Hard and Soft Spherical-Bound Stack decoder for MIMO systems

1 Hard and Soft Spherical- Bound S tack decoder for MIMO sys tems Ghaya Rekaya Ben-Othman, IEEE Member , Rym Ouertani, IEEE Student Member and Abdellatif Salah, IEEE Student M ember ⋆ Abstract Classical ML deco ders of MIMO systems like the spher e decoder, the Schno rr-Euchner algo rithm, the Fano and the stack decoders suf fer of high comp lexity for hig h number of antennas and large constellation sizes. W e prop ose in this paper a novel sequential algorith m wh ich combines the stack algorithm search strategy and the sphere decoder search region. The prop osed decod er that we call the Sph erical-Boun d- Stack decoder (SB-Stack) can then be used to re solve la ttice and large size constellation s decoding with a reduced com plexity comp ared to the classical ML d ecoders. The SB-Stack decod er will b e fu rther extended to support soft-output detection over linear chann els. It will be shown that the soft SB-Stack deco der o utperf orms other MIMO soft decod ers in term o f perfor mance and co mplexity . I N T RO D U C T I O N In this work, we are interested in the d ecoding of multi-antenna sys tems us ing spatial multiplexing [1] and linear space time block codes (STBC). In [2], a latti ce represen tation of MIMO sy stems is proposed . The decoding p roblem can then be see n as a closest lattice p oint se arch p roblem (CLPS) and the lattice dec oders are then used. For MIMO systems , we find in the literature ma inly two classes of decoding strategies. On one ha nd, there a re the optimal dec oders (ML) such as the sphe re decod er and the Schnorr-Euchner algo rithm, which have a complexity that increas es drastically with the lattice dimension and the constellation size . In the other ha nd, there are the sub -optimal decoders like the ZF , ZF -DFE, ⋆ Ghaya Rekaya Ben-Othman, Rym Ouertani and Abdellatif Salah are wit h TE LECOM ParisT ech , 46 rue Barrault, 75013 Paris FRANCE. Phone: +33 1 45 8 1 76 3 3 / + 33 1 45 81 78 4 0, Fax: +33 1 45 80 40 36, Emails: rek aya, ouerta ni, salah@telecom-par istech.fr Nov ember 22, 2021 DRAFT 2 MMSE and MMSE-DFE decode rs which have a lo w complexity but poor pe rformances. In p ractice, either performance or comp lexity can be advantaged. Unfortunately , with these decoders, there is a g reat margin betwee n performanc e and complexity and no trade-off is pos sible. Sequen tial decoders are a lso us ed to de code MIMO sy stems, espe cially in [3] w here the au thors studied the Fano d ecode r and a g eneral tree sea rch framework for de coding MIMO systems was es tablished. Another kind of sequen tial dec oders exists in the literature which is the s tack d ecode r introduc ed by Zigangirov in [4]. Both the stack and the Fano algorithms adopt the s ame tree search strategy , however their implementation and complexities are quite different. In [5], the a uthors focuse d on the Fano decoder and stud ied its performances. In this pa per , we are interested in the stack one. W e propos e here an enha nced stac k decode r that we ca ll the Sphe rical-Bound Stac k decode r (SB- Stack). This one co nstraints the stac k sea rch region to a sphere but conserves the original search strategy of the stack algorithm . W e will give two versions of the SB-Stack decode r: • The hard output SB-Stack d ecode r to decod e lattice and finite constellations: we s how that the proposed decode r outperforms the c lassical MIMO ones in term of complexity wh ile keep ing ML performances. Furthermore, we s how that the SB-Stack deco der can be sub -optimal under some constraints of fering performances from ML to ZF -DFE with prop ortional comp lexities, which make it possible to get diff erent performance/co mplexity trade-off s. • The soft output SB-Stack dec oder: we sh ow that the proposed d ecode r is less complex tha n the most known so ft MIMO ones. It will be o bserved that the prop osed soft SB-Stack of fers an interesting improvement of the performances and exhibits a ga in up to 2dB . The p aper is c omposed of three parts. In the fi rst pa rt, we introdu ce the s ystem model, the n otations, an overview of tree search strategies a nd a description of the sequen tial a lgorithms. In the sec ond part, we prese nt the hard output SB-Stack decod er and its a dapted version in the ca se of finite cons tellations. This latter will be extended to a soft-output de coder in part III a nd compa red to the most known soft algorithms for MIMO systems. Nov ember 22, 2021 DRAFT 3 P A RT I : A N OV E RV I E W O F T R E E S E A R C H D E C O D I N G 1 . S Y S T E M M O D E L A N D N O TA T I O N S A. MIMO scheme with s patial multiplexing (SM) Let us con sider a MIMO system with M transmit an d N recei ve antenna s us ing spatial multiplexing scheme . The cha nnel is as sumed to be qu asi-static, and the rec eiv ed sign al is g i ven by y c N = H c N × M · x c M + w c N (1) where H c is the cha nnel transfer matrix with co mplex entries h ij representing the fading coe f ficients between the i th receiv e and the j th transmit antenna s and are modeled by ind epende nt Gauss ian random variables of zero-mean a nd variance 0.5 pe r co mponent. The c omponen ts of the information transmitted vector x c are carved in Z [ i ] o r in a q-QAM cons tellation and w c represents the i.i.d comp lex a dditiv e white Gaussian noise vector with z ero-mean a nd variance σ 2 . A rep resentation of the multi-antenna scheme by a lattice packing was proposed in [6]. This one is obtained by separating the real and imaginary parts as y =   ℜ ( y c ) ℑ ( y c )   =   ℜ ( H c ) −ℑ ( H c ) ℑ ( H c ) ℜ ( H c )   ·   ℜ ( x c ) ℑ ( x c )   +   ℜ ( w c ) ℑ ( w c )   (2) y = H · x + w (3) H is therefore the equiv a lent lattice gene rator matrix of dimens ion 2 N × 2 M . B. MIMO scheme with ST BC W e now c onsider a code d syste m using a linear sp ace time block code [7], such as the T AST co des [5] and the perfect codes [8][9]. The received s ignal matrix is then given by Y c N × T = H c N × M · C c M × T + W c N × T (4) where C c M × T represents the codeword matrix. W e consider a MIMO symmetric system, i.e , M = N , and a s quare c ode, which means that the temporal code length T is equ al to M . The la ttice rep resentation Nov ember 22, 2021 DRAFT 4 is obtaine d here by vectorization a nd sepa ration of the real and imagina ry parts of the rece i ved signal Y c N × T [8]. The equation (4) becomes therefore y c N · T =       H c N × M 0 . . . 0 H c N × M       ·       φ 11 . . . φ 1 ,M · T . . . . . . . . . φ M · T , 1 . . . φ M · T ,M · T       ·       x 1 . . . x M · T       +       w 1 . . . w N · T       W e then get an equiv a lent sys tem to (1) y c N · T = H c 1 ,N · T × M · T · φ c M · T × M · T · x c M · T + w c N · T = H ′ c N · T × M · T · x c M · T + w c N · T (5) Then, the separation of the real and imag inary pa rts is app lied on the former e quation as in (2), and the coded system is therefore gi ven by y n = H ′ n × n · x n + w n = H · x + w (6) where we define by H = H ′ n × n the equiv alent lattice gene rator matrix of dimens ion n = 2 M 2 . C . Decoding scheme Under the as sumption of a perfect k nowledge of the cha nnel s tate information a t the receiver , the ML decoding rule is giv en b y ˆ x = arg  min x ∈ Z n or x ∈ ( q − QAM ) n k y − H · x k 2  (7) T he obtained system model ca n then be decode d by either sub -optimal decoders like the ZF , ZF-DFE, MMSE dec oders [10][11], or o ptimal o nes su ch as the s phere de coder a nd the Schno rr -Euchner algorithm [12][13] which are basically tree-search algorithms. More generally , to apply tree-search algorithms, we need first to expose the tree s tructure. A QR or a Cho lesky decomposition can be then applied on the lattice gen erator matrix H . Th ese two method s are qu ite equivalent, howe ver the QR is more complex than the Cholesky d ecompo sition but it is more stable numerically [14]. For our s imulations, we then choos e to apply the QR dec omposition. Then , the Nov ember 22, 2021 DRAFT 5 system can be written as y = Q · R · x + w (8) where Q is an orthogon al matrix and R an upp er triangular one. The multiplication of both side s of (8) by the transpose of Q does not chang e the dec oding prob lem a nd we get z = Q † · y = R · x + Q † · w Exploiting the upper triangular form of R , o ne c an s olve the decoding problem us ing a tree se arch algorithm and the ML metric to be minimized is then giv en by k z − R · x k 2 / x ∈ Z n or x ∈ ( q − QAM ) n (9) Throughou t this p aper , we cons ider a tree rooted at a fic ti ve node x r oot . The no de at lev el k is de noted by the vector x ( k ) = ( x n , x n − 1 , ..., x k ) wh ere x j , j = 1 , . . . , n are the compon ents of x . Moreover , the branches of the tree at level k d efine all the possible values that ca n be ta ken by x k , and each node x ( k ) is assoc iated with the squa red distanc e f ( x ( k ) ) = n X i = k f i ( x i ) (10) where f i ( x i ) =    z i − P n j = i r i,j x j    2 . W e call f ( x ( k ) ) the co st function of the nod e x ( k ) . It rep resents the ”sub-distanc e” betwee n the received and the transmitted signal at the lev el k . The tree search co nsists in exploring the tree nodes in order to find the path ( x n , x n − 1 , ..., x 1 ) with the least cost. In the literature, we find dif fere nt tree s earch strategies. In the next sec tion, we will presen t the most known ones. 2 . A N OV E R V I E W O F T R E E S E A R C H S T R A T E G I E S A. Breadth F irst Se arc h (BrFS) The breadth first searc h algorithm is a tree search algorithm that starts from a root node x r oot and explores all its neighbo ring. Then, for e ach of the se nodes, it explores all the ir unexp lored neighbo rs, and so on un til it hits the e nd of the tree. The BrFS is then an exhaustive tree search algorithm. It moves Nov ember 22, 2021 DRAFT 6 level 0 level 4 level 1 level 3 level 2 x ro ot x 2 4 x 3 x 2 x 1 x 1 4 x (1) = ( x 4 , x 3 , x 2 , x 1 ) Fig. 1. Example of a tree structure with a dimension of n=4 from level k + 1 to the level k until it explores all the no des in the first one. The solution found is then the ML one. B. Depth F irst Searc h (DFS) Unlike the BrFS, the DFS a lgorithm s tarts from the root nod e, explores its first child nod e and proce eds by going d eeper and dee per until the end o f the tree o r until it hits a nod e that has no children. Then, the algorithm back tracks and returns to the most recen t node being expanded . W e note that this algorithm needs more memory to ’ remembe r’ which no des having been already v isited. Howe ver , since it explores all the possible paths in the tree, the DFS is an exact-ML algo rithm. C. Best F irst S earch (BeFS) One can see the BeFS as an optimization of the B rFS. In fact, the B eFS s trategy a ims to find the best path in the tree b y expanding o nly the most promising nodes chos en acc ording to some rule. In gen eral, the BeFS uses an evaluation function and selects the next nod e to expand with the be st score. In fact, starting from a given node, the algorithm ev aluates fi rst all its su ccess ors a nd selects the one to expand with the best score and so on until finding the final node. D. Branch and Bound algorithm V isiting all the tr ee node s to find the one with the shortest path, using one of the three s trategies described a bove, is p rohibiti vely co mplex. Howe ver , this co mplexity can be reduc ed u sing the Branch and Bo und algorithm (BB) wh ich come s to establish co nstraints on the tree se arch by using a bound ing Nov ember 22, 2021 DRAFT 7 function . This means that the algorithm choos es the nodes to expand by c omparing their s cores ag ainst this function. If the cos t nod e is within the define d bound ers, the nod e will be explored, e lse the nod e will be jumped, which allo ws to limit the expanding of some unnec essa ry nodes and advantages the mos t promising ones. The sphe re dec oder and the S chnorr-Euchner algorithm are both BB algorithms using a d epth-first- search s trategy . In fact, they start from the upper le vel in the tree and fi rst consider one possible value ˜ x n inside the b ounde d region and conduc ts a d epth search over the sub-trees { x /x n = ˜ x n } before going back to another sub-trees { x /x n 6 = ˜ x n } , and so on. The seq uential decode rs a lso cond ucts BB algorithms using a Be FS strategy . In the follo wing, we will focus on the most known ones, namely the Fano and the stack algorithms. A des cription of these sequen tial decod ing algorithms is therefore given. 3 . S E Q U E N T I A L D E C O D E R S The se quential decode rs were originally proposed to de code binary trellis c odes[15]. The mos t used one is the Fano decode r introduce d in 1963[16]. Later , Zigang irov prop osed in 19 66 a sequential algorithm using a stack storage (or me mory). In the 19 60s, memory allocation represented an add itional problem, that’ s why the Fano dec oder was more s uitable for hardware impleme ntation and so far the stac k decoder has no t b een widely u sed. Nowadays, the price o f me mories is continually dropp ing and the stack dec oder is therefore being of great interest. In [3], the authors have rediscovered the Fano d ecoder and applied it to d ecode MIMO scheme s. In this work, we will focus on the stack deco der a nd bring the necess ary mo difications to decode lattice and finite constellations. Be fore presenting the mo dified stack decoder , we recall the principal of the original Fano and stac k ones. A. F ano decode r W e will detail the sea rch strategy of the Fano decoder . Let us su ppose that the de coder is at a some node x k of a level k in the tree. The decod er ca n cho ose between 3 op tions: proce ed forward to the next node at lev el k − 1 , move back to a pred ecess or n ode x k +1 , or move laterally to a neighbor nod e at the same level k . W e call f look and f pr e the costs of respec ti vely the succe ssor and the pred ecess or node. At eac h step, the dec oder looks forward to the bes t next node . The b est node is the one having the Nov ember 22, 2021 DRAFT 8 least metric. The deco der can visit a node if its metric is smaller than a certain threshold Υ . Each time a new nod e is vis ited, Υ is lo wered by a step-size ∆ . If the succes sor’ s cost is larger than the current threshold, the deco der looks a t the predecess or no de. If f pr e ≤ Υ , the decoder moves backward to this node and looks forward to its next best succ essor if it exists. Otherwise, that mea ns that all the node s which costs a re s maller than the threshold are explored. The dec oder inc reases then Υ , tries to move forward to the n ext no de, and so on. The algorithm terminates when a leaf node is reache d. W e no te that, from a cu rrent node , the Fano algorithm moves either to its predeces sor or to its succ esso r , but never jumps. However , since the dec oder requires no memory , it may return to the s ame node several times which includes additional computations neede d to decode a given sequen ce. B. Stack de coder For the stack algorithm, the search principle and strategy remain the same. Howe ver , the main dif ference is that the stack stores a ll the paths crosse d by the algorithm in an o rdered list called ”stac k” or ”memory”, whereas the Fano only retains the bes t path. In fact, starting from a root node x r oot , the stack algorithm generates all the child node s of x r oot . W e call a ch ild n ode the s ucces sor node . Th e algorithm c omputes then the res pective c osts of thos e nodes according to the c ost function in e quation (10 ) and stores them in an increasing order in the memory , s o that the top n ode of the stack is the one having the lea st c ost. Afterwards, the a lgorithm gene rates the children o f the top , computes their cos ts, plac es them in the memory and removes the top n ode being just extended. The algorithm reorders again the stack, generates the child nodes of the current top node, and so on. The algorithm terminates when a path of len gth n is found on the top of the stack, in other words, whe n a leaf n ode reache s the top of the list. The flowchart of the stack a lgorithm is presented in F ig.2, wh ere we de fine by Gen ( x ( k ) ) the function that generates all the children no des of x ( k ) and by S or t () the function that reorders the s tored no des in the stack. Note that, since the s tack de coder requires a memory , it visits fewer node s than the Fano which may return to a node having already b een visited. Con sequen tly , the stack dec oder is much faster a nd so les s complex than the Fan o. In the s equel, we will propose a modified stack dec oder tha t is less complex than the original one. Nov ember 22, 2021 DRAFT 9 No Yes Exit I N P U T : y, H z = Q T · y [ Q, R ] = QR ( H ) List =[] Sort( List ) Put x ro ot in List Remov e x ( k ) Store ( x 1 k − 1 , . . . , x j k − 1 ) in List x ( k ) =top( List ) [ x 1 k − 1 , . . . , x j k − 1 ]=Gen( x ( k ) ) x ( k ) =leaf node? Return x ( k ) Fig. 2. Flowch art of the stack decoder P A RT I I : H A R D D E C O D I N G U S I N G T H E S TAC K A L G O R I T H M 1 . L AT T I C E D E C O D I N G Stack dec oding was originally designed to dec ode binary trellis code s, wh ere the codeword is taken in a finite alphab et. Howev er , co nsidering a lattice, the codeword is taken in the infinite field Z n which leads to an infinite tree structure. Ap plying the stac k de coder se ems to be impossible in this cas e. Our purpose is then to propose a modified version of the s tack a lgorithm in order to decode lattice. A. 1 st appr oach T o app ly the stack d ecode r , we should sea rch for the closes t po int in a finite region Λ ⊂ Z n . Unfortu- nately , the trunca tion of the tree will af fect the pe rformances. In fact, if the transmitted co dew ord be longs to Λ , the decoder will systema tically find it, howev er an error is occurred when the codeword is out of the search region. Th e main c hallenge is then how to choos e the optimal region Λ . Nov ember 22, 2021 DRAFT 10 Y et, the triangular form of the lattice bas is reminds us the Schnorr Eu chner enumeration strategy[17]. The key idea of this algorithm is to view the lattice as a sup erposition of n hy perplanes and to start the search by projecting the received vector o n the nearest hyperplane. The resu lting point is then recursiv ely projected on the following n − 1 hyperplanes . Th e po int fou nd is the ”ba bai po int” and it corresp onds to the ZF-DFE p oint[6]. Similarly to the S chnorr Euchner , the proposed search a lgorithm is based o n the babai point, howe ver the se arch strategy and the con struction of the tree a re q uite dif ferent. In fact, the Schn orr Euchner cons ists in enumerating all the possible node s ins ide a bou nded region by zigzagging around the babai point using a DFS strategy . W e propose here an algorithm that constructs a tree ce ntered at the babai p oint u . At eac h lev el, it enumerate s the neighbor lattice points define d as u ± t = ( u 1 ± t 1 , u 2 ± t 2 , ..., u n ± t n ) where t is a vector in Z n , a Be FS strategy is further applied on this tree. Applying this a lgorithm, we ca n delimit the s ize of the cons tructed tree by cho osing the number of the neighboring lattice nodes of the babai point that we would c onsider . Ho we ver , the ML p oint is not guaranteed to be included in the cons idered tree. T o reac h it, we sh ould e nlarge the search region. Meanwhile, that implies to have a d enser tree which leads to a more complex dec oding tas k. In Fig.3, we plot the symbol e rror rate a s a fun ction of the signal to noise ratio (SNR) gi ven in dB scale, for a 4 × 4 MIMO system us ing a SM. First, we p roceed by cons idering the search region defined as Λ a = { u i − 1 , u i , u i + 1 , i = 1 , . . . , n } . T his induces that the lattice points con cerned with the se arch algorithm are on ly the immediate neighbors of the babai po int. Nev ertheless, in bad c hannel conditions, the ML po int may b e far -off and the n unreachab le in this case. This is s hown in the curve (a), where the pe rformances are sub-op timal and exhibit a loss o f 2 dB from ML. For the same system, we have prog ressively enlarged the se arch region and observed the algorithm’ s behavior . T he curves (a)-(d) report the pe rformances obtained by respectiv ely considering the s earch regions Λ a = { u i − 1 , u i , u i + 1 } , Λ b = Λ a S { u i − 2 , u i + 2 } , Λ c = Λ b S { u i − 3 , u i + 3 } and Λ d = Λ c S { u i − 4 , u i + 4 } . As sh own in Fig.3, the decod er p rovides sub-optimal p erformances, but it a pproache s the ML as well a s we cover a lar ger se arch region. Howev er , the complexity increases with the performances . Th erefore, a compromise may be do ne and this decod ing a lgorithm can be of great interest. In fact, at the start of the algorithm, one only need s to choo se the p erformance-complexity trade-off to reach, to defin e the appropriate se arch region. Nov ember 22, 2021 DRAFT 11 In simulations, w e have c onsidered a un iform vector t . One can think to u se a vector t with large t i for first co mponents and small t i for the las t o nes. Th is ch oice ma y be efficient due to the problem of error propagation in the tree search. 9 10 11 12 13 14 15 16 17 18 10 −3 10 −2 10 −1 SNR (dB) Symbol Error Rate Stack, (a) Stack, (b) Stack, (c) Stack, (d) ML Fig. 3. Performances of a MIMO system using a SM with M = N = 4 , obtained for differe nt search regio ns B. 2 nd appr oach (SB-Stack decod er) In the 1 st approach , the sea rch region is cen tered on the b abai point. Howe ver , this latter is generally a rough estimation of the transmitted codeword, then centering the search region on it is n ot the b est way to do sinc e the ML solution may not be reached, as shown in Fig.4. Therefore, we propose here a secon d approach for the lattice de coding inspired by the sphere de coder . The principle of this latter is to enumerate all the lattice points found in a s phere of a radius C centered on the received po int. Ea ch time a point is found, the radius is upd ated, which limits the nu mber of the enumerated points but also ens ures the c losest point criterion. T he sph ere deco der uses the DFS strategy . Like the sphere d ecode r , the SB-Stac k algorithm only explores the lattice points inside the sp here with the radius C using the Be FS strategy , which induc es to the d efinition of an u pper and a lower bounds for each lattice point comp onent (the tree nod es). Starting from the root node, the algorithm then computes the up per and lower bou nds of the first c omponen t x n , respe ctiv ely deno ted b inf ,n and b sup,n and g enerates a ll the nodes within these bound s, which rep resent the branche s of the tree at lev el n . Nov ember 22, 2021 DRAFT 12 region search ML point u + t Babai p oin t u Fig. 4. Example of a latti ce defined in Z 2 , the search region does not contain the ML point The algorithm then computes the cost of each node x n and stores them with their respective c osts in the stack memory . After that, it reo rders the nod es in the memory in a n increasing order acco rding to their costs, s elects the top node, then computes the bo unds o f the next level ( n − 1) us ing the value o f the top node x n , g enerates all its poss ible children an d stores them in the memory after removing the top node. W e call ch ildren of x n all the n odes x n − 1 within the b ounds b inf ,n − 1 and b sup,n − 1 . This proc edure is repeated until a leaf node reache s the top of the memory . Note that, although the a pparent similarity between the traditional sphere dec oder and the SB-Stack algorithm, these two s earch a lgorithms rais e great d if ferences. In this way , unlike the sp here d ecode r the radius in the SB-Stack dec oder rema ins u nchang ed during all the deco ding p rocess, while for the sph ere decode r the radius is being upd ated each time a point is found. In fact, the sphere dec oder proc eeds by first searching one candida te solution by pe rforming a DFS. Th is candida te is then an e stimation o f the closest solution, which means tha t, the ML point is at lea st from this distance to the received point, an d it is then unnece ssary to look for candidates beyond this metric. Cons equen tly , the s phere ra dius ca n be reduced to this new distance . Howe ver , the search s trategy for the stack algorithm is dif ferent. As desc ribed ab ove, at each step the algorithm ma y backtrack to a higher n ode before rea ching the end of the tree which correspond s to a candidate solution. Thus, there is no po ssibility to evaluate the ML distance in prog ress of the algorithm, and the radius must therefore be fixed. For mo re clarity , we will now detail the bounds calculation which is the s ame as those of the sphere decode r in [12]. Nov ember 22, 2021 DRAFT 13 Bounds calculation: First, remind that the distance to minimize is gi ven by k z − R · x k 2 . Let us write z = R · ρ , whe re ρ is the ZF point. It represents the co ordinate of the vec tor z in the new lattice generated by R . The euclidean distanc e is now written in the lattice system as: k R · ( ρ − x ) k 2 = k R · ξ k 2 , where ξ defines the coordinate of the translated point x . In this cas e, the lattice points cons idered in the metric minimization are those within the distan ce C from the receiv ed po int z , w e then have k R · ξ k 2 = n X i =1  P n j = i r ij ξ j  2 ≤ C 2 (11) Let now de fine by: q 1 ii = r 2 ii for i = 1 , . . . , n , and q 1 ij = r ij r ii , i = 1 , . . . , n for i = 1 , . . . , n , and j = i + 1 , . . . , n . The equation (11) is rewritt en as k R · ξ k 2 = n X i =1 q 1 ii  ξ i + P n j = i +1 q 1 ij ξ j  2 ≤ C 2 (12) By working b ackward, we defi ne the b ounds at any level i by b inf ,i = & − s T i q 1 ii + S i ' ≤ x i ≤ $ s T i q 1 ii + S i % = b sup,i (13) where we refer to T i and S i by T i − 1 = C 2 − n X l = i q 1 ll ξ l + n X l = i q 1 lj ξ j ! 2 S i = ρ i + n X l = i +1 q 1 il ξ l (14) The flowchart of the SB-Stack decod er is given in Fig.5. Note that the bounds of a gi ven node x i are depe ndent of thos e of the former nod e x i +1 as shown in the equation (14). Therefore, the SB-Stack algorithm also stores the bounds of each generated node, that will be use d to c ompute its children n odes when this latter should be extended. C onsequ ently , the SB-Stack stores more information than the clas sical stack algorithm an d the price to pa y is an inc reasing memory size. But the search region limitation allo ws to visit fewer nod es and so to c on verge more quickly an d ensures to obtain the ML solution. Nov ember 22, 2021 DRAFT 14 Yes No No b inf,i = l − q T i q 1 ii + S i m − 1 b sup,i = j q T i q 1 ii + S i k [ Q, R ] = Q R ( H ) z = Q T · y Sort( List ) b inf,i = b inf,i + 1 , i = 1 , . . . , n , j = i + 1 , . . . , n R − 1 = inv ( R ) ρ = R − 1 · y E X I T i = leng th ( x ) T i = T pr e − q 1 i +1 ,i +1 ( S pr e − x i +1 ) 2 S i = ρ i + P n j = i +1 q 1 i,j ( ρ j − x j ) q 1 i,i = r 2 i,i q 1 i,j = r i,j r i,i C + + List empt y ? b inf,i < b sup,i List T = [ List T ; T i ] List S = [ List S ; S i ] List = [ List ; x c ] x c = [ b inf,i x ] x =T op(List) T pr e = List T ( k ) S pr e = List S ( k ) Retur n x T i = C i = n S i = ρ List = [] List T = [] , List S = [] x = [] x = leaf no de I N P U T : y, H, C k as x = List ( k , :) Fig. 5. Flowch art of the Spherical-Bound-Stack decoder Nov ember 22, 2021 DRAFT 15 Once the bo unds are comp uted, the algorithm gene rates all the node s corresponding to the dif ferent values of x i . If there is no valid value o f x i (i.e. b inf ,i > b sup,i ), that means that x i +1 has no children and thus this intermedia te n ode h as no chance to y ield to a leaf no de. It will be then removed from the stack list. Howe ver if the stac k is e mpty , that mean s tha t n o lattice point was foun d ins ide the s phere. In this case, the sphere radius must be increase d and the algorithm is then restarted. From this observation, it is clear tha t the complexity of the algorithm depe nds a lso on the sphere radius. In fact, if C is too large, we obtain too ma ny points, and so a lar ge tree search. Bu t if C is too small, we obtain no points. Fu rthermore, for small SNR, the rec eiv ed s ignal is much affected by the noise and a large radius is neede d, while for h igh SNR, the ML point is c lose to the received s ignal and a sma ll radius is sufficient. In [18], a formula to choose the optimal radius as a function of the SNR was first reported by Hassibi and al. as C 2 = 2 .n.σ 2 (15) In the other ha nd, the lattice ma trix H is the produ ct of the fading matrix H 1 and the unitary code generator matrix φ (equation (5)). Consequ ently , in pres ence o f a deep fading the lattice can b e much distorted an d may be more stretch ed from some axes tha n others . It is then more s uitable to compute the sph ere radius according to the fading too. The refore, a formula taken into accoun t both the noise variance and the fading matrix was propos ed in [19 ] C 2 = min (2 .n.σ 2 , min( diag ( H T · H ))) (16) As bo th the sph ere and the SB-Stac k deco ders are ML, we will foc us our co mparison on the c omplexity which we count as t he total number of multipl ications of the search phase. In Fig.6, we pl ot the complexity as a function of the SNR for a 2 × 2 a nd a 4 × 4 MIMO systems using SM. W e only consider the complexity of the search phase as the same pre-process ing pha se (QR de compos ition) is made for both de coders. W e can see that the SB-Stack of fers a cons iderable complexity reduction for dif ferent lattice dimensions, which is about 40% than the sphe re decoder for the 2 × 2 sys tem and 50% for the 4 × 4 one at lo w SNR. This important c omplexity reduc tion is due to the search strategy of the SB-Stack decode r which allows to look inside the sp here only for the most p romised points, unlike the s phere dec oder which checks a ll the lattice points inside the sphere . Nov ember 22, 2021 DRAFT 16 0 2 4 6 8 10 12 14 16 18 0 500 1000 1500 2000 2500 3000 3500 4000 SNR (dB) Complexity SD 4x4 SB−Stack 4x4 SD 2x2 SB−Stack 2x2 Fig. 6. Performance and complexity of the SB- Stack decod er for a 2 × 2 and a 4 × 4 MIMO systems using a SM In p ractical transmission schemes , we do n ot con sider information s ymbols in Z n but in finite c onstel- lations. The most used one s for MIMO sche mes are the QAM co nstellations. W e propose in the sequel to modify the stack and the SB-stack decode rs to take into accoun t the finite QAM constellations. 2 . D E C O D I N G FI N I T E C O N S T E L L A T I O N S In this section, we will focus on the decoding of fi nite co nstellations using the stack de coder . As in sec tion II.1, we propose two a pproache s: the first ap proach is largely inspired by the original stac k decode r , while the second one is a readjus tment o f the proposed SB-Stack a lgorithm des cribed ab ove. A. 1 st appr oach Using finite constellations tasks the decoding problem in its original context whe re the stac k decoder was a pplied to decode binary co des. In o ur cas e, the tree is n o longe r bina ry even though it remains finite. As a first and natural approach, we prop ose to use the original stack decode r , but instead of the binary values of the tree no des, we consider the correspond ent interval of the q − QAM con stellation. For example, for a 16 − QAM c onstellation, each tree node be longs to the set I c = { ± 1 , ± 3 } , then we have a 4 − ar y tree . More generally , for a q − QAM , the node s to consider are in  ± 1 , ± 3 , . . . , ± √ q − 1  . Consequ ently , the tree s tructure is directly linked to the used c onstellation, an d for lar ge s izes, the information set to which b elong the s ymbols is too large which lea ds to a n excessively dense tree and Nov ember 22, 2021 DRAFT 17 6 8 10 12 14 16 18 20 22 24 26 0 2 4 6 8 10 12 14 x 10 4 SNR (dB) Complexity Stack, 64−QAM SD, 64−QAM Stack, 16−QAM SD, 16−QAM Fig. 7. Comparison of t he complexity of the stack decoder and the sphere decoder using dif f erent Q AM constellations, for a MIMO system with SM, M = N = 4 thus to a very complex decod ing. T o illustrate this, we repres ent in Fig.7 the complexity of the stac k decode r for 16 and 64 − QAM constellations as a function of SN R for a 4 × 4 MIMO s ystem us ing SM. As we can see, the complexity increases as the co nstellation size increases . W e also note that, for large sizes, the stack deco der is much more comp lex tha n the sphere dec oder . In fact, for each level the stack decode r generates all the pos sible no des of the constellation, while the sphe re dec oder only selec ts the closer one s. Moreover , this complexity is e spec ially high for lo w SNR where the decoder crosse s more nodes to reach the optimal solution. The comp lexity therefore comes from the high n umber of v isited no des. In [20], a tree s earch a lgorithm was propo sed. Th is one pe rforms a stack algorithm b ut it limits the s ize o f the stack so tha t it only retains K nod es at ea ch level of the tree. Nevertheless, the nod es se lected in the s tack may not lead to the sh ortest path but those corresponding to the smalles t metric may have been disca rded in higher lev els in the tree. Consequ ently , the ML solution is not reac hed. So, this a lgorithm allows to reduce the complexity by limiting the number of the generated nodes but at the price of a pe rformance loss . W e prop ose in the seq uel to us e the SB-Stack dec oder described in section II.1. W e then gi ve in the follo wing the necess ary modifications to ada pt it to dec ode fi nite c onstellations. Nov ember 22, 2021 DRAFT 18 6 8 10 12 14 16 18 20 22 24 26 28 1200 1400 1600 1800 2000 2200 2400 SNR (dB) Complexity SD SB−Stack Fig. 8. Complexity of the SB-S tack decoder vs the sphere decode r for a 4 × 4 MIMO system with SM, using a 16 − QA M constellation B. 2 nd appr oach using the SB-Stack algorithm The SB-Stack algorithm as presen ted above is expec ted to decode lattice. For information sy m- bols taken in a q − QAM constellation, each c omponen t of x be longs to the fin ite interval I c =  ± 1 , ± 3 , . . . , ± √ q − 1  ⊂ Z n . The no des conce rned with the search algorithm inside the sphere a re only the o nes that be long to the cons tellation and so thos e which are within I c . Furthermore, in orde r to restrict the sea rch to a se t o f Z n , we conside r the transformation u i = ( x i + √ q − 1) / 2 . The n ew bounde rs o f the constellation are then given by I c, Z =  0 , 1 , 2 , . . . , √ q − 1  . Cons equently , the nod es that we look for are taken in the interval  sup ( b inf ,i , 0) , inf  b sup,i , √ q − 1  instead o f the interval [ b inf ,i , b sup,i ] compu ted in (13). In Fig.8, we plot the complexity of this mo dified SB-Stack dec oder for a 4 × 4 MIMO syste m us ing a 16 − QAM co nstellation. As we can s ee, this latter exhibits a complexity reduction of at least 30% compared to the sphere decode r while maintaining the ML performanc es. C. Sub-optimal SB-Stack dec oder The SB-Stack algorithm propose d above s earches for the shortest p ath ( x n , x n − 1 , . . . , x 1 ) that mini- mizes the metric f ( x ( n ) ) . T he s olution to this prob lem is then the ML one. Howev er , by introducing a Nov ember 22, 2021 DRAFT 19 parameter in the cost function as defined in [3], we can re write (10) as f ( x ( k ) ) = n X i = n − k +1 f i ( x i ) − b · k (17) where b ∈ R + is called the bias. Ea ch path found in the tree is then weighted with a negative value − b · k . Und er this co nstraint, the algorithm a dvantages the paths of lar ge r lengths since the smalles t metrics correspon d to the deepes t p aths in the tree. Hence, this parametrized version allo ws the SB-Stack decode r to visit less nod es. C onsequ ently the complexity is re duced a nd the d ecode r c on verges more rapidly , h owe ver the so lution is n ot gu aranteed to be ML but depe nds on the value of b . In Fig.9.a, we plot the performances define d as the symbo l error rate as a function of the SNR, for a 4 × 4 MIMO sy stem for dif ferent values o f the bias. In Fig.9.b, we plot the correspon dent c omplexities. W e show that the performance decreases as b increas es, an d for a high value of b , it approach es the ZF-DFE. In fact, since the de coding rule is no longe r the minimization o f the euc lidean distance as in (10), the pe rformances obta ined a re not o ptimal. Th ough, at small values of b , the second term − b · k is negligible in equation (17 ), the c ost func tion is a pproximately e qual to (10) an d then nea r -ML performances are achieved. The co mplexity howev er d ecrease s continuou sly with b . This p arametrized version of the SB-Stack de coder is therefore very interesting since it allows to obtain many ran ges o f performances with reduced complexities by only cha nging the value of the bias. P A RT I I I : S O F T D E C O D I N G U S I N G T H E S T A C K A L G O R I T H M In p art II, lattice decoding for MIMO schemes is c onsidered and a less co mplex sta ck dec oder is proposed . This one is prese nted in the case of hard ou tputs. However , wh en a channe l coding is integrated to the trans mission sch eme, the s oft dec oding become s necess ary . I n this part, soft o utput de tection is then in vestigated and we s how that the propose d SB-Stack dec oder c an be extended to supp ort so ft-outputs. 1 . O V E R V I E W O F S O F T D E C O D I N G In the literature, so ft-output MIMO decoding was s tudied. Some solutions to this issue ha ve been proposed in [21],[22] and the so c alled ’li st’ or ’candidate list’ was introduced. The most known soft- output lattice decod er for MIMO sys tems is the List Sphe re De coder (LS D). W e are interested here in the propose d stac k d ecode r with spherical b ounds , which off ers a n interes ting structure and a great advantage to provide a se lected list. Nov ember 22, 2021 DRAFT 20 8 12 16 20 24 28 32 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR (dB) Symbol Error Rate ZF−DFE SB−Stack, b=10 SB−Stack, b=1 SB−Stack, b=0.5 SB−Stack, b=0.1 SB−Stack, b=0.05 SB−Stack, b=0.02 SB−Stack, b=0.01 SB−Stack, b=0 (ML) (a) 8 12 16 20 24 28 32 205 210 215 220 225 230 SNR (dB) Complexity SB−Stack, b=10 SB−Stack, b=1 SB−Stack, b=0.5 SB−Stack, b=0.1 SB−Stack, b=0.05 SB−Stack, b=0.02 SB−Stack, b=0.01 SB−Stack, b=0 (ML) (b) Fig. 9. Performance and complexity of t he SB-Stack decoder for a 2 × 2 MIMO system w ith SM using a 16 − QAM constellation, for different v alues of the bias Soft d ecoding can be realized us ing a pos teriori probab ility technique s. A p osteriori probab ility (APP) techniques are a jud icious c hoice for high pe rformance rec eiv ers with reas onable c omplexity . Max imizing the APP for a giv en bit minimizes the prob ability of making an error on that bit. The A PP is usually expressed a s a log-likelihood ratio (LLR) value. A de cision is mad e from a LLR value by using its sign to tell whethe r the bit is one or zero. The magnitude of the LLR value ind icates the reliability of the decision. LLR values ne ar z ero corres pond to unreliable bits. In the following, the logical zero for a bit Nov ember 22, 2021 DRAFT 21 is rep resented by amplitude level b k = − 1 and logica l one by b k = +1 . The modu lator map s each lay er of the bits into data symbols through the mapping f : {− 1 , + 1 } 1 × B → C where C deno tes the d ata s ymbol cons tellation and B = log 2 | C | is the n umber o f bits represen ted by each data symbol. Let’ s K denotes the number of symbols belonging to eac h c odeword transmitted in each channe l use. The LLR of the i th bit, where i ∈ [1 , B K ] , is defi ned a s LLR ( b i ) = log Pr ( b i = +1 / y , H ) Pr ( b i = − 1 / y , H ) (18) One c an as sume eq ual probab ility for e ach da ta bits (an interleav er at the e ncode r can be used to scra mble the bits). Using Bayes theorem, the bit metric can be written as LLR ( b i ) = log P b ∈D i, +1 Pr ( y / b , H ) P b ∈D i, − 1 Pr ( y / b , H ) . (19) where D i, +1 and D i, − 1 are the s et of 2 B K − 1 bit vectors b with b i being +1 and − 1 . If we a ssume a n additiv e zero me an white circularly symmetric complex Gauss ian noise, the equation (19) can be written as LLR ( b i ) = log P b ∈D i, +1 e − 1 σ 2 k y − H · x ( b ) k 2 P b ∈D i, − 1 e − 1 σ 2 k y − H · x ( b ) k 2 . (20) In orde r to reduce the corresponding computational comp lexity , on e can employ the max-log approxi- mation [23] to g et LLR ( b i ) ≈ max b ∈D i, +1  − 1 σ 2 k y − H · x ( b ) k 2  − max b ∈D i, − 1  − 1 σ 2 k y − H · x ( b ) k 2  = 1 σ 2  min b ∈D i, − 1 k y − H · x ( b ) k 2 − min b ∈D i, +1 k y − H · x ( b ) k 2  (21) Soft-output d etection on MIMO channels ca n be achieved via an exhaustive list as in [24] or a limited size list of spherica l shape as in [21] and [22]. The APP d etector ba sed on an exha ustiv e has a relatively large complexity exponen tial in the number of transmit antennas and the number of bits per modulated symbol. In other hand, a non-exhaustive list APP detector is sub-optimal but has a low complexity which is proportional to the list s ize. Several list Nov ember 22, 2021 DRAFT 22 decode rs were already proposed. W e reca ll in the follo wing the most kn own ones a nd we propo se a new soft-decoder based on the SB-Stack. A. List Sphere De coder (LSD) An exhaus ti ve search need s to examine all the constellation points. the sphe re decoder a voids a n exhaustive search by examining only the points that li e inside a sphe re with a gi ven radius C . The performance of the algo rithm is closely tied to the choice of the initial radius C . If C is chos en too small, the algorithm cou ld fail to find a ny point inside the sph ere, requiring that C be increa sed. Howe ver , the larger C is chos en, the lar ger the s earch will spend time. In [22], a simple modification to the sphe re decode r w a s introduced . The sphere decoder gene rates a li st L of N p points. These points make || y − Hx || 2 smallest among all the points inside the sphere. The list, b y definition, must include the ML point. T o create L , the sp here dec oder nee ds to be modified in two ways : when a candida te is found inside the sphere, the radius C sho uld not be red uced. In addition, the can didate is added to the list if one of the follo wing con ditions is satisfied : either the list is not full or a t lea st one ca ndidate in the list h as a higher c ost tha n the new ca ndidate. In this last case, the new ca ndidate replac es the one having the lar ge euclidean d istance to the received point. Thus, the c onstructed list c ontains the ML point and N p − 1 neighbors for which the square error is smallest. T he soft information abo ut any given bit b k is es sentially contained in L : if there a re more entries in L with b k = 1 than those with b k = − 1 , the n it can be conclude d that the likely value for b k is +1 , wherea s if there are fewer entries in L with b k = 1 , then the likely value is − 1 . A larger radius C gen erally allows for a larger N p which ma kes the list more reliable. There is also a trade-off between the ac curacy and the d ecoding delay of the LSD. Find ing N p points is gen erally slower than just finding one point, be cause the s earch radius a lw ays stay s fixed and does not decreas e with each found po int. One problem of this algorithm is the variable s ize of the list. In[22], a radius, function of the desired number of points, is propose d. The deco ding error can be written as k y − H · x k 2 = k w k 2 σ 2 χ 2 2 N p , (22) where χ 2 2 N p is a chi-square random variable with 2 N p degrees of freedom. The expected value of this Nov ember 22, 2021 DRAFT 23 Sphere centered on ML point ML point Received point Sphere centered on received point Fig. 10. Sphere centered on the ML point and the sphere centered on the receiv ed point random variable is σ 2 E [ χ 2 2 N p ] = 2 σ 2 N p . One possible choice of the radius is C 2 = 2 σ 2 ζ N p − y †  I − H  H † H  − 1 H †  y (23) where ζ > 1 is chose n so that on e ca n be reaso nably s ure, a s measured by a confid ence interval for the χ 2 2 N p random variable, that the true transmitted x will be found. The impo rtant we ak point in the LSD is the instab ility of the list size. The numbe r of visited points before rea ching the ML point can not be fixed exactly , on ly an app roximate n umber c an be provided. T he sphere rad ius is selected to give n early the nee ded number . Moreover , the cons tructed list is not cen tered at the ML point. A Shifted Spherical List Decod er was proposed in [21] to resolve this problem. B. Shifted Spherica l Decode r (SSD) The APP detector starts by a pplying a sp here d ecode r to find the ML point, then a spherica l list centered around the ML point is built. This list dep ends o n the ML p oint p osition an d the channe l state. The trick behind this idea is to c enter the spherical list L on the ML point instead of the ZF point. F ig.10 shows in two dimens ional lattice the s phere centered o n the ML point compared to the o ne c entered o n the ZF point. Usually the rec eiv ed po int y is outside the cons tellation, specially when c onsidering lar ge dimen sions n . Th e sphe re deco der centered on the received point visits a lot o f lattice points to fin d a sma ll nu mber of constellation points. Howe ver , wh en the sphe re is cen tered on the ML point, the number of enumerated points is reduce d a nd higher likelihood constellation points are considered. But to guarantee a high stability for the numbe r o f po ints required in the list, one s hould be careful for the choice of the shifted list radius. This rad ius should take into acco unt the nu mber of p oints to c reate the list. In [22], an Nov ember 22, 2021 DRAFT 24 approximation is ma de: the volume of the sphe re con taining N p points is equal to the volume of N p fundamental parallelotopes. As a result, the radius C can be comp uted as C ≈  N × v ol (Λ) V  1 n , (24) where v ol (Λ) = | det ( H ) | , H is the lattice generator matrix, n is the dimension o f H and V is the u nit radius sphe re volume in the real space R n , V = π M M . This me thod has the dis advantage of be ing stable only for high values of N p . If we a ssume N 0 the eff ectiv e number of points found inside the list L , one can check that lim N p →∞ N 0 N p = 1 (25) But when con sidering a fin ite constellation, N 0 will dec rease beca use o f the limited sha pe of the intersection b etween the sphere an d the constellation. T his de pends on the ML point position inside the constellation an d the s hape of this constellation. As a result, the radius C of the shifted spherical list for the constellation can be giv en by C =  α [ n hy p ] × µ γ × N p × v ol (Λ) V  1 n , (26) α is a n expans ion f actor of the list s ize which de pends on the number of hyperplane s n hy p at the constellation boundaries pa ssing through the ML point. µ γ is an additional expansion factor depending on the shape of the constellation [21]. 2 . S O F T D E C O D I N G U S I N G T H E S T A C K D E C O D E R W I T H S P H E R I C A L B O U N D ( S O F T S B - S T A C K ) A. Principle W e p ropose h ere a n extens ion of the classica l stack de coder an d the new proposed SB-Stac k de coder to get soft information outputs. W e have modifie d this a lgorithm to generate soft-output information in the form of LL R. Stac k de coders hav e the capability of g enerating a can didate list in their original algo rithm. In ea ch iteration, children nod es are g enerated and stored in the stac k ordered in function of their costs. At the en d of the algorithm, the fi rst le af node rea ching the top of the stack is the ML point. In this work, we improve the SB-Stack algorithm to make it suitable for a soft output by cons tructing a list inste ad o f only the ML point. In fact, after the end of the proce ss, o ne can remark that stack is still full of nodes with dif ferent sizes (with d if ferent le vels in the tree) and no one among them is reach ing the top of the Nov ember 22, 2021 DRAFT 25 stack. The mo st straightforward idea is to extract the ML p oint from the original stack , put it in another stack and continue the se arching ph ase. T he n ext node reach ing the top o f the s tack is also removed an d putted in the secon d stack with its corresp onding cos t. Th ere is two po ssibilities to s top the algorithm: • either we fix the number of points in the list (the size of the s econd stack). In this c ase the algorithm continues in this manner until the secon d stack will be full. • anothe r pos sible criterion is to fix a lower bound on the node costs (worst c ost to be admissible), and wh en the cost falls b elow this limit value, the algorithm giv es up. Thus, on ly the no des stored in the sec ond stack will contrib ute to the soft de cision. This n ew Soft SB- Stack d ecoder is a n extens ion of the first one and aims a t generating more leaf no des. The s econd stack is used later to generate the LLR. The main advantages of this algorithm are • the stab ility: the algo rithm will stop as soon as the n umber of candidates is reach ed. Th e issue regarding the computa tion or es timation of the ide al ra dius value is removed. • the list is centered at the ML point. In other words, the list is filled up with the clos est points only in an ascending c ost order , lead ing to a n optimal LLR computation for a gi ven list size . • a lo w comp lexity since we only pursue the stac k algorithm with no a dditional s earch method and exploit the node s being alread y computed a nd still in the stack. The disadvantage of all the previous soft decod ers is their inability to provide soft outpu ts with low complexity , and the worst c ase corresponds to the exhaustive search. The Soft SB-Stack de coder provides less complexity than these latters. Moreover , we can apply the bias pa rameter as in the eq uation (17). This leads to a trade-off of p erformance-complexity with soft outputs, wh ich was never done before. In the ca se of a non-null bias, the ML point is reach ed later and belongs to the list. One can even impo se a n aggregate run time con straint. A straightforward conclusion is the flexibility of the new Soft S B-Stack dec oder with practical con- straints that system engineers can be faced with the d esign of the receiv er . B. Simulations and r esults In this part, we illustrate the application of the new Soft SB-Stack dec oder for MIMO sp ace time transmission. The bina ry information is enco ded us ing a rate R-con volutional code. The code d bits are fed to a q-QAM mapp er (Gray mapping) that generates symbols. The sp ectral efficiency in the V -BLAST case is R × B × M bits per cha nnel us e, B = log 2 ( q ) . Nov ember 22, 2021 DRAFT 26 −10 −8 −6 −4 −2 0 2 4 6 8 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 LLR Density Distribution 6.2 10.4 SNR =0 dB SNR = 3 dB Fig. 11. LLR Density Distri bution for SNR=0 dB and SNR=3 dB Conv. Encoder Modulator Space − Time Encoder Soft Space − Time Decoder Demodulator Viterbi Algorithm Data Data Source Sink Fig. 12. Diagram of transmitter and recei ver wit h soft space time decoding Fig.11 shows the LLR distribution of the c andidates found inside the s tack for S N R = 0 dB and S N R = 3 dB . It can b e observed that when SN R increases, the L LR distrib ution c urve is g oing to g et a concave shap e with a c avity aroun d zero. This can be expec ted since z er o − LLR me ans amb iguity in the decision which is diminished when SNR increases . For high SNR, the LLR v alues stretch to infinity and in prac tice they are saturated to a high chosen value. The LLR distrib ution curves provide us with information abo ut the intervals to which LLR belong. Nov ember 22, 2021 DRAFT 27 LLR will be s ampled into 2 m − 1 levels o f their interval distribution a nd then qua ntized to m bits to serve as input for the so ft V iterbi dec oder . Fig.12 shows the diag ram of the s imulated transmitter an d rece i ver . W e con sider the deco ding of 200 information bits per frame, a rate 1 2 -con volutional code modu lated by a 4 − QAM constellation. The generator G = (7 , 5) (in o ctal notation) has the memory of T = 3 . T wo transmitting and two rece i ving antennas are use d and symbols are sp atially multiplex ed. In the reception side, for soft input V iterbi decode r , LLR are provided by the space time decode r using a list of candida tes. In F ig.13.a, we plot the BER (Bit Error Rate) as a func tion of E b / N 0 under a Ray leigh flat fading channe l. Monte Ca rlo s imulations a re us ed with 10 6 distinct channel realizations, where the channel realization rema ins con stant over one s ignal block an d change s randomly from bloc k to block. W e assume that the channe l state information is av ailable at the rec eiv er . This figure shows a comparison b etween the different soft deco ders cited above. In this s imulation, the size of the c andidate list of the unc onstrained L SD (ULSD) is not pred etermined as in the original LSD. One ca n fix a sphe re radius which determines the average size of the c andidate list which is not constrained, therefore it is referred to this as Uncons trained List Sphere Decod er (ULSD) [25]. For the SSD [21] a nd the S oft SB-Stack d ecode rs, we take a list of 6 c andidates . W e ca n see that he Soft SB-Stack deco der ou tperforms the other d ecode rs in term of p erformance and exhibits a g ain over 1dB co mpared to the SSD. T he achieved improvement is u p to 2 d B compared to the LSD. This is due to the fact tha t the SB-Stack decode r is more flexible for increasing the list size and the algorithm can continue running to ge t more c andidates which is not the case of the ULSD a nd the SSD which are constrained by the chose n radius. Next, we compare the complexity o f the dif ferent soft deco ders. In Fig.13.b, we plot the number of multiplications n eeded to deco de one transmitted codeword. As migh t be expected, the Soft SB-Stack decode r enjoys a n advantageous average co mplexity compared to the S SD. Howe ver , it is o utperformed by the L SD in term of complexity but the performance of this later is worse. The improved performance of the SB-Stack decod er comp ared to the LSD co mes a t a c ost of additional complexity requiremen ts. C O N C L U S I O N The main pu rpose o f this work was to apply se quential de coders, espec ially the stack o ne, for multi- antenna syste ms over linear Gaus sian c hanne ls. W e saw that the MIMO decod ing can be tasked on a Nov ember 22, 2021 DRAFT 28 0 2 4 6 8 10 12 10 −4 10 −3 10 −2 10 −1 10 0 Eb/No (dB) BER LSD N p =6 Soft Stack N p =6 Shifted SD N p =6 Exhaustive (a) 0 5 10 15 200 250 300 350 Eb/No (dB) Complexity : Multiplications/Symbol Exhaustive Soft Stack N p =6 Shifted SD N p =6 LSD N p =6 (b) Fig. 13. Performance on 2 × 2 ergodic MIMO with 4 − QA M , rate 1 2 con volutional encod er CLPS p roblem. T ow a rd this en d, we proposed ap proaches to apply the stack dec oder to decode lattice. Our main contribution was to introduce a novel version of the stac k decoder for MIMO systems with a reduced c omplexity . Th e propose d deco der , that we called the Spherical Bound S tack d ecode r , cons ists on a modified stack decoder comb ining both the s tack a lgorithm se arch strategy and the sphere d ecode r properties. In a first ti me, this modified deco der was introduced to decode lattice. In a se cond time, we brought the nec essary modifica tions to app ly it in the cas e of finite con stellations a nd we s howed Nov ember 22, 2021 DRAFT 29 by simulations that the SB-Stack dec oder outperforms the c lassical lattice a nd MIMO ones in term of complexity . As a seco nd part, we extende d the SB-Stack deco der to support so ft outpu t MIMO detection. By exploiting the advantage of the memory use to deli ver a soft output, a good improvement in the per- formances is distinguished . Th e simulation results s how that the Soft SB-Stack de coder outperforms the known List S phere Decode r a nd the Shifted Sph ere Decoder and is a t 2dB from the ML performances. Moreover , due to the stack properties, the Soft SB-Stack d ecode r is a lso easily implemented an d is more flexible for an increas e of ca ndidates list s ize. R E F E R E N C E S [1] P . W . W olniansk y , G. J. Foschini, G. D. Golden, and R. A. V alenzuela, “V-BLAS T: an architecture for realizing very high data rates ov er the rich-scattering wireless channe l, ” Bell Labs, Lucent T echnologies, Crawford Hill Laboratory 791 Holmdel-K eypo rt RD., Holmdel, NJ0773 3. [2] M. O. Damen, A. Chkeif, and J.-C. Belfiore, “Lattice code decoder for space-time codes, ” IEEE Communications L etters , vol. 4, pp. 161–16 3, May 2000. [3] A. D. Murug an, H. ElGamal, M. O. Damen, and G. Caire, “A unified framew ork for tree search decoding: rediscovering the sequential decoder , ” IEEE T ransactions On Information Theory , vol. 52, pp. 933–953, March 2006. [4] K. S. Zi gangiro v , “Some sequential decoding procedures, ” Pr obl. P er edach. Inform. , v ol. 2, pp. 13–25, 1966. [5] H. ElGamal and M. O. Damen, “Universal space-time coding, ” IEEE Tr ansactions On Information Theory , May 2003 . [6] M. O. Damen, H. ElGamal, and G. Caire, “On maximum likelihoo d detection and the search for the closest latti ce point, ” IEEE T ransactions on Information Theory , pp. 2389–2402, October 2003. [7] B. Hassibi and B . M. Hochwald, “Linear dispersion codes, ” P r oceedings of the IEEE International Confer ence on Information Theory , p. 325, June 24-29 2001. [8] J.-C. Belfiore, G. Rekaya, and E. V iterbo, “The Golden Code: a 2x2 full rate space-time code with non-v anishing determinants, ” IE EE Tr ansactions on Information Theory , vo l. 51, pp. 1596–1600, Nov ember 2004 . [9] F . Oggier, G. R ekaya, J.-C. Belfiore, and E. V iterbo, “Perfect space-time block codes, ” IE EE T ransctions on Information Theory , vol. 52, pp. 3885 –3902, September 2006 . [10] J. M. Ciof fi and G. D. Forne y , “Generalized decision feedb ack equalization for packet transmission with ISI and Gaussian noise, ” i n Communications, Computation, Contr ol, and Signal Pr ocessing , vol. 79, p. 127, 1997. [11] G. D. Forney , “On the role of MMSE esti mation in approaching the information-theoretic limits of linear Gaussian channels: Shannon meets W iener , ” i n Proc . 41th A nnu. Allerton Conf. Communications, Contro l, and Computing, Monticello , October 2003. [12] J. Boutros and E. V iterbo, “A univ ersal l attice code decod er for fading channels, ” IEEE T ransactions On Information Theory , vol. 45, pp. 1639 –1642, July 1999 . Nov ember 22, 2021 DRAFT 30 [13] C . Schnorr and M. Euchner , “Lattice basis reduction: improved practical algorithms and solving subset sum problems, ” Mathematical Pr ogramming , vol. 66, pp. 181–19 9, September 1994. [14] “Numerical Recipes, ” http://www .nr .com . [15] R . Johannesson and K. S . Zigangirov , “Fund amentals of con volution al coding, ” IEEE series on digital and mobile communication , pp. 269–315, 1999. [16] R . M. Fano, “A heuristic discussion of probabilistic decod ing, ” IEEE T rans actions On Information Theory , vol. 9, pp. 64– 74, April 1963 . [17] E . Agrell, T . Eri ksson, A. V ardy , and K. Zeger , “Closest point search in latti ces, ” IEEE Tr ansactions On Information Theory , vol. 48, pp. 2201 –2214, August 2002. [18] B . Hassibi and H. V ikalo, “On the expected complexity of sphere decoding, ” in Pr oceedings of Asilomar Confer ence on Signals, Systems, and Computers , vol. 2, pp. 1051–1055, 2001. [19] G. Rekaya, "Nouvelles con structions algébriqu es de codes sp atio-tempor els atteignant le compr omis multiplex ag e-divers ité" . PhD thesis, Ecole Nationale Superieure des T elecommunications, December 2004. [20] W . Chin, “QRD Based Tree Search Data Detection fo r MIMO Communication S ystems, ” V ehicular T echnolog y Confer ence , VTC 2005 -Spring . 2005 IE EE 61st , vol. 3, pp. 1624–162 7, June 2005. [21] J. Boutros, N. Gresset, L. Brunel, and M. Fossorier , “Soft-input soft-output lattice sphere decoder for l inear channels, ” IEEE Global T elecommunications Confer ence GLOBECOM , 2003. [22] B . M. Hochwald and S. ten Brink, “ Achie ving near-capacity on a multiple-antenna channel, ” IEEE T ran sactions on Communication , v ol. 53, pp. 389–399, March 2003. [23] J. Hagenau er , E. Offer , and L. Papke, “Iterati ve de coding of binary block and con volution al codes, ” IEE E T ransa ctions on Information Theory , vol. 42, pp. 429–445, March 1996. [24] J. B outros, F . Boixadera, and C. Lamy , “Bi t-interleave d coded modulations for multiple-output channels, ” IEEE 6th Int. Symp. on Spr ead Spectrum T echniqu es and App. , v ol. 2, pp. 13–25, September 2000. [25] P . Radosa vljev ic and J. R. Cava llaro, “Soft sphere detection with bounded search for high-throughput MIMO receiv ers, ” Signals, Systems a nd Computers, 2006. ACSSC ’06. F ortieth Asilomar Con fer ence, P acific Gro ve, CA , USA , pp. 1175–1179, March 2006 . Nov ember 22, 2021 DRAFT