A New Class of Iterative Equalizers for Space-Time BICM over

MIMO Block Fading Multipath AWGN Channel

Raphaël Visoz, Member, IEEE, Antoine O. Berthet, Member, IEEE,and Sami Chtourou, Student Member, IEEE∗

November 19, 2005

Abstract

This paper addresses the issue of advanced equalization methods for space-time communications over

multiple-input multiple-output block fading channel with intersymbol interference. Instead of resorting

to conventional multiuser detection techniques (based on the straightforward analogy between antennas

and users), we adopt a different point of view and separate time equalization from space equalization,

thus introducing a higher degree of freedom in the overall space-time equalizer design. Time domain

equalization relies on Minimum Mean Square Error criterion and operates on multidimensional modu-

lations symbols, whose individual components can be detected in accordance with another criterion. In

particular, when the optimum Maximum A Posteriori criterion is chosen, substantial performance gains

over conventional space-time turbo equalization have been observed for different transmission scenarios,

at the price of an increased, albeit manageable, computational complexity.

1 Introduction

1.1 Research context

Since the last few years, space-time coding has been the scene of considerable attention and progress as a wayof boosting the capacity of wireless mobile radio channel [1] [2]. In particular, the design of efficient space-time codes remains one of the most challenging topics in the research community, especially for channelswith InterSymbol Interference (ISI). For single antenna systems, Bit-Interleaved Coded Modulation (BICM)[3] [4] was shown to offer remarkable diversity gain on an ergodic Single-Input Single-Output (SISO) fadingchannel, while simultaneously guaranteeing a large minimum Euclidean distance for the AWGN channel,through labeling optimization and Iterative Decoding (ID) [5]. Driven by those results, it seemed natural

to investigate how BICM-ID would behave in SISO block fading [6] and Multiple-Input Multiple-Output(MIMO) environments [7]. Space-Time BICM (STBICM) is the straightforward multiple-antenna extensionof BICM, when interleaving also acts in space dimension [8] [9]. On an ergodic MIMO fading channel, itwas recently confirmed that STBICM-ID could perform very close to the Shannon limit [10]. In parallel,several other contributions have tested the potential of STBICM-ID in non-ergodic more realistic scenarios,∗This work was supported by France Télécom under grant no. 42271470 (0612812002). Parts of the paper were presented at

IEEE ITW’03, Paris, France, May 2003, IEEE ICC’2004, Paris, France, June 20-24, 2004 and at IEEE PIMRC’2004, Barcelona,Spain, Sept. 5-7 2004.Raphaël Visoz and Sami Chtourou are with France Telecom R&D, Issy Les Moulineaux, France.Antoine O. Berthet is with Ecole Supérieure d’Electricité (SUPELEC), Department of Telecommunications, Gif-sur-Yvette,France.

1

e.g., MIMO block fading channels with ISI [11] [12] [13] [14], and observed excellent performance comparedto other existing space-time coding schemes.

1.2 Problem statement

Unfortunately, the seeming simplicity of STBICM has a serious price, which is magnified when the MIMOchannel is frequency-selective. Indeed, when neither space nor time dimensions are orthogonalized at thetransmitter, strong ISI and Multiple Antenna Interference (MAI) come as a result and have to be compen-sated for at the receiver. Moreover, the interleaver between the channel code and the modulator, source ofdesign flexibility, precludes any brute force optimum joint decoding of STBICM at the receiver and compelsto adopt a sub-optimal two-step decoding procedure: MIMO detection on one hand, outer decoding on theother hand. Clearly, the most demanding task is the MIMO detection. Moreover, to the best of the Authors’knowledge, resorting to the turbo principle is the only option to recover the Maximum A Posteriori (MAP)ensemble performance. Since several iterations will be required to converge towards optimum results, it is

even more crucial to find very low-complexity algorithms to realize symbol digit detection and decoding.

1.3 In this paper...

We propose a class of efficient iterative (turbo) equalizers for interleaved Space-Time Codes (STC) transmit-ted over MIMO block fading multipath channel and apply them to the decoding of STBICM. Our concern isto solve the problem of ISI and MAI cancellation with polynomial (at most cubic) complexity in all systemparameters, while performing as close as possible from the theoretical available benchmarks, namely the

(coded) Matched-Filter Bound (MFB) and the channel outage capacity. Our equalizing strategy is inspiredfrom the seminal papers by Wang and Poor [15] and Tüchler et al. [18] and motivated by the observationthat, far from being trivial, the space-time generalization offers interesting degrees of freedom in the receiverdesign. Indeed, in contrast with many others capitalizing on the analogy between MultiUser Interference(MUI) and MAI (e.g., [19]), the original approach adopted here views the MIMO signaling as nothing but thetransmission of a multidimensional space modulation on a SIMO channel. This shift in viewpoint naturallycalls for decoupling ISI cancellation, performed in Minimum Mean Square Error (MMSE) multidimensionalsense, from MAI resolution. Such a functional split is the core contribution of the paper. Incidentally, weverify that, for the resulting class of space-time equalizers, which assume finite filter order and discrete inputsignaling, the so-called Genie-Aided Decoding (GAD) coincides with the (coded) MFB1. The paper is orga-

nized as follows. In Section II, the communication model is briefly reviewed and the notation introduced.We also deal with information-theoretic aspects related to the outage capacity in order to identify a suitableabsolute performance reference. In Sections III and IV, we discuss iterative decoding strategies, startingfrom the exact belief propagation decoder and then deriving successive simplification stages. Section IVdepicts our novel approach, in which ISI and MAI cancellation tasks are decoupled. Section V is devoted tonumerical results and comments. Section VI concludes the paper.Notation

• The superscripts ∗,| and † indicate conjugate, transpose and Hermitian transpose, respectively.

• diag., tr., det. denote diagonal, trace, and determinant operators on square matrices, respec-tively.

1This is a major advantage compared to the classical MMSE Decision-Feedback Equalizer (MMSE-DFE) [20] [21] [22] whichwas demonstrated to be information lossless only under severe assumptions (GAD, infinite interleaver depth, infinite filter lengthand Gaussian signaling).

2

• Matrices are denoted by capital lettersM, with ith row mi and jth column mj . Entry (i, j) is denotedmi,j or equivalently [M]i,j . The n× n identiy matrix is denoted by In while 0n and 1n stand for theall-zero or all-one n-dimensional column vectors, respectively.

• We use the proportionality symbol ∝ in order to indicate that the quantity in the RHS is defined up toa multiplicative factor chosen to produce a true probability mass function (pmf) or probability densityfunction (pdf).

• E . denotes expectation. Unless otherwise stated, bolt Greek letters Θ and Ξ are employed forexpressing covariance matrices, respectively, i.e., Θxy = E

n(x− E x) (y− E y)†

oand Θx =

En(x− E x) (x− E x)†

ofor x and y random vectors.

2 Communication model

The proposed class of iterative (turbo) equalizers is preferably derived in the context of a Single-User (SU)point-to-point transmission scenario over a MIMO NB-block fading ISI AWGN channel with NT transmitand NR receive antennas and memory M . Due to some delay constraint, the transmission of a code wordspans over a finite number of independent fading realizations. Fading realizations are thought as separated

both in time and/or frequency and may be correlated or not. Channel State Information (CSI) is perfectlyknown at the receiver and unknown at the transmitter [23]. We focus on this communication model for,at least, two reasons. First, it encompasses many others as subcases. Second, it is well suited to representthe reality of a slowly time-varying fading process where fading blocks may result from frequency-hoppingin TDMA systems. If ISI and MAI are perfectly removed (GAD assumption), the MIMO block fadingmultipath channel decomposes into a virtual SIMO NB × NT -block fading multipath channel, a propertyactually used to design the space-time interleaver, as explained in the next section.

2.1 Space-time bit-interleaved coded modulation

Let C be a linear code of dimension ko, length no and rate ρo over F2 which transforms a messagem ∈ Fko2 intoa code word c ∈ Fno2 . The produced code word enters a semi-random bitwise interleaver I, whose output issegmented into a collection D =

©Db : b = 0, . . . , NB − 1

ªof NB binary matrices Db of dimension qNT ×L,

where q is the number of bits per constellation symbol per transmit antenna and L the block length in channeluse (c.u). The interleaver design meets two different rules: first, consecutive coded bits must be diagonallydistributed among the NB ×NT fading blocks for properly exploiting the available space-time diversity [24].Second, consecutive coded bits must also be distributed among different channel uses in order to maximize thegirth of the underlying STBICM factor graph [25] (cf. section III) and, thus, to ensure a better convergence

in iterative decoding (see, for example, [26] for the fading flat subcase). Columns of matrices Db are vectorsdbk ∈ FqNT

2 , k = 0, . . . , L − 1 containing one subvector dbt,k per channel input t = 0, . . . , NT − 1 with qstacked binary components. Within each column dbk, all subvectors dbt,k are mapped, through a labelingrule ϕt : F

q2 → X ⊆ C, into a complex symbol xbt,k belonging to the complex signal set X of cardinality

|X | = 2q (average normalized enerngy). Note that we assume identical constellations on each transmitantenna, but authorize different labeling rules. After signal mapping, the matrix symbol digit Db is thustransformed into a complex matrix Xb of dimension NT × L. Coding, interleaving, and per-antenna signalmappings can be regarded as a global coding-modulation process Ψ : Fko2 → S ⊆ XNT×L×NB which mapsthe binary message m into the codeword X =

©Xb : b = 0, . . . ,NB − 1

ª. Since the labeling functions ϕt

3

are, in general, non-linear, so is Ψ. Falling into the general class of STCs, this architecture offers a spectralefficiency η = ρoqNT bits per channel use (b/c.u) under ideal Nyquist band-limited filtering assumption.

2.2 MIMO block fading multipath channel

LetHb∈CNR×NT×(M+1) denote the MIMO fading multipath channel block b andH =©Hb : b = 0, . . . , NB − 1

ªthe collection of all channel blocks. Let Y =

©Yb∈CNR×L : b = 0, · · · , NB − 1

ªbe the collection of all re-

ceived matrices. The discrete-time base-band equivalent vector channel output ybk ∈ CNR at time k =0, . . . , L− 1 +M can be written as

ybk =MXm=0

Hbmx

bk−m +w

bk (1)

where xbk ∈ XNT is the vector constellation symbol transmitted at time k, Hbm ∈ CNR×NT is the mth matrix

tap of the channel Finite Impulse Response (FIR), wbk ∈ CNR is the vector of additive complex noise. Thevectors of additive complex noise wbk are assumed random independent identically distributed (i.i.d) zero-mean circularly-symmetric complex Gaussian and thus follow the pdf CN

³0NR ,Θwb

k= σ2INR

´. Channel

Hb, constant along the corresponding block duration L (in c.u), has FIR of length M + 1, whose symbol-spaced tapsHb

0, ...,HbM are NR×NT matrices with zero-mean circularly-symmetric complex Gaussian entries

satisfying the normalization mean power constraint

E

"diag

(MXm=0

HbmH

b†m

)#= NT INR (2)

Due to the absence of CSI at transmitter, we further assume an equal-power system, i.e.,

Θxbk = Enxbkx

b†k

o= INT

(3)

2.3 Matrix formulation

The previous convolution model (1) has a blockwise matrix equivalent.

yb =Hbxb +wb (4)

where xb,yb,wb are stacked vectors

xb =hxb|L−1 · · · xb|0

i>∈ CNTL

wb =hwb|L−1+M · · · wb|0

i>∈ CNR(L+M)

yb =hyb|L−1+M · · · yb|0

i>∈ CNR(L+M)

(5)

and where Hb is the Sylvester channel matrix

Hb =

HbM...

. . .

Hb0 · · · Hb

M

. . .. . .

Hb0 · · · Hb

M

. . ....Hb0

(6)

4

of dimension NR (L+M) × NTL. In practice, this block model can be well approximated by a shorterlength-LF sliding window version, defined as

ybk=Hbxbk +w

bk (7)

where xbk,ybk,wb

k are stacked vectors

xbk =hxb|k+L1 · · · x

b|k−L2−M

i|∈ CNT (LF+M)

ybk=hyb|k+L1 · · · y

b|k−L2

i|∈ CNRLF

wbk =hwb|k+L1 · · · w

b|k−L2

i|∈ CNRLF

(8)

with LF = L1 + L2 + 1 and where Hb is the Sylvester channel matrix

Hb =

Hb0 · · · Hb

M

. . .. . .

Hb0 · · · Hb

M

(9)

of dimension NRLF ×NT (LF +M).

2.4 Information-theoretic aspects

It is well known that the capacity in the strict Shannon sense is not meaningful for a block fading channel[2]. Instead, one can invoke a relationship between outage capacity probability and supportable rate [27][28]. In [6, Theorem 2, p. 776], it was proved that the outage capacity of a MIMO NB-block (NB < ∞)fading flat channel was the asymptotic limit for the random coding upper and lower bounds on the BLock

Error Rate (BLER) when L→∞. We conjecture that this result is also valid for block fading channels withISI and, for some transmission rate η, define the outage capacity (i.i.d input, Gaussian) as [29] [30] [31] [32]

PGout (η, γ) , PÃ1

NB

NB−1Xb=0

IG¡xb;yb

¯Hb, γ

¢< η

!(10)

where

IG (x;y |H,γ ) =Z 1

0

log2

µdet

½INR +

γ

NTH(θ)H†(θ)

¾¶dθ (11)

is the maximum average mutual information obtained for x being circularly-symmetric complex Gaussianrandom vector with zero-mean and covariance γ

NTINT . In (11), γ is the average Signal-to-Noise (SNR) per

receive antenna and H(θ) is the Discrete Fourier Transform (DFT) of the MIMO channel FIR. On Fig. 1,the capacity is plotted as a function of the SNR for an outage probability of 1% and a quasi-static (NB = 1)2× 2 MIMO channel with increasing M (equal-energy taps). The frequency selectivity of the channel actsas an additional ”intrinsic” source of diversity which translates into an increase of the capacity. We observethat the improvement is especially significant between M = 0 (frequency-flat channel) and M = 1, while

it seems to saturate for M ≥ 2. In order to obtain a more precise performance limit, we can compute theinformation rate assuming that the channel input x belongs to a specified discrete alphabet D = XNT andsimilarly introduce the information outage probability

PDout (η, γ) , PÃ1

NB

NB−1Xb=0

ID¡xb;yb

¯Hb, γ

¢< η

!(12)

5

To the best of the Authors’ knowledge, there exists no closed form expression of the (conditional) averagemutual information for ISI channels. Recently, for an i.i.d input with uniform distribution, Loeliger et al.[33] have proposed a computationally efficient method to estimate the differential entropy h (y |H,γ ) in

ID (x;y |H,γ ) , h (y |H, γ )− h (y |x,H,γ ) (13)

as h (y |H,γ ) = limL→∞ h (y0, . . . ,yL−1 |H,γ ). A single forward recursion of the forward-backward (orBCJR) algorithm [34] yields 1

L log p (y0, . . . ,yL−1 |H,γ ) which closely approximates h (y |H,γ ) for large L(Shannon-MacMillan-Breinman theorem [35, 15.7]). The complexity of the method, adapted to our commu-nication model, increases as O(2qNTM ), so that huge computational efforts would be required to numericallyevaluate PDout (η, γ) for high-order constellations and large MIMO systems with moderate to large M . For

weakly loaded systems (i.e., low-rate outer codes at fixed D), PGout (η, γ) can be used as a lower bound.However, it is well known (at least for fading flat MIMO channels) that this lower bound looses its tightnesswhen the outer code rate augments and comes close to 1 (at fixed D). Here, we confine ourselves to providesome insights of what happens in the case of frequency-selective MIMO channels. On Fig. 2, the gap betweenPGout (η, γ) and PDout (η, γ) is depicted as a function of the channel memory M . PDout (η, γ) is plotted for aBPSK i.i.d. discrete input, an outer code rate ρo = 3/4 (η = 1.5 b/c.u) and a quasi-static 2 × 2 MIMOchannel with M ranging from 0 to 2 (equal-energy taps). Our numerical results clearly indicate that thegap between PGout (η, γ) and PDout (η, γ) fades as M increases. An intuitive explanation is that ISI, acting asa linear precoding, induces some kind of ”Gaussianisation effect”. In Fig. 3, the gap between PGout (η, γ)and PDout (η, γ) is instead depicted as a function of the constellation size. PDout (η, γ) is plotted for an outercode rate ρo = 3/4, a quasi-static 2 × 2 MIMO channel with M = 1 (equal-energy taps), and a BPSK(resp. QPSK) i.i.d. discrete input (η = 1.5 or 3 b/c.u). Again, the gap between PGout (η, γ) and PDout (η, γ)rapidly vanishes as we expand the constellation. In all cases, we chose L = 104 c.u to numerically evaluateh (y |H, γ ) (taking into account the analysis of the convergence behavior in [33, Fig. 2]). On the basis ofthose observations, it seems reasonable to conclude that PGout (η, γ) is a valid approximation of PDout (η, γ) forthe highly-loaded transmission scenarios considered in the section V.

3 Joint equalization and decoding of STBICM

Despite its title, the formalism retained in this section encompasses any kind of STC provided that it includes

an interleaver either inside the code structure (e.g., bit-interleaving for STBICM) or between the code andthe MIMO ISI channel (i.e., symbolic space-time interleaving).

3.1 Optimal approach and factor-graph representation

In the sequel, conditioning of all pmfs and pdfs with respect to H is implicit (perfect CSIR assumption).From Bayesian decision theory, we know that an overall MAP decoding, if feasible, would provide the

minimum BLER, i.e., an optimal solution in the context of data packet transmission. The SU-MIMO NB-block fading ISI channel with coding is fully described by the A Posteriori Probability (APP) P [m |Y ].Using the independence between blocks and the uniform a priori probability of the messages, this APP canbe expanded as

P [m |Y ] ∝ p(Y |m)P [m] ∝ P(X |m)NB−1Yb=0

p(Yb¯Xb ) (14)

where the indicator function P(X |m) , 1 X = Ψ (m) acts as a code constraint function, and where factorsp(Yb

¯Xb ) act as channel transition functions for each block. The Factor-Graph (FG) for P [m |Y ] represents

6

the factorization (14) under the form of a bipartite graph where variables ml (information bits) and xbt,k(components of vectors constellation symbols) are called variables nodes and previous functions are calledfunction nodes [25]. Each function node is connected to all variable nodes which intervene as arguments init. Fig. 4 illustrates the FG for a SU-MIMO channel with NB = 2, NT = 2 and q = 2 (QPSK symbols). OnFig. 4, the part of the FG in gray details the underlying bit-level structure of the STBICM (interleaving,bit-to-symbol labeling), that our formalism ignores. The MAP message estimate is defined as

bm = arg maxm∈Fko2

P [m |Y ] (15)

Computing the APP of messages by brute-force is intractable for practical message sizes ko. Instead, we

resort to efficient sub-optimum approaches.

3.2 Belief propagation joint decoder

Let ml be the lth information bit in the message m. The optimal MAP detection rule minimizing the BitError Rate (BER) is given by

bml = arg maxm∈F2

P [ml = m |Y ] (16)

where P [ml = m |Y ] denotes a marginal APP formally expressed asP [ml = m |Y ] =

Xm∈Fko2 :ml=m

P [m |Y ] (17)

As before, computing the marginal APPs on information bits by brute-force is impossible. A general methodfor approximating those APPs consists in applying the Belief Propagation (BP) algorithm [36] to the FG.The BP algorithm (also called sum-product) relies on the definition of some computation building blocks

that exchange messages in the form of probability distributions between adjacent variables nodes. Moreprecisely, if a variable node is connected to a function node by an edge, the messages exchanged along thatedge in either directions are functions of that node. Following [37], we let Qb

t,k , Qbt,k(a) : a ∈ X andPbt,k , P bt,k(a) : a ∈ X the messages sent to the variable node xbt,k calculated at the bth channel transitionfunction node and at the code constraint function node respectively (a denotes a dummy variable taking onvalues in X ). Clearly, Qb

t,k and Pbt,k play the role of pmfs defined over X for xbt,k. The general BP rules in

[25] and [36], applied to the present case, yield several types of computations. Since the FG contains cycles(due to the coding modulation scheme), those computations are iterative (iteration index omitted).

• Computation at the bth channel transition function node of :Qbt,k(a) ∝

XA∈XNT×L:at,k=a

p(Yb |A)Y

(t0,k0)6=(t,k)P bt0,k0(at0,k0) (18)

for all a ∈ X . In the terminology of turbo coding, the message Qbt,k, delivered by the MIMO detector,

is referred to as ”extrinsic” pmf on xbt,k given the a priori pmfs Pbt0,k0 .

• Computation at the code constraint function node of :P bt,k(a) ∝

XA∈S:abt,k=a

Y(b0,t0,k0)6=(b,t,k)

Qb0t0,k0(a

b0t0,k0) (19)

for all a ∈ X . In the terminology of turbo coding, the message Pbt,k, delivered by the decoder, isreferred to as ”extrinsic” pmf on xbt,k given the a priori marginal pmfs Q

b0t0,k0 . The APP on x

bt,k is

simply proportional to P bt,k(a)Qbt,k(a) for all a ∈ X .

7

• Computation at the variable nodes xbt,k (degree 2), which, in fact, act as relays, i.e., xbt,k sendsQbt,k to the code constraint node function node and P

bt,k to the channel transition function nodes.

A decision on each information bit ml is made after a fixed number of iterations, according to the rule:

bml = sign ln

PA=Ψ(m)∈S:ml=1

Q(b,t,k)Q

bt,k(a

bt,k)P

A=Ψ(m)∈S:ml=0

Q(b,t,k)Q

bt,k(a

bt,k)

(20)

On the basis of (20), a block error is declared as soon as at least one bit decision is wrong in the message

m. It can be shown [25] that the decision rule (20) coincides with the MAP decision (16) provided thatthe code word length is large enough and the number of iterations sufficient [37]. As already mentioned,the computation of (18) is always the most critical part of the BP algorithm, due to multiple sums ondiscrete alphabets. It can be efficiently performed by the forward-backward (or BCJR) algorithm [34], whichdynamically exploits the Markovian structure of the MIMO ISI. This option comprising both algorithmicchoices and scheduling was first reported in [11, quasi-static channel]. Unfortunately, the complexity ofthe forward-backward algorithm, being directly related to the number of states of the MIMO ISI channeltrellis (i.e., the hypertrellis constructed as a Cartesian product of the NT individual ISI channel trellises) itproceeds on, increases asO(2qNTM ) and quickly explodes. Some previous works have attempted to tackle thisimpediment by resorting to massive trellis state reduction together with Per Survivor Processing (PSP) [39]

[13] or sequential algorithms [40] . To further decrease the complexity and/or mitigate the error propagationinduced by PSP, the channel FIR may be reshaped with MIMO pre-filtering [41] [42] [43] [44]. However, forlarge MIMO systems and/or high-order modulations, the number 2qNT of transitions per state becomes thetrue limiting factor of such techniques. Another possible approach to compute (18) could consist in employinga list-APP version [10] of the Sphere-Decoder (SD) (see also improvements in [45] [46] or variations [47])derived for the communication model (4) [50]. Indeed, seeing the convolution with the MIMO ISI channelas a linear precoding Hb over C, the problem becomes similar to a sphere-constrained search of the closestpoint in the lattice generated by this linear precoding. Although the complexity of the SD can be greatlyreduced by exploiting the Toeplitz structure of the matrix Hb, we believe that, for large L and low SNRs,this approach would require very huge computational efforts to obtain lists of codeword candidates, and thus

does not seem especially suited to STBICM-ID.

4 Simplifying the exact BP joint decoder

4.1 A new class of space-time equalizers

In order to simplify the exact BP joint decoder, a new algorithm is proposed, based on a two-stage process:

a multidimensional Wiener filtering stage to remove the ISI corrupting a group of constellation symbolsin the vector xbk ∈ Xb followed by a detection stage to deal with the residual MAI within each of thosegroups2. Since the same treatment is performed in each fading block, we omit index b for the sake of sim-plicity. Let ∆i : i = 0, . . . ,NI − 1 be a partition of the symbol component indices in xk with NI ≤ NT ,∆i ,

h∆(i)min, . . . ,∆

(i)max

ibeing an index interval with cardinality |∆i|. By a slight abuse of notation, let

x∆i,k be the subvector of xk associated with ∆i. The optimal MAP estimation of vector xk is replacedby a sub-optimum biased linear MMSE estimation of subvectors x∆i,k : i = 0, . . . , NI − 1 in xk given theobservation vector y

kand relevant a priori pmfs Pt0,k0 : (t0, k0) 6= (∆i, k) on neighboring vector constel-

lation symbols. In compliance with the definition of ∆i and with the sliding-window model (7), notation

2This idea was also independently proposed in [48] [49] in the context of symbolic interleaved layered space-time trellis codestransmission.

8

Pt0,k0 : (t0, k0) 6= (∆i, k) means the set of a priori pmfs Pt0,k0 for all index couples of symbol components0 ≤ t0 ≤ NT − 1 and symbol intervals k − L2 −M ≤ k0 ≤ k + L1, except the couples such that t0 ∈ ∆i andk0 = k (for which uniform distributions apply). Biased linear MMSE estimates z∆i,k take the form of

z∆i,k =G∆i,kx∆i,k + ξ∆i,k (21)

in whichG∆i,k is a square matrix of dimension |∆i| to be detailed hereafter and ξ∆i,k is a zero-mean complexrandom vector with conditional covariance matrix Θξ∆i,k . On the basis of (21), we then compute messagesfor symbol components in z∆i,k by resorting to a Gaussian Approximation (GA) on the compound residualISI + thermal noise term ξ∆i,k (i.e., ξ∆i,k is assumed complex circularly-symmetric Gaussian distributed).The validity of this approximation stems from the fact that, in the large system limit M →∞, the output(21) of the |∆i|-dimensional Wiener filter F∆i,k derived for group ∆i and symbol interval k converges

almost surely to a conditionally Gaussian random vector. This becomes a quite appealing MIMO detectionscenario for which efficient algorithms have already been proposed in the literature, ranging from the optimalMAP MIMO detector to the very suboptimal Single-User Matched-Filter Interference Canceller (SUMF-IC).Prominent among them are the list-APP SD and those derived from Markov Chain Monte Carlo (MCMC)methods [51] [52].

4.2 Conditional MMSE-based ISI cancellation

In the sequel, we drop index i for notation ease. Given the messages Pt,k, we can compute the (conditional)mean ext,k , E xt,k |Pt,k and the (conditional) variance σ2xt,k , E |xt,k − ext,k|2 |Pt,k of each vectorconstellation symbol xt,k. In compliance with sliding-window model (7), let us introduce the stacked vectorsexk , E xk |Pt,k and exk|∆ , E xk |Pt0,k0 : (t0, k0) 6= (∆, k) and let E∆ be the matrix of dimensionNT (LF +M)× |∆| defined as

E†∆ , [· · · 0|∆| · · ·| z NTL1+∆min−1

I|∆| · · · 0|∆| · · · ]| z NT (L2+M+1)−∆max

(22)

The achieving of the biased MMSE estimate z∆,k results from the following canonical decomposition.First step: The stacked observation vector y

kis rendered zero-mean by subtracting the mean ey

k|∆conditioned to Pt0,k0 : (t0, k0) 6= (∆, k), expressed as

eyk|∆ ,Hexk|∆ = H³exk −E∆E†∆exk´ (23)

Second step: The biased estimate z∆,k of the vector x∆,k is simply given by the output of the |∆|-dimensional Wiener filter F∆,k applied to the zero-mean observation yk−eyk|∆ and minimizing the conditionalMean Square Error (MSE)

E½hx∆,k −F∆,k

³yk− ey

k|∆

´i hx∆,k −F∆,k

³yk− ey

k|∆

´i†|Pt0,k0 : (t0, k0) 6= (∆, k)

¾(24)

in the sense of the stochastic matrix inner product hx,yi , E©xy† |P (•)ª = Θx,y 3. Using the projectiontheorem associated to this inner product and operating on the projection space generated by y

k− ey

k|∆, weobtain the following expression for F∆,k

F∆,k = E†∆H

†hHΘxk|∆H

† + σ2INRLF

i−1(25)

3Note that such a MSE implies the minimization of the MSE associated with the conventional stochastic innerproducthx,yi , E©x†y |P (·)ª = E©tr ©xy† |P (·)ªª.

9

in which Θxk|∆ is the conditional covariance matrix

Θxk|∆ , E½³xk − exk|∆´³xk − exk|∆´† |Pt0,k0 : (t0, k0) 6= (∆, k)¾ (26)

It is also convenient to define the conditional covariance matrix

Θxk , En(xk − exk) (xk − exk)† |Pt,ko (27)

If perfect space-time interleaving is assumed, both time independence between vectors symbol and spaceindependence between vector components hold4. As a consequence, conditional covariance matrices Θxk andΘxk|∆ are diagonal. More precisely

Θxk = diagnσ2x1,k+L1

, . . . ,σ2xt,k , . . . ,σ2xNT ,k−L2−M

o(28)

and

Θxk|∆ = Θxk +E∆hI|∆| −E†∆ΘxkE∆

iE†∆ (29)

The output z∆,k of the |∆|-dimensional Wiener filter F∆,k can be re-written as

z∆,k = F∆,k(yk − eyk|∆) = F∆,kHE∆x∆,k + ξ∆,k =G∆,kx∆,k + ξ∆,k (30)

Let ν∆,k , x∆,k−z∆,k be the error vector on z∆,k and Θν∆,k and Θξ∆,k the conditional covariance matricesof vectors ν∆,k and ξ∆,k given Pt0,k0 : (t0, k0) 6= (∆, k). It follows from the orthogonality principle that

Θν∆,k = I|∆|−G∆,k (31)

Besides, Θν∆,k is directly related to Θξ∆,k as

Θν∆,k =¡G∆,k − I|∆|

¢ ¡G∆,k − I|∆|

¢†+Θξ∆,k . (32)

Combining (31) and (32) yields

Θξ∆,k = (I|∆|−G∆,k)G†∆,k (33)

If the time correlation is purely neglected in the message computation at the channel transition functionnodes, the spatial correlation is not and may play an essential role, as demonstrated in the simulationsection. Note that step 2 maximizes the |∆| Signal-to-Interference-plus-Noise Ratios (SINRs) at the inputof the (residual) MAI detector. Those SINRs are expressed ∀t ∈ ∆, ∀k as

β(∆)t,k =

hG†∆,kΘ

−1ξ∆,k

G∆,k

it,t

(34)

Left multiplication of the filter F∆,k by any constant square invertible matrix of dimension |∆| does notmodify them.Assuming GAD of the multidimensional ISI, i.e.,

exk|∆ = xk −E∆E†∆xk and Θxk|∆ = E∆E†∆, (35)

the |∆|-dimensional Wiener filter (25) turns into

FGAD∆ = E†∆H†hHE∆E

†∆H

† + σ2IRLF

i−1(36)

4 In practice, this independence property is proved even for finite (relatively small) interleaver depths.

10

Using the well-known identity X†hXX† + σ2I

i−1=£X†X+ σ2I

¤−1X† yields

FGAD∆ =hE†∆H

†HE∆ + σ2I|∆|i−1

E†∆H† (37)

By further extending the GAD assumption to MAI resolution, the |∆| components of z∆,k in (21) becomeproportional to the outputs of the |∆| parallel scalar flat fading AWGN channels γtxt,k + wt,k : t ∈ ∆ withwt,k ∼ CN

¡0,σ2

¢and

γt =

vuut" MXm=0

H†mHm

#t,t

(38)

Hence, if ISI and MAI are perfectly removed (GAD assumption), the Matched-Filter (MF) Signal-to-NoiseRatios (SNRs)

©γt/σ

2 : t ∈ ∆ª are reached at the |∆| outputs of the proposed space-time equalizer, provingits capacity to perfectly recover both multipath and receive antenna diversity.

4.3 Unconditional MMSE-based ISI cancellation

The conditional |∆|-dimensional Wiener filter F∆,k has to be computed for each symbol interval in eachblock at each iteration. A simplification consists in replacing the conditional covariance matrix Θxk|∆ in(25) by its ensemble average formally expressed as

Ξx|∆ = E©Θxk|∆

ª(39)

as first proposed in [58]. The resulting biased estimate z∆,k becomes the output of the |∆|-dimensionalWiener filter F∆ which minimizes the so-called unconditional Mean Square Error (MSE)

E½hx∆,k −F∆

³yk− ey

k|∆

´i hx∆,k −F∆

³yk− ey

k|∆

´i†¾(40)

and is given by

F∆ = E†∆H

†hHΞx|∆H

† + σ2INRLF

i−1(41)

In this way, the |∆|-dimensional Wiener filter must be recalculated at each iteration but remains constantfor all symbol intervals. Ξx|∆ is obviously diagonal. Again, resorting to

Ξx = E©Θxk

ª= diag

nEnσ2x1,k+L1

o, . . . ,E

nσ2xt,k

o, . . . ,E

nσ2xNT ,k−L2−M

oo(42)

yields

Ξx|∆ = Ξx +E∆hI|∆| −E†∆ΞxE∆

iE†∆ (43)

In practice, the true mean involved in the definition of Ξx can be replaced by the empirical mean

Ξx ≈ bΞx , 1

L

L−1Xk=0

Θxk (44)

The output of the |∆|-dimensional filter F∆ can be now expressed as

z∆,k =G∆x∆,k + ξ∆,k (45)

where G∆ = F∆HE∆ and Θξ∆ = (I|∆|−G∆)G†∆.

11

4.4 Computational complexity issues

For each group of symbol components indexed by ∆i, the conditional MMSE-based ISI cancellation requiresone matrix inversion of dimension NRLF per symbol interval k to compute the |∆i|-dimensional Wiener filterF∆i,k. For practically relevant system sizes, this algorithm seems prohibitively complex. On the contrary,

the unconditional MMSE-based ISI cancellation, which involves only one matrix inversion of dimensionNRLF for all symbol intervals, is easily implementable. When NI ≥ 2, it is straightforward to prove thatthe NI |∆i|-dimensional Wiener filter can all be deduced from a single matrix inversion of dimension NRLF(contrary to what initially stated in [63] [54, p. 1807]). The proof does not depend on the particular groupindex i and is done for the conditional case. Given the relation (29) between Θxk|∆ and Θxk , and setting

Ak =HΘxkH† + σ2IRLF (46)

B∆,k = I|∆| −E†∆ΘxkE∆ (47)

the Wiener filter F∆,k can be equivalently re-written as

F∆,k = E†∆H

†hAk +HE∆B∆,kE

†∆H

†i−1

(48)

Using the matrix inversion lemma, we get

F∆,k = C−1∆,kE

†∆H

†A−1k (49)

where C∆,k is the ∆-dependent square matrix of dimension |∆| expressed as

C∆,k = I|∆| +E†∆H

†A−1k HE∆B∆,k (50)

The matrix Ak is ∆-independent and its inverse A−1k intervenes in all multidimensional conditional Wiener

filters derived for symbol interval k. The highly structured nature of Ak allows to employ efficient inversionalgorithms [59] [60]. On the contrary, the matrix C−1∆,k is specific to filter F∆,k and obviously requiresone additional matrix inversion of dimension |∆|. However, since |∆| ≤ NT << NRLF , the computationcomplexity of the Wiener filters is primarily dictated by the inversion of Ak and is not very sensitive to thechoice of the partition ∆i : i = 0, . . . , NI − 1. The same kind of derivations and conclusions hold for theunconditional variant, except that Wiener filters are calculated only once for all symbol intervals.

4.5 Interest and link with other existing algorithms

Different choices of criteria for ISI cancellation and MAI resolution, as well as different partitions yield afamily (or class) of space-time turbo equalizers. The new degree of freedom on the criterion chosen for MAIresolution clearly impacts the behavior of the global iterative algorithm (convergence speed and asymptoticperformance) and allows to adapt the space-time equalizer to a particular transmission scheme, to thesystem load and/or to the channel selectivity (complexity/performance trade-off optimization). Prominentamong the possible partition choices is the particular case corresponding to NI = NT and ∆i = i ,i = 0, . . . , NT−1. In section V, this option will be considered as a reference and coined ”MMSE joint”. There,antennas are seen as distinct users and the problem becomes formally equivalent to the one of MultiUserDetection (MUD) in the presence of ISI [15, section V], a fruitful analogy especially relevant for MU-MIMOchannels, albeit conceptually less justified for SU-MIMO channels. Space-time turbo equalizers belonging tothis subclass are described in [19] [53] (conditional Wiener filters) and [54] (unconditional Wiener filters). Insection V, we rather focus on another important variant for which NI = 1 and ∆0 = ∆ = 0, . . . , NT − 1.

12

MAP-based and Max SNR-based criteria for MAI resolution are presented in [63], whereas MMSE-basedcriterion for MAI resolution is derived in [55]. This latter combination (coined ”MMSE/MMSE”) comes downto being strictly equivalent to the conventional MMSE joint approach as shown in the Appendix. Finally,we point out that this idea of solving ISI and MAI separately has certain similarities with Varanasi and al.canonical description of group detection derived in the context of MUD for overloaded CDMA systems [56],except that, here, group detection is performed on superimposed multidimensional symbol sets.

4.6 Extending the concept to higher dimensions

Nothing prevents to equalize vector constellation symbols by groups of NS > 1, thus extending the conceptto higher dimensions. Such an extension may take into account time dependencies between consecutive

NT -dimensional vector constellation symbols (e.g., induced by any linear dispersive precoder). We redefinestacked vectors xbk,y

bk,wbk as

xbk =hxb|kS+S−1 · · · xb|kS

i|∈ CNTNS

ybk=hyb|kS+S−1 · · · yb|kS

i|∈ CNRNS

wbk =hwb|kS+S−1 · · · wb|kS

i|∈ CNRNS

(51)

and introduce the vector sbk of dimension NTNS whose components are linked to the ones of xbk according

to the rule

sbk =hs1,k , x1,kS+S−1 · · · sNSNT ,k , xNT ,kS

i|∈ CNTNS (52)

From convolutional model (1), we express the matrix model

ybkS+S−1

...ybkS

=Hb0 · · · Hb

M

. . .. . .

Hb0 · · · Hb

M

xbkS+S−1...xbkS...

xbkS−M

+

wbkS+S−1

...wbkS

(53)

In this model, the Sylvester channel matrix Hb of dimension NRNS × NT (NS +M) can be sliced intoMS = dM/NSe distinct matrices Hb0, . . . ,Hb

MSof dimension NRNS ×NTNS , Hb

0 being built with the firstNTNS columns of H

b, Hb1 being built with the next NTNS columns of H

b, and so on, until HbMS

built withthe last remaining columns of Hb completed with 0NRNS columns. The convolutional model (1) is turnedinto a new convolutional model

ybk=

MSXm=0

Hbmx

bk−m +w

bk (54)

describing a virtual MIMO ISI channel with NTNS transmit antennas, NRNS receive antennas and memoryMS . In (54), the stacked vector xbk−m can be replaced by sbk ≡ xbk−m, so that

ybk=

MSXm=0

Hbms

bk−m +w

bk (55)

All previous derivations (e.g., filters expressions) describing the space-time equalizer apply to this new model.Our aim is now to detect the NTNS components in vector sbk with the freedom to partition them into NI

13

distinct groups. Higher potential dimensions |∆i| ≤ NTNS for those groups entail resorting to list-APPSD for post MAI resolution in MAP sense. It is worth pointing out that, in this particular case, thestructure of the proposed space-time equalizer naturally solves the problem of adapting the SD to a MIMOcommunication model with ISI. The efficiency of the resulting class of algorithms should be compared withthe recent approach followed by Vikalo, Hassibi and Mitra [57], where the sphere-constrained search of theSD is directly combined with the dynamic programming principles of the Viterbi algorithm.

5 Numerical results

5.1 Simulation setting

In [6, pp. 778-779], it is shown that terminated convolutional codes with 4-PSK (and Gray labeling) are goodblock codes for block fading flat channels provided that the code length is kept rather small (no ≤ 2000).Driven by those practical conclusions, we choose to consider families of STBICM made of identical PSK(or QAM) for all transmit antennas and 64-state Non-Recursive Convolutional (NRC) outer codes withvarious rates and maximal dfree. In all simulations, we plot the (coded) MFB as a compelling benchmarkto measure the turbo-equalizer efficiency at fixed coding-modulation scheme. The coded MFB is a semi-

analytic bound based on the statistics given by (38) and on a simulated iterative demapping/decoding of theSTBICM. We also plot the outage capacity as an absolute limit, since the outage capacity turns out to bea tight estimate of the average BLER even for finite L around several hundred c.u [6, p. 778] . Taking intoaccount the upper layer ARQ protocol integrated in most existing data transmission systems the averagethroughput η (1− BLER) is the relevant measure of efficiency whose maximization implied fairly large ratesη and thus rather high BLER [62]. In practice, we focused on BLER above 10−3. In this BLER region ofinterest, we found that a simple and identical Gray labeling for each transmit antenna turned out to bethe best compromise between the first and the last iteration performance. Note that the Eb/N0 appearingin all simulations refers to the signal to noise ratio per receive antenna and per useful transmitted bit.Furthermore, all MIMO channel taps are supposed to be perfectly uncorrelated.

5.2 Main objective

Our main objective is to demonstrate that the BLER performance of the conventionalMMSE joint space-time turbo equalizer (NI = NT , ∆i = i , ∀i = 0, . . . , NT − 1) [54] can be significantly outperformed bythe MMSE/MAP space-time turbo equalizer NI = 1, ∆0 = ∆ = 0, . . . , NT − 1 with MAP criterionfor MAI resolution. We are well aware that the comparison is unfair in terms of complexity, since MAPMIMO detection on model (21) potentially requires huge computation efforts. This is the reason why we

put emphasis on transmission scenarios where the gain in performance motivates the price in complexity.In other words, little attention is paid to low-rate STBICM for which the two space-time turbo equalizersconverge towards the same performance. Also, as a first and very basic experiment, the conditional variantof each space-time turbo equalizer was compared to its unconditional counterpart for various transmissionscenarios. We always observed quite similar BLER performance, which proves the validity of approximation(39). In the rest of this section, all simulations assume unconditional Wiener filters.

5.3 Gaussian approximation

We start by defining an STBICM employing a 8-PSK constellation (Gray labeling) and a rate-1/3 64-state NRC code with generator polynomials (1338, 1458, 1758). The code length is fixed to no = 1800 bits

14

(including tail). The MIMO channel is quasi-static (i.e., NB = 1) with the smallest possible number ofantennas (NT = NR = 2) and the smallest possible dispersion (M = 1). Taps have EQual energy (EQ2channel). The spectral efficiency is η = 2 b/c.u. Both MMSE joint and MMSE/MAP space-time turboequalizers are presented. For the computation of the unconditional Wiener filters, the length of the slidingwindow is chosen equal to LF = 9 (L1 = L2 = 4). The corresponding BLER performance are depicted onFig. 5 and compared with the performance of the exact BP iterative joint decoder (6 iterations). It is

a remarkable fact that, even with such limited system dimensions, the GA in (21) seems justified. As aconsequence, the performance of the exact BP iterative joint decoder is closely approached by much simplerMMSE-based versions, at the price of a few additional iterations. Although the difference in terms of finalperformance between the MMSE joint and the MMSE/MAP space-time turbo equalizers is negligible, thelatter benefits from a higher convergence speed. This result turns out to be general for loads up to 1 bit/c.uper transmit antenna.

5.4 Diversity capture on block fading channel

It is well known that the (ST)BICM is a particularly interesting coding scheme to recover the block fadingchannel diversity. Indeed, under the assumption of a well designed interleaver, the blockwise Hammingdistance δβ of a (ST)BICM of rate ρo transmitted over a NR × NT MIMO NB-block can reach the fun-damental upper bound dictated by the Singleton Bound (SB) δSB , (1 + bNBNT (1− ρo)c) [65] [24] [26].The latter coincides with the maximum possible diversity δmax = NBNT if ρo ≤ 1/(NBNT ). To illustratethis phenomenon, we consider the transmission of a low-rate STBICM employing an 16-QAM constellation(Gray labeling) and a rate-1/4 64-state NRC code with generator polynomials (1178, 1278, 1558, 1718) overa 2× 2 MIMO 2-block EQ2 fading channel. We simulate the decoding performance of the MMSE joint andthe MMSE/MAP space-time turbo equalizers (6 iterations each). The results are depicted on Fig. 6. In

both cases, the (coded) MFB is very closely approached. Moreover, since the slope of the MFB is parallelto the one of the outage probability, all the promised diversity gain δmaxNR(M + 1) has been captured.

5.5 Impact of the chosen criterion for MAI resolution

In the remaining sections, we turn to high-rate codes and introduce a new STBICM employing an 16-QAMconstellation (Gray labeling) and a rate-1/2 64-state NRC code with generator polynomials (1338, 1718). Thecode length is fixed to no = 1800 bits (including tail). Transmission occurs on a quasi-static 2×2MIMO EQ2fading channel. The spectral efficiency is η = 4 b/c.u. On Fig. 7, the BLER performance is plotted for bothMMSE joint and MMSE/MAP space-time turbo equalizers (6 iterations each). For the computation of theunconditional Wiener filters, the length of the sliding window is chosen equal to LF = 13 (L1 = L2 = 6). Wenow observe a significant difference between the curves, especially in terms of slope. This gap is even morepronounced in Fig. 8 where all parameters are kept unchanged except the outer code: a rate-3/4 64-stateNRC code with dfree = 6 (spectral efficiency η = 6 b/c.u). Note that this configuration cannot achievefull diversity, i.e., δSB < δmax. The two experiments highlight the benefit of being able to adapt the MAIresolution criterion to the transmission scenario. For high system loads (typically ≥ 2 b/c.u per transmitantenna), where the extrinsic information fed back by the outer decoder is of poor quality, MMSE-basedMAI cancellation (together with ISI cancellation) fails to provide satisfying results, entailing the need to

upgrade the criterion chosen for MAI resolution, i.e., to implement the MAP criterion. A formal analysisof the fine dynamics governing message passing in the FG for both space-time turbo equalizers seems notobvious and we are currently investigating tools to better understand how the different functions interact.

15

5.6 Effect of the channel ISI profile

In this section, we study the influence of the channel ISI profile. We still consider a STBICM employinga 16-QAM constellations and a rate-3/4 64-state NRC code (spectral efficiency η = 6 b/c.u). On Fig.9, transmission occurs on a quasi-static 2 × 2 MIMO EQ4 fading channel (strongly frequency-selective),

whereas Fig. 10 addresses the transmission over a quasi-static 2 × 2 MIMO fading channel with M = 3

and EXPonential decreasing taps (EXP4 channel): on each link (r, t), the M + 1 channel coefficients arei.i.d circularly-symmetric complex Gaussian following pdfs CN ¡

0,σ2m¢with σ2m ∝ exp(−αm), α = 2, m =

0, . . . ,M . For the computation of the unconditional Wiener filters, the length of the sliding window is chosenequal to LF = 25 (L1 = L2 = 12). The slope of the performance curve corresponding to MMSE/MAP space-time turbo equalization (last iteration) is far steeper that the one corresponding to conventional MMSE jointspace-time turbo equalization, revealing a much better capacity to capture the available diversity, whichimmediately translates into significant performance gains: 3.5 dB gain on Fig. 10 versus 1.0 dB gain on Fig.9 at BLER 10−2. As a rule of thumb, the more the matrix

PmH

†mHm has powerful non-diagonal elements,

the more the MAP MIMO detection is expected to bring gains. Inversely, when the non-diagonal elementsare weak the MAI resolution criterion can be relaxed to Max-SNR without significant performance penalty(see [63, Fig. 2 and Fig. 4]).

5.7 Final comments

We conclude the section by few additional comments:

• In [66] [67], it is demonstrated through Monte-Carlo simulations that if the channel is dispersive enough,the number of antennas sufficiently large, and the load per antenna typically less or equal to 1 bit/c.u,a bank of simple matched filters may eventually replace the NT Wiener filters during the courseof iterations without notable degradation in terms of performance. Our proposed equalizer designenables low-complexity configurations that are not reachable by the previously proposed approaches,i.e., Max-SNR-based ISI cancellation (in a multidimensional sense) followed by either MMSE-based orMAP-based MAI resolution.

• We chose the natural iterative scheduling: one pass of ISI and MAI cancellations followed by onepass of decoding. For this specific scheduling, the Appendix demonstrates the equivalence betweenMMSE/MMSE and joint MMSE. It is worth mentioning, however, that the separation between ISIand MAI cancellations allows some additional flexibility in terms of scheduling, e.g., one can choose toprioritize iterations between MAI cancellation and decoding.

• In this paper, a time domain implementation of the multidimensional Wiener filters is chosen anda sliding-window approach emphasized in order to reduce its complexity. Another low-complexity

alternative presented in [64] would be a blockwise frequency domain implementation relying on theinsertion of a cyclic prefix at the transmitter.

• The principles presented here can be expanded to deal with imperfect CSIR. Another iterative loopdealing with channel estimation (and reestimation) can easily complete the receiver structure [55].

6 Conclusion

In this paper, the principles of iterated-decision linear equalization have been extended to STBICM overMIMO block fading multipath AWGN channel. Contrary to the classical approach inherited from MUD,

16

where ISI and MAI cancellations are jointly performed in MMSE sense for each distinct antenna, our approachdecouples the two tasks, allowing an additional degree of freedom in the receiver design. Time domainequalization still relies on MMSE criterion and operates on multidimensional modulations symbols, whoseindividual components can be detected in accordance with another criterion. When the optimum MAPcriterion is chosen, substantial performance gains over conventional space-time turbo equalization have beenobserved for different transmission scenarios, at the price of an increased computational complexity (the

latter can be managed, e.g., by resorting to the list-APP SD). Despite those performance improvements,simulations have also revealed that the distance with the (coded) MFB could stay significant for very highloads (very weak outer codes, large constellations). As a final point, it is worth stressing that the generalconcept developed in this paper may be essential to conceive space-time turbo equalizers able to fully exploitmultidimensional design optimization attempts at the transmitter (e.g., labeling or linear precoding) or todeal with space correlation between antennas. This will form the subject of further investigations.

7 Appendix

In this Appendix, we proove the equivalence of the two following scenarios.

Scenario 1: For some particular subgroup ∆ of symbol components, the space-time equalizer providesa biased MMSE estimate z(1)∆,k of the subvector x∆,k by directly applying the |∆|-dimensional Wiener filter

F∆,k = E†∆H

†hHΘxk|∆H

† + INRLF

i−1(56)

to the signal yk− ey

k|∆ = yk −Hexk|∆.Scenario 2: For some particular subgroup ∆0 ⊃ ∆ of symbol components, the space-time equalizer first

provides an intermediate biased MMSE estimate z∆0,k of the subvector x∆0,k by applying the |∆0|-dimensionalWiener filter

F∆0,k = E†∆0H

†hHΘxk|∆0H† + INRLF

i−1(57)

to the signal yk− ey

k|∆0 = yk −Hexk|∆0 and then provides a biased MMSE estimate z(2)∆,k of the subvectorx∆,k by applying the second |∆|-dimensional Wiener filter

Φ∆,k = E†∆E∆0G†

∆0,k

hG∆0,kΘx∆0,k|∆G

†∆0,k +Θξ∆0,k

i−1(58)

to the signal z∆0,k − ez∆0,k|∆ = z∆0,k −G∆0,kex∆0,k|∆ where, as shown in previous sections,

G∆0,k = F∆0,kHE∆0 (59)

Θξ∆0,k =¡I|∆0| −G∆0,k

¢G†∆0,k (60)

and where the conditional covariance matrix

Θx∆0,k|∆ , En¡x∆0,k − ex∆0,k|∆

¢ ¡x∆0,k − ex∆0,k|∆

¢† |Pt,k : t ∈ ∆0/∆o (61)

We first establish a condition which ensures that z(1)∆,k = z(2)∆,k. Starting from

z(2)∆,k = Φ∆,k

hF∆0,k

³yk−Hexk|∆0

´−G∆0,kex∆0,k|∆

i=

Φ∆,k

hF∆0,k

³yk−H

³exk −E∆0E†∆0exk´´−G∆0,k

³E†∆0exk −E†∆0E∆E

†∆E∆0E†∆0exk´i (62)

17

and using the matrix identity

E∆0E†∆0E∆E†∆E∆0E†∆0 = E∆E

†∆ (63)

we get

z(2)∆,k = Φ∆,kF∆0,k

hyk−H

³exk −E∆E†∆exk´i (64)

And since

z(1)∆,k = F∆,k

hyk−H

³exk −E∆E†∆exk´i (65)

we conclude that

z(1)∆,k = z

(2)∆,k ⇔ F∆,k = Φ∆,kF∆0,k (66)

We now prove the RHS of this equivalence. Applying the matrix inversion lemma on

Φ∆,k = E†∆E∆0G†

∆0,k

hG∆0,kΘx∆0,k|∆G

†∆0,k +

¡I|∆0| −G∆0,k

¢G†∆0,k

i−1(67)

yields

Φ∆,k = E†∆E∆0

hΘx∆0,k|∆ − I|∆0| −G−1∆0,k

i−1G−1∆0,k (68)

Let us introduce the matrix Ω∆0,k defined as

Ω∆0,k = Θxk|∆0 −E∆0E†∆0 (69)

= Θxk|∆ −E∆0Θx∆0,k|∆E†∆0 (70)

Using (69) and applying the matrix inversion lemma, we get

G−1∆0,k = I|∆0| +B−1∆0,k (71)

with

B∆0,k = E†∆0H

†A−1∆0,kE∆0H (72)

A∆0,k =hHΩ∆0,kH

† + INRLF

i(73)

Combining (68) and (71) yields

Φ∆,k = E†∆E∆0

hΘx∆0,k|∆ +B

−1∆0,k

i−1 hI|∆0| +B−1∆0,k

i(74)

Besides, again exploiting (69) and matrix inversion lemma, it is straightforward to show that

F∆0,k =hI|∆0| +B−1∆0,k

i−1B−1∆0,kE

†∆0H

†A−1∆0,k (75)

so that

Φ∆,kF∆0,k = E†∆E∆0

hΘx∆0,k|∆ +B

−1∆0,k

i−1B−1∆0,kE

†∆0H

†A−1∆0,k (76)

The last step involves some basic algebra on F∆,k. Invoking (70) and the matrix identity

E†∆ = E†∆E∆0E†∆0 (77)

18

and applying the matrix inversion lemma, we get

F∆,k = E†∆E∆0

·I|∆0| −B∆0,k

hΘ−1x∆0,k|∆ +B∆0,k

i−1¸E†∆0H

†A−1∆0,k (78)

which is equivalent to

F∆,k = E†∆E∆0

hΘx∆0,k|∆ +B

−1∆0,k

i−1B−1∆0,kE

†∆0H

†A−1∆0,k (79)

This completes the proof. ¥Hence, the conventional turbo-equalization [19] [53] [54] has a nice interpretation in terms of a two-stage

process where the ISI cancellation would be first performed in MMSE sense on NT -dimensional symbols

and where the residual MAI within vector symbols would be treated afterwards by again resorting to theMMSE-IC [55].

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23

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

-10 -8 -6 -4 -2 0 2 4 6 8 10

SNR (dB)

Cap

acity

(bi

ts/c

.u)

2x2 M =0 1%

2x2 M =1 1%

2x2 M =2 1%

2x2 M =3 1%

Fig. 1. Achievable information rates quasi-static 2× 2 MIMO EQ channel

24

1,0E-03

1,0E-02

1,0E-01

1,0E+00

-4 -2 0 2 4 6 8 10 12 14

SNR (dB)

BLE

R

G auss 2x2 M =0

G auss 2x2 M =2

B P SK 2x2 M =0

B P SK 2x2 M =2

Fig. 2. Comparison of PGout (η, γ) and PDout (η, γ) quasi-static 2× 2 MIMO EQ channel

25

1,0E-03

1,0E-02

1,0E-01

1,0E+00

-4 -2 0 2 4 6 8 10 12

SNR (dB)

BLE

R

G auss 1.5 bits/c.u

G auss 3 bits/c.u

B P SK 1,5 bits/c.u

Q P SK 3.0 bits/c.u

Fig. 3. Comparison of PGout (η, γ) and PDout (η, γ) quasi-static 2× 2 MIMO EQ2 channel

26

Channel transition function node block no.1

Information bit nodes

Const. symbol nodes (vector components)

Code constraint function node

Channel transition function node block no.0

Coded bit nodes

Labeling function nodes

Symbol digit nodes

Interleaver connections

Global code constraint function node Ψ

Fig. 4. Factor graph of a STBICM on a MIMO block fading ISI channel

27

1,0E-03

1,0E-02

1,0E-01

1,0E+00

-4 -2 0 2 4 6

Eb/No (dB)

BLE

RO utage P rob.

M FB

exact B P it0

exact B P it1

exact B P it2

M M SE/M A P it0

M M SE/M A P it1

M M SE/M A P it5

M M SE joint it0

M M SE joint it1

M M SE joint it5

Fig. 5. Performance 8-PSK, rate-1/3, quasi-static 2× 2 MIMO EQ2 channel, η = 2 b/c.u

28

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

-4 -2 0 2 4 6

Eb/No (dB)

BLE

R

O utage P rob.

M FB

M M SE joint it0

M M SE joint it1

M M SE joint it5

M M SE/M A P it0

M M SE/M A P it1

M M SE/M A P it5

Fig. 6. Performance 16-QAM, rate-1/4, 2× 2 MIMO 2-block EQ2 channel, η = 2 b/c.u

29

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

0 2 4 6 8 10 12 14 16

Eb/No (dB)

BLE

R

O utage P rob.

M FB

M M SE joint it0

M M SE joint it1

M M SE joint it5

M M SE/M A P it0

M M SE/M A P it1

M M SE/M A P it5

Fig. 7. Performance 16-QAM, rate-1/2, quasi-static 2× 2 MIMO EQ2 channel, η = 4 b/c.u

30

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

0 2 4 6 8 10 12 14 16 18 20 22

Eb/No (dB)

BLE

R

O utage P rob.

M FB

M M SE joint it0

M M SE joint it1

M M SE joint it5

M M SE/M A P it0

M M SE/M A P it1

M M SE/M A P it5

Fig.8. Performance 16-QAM, rate-3/4, quasi-static 2× 2 MIMO EQ2 channel, η = 6 b/c.u

31

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

0 2 4 6 8 10 12 14 16 18

Eb/No (dB)

BLE

R

O utage P rob.

M FB

M M SE joint it0

M M SE joint it1

M M SE joint it5

M M SE/M A P it0

M M SE/M A P it1

M M SE/M A P it5

Fig.9. Performance 16-QAM, rate-3/4, quasi-static 2× 2 MIMO EQ4 channel, η = 6 b/c.u

32

1,0E-04

1,0E-03

1,0E-02

1,0E-01

1,0E+00

0 2 4 6 8 10 12 14 16 18

Eb/No (dB)

BLE

R

O utage P rob.

M FB

M M SE joint it0

M M SE joint it1

M M SE joint it5

M M SE/M A P it0

M M SE/M A P it1

M M SE/M A P it5

Fig. 10. Performance 16-QAM, rate-3/4, quasi-static 2× 2 MIMO EXP4 channel, η = 6 b/c.u

33