LR-aided MMSE lattice decoding is DMT optimal for all approximately universal codes

LR-aided MMSE lattice decoding is DMT o ptimal for all approximately uni v ersal codes Joakim Jald ´ en Institute of Communica tions and Radio-Frequency Engineer ing V ienna Un i versity o f T echnolo gy V ienn a, Austria Email: jjalden@nt.tuwien. ac.at Petros Elia Departemen t of Mobile Commu nications EURECOM Sophia Antipolis, France Email: elia@eurecom.f r Abstract —Currently fo r th e n T × n R MIMO channel, any explicitly constructed space-time (ST) designs that achieve opti- mality with respect to t he diversity multi plexing tradeoff (DM T) are known to do so only when decoded using maximum like- lihood ( ML) decoding, which may in cur prohibitive decoding complexity . In this paper we pro ve th at MMSE regularized lattice decoding, as well as the computationally efficient lattice reduction (LR) aided MMSE d ecoder , allows for efficient and DMT optimal decodin g of any app rox imately univers al lattice- based code. The res ul t identifies f or the first time an explicitly constructed encoder and a computationally effi cient decoder that achiev e DMT optimality for all m ult iplexing gains and all channel dimensions. The results hold irrespectiv e of the fadin g statistics. I . I N T RO D U C T I O N The introdu ction of MIMO-related scenarios such as MIMO-OFDM an d co operative-di versity has introd uced the need for multi-dimen sional enc oding schem es which can be efficiently deco ded an d which can guar antee good error prob- ability perfor mance under a plethora of channe l topolog ies and statistics. T ow ards addressing this need, substantial amo unts of research h as looked to improve and analy ze the err or pro babil- ity performa nce and decoding c omplexity of different MIMO encodin g/decodin g schemes, with such work of ten focusing on applying specific error p robability perform ance measures to analyze the behavior of specific transmission sche mes as they are decoded by different, o ptimal and sub optimal decode rs. A. Related work From the encoding point o f view , substantial resear ch has aimed towards provid ing space- time (ST) codes with structure that allo ws for go od error prob ability per forman ce and ef ficient decodin g. Such work include [ 1] which provides codes based on Clifford algeb ras that can be seen as gen eralizations of orthog onal design s an d which have goo d maximum likelihood (ML) decodin g co mplexity . Fu rthermo re, the work in [2] describes co des that take advantage of channel asymmetr y ( n R < n T ) to achieve good error probability perfor mance with reduced decod ing complexity . From the point of v ie w o f detection an d decod ing, ML- based decod ers ar e known to provid e optimal per forman ce but often do so o nly with pr ohibitive c omputation al comp lex- ity . Dif fere nt comp utationally efficient sub-op timal receiver architecture s were introduced , with work focusing o n linear receivers (MMSE and Z F) and their d ecision feed back equal- ization (DFE) v arian ts [3 ], as we ll as lattice reduction (LR) aided v ersio ns of these [4 ], [5]. Substantial work further loo ked to analyze th e perfo rmance of such r eceiv ers. For examp le, the work in [6] showed that L R-aided ZF decoding c an achieve maximal receiv e diversity for uncod ed V -BLAST . W ith the rece nt emergenc e of the d iv ersity multiplexing tradeoff (DMT) [7] descr ibing, in a u nified m anner, the f unda- mental perfor mance limits of outage-limited MIMO comm uni- cations, research h as fo cused o n establishin g the DMT perfor- mance of dif fer ent enco der/decod er architectures. The work in [8] p roved that the naive lattice decoder fails to achie ve the diversity mu ltiplexing trad eoff in gen eral. Furthermor e, in [9], DMT a nalysis reveals that both ZF and MMSE linea r receivers are suboptimal in terms of th eir achievable diversity . An importa nt step towards establishing that DMT optimality can b e achieved with computatio nally e ffi cien t encoders and decoder s was presented in [10 ]. By using an ensemble of lattice codes, an MMSE p re-proc essing step, an d an optimal lattice translate, it was shown that there exists lattice codes that, when d ecoded using lattice decodin g, achiev e optimal DMT perform ance over the i.i.d. Rayleigh fading chan nel. B. Contributions o f present work In this work, we extend the results in [10] b y bypassing random ensemble argume nts to show that DMT optimality is achiev able for all multiplexing gains, b y employin g explicitly constructed enco ders and computation ally efficient deco ders. Specifically , we con sider explicitly constructed appro ximately universal codes [11]–[1 3], and regularized lattice dec oding. It is shown th at DMT optimality ho lds for all fading statistics. The key to DMT optimality , as will be shown later, is the MMSE regular ization of th e decoding metric. W e also establish the DMT op timality of th e co mputation ally efficient LLL based LR-aided MMSE decode r [5] . I I . S Y S T E M M O D E L A N D S P AC E - T I M E C O D I N G W e con sider the quasi-static n T × n R MIMO c hannel mod el Y = HX + W (1) where Y ∈ C n R × T , H ∈ C n R × n T , X ∈ C n T × T for T ≥ n T , W ∈ C n R × T , and vec( W ) ∼ N C ( 0 , I ) . Her e, we use vec( W ) to denote the column -by-co lumn vectorization of W , and N C ( 0 , I ) to denote a rotatio nally inv arian t circu larly symmetric complex normal random vector with unit variance. The co de matrices X are d rawn f rom a spac e-time block co de X , satisfying the power co nstraint 1 T k X k 2 F ≤ ρ ∀ X ∈ X . (2) A. The diversity multiplexing tradeoff The rate of an ST co de X is gi ven b y R , T − 1 log |X | and a sequence of codes o r scheme , ind exed b y ρ , is said to have a multiplexing gain of r if (c.f . [ 7]) r = lim ρ →∞ R log ρ . When ˆ X is th e outp ut of the d ecoder (not necessarily ML) giv en Y and H , the diversity gain d of the scheme is d = − lim ρ →∞ log P( ˆ X 6 = X ) log ρ . The c entral r esult of [7] is that for a fixed multiplexing g ain r , there is a fundam ental limit to the diversity gain: d ≤ d out ( r ) , − lim ρ →∞ log P(log det( I + ρ HH † ) ≤ ¯ R ) log ρ , (3) where ¯ R = r log ρ and where H † denotes the He rmitian tra ns- pose of H . In the case of i.i.d. Rayleigh fading, d out ( r ) is g iv en by th e piecewise linear curve conne cting ( n R − k )( n T − k ) for k = 0 , 1 , . . . , min( n R , n T ) [7]. A scheme which satisfies (3) with equality for som e r is said to b e DMT o ptimal for this multiplexing ga in. W e will in the following make use of the . = notation (c.f. [7]) fo r exponential equalities wh ere f ( ρ ) . = ρ b is taken to mean lim ρ →∞ log f ( ρ ) / log ρ = b . The symb ols . ≤ and . ≥ are defined similarly . Let O ǫ denote the ǫ -n o-outag e set given b y O ǫ , { H | lo g det( I + ρ HH † ) > ( r + ǫ ) log ρ } . (4) As noted in [ 14] (see also [7]), a sufficient condition for DMT optimality , regard less of the fading statistics, is that P( ˆ X 6 = X | H ∈ O ǫ ) . = ρ −∞ (5) for all ǫ > 0 , i.e. that the co nditional probab ility of d ecoding error vanishes exponen tially fast f or channels in O ǫ . B. Appr oximately universal la ttice space-time codes In this paper, we consider a sequence of lattice ST co des: X = { X = Ma t( θ Gs ) | s ∈ S r } (6) where θ ∈ R + , G ∈ C n T T × κ for some κ ∈ N , and where S r , { s ∈ Z κ G | k s k 2 ≤ ρ rT κ } , (7) where Z G = Z + i Z denotes the set of Gaussian integers 1 . Mat( x ) den otes the n T × T matrix f ormed via co lumn-b y- column stackin g of consecutive n T -tuples o f x ∈ C n T T . E ach 1 Extensions of our main results to other constellati ons, such as the HEX constel lations, is straight forward and will appear in a journa l version of this work. It is omitted here due to lack of space. codeword is thus associated, via th e lattice generator matrix G , to a unique data vector s ∈ S r ⊂ Z κ G . The ch oice of S r in (7) ensu res a multiplexing gain r and ch oosing θ in ord er to satisfy the p ower constraint (c.f. (2)) with eq uality implies th at θ 2 . = ρ 1 − rT κ . W e assume throu ghout that the lattice g enerator matrix G is indepen dent of ρ and r . A key feature of lattice ST codes is tha t they may b e decoded by a class of deco ders known as lattice d ecoders [10] . T o this end we note that the in put-outp ut relation from s to y , vec( Y ) is y = Fs + w (8) where w , vec( W ) , an d where the effective chan nel matrix is F , θ ( I T ⊗ H ) G . (9) The ML decod er is thus equivalent to (c.f. [10]) ˆ s ML = arg min ˆ s ∈S r k y − F ˆ s k 2 , (10) and may be app roximated by a lattice decoder, wher eby the constellation boundary imposed by S r is ignored [10]. The decodin g o f the lattice ST codes will be discussed in greater detail in Sections III and IV. Let µ i ( A ) denote th e i th eigenv alue o f a Hermitian matrix A ∈ R q × q , o rdered such that µ 1 ( A ) ≤ . . . ≤ µ q ( A ) . Fur ther , let n , min( n T , n R ) . A sequen ce of ST co des (n ot necessarily lattice codes) is a ppr oximately universal [14] over the n T × n R channel if and only if (c.f. [11 ]) n Y i =1 µ i ( ∆∆ † ) . ≥ ρ n − r (11) for a ll codew ord difference matrices ∆ = X 1 − X 2 , w here X 1 , X 2 ∈ X , and X 1 6 = X 2 . It is known that app roximate universality is a sufficient con dition for DMT optim ality for any fadin g statistics, assum ing ML decoding [14 ]. For approx imately universal codes we hav e the following lemma that follows directly from [1 4, Eq uation (21)] 2 . Lemma 1: Let ∆ = X 1 − X 2 for X 1 , X 2 ∈ X , X 1 6 = X 2 . If X is approxim ately universal over th e n T × n R channel and H ∈ O ǫ it follows th at k H ∆ k 2 F . ≥ ρ ǫ n . For appr oximately universal lattice ST codes we may a lso giv e th e following corollar y to Lemma 1 . The proof is giv en in the append ix. Cor o llary 2: Let X be a lattice ST code of the form (6) which, fo r a fixed lattice ge nerator m atrix G , is appro ximately universal for all multip lexing gains in a ne ighborh ood of r . Then, for s 1 , s 2 ∈ S r + ζ , s 1 6 = s 2 , a nd H ∈ O ǫ it holds that k F ( s 1 − s 2 ) k 2 . ≥ ρ ǫ − ζ n − ζ T κ (12) for sufficiently small ζ , 0 < ζ < ǫ . 2 In relation to the result presented here, we point out a small typo in equati ons (20) and (21) in [14], where in (20) 2 R (1+ ǫ ) ( | λ 1 | · · · | λ n m | ) 2 /n m should be replaced by (2 R (1+ ǫ ) | λ 1 | 2 · · · | λ n m | 2 ) 1 /n m , c.f. (17) in the same paper . Note also the slightly dif ferent definition of O ǫ in our paper and O ǫ in [14], where in the definition of O ǫ we use ( r + ǫ ) in place of r (1 + ǫ ) . Approx imately universal lattice ST c odes, which satisfy the conditions of Corollary 2 , a re k nown to exist f or any ( n R , n T ) -tuplet an d multip lying gain r , see e.g. [1 1]. The codes in [11] are in fact, for a fixed G , appr oximately un iv ersal over all r ∈ [0 , n ] . In wh at f ollows we only consider co des for which Corollary 2 applies. W e also point out that in the definition of approxima tely universal lattice ST co des we require that the set o f data symbols S r is giv en by the Gaussian integers within a h yper-sphere of radiu s ρ rT 2 κ . It is read ily seen that pre sented analysis carries over (at the expense o f extra notational comp lexity) to the more practica l case wher e the constellation is cubic, i.e. |ℜ ( s k ) | , |ℑ ( s k ) | ≤ ρ rT 2 κ , w hich also maintains the scheme’ s multiplexing gain . I I I . R E G U L A R I Z E D L A T T I C E D E C O D I N G As noted in Section II, ML decod ing is equiv alent to solving (10). The naive lattice decoder (c.f. [10]) is obtained by simply ignorin g the constellation boun dary of S r ⊂ Z κ G : ˆ s 0 = ar g min ˆ s ∈ Z κ G k y − F ˆ s k 2 . (13) W e count th e ev ent when the decoder decides in fa vor of a codeword not in the constellation as an erro r . The benefit of using (13) in place of (10) is that one may a void the potentially complicated bound ary control, an d ap ply too ls f rom lattice reduction theory for solving (13). However , as argued in [10] and subsequ ently proved in [8], th e naive lattice decoder is not in g eneral DM T o ptimal. I t was howe ver also shown in [10] that the problem is not with lattice coding and decod ing per se, but r ather with the naive imp lementation. Intuitively , as the ML decoder (c.f. (10)) is DMT o ptimal for approx imately u niv ersal codes and the naiv e lattice d ecoder is not, the sub-o ptimality o f the naiv e lattice decode r m ust stem from the fact that it decide s, with high pro bability , in fav or of codewords that do no t belong to the constellation S r . Note h ere that s / ∈ S r , s ∈ Z κ G , implies k s k 2 > ρ rT κ . Thu s, h aving the decoder penalize vectors s with large nor m, on e c an expect to red uce the prob ability of out- of-con stellation errors. Th is amounts to regularization of the dec oding metric and we let the α -regularized lattice decoder be given by ˆ s α = arg min ˆ s ∈ Z κ G k y − F ˆ s k 2 + α k s k 2 . (14) Clearly , fo r α = 0 the regularized lattice decoder coincides with the n ai ve lattice deco der . W e will ho wever in what follo ws show that by choosing α app ropriately , o ne can achieve DMT optimality for any app roximately u niversal co de. Th e resu lt is captured by the following theor em. Theor em 3: App roximately univ ersal lattice codes, decod ed using the α -regularized decod er with α = ρ − rT κ , achiev e DMT optimality and do so irrespective of the fading statistics. Pr o of: W e will show that when s is the data vector c orre- sponding to the transmitted codeword o f an approximate ly universal code, wh en α = ρ − rT κ and w hen ǫ > 0 , using the α - regularized decoder in (14 ) implies that P  ˆ s α 6 = s | H ∈ O ǫ  . = ρ −∞ . I n other words, the co nditional pr obability of error van- ishes expo nentially fast for channels (strictly) not in outage, establishing DMT optimality . T o wards this en d, gi ven ǫ > 0 , choose ζ an d δ such that 0 < ζ < ǫ , wh ere ζ is sufficiently small f or Corollary 2 to apply , and such that (c.f. (1 2)) ǫ − ζ n − ζ T κ > δ > 0 and ζ T κ > δ > 0 . (15) This can always b e done. Assume also that H ∈ O ǫ and that the noise vector w satisfies k w k 2 ≤ ρ δ . Consider first th e α -regularized metric (c.f. (1 4)) for the transmitted data vector s ∈ S r . As (c.f. (8)) k y − Fs k 2 = k w k 2 it follows that k y − Fs k 2 + α k s k 2 ≤ ρ δ + αρ rT κ . = ρ δ (16) where we used that k w k 2 ≤ ρ δ , tha t α = ρ − rT κ and that s ∈ S r which implies k s k 2 ≤ ρ rT κ (c.f. (7)). For any data vector ˆ s ∈ S r + ζ , ˆ s 6 = s , we note that k y − F ˆ s k = k F ( s − ˆ s ) + w k ≥ k F ( s − ˆ s ) k − k w k . As k F ( s − ˆ s ) k . ≥ ρ 1 2 ( ǫ − ζ n − ζ T κ ) by Corallary 2 and as k w k ≤ ρ 1 2 δ , it follows by (15) th at k y − F ˆ s k 2 . ≥ ρ ǫ − ζ n − ζ T κ and k y − F ˆ s k 2 + α k ˆ s k 2 . ≥ ρ ǫ − ζ n − ζ T κ , (17) for any ˆ s ∈ S r + ζ , ˆ s 6 = s . For ˆ s / ∈ S r + ζ , ˆ s ∈ Z G , it holds that k ˆ s k 2 > ρ ( r + ζ ) T κ (c.f. (7)) by which it follows th at k y − F ˆ s k 2 + α k ˆ s k 2 ≥ αρ ( r + ζ ) T κ ≥ ρ ζ T κ . (18) By defining ξ , min  ǫ − ζ n − ζ T κ , ζ T κ  where ξ > δ du e to (1 5), and comb ining (17) and (18) it follows that k y − F ˆ s k 2 + α k ˆ s k 2 . ≥ ρ ξ (19) for any ˆ s ∈ Z κ G , ˆ s 6 = s . As δ < ξ (i.e. δ is strictly sm aller th an ξ ), it f ollows by (16) and (19) that there is ρ 0 such that k y − Fs k 2 + α k s k 2 < k y − F ˆ s k 2 + α k ˆ s k 2 for any ˆ s ∈ Z κ G , ˆ s 6 = s , and ρ ≥ ρ 0 . T his implies th at the α -regularized decoder will make a cor rect decision. In oth er words, if H ∈ O ǫ , it follows that k w k 2 > ρ δ constitutes a nec- essary condition for an er ror to occur when ρ ≥ ρ 0 . Howe ver , as P  k w k 2 ≥ ρ δ  . = ρ −∞ due to th e exponential tails of the Gaussian distribution, we see th at P  ˆ s α 6 = s | H ∈ O ǫ  . = ρ −∞ and the claim of Theor em 3 follows.  The metric in (1 4) is not identical to th e metric used in the MMSE-GDFE d ecoder co nsidered in [1 0], althou gh the two metrics share some key features. In particular, if the lattice translate is omitted, it can be shown th at the metric in [10] is equiv alent 3 to (c.f. (14)) k y − Fs k 2 + ρ − 1 k θ Gs k 2 , (20) i.e. the regularization is app lied to the vectorized cod e word x = θ Gs instead o f s . It is a straightfo rward exercise to repeat the pr oof of Theorem 3 and sho w that dec oding with respect to (20) is also DMT optimal. T o th is end, note that θ 2 ρ − 1 . = ρ − rT κ . In fact, when G is an orth ogonal matrix, as is the case for perfec t codes [12], (20) redu ces to (14). This confirms th e observation mad e in [10] that the “m agic” ingredien t o f the GDFE-MMSE deco der , in terms of DMT optimality , is MMSE pre-pro cessing. Similarly , it reveals that α = ρ − rT κ is the correspondin g “magic” parameter f or the regularized lattice deco der which mo ti vates us to refer to the regularized lattice d ecoder with α = ρ − rT κ as the MMSE regularized lattice deco der . It sh ould howe ver be no ted that the choice of α = ρ − rT κ can n aturally also be directly obtained from th e lin ear MMSE filter for s gi ven the observation y (c.f . (8) and Section IV). I V . L A T T I C E R E D U C T I O N A I D E D D E C O D I N G By “completing the squares”, the α -regularized metric may equiv alently b e written as k y − F ˆ s k 2 + α k ˆ s k 2 = k z − R ˆ s k 2 + c (21) where R ∈ C κ × κ is a square root factor of F † F + α I , i.e. R † R = F † F + α I , (22) where z , R −† F † y , a nd where c , y †  I − F † ( F † F + α I ) − 1 F  y ≥ 0 . (23) The α -regularized d ecoder can thus be expressed as ˆ s α = arg min ˆ s ∈ Z κ G k z − R ˆ s k 2 . (24) The op timization prob lem in (2 4) howev er still require th e solution to a closest vector problem (CVP), which is NP-hard in gener al. This makes sub-o ptimal solutions appealin g. T o this end, consider the deco der given by ˆ s α, MMSE = arg min ˆ s ∈ Z κ G k R − 1 z − ˆ s k 2 . (25 ) The d ecoder in (25) is easily implemented b y co mponen t-wise round ing o f R − 1 z to the nearest integer vector . I t is relativ ely straightfor ward to verify tha t R − 1 z = ( F † F + α I ) − 1 F † y which implies th at the so lution to ( 25) correspon ds to the standard linear MMSE decod er . Y ao an d W ornell [ 4] suggested the use of lattice red uction to improve the app roximation qua lity when replacing ( 24) by 3 Note also that the metric in [10] is expre ssed in a real val ued form which allo ws for more general code designs. The real value d reformulation will be considere d in a journal version of this wor k. (25). The key idea b ehind this ap proach is to note that (24) is equiv alent to min ˜ s ∈ Z κ G k z − R T ˜ s k 2 (26) where T is a un imodular matrix , i.e. T is a one-to-o ne map from Z κ G to Z κ G or equivalently , T ∈ Z κ × κ G and | det( T ) | = 1 . W e write ˜ R = R T in wh at follows, an d refer to ˜ R as th e lattice redu ced chann el. The p rocess of finding T , giv en R , is known as lattice redu ction. The sub-o ptimal solution corre sponding to (26) is giv en b y ˜ s α, LR − MMSE = arg min ˜ s ∈ Z κ G k ˜ R − 1 z − ˜ s k 2 (27) where ˜ R = R T and the appr oximate solutio n to (2 4) is ˆ s α, LR − MMSE = T ˜ s α, LR − MMSE . (28) The ke y observation o f [ 4] is that by making ˜ R well con- ditioned (by the app ropriate choice of T ), the quality o f the approx imation may be significan tly improved. The resulting decoder ( defined by (27) and (2 8)) is k nown as th e LR-a ided MMSE decoder [5]. The most c ommonly con sidered lattice redu ction algorithm is the computatio nally efficient LLL alg orithm [15]. T he LLL algorithm is also known to provide maximum recei ve di versity , at multiplexing gain r = 0 and u nder i.i.d. Rayleigh fading, for u ncoded V -BLAST transmissions [ 6]. In wh at fo llows we prove that LLL based LR-aided d ecoding can in fact achie ve the most g eneral d iv ersity-r elated optimality , by showing that the LLL based LR-aide d MMSE decod er can, in the context of lattice co des, achieve th e maxima l div ersity gain f or all multiplexing gain s r and fading statistics. Theor em 4: App roximately universal lattice codes, wh en decoded using the LLL based LR-aided MMSE decod er , achieve the optimal DMT tradeoff, and d o so irrespective of fading statistics. Pr o of: T o prove the above, we will demo nstrate that P  ˆ s α, LR − MMSE 6 = s | H ∈ O ǫ  . = ρ −∞ . T o th is end, let ˜ R = R T b e the LLL lattice redu ced channel matrix. It follows by the bo unded o rthogon ality defect of L LL r educed bases (c.f. [15] and the pr oof in [6]) that there is a co nstant K κ > 0 , indepen dent of R , for which σ max ( ˜ R − 1 ) ≤ K κ λ ( R ) . (29) where σ max ( ˜ R − 1 ) is the largest sing ular v alue of ˜ R − 1 and where λ ( R ) , min c ∈ Z κ G \{ 0 } k Rc k (30) denotes the shor test vector in the lattice gen erated b y R . Although th e pr oof in [6] was given for real valued bases it straightfor wardly extends to the com plex case, c.f . [1 6]. Assume, as in the proof of Theorem 3, that H ∈ O ǫ and k w k 2 ≤ ρ δ . For ˆ s ∈ Z κ G , ˆ s 6 = s , it fo llows that k z − R ˆ s k = k ( z − Rs ) + R ( s − ˆ s ) k ≤ k R ( s − ˆ s ) k + k z − Rs k and k R ( s − ˆ s ) k ≥ k z − R ˆ s k − k z − Rs k . ≥ ( ρ ξ − c ) 1 2 − k z − Rs k (31) where the last inequ ality follows by combining (19) and (21). As c . ≤ ρ δ and k z − R s k 2 . ≤ ρ δ by (16) and (21), and sin ce ξ > δ , we may conclud e from (31) that k R ( s − ˆ s ) k 2 . ≥ ρ ξ , f or any ˆ s ∈ Z κ G , ˆ s 6 = s . By id entifying c = s − ˆ s ∈ Z G \{ 0 } in (30) it follows tha t λ 2 ( R ) . ≥ ρ ξ and by (29) that σ 2 max ( ˜ R − 1 ) . ≤ ρ − ξ . (32) From (27) and (28) it m ay be seen that ˆ s α, LR − MMSE 6 = s if and only if ˜ s α, LR − MMSE 6 = ¯ s where ¯ s = T − 1 s . Th e metric in (27), ev aluated for ˜ s = ¯ s , satisfies k ˜ R − 1 z − ¯ s k 2 = k ˜ R − 1 ( z − ˜ R ¯ s ) k 2 ≤ σ 2 max ( ˜ R − 1 ) k z − ˜ R ¯ s k 2 . ≤ ρ δ − ξ (33) where the last ineq uality f ollows by (32) together with k z − Rs k 2 . ≤ ρ δ and Rs = ˜ R ¯ s . For ˜ s ∈ Z κ G , ˜ s 6 = ¯ s , it follows that k ˜ R − 1 z − ˜ s k = k ˜ R − 1 z − ¯ s + ( ¯ s − ˜ s ) k ≥ k ¯ s − ˜ s k − k ˜ R − 1 z − ¯ s k By no ting that k ¯ s − ˜ s k 2 ≥ 1 if ¯ s 6 = ˜ s , th at k ˜ R − 1 z − ¯ s k 2 . ≤ ρ δ − ξ (c.f. (33)) and that δ − ξ < 0 , it f ollows that k ˜ R − 1 z − ˜ s k 2 . ≥ ρ 0 (34) for ˜ s ∈ Z κ G , ˜ s 6 = ¯ s . Combin ing (3 3) and (34) y ields k ˜ R − 1 z − ¯ s k 2 < k ˜ R − 1 z − ˜ s k 2 for all ˜ s ∈ Z κ G , ˜ s 6 = ¯ s , and sufficiently large ρ imp lying that the decision of the LR- aided MMSE decoder (c.f. (27) and (2 8)) is c orrect. As in the pr oof of Th eorem 3, we see that given H ∈ O ǫ it m ust hold that k w k 2 > ρ δ for an err or to occur, which implies P  ˆ s α, LR − MMSE 6 = s | H ∈ O ǫ  . = ρ −∞ .  V . C O N C L U S I O N In this paper, we co nsider the prob lem o f efficiently d e- coding appr oximately univ ersal lattice ST c odes. W e show that MMSE regularized lattice decodin g in gen eral, and the computatio nally efficient LLL based LR-aided MMSE d ecoder in particular, realize the maximum rec ei ve di versity a nd th us DMT optimality for app roximately uni versal lattice codes. The result ho lds for any fadin g statistics and confirms that the key to achieving DMT optimality is the regularizatio n o f the decodin g metric provid ed by the MMSE decod er . A P P E N D I X Pr o of of Cor ollary 2: By th e eq uiv alent chan nel mod el (c.f. (6), (8) and (9)) and Lemm a 1 it fo llows that k H ( X 1 − X 2 ) k 2 F = k F ( s 1 − s 2 ) k 2 . ≥ ρ ǫ n for s 1 , s 2 ∈ S r , s 1 6 = s 2 , g iv en that H ∈ O ǫ . For the un - normalized equiv alent ch annel ¯ F , ( I T ⊗ H ) G we have θ 2 k ¯ F ( s 1 − s 2 ) k 2 = ρ 1 − rT κ k ¯ F ( s 1 − s 2 ) k 2 . ≥ ρ ǫ n , where ¯ F is indepen dent of ρ and r ( note that F = θ ¯ F ). Consider now the applicatio n of Lemma 1 to a scheme with multiplexing gain r ′ = r + ζ , where 0 < ζ < ǫ . By the assumption that H ∈ O ǫ it follows that log det( I + ρ HH † ) > ( r + ǫ ) log ρ = ( r ′ + ǫ − ζ ) log ρ which by the application of Lemm a 1 imp lies tha t ρ 1 − r ′ T κ k ¯ F ( s 1 − s 2 ) k 2 . ≥ ρ ǫ − ζ n (35) for s 1 , s 2 ∈ S r ′ , s 1 6 = s 2 . Rewriting (35) it term s of r yields ρ 1 − rT κ k ¯ F ( s 1 − s 2 ) k 2 . ≥ ρ ǫ − ζ n − ζ T κ or eq uiv alently k F ( s 1 − s 2 ) k 2 . ≥ ρ ǫ − ζ n − ζ T κ for any s 1 , s 2 ∈ S r ′ = S r + ζ , s 1 6 = s 2 .  A C K N OW L E D G M E N T This work was supported by the Euro pean Com mission throug h the FP6 STREP project MASCO T (IST -026 905) and in the f ramework of the FP7 Network o f Excellence in W ireless COMmunication s NEWCOM++ (IST -2167 15). R E F E R E N C E S [1] S. Karmakar and B. S. Rajan, “Mult igroup-decoda ble ST BCs from Clif ford algebras, ” IEEE T ransact ions on Information Theory , vol. 55, no. 1, pp. 223–231, Jan. 2009. [2] C. Holla nti and K. Ranto, “ Asymmetric space -time block codes for MIMO systems, ” in Pr oc. IEEE Information Theory W orkshop on Informatio n Theory for W ir eless Networks , July 2007, pp. 101–105. [3] D. Ts e and P . V iswanath, Fundamentals of wirel ess communicati on . Cambridge Unive rsity Press, 2005. [4] H. Y ao and G. W . W ornell, “Lattice- reduction-aided detec tors for MIMO communic ation systems, ” in P r oc. IEEE Global Communi cations Confer ence (GLOBECOM) , vol. 1, Nov . 2002, pp. 424–428. [5] D. W ¨ ubben, R. Bohnke, V . Kuhn, and K.-D. Kammeyer , “Near - maximum-lik elihood detection of MIMO systems using MMSE -based latti ce reduction, ” in Pr oc. IEEE Internation al Confer ence on Commu- nicati ons (ICC) , vol. 2, June 2004, pp. 798–802. [6] M. T aherzade h, A. Mobasher , and A. K. Khandan i, “LLL reduction achie ves the recei ve di versity in MIMO decoding, ” IEEE T rans. Inform. Theory , vol. 53, no. 12, pp. 4801–4805, Dec. 2007. [7] L. Zheng and D. N. C. Ts e, “Div ersity and multiple xing: A fundamental tradeof f in multiple -antenna chan nels, ” IEEE T rans. Inform. Theory , vol. 49, no. 5, pp. 1073–1096, May 2003. [8] M. T aherzad eh and A. K. Khandani, “On the limitations of the naiv e latti ce decoding, ” in P r oc. IEE E International Syposium on Information Theory (ISIT) , June 2007, pp. 201–204. [9] K. R. Kumar , G. Caire, and A. L. Moustakas, “The di versity- multiple xing tradeof f of linear MIMO recei vers, ” in Proc . IEEE Infor - mation Theory W orkshop (ITW) , Sept. 2007, pp. 487–492. [10] H. E l Gamal, G. Caire, and M. O. Damen, “Lattic e coding and decoding achie ve the optimal di versity-mul tiplexi ng tradeof f of MIMO channels, ” IEEE Tr ans. Inform. Theory , vol. 50, no. 6, pp. 968–985, June 2004. [11] P . Elia, B. A. Sethuraman, and P . V . Kumar , “Perfect space-time codes for any number of transmit antennas, ” IEEE T rans. Inform. Theory , vol. 53, no. 11, pp. 3853–3868, Nove mber 2007. [12] F . Oggier , G. Rekaya, J. C. Belfiore, and E. V iterbo, “Perfect space-time block cod es, ” IEEE T rans. Inform. Theory , v ol. 52, no. 9, pp. 3885–3902, Sept. 2006. [13] K. R. Kumar and G. Caire, “Constructio n of structures LaST codes, ” in Pr oc. IEEE International Syposium on Information Theory (ISIT) , July 2006, pp. 2834–2838. [14] S. T avildar and P . V iswana th, “ A pproximat ely univ ersal codes ov er s lo w- fadi ng channels, ” IEEE T rans. Inform. Theory , vol. 52, no. 7, pp. 3233– 3258, July 2006. [15] A. K. Lenstra, H. W . Lenstra, and L. Lov ´ asz, “Fact oring polynomials with ration al coef ficient s, ” Matematisc he Annalen , vol. 261, no. 4, pp. 1432–1807, Dec. 1982. [16] Y . H. Gan and W . H. Mo w , “Comple x lattice reduction algorit hms for lo w-complexi ty MIMO detection , ” in Proc. IEEE Global Communica- tions Confere nce (GLOBECOM) , vol. 5, Nov . 2005, pp. 2953–2957.