Space-time coding techniques with bit-interleaved coded. modulations for MIMO block-fading channels

Transcription

1 Space-time coding techniques with bit-intereaved coded moduations for MIMO boc-fading channes Submitted to IEEE Trans. on Information Theory - January Draft version 30/05/2006 Nicoas Gresset, Loïc Brune and Joseph Boutros ENST Paris, 46 rue Barraut, Paris, France Mitsubishi Eectric ITE-TCL, 1 aée de Beauieu, Rennes, France gresset@enst.fr, brune@tc.ite.mee.com, boutros@enst.fr Abstract The space-time bit-intereaved coded moduation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a boc-fading MIMO channe. Its maximumieihood (ML) performance is computed under idea intereaving conditions, which enabes a goba optimization taing into account channe coding. Thans to a diversity upperbound derived from the Singeton bound, an appropriate choice of the time dimension of the spacetime coding is possibe, which maximizes diversity whie minimizing compexity. Based on the anaysis, an optimized intereaver and a set of inear precoders, caed dispersive nuceo agebraic (DNA) precoders are proposed. The proposed precoders have good performance with respect to the state of the art and exist for any number of transmit antennas and any time dimension. With turbo codes, they exhibit a frame error rate which does not increase with frame ength. Index terms Mutipe antenna channes, bit-intereaved coded moduation, space-time coding, Singeton bound, intereaving 1

2 1 Introduction The wide pane of today s wireess transmission contexts maes impausibe the existence of a miracuous universa soution which aways exhibits good performance with ow compexity. Different scenarios (indoor, outdoor with ow veocity, outdoor with high veocity) correspond to different amounts of time and frequency diversity. The success of the muti-carrier moduation as a soution for future wireess systems is in part due to the ow receiver compexity even over arge frequency bands. In this paper, we focus on an indoor environment and design a system approaching the optima performance taught by information theory. In a wireess indoor environment, both time and frequency diversities may be poor due to sma termina veocity and possiby very short channe impuse response. These particuary tricy ow-diversity channes are modeed as bocfading channes. Over ow-diversity mutipe-input mutipe-output (MIMO) channes, space-time coding techniques often enabe transmission with improved data rate and diversity, within a imit given by the ran of the MIMO system [1][18][19][21]. As a first step in providing increased data rates in future generations of indoor wireess oca access networs (WLANs), we study how to appropriatey choose the channe coding, the channe intereaving and the space-time coding. For frame sizes of practica interest, coded moduations have to be considered since space-time codes empoyed with uncoded moduations exhibit a frame error rate (FER), which is dramaticay degraded [23, Annex A]. Thus, we focus on the bit-intereaved coded moduation (BICM) structure, which is the concatenation of a channe encoder, an intereaver and a moduator. The anaysis of the BICM maximum-ieihood (ML) performance is tractabe and eases the coded moduation design. Furthermore, thans to the intereaver, iterative processing at the receiver achieves quasi-ml performance with reduced compexity. On a MIMO channe, the BICM may be concatenated with a simpe fu-rate space division mutipexing scheme (SDM) [21]. In this paper, we improve performance of this space-time BICM (ST-BICM) by repacing the SDM by a more efficient fu-rate inear space-time code: a inear precoding or equivaenty a space-time spreading. Linear precoding is performed by mutipying the compex mutipe-antenna signa by 2

3 a square compex space-time matrix. The space-time matrix enhances the diversity by mixing the symbos of different time periods and antennas together. The choice of the ST-BICM structure may aso be expained as foows: We aim at optimizing a fu-rate space-time code based on inear precoding, taing into account the structure of the whoe transmitter, which inevitaby incudes an error correction code, an intereaver and a symbo mapper. Usuay, space-time codes are designed independenty from the other eements of the transmitter. However, frames of bits are ined through the error correction code and optimizing the space-time code taing into account the whoe transmitter is equivaent to optimizing a BICM concatenated with a space-time code, i.e., an ST-BICM. On an ergodic channe, the achieved diversity order is equa to the code minimum distance mutipied by the number of receive antennas. In most cases, the minimum distance is high enough and increasing diversity through inear precoding does not bring much improvement. For a boc-fading channe, the diversity is upperbounded by the number of channe reaizations in a codeword mutipied by the number of transmit antennas and the number of receive antennas. Using the Singeton bound, we wi exhibit an additiona upperbound on the diversity order, which may be very imiting without precoding. Hence, in this paper, we wi study ST-BICM with inear precoding, focusing on the boc-fading channe and optimize the inear precoding using the ST-BICM ML performance in order to achieve fu-diversity and maximum coding gain. First, we derive the coding gain of an idea ST-BICM. It is reated to the notion of Shannon code and sphere-hardening [38]. Indeed, the idea Shannon code for additive white gaussian noise (AWGN) channes is ocated near a sphere, caed the Shannon s sphere. Thans to the intereaver, the squared Eucidean norm of BICM codewords has ow variance, which impies that codewords ie cose to the Shannon s sphere. The BICM may be seen as a quantization of this sphere, which shoud be as uniform as possibe to maximise the size of Voronoi regions. On MIMO fading channes, the Shannon s sphere becomes a Shannon s eipsoid [20] and BICM codewords are randomy 3

4 ocated cose to the eipsoid. We show that the idea BICM configuration maximizes the Voronoi region voume whatever the channe reaization. We present a practica system that approaches the idea BICM configuration incuding the so-caed dispersive nuceo agebraic (DNA) precoder and compare its performance to the idea ST-BICM performance. The DNA precoder exists for any numbers of space and time dimensions. We finay design a practica intereaver, which approximates the idea intereaving conditions. The paper is organized as foows: In section 2, the ST-BICM transmitter and the associated iterative receiver are presented. In section 3, we derive the anaytica ML performance under idea intereaving assumption for an ergodic channe without precoding and a boc-fading channe with and without precoding. Using the Singeton bound, we show in section 4 that idea intereaving conditions cannot be achieved on a boc-fading channe with any ind of parameters and that inear precoding may be mandatory in some configurations. Section 5 describes the inear precoding optimization for a boc-fading channe and section 6 the intereaver design for convoutiona codes and its appication to turbo codes. Finay, simuation resuts are presented in section 7, which confirm the behavior which was expected from the anaytica study. Furthermore, they show the good performance of the DNA precoders and the advantage of using turbo codes to get a non-increasing FER when the frame size increases. 2 System mode and notations 2.1 Transmitter scheme The transmitter scheme is buit from the foowing fundamenta boc concatenation: A binary error-correcting code C foowed by a deterministic intereaver Π, a symbo mapper (e.g., for a quadrature ampitude moduation (QAM)), a fu-rate space-time spreader S (i.e., a inear precoder) and a set of n t transmit antennas. Fig. 1 iustrates the BICM transmitter structure. 4

5 Without oss of generaity, we assume that the error correcting code C is a convoutiona code with rate R C. The encoder associates with the input information word b the codeword c C. Sequence b (resp. c) hasengthk C L C (resp. N C L C )bits,wherel C is the codeword ength in treis branches. The intereaver Π, which scrambes the L C N C coded bits, is a crucia function in the BICM structure, as it aows the receiver to perform iterative joint detection and decoding. Indeed, it ensures independence between extrinsic and a priori probabiities, in both the detector and the decoder. Furthermore, when maximum-ieihood (ML) decoding is tractabe, intereaving prevents erroneous bits of a same error event from interfering to each other in the same precoded symbo. The intereaver Π may be pseudo-random (PR) or semi-deterministic with some deterministic constraints as described in section 6. In the symbo mapper, m consecutive intereaved coded bits are mapped together onto a moduation symbo, according to a bijection between bit vectors and moduation symbos caed mapping or abeing. The number of moduation symbos is equa to M =2 m. For each channe use, i.e., in each time period, the mapper reads mn t coded bits and generates n t moduation symbos. To mae the reading easier, the obtained n t -dimensiona consteation Ω wi denote both the set of symbos and the set of binary abeings. A aong this paper, we wi consider QAM moduations as they achieve a good compromise between spectra efficiency (in bits/s/hz or bits/dim) and performance. Moreover, with QAM moduation, the system is easiy modeed using a attice consteation structure [10], which gives access to the attice theory toobox, both for transmitter and receiver optimizations. We assume that the QAM moduation has unit energy. The inear precoder S spreads the QAM symbos over s time periods. It converts the n t n r vector channe into an N t N r vector channe, where N t = n t s and N r = n r s. The N t N t matrix S mutipies a vector of N t QAM symbos z =(z,1,z,2,...,z,nt )atthe mapper output, generating N t symbos to be transmitted during s time periods. Vector z is the th vector to be precoded. The precoder S spreads the transmitted symbos over a higher number of channe states to expoit diversity. S is normaized as foows: N t N t Si,j 2 = N t (1) i=1 j=1 5

6 In this paper, we assume a boc-fading channe with n c distinct channe reaizations during a codeword. We denote n s the number of distinct channe reaizations during a precoded symbo. To simpify notations, we assume that n s divides n c. We wi ca channe state the 1 n r SIMO channe associated with one of the n t transmit antennas and one of the n c channe reaizations. The channe experienced by precoded symbo is represented by a N t N r boc-diagona matrix H with s bocs of size n t n r. We assume that each of the n s channe reaizations is repeated s/n s times. Matrix H is organized as foows: ( H =diag H [1],...,H[1],H[2],...,H[2],...,H[ns] )...,H [ns] (2) where H [t] denotes the n t n r compexmatrixrepresentingthet-th channe reaization experienced by precoded symbo. H [t] is repeated s/n s times. Eements of H [t] are independent compex Gaussian variabes with zero mean and unit variance. Let H denote the set of channe reaizations observed during the transmission of a codeword. Thans to the extended channe matrix, we write the channe input-output reation as: y = x + η = z SH + η (3) where y C Nr and each receive antenna is perturbed by an additive white compex Gaussian noise η,j, j =1...N r, with zero mean and variance 2N 0. We define the signa-to-noise ratio E b /N 0, where E b is the tota energy of an information bit at the receiver. Thans to inear precoding, the n t n r MIMO n c -boc-fading channe is converted into an N t N r MIMO N c -boc-fading channe where N c = n c /n s. If n s = 1, the precoder experiences a quasi-static n t n r MIMO channe. In the foowing, index wi be omitted if a singe precoded symbo is considered and precoding time period wi refer to a transmission over SH, i.e., over s time periods. The concatenation of the binary error correcting code C, the intereaver Π, the mapper Ω, the inear precoder S and the channe describes a goba Eucidean code C E which converts L C K C 6

7 information bits into a compex L C N C /m-dimensiona point. 2.2 Iterative receiver scheme An idea BICM receiver woud directy perform an ML decoding on the set C E of transmitted codewords. However, it requires an exhaustive search among the 2 KCLC codewords, which is intractabe. A existing receivers use the concatenated structure of the BICM to spit the reception into severa steps. In this paper, we assume perfect synchronization and channe estimation. Thus, the receiver, as depicted on Fig. 2, is divided in two main eements: a soft-input soft-output (SISO) APP QAM detector, which acts as a soft-output equaizer for both the space-time spreader and the MIMO channe, converting the received point y into information on the coded bits in the estimated coded sequence ĉ, and a SISO decoder for C, improving the information on coded bits and estimating the information bit sequence ˆb. The depicted iterative joint detection and decoding process is based on the exchange of soft vaues between these two eements. The SISO detector computes extrinsic probabiities ξ(c ) on coded bits thans to the conditiona ieihoods p(y /z) and the aprioriprobabiities π(c ) fed bac from the SISO decoder: ξ(c )= z Ω(c =1) [e y z SH ] 2 2N 0 r π(c r) z Ω [e y zsh 2 ] (4) 2N 0 r π(c r) where Ω is the Cartesian product (M-QAM) Nt, i.e., the set of a vectors z generated by the QAM mapper, Ω =2 mnt. The subset Ω(c =1),for =0, 1,...,mN t 1, is restricted to the vectors z in which the -th coded bit is equa to 1. The detector independenty computes the soft outputs for each precoding time period. At the first iteration, no a priori information is avaiabe at the detector input. Through the iterations, the a priori probabiity on consteation points computed from the probabiities fed bac by the SISO decoder becomes more and more accurate. Idea convergence is achieved when a priori probabiities provided by the decoder are perfect, i.e., equa to 0 or 1. The decoder uses a forward-bacward agorithm [2], which computes the exact extrinsic probabiity using the treis structure of the code. 7

8 3 Theoretica performance for ideay intereaved BICM Heavy wor has been made to estimate the frame or bit error rate of the BICM with ML decoding, in particuar using Gaussian approximations or numerica integrations [5], but a cosed-form expression of the pairwise error probabiity had not been derived yet. This section first describes an accurate computation of bit and frame error rates of BICM ML performance over ergodic MIMO channe with idea intereaving and without precoding. A more detaied description of the derivation may be found in [22] and [23]. Under idea intereaving condition, we are abe to derive a cosed form expression of the probabiity density of the og ieihood ratio (LLR) at the output of the detector and then a cosed form expression of the pairwise error probabiity at the output of the decoder. It is then straightforward to use we-nown techniques to estimate the bit or frame error rate of a coded moduation from pairwise error probabiity. This subject has been extensivey discussed for coded moduations over AWGN channes. Exampes are the union bound on the transfer function of a convoutiona code and the more accurate tangentia sphere bound [35] for spherica consteations. Subsequenty, we extend the study to the boc-fading MIMO channe when inear precoding is used at the transmitter. Note that the method is aso vaid for correated MIMO channes. We extract from the bit error rate expression some design criteria on BICM precoder, intereaver, and error correcting code. 3.1 Idea intereaving condition The evauation of the bit error rate (BER) or frame error rate (FER) of a coded moduation is usuay based on the derivation of an upper bound on the actua performance obtained by a baanced summation of pairwise error probabiities. Each pairwise error probabiity invoves the Eucidean distance between two codewords with a Hamming distance w. With an n t n r MIMO boc-fading channe with n c bocs, the minimum diversity recovered at the detector output, and thus at the decoder output, is aways equa to the reception diversity n r. Let us consider an error event with w erroneous bits. Assume that the maximum diversity 8

9 order is Υ max. If w Υ max, we achieve fu diversity if each of the Υ max independent fading random variabes is experienced by at east one bit among w. In a precoding time period in which at east an erroneous bit is transmitted, the transmitted and competing points are caed x = z SH and x = z SH. When performing ML decoding or APP detection, we are interested in the equivaent Binary Shift Keying (BSK) moduation defined by the two points x and x.the vector (z z )SH has sn r circuar symmetric Gaussian components. Thus, whatever the number of erroneous bits on a precoding time period, the obtained diversity is imited to sn r. Having severa erroneous bits per precoding time period is useess. On the contrary, if the erroneous bits are ocated on different precoding time periods and experience different fading random variabes, a higher diversity is achieved. This is what we ca the non-interference property. Furthermore, we wi see in section 5 that an equi-distribution of erroneous bits on channe states is required to achieve a maximum coding gain. We ca it the equi-distribution property. The idea intereaver is defined as foows: Definition 1 (Idea Intereaving) For any pair of codewords with w different bits at positions i 1,...,i,...,i w, an idea intereaver aocates the bits to transmitted symbos as foows: Non-interference property: i,i, bits at positions i and i are transmitted on different precoding time periods, Equi-distribution property: the bits at positions i 1,...,i,...,i w are as equiprobaby distributed over a channe states as aowed by w. In practice, such an intereaver does not aways exist. We wi see in the foowing that the Singeton bound gives an existence condition of the idea intereaver. In section 6, we present optimized intereavers that approach the idea condition. 9

10 3.2 Exact pairwise error probabiity for ergodic channes without precoding In [22], we have estabished a cosed form expression for the conditiona pairwise error probabiity on ergodic MIMO channes under ML decoding of the BICM and idea channe intereaving. The mathematica derivation in this subsection foows [22]. Transmitted symbos are not precoded: s =1,S = I nt the n t n t identity matrix. Thus, (2) reduces to H = H [1]. Consider the pairwise error probabiity that a codeword c Cis transmitted and a codeword c Cis decoded. The w different bits between the two codewords are transmitted in w different time periods, compementing one bit in the mapping of one of the n t QAM symbos. The transmitted noiseess vectors corresponding to the two codewords (c, c ) ony differ in w positions. Let us define Z = {z 1,...,z w } and Z = {z 1,...,z w } the w-dimensiona vectors corresponding to these positions and X = {x 1,...,x w } and X = {x 1,...,x w} the transmitted vectors corresponding to Z and Z. We define d = z z. The Eucidean distance X X depends on both the set of distances {d 1,...,d,...,d w } and the set of channe reaizations H. Let D denote the set of a Eucidean distances obtained by fipping one bit in the consteation Ω. Define the set Δ = {δ 1,...,δ nd } D with distinct eements from the sequence (d 1,d 2,...,d w ) Δ w D w, i.e., the Eucidean distance d taes its vaues from the set Δ. Obviousy, n d = Δ D. Let the integer λ denote the frequency of δ in the sequence (d 1,d 2,...,d w ), n d n=1 λ n = w and Λ = {λ 1,...,λ nd }. The pairwise error probabiity conditioned on the channe reaization set H and the Hamming weight w is expressed as ( w ) P w,h (c c )=P w,h (X X )=P LLR < 0 =1 (5) where LLR is the -th LLR, corresponding to the -th error position, and is equa to LLR = y x 2 y x 2 2N 0 N ( R, R ) 2N 0 N 0 (6) 10

11 R = (z z )H 2 has a chi-square distribution of order 2n r. Averaging over H, wecacuate the characteristic function of E H [ w =1 LLR ]: ( w ( ) d 2 nr ) ψ(jν)= 2N 0 =1 n d n= n d,n 0 [jν + β n ] nrλ n (7) where ( n>0, β n = ( n<0, β n = N0 δ 2 n 1+ 8N0 δ 2 n ) ) (8) Appying a partia fraction expansion, we obtain the expression of the pairwise error probabiity: P w (X X )=P w (Δ, Λ) = w =1 ( 2N ) nr n d 0 d 2 n rλ n n=1 i=1 ( α n,i 1+ 8N0 δ 2 n ) i (9) where the coefficients α n,i are given by an identification of the coefficients of two series expansions in ɛ as in [22]. We compute the asymptotic expression when the noise eve is ow. Indeed, the coding gain and diversity are measured for high signa-to-noise ratios, where the performance has a inear asymptote on ogarithmic scaes. P w (Δ, Λ) N0 0 ( ) 2nr w 1 w ( ) nr ( )( ) wnr 2N0 2nr w 1 2N 0 = (10) n r w n r w G ergo (Δ, Λ) =1 d 2 with ( n ) = n!/(!(n )!). The diversity associated with the considered pairs of Hamming weight w is the exponent of 2N 0,equatown r. We define the coding gain or coding advantage as the coefficient dividing 2N 0, i.e., ( w 1/w G ergo (Δ, Λ) = d) 2 (11) =1 A sequences (d 1,...,d w ) corresponding to the same pair (Δ, Λ) yied the same pairwise error probabiity. By averaging over a possibe pairs (c, c ) or equivaenty over a sets of distances D w,weobtainp w = E D w [P w (Δ, Λ)], the conditiona probabiity that an error event of Hamming 11

12 weight w occurs. From this pairwise error probabiity, it is easy to estimate the FER or BER of the BICM with idea intereaving thans to a cassica union bound on the weight enumeration function of the error correcting code. Moreover, we may derive a design criterion of the BICM from the coding gain G ergo (Δ, Λ) expression. In the foowing, we derive the coding gain for boc-fading channes and inear precoding in order to obtain the ML design criterion of the ST-BICM. 3.3 Exact pairwise error probabiity for MIMO boc fading channes without precoding We assume that Definition 1 is satisfied. For a boc-fading channe with n c independent reaizations in a frame, the decision variabe between X and X is sti given by (5). However, the invoved channe matrices are not independent as for an ergodic channe. The conditions of independence are the foowing: If two LLR random variabes depend on two different channe reaizations, they are independent. If two LLR random variabes depend on the same channe reaization but on different transmit antennas, the random variabes are independent. The maximum number of independent LLR variabes is n c n t, the transmit diversity order. We choose the error correcting code so that w n t n c. We now group the w random variabes LLR into min(n t n c,w)=n t n c independent bocs. Let LLR,,i be the i-th og-ieihood ratio corresponding to the BSK transmission on the -th antenna of the -th boc, =1...n c, = 1...n t and i =1...κ,,whereκ, is the number of bits transmitted on the -th antenna of the -th boc. We have n c nt =1 =1 κ, = w. Finay, LLR is the sum of the n t n c independent random variabes LLR, = κ, i=1 LLR,,i. Letd,,i denote the distance associated with LLR,,i, and define γ 2, = κ, i=1 d2,,i the distance associated with LLR,. Wehave LLR, N ( R,, R ), 2N 0 N 0 (12) 12

13 where R, = γ 2, H () 2 and H () isthe-th row of H. For a i, LLR,,i are transmitted over the equivaent 1 n r SIMO channe defined by H (), which is chi-square distributed with degree 2n r.thellr, variabes are transmitted on independent channe states, as for the ergodic channe case, we directy appy (9) and obtain the conditiona pairwise error probabiity cosedform expression P w (X X )=P w (Δ, Λ) = n c n t =1 =1 ( 2N 0 γ 2, ) nr nd n rλ n n=1 i=1 ( α n,i 1+ 8N0 δ 2 n ) i (13) where δ n Δand(Δ, Λ) is the pair of sets representing the sequence (γ 1,1,...,γ nt,n c ). The α n,i coefficients are computed as for (9). The asymptotic expression of P w (Δ, Λ) is P w (Δ, Λ) N0 0 ( 2nr n t n c 1 n r n t n c ( n t ) nc =1 =1 2N 0 γ 2, ) nr (14) The diversity associated with the considered pairs of Hamming weight w is then equa to the exponent n t n c n r. The coding gain is given by the geometrica mean of the γ, 2 and is equa to G bf (Δ, Λ) = ( nc n t κ, ) 1/(ntn c) d 2,,i =1 =1 i=1 (15) We wi see in the foowing how to use this coding gain as a design criterion for the ST-BICM optimization. We now consider an equivaent computation of the coding gain for a ineary precoded ST-BICM. 3.4 Exact pairwise error probabiity for MIMO boc fading channes with precoding When a inear precoder S of size N t N t is used, the detector computes soft outputs on the N t transmitted symbos using the equivaent channe matrix SH of size N t N r. The structure of 13

14 H is described in (2). SH can be seen as a correated MIMO channe [42]. Under the idea intereaving condition, we consider at most a singe erroneous bit per boc of s time periods in position 1 mn t inside the binary mapping of the transmitted symbo z, eading to symbo z. For simpicity reasons, we assume that the error weight w satisfies w N t N c. Moreover, we assume that the mapping is mono-dimensiona: the BSKs are transmitted on a singe seected input of the matrix SH. Let LLR,,i be the i-th variabe among κ,, corresponding to the transmission of a BSK on the equivaent 1 N r channe S H,whereS corresponds to the -th row of S. We have N c Nt =1 =1 κ, = w. Let d,,i denote the BSK distance associated with LLR,,i. We can use the factorization LLR, = κ, i=1 LLR,,i of a the LLR variabes fitered with S H : LLR, N ( R,, R ), 2N 0 N 0 (16) where R, = V, H 2, V, = γ, S and γ 2, = κ, i=1 d2,,i.thevariaber, is a generaized chisquare random variabe with 2N r correated centered Gaussian components. The random variabe LLR = N t =1 LLR, satisfies LLR N ( Nt =1 R, 2N 0, ) Nt =1 R, N 0 (17) From appendix A, we get the foowing characteristic function: E H [Ψ LLR (jν)] = n s n t t=1 u=1 ( 1 nr ν(j ν) ϑ [t] 2N,u) (18) 0 where ϑ [t],u is the u-th eigenvaue of Σ [t] = Nt =1 γ,s 2 [t] S [t] = N t s/n s γ, 2 =1 i=1 S [t][i] S [t][i] = M [t] M [t] (19) 14

15 M [t] is as an n t n t Hermitian square root matrix of Σ [t] s/n s n t matrices S [t] are defined from S as foows: and row vectors S[t][i] of size n t and S = S [1] (1 N t /n s ) S [ns] {}}{{}}{ S [1][1] 1 S [1][s/ns] 1 S [ns][1] 1 S [ns][s/ns] 1 S [1][1] 2 S [1][s/ns] 2 S [ns][1] 2 S [ns][s/ns] 2.. S [1][1] N }{{ t S [1][s/ns] N }} t S [ns][1] N {{}}{{ t S [ns][s/ns] N }} t {{} N t /s = n t coefficients (20) and S [t] = S [t][1] S [t][2].. S [t][s/ns] (21) The set of eigenvaues ϑ [t],u is a function of the precoding matrix S and the BSK distances set D w. Thans to the independence of channe reaizations for different vaues, we can mutipy the characteristic functions: Ψ(jν)= N c n s n t =1 t=1 u=1 ( 1 nr ν(j ν) ϑ [t] 2N,u) (22) 0 Denote Δ = {δ v } the set of n δ square-roots of non-nu eigenvaues extracted from the sequence defined by the ϑ [t],u vaues. Each eigenvaue δ2 v is repeated λ v times. Observe that n δ n c n t. Finay, using the partia fraction expansion of Ψ(jν) as for (9), we obtain the exact pairwise error probabiity P w (Δ, Λ) conditioned on d H (c, c )=w: P w (Δ, Λ) = n δ v=1 ( 2N ) λvn r n δ 0 δv 2 n rλ v v=1 i=1 ( α v,i 1+ 8N0 δ 2 v ) i (23) 15

16 The asymptotic expression of P w (Δ, Λ) is P w (Δ, Λ) N0 0 ( 2nr N δ 1 n r N δ ) nδ ( ) λvn r 2N0 (24) δ 2 v=1 v where N δ = n δ v=1 λ v is the tota number of non-nu eigenvaues. The diversity associated with the considered pairs of Hamming weight w is the exponent equa to n δ v=1 λ vn r = N δ n r. The coding gain is given by G s,ns (Δ, Λ) = ( nδ v=1 δ 2λv v ) 1/Nδ (25) We have derived for any signa-to-noise ratio an exact expression of the pairwise error probabiities of a BICM with inear precoding, which is usefu for a tight BER and FER estimation. The asymptotic expression eads to the we-nown ran and determinant criteria [39][19] for space-time code optimization over MIMO boc-fading channes, where the considered space-time code is the whoe BICM structure. As a remar, the asymptotic design criterion is usuay derived by first upperbounding the Q(x) function by exp( x 2 /2)/2 and then averaging over the channe reaizations. The obtained asymptotic expression has a mutipying coefficient different from ( 2n rn δ 1 n rn δ ), which is inexact but provides the same design criterion. Moreover, we notice that appying the Taroh criterion [39] on the ran and determinant to the precoder aone does not ead to the whoe BICM optimization. Quasi-optima inear precoders wi be designed to achieve fu diversity and approach optima coding gain in section Evauation of the Frame Error Rate For ergodic channes, the frame error rate is easiy computed via a truncated union bound. Indeed, ony error events with minimum Hamming distance impact the error rate for a high signa-to-noise ratio and the observed diversity is equa to n r d Hmin. For boc-fading channes, the frame error rate computation is much more tricy since each pairwise error probabiity is supposed to have 16

17 the fu-diversity order n c n t n r. Due to the random nature of each eigenvaue in (25), it is difficut to now the impact of each distance configuration on the fina FER. However, one may assume that for a sufficienty high signa-to-noise ratio, the FER satisfies the foowing expression: FER w A w E (Δ,Λ w) [P w (Δ, Λ)] (26) where A w is weighting the impact of pairwise error probabiities with Hamming weight w in the goba error probabiity and the expectation on (Δ, Λ) is aowed by the intereaver random structure. Let us define G the goba coding gain. Since each pairwise error probabiity is supposed to have fu diversity, we write ( )( ) nrn 2nr n t n c 1 tn c 2N0 FER (27) n r n t n c G and G nrntnc = w A w E (Δ,Λ w) [ G(Δ, Λ) n rn tn c ] (28) where G(Δ, Λ) is the coding gain associated with one pair of codewords. We note that optimizing independenty a pairwise error probabiities, which wi be done in the foowing, enhances the goba performance. Moreover, we observe that the number of receive antennas does not affect the coding gain of a singe pairwise error probabiity. The effect of the receive diversity appears in the expression of the goba coding gain (28). As n r n t n c grows, the smaest coding gains have more impact on the fina performance. Asymptoticay, if n r n t n c +, ony the nearest neighbors in the Eucidean code have an infuence on the FER, as for AWGN channes. We wi see in section 5.1 that the best coding gain is achieved when a eigenvaues ϑ [t],u are equa. In this idea configuration, the coding gain is shown to be the same as with the same coded moduation transmitted on a 1 n c n t n r quasi-static SIMO channe. Simuating this atter case is 17

18 ess compex: the performance curve is semi-anayticay computed using a reference curve on an AWGN channe. Aternativey, performance may be obtained by computing the Tangentia Sphere Bound for spherica moduations [26]. In the foowing, idea BICM wi refer to the performance of the idea configuration, which wi be drawn on simuation resuts. This ower bound has the advantage to tae the moduation and error correcting code into account and wi be usefu to evauate the optimaity of both the inear precoder and the channe intereaver. 4 The Singeton bound with inear precoder Definition 1 ensures that any pair of codewords benefits from a fu diversity order. In this section, we derive a condition on the existence of a practica intereaver that coud achieve the conditions of Definition 1. Let us first mae the foowing assumption: Assumption 1 The detector perfecty converts the N t N r correated MIMO N c -boc-fading channe SH with QAM input into a 1 sn r SIMO n t n c /s-boc-fading channe with BSK input, assuming that s is a divisor of n t n c. We wi present in section 5 inear precoders that satisfy Assumption 1. Under this condition, the detector coects an amount of diversity equa to sn r. The fu diversity n t n c n r is coected by the detector when s = n t n c, but unfortunatey, the APP signa detection has an exponentia compexity in s. On the other hand, the BICM channe decoder is aso capabe of coecting a arge amount of diversity, but the atter is sti imited by the Singeton bound [28][29][33]. Hence, the owest compexity soution that reaches fu diversity is to draw advantage of the whoe channe code diversity and recover the remaining diversity by inear precoding. The best way to choose the spreading factor s is given by the Singeton bound described hereafter. The studied ST-BICM is a seria concatenation of a rate R C binary convoutiona code C, an intereaver of size N C L C bits, and a QAM mapper foowed by the precoder as described in 18

19 section 2. When S is the identity matrix, the ST-BICM diversity order is upper-bounded by [29]: Υ n r ( n c n t (1 R c ) + 1) (29) The maxima diversity given by the outage imit under a finite size QAM aphabet aso achieves the above Singeton bound [24]. With a vanishing coding rate, i.e., R c 1/(n c n t ), it is possibe to attain the overa system diversity order n r n c n t produced by the receive antennas, the transmit antennas and the distinct channe states. Unfortunatey, this is unacceptabe due to the vanishing transmitted information rate. Precoding is one means to achieve maximum diversity with a non-vanishing coding rate. The integer N b = n c n t /s is the best diversity mutipication factor to be coected by C. The ength of a C codeword is L C N C binary eements. Let us group L C N C /N b bits into one non-binary symbo creating a non-binary code C. Now, C is a ength-n b code buit on an aphabet of size 2 LCNC/N b. The Singeton bound on the minimum Hamming distance of the non-binary C becomes D H N b N b R C + 1. Mutipying the previous inequaity with the Naagami aw order sn r yieds the maximum achievabe diversity order after decoding: nc n t Υ sn r s (1 R C)+1 (30) Finay, since Υ is upper-bounded by the channe intrinsic diversity and the minimum Hamming distance d Hmin of the binary code, we can write Υ min j ncn t sn r s (1 RC)+1 ; n tn cn r; sn rd Hmin =Υ max (31) If d Hmin is not a imiting factor (we choose C accordingy), we can seect the vaue of s that eads to a modified Singeton bound greater than or equa to n t n c n r. Proposition 1 Considering a BICM with a rate R C binary error-correcting code on an n t n r 19

20 MIMO channe with n c distinct channe states per codeword, the spreading factor s of a inear precoder must be a divisor of n t n c and must satisfy s R C n c n t in order to achieve the fu diversity n t n c n r for any pair of codewords. In this case, the idea intereaving conditions can be achieved with an optimized intereaver. The smaest integer s opt satisfying the above proposition minimizes the detector s compexity. If R C > 1/2, then s opt = n c n t which invoves the highest compexity. If R C 1/(n c n t ), inear precoding is not required. Tabes 1 and 2 show the diversity order derived from the Singeton bound versus s and n t,for n c =1andn c = 2 respectivey. The vaues in bod indicate fu diversity configurations. For exampe, in Tabe 1, for n t =4,s = 2 is a better choice than s = 4 since it eads to an identica diversity order with a ower compexity. 5 Linear precoder optimization Many studies have been pubished on space-time spreading matrices introducing some redundancy, we-nown as space-time boc codes. On one hand, some of them are decoded by a ow-compexity ML decoder, but they sacrifice transmission data rate for the sae of high performance. Among them, the Aamouti scheme [1] is the most famous, but is ony optima for a 2 1 MIMO channe. The other designs aowing for ow ML decoding compexity are based on an extension of the Aamouti principe (e.g., DSTTD [41]) but aso sacrifice the data rate. On the other hand, fu rate space-time codes have recenty been proposed [4][11][12][13][14][18][34]. However, their optimization does not tae into account their concatenation with an error correcting code. In this section, we describe a near-idea soution for inear precoding in BICMs under iterative decoding process. Our strategy is to separate the coding step and the geometry properties in order to express some criteria aowing the construction of a space-time spreading matrix for given channe parameters n t, n r and n c. The incusion of rotations to enhance the BICM performance over singe antenna channes has been proposed in [31]. Our soution uses this concept for designing a space-time code 20

21 incuding a powerfu error correcting code. When the channe is quasi-static or boc-fading with parameter n c, the diversity is upper bounded by n c n t n r which may be more imiting than n r d Hmin (e.g., n t =2,n r =1,n c =1). We introduce a new design criterion of space-time spreading matrices that guarantees a diversity proportiona to the spreading factor, within the upper-bound, and a maxima coding gain at the ast iteration of an iterative joint detection and decoding. 5.1 Coding gain under both idea intereaving and precoding First we oo for the best achievabe coding gain for the fixed parameters n t, n r, n c, R C and the appropriate way to choose the error correcting code, the binary mapping, the inear precoder and its parameters s and n s to achieve the idea coding gain. We want to achieve fu diversity under ML decoding or iterative joint detection and decoding, this induces that there are n c n t non-nu eigenvaues ϑ [t],u (see (24)): G s,ns (Δ, Λ) = ( Nc n s n t =1 t=1 u=1 ϑ [t],u ) 1/(ncn t) (32) Furthermore, we want to maximize the G s,ns (Δ, Λ) expression. Assuming that each row S is normaized to 1, we get N c n s n t =1 t=1 u=1 Nc ϑ [t],u = N t =1 =1 κ, N c N t γ, 2 = d 2,,i (33) =1 =1 i=1 Under this condition, the idea coding gain is achieved when a eigenvaues are equa Nc ϑ [t],u = N t =1 =1 γ 2, n t n c (, t, u) (34) 21

22 which eads to N c N t G idea (Δ, Λ) = =1 =1 γ 2, n t n c = w i=1 d 2 i n t n c (35) The exact pairwise error probabiity expression simpifies to the cassica expression of the performance of a BPSK with distance w j=1 d2 j over a diversity channe with order n cn t n r [36]: 0 P w,idea (Δ, Λ) 1 ψ1+ 8N0ntnc P w j=1 d2 j! 1/2 1n cn tn r ncn X tn r 1 A =0 ncntn r ψ1+ 8N0ntnc P w j=1 d2 j! 1/2 1 A As stated in the introduction, in an ST-BICM, precoded moduation symbos quantify the Shannon sphere and best quantization is obtained by uniformy distribute them on the sphere. After transmission on a fading channe, vectors beong to an eipsoid obtained by appying an homothety on the sphere. From (34) and (35), we see that the idea coding gain is obtained by equay distributing the Eucidean distance between two codewords among the n t n c channe states. Hence, the Eucidean distance varies as a n t n c n r Naagami distribution, according to the square norm of the eipsoid axes. Thus, an idea ST-BICM aims at uniformy distributing the precoded moduation symbos, whatever the channe reaization, i.e., whatever the homothety. The idea coding gain is a fundamenta imit which cannot be outperformed. It is usefu to evauate how optima the practica design of a BICM is. We aim at finding the best design, corresponding to eigenvaues which are as cose to each other as possibe. The more different from each other the eigenvaues are, the ower the product in (32) and the coding gain are. From (35), we see that (36) the idea coding gain is the same as for the same coded moduation transmitted on a 1 n c n t n r singe-input mutipe-output (SIMO) channe, appying the appropriate E b /N 0 normaization. Without inear precoding, the idea coding gain is ony achieved if a γ, are equa. Remember that each γ, is a sum of κ, distances d,,i. Thans to the second point in Definition 1, the κ, vaues are cose to w/(n t n c ) and their variance decreases when w increases. Thus, with a powerfu error correcting code having minimum Hamming distance much greater than n t n c and D, eachγ, vaue is amost equa to the average N c Nt κ, =1 =1 i=1 d2,,i /(N tn c )ofd,,i vaues 22

23 and quasi-idea coding gain is observed. If the error correcting code is not powerfu enough to achieve the idea coding gain, i.e., the γ, vaues are very different, the inear precoder provides an additiona coding gain by averaging the γ, vaues, as we wi see in the foowing. First, we derive the optima coding gain which can be achieved using an idea inear precoder for a given binary abeing and error correcting code. Variabes γ, for different vaues correspond to independent channe reaizations H which are not ined by the inear precoder. Thus, random variabes n s nt t=1 u=1 ϑ[t],u are independent for distinct vaues of. The optima coding gain with inear precoding is G s,ns,opt(δ, Λ) = N c =1 ( Nt =1 γ 2, n t n s ) 1/Nc (37) Equation (37) means that an optima inear precoder is capabe of maing eigenvaues equa for asame. However, for different vaues of, eigenvaues ϑ [t],u are different, which induces a coding gain oss. When the mapping and error correcting code are given and the intereaving is idea, the choice of inear precoding parameters impacts on optima coding gain. Let us consider codewords that are equidistant from the transmitted codeword, i.e., a set of distance configurations corresponding to a same vaue of, γ2,. The variance of N t =1 γ2, /(n tn s ) over this set decreases when n s increases, as the number of distances buiding each γ, is higher. The ower the variance of eigenvaues, the higher the coding gain. Thus, G s,ns,opt(δ, Λ) is an increasing function of n s and, for a given s, we shoud choose n s =min(s, n c ). The optima coding gain G s,min(s,nc),opt(δ, Λ) is an increasing function of s. Ifn s = n c, which impies s = n t n c, the idea coding gain is achieved by the optima precoder. Finay, we can surround the coding gain at fu diversity as foows: s G idea (Δ, Λ) G s,min(s,nc),opt(δ, Λ) G s,1,opt (Δ, Λ) G 1,1,opt (Δ, Λ) G bf (Δ, Λ) (38) If, for any pairwise error probabiity, G bf (Δ, Λ) G idea (Δ, Λ), the inear precoder optimization is useess from a coding gain point-of-view. However, obtaining near-idea coding gain without 23

24 precoding requires an optimization of the error correcting code and mapping for any pairwise error probabiity, which is intractabe for non-trivia moduations and codes. Furthermore, the first objective of inear precoding is the diversity contro, which has a high infuence on the performance even at medium FER ( ), especiay for ow diversity orders. Therefore, precoding is oftenusefuinthebicmstructure. After the impact of the inear precoding for a given pairwise error probabiity, et us consider the behavior of the goba performance under inear precoding. As stated in section 3.5, if n r n t n c grows, the pairs of codewords providing the smaest coding gains have more impact on the fina performance. Since the inear precoder provides a more substantia gain for the ow Hamming weight configurations, the coding gain of the inear precoder wi be magnified as the diversity grows. Exampe of idea coding gain: In order to iustrate the roe of the inear precoding in the coding gain optimization, we consider a 2 1 quasi-static MIMO channe and a pairwise error probabiity between two codewords separated by a Hamming distance of w bits.fig.3represents the distribution of the two γ 1 and γ 2 vaues over the two transmit antennas without inear precoding. Bits transmitted on antennas 1 and 2 are transmitted on the sets of time periods T 1 and T 2, respectivey. Thans to idea intereaving, T 1 T2 =. This iustrates the factorization of the distances into the γ vaues. The instantaneous coding gain is equa to γ1 2γ2 2.Nowetusconsider a specific inear precoder, which spreads the vaues γ 1 and γ 2 as presented in Fig. 4 over two time periods and two transmit antennas dividing the squared distance in two equa parts γ1 2/2andγ2 2 /2 respectivey. The average vaue (γ1 2 + γ2)/2 2 is transmitted on each antenna, the coding gain is optima and equa to (γ1 2 + γ2)/2. 2 For exampe, consider a BPSK moduation and a pairwise error probabiity with Hamming weight 3. With optima inear precoding, the idea intereaving provides for exampe γ 2 1 =2 22 and γ 2 2 =1 22. With optima inear precoding, we have a distance ( )/2 associated with each antenna. The ratio between the two averaged coding gains 24

25 is equa to 9/8, i.e., we expect a gain of 0.26 db when using inear precoding. With w =5and w = 11, the coding gain becomes 10 og 10 ( 24/25) 0.09 db and 10 og 10 ( 120/121) 0.02 db, respectivey. The higher the Hamming weight invoved in the pairwise error probabiity is, the ess the coding gain provided by inear precoding is. We see on Tabe 3 the best gain to be provided by inear precoding for a quasi-static channe with BPSK input with respect to a fu diversity unprecoded scheme. These gains are particuary ow because the error correcting code aims at recovering a arge amount of coding gain. This iustrates that BICMs are very efficient transmission schemes. As a remar, if a moduation with higher spectra efficiency is used with Gray mapping, the nearest neighbor in the Eucidean code has the same distance configuration as if a BPSK moduation was used. Moreover, for high diversity orders, the goba error rate for high E b /N 0 wi be dominated by the neighbors and the gain provided by inear precoding wi be very cose to the ones shown in Tabe 3. However, if the diversity is ow, the gains provided by inear precoding may be much more important. Assume that a 16-QAM moduation with Gray mapping is transmitted on a n t = 2 quasi-static channe. For instance, if w = 5, there exists a neighbor with distance configuration (3A, 3A, 3A, A, A) (e.g., see [22]), and γ1 2 =9A2 +9A 2 +9A 2, γ2 2 = A2 + A 2. The gain to be provided by inear precoding is equa to 10 og 10 (29/2/ 54) = 2.95 db. As aready stated, the fina coding gain is equa to a weighted sum of a the coding gains, where the weighting coefficients cannot be easiy computed in the case of ow diversity orders. Even if inear precoding does not aways provide a substantia coding gain, its prior aim is the diversity order contro. Thus, we wi focus on the design of inear precoders aiming at reaching fu diversity and maximizing the coding gain for any set of parameter (n t,s,n s ). 25

26 5.2 A new cass of inear precoders Under inear precoding, the optima coding gain is achieved if a ϑ [t],u variabes are equa for a same. Let us first consider the eigenvaues associated with the independent reaizations in the spreading matrix, indexed by t. First, two matrices M [t1] and M [t2], as introduced in (19), shoud have the same eigenvaues, which is satisfied if (t 1,t 2 ),M [t1] = R t1,t2 M [t2] R t1,t2,wherer t1,t2 is a unitary matrix, for exampe a rotation. Hence, (t 1,t 2 ),S [t 1] = S [t 2] R t1,t2. The precoding sub-part S [t1], with spreading factor s = s/n s, experiences a quasi-static channe. We assume that s is an integer, divisor of n t. It is sufficient to design the first sub-part of the precoder matrix rows for a quasi-static channe and rotate it to compute the other sub-parts. Furthermore, any choice of R t1,t2 eads to the same performance because the eigenvaues remain unchanged. The condition simpifies to S [t 1] = S [t 2]. Let us now optimize for a given index t the equivaent precoder over the quasi-static channe ( ) diag H [t],...,h[t],inwhichh [t] is repeated s times. If a the eigenvaues of M [t] M [t] are equa, M [t] [t] and M are weighted unitary matrices and M [t] M [t] = M [t] M [t] Nt = s γ, 2 =1 i=1 S [t][i] S [t][i] (39) Matrix S [t][i] S [t][i] has ran one and matrix s i=1 S[t][i] S [t][i] has maximum ran s. If s <n t, it can be shown that it is impossibe to get a eigenvaues equa to N t =1 γ2, /n tn s as required to achieve the optima coding gain. However, in order to insure that M [t] M [t] has a ran n t and that the eigenvaues are as equa as possibe, we group ss vaues γ, together and associate them with one of the n t /s groups of s eigenvaues: we denote S [t][i][j] the j-th sub-part of size s of S [t][i] and { 2, 1 } the index of the ( 2 1)ss + 1 -th row of S, where 2 [1,n t /s ], 1 [1,ss ]. Let us assume that S [t][i] { 2, 1} has ony one non-nu sub-part in position 2, i.e., j 2 S [t][i][j] { 2, 1} =[0,...,0] (40) 26

27 Considering such a structure is equivaent to considering separate precoding on n t /s distinct groups of s transmit antennas. We have N t =1 n t/s γ,s 2 [t] S [t] = = n t/s ss 2=1 1=1 ss s γ,{ 2 2, 1} 2=1 1=1 i=1 s γ,{ 2 D 2, 1} 2 S [t][i] { 2, 1} S[t][i] { 2, 1} (41) S [t][i][2] { 2, 1} S[t][i][2] { 2, 1} i=1 (42) where D 2 (A) is a boc diagona matrix with ony one non-nu boc A in position 2.Wechoose S [t][i][2] { 2, 1} proportiona to the i-th row of a s s unitary matrix, such as S [t][i][2] { 2, 1} 2 =1/s: N t =1 γ,s 2 [t] S [t] = = 1 s n t/s ss 2=1 1=1 ss 1=1 γ 2,{ 2, 1} D 2 ( 1 s I s ) ) diag (γ,{1,1} 2 I s,...,γ2,{nt/s,1} I s (43) (44) which eads to 2 n t /s,u s, ϑ [t],( 2 1)ss +u = 1 s ss 1=1 γ 2,{ 2, 1} (45) The random variabes γ,{ 2 2, 1} are independent and identicay distributed for different vaues of 1 and 2, the coding gain is G s,ns (Δ, Λ) = N c n t/s =1 2=1 ss 1=1 γ 2,{ 2, 1} s s /(N cn t) (46) For any vaue of n s, the gain expressed in (46) is a geometric mean of order n t n c /s. For a given reaization {d 1,...,d w },agivens and for any n s, N c nt/s ss =1 2=1 1=1 γ2,{ 2, 1} and thus ss 1=1 γ2,{ 2, 1} are constant, ensuring the same coding gain. However, such a precoder does not achieve the optima coding gain for any vaue of s. The summation is made over ss different vaues whereas the optima coding gain in (37) necessitates a summation over sn t vaues. Ony if ss is high enough, the obtained coding gain is amost optima. If s = n t, the compete spatia transmit diversity is coected by the detector and the optima coding gain is achieved. 27

28 Proposition 2 Dispersive Nuceo Agebraic (DNA) Precoder Let S be the N t N t precoding matrix of a BICM over a n t n r MIMO n c -boc-fading channe. Assume that S precodes a channe boc diagona matrix with s bocs and n s channe reaizations. We denote s the spreading factor, N t = sn t and s = s/n s.lets [t] be the i-th sub-part of size n t of S [t].lets [t][i][j] be the t-th sub-part of size N t /n s of the -th row of S. LetS [t][i] be the j-th sub-part of size s of S [t][i].thesubpart S [t][i][j] is caed nuceotide. The inear precoder guarantees fu diversity and quasi-optima coding gain at the decoder output under maximum ieihood decoding of the BICM if it satisfies the two conditions of nu nuceotides and orthogona nuceotides for a t [1,n s ],i [1,s ], 1 [1,ss ], 2 [1,n t /s ] and { 2, 1 } =( 2 1)ss + 1 : j 2,j [1,n t /s ], S [t][i][j] { =0 2, 1} 1 s Nu Nuceotide condition i i, i [1,s ], S [t][i][2] { 2, 1} S[t][i ][ 2] { 2, 1} = 1 s d(i i ) Orthogona Nuceotide condition (47) where d(0) = 1 and d(x 0)=0. Let us tae for exampe n t =4,n s =1ands = 2. A DNA matrix woud have the foowing structure: DNA(n t =4,n s =1,s=2)= S [1][1][1] {1,1} 0 S [1][2][1] {1,1} 0 S [1][1][1] {1,2} 0 S [1][2][1] {1,2} 0 S [1][1][1] {1,3} 0 S [1][2][1] {1,3} 0 S [1][1][1] {1,4} 0 S [1][2][1] {1,4} 0 0 S [1][1][2] {2,1} 0 S [1][2][2] {2,1} 0 S [1][1][2] {2,2} 0 S [1][2][2] {2,2} 0 S [1][1][2] {2,3} 0 S [1][2][2] {2,3} 0 S [1][1][2] {2,4} 0 S [1][2][2] {2,4} (48) 28