechanisms to im prove this issue, wher eas the latter are o utperfo rmed ev en by the conventional TV LE-IC. Both schemes ar e extended to time-variant sof t f e edback structur es in this paper, with d ifferent filter compu tation hyp otheses. W e show that an exact appr oach ju stified with sequential Bayesian MMSE estimators (DFE-IC APP) outpe r forms o ther APP feedback alternatives. Howe ver , due to th e use of posterio r estimate s, this structure does not fit within the turbo pr in ciple wh ic h requ ires th e exchange of extrinsic inform ation. Consequ ently , we focus our discourse o n the deriv ation of FIR DFE within the exp ectation propag ation fr amew ork, which allows the co mputation of a novel type of extrinsic feedba ck from the d e mapper to the equalizer . Build ing up on the emergin g tr e nd on self- iter ated S ¸ AH ˙ IN et al. - ITER A TIVE EQU ALIZA TION WITH DECISION FEE DBA CK BASED ON EXPECT A TION PROP AGA TION 13 13 14 15 16 17 18 19 20 21 22 23 10 −4 10 −3 10 −2 10 −1 E b / N 0 (dB) Bi t E rror Rate (BER) 16 18 20 22 24 26 28 30 32 34 36 10 −4 10 −3 10 −2 10 −1 E b / N 0 (dB) Bi t E rror Rate (BER) SI BLE-IC [36] SI LE-IC [36] SI DF E-IC EP 0 Self. I ter. 1 Self. I ter. 2 Self. I ter. Rate 1/2 Rate 3/4 Fig. 12. SI LE-IC and DFE-IC in Proakis C with LDPC coded 16-QAM, with 5 turbo iterations. 16 18 20 22 24 26 28 30 32 34 36 38 40 10 4 10 5 10 6 E b / N 0 (dB) requi re d f or BLER=10 − 2 Complex it y (FLO Ps p er equali zed sy m bol) 0 Self. I ter . 1 Self. I ter . 2 Self. I ter . SI BLE-IC SI LE-IC SI DF E-IC EP 16-Q AM 1/2 16-Q AM 3/4 Fig. 13. Performance complexity trade-of f for self-iter ations in LDPC coded Proakis C. EP-based equalizer s, the propo sed DFE-IC ca n be self-iter ated to further improve perfo rmances. Thanks to finite-leng th an d asymp totic analysis, DFE-IC EP , with SI or no t, is shown to set n ew upper limits in achievable perfor mance amon g FIR tu rbo receivers. At high data rates, ev en exact self-iterated b lock line ar r eceiv ers fall over 3 dB behind the propo sal. Finally , the gap of ach iev able rates b y turbo DFE-I C to the channel capacity remain s still significan t at very hig h spec tr al efficiencies. Bidirectional extension of T V DFE-EP sho uld be explored to try to close th is ga p . A P P E N D I X A. Derivatio n of MMSE F IR with IC In this appen dix, FIR equalizatio n with interfe rence cancel- lation is der iv ed b y minimizin g the Bayesian MM SE criterio n J = E A k [ | x k − x e k ′ | 2 ] , where x e k ′ = f ′ k T y k + g ′ k is the equalized linear e stimate, and A k is a joint multiv ariate Gaussian p r ior d istribution on x k defined with m eans ¯ x fir k and variances ¯ v fir k (see sec. II- B). E A k [ · ] and Cov A k [ · ] respectively denote the expectation and the covariance with resp ect to distribution A k . Solution to this is gi ven by E A k [ x k | y k , H k ] , i.e. th e sym bol mean with respect to p A k ( x k | y k , H k ) . Th is distribution is the margin a lization of the con jugate Gau ssian posterior p A k ( x k | y k , H k ) , i.e. of likelihood p ( y k | x k , H k ) and prior A k . Hence, x e k ′ is deduced by multiplying the MM SE estimator of x k [41] by e k : f ′ k = e H k Cov A k [ y k , x k ]( V a r A k [ y k ]) − 1 , (37) g ′ k = e H k E A k [ x k ] − f ′ k T E A k [ y k ] , (38) by developing expectations ab ove with prior statistics, it holds f ′ k = ¯ v fir k h H k ( Σ fir k ) − 1 , (39) g ′ k = ¯ x fir k − f ′ k T ¯ x fir k , (40) with Σ fir k = k w σ 2 w I N + H k V fir k H H k and V fir k = diag ( v fir k ) . This recei ver is b ia sed , as its MMSE estimators’ nature: E A k [ x e k ′ | x k = x ] = (1 − ¯ v fir k ξ fir k ) ¯ x fir k + ¯ v fir k ξ fir k x, with ξ fir k = h H k Σ fir k − 1 h k . Removin g additive an d multip lica tive biases with x e k = ( x e k ′ − (1 − ¯ v fir k ξ fir k ) ¯ x fir k ) / ( ¯ v fir k ξ fir k ) yield s the estimator giv en in ( 4), w h ich completes the proo f . R E F E R E N C E S [1] C. Douilla rd, M. J ´ ez ´ equel, C. 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