Asymptotic Analysis of the Performance of LAS Algorithm for Large-MIMO Detection

Asymptotic Analysis of the P erformance of LAS Algorithm for Large-MIMO Detection ∗ Saif K. Mohamm ed , A. Cho c k alingam, and B. Sundar Ra jan Departmen t of ECE, Indian Institute of Science, Bang alore 560012, India Abstract In our recen t w ork, w e rep orted a n exhaus tiv e study on t h e sim ulated bit error rate (BER) p erformance of a l ow-co m plexit y likel ih oo d ascent searc h (LAS) algorithm f or detection in large m ultiple-input m u ltiple-o u tput (MIMO) systems with la r ge n um b er of antennas t h at ac hiev e high sp ectral efficiencies. Though the algorithm w as sho wn to ac hieve increasingly closer to n ear maxim um-lik eliho o d (ML) p erforman ce thr ough sim ulations, no BER analysis w as rep orted. Here, w e extend our w ork on LAS and rep ort an asymptotic BER analysis of the LAS algorithm in the large system limit, where N t , N r → ∞ with N t = N r , where N t and N r are the num b er of transmit and receiv e an tennas. W e pro ve that the error p erformance of the LAS detector in V-BLAST with 4-QAM in i.i.d. Ra yleigh f ading conv erges to that of the ML detector as N t , N r → ∞ . Keywor ds – High sp e ctr al efficiencies, lar ge-MIMO dete ction, likeliho o d asc ent se ar ch. 1 In tro du ction Multiple-input m ultiple-o utput (MIMO) systems that emplo y larg e num b er of transmit and receiv e antennas can offer ve ry high spectral effic iencies of the order of tens to hundreds of bps/Hz [1],[2]. Ac hieving near-o ptimal signal detection at low complexities in suc h large- dimension systems has b een a c hallenge. In our recen t works, w e ha v e sho wn that certain algorithms from machine learning/artificial in telligence ac hiev e near- optimal p erformance in large-MIMO systems that emplo y tens of transmit a nd receiv e an tennas using V- BL AST and non-orthogo nal space-time blo c k co des (STBC) [3] with tens to h undreds of dimensions in space and time, at lo w complexities. Suc h algorithms include lo cal neigh b orho o d searc h based algorithms like a likeliho o d asc ent se ar ch (LAS) algorithm [4],[5] a nd a r e active tabu se ar ch (R TS) a lgorithm [6], and algorithms based on pr ob abilistic data ass o ciation (PDA) [7] and b elief pr op agation (BP) [8],[9 ]. Similar algorithms hav e b een earlier rep orted in the con- text o f multius er dete ction [10]-[16]. In [4]-[9], through detailed sim ulations, w e ha ve show n that LAS and R TS alg orithms ac hieve increasingly closer t o maximum - lik eliho o d (ML) p er- formance and that PD A and BP algorithms ac hieve near maxim um a p osteriori probability ∗ This pap er in part w a s presen ted in IEEE PIMRC’2008, Cannes, F ranc e , Septem b er 2008 . 1 (MAP) p erformance for increasing n umber of dimensions in lar g e-MIMO systems. F o r e.g., in [5], the BER perfo r mance of the basic LAS algorithm (whic h uses a single sym b ol up date based neighborho o d definition) and its generalized vers io n ( whic h uses a m ultiple sym b ol up date based neigh b orho o d definition) has b een exhaustiv ely studied through sim ulations. Ho w eve r , BER p erformance ana lys is of the LAS algorithm for larg e-MIM O detection has not b een rep orted. In t his corresp ondence, we fill some of this gap by presen ting an asymp- totic BER analysis o f the LAS algorithm in the large system limit, where N t , N r → ∞ with N t = N r , where N t and N r denote the n umber of transmit and receiv e an tennas, r esp ectiv ely . Asymptotic p erformance analysis of larg e sy stems in the con text of m ultiuser detection and MIMO comm unicatio n ha ve b een rep orted in the lit erature [17]-[22], using random matrix theory (e.g., [17],[18]), replica metho d (e.g., [19],[20],[21 ]), and free probabilit y theory (e.g., [22]). W e, in this correspondence, presen t an asymptotic BER analysis of the LAS algorithm in t he lar g e system limit. Sp ecifically , w e presen t an a nalytical pro of t ha t t he error p erfor- mance of the L AS detector for V-BLAST with 4-QAM in i.i.d. Ra yleigh fading conv erges to that of the ML detector as N t , N r → ∞ with N t = N r , whic h is an analytical result that has no t b een rep orted so far. The rest of this corr esp ondence is org a nize d as follo ws. The MIMO system mo del a nd LAS detection alg orithm are summarized in Section 2. The asymptotic analysis of the LAS algorithm is presen ted in Section 3. Lengthy pro ofs of lemmas a nd theorems are mov ed to the a pp endices. Sim ulation results and discussions ar e presen ted in Section 4. Conclusions are giv en in Section 5. 2 System Mo de l Consider a V-BLAST system with N t transmit an tennas and N r receiv e antennas, N t ≤ N r . Let x c ∈ C N t × 1 denote 1 the sym b ol vec to r transmitted, and H c ∈ C N r × N t denote the channel matrix suc h that its ( i, j )th en try h i,j is the complex channe l gain from the j th tr a nsm it an tenna to the i th receiv e a n tenna. Assuming ric h scattering, we mo del the en tries of H c as i.i.d. C N (0 , 1). L et y c ∈ C N r × 1 and n c ∈ C N r × 1 denote the receiv ed signal v ector and the noise v ector, resp ectiv ely , at the receiv er, whe re the en tries o f n c are mo deled a s i.i.d 1 V ector s a re denoted by b oldface lo wercase le tters, and matr ices ar e denoted by b oldface upperca se letters. [ . ] T and [ . ] H denote transp ose and conjugate transpo se op erations, resp ectiv ely . || . || denotes E uclidean distance. 2 C N (0 , σ 2 ). The receiv ed signal ve ctor can then b e written a s y c = H c x c + n c . (1) Let y c , H c , x c , and n c b e decomp osed in to real and imaginar y part s as follow s: y c = y I + j y Q , x c = x I + j x Q , n c = n I + j n Q , H c = H I + j H Q . (2) F urther, w e define H r ∈ R 2 N r × 2 N t , y r ∈ R 2 N r × 1 , x r ∈ R 2 N t × 1 , and n r ∈ R 2 N r × 1 as H r =  H I − H Q H Q H I  , y r = [ y T I y T Q ] T , x r = [ x T I x T Q ] T , n r = [ n T I n T Q ] T . (3) No w, (1) can b e written as y r = H r x r + n r . (4) Henceforth, we shall work with the real- v alued signal mo del of the system in (4). F or notational simplicit y , w e dr o p subscripts r in (4) and write y = Hx + n , (5) where H = H r ∈ R 2 N r × 2 N t , y = y r ∈ R 2 N r × 1 , x = x r ∈ R 2 N t × 1 , n = n r ∈ R 2 N r × 1 . In this real-v alued system mo del, the real-part of the complex data sy mbols will b e mapp ed to [ x 1 , · · · , x N t ] and the imaginary-part of these sym b ols will b e mapp ed to [ x N t +1 , · · · , x 2 N t ] . F or M -QAM, [ x 1 , · · · , x N t ] can b e view ed to b e from an underlying M -P AM signal set a nd so is [ x N t +1 , · · · , x 2 N t ]. Let A i denote the M -P AM signal set from whic h x i tak es v alues, i = 1 , 2 , · · · , 2 N t ; e.g., for 4 - QAM, A i = { 1 , − 1 } for i = 1 , 2 , · · · , 2 N t . No w, define a 2 N t - dimensional signal space S to b e the Cartesian pro duct of A 1 to A 2 N t . The ML solution v ector, d M L , is g iv en by d M L = arg min d ∈ S k y − H d k 2 = arg min d ∈ S  d T H T Hd − 2 y T Hd  . (6) In the following subsection, we summarize the low-comple xity LAS alg orithm, using a neigh- b orho o d definition based o n 1- sy mbol up dates, presen ted in [5 ] for large-MIMO detection for M -QAM. The channe l matrix H is assumed to b e kno wn p erfectly a t the receiv er. 2.1 LA S Algorithm for L arge-MIMO Detection The LAS algorithm starts with an initial v ector d (0) , giv en by d (0) = By , where B is the initial solution filter, whic h can b e a matc hed filter (MF) or zero-forcing (Z F) filter or MMSE 3 filter. The index m in d ( m ) denotes the itera t io n num b er in a g iven searc h stage. The ML cost f unction a fter the k th itera t ion in a giv en searc h stage is giv en by C ( k ) = d ( k ) T H T Hd ( k ) − 2 y T Hd ( k ) . (7) The d vec to r is up dated from k th to ( k + 1)th iteratio n b y up dating one sym b ol, sa y , the p th sym b ol, as d ( k +1) = d ( k ) + λ ( k ) p e p , (8) where e p denotes the unit v ector with its p th en t ry only a s one, and all other en tries as zero. Since d ( k ) and d ( k +1) should b elong to S , λ ( k ) p can take o nly certain in teger v al- ues. F or example, for 16-QAM, A p = {− 3 , − 1 , 1 , 3 }  , and λ ( k ) p can take v alues only fr om {− 6 , − 4 , − 2 , 0 , 2 , 4 , 6 } . Using (7) and ( 8), and defining a matrix G as G △ = H T H , (9) w e can write the cost difference C ( k +1) − C ( k ) as F ( l ( k ) p ) △ = C ( k +1) − C ( k ) = l ( k ) 2 p a p − 2 l ( k ) p | z ( k ) p | , where z ( k ) p is t he p th en t r y o f the z ( k ) v ector giv en b y z ( k ) = H T ( y − Hd ( k ) ) , a p △ = ( G ) p,p is t he ( p , p ) t h en try of the G matrix, and l ( k ) p = | λ ( k ) p | . The v alue of l ( k ) p whic h giv es the largest descen t in the cost function fro m t he k th to t he ( k + 1)th iteration (when sym b ol p is up dated) is obtained as l ( k ) p,opt = 2 $ | z ( k ) p | 2 a p ' , (10) where ⌊ . ⌉ denotes the rounding op eration. If d ( k ) p w ere up dated using l ( k ) p,opt , it is p ossible t ha t the up dated v alue do es not b elong to A p . T o a v o id this, w e adjust l ( k ) p,opt so that the up dated v alue of d ( k ) p b elongs to A p . Let s = arg min p F ( l ( k ) p,opt ) . (11) If F ( l ( k ) s,opt ) < 0 , the up date for the ( k + 1 )th iteration is d ( k +1) = d ( k ) + l ( k ) s,opt sgn( z ( k ) s ) e s (12) z ( k +1) = z ( k ) − l ( k ) s,opt sgn( z ( k ) s ) g s , (13) where g s is the s t h column o f G . If F ( l ( k ) s,opt ) ≥ 0, then the searc h terminates, and d ( k ) is declared as the detected data vec t o r. 4 3 Asymptotic An alysis of LAS Algorithm In this section, we prov e the asymptotic conv ergence of t he error probability of the LAS detector to that of the ML detector for N t , N r → ∞ with N t = N r in V-BLAST. Consider 4-QAM, i.e., S ∈ { +1 , − 1 } 2 N t , and let N t = N r . An n -sym b ol up date on a data v ector d ∈ S transforms d to ( d − ∆ d n ) suc h that ( d − ∆ d n ) ∈ S . F urther, ( d − ∆ d n ) is obtained b y c hanging n symbols in d at distinct indices giv en b y the n -tuple u n △ = ( i 1 , i 2 , · · · , i n ), 1 ≤ i j ≤ 2 N t , ∀ j = 1 , · · · , n and i j 6 = i k for j 6 = k . Therefore, we can write ∆ d n as ∆ d n = n X k =1 2 d i k e i k , (14) where d i k is the i k th elemen t of d . Let L n ⊆ S denote the set of data vec to rs suc h that for an y d ∈ L n , if a n -symbol up date is p erformed o n d resulting in a ve ctor ( d − ∆ d n ), then || y − H ( d − ∆ d n ) || ≥ || y − Hd || . Our main result in this section is Theorem 2. T o pro ve Theorem 2, we need the follow ing Lemmas 1 t o 5, Slutsky’s theorem [23], and Theorem 1. Lemma 1 L et d ∈ S . T hen, d ∈ L n if and o n ly if, for any n -up date on d , n ∈ [1 , 2 · · · , 2 N t ] ,  y − Hd + 1 2 H ∆ d n  T  H ∆ d n  ≥ 0 . (15) Pr o of: By definition, if d ∈ L n , then no n -sym b ol up date can result in a reduction in the ML cost function. Using this, we can write k y − H ( d − ∆ d n ) k 2 ≥ k y − Hd k 2 . (16) Simplifying (16), w e get (15). Since the c ho ice o f the indices in u n is arbitrary , the lemma holds true for all possible n -tuples of distinct indices. F or the con ve rse, if d satisfies (15) for all possible u n for a given n , then, since (15) and (16) are equiv alent, d also satisfies (16) for all p ossible u n . This implies that d ∈ L n .  If d ∈ L 1 , then using Lemma 1 and (5), we can write  n + H ( x − d ) + h p d p  T  h p d p ) ≥ 0 , ∀ p = 1 , · · · , 2 N t , (17) where h p is the p th column of H . Lemma 2 Assuming uniqueness of the ML ve ctor d M L in (6), a symb ol ve ctor d ∈ S is the ML ve ctor if and only if the noise ve ctor n sa tisfi e s the f o l lowing set of e quations  n + H ( x − d ) +  n X j =1 h i j d i j  T  n X j =1 h i j d i j  ≥ 0 , (18) ∀ n = 1 , · · · , 2 N t , and fo r al l p ossible n -tuples ( i 1 , · · · , i n ) for e a c h n . 5 Pr o of: If d is the unique ML v ector, then from the definition of the ML criterion in ( 6 ), it m ust be true that an y n -up date on d will not resu lt in an y decrease in the ML cost function. Therefore, d ∈ L n , ∀ n = 1 , 2 · · · , 2 N t . Hence, by Lemma 1 , it m ust b e true that d satisfies (15) for all n = 1 , 2 , · · · , 2 N t and for all p ossible u n for eac h n . Substituting y = Hx + n in (15), w e get (18). This pro ve s the direct result. T o pro v e the con verse , let the noise v ector n satisfy (18) for some v ector d . Since y = Hx + n , the conditions in (18) imply the conditions in (15) for all n = 1 , 2 , · · · , 2 N t and for a ll p ossible u n for eac h n . Therefore, by Lemma 1, d ∈ L n for a ll n = 1 , 2 , · · · , 2 N t , whic h then implies that d indeed is the ML v ector.  Definition: F o r each d ∈ S and for eac h integer m , 1 ≤ m ≤ 2 N t , we asso ciate the set of v ectors R d m = n v | v ∈ R 2 N t and  v + H ( x − d ) +  P n j =1 h i j d i j  T  P n j =1 h i j d i j  ≥ 0 , ∀ n = 1 , · · · , m, and for all p ossible n -tuples ( i 1 , · · · , i n ) for each n o , and define R d △ = R d 2 N t . Lemma 3 If the noise ve ctor n ∈ R d , then d is the ML ve ctor. L et d i , d j ∈ S and d i 6 = d j . Then R d i and R d j ar e disjoint. Pr o of: F rom Lemma 2 and the definition of R d , it is clear that d is the ML v ector if and only if n ∈ R d . The disjointnes s of R d i and R d j , i 6 = j , can b e sho wn b y contradiction. If R d i and R d j are not disjoin t, then there exists some vec t o r v b elonging to b oth R d i and R d j . If v were to b e the noise v ector n , then, v w o uld satisfy the set of equations in (18) for b oth d = d i and d = d j , since v b elongs to b oth R d i and R d j , This, by L emma 2, implies that b oth d i and d j are ML v ectors, which is a con tr adiction b ecause of the uniqueness of the ML vec t o r.  Lemma 4 L et h ∈ R 2 N t b e a r andom ve ctor with i.i.d entries distribute d as N ( 0 , 0 . 5) . L et { h i } , i = 1 , 2 , · · · , m b e a set of ve ctors, with e ach h i ∈ R 2 N t and having i.i.d entries distribute d as N (0 , 0 . 5 ) , E [ h i h T j ] = 0 f o r i 6 = j , and E [ hh T j ] = 0 f o r j = 1 , · · · , m . Then lim N t →∞ P m k =1 h T h k mN t = 0 . (19) Pr o of: Let ˜ h △ = 1 √ m P m k =1 h k . Then, ˜ h ∼ N ( 0 , I 2 ). Therefore, w e ha ve lim N t →∞ P m k =1 h T h k mN t = lim N t →∞ h T ˜ h √ mN t . (20) W e can write lim N t →∞ h T ˜ h N t = lim N t →∞ P 2 N t k =1 h k ˜ h k N t , (21) 6 where h k and ˜ h k are the k th elemen ts of h a nd ˜ h , respectiv ely . The r.v’s h k ˜ h k , k = 1 , · · · , 2 N t are i.i.d with mean zero. F rom the strong law of larg e n umbers [23 ], it follow s that lim N t →∞ P 2 N t k =1 h k ˜ h k 2 N t = 0. Using this in (20) completes the pro of.  Before w e presen t the next lemma, we prese nt the Slutsky’s theorem on con v ergence of random v ariables, whic h is used to prov e Lemma 5 and Theorem 1. Slutsky’s The or em [23]: Let { X m } a nd { Y m } b e sequences of random v ariables. If { X m } con v erges in distribution to a random v ariable X , a nd { Y m } conv erges in proba bilit y to a constan t c , then it is true that i ) { X m + Y m } con v erges in distribution to X + c , ii ) { X m Y m } con v erges in distribution to c X , and iii ) n X m Y m o con v erges in distribution to X c . Lemma 5 F or a given u n and a g i v en d ∈ S , de fine a r.v z u n , d as z u n , d △ = P n k =1 P n j = k +1 h T i j h i k d i j d i k P n j =1 k h i j k 2 , (22) wher e i j ∈ u n , j = 1 , · · · , n . F or an y u n and any d ∈ S , z u n , d c onve r ges to zer o in pr ob ability as N t → ∞ , i.e., z u n , d p − → 0 as N t → ∞ , ∀ n = 2 , 3 , · · · , 2 N t . Pr o of: Pro of of this Lemma is give n in App endix A.  In Fig. 1, w e plot the sim ulated p df of z u n , d for n = 2 N t for differen t v alues of N t = N r for a certain u n and d (the p df was observ ed to b e same for differen t u n and d ). W e observ e that with increasing N t = N r , the p df of z u n , d tends tow a rds t he Dirac delta function at zero. This implie s tha t z u n , d tends to zero in dis t r ibutio n, and hence in probabilit y , for large N t = N r , whic h is formally prov ed in Lemma 5. Theorem 1 L et d ∈ S and n ∈ R d 1 . Then n ∈ R d in pr ob ability as N t → ∞ , i.e., for a ny δ , 0 ≤ δ ≤ 1 , ther e exists an inte ger N ( δ ) such that for N t > N ( δ ) , p ( n ∈ R d ) > 1 − δ . Pr o of: Pro of of this theorem is giv en in App endix B.  Theorem 2 The data ve ctor/bit err or pr ob ability of the LAS dete c tor c onver ges to that of the ML de te ctor as N t , N r → ∞ with N t = N r . Pr o of: Let d LAS b e the final output sym b ol v ector of the LAS algorithm given x , H and n . The algorit hm terminates if and only if no 1- updat e results in an y further decrease of the 7 cost function. This implies that for the g iv en x , H and n , d LAS ∈ L 1 , and therefore it must b e true that n satisfies (1 7) with d replaced by d LAS . These set of equations are the same whic h define the r egio n R d 1 . Therefore, replacing d b y d LAS , w e can equiv alently claim that n ∈ R d 1 LAS . Using Theorem 1, w e can further claim that asymptotically as N t → ∞ , n ∈ R d LAS in probabilit y . F rom Lemma 3 , w e know that if n ∈ R d LAS , then d LAS is indeed the ML ve ctor for the g iven x , H and n . Therefore, w e can state that a symptotically as N t → ∞ , d LAS is indeed the ML ve ctor in pro babilit y . That is, for any δ , 0 ≤ δ ≤ 1, there exists an in teger N ( δ ) suc h that for N t ≥ N ( δ ) P ( d LAS is the ML ve ctor ) > (1 − δ ) . (23) Therefore, w e can write that for N t ≥ N ( δ ) P LAS ( er r or ) = P ( d LAS 6 = x ) = P ( d LAS 6 = x | d LAS = ML ve ctor ) P ( d LAS = ML v ector) + P ( d LAS 6 = x | d LAS 6 = ML v ector ) P ( d LAS 6 = ML ve ctor ) . (24 ) F rom (23), w e ha v e P ( d LAS 6 = ML v ector ) ≤ δ . Also, P ( d LAS 6 = x | d LAS = ML ve ctor ) is the probability of error for the ML detector, whic h w e denote by P M L ( er r or ) . Using these, w e can b ound t he pro ba bilit y of error fo r the LAS detector as P LAS ( er r or ) ≤ P M L ( er r or ) + δ P ( d LAS 6 = x | d LAS 6 = ML v ector) ≤ P M L ( er r or ) + δ. ( 2 5) Since δ can b e arbitrarily small, we can conclude from (25) that indeed as N t → ∞ , the sym b ol v ector error probabilit y of the LAS detector con v erges t o that of t he ML detector. This pro of can b e adapted to sho w that apart from the sym b ol v ector error probability , the bit error probability of the LAS detector also con v erges to that of the ML detector. The pro of for the bit error probabilit y conv ergence is along the same lines as (24) and (25) , except that instead of defining the error ev ent as d LAS 6 = x , we de fine error ev ents for eac h bit. F or example, for t he p th bit, the error ev ent is defined as d p LAS 6 = x p .  4 Sim ulation Results and Discu ssions In F ig. 2 we show the simulated BER p erformance of the LAS detector for V-BLAST with 4-QAM and MMSE initial ve ctor for increasing N t = N r . Since an analytical expression fo r ML p erformance in the large MIMO system limit is not a v ailable and sim ulating the ML p erformance f or large dimensions inv olves prohibitiv ely high complexit y , w e plot t he SISO 8 A W GN p erformance as a lo w er b ound for comparison. It can b e seen that for incre a sing N t = N r , the BER perfo rmance o f the LAS detector appro ac hes the SISO A W G N p erformance at high SNRs. Figure 3 sho ws t he a verage SNR required to ac hiev e a BER of 10 − 3 for increasing N t = N r and 4 -QAM. It can b e seen that, f or large N t = N r , the required SNR gets increasingly closer to that required in SISO A W G N for increasing N t = N r . A similar b ehav ior can b e observ ed in Fig. 4 for 16-Q AM as w ell. In Figs. 3 and 4, we also see that there is an initia l degradation in p erformance for increasing n um b er of an tennas (for N t < 10). This sho ws that the LAS detector is suboptimal for small sys tems with small n um b er of an tennas 2 , and b ecomes optimal in the large sy stem limit (as pro v ed in the previous section). L AS detector ac hieve s close to lar g e sy stem limit p erformance in systems with large n um b er of dimensions (e.g., h undreds of dimens ions in Figs. 3 and 4). Suc h large n um b er o f dimensions need not b e realized in spatial dimension alo ne, as in V-BLAST. As sho wn in [5], exploiting time dimension in addition to space dimension, large non-o rthogonal STBC MIMO systems can render larg e dimensions with less num b er of transmit antennas that can b e implemen ted in pr a ctice . A 16 × 16 non-orthog onal STBC from cyclic division algebra [3 ] with complex data sym b ols has 5 12 real dimensions; with 64- QAM and rate-3 /4 turb o co de, this STBC ac hiev es a sp ectral efficiency of 72 bps/Hz. In [5], LAS a lg orithm has b een shown to a chiev e near- capacit y p erformance in 16 × 16 STBC MIMO systems eve n in the presence of spatia l correlation and with estimated channel matrix. F urther, considering that NTT DoCoMo has demonstrated a 12 × 12 V-BLAST MIMO sys tem op erating at 5 Gbps a t a sp ectral efficiency of 50 bps/Hz at 10 Km/ hr mobile sp eeds [24 ], the av ailability of lo w-complexit y large-MIMO detection algorithms lik e the LAS algor ithm analyzed in this corresp ondence can motiv at e the adoption of 16 × 16 and 24 × 24 MIMO systems op erating at sp ectral efficiencies in excess of 5 0 bps/Hz in emerging wireless standards lik e IEEE 80 2 .11 VHT and IEEE 80 2.16/L TE-A. 5 Conclus ions W e conclude with the following tw o remarks: i ) The deriv ation of analytical BER expres- sions for the ML p erformance in the large MIMO system limit for differen t signal sets is an op en problem. Since la rge MIMO systems can b e viable in practice due to the a v ailabil- 2 W e do not have a theoretical explanation for this small system b e havior of the LAS detector, whereas we are able to pro ve its asymptotic large system behavior. 9 it y of low-complex ity detectors lik e the LAS detector, analytical BER expres sions for the ML p erformance in the large MIMO system limit would b e quite useful as a b enc hmark for comparing the p erformance o f practical detectors in larg e-MIM O systems. The statistical mec hanics approach emplo y ed in [19] for la r ge CDMA system BER analysis can b e in v es- tigated for suc h an analysis. ii ) While w e are able to prov e the asymptotic con v ergence of LAS p erformance to ML p erformance f or 4- Q AM here, our sim ulation results f o r higher order QAM (e.g., 16-QAM; see Fig. 4) show similar b eha vioral trend lik e that for 4-QAM. Conse- quen tly , w e conjecture that suc h a conv ergence holds for general M -QAM and an analytical pro of to sho w this can b e attempted as an extension to this w ork. App end ix A: Pro o f of Lemma 5 W e presen t the pro of o f L emm a 5 in this a ppendix. The pro of is b y mathematical induction on n . Base Case: F or n = 2, w e ha ve to sho w that d p d q h T p h q k h p k 2 + k h q k 2 p − → 0 as N t → ∞ , ∀ p, q = 1 , 2 , · · · , 2 N t , p 6 = q . (26) W e can write the rando m v ariable h T p h q k h p k 2 + k h q k 2 as h T p h q / (2 N t ) ( k h p k 2 + k h q k 2 ) / (2 N t ) . (27) As N t → ∞ , b y strong la w of large nu mbers, the denominator of (27) con ve rg es to 1 almost surely . Also, the n umerator of (2 7) can b e written as h T p h q 2 N t = P 2 N t k =1 h p,k h q ,k 2 N t , (28) where h p,k and h q ,k refer to the k th en try of the vec to rs h p and h q , resp ectiv ely . Eac h h p,k h q ,k term in the s ummatio n in (28) has the same distribution and has mean 0. Therefore, b y strong law of large num b ers, w e can see that h T p h q 2 N t con v erges to 0 almost surely . This also implies that h T p h q 2 N t con v erges in distribution to the constant 0, and hence b y Slutsky’s theorem, h T p h q k h p k 2 + k h q k 2 con v erges in distribution to 0 . Since, if a sequence of r.v’s conv erges in distribution to a constan t then the sequence con ve rg es in probability to that constant, we conclude that indeed h T p h q k h p k 2 + k h q k 2 con v erges in probability to 0. This prov es the the base case. Induction Hyp othesis: Let z u n , d p − → 0 as N t → ∞ , ∀ n = 2 , 3 , · · · , m . 10 Induction S tep: Pro o f for n = m + 1: W e hav e z u ( m +1) , d = P m +1 k =1 P m +1 j = k +1 h T i j h i k d i j d i k P m +1 j =1 k h i j k 2 (29) = P m k =1 P m j = k +1 h T i j h i k d i j d i k + P m k =1 h T i ( m +1) h i k d i ( m +1) d i k k h i ( m +1) k 2 + P m j =1 k h i j k 2 (30) = P m k =1 P m j = k +1 h T i j h i k d i j d i k P m j =1 k h i j k 2 + P m k =1 h T i ( m +1) h i k d i ( m +1) d i k P m j =1 k h i j k 2 1 + k h i ( m +1) k 2 P m j =1 k h i j k 2 . (31) Using Slutsky’s theorem and the strong law of large n umbers, it can b e sho wn t ha t the denominator in (31 ) con verges to (1 + 1 m ) in probability . Also, from the induction h y- p othesis, the term P m k =1 P m j = k +1 h T i j h i k d i j d i k P m j =1 k h i j k 2 in the n umerator of (31) con v erges in proba bil- it y to 0. Therefore, the n umerator in (31) con v erges to the same distribution that the term P m k =1 h T i ( m +1) h i k d i ( m +1) d i k P m j =1 k h i j k 2 con v erges t o. Also, the term P m k =1 h T i ( m +1) h i k d i ( m +1) d i k P m j =1 k h i j k 2 is the same as ( P m k =1 h T i ( m +1) h i k d i ( m +1) d i k ) / ( mN t ) ( P m j =1 k h i j k 2 ) / ( mN t ) . F urther, from the strong law of large n umbers, the term ( P m j =1 k h i j k 2 ) / ( mN t ) con v erges almost surely to 1. Therefore, from Slutsky’s theorem, w e kno w that ( P m k =1 h T i ( m +1) h i k d i ( m +1) d i k ) / ( mN t ) ( P m j =1 k h i j k 2 ) / ( mN t ) con v erges in distribution to the distribution to whic h the t erm ( P m k =1 h T i ( m +1) h i k d i ( m +1) d i k ) / ( mN t ) con v erges. F or a given vec to r d , h i k d i k is a ra ndom ve ctor whose distribution is the same as that of h i k . Therefore, applying Lemma 4, w e see that the term ( P m k =1 h T i ( m +1) h i k d i ( m +1) d i k ) / ( mN t ) con v erges almost surely to 0. Hence, the numerator in (31) conv erges in probability to the constan t 0 . Therefore, z u ( m +1) , d p − → 0 as N t → ∞ . This prov es the induction step and completes t he pro of of Lemma 5.  App end ix B: Pro o f of Theorem 1 W e presen t the pro of o f Theorem 1 in this app endix. W e shall pro ve thr o ugh induction t ha t if n ∈ R d 1 , then n ∈ R d m in probability , ∀ m = 2 , · · · , 2 N t , as N t → ∞ . Base Case ( m = 2 ): Let n ∈ R d 1 . Therefore, from the definition of R d m , n satisfies (17). W e sho w that n ∈ R d 2 in pro babilit y as N t → ∞ . F or n to b elong to R d 2 , in additio n to satisfying (17), n mus t also satisfy the fo llo wing equation ∀ p, q = 1 , · · · 2 N t , p 6 = q :  n + H ( x − d ) + h p d p + h q d q  T  h p d p + h q d q  ≥ 0 , (32) 11 whic h can b e rewritten a s  n + H ( x − d )  T h p d p +  n + H ( x − d )  T h q d q ≥ −k h p k 2 − k h q k 2 − 2 d p d q h T p h q . (33) Since n satisfies ( 17), it satisfies the follow ing t wo equations:  n + H ( x − d )  T h p d p ≥ −k h p k 2 ,  n + H ( x − d )  T h q d q ≥ −k h q k 2 . (34) Comparing (34) and (33), we notice that if h p and h q are orthogonal, then n trivially satisfies (33) f o r all N t . Therefore, when h p and h q are non-or t hogonal, the only extra term in the RHS of (33) is 2 d p d q h T p h q . Applying Lemma 5, with n = 2, w e see that as N t → ∞ , the r.v. h T p h q k h p k 2 + k h q k 2 con v erges to zero in proba bilit y . Then, w e can write, for any ǫ , 0 ≤ ǫ ≤ 1 p | h T p h q | k h p k 2 + k h q k 2 > ǫ ! < ǫ, ∀ N t > f ( ǫ ) . (35) No w, let us a nalyze p ( n ∈ R d 2 ) for the case of d p d q = +1 (a similar analysis holds for d p d q = − 1). Consider t w o disjoint even ts E 1 = n | h T p h q | k h p k 2 + k h q k 2 < ǫ o and E 2 = n | h T p h q | k h p k 2 + k h q k 2 > ǫ o . Then, w e can write p ( n / ∈ R d 2 ) = p ( n / ∈ R d 2 | E 1 ) p ( E 1 ) + p ( n / ∈ R d 2 | E 2 ) p ( E 2 ) . (36) The ev ent E 1 can b e further split into t wo disjoin t ev en ts E 11 and E 12 , giv en b y E 11 =  0 < h T p h q < ǫ ( k h p k 2 + k h q k 2 )  and E 12 =  0 > h T p h q > − ǫ ( k h p k 2 + k h q k 2 )  . Also, from (35), p ( E 1 ) > 1 − ǫ and p ( E 2 ) < ǫ . Therefore, using (36), w e can write p ( n / ∈ R d 2 ) < p ( n / ∈ R d 2 | E 1 ) p ( E 1 ) + ǫ < p ( n / ∈ R d 2 | E 11 ) p ( E 11 ) + p ( n / ∈ R d 2 | E 12 ) p ( E 12 ) + ǫ < p ( n / ∈ R d 2 | E 11 ) + p ( n / ∈ R d 2 | E 12 ) + ǫ. (37) If ev en t E 11 is tr ue, then −  k h p k 2 + k h q k 2  > −  k h p k 2 + k h q k 2 + 2 h T p h q  > −  k h p k 2 + k h q k 2  (1 + 2 ǫ ) . ( 3 8) Since n ∈ R d 1 , n satisfies (3 4), and hence satisfies the follow ing equation: ( n + H ( x − d )) T ( h p d p + h q d q ) ≥ −  k h p k 2 + k h q k 2  . (39) 12 Using (38) and (39), w e see that n satisfies (33 ), and therefore n ∈ R d 2 . Hence, w e can conclude that p ( n / ∈ R d 2 | E 11 ) = 0 . No w, w e can rewrite (3 7) as p ( n / ∈ R d 2 ) < p ( n / ∈ R d 2 | E 12 ) + ǫ. (40) If ev en t E 12 is tr ue, then −  k h p k 2 + k h q k 2  (1 − 2 ǫ ) > −  k h p k 2 + k h q k 2 + 2 h T p h q  > −  k h p k 2 + k h q k 2  . (41) Using (3 3) and (41), w e can write tha t p ( n / ∈ R d 2 | E 12 ) = p   n + H ( x − d )  T h p d p +  n + H ( x − d )  T h q d q ≤ −k h p k 2 − k h q k 2 − 2 h T p h q    E 12  < p   −  k h p k 2 + k h q k 2  ≤ ( n + H ( x − d )) T ( h p d p + h q d q ) ≤ −  k h p k 2 + k h q k 2  (1 − 2 ǫ )    E 12  . (42) Define R ǫ to b e a set o f ve ctors in R 2 N t , as R ǫ △ = n v   −  k h p k 2 + k h q k 2  ≤ ( v + H ( x − d )) T ( h p d p + h q d q ) ≤ −  k h p k 2 + k h q k 2  (1 − 2 ǫ ) o . (43) Also, define a function f 2 as f 2 ( ǫ ) △ = p ( n ∈ R ǫ | E 12 ) . (44) Using the ab ov e definitions, (42) can rewritten as p ( n / ∈ R d 2 | E 12 ) < f 2 ( ǫ ) . (45) Let ǫ 1 , ǫ 2 ∈ R , ǫ 1 , ǫ 2 > 0, and ǫ 1 > ǫ 2 . F rom the definition of R ǫ in (43), it can b e seen that R ǫ 2 ⊂ R ǫ 1 . This implies that f 2 ( ǫ 1 ) > f 2 ( ǫ 2 ). Hence f 2 is a monotonically increasing function. Using (45), w e can rewrite (40) as written as p ( n / ∈ R d 2 ) < f 2 ( ǫ ) + ǫ. (46) Therefore, p ( n ∈ R d 2 ) > 1 − ( f 2 ( ǫ ) + ǫ ) . (47) No w define g 2 ( ǫ ) △ = f 2 ( ǫ ) + ǫ . So g 2 is a monotonic function and is therefore inv ertible. Let δ = g 2 ( ǫ ). Using (35) and the ab ov e definitions, w e can write that N t > f ( ǫ ) > f  g − 1 2 ( δ )  > N 2 ( δ ) , (48) 13 where N 2 △ = f ◦ g − 1 2 . W e can then write (47) as p ( n ∈ R d 2 ) > 1 − δ. (49) Since g 2 is a con tinuous monotonic function, for an y δ , 0 ≤ δ ≤ 1, t here exists an integer N 2 ( δ ) suc h that for N t > N 2 ( δ ), p ( n ∈ R d 2 ) > 1 − δ . Therefore, n ∈ R d 2 in pr o babilit y as N t → ∞ , thus prov ing the base case. Induction Hyp othesis: Let n ∈ R d m − 1 in probabilit y as N t → ∞ . Induction S tep: W e need t o pro ve that n ∈ R d m in probabilit y as N t → ∞ . F or n to b elong to R d m , n mu st satisfy the follo wing equation for all p ossible m -tuples ( i 1 , i 2 , · · · , i m ):  n + H ( x − d ) +  m X j =1 h i j d i j  T  m X j =1 h i j d i j  ≥ 0 , (50) whic h can b e written as  n + H ( x − d )  T  m − 1 X j =1 h i j d i j  +  n + H ( x − d )  T h i m d i m ≥ −k P m − 1 j =1 h i j d i j k 2 − k h i m k 2 − 2  P m − 1 j =1 h i j d i j  T h i m d i m . (51) Ho w eve r , w e kno w from the induction h yp othesis that ( n + H ( x − d )) T ( P m − 1 j =1 h i j d i j ) ≥ −k P m − 1 j =1 h i j d i j k 2 . Also, since n ∈ R d 1 , w e know that ( n + H ( x − d )) T h i m d i m ≥ −k h i m k 2 . Therefore, if the term 2( P m − 1 j =1 h i j d i j ) T h i m d i m in the RHS of (51) w ere 0, then (50) w ould ha ve b een trivially satisfied. W e now sho w that the con tribution of the term 2 ( P m − 1 j =1 h i j d i j ) T h i m d i m when compared to the other tw o terms in the RHS (5 1 ) con v erges to 0 as N t → ∞ . Define a r.v. v m △ = 2( P m − 1 j =1 h i j d i j ) T h i m d i m k P m − 1 j =1 h i j d i j k 2 + k h i m k 2 . Our ob jectiv e is to show that as N t → ∞ , v m → 0 in probability . This is equiv alent to proving that w m △ = v m + 1 = k P m j =1 h i j d i j k 2 k h i m k 2 + k P m − 1 j =1 h i j d i j k 2 con v erges to one in probability as N t → ∞ . W e can write w m as w m = k P m j =1 h i j d i j k 2 P m j =1 k h i j k 2 k h i m k 2 P m j =1 k h i j k 2 + k ( P m − 1 j =1 h i j d i j ) k 2 P m j =1 k h i j k 2 . (52) F rom Lemma 5, w e know that for an y in teger m , 1 ≤ m ≤ 2 N t , it is true t ha t P m k =1 P m j = k +1 h T i j h i k d i j d i k P m j =1 k h i j k 2 con v erges to 0 in probabilit y as N t → ∞ . By Slutsky’s theorem, this is equiv alent t o 2 P m k =1 P m j = k +1 h T i j h i k d i j d i k P m j =1 k h i j k 2 + 1 = k P j = m j =1 h i j d i j k 2 P m j =1 k h i j k 2 p − → 1 (53) 14 as N t → ∞ . W e shall use this result t o prov e the conv ergence of w m in (52). Using (53), it can b e seen that the n umerator of w m in (52) con v erges to 1 as N t → ∞ , i.e., k P m j =1 h i j d i j k 2 P m j =1 k h i j k 2 p − → 1 , as N t → ∞ . (54) In t he denominator of (5 2), it can b e sho wn that the term k h i m k 2 P m j =1 k h i j k 2 p − → 1 m , as N t → ∞ . (55) The 2nd term in the denominator o f (52 ) can b e rewritten a s k ( P m − 1 j =1 h i j d i j ) k 2 P m j =1 k h i j k 2 = k ( P m − 1 j =1 h i j d i j ) k 2 P m − 1 j =1 k h i j k 2 k h i m k 2 P m − 1 j =1 k h i j k 2 + 1 . (56) Similar to the deriv ation of (53 ) , we can claim that the nume ra tor in (56) con ve rg es to one in probability . F rom Slutsky’s theorem, it can b e sho wn that k h i m k 2 P m − 1 j =1 k h i j k 2 con v erges to 1 m − 1 in probabilit y . Using t his and Slutsky’s theorem, it can b e shown that (56) conv erges to m − 1 m in pro babilit y . 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T aricco , a nd A. T ulino , “Performance of space-time codes for a la r ge num b er of antennas,” IEEE T r ans. Inform. The ory , v ol. 48 , no. 7, pp. 1794-180 3, July 2002. [23] A. K. Basu, Me asur e The ory and Pr ob ability , P ren tice-Hall of India, 2004 . [24] H. T aok a a nd K . Higuchi, “Field ex periment on 5-Gbit/s ultra-high-s p eed pa c ket tra ns mission using MIMO multiplexing in bro adband pac ket radio access,” N TT DoCoMo T e ch. Journ., vol. 9, no. 2, pp. 25-31 , September 2007. 16 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20 25 Nt = Nr = 50 Nt = Nr = 200 Nt = Nr = 600 Figure 1: Simulated p df of z u n , d for n = 2 N t for increasing N t = N r . 4-QAM. The p df tends to wards Dira c delta f unction a t zero. 7 7.5 8 8.5 9 9.5 10 10.5 11 10 −4 10 −3 10 −2 10 −1 Average Received SNR (dB) Bit Error Rate Nt = Nr = 1 Nt = Nr = 10 Nt = Nr = 50 Nt = Nr = 100 Nt = Nr = 200 Nt = Nr = 600 SISO AWGN BER improves with increasing Nt = Nr V−BLAST, 4−QAM MMSE initial vector Figure 2: Sim ulated BER p erformance of the LAS detector f o r V- BLAST as a function of a ve ra ge receiv ed SNR for increasing v alues of N t = N r . MMSE initial v ector, 4-QAM . LAS detector a c hiev es near SISO A W G N p erformance at high SNRs for large N t = N r . 17 10 0 10 1 10 2 10 3 8 10 12 14 16 18 20 22 24 26 28 Number of antennas, Nt = Nr Average received SNR required (dB) LAS Detector SISO AWGN Near SISO AWGN Performance V−BLAST, 4−QAM Nt = Nr, MMSE initial vector Target BER = 10 −3 Figure 3: Av erage receiv ed SNR r eq uired to ac hiev e a tar g et BER of 10 − 3 in V-BLAST for increasing v alues of N t = N r for 4-QAM . LAS detector with MMSE initial v ector. LAS detector a c hiev es near SISO A W G N p erformance for large N t = N r . 10 0 10 1 10 2 10 3 10 4 15 20 25 30 35 40 45 50 Number of antennas, Nt = Nr Average received SNR required (dB) LAS Detector SISO AWGN Target BER = 10 −4 V−BLAST, 16−QAM Nt = Nr, MMSE initial vector Figure 4: Av erage receiv ed SNR r eq uired to ac hiev e a tar g et BER of 10 − 4 in V-BLAST for increasing v alues of N t = N r for 16-QAM . LAS detector with MMSE initia l vec to r. LAS detector p erformance a pproac hes SISO A WGN p erformance for large N t = N r . 18