Achieving a vanishing SNR-gap to exact lattice decoding at a subexponential complexity

Achie ving a v anishing SNR-gap to e xact lattice decoding at a subexponential comple x ity Arun Singh, Petros Elia and Joakim Jald ´ en Abstract —The work identi fies the first l attice decodin g solution that achiev es, in the general outage-limited MIMO setting and in the high-rate and high-SNR limit, both a vanishing gap to the error -p erfo rmance of the (DMT optimal) exact solution of prepr ocessed lattice decoding, as well as a comput ational complexity that is subexponenti al in the numb er of codeword bits. The proposed solution employs lattice reduction (LR)-aided regularized (lattice) spher e decodin g an d p roper timeout policies. These perf ormance and complexity guarantees hold f or most MIMO scenarios, all reasonable fadin g statistics, all channel dimensions and all full-rate lattice codes. In sharp contrast to th e abov e very manageable complexity , the complexity of other standard preproces sed lattice decoding solutions is rev ealed here to be extremely h igh. Specifically the work is first t o quantify the complexity of these lattice (sphere) decoding solutions and to prov e th e surprising result that the complexity r equired to achieve a certain rate-reliability perfo rmance, i s exponential in the lattice d imensionality and in the number of codeword bits, and it in fact matches, in common scenarios, the complexity of ML-based solutions. Through thi s sharp contrast, the work was able to, for the first time, rigorously demonstrate and quantify the pivo tal role of lattice r eduction as a special complexity reducing ingredient. Finally the work analytically refines transceiv er DM T analysis which generally fails to address potentially massiv e gaps between theory and practice. Instead the adopted vanishing gap condition guarantees that the decoder’ s error curve is arbitrarily close, giv en a sufficiently high S NR, t o th e optimal error curve of exact solutions, whi ch is a much stronger condition than DMT optimality whi ch only guarantees an error gap that is subpolynomial in SNR, and can thus be unboun ded and g enerally unacceptable fo r practical implementations. I . I N T RO D U C T I O N The work applies to th e gene ral setting of outage-lim ited MIMO commu nications, wher e M IMO techniqu es offer sig - nificant advantages in terms of incre ased through put an d reliability , although at a cost of a potentially much h igher computatio nal comp lexity fo r de coding at th e receivers. This high comp lexity brings to th e fore the need for e fficient decoder s that tradeoff erro r-performance with complexity in a better manner than compu tationally expensive d ecoders like the strictly optimal max imum-likeliho od (ML) decoder . The researc h leading to these results has recei ved funding from the European Researc h Council under the European Community’ s Sev enth Frame- work Programme (FP7/2007-2013) / ERC grant agreement no. 228044, from the Swedish Foundati on for Strategic Research (SSF) / grant ICA08-0046, from FP7/2007-2013 grant agreement no. 257616 (CONECT), and from the Mitsubishi Electric R & D Centre E urope project Home-eNodeBS. A. Singh and P . Elia are with the Mobile Communications Department, EURECOM, Sophia Antipolis, France (email: { singhak, elia } @e urecom.fr) J. Jald ´ en is wit h the A CCESS Linna eus Cente r , Signal Processing Lab, KTH Roya l Institute of T echnol ogy , Stoc kholm, Sweden (email: jalden@kth.se) Specifically in terms of ML-based decoding, the use of th e brute-f orce M L decoder, introdu ces a comp lexity th at scales exponentially with the number of codew ord bits. If on the other hand, a small gap to the exact M L performance is acceptable, then different b ranch- and-b ound algorithms such as the sphere decoder (SD) have been kn own to accept red uced com puta- tional resources. Despite the r educed c omplexity of spher e decodin g, r ecent work in [1] has revealed that, to achieve a vanishing error-gap to optimal ML solution s, even such branch -and-b ound alg orithms generally req uire computation al resources that, albeit significan tly smaller than tho se required by a brute-f orce ML decode r , ag ain gr ow exp onentially in the rate and the d imensiona lity , an d remain pro hibitive for se veral MIMO scenario s. This high comp lexity required by ML- based de coding solu- tions, serves as f urther motivation fo r exploring other families of decoding meth ods. A natu ral alternati ve is lattice decoding obtained by simply re moving the co nstellation bo undar ies of the ML-based search , an action that loosely speakin g explo its a certain symmetry which in turn may yield faster implemen- tations. It is the case thoug h th at even with lattice decodin g, the compu tational comp lexity can be pro hibitive: finding the exact solution to the lattice decod ing problem is generally an NP hard problem (cf. [2]). At the same time though, the other extreme of very early termin ations of lattice deco ding, such as linear so lutions, h av e b een kn own to achieve computation al efficiency at th e expense th ough of a very sizab le, an d often unbou nded, gap to th e exact so lution of the lattice deco ding problem . In th is work we explore lattice dec oding solution s that, in conjunc tion wit h terminating policies, strike the proper balance between this expo nential complexity and expone ntial gap. A. System model W e conside r the g eneral m × n point-to-p oint multiple-inpu t multiple-ou tput model giv en by y = √ ρ Hx + w (1) where x ∈ R m , y ∈ R n and w ∈ R n respectively denote the tran smitted cod ew ords, the received sign al vectors, and the add iti ve white Gau ssian noise with u nit variance, wh ere the parameter ρ takes th e role of the signal to noise ratio (SNR), and where the fading matrix H ∈ R n × m is assumed to be ran dom, with elements dr awn fr om arbitrary statistical distributions. W e co nsider that o ne u se o f (1) corresp onds to T u ses of som e under lying “p hysical” channe l. W e fur ther as- sume th e transmitted code words x to be u niform ly d istributed over some codeb ook X ∈ R m , to be statistically inde penden t of the chann el H , and to satisfy th e power constraint E {k x k 2 } ≤ T . (2) B. Rate, reliability a nd com plexity in o utage-limited MIMO communica tions In terms of error performan ce, we let P e denote the proba- bility of codeword error , and we con sider the rate, R = 1 T log |X | , (3) in bits per chann el use (bpcu) , where |X | deno tes the cardi- nality of X . Regarding complexity , we let N max describe the compu- tational reso urces, in floating poin t opera tions (flop s) per T channel uses, tha t the transceiver is endowed with, in the sense that after N max flops, th e transceiver m ust simply ter minate, potentially prematur ely and befo re co mpletion of its task. W e note that n aturally , N max is intimately intertwined with th e desired P e and R , and th at any attemp t to significantly r educe N max may be a t the expe nse of a sub stantial degradatio n in error-perfor mance. In th e high SNR regime, a g iv en encod er X r and deco der D r are said to achie ve a multiplexing g ain r (cf. [3]) and diversity gain d ( r ) if lim ρ →∞ R ( ρ ) log ρ = r, and − lim ρ →∞ log P e log ρ = d ( r ) . (4) In the same high SNR regime, the complexity is here chosen to take the form c ( r ) := lim ρ →∞ N max log ρ , (5) which is hence forth denoted as the complexity exponen t . Noting th at R = r log ρ , we observe th at c ( r ) > 0 im plies a com plexity that is expone ntial in the rate . Remark 1: A reasonab le question at this point wou ld per- tain as to why the com putational resources N max scale with ρ and are dependent on r , to which we n ote th at the complexity of d ecoding is generally d epende nt on the density of the codebo ok, wh ich in turn d epends o n ρ an d R . Furtherm ore this de penden ce of th e c omplexity exponen t (and by extensio n of N max ) on r , re flects a p otential ability to regulate the computatio nal resour ces dep ending on the rate. Finally the fact that both P e and N max are repre sented as p olyno mial function s of ρ , simply stems fro m the fact th at both P e and |X | naturally scale a s poly nomial functions of ρ . Specifically we quickly note tha t c ( r ) c aptures th e entire complexity r ange 0 ≤ c ( r ) ≤ r T of all reaso nable transceivers, with c ( r ) = 0 co rrespon ding to the fastest p ossible tra nsceiv er (req uiring a su bexponen tial number o f flops per T channel uses), and with c ( r ) = r T correspo nding to the optimal but argu ably slowest, f ull-search uninterr upted ML decoder 1 in the presence of a canonical code with multiplexing gain r , i.e., with |X r | = 2 RT = ρ r T . If this cano nical code thou gh is linear, search ing the entire codebo ok can be avoided b y alg orithmic so lutions like th e sphere decod er (SD) which can p rovide substantial com plexity reduction s at a po tential small loss in error per forman ce. Such solutions take advantage of the line ar n ature of th e code th at is defined by a generator matrix G and a shaping r e g ion R ′ . Specifically f or r ≥ 0 , a (seq uence of) full-ra te linear (lattice) code(s) X r is given by X r = Λ r ∩ R ′ where Λ r , ρ − rT κ Λ and Λ , { Gs | s ∈ Z κ } , where Z κ denotes the κ = min { m, n } dimensiona l integer lattice, where R ′ is a comp act conv ex subset of R κ that is indepen dent of ρ , and where G ∈ R m × κ is f ull r ank an d in depend ent of ρ . For the class of lattice co des considered here, th e cod ew ords take the fo rm x = ρ − rT κ Gs , s ∈ S κ r , Z κ ∩ ρ rT κ R , (6) where R ⊂ R κ is a na tural bijectio n of the shaping region R ′ that preserves the code, and wh ere R contains the all zero vector 0 . As n oted before , despite the redu ced comp lexity of sphere decodin g of such lattice codes (as comp ared to b rute-fo rce ML decoding), recent work in [1 ] h as r ev ealed that e ven such branch -and-b ound alg orithms generally req uire computation al resources that grow exponen tially in the numb er of co dew ord bits and the dimension ality . As an indicative example of this high co mplexity , we n ote that the work in [1] showed that such SD algorithms, when a pplied for decod ing a large family of h igh-pe rformin g co des includin g all known f ull-rate DMT optimal cod es, over the n T × n R quasi-static MIMO chann el with Rayleigh fading and n R ≥ n T , intro duce a complexity exponent 2 of the form c ( r ) = T n T  r ( n T − ⌊ r ⌋ − 1) + ( n T ⌊ r ⌋ − r ( n T − 1)) +  . (7) In the above, ⌊ r ⌋ deno tes the largest integer not gr eater th an r . T he expo nent, which simplifies to c ( r ) = T n T r ( n T − r ) for integer v alues of r , r eaches at r = n T / 2 (for even values of n T ) an overall m aximum value of n T T / 4 which, for the afo rementio ned codes is equ al to κ/ 8 , co rrespon ding to com plexity in the order of 2 1 8 κ log ρ = ρ κ/ 8 = p |X | . At any fixed m ultiplexing g ain, these req uired compu tational resources can be seen to be in the or der of 2 RT ( n T − r n T ) flops which reveals a complexity that is expon ential in the num ber of codeword bits, a nd a corre sponding exponential slope of n T − r n T . 1 W e her e note that stric tly speaking , X r , D r may potentiall y introduce a comple xity expon ent larger than r T . In such a case though, X r , D r may be substit uted by a lookup table implement ation of X r and an unre stricte d ML decoder . This encoder-d ecoder will jointly require resources that are a constant multipl e of | X r | . = ρ r T as it has to construct an d visit all possible |X r | co de words, at a computatio nal cost of a bounded numbe r of flops per code wo rd visit. It is noted that the number of flops per visited code word is natural ly independent of ρ . 2 Although premature at this point, we hasten to note for the expert reader that th is compl exit y indeed hold s irrespe cti v e of th e radi us updatin g policy , irrespec ti ve of the decoding ordering, and as we will see later on, holds ev en in the presence of MMSE preprocessing . C. T ransition to la ttice decod ing for r educing co mplexity As men tioned, this h igh comp lexity o f ML b ased (co n- strained) d ecoder s, mo tiv ates co nsideration of other d ecoder families, with a natur al altern ativ e being the u nconstra ined (naive) lattice deco der which takes the general for m ˆ x L = a rg min ˆ x ∈ Λ r k y − √ ρ Hˆ x k 2 . (8) Naturally whe n ˆ x L / ∈ X r , the d ecoder declar es an er ror . The use o f lattice deco ding, and specifically of pr eprocessed lattice deco ding in MIMO com munication s h as received sub- stantial attention fro m works like [4], [5] and [ 6], wher e the latter proved tha t lattice decod ing in the presence of MMSE prepro cessing ac hieves the optimal DMT fo r specific MIMO channels and statistics, and fo r DMT -optimal rand om co des. The use of lattice decoding as an alternative to comp utationally expensiv e ML based solution s, was recen tly further validated on the one ha nd by the aforem entioned work in [1], [7] wh ich revealed the large computatio nal disadvantages of ML based solutions, and o n the o ther ha nd by the work in [8] which further confirmed the perform ance advantages of lattice decod- ing by showing that r egularized (MMSE-prep rocessed) 3 lattice decodin g achieves th e op timal DMT perfor mance, for a lmost all MIMO scenarios a nd fading statistics, and all non-ran dom lattice cod es, irrespective o f the codes’ ML perfor mance. It is the case thou gh that the aforementione d extreme complexity of exact lattice decoding solutions, in co njunctio n with the poten tially u nbou nded erro r-performanc e degradation (gap) o f very ear ly term inations (as opposed to exact imple- mentations) of lattice d ecoding , bring to the fore the ne ed for balanced app roximatio ns of lattice deco ding solutions that bet- ter balan ce th e very sizable c omplexity and gap. Specifically for any simplified variant D r of th e baselin e (exact) MMSE- prepro cessed lattice de coder, th is gap can, in the high SNR regime, be qu antified as g L ( c ) , lim ρ →∞ P e P ( ˆ x 6 = x ) (9) where P ( ˆ x 6 = x ) descr ibes the pr obability of error of the exact MMSE-prep rocessed lattice decoder, where P e denotes the probab ility of error of D r , and where c (i.e., c ( r ) ) is the complexity exponent that describ es the (asymptotic r ate of increase of th e) compu tational resour ces requ ired to achieve this pe rforma nce gap. Generally a smaller co mputation al com- plexity expo nent c imp lies a larger gap g L ( c ) . The clear task h as remained for some time to construc t decod ers that optimally traverse this tradeoff between g and c , i. e., that reduce the perfor mance g ap to the exact lattice decod ing solu- tion, with reasonable c omputatio nal comp lexity . Equ iv alently for N max ( g ) deno ting the computation al reso urces in flops required to achieve a certain gap g to the b aseline exact 3 W e will interchangea bly use MMSE-prepr ocessed decoder and re gularized decode r , with the first term being more commonly used, and with the second implying a more ge neral family of dec oders (cf. [8] where the e qui v alenc e betwee n the two decoders is discussed.). Even though in the asymptotic setting of interest, the two acce pt the same results throughout the paper , some ext ra error-pe rformance gains can be achie ved by proper optimiz ation of the regul arize d deco der (cf. [9]). MMSE-prep rocessed lattice d ecoder, the above task can be described, in th e hig h SNR regime, as tr ying to m inimize lim ρ →∞ log N max ( g ) log ρ . This will be achieved later on. D. Contrib utions W e first show th at the co mputation al com plexity required by the MMSE-prepr ocessed (unconstrain ed) lattice sphere decoder, asym ptotically matches the c omplexity of th e (con - strained) ML-b ased (MMSE-p reproce ssed or not) sphere de- coders, and is commonly exponential in the d imensionality and the number of co deword b its. This is established f or a large class o f codes of arb itrary error-perfo rmance, a large class of fading statistics, a nd specifically for th e qua si-static MIMO channel – fo r examp le the comp lexity required for DMT o ptimal lattice sphere decod ing, in the pre sence of a large family of DMT optimal c odes, takes the p reviously seen simp le p iecewise linear form in ( 7). In a parenth etical note, and d eviating slightly fro m the spirit o f this paper, we also pr ovide a un iv ersal u pper b ound on the complexity of regularized lattice sphere decoding , which holds irrespectiv e of the lattice cod e applied and irr espective of the fading statistics. This up per bo und ag ain takes th e form in (7), matchin g that in th e case of co nstrained ML-b ased sphere deco ding, thu s revealing the surprising fact that there exists no statistical channel behavior that will allow the r emoval of th e boundin g region to cause unbou nded in creases in the complexity o f the decoder 4 . W ith p rovable e vidence of the very h igh co mplexity o f regularized lattice deco ding, we turn to the powerful too l of lattice reduction and seek to under stand its effects o n computatio nal comp lexity . While there has existed a general agreemen t in the co mmunity th at lattice redu ction do es red uce complexity , cf. [10], this has not yet be en suppo rted analyti- cally in any relev ant commun ication settings. In fact, and quite opposite to commo n wisdom, it was recen tly shown that for a fixed- radius 5 sphere d ecoding implem entation of th e naive lattice decoder [11], LR does not impr ove the sphere decoder complexity tail exponen t. What our present work sho ws is that lattice reduction reduces an ML-like expon entially increasing complexity , to very manag eable subexpon ential values. W e spec ifically pro- ceed to p rove tha t the LR-aided r egularized lattice deco der, implemented by a fixed-rad ius sph ere decoder an d tim eout policies that o ccasionally ab ort deco ding and declare an error, achieves g L ( ǫ ) = 1 , lim ρ →∞ log N max ( g ) log ρ = 0 ∀ ǫ > 0 , g ≥ 1 , i.e., achieves a vanishing gap to the exact implemen tation of regularized lattice decodin g a nd does so with a complexity 4 In other words, this comple xity bound holds e ven if the channel statistics are such that the channel realizat ions cause the decoder to always have to solve the hardest possible lattice search problem. 5 The radius here is conside red fixed in the sense that it does not vary with respect to the channel realizati on and rate. exponent that vanishes to zero , which in turn implies sub ex- ponen tial co mplexity in the sense that th e com plexity scales slower than any co nceiv able expo nential fun ction. It is finally noted tha t this vanishing gap approach serves the practical purpo se of an analytical r efinement over basic d iv ersity anal- ysis which generally fails to address potentially massi ve gaps between theo ry and pr actice. E. Notation W e use . = to denote the e xponential equality , i.e., we write f ( ρ ) . = ρ B to denote lim ρ →∞ log f ( ρ ) log ρ = B , an d . ≤ , . ≥ are similarly defined. W ith this notation, we can write P e . = ρ − d ( r ) (cf. (4)). In this paper we use p • q to denote the smallest inte ger not smaller than the ar gument, x • y to denote the largest integer not larger th an the a rgument, ( • ) H to deno te th e conjug ate transpose of ( • ) , ( • ) + to d enote max { 0 , ( • ) } and v ec ( • ) to denote the o peration whereb y the co lumns of the argument ( • ) are stacked to form a vector . I I . M M S E - P R E P RO C E S S E D L A T T I C E S P H E R E D E C O D I N G C O M P L E X I T Y W e p roceed to de scribe the pr eprocessed lattice deco der, its sp here d ecoding im plementation , and for a practical set- ting of in terest th at in cludes th e q uasi-static MIMO ch annel and common cod es, to establish th e deco der’ s comp utational complexity . A. Lattice sphere decoding Combining (1) and ( 6) yields th e equiv alent model y = M r s + w (10) where M r = ρ 1 2 − rT κ HG ∈ R n × κ (11) is a f unction of the mu ltiplexing gain 6 r . Consequently the corr espondin g naive lattice decoder in (8) takes the for m (see for example [8], also [10]) ˆ s L = a rg min ˆ s ∈ Z κ k y − M ˆ s k 2 . (12) As a result tho ugh of neglecting the bo undar y region, the above d ecoder declares additional errors if ˆ s L / ∈ S κ r , resulting in possible perform ance costs. These costs motivated the use of MM SE preproce ssing which essentially regularizes the decision metric to p enalize vector s outside the bo undar y constraint S κ r (cf. [8]). Spe cifically th e M MSE-prep rocessed lattice decoder is obtained by imp lementing an unconstrained search over the MMSE-prep rocessed lattice, and takes th e form ˆ s r − ld = a rg min ˆ s ∈ Z κ k Fy − R ˆ s k 2 , (13) where F an d R are r espectively the MMSE f orward an d feedback filters such that F = R − H M H , where R H R = M H M + α 2 r I , (14) 6 For simplicity of notati on we will, in most cases, denote M r with M . where α r = ρ − rT κ and where R is an upper-triangular m atrix (more details can be foun d in Append ix D). For r , Fy , th e model transitions fr om (10) to r = R − H M H Ms + R − H M H w = R − H ( R H R − α 2 r I ) s + R − H M H w = Rs − α r 2 R − H s + R − H M H w = Rs + w ′ (15) where w ′ = − α 2 r R − H s + R − H M H w (16) is the equivalent noise th at includes self-interf erence (first summand ) and color ed Gaussian n oise. Consequ ently the correspo nding regularized lattice decoder takes the form ˆ s r − ld = a rg min ˆ s ∈ Z κ k r − R ˆ s k 2 , (17) which is th en solved by the sphere d ecoder wh ich recursiv ely enumera tes all lattice vectors ˆ s ∈ Z κ within a giv en sph ere o f radius ξ > 0 , i.e., which identifies as candidates the vectors ˆ s that satisfy k r − R ˆ s k 2 ≤ ξ 2 . (18) The algorithm specifically uses the upper-triangular nature of R to r ecursively id entify partial symbol vectors ˆ s k , k = 1 , · · · , κ , for wh ich k r k − R k ˆ s k k 2 ≤ ξ 2 , (19) where ˆ s k and r k respectively den ote the last k co mpon ents o f ˆ s and r , an d where R k denotes the k × k lower -right submatr ix of R . Clearly any set o f vectors ˆ s ∈ Z κ , with comm on last k co mpon ents that fail to satisfy (19), may be exclu ded from the set of cand idate vectors that satisfy (18). The enumera tion o f partial symbo l vector s ˆ s k is equiv alent to the tr av ersal of a regular tree with κ layers – o ne layer p er symbol compon ent of th e sy mbol vectors, su ch that layer k correspo nds to th e k th compon ent o f the tr ansmitted symb ol vector 7 s . Th ere is a on e-to-on e correspon dence be tween th e nodes at layer k and the par tial vectors ˆ s k . W e say that a no de is visited b y the sphere decod er if and on ly if the correspo nding partial vecto r ˆ s k satisfies (19), i. e., there is a bijection between th e visited nodes at layer k and the set N k , { ˆ s k ∈ Z k | k r k − R k ˆ s k k 2 ≤ ξ 2 } . (20) B. Complexity of MMSE-p r epr oce ssed lattice spher e deco ding Consequently the total num ber of visited no des (in all layer s of the tree) is given by N S D = κ X k =1 N k , (21) where N k , |N k | is the nu mber of visited nodes at la yer k o f the search tree. The total number of visited nodes is comm only 7 W e will henceforth refer to the symbol vector s ∈ S κ r correspond ing to the transmitte d code w ord x = ρ − rT κ Gs (cf. (6)) , simply as the transmitted symbol vector . taken as a measure of the sph ere decoder co mplexity . It is easy to show that in the scale of inter est the SD co mplexity exponent c ( r ) wou ld not chang e if instead of co nsidering the number of visited nodes, we considered the numbe r of flops spent by th e dec oder 8 . Naturally the total num ber o f visited no des is a function of the search radiu s ξ . W e here use a fixed r adius, which may result in a non-zero probability that th e transmitted symbol vecto r s is n ot in N κ . Consequen tly we must cho ose a radius that strikes the pro per ba lance b etween decreasing the aforemen tioned p robab ility and a t the same time sufficiently decreasing the size of N κ . T ow ards this we note th at for the transmitted sym bol vector s , the m etric in (17) satisfies k r − Rs k 2 = k w ′ k 2 , which m eans that if k w ′ k > ξ , th en the transmitted symbol vector is excluded from the search, resultin g in a decod ing error . As Lemma 2 will later argue tak ing into consider ation the self -interfere nce an d n on-Gau ssianity of w ′ , we can set ξ = √ z log ρ , for some z > d ( r ) such that P  k w ′ k 2 > ξ 2  ˙ < ρ − d ( r ) , which implies a vanishing probab ility of exclud ing the tran s- mitted in forma tion vector from th e search , and a vanishing degradation of erro r perform ance. W e here no te th at the MMSE-prep rocessed lattice spher e decoder differs from its ML-ba sed equivalent in two aspe cts: the presen ce of M MSE prepr ocessing and the a bsence of a bound ing region to constrain the sear ch. Th ese two aspects are gener ally perceived to have an opp osite e ffect o n the complexity . On the one han d, MMSE prep rocessing, which we recall f rom (2 0) to in troduce unpru ned sets N k , { ˆ s k ∈ Z k | k r k − R k ˆ s k k 2 ≤ ξ 2 } , k = 1 , · · · , κ, is associated to r educed c omplexity in lattice-based SD solu- tions (cf. [11]) du e to the resultin g p enalization of faraway lattice p oints (cf . [8]). On the other hand, the absence o f bound ary con straints can be associated to increased complex- ity as it in troduc es an unbou nded number of cand idate vectors. W e p roceed to show that in ter ms of the comp lexity exponent, under co mmon MIMO scenarios and codes, th ese two aspec ts exactly cancel each other o ut, and that consequ ently MMSE- prepro cessed lattice sph ere d ecodin g introduces a co mplexity exponent th at matches that of ML -based sphere decodin g (cf. [1]), which it self is shown h ere to also match the complexity exponen t o f ML -based SD in the presence of MMSE pr eprocessing 9 . Before p roceed ing w e note th at this an alysis is specific to sphere decod ing, and th at it d oes not accoun t fo r any other ML b ased solu tions that could, under some (arguably rare) circumstances, be more e fficient. A classical example o f such rare circumstances would be a MIMO scenario, or equiv alently 8 T o see this, we c onsider tha t the co st of visiting a node, is inde pendent of ρ . Once at a visited node, this same bounded cost include s the cost of establi shing which children- nodes not to visit in the next layer . 9 W e clarify that ML-based SD in the presenc e of MMSE preprocessing, correspond s to unpruned sets N k ∩ S k r where S k r is the k -dimensional set resultin g from the natural reduction of S κ r from (6). a set of fade statistics, that always generate d iagona l channel matrices. Another example would be having codes dr awn from orthog onal designs which introdu ce very sma ll d ecodin g com- plexity , but wh ich a re p rovably sho wn to be highly sub optimal except f or very few un ique cases like the n T = 2 , n R = 1 quasi-static case [12]. I n light of this, in this section only , we mainly fo cus on the widely co nsidered n T × n R ( n R ≥ n T ) i.i.d. and quasi-static MIMO setting and on the large but specific family o f full-rate ( κ = 2 min { n T , n R } T = 2 n T T ) threaded codes (cf. [13]–[16]), which includes all known DMT optimal cod es as well as un coded transmission (V -BLAST ). W e pr oceed with the main Theore m of the section, wh ich applies un der natural d etection o rderin g (cf. [ 1], [5]), a nd under the assum ption of i.i.d. regular fading statistics 10 . Theor em 1: The comp lexity e xponen t for MMSE- prepro cessed lattice spher e decodin g any full-rate thr eaded code over th e quasi-static MIMO chan nel with i.i.d . regular fading statisti cs, is equal to the comp lexity exponent o f ML-based SD with or without MMSE p repro cessing. Pr o of: See Append ix A. W e clarify that even thoug h all thr ee decod ers are DMT optimal, th e ab ove result incorp orates more tha n just DMT optimal decod ing, in the sense that any tim eout policy will tradeoff d ( r ) with c ( r ) id entically f or ML-based an d lattice- based sphere decoding. In other w ords the three decoders share the same d ( r ) a nd c ( r ) cap abilities, irrespective of the timeout policy . Furthermo re, conside ring d ifferent SD d etection ord erings (cf. [5]), the f ollowing extends the rang e of c odes f or which the ML-b ased and lattice-based SD sha re a similar c omplexity . The proof follo ws from the proof of Theorem 1 in Appendix A, and fro m Theorem 4 in [1]. Cor o llary 1a : Given any f ull-rate code of arb itrary DMT perfor mance, there is always at least one non -rando m fixed permutatio n o f the colu mns of G , for wh ich the comp lexity exponent of th e MM SE-prepr ocessed lattice sphere decoder matches that of the M L ba sed sphere d ecoder . The following f ocuses on a specific example of practical interest. Cor o llary 1b : Th e complexity expon ent for DMT o ptimal MMSE-prep rocessed lattice sphere decoding of minimum de- lay ( T = n T ) DMT op timal threaded codes over th e quasi- static MI MO channe l with i.i.d . regular fading statistics, takes the following form c r − ld ( r ) = r ( n T − ⌊ r ⌋ − 1) + ( n T ⌊ r ⌋ − r ( n T − 1)) + , (22) which simplifies to c r − ld ( r ) = r ( n T − r ) (23) for integer values of r . 10 The i.i.d. regul ar fading statistics satisfy the general set of conditi ons as describe d in [17], where a) the near-z ero behavior of the fading coeffic ients h is bounded in pro babili ty as c 1 | h | t ≤ p ( h ) ≤ c 2 | h | t for some positi ve and finite c 1 , c 2 and t , where b) the tail behavi or of h is bounded in probabi lity as p ( h ) ≤ c 2 e − b | h | β for some positi v e and finite c 2 , b and β , and where c) p ( h ) is upper bounded by a constant K . Pr o of: See Append ix B. Further evidence that co nnects th e complexity behavior of MMSE-prep rocessed lattice- based SD, with that o f its ML- based c ounterp art, now comes in the form of a non-trivial universal bound that is shar ed by the two meth ods. This is pa r- ticularly relevant because unconstrain ed lattice decoding cou ld conceiv ably req uire unbo unded com putation al resource s g iv en the u nbou nded numbe r of ca ndidate lattice poin ts. Specifically the fo llowing un iv ersal upper bound on the complexity of regularized lattice-based SD, match es th e up per boun d in [1] for th e ML case, an d it holds irre spectiv e of th e full-r ate lattice code applied and irrespe ctiv e of the fading statistics. The gen erality with respect to the fading statistics is im portan t because it guara ntees that no set of fadin g statisti cs, even those that alw ays generate infinitely dense lattices, can cau se a n unbou nded in crease in th e com plexity due to removal of the bound ary constraints. Cor o llary 1c: Irrespective of the fadin g statistics a nd o f the full-rate lattice code applied, the comp lexity exponents of MMSE-p reproce ssed lattice SD and o f ML-based SD, are upper bou nded by c ( r ) = T n T  r ( n T − ⌊ r ⌋ − 1) + ( n T ⌊ r ⌋ − r ( n T − 1)) +  (24) which simplifies to c ( r ) = T n T r ( n T − r ) (25) for integer r . Pr o of: See Append ix B. The above results revealed the very h igh, ML -like comp lex- ity of MMSE-p repro cessed lattice d ecoding . Coming b ack to the main fo cus of the paper, an d after reverting to th e m ost general setting of MIMO scenarios, statistics and full-rate lattice codes, we proc eed to show how prope r utilization o f lattice spher e decoding and LR techn iques can inde ed reduce the co mplexity exponen t to zero, at an error-performance cost that vanishes in the high SNR limit. I I I . L R - A I D E D R E G U L A R I Z E D L A T T I C E S P H E R E D E C O D I N G C O M P L E X I T Y Lattice r eduction techniqu es have been typically used in the MIMO setting to improve the error pe rforma nce of suboptima l decoder s (cf . [ 18], [19], see also [20], [21]). In the cur rent setting the LR alg orithm, wh ich is emp loyed at the receiver after th e action of MMSE prep rocessing, modifies the search of the MMSE-pr eprocessed lattice decoder, from ˆ s r ld = arg min ˆ s ∈ Z κ k r − R ˆ s k 2 (cf. (17)), to the new ˜ s lr − r ld = a rg min ˆ s ∈ Z κ k r − R T ˆ s k 2 , (26) by accepting as input the MM SE-prepr ocessed lattice genera- tor matrix R , and p roducin g as o utput the matrix T ∈ Z κ × κ which is u nimodu lar m eaning that it has integer coefficients and u nit-nor m determinant, and which is designed so that R T is (loo sely speakin g) mor e ortho gonal than R . As a result of this unimod ularity , we have that T − 1 Z κ = Z κ , and consequ ently the new search in (26) corr esponds to yet another la ttice decod er , ref erred to as the LR-a ided MM SE- prepro cessed lattice decod er , which operates over a g enerally better con ditioned channel matrix R T . Finally with spher e decod ing in mind, the LR alg orithm is followed by th e QR decomp osition 11 of the new lattice- reduced MMSE-prep rocessed matrix R T , resultin g in a new upper-triangu lar mo del ˜ r = ˜ R ˜ s + w ′′ (27) and in the n ew LR-aided MMSE-p repro cessed lattice search, which accepts the application of the sphere decoder, and which takes the for m ˜ s lr − r ld = a rg min ˆ s ∈ Z κ    ˜ r − ˜ R ˆ s    2 , (28) where ˜ Q ˜ R = R T correspo nds to the QR-decom position o f R T , where ˜ R is upper-triangular, where ˜ r , ˜ Q H r , ˜ s = T − 1 s , and wher e w ′′ = ˜ Q H w ′ . At the very end, ˆ s lr − r ld = T ˜ s lr − r ld , (29) allows for calculation of the estimate of the transmitted symbol vector s in (1 0). W e note here tha t this (exact) solution o f the LR-aid ed MMSE-prep rocessed lattice decoder defined b y (28), ( 29), is iden tical to the exact solu tion of the MMSE-pre processed lattice deco der given b y (17), b ecause min ˆ s ∈ Z κ k r − R ˆ s k 2 = min ˆ s ∈ Z κ   r − R TT − 1 ˆ s   2 ( a ) = min ˆ s ∈ Z κ    r − ˜ Q ˜ RT − 1 ˆ s    2 ( b ) = min ˆ s ∈ Z κ    ˜ r − ˜ RT − 1 ˆ s    2 = min ˆ s ∈ T − 1 Z κ    ˜ r − ˜ R ˆ s    2 ( c ) = min ˆ s ∈ Z κ    ˜ r − ˜ R ˆ s    2 , (30) where ( a ) f ollows from the fact th at ˜ Q ˜ R = R T , ( b ) follows from the rotationa l inv ariance of the Euclidean norm, and ( c ) follows from the fact that T − 1 Z κ = Z κ . While thou gh the two lattice deco ding solutions (w ith and without LR) pr ovide id entical erro r p erform ance in th e setting of exact imp lementation s, we proceed to show that, in terms of co mplexity , lattice reduction techniq ues, and specifically a p roper utilization o f the L LL algorithm [2 2], ca n p rovide dramatic improvements. A. Complexity of the LR-aid ed r e gularized lattice sphere de - coder W e are here intere sted in establishing the com plexity of the LR-aid ed regularized lattice sph ere deco der . Given th at 11 A more proper statement would be that the QR decomposi tion is performed by the LR algorithm it self. the costs of implemen ting MM SE prep rocessing and of im - plementing the linear transfor mation in (2 9) are negligible in th e scale of interest 12 , we limit our focus on establishing the cost of lattice redu ction, and then th e cost of the SD implementatio n of the search in (28). Starting with the SD complexity , a s in (20), we identify the correspo nding unpruned set at lay er k to b e N k , { ˆ s k ∈ Z k | k ˜ r k − ˜ R k ˆ s k k 2 ≤ ξ 2 } , (31) and in bo undin g the size of the above, we first focu s on understan ding the statistical beh avior of the k × k lower - right submatrices ˜ R k of matrix ˜ R ( k = 1 , · · · , κ ), where we recall that ˜ R is the u pper triangu lar code-cha nnel matrix, after MMSE prepro cessing and LLL lattice reduction. T ow ards this, and f or d L ( r − ǫ ) d enoting the diversity gain of the exact implem entation of the regularized lattice decoder at multiplexing gain r − ǫ , we have the following lemma on the sma llest singular value o f ˜ R k . Th e p roof app ears in Append ix C. Lemma 1: The smallest singular value σ min ( ˜ R k ) of sub- matrix ˜ R k , k = 1 , · · · , κ , satisfies P  σ min ( ˜ R k ) . < ρ − ǫT κ  . ≤ ρ − d L ( r − ǫ ) , for a ll r ≥ ǫ > 0 . (32) T o b ound the cardina lity N k of N k (cf. (31)), and eventually the total num ber N S D = P κ k =1 N k of lattice poin ts visited by the SD, we proceed along the lines of the work in [1], m aking the pro per mod ifications to accou nt for MMSE prepro cessing, fo r th e removal o f the bou nding region, and for lattice reductio n. T owards this we see th at, after removing the bo undar y constraint, Lemm a 1 in [1] tells us that N k , |N k | ≤ k Y i =1  √ k + 2 ξ σ i ( ˜ R k )  , where σ min ( ˜ R k ) = σ 1 ( ˜ R k ) ≤ · · · ≤ σ k ( ˜ R k ) are the singular values of ˜ R k . Conseq uently we h ave th at N k ≤  √ k + 2 ξ σ min ( ˜ R k )  k . (33) As a r esult, for any ˜ R k such that σ min ( ˜ R k ) . ≥ ρ − ǫT κ , (34) and given that ξ = √ z log ρ for so me finite z , then N k . ≤ √ k + 2 √ z log ρ ρ − ǫT κ ! k . = ρ ǫT k κ , (35) 12 Even thou gh the work here focuses on decodi ng, we can also quickl y state the obvious fact that the cost of constructing the code w ords is also negligib le in the scale of interest because it again only inv olv es a finite-dimensi onal linea r transformation (cf. (6)). which gu arantees th at the total numb er of visited lattice p oints is upp er bound ed as N S D = κ X k =1 N k . ≤ κ X k =1 ρ ǫT k κ . = ρ ǫT . (36) Consequently , d irectly from L emma 1, we have that P  N S D ˙ ≥ ρ ǫT  ˙ ≤ ρ − d L ( r − ǫ ) . ( 37) A similar appr oach deals with the complexity of the LLL algo- rithm, which is kn own (cf . [2 3]) to be gener ally unbo unded . Specifically drawing fro m [ 8, Lemma 2 ], und er the natural assumption of power-limited ch annels 13 (cf. [8]), und er the natural assumption that d L ( r − ǫ ) > d L ( r ) for all ǫ > 0 , and for N LR denoting the number of flops sp ent by th e LLL algorithm , one can read ily conclude that P ( N LR ≥ γ log ρ ) ˙ ≤ ρ − d L ( r − ǫ ) , (38) for any γ > 1 2 ( d L ( r − ǫ )) . Consequ ently the overall complexity N . = N S D + N LR , in flop s, for the L R-aided MMSE prep rocessed lattice sph ere decoder, satisfies the following P  N ˙ ≥ ρ ǫT  . = P  { N S D ˙ ≥ ρ ǫT } ∪ { N LR ˙ ≥ ρ ǫT }  . ≤ ρ − d L ( r − ǫ ) . (39) Now g oing back to (5), and ha ving in mind appropria te timeou t policies th at bo und N max while at the same time sp ecifically guaran tee a vanishin g err or performance g ap to the exact solution of regularized lattice deco ding, we can see that the complexity exponent c ( r ) takes the equivalent form r ecently introdu ced (for the ML case) in [1] c ( r ) = inf { x | − lim ρ →∞ log P ( N ≥ ρ x ) log ρ > d L ( r ) } . (40) T o see this we quick ly note that for N max = ρ x where x = c ( r ) − δ for any δ > 0 , it is the case that (cf. (9)) lim ρ →∞ P( N ≥ ρ x ) P( ˆ x L 6 = x ) → ∞ . Finally applyin g (39) we see tha t f or any p ositiv e ǫ 1 < ǫ , it is the case that c ( r ) = inf { ǫ | − lim ρ →∞ log P  N ≥ ρ ǫT + ǫ 1  log ρ > d L ( r ) } (4 1) which vanishes ar bitrarily close to zero, r esulting in a zero complexity exponent. What r emains is to consider th e e rror-perfor mance gap in the presence the LR-a ided r egularized lattice SD with a time- out po licy th at interru pts at N max = ρ x for any vanishingly small x > 0 . 13 This is a moderate assumption that asks that E  k H k 2 F  . ≤ ρ . W e note that this holds true for any telecommunic ations settin g. B. Gap to the exact solution o f MMSE-prepr ocessed lattice decodin g W e h ere prove that the LR-aided regularized lattice spher e decoder and the associated time- out policies th at guaran tee a vanishing complexity e xponent, also guarantee a v anishing gap to the er ror perfo rmance of th e exact la ttice decodin g imple- mentation. This result is m otiv ated by potentially exponential gaps in the perf ormance of othe r DMT optim al decoder s (cf. [8]), wh ere th ese gap s may grow exponen tially up to 2 κ 2 (cf. [24]) or m ay pote ntially be un boun ded [25]. T owards establishing this g ap, we r ecall that the exact MMSE-prep rocessed lattice deco der in ( 13) makes er rors when ˆ s r − ld 6 = s . On the other han d the LLL-reduced MMSE- prepro cessed lattice sphere decoder with ru n-time constraints, in addition to mak ing the same e rrors ( ˆ s r − lr − ld 6 = s ) , a lso makes errors when the run-time limit of ρ x flops b ecomes activ e, i.e., when N ≥ ρ x , as well as when a small search radius causes N κ = ∅ . Consequently the corre spondin g perfor mance g ap to the exact regularized decod er , takes the form g L ( x ) = lim ρ →∞ P ( { ˆ s r − lr − ld 6 = s } ∪ { N ≥ ρ x } ∪ {N κ = ∅} ) P ( ˆ s r − ld 6 = s ) . T o bo und the above gap , we ap ply th e unio n b ound a nd the fact that P ( N κ = ∅ ) ≤ P ( k w ′′ k > ξ ) to ge t that g L ( x ) ≤ lim ρ →∞ P ( ˆ s r − lr − ld 6 = s ) P ( ˆ s r − ld 6 = s ) + lim ρ →∞ P ( N ≥ ρ x ) P ( ˆ s r − ld 6 = s ) + lim ρ →∞ P ( k w ′′ k > ξ ) P ( ˆ s r − ld 6 = s ) . (42) Furthermo re from (30) we observe that P ( ˆ s r − lr − ld 6 = s ) = P ( ˆ s r − ld 6 = s ) , (43) and fro m (39) we recall that P  N ˙ ≥ ρ ǫT  . ≤ ρ − d L ( r − ǫ ) which implies that for any x > 0 it ho lds that lim ρ →∞ P ( N ≥ ρ x ) P ( ˆ s r − ld 6 = s ) = 0 . (44) Finally the last term in (42) relates to the sear ch radiu s ξ , and to the behavior of the no ise w ′′ which was shown in (16), (27) to take the fo rm w ′′ = ˜ Q H  − α 2 r R − H s + R − H M H w  . (45) The following lemma, wh ose pro of is fou nd in Ap pendix D, accounts for the fact th at w ′′ includes self-interf erence and colored noise, to bo und the last term in (42). Lemma 2: There exist a finite z > d L ( r ) for which a search radius ξ = √ z log ρ guar antees that lim ρ →∞ P ( k w ′′ k > ξ ) P ( ˆ s r − ld 6 = s ) = 0 . (46) Consequently co mbining (43), (44) and (46) gives that g L ( x ) = 1 , ∀ x > 0 . The following directly hold s. Theor em 2: LR-aided MMSE-pr eprocessed lattice sphe re decodin g with a c omputatio nal constraint activ ated at ρ x flops, allows for a vanishing g ap to the exact solu tion of MMSE - prepro cessed lattice decoding, f or any x > 0 . Equi valently the same LR-aided d ecoder gu arantees that g L ( ǫ ) = 1 and lim ρ →∞ log N max ( g ) log ρ = 0 ∀ ǫ > 0 , g ≥ 1 , for all fading statistics, all MIMO scenarios, and a ll full-r ate lattice cod es. I V . C O N C L U S I O N S The work identified the first lattice decod ing solutio n that achieves, in the most gen eral outage- limited MIMO setting and the h igh rate and high SNR limit, b oth a vanishing gap to the erro r-performan ce of the (DMT o ptimal) e xact solu tion of prepr ocessed lattice de coding , as well as a com putation al complexity that is subexponential in the number of codeword bits. The p roposed solution employs lattice red uction ( LR)- aided regularized lattice spher e decodin g and prop er timeou t policies. As it turns o ut, lattice r eduction is a spec ial ing re- dient that allows f or complexity red uctions; a role that was rigoro usly d emonstrated he re for th e first time, b y pr oving that without lattice r eduction , for most co mmon codes, the complexity cost for asymptotically op timal regularized lattice sphere decod ing is expo nential in the n umber of c odeword bits, and in many cases it in fact matches the co mplexity cost of ML sphere decod ing. In light of the fact that, prio r to this work, a vanishing error perfor mance gap was g enerally attributed o nly to near -full la t- tice sear ches th at have exponential co mplexity , in conjunction with the fact that su bexponen tial complexity was g enerally attributed to early-te rminated (linear) solution s which have though a perfor mance gap that can be up to expone ntial in dimension an d/or rate, the work constitutes the first proo f that sub exponential comp lexity need not co me at the cost of exponential reduction s in lattice decod ing error perf ormanc e. A P P E N D I X A P R O O F F O R T H E O R E M 1 A N D C O R O L L A RY 1 A In the fo llowing we begin by p roviding an upper b ound on the co mplexity expon ent o f MMSE-prep rocessed (u ncon- strained) lattice sphere de coding, where this boun d ho lds for the gener al quasi-static MI MO chann el, for all fading statistics and for any full-rate lattice code. W e will then proceed to provide a l ower bound on the complexity exponent of the same decoder, where th is b ound , u nder the extra assumptions of regular i.i.d. fading statistics an d of layer ed codes, will in fact match the above men tioned upper bou nd to p rove the theo rem and the associated corollaries. Before proceed ing with the bound s, we describe the n T × n R ( n R ≥ n T ) quasi-static point- to-poin t MIMO chan nel, and its corresp onding association to the gener al MIMO channel mod el in (10) and m etric in (1 7). The a foremen tioned qua si-static chann el model ta kes the form Y C = √ ρ H C X C + W C , (47) where X C ∈ C n T × T , Y C ∈ C n R × T and W C ∈ C n R × T represent th e transmitted , received and noise sig nals over a period of T tim e slots, and wher e H C ∈ C n R × n T represents the matrix of fade coefficients. Th e real-v alued representation of (47) can be written as Y R = √ ρ H R X R + W R , (48) where Y R =  ℜ{ Y C } −ℑ{ Y C } ℑ{ Y C } ℜ{ Y C }  , H R =  ℜ{ H C } −ℑ{ H C } ℑ{ H C } ℜ{ H C }  , X R =  ℜ{ X C } −ℑ{ X C } ℑ{ X C } ℜ{ X C }  and W R =  ℜ{ W C } −ℑ{ W C } ℑ{ W C } ℜ{ W C }  , and subsequen t vectorization giv es the rea l-valued m odel y = √ ρ ( I T ⊗ H R ) x + w (49) where y = v ec ( Y R ) , x = v ec ( X R ) , and w = v ec ( W R ) . The system mo del in (4 9) is o f the familiar form y = √ ρ Hx + w (50) as in (1) with m = 2 n T T , n = 2 n R T , and where H = I T ⊗ H R . (51) As b efore the vectorized codew ords x , associated to the full- rate co de, take the fo rm x = ρ − rT κ Gs , s ∈ Z κ ∩ ρ rT κ R , (52) where κ = 2 min { n T , n R } T = 2 n T T = m , which allows u s to rewrite the mod el as y = Ms + w , ( 53) for M = ρ 1 2 − rT κ HG = ρ 1 2 − rT κ ( I T ⊗ H R ) G . (54) Finally the correspo nding co herent MMSE-prep rocessed lat- tice decod er fo r the transmitted sym bol vector s , can b e expressed to be ( cf. (17)) ˆ s r − ld = a rg min ˆ s ∈ Z κ k r − R ˆ s k 2 , (55) where r = Q H 1 y an d R ∈ C κ × κ is the upper-triangular matrix , where furth ermore b oth Q 1 and R resu lt from the thin QR decomp osition of the ( n + κ ) × κ dimensio nal prepr ocessed channel matrix M r eg ,  M α r I  = QR =  Q 1 Q 2  R (56) and wher e as bef ore α r = ρ − rT κ . A. Upper boun d on complexity of re gu larized lattice SD In estab lishing th e up per bo und, we consider Lemma 1 in [1], which we p roperly modify to account for MMSE prepro - cessing and fo r the removal of the con stellation bo undar ies, and g et that the number N k of n odes visited at layer k by the MMSE-prep rocessed lattice sphere decoder , is uppe r boun ded as N k = |N k | ≤ k Y i =1  √ 2 k + 2 ξ σ i ( R k )  , (57) where σ i ( R k ) , i = 1 , · · · , k deno te the singular values of R k in increasing order . T owards lower bounding σ i ( R k ) , we note that σ i ( R k ) ≥ σ i ( R ) = σ i ( M r eg ) = q α 2 r + σ i ( M H M ) , (5 8) where the first in equality makes use o f the in terlacing pro perty of singular values of sub-matr ices [26]. Furthermo re for µ j , − log σ j ( H H C H C ) log ρ , j = 1 , · · · , n T (59) and µ 1 ≥ · · · ≥ µ n T , we see that σ j ( H C ) = ρ − 1 2 µ j , and from (54) that σ i ( M ) ≥ ρ 1 2 − rT κ σ min ( G ) σ ( i ) ( I T ⊗ H R )) . = ρ 1 2 − rT κ σ l 2 T ( i ) ( H C ) = ρ − rT κ + 1 2 (1 − µ l 2 T ( i ) ) , (60) where l T ( i ) ,  i T  , and where the asymp totic eq uality is due to the fact that σ min ( G ) . = ρ 0 . Sub stituting from (6 0) in ( 58) we now have tha t σ i ( R k ) . ≥ ρ − rT κ + 1 2 (1 − µ l 2 T ( i ) ) + , i = 1 , · · · , κ. (61 ) Correspon ding to (57) we see that  √ 2 k + 2 ξ σ i ( R k )  . ≤ ρ ( rT κ − 1 2 (1 − µ l 2 T ( i ) ) + ) + , for any i = 1 , · · · , 2 n T T , and from (57) we h av e that N k ( µ ) . ≤ ρ P k i =1 ( rT κ − 1 2 (1 − µ l 2 T ( i ) ) + ) + , (62) where µ = ( µ 1 , · · · , µ n T ) . It follows that N S D ( µ ) = κ X k =1 N k ( µ ) . ≤ κ X k =1 ρ P k i =1 ( rT κ − 1 2 (1 − µ l 2 T ( i ) ) + ) + . = ρ P κ i =1 ( rT κ − 1 2 (1 − µ l 2 T ( i ) ) + ) + . = ρ T P n T j =1  r n T − (1 − µ j ) +  + , (63) where th e last asymptotic equality is du e to the multiplicity of the singular values. Now consider the set T ( x ) ,    µ | T n T X j =1  r n T − (1 − µ j ) +  + ≥ x    , (64) and note that for any y < x , then (63) and µ / ∈ T ( y ) jointly imply th at N S D < ρ x , which in turn implies that P ( µ / ∈ T ( y )) ≤ P ( N S D < ρ x ) and c onsequen tly that − lim ρ →∞ log P ( N S D ≥ ρ x ) log ρ ≥ − lim ρ →∞ log P ( µ ∈ T ( y )) log ρ . (65) In ev aluating the r ight han d side o f (65) we note that T ( y ) is a closed set and thus, ap plying th e lar ge deviation p rinciple (cf . [27]), we have that − lim ρ →∞ log P ( µ ∈ T ( y )) log ρ ≥ inf µ ∈T ( y ) I ( µ ) (66) for some ra te functio n I ( µ ) . Consequen tly from (65) and (6 6), it follows that − lim ρ →∞ log P ( N S D ≥ ρ x ) log ρ ≥ inf µ ∈T ( y ) I ( µ ) . (67) This lower bound specified in (67) holds for any y < x . Consequently to get the tightest po ssible boun d, we need to find sup y d L ( r ) } = sup { x | inf µ ∈T ( x ) I ( µ ) ≤ d L ( r ) } = max { x | inf µ ∈T ( x ) I ( µ ) ≤ d L ( r ) } (69) where the above follows fro m the afo rementio ned fact that − lim ρ →∞ log P( N S D ≥ ρ x ) log ρ (and by e xtension also inf µ ∈T ( x ) I ( µ ) ) is continu ous and nond ecreasing in x , an d fr om the fact th at T ( x ) is a c losed set. Con sequently c r − ld ( r ) takes the for m c r − ld ( r ) , max µ x (70a) s.t. T n T X j =1  r n T − (1 − µ j ) +  + ≥ x, (7 0b) I ( µ ) ≤ d L ( r ) , (70c) µ 1 ≥ · · · ≥ µ n T ≥ 0 . (70d) Furthermo re since T ( x ) is a closed set, the maxim um x in (70) must be such tha t (70b) is satisfied with equ ality , in which case c r − ld ( r ) can be obtaine d as the so lution to a co nstrained maximization pr oblem according to c r − ld ( r ) , max µ T n T X j =1  r n T − (1 − µ j ) +  + (71a) s.t. I ( µ ) ≤ d L ( r ) , (71b) µ 1 ≥ · · · ≥ µ n T ≥ 0 . (71c) Equiv alently for µ ∗ = ( µ ∗ 1 , · · · , µ ∗ n T ) be ing one of the maximizing v ectors 14 , i.e., such that µ ∗ ∈ T ( x ) and I ( µ ∗ ) = d L ( r ) , then c r − ld ( r ) takes the form c r − ld ( r ) = T n T X j =1  r n T − (1 − µ ∗ j ) +  + . (72) 14 In gen eral, (71) doe s not have a unique optimal point beca use ( a ) + is constant in a for a ≤ 0 . As we will now show , the above bound is also sha red by th e ML-based spher e decod er , with or witho ut MMSE p reproc ess- ing, irrespectiv e of the full-r ate code an d the f ading statistics. Directly fr om [ 1, Th eorem 2] , an d tak ing into co nsideration that M MSE-prep rocessed lattice d ecodin g is DMT optimal fo r any co de [8], we recall th at th e equivalent u pper bo und for the ML-based sph ere decoder, withou t MMSE prep rocessing, takes the for m c ml ( r ) , max µ T n T X j =1 min  r n T − 1 + µ j , r n T  + (73a) s.t. I ( µ ) ≤ d L ( r ) , (73b) µ 1 ≥ · · · ≥ µ n T ≥ 0 . (73c) Comparing ( 71) and (73) we ar e able to conclu de tha t b oth the ob jectiv e functio ns (71a) and ( 73a) as w ell as bo th p airs of co nstraints are identical. T o see this, we first note that for 0 ≤ µ j ≤ 1 , then min  r n T − 1 + µ j , r n T  + =  r n T − 1 + µ j  + ,  r n T − (1 − µ j ) +  + =  r n T − 1 + µ j  + , and fur thermo re we note th at for µ j > 1 , then min  r n T − 1 + µ j , r n T  + =  r n T − (1 − µ j ) +  + = r n T , which proves that c ml ( r ) and c r − ld ( r ) are identical. In consider ing the case of M MSE-prep rocessed ML SD, it is easy to see that the summands in th e objec- ti ve function in (7 3a) will b e modified to take th e f orm min  r n T − (1 − µ j ) + , r n T  + which can be seen to match (71a) for all µ j ≥ 0 , which in tu rn con cludes the pro of that the upper boun d c r − ld ( r ) for MMSE-prep rocessed lattice SD is also shared by the ML-based sphere de coder, with or without MMSE prepro cessing, irr espective of the full-rate code, an d for all fade statistics represented b y mo noton ic rate fu nctions. B. Lower boun d on complexity of r e gularized lattice SD W e will here , un der the extra assump tions of regular i. i.d. fading statistics and of layer ed cod es with natural decodin g order, p rovide a lower b ound th at matches the upp er bo und in (72). Th e same bou nd and tightness will also apply to any full-rate code, under the assumption of a fixed, worst case decodin g ordering. The g oal here is to show th at at layer k = 2 q T , for some q ∈ [1 , n T ] , the sp here decode r visits close to ρ c r − ld ( r ) nodes with a p robab ility that is large comp ared to the pro bability of decodin g error P ( s L 6 = s ) . = ρ − d L ( r ) , wh ich from the expression of the complexity expon ent (40), will p rove that c r − ld ( r ) = c r − ld ( r ) . Going b ack to (72), we let q be the largest integer fo r which r n T − (1 − µ ∗ q ) + > 0 , (74) in whic h case ( 72) takes the fo rm c r − ld ( r ) = T q X j =1 r n T − (1 − µ ∗ j ) + . (75) W e recall from (59) that µ j = − log σ j ( H H C H C ) log ρ , j = 1 , · · · , n T , and that µ ∗ ∈ T ( x ) satisfies I ( µ ∗ ) = d L ( r ) an d maximizes (71a). W e also n ote that withou t loss of generality we can assume that q ≥ 1 as othe rwise c r − ld ( r ) = 0 (cf. (72)). Conseque ntly it is the case that µ ∗ j > 0 f or j = 1 , · · · , q . Furthermo re given the mo notonicity o f the rate function I ( µ ) , and the fact tha t the objectiv e f unction in ( 71) does not increase in µ j beyond µ j = 1 , we m ay also assume witho ut loss of generality that µ ∗ j ≤ 1 for j = 1 , · · · , n T . As in [1] we proceed to define two events Ω 1 and Ω 2 which we will prove to b e jo intly sufficient so th at, at layer k = 2 q T , the sphere deco der visits close to ρ c r − ld ( r ) nodes. The se ar e giv en by Ω 1 , { µ ∗ j − 2 δ < µ j < µ ∗ j − δ, j = 1 , · · · , q 0 < µ j < δ, j = q + 1 , · · · , n T } , (76) for a given sma ll δ > 0 , a nd Ω 2 , { σ 1  ( I T ⊗ V H p ) G | p  ≥ u } , (77) for some gi ven u > 0 , where fo r p , n T − q then G | p denotes the first 2 pT columns of G , and wher e V p denotes the last 2 p column s of V obtained by ap plying the singular value decomposition on H R , i.e., H R = UΣV H , wher e Σ , diag { σ 1 ( H R ) , · · · , σ 2 n T ( H R ) } with σ 1 ( H R ) ≤ · · · ≤ σ 2 n T ( H R ) and VV H = I . Hence, V H p correspo nds to the 2 p largest singular values of H R . Note also that by c hoosing δ sufficiently sma ll, and using the fact that µ ∗ i > 0 fo r i = 1 , · · · , q , we ma y witho ut loss of gener ality assume that Ω 1 implies tha t µ j > 0 f or all j = 1 , · · · , n T . Modifyin g th e appr oach in [1, Th eorem 1] to a ccount fo r MMSE pr eprocessing and un constrained decod ing, the lower bound on the number of nodes visited at layer k by th e sp here decoder, is given by N k ≥ k Y i =1  2 ξ √ k σ i ( R k ) − √ k  + . (78) In the following, and up until (84), we will work to w ards upper bound ing σ i ( R k ) so that we ca n then lower bound N k . T owards this let M r eg | p ,  ρ 1 2 − rT κ HG | p α r I | p  ∈ R 2( n R + n T ) T × 2 pT contain the first 2 pT co lumns o f M r eg from (56), and note that ( M r eg | p ) H M r eg | p = ρ 1 − 2 rT κ G H | p H H HG | p + α 2 r I , and that from (5 1) we get ( M r eg | p ) H M r eg | p = ρ 1 − 2 rT κ G H | p ( I T ⊗ H H R H R ) G | p + α 2 r I . Since H H R H R = V (diag { σ 1 ( H H R H R ) , · · · , σ 2 n T ( H H R H R ) } ) V H = V (diag { σ 1 ( H H R H R ) , · · · , σ 2 n T ( H H R H R ) } − σ (2 q +1) ( H H R H R )diag { 0 , · · · , 0 | {z } 2 q , 1 , · · · , 1 | {z } 2 p } ) V H + σ (2 q +1) ( H H R H R ) V (diag { 0 , · · · , 0 | {z } 2 q , 1 , · · · , 1 | {z } 2 p } ) V H , we have that H H R H R  σ (2 q +1) ( H H R H R ) V (diag { 0 , · · · , 0 | {z } 2 q , 1 , · · · , 1 | {z } 2 p } ) V H = σ (2 q +1) ( H H R H R ) V (diag { 0 , · · · , 0 | {z } 2 q , 1 , · · · , 1 | {z } 2 p } ) (diag { 0 , · · · , 0 | {z } 2 q , 1 , · · · , 1 | {z } 2 p } ) V H = σ (2 q +1) ( H H R H R ) V p V H p where the last equ ality fo llows f rom the fact that V p contains the last 2 p column s of V and where A  B den otes th at A − B is positive-semidefinite. Since σ i ( H H H ) ∈ R and since the Kro necker prod uct induces singular value multiplicity , it follows that ( M r eg | p ) H M r eg | p  ρ 1 − 2 rT κ σ (2 q +1) ( H H R H R ) G H | p ( I T ⊗ V p V H p ) G | p + α 2 r I . W ith respect to the smallest singu lar value o f ( M r eg | p ) H M r eg | p we have σ 1 (( M r eg | p ) H M r eg | p ) ≥ ρ 1 − 2 rT κ σ (2 q +1) ( H H R H R ) · σ 1  G H | p ( I T ⊗ V p V H p ) G | p  + α 2 r and con sequently , given th at H R ∈ Ω 2 , we have th at σ 1 ( M r eg | p ) ≥ ρ − rT κ q u 2 ρσ l 2 (2 q +1) ( H H C H C ) + 1 . = ρ − rT κ ρ 1 2 (1 − µ q +1 ) + ≥ ρ − rT κ + 1 2 (1 − δ ) + , (79) where th e first inequality f ollows fr om (77), the exponential equality f ollows fr om (5 9) an d fro m the fact th at u > 0 is fixed and ind ependen t o f ρ , and the last inequality fo llows from (76). From (54) we have that σ i ( M r eg ) ≤ ρ − rT κ q (1 + ρ ( σ κ ( G ) σ l 2 T ( i ) ( H C )) 2 ) . = ρ − rT κ + 1 2 (1 − µ l 2 T ( i ) ) + , i = 1 , · · · , 2 n T T , (80) where the asymp totic equ ality follows fro m the fact that σ κ ( G ) is fixed and indepe ndent of ρ . Furthermo re (76) giv es that for i = 1 , · · · , 2 q T then σ i ( M r eg ) . ≤ ρ − rT κ + δ + 1 2 (1 − µ ∗ l 2 T ( i ) ) + , (81) where we h av e made use of the fact that µ ∗ j ≤ 1 fo r j = 1 , · · · , n T . Giv en that µ ∗ j > 0 for j = 1 , · · · , q , then for sufficiently small δ and for i = 1 , · · · , 2 q T , we h ave th at − rT κ + 1 2 (1 − δ ) + ≥ − rT κ + δ + 1 2 (1 − µ ∗ l 2 T ( i ) ) + , which means that for sufficiently small δ , a co mparison of (79) and (81) y ields σ i ( M r eg ) < σ 1 ( M r eg | p ) , for i = 1 , · · · , 2 q T . Th e above inequ ality allows u s to apply Lemma 3 in [1], which in turn g iv es that σ i ( R k ) ≤ " σ κ ( M r eg ) σ 1 ( M r eg | p ) + 1 # σ i ( M r eg ) , (82) for i = 1 , · · · , 2 q T . Setting i = κ in (80) u pper b ound s the maximu m sing ular value of M r eg as σ κ ( M r eg ) . ≤ ρ − rT κ + 1 2 (1 − µ n T ) + ≤ ρ 1 2 − rT κ , (83) where th e last inequality is due to the fact that µ j ≥ 0 . Consequently co mbining (83) and (79) g iv es that " σ κ ( M r eg ) σ 1 ( M r eg | p ) + 1 # . ≤ ρ 1 2 δ , which toge ther with (8 1) and (8 2) gives th at σ i ( R k ) . ≤ ρ − rT κ + 3 2 δ + 1 2 (1 − µ ∗ l 2 T ( i ) ) + , i = 1 , · · · , 2 q T . (8 4) Consequently , g oing back to (7 8), we h av e that  2 ξ √ k σ i ( R k ) − √ k  + . ≥ ρ ( rT κ − 3 2 δ − 1 2 (1 − µ ∗ l 2 T ( i ) ) + ) > 0 (85) and furthermo re for i = 1 , · · · , 2 q T , we hav e that r T κ − 3 2 δ − 1 2 (1 − µ ∗ l 2 T ( i ) ) + > 0 directly fro m definition of q and fo r sufficiently small δ . As a result, fo r k ≤ 2 q T we have that N k . ≥ k Y i =1 ρ ( rT κ − 3 2 δ − 1 2 (1 − µ ∗ l 2 T ( i ) ) ) (86) = ρ P k i =1 ( rT κ − 1 2 (1 − µ ∗ l 2 T ( i ) ) + ) − 3 2 kδ , (87 ) and setting k = 2 q T we hav e that N 2 qT . ≥ ρ ( P 2 qT i =1 ( rT κ − 1 2 (1 − µ ∗ l 2 T ( i ) ) + ) − 3 qT δ ) (88) = ρ ( T P q j =1 ( rT κ − (1 − µ ∗ j ) + ) − 3 qT δ ) (89) = ρ ( c r − ld ( r ) − 3 qT δ ) , (90) where the last eq uality follows from (75). Con sequently N S D ≥ N 2 qT . ≥ ρ c r − ld ( r ) − 3 qT δ , for small δ > 0 . Given that δ can be ch osen arbitrarily small, and g iv en that ev ents Ω 1 and Ω 2 occur, then the nu mber of nodes visited by th e SD at layer 2 q T is arbitrar ily clo se to the upper bou nd of ρ c r − ld ( r ) . Now to show that c r − ld ( r ) ≥ c r − ld ( r ) − 3 q T δ , we just h av e to prove that − lim ρ →∞ P  N S D . ≥ ρ c r − ld ( r ) − 3 qT δ  log ρ < d L ( r ) . T oward this we no te that as ( 76) and (77) imply that N S D . ≥ ρ c r − ld ( r ) − 3 qT δ , it follows that P  N S D . ≥ ρ c r − ld ( r ) − 3 qT δ  ≥ P (Ω 1 ∩ Ω 2 ) = P (Ω 1 ) P (Ω 2 ) where the equ ality follows fro m the i.i.d. assump tion o n the entries in H C , wh ich makes the singu lar values o f H H C H C indepen dent of th e singu lar vectors o f H H C H C [28], [ 29], an d which in turn also implies indepen dence of the sing ular values of H H C H C (event Ω 1 ) f rom the sing ular vectors of H H R H R (event Ω 2 ). W e now turn to [1, Lemma 2 ] a nd rec all that for the layer ed codes assumed here , as well as f or any full-rate design an d some non -rand om fixed decoding order ing (corresp onding to a permutatio n of the columns of G ), t here exists a unitary matrix V ′ p such that rank  ( I T ⊗ ( V ′ p ) H ) G | p  = 2 pT i.e., that σ 1  ( I T ⊗ ( V ′ p ) H ) G | p  > 0 . Howe ver , by continuity of sing ular values [26] it follows for sufficiently small u > 0 (cf .(77)) that P (Ω 2 ) > 0 , which implies 15 that P (Ω 2 ) . = ρ 0 as Ω 2 is independent of ρ . T his in turn imp lies that P  N S D . ≥ ρ c r − ld ( r ) − 3 qT δ  . ≥ P (Ω 1 ) . (91) W ith Ω 1 being an open set, we have that − lim ρ →∞ P (Ω 1 ) log ρ ≤ inf µ ∈ Ω 1 I ( µ ) , = q X j =1 ( | n T − n R | + 2 j − 1)( µ ∗ j − 2 δ ) , = d L ( r ) − 2( | n T − n R | + q ) q δ, < d L ( r ) , (92) where the above follows fr om the monoto nicity of the rate function I ( µ ) = n T X j =1 ( | n T − n R | + 2 j − 1) µ i + n R n T t 2 µ n T , ev aluated at { µ ∗ 1 − 2 δ · · · , µ ∗ q − 2 δ, 0 , · · · , 0 } = arg inf µ ∈ Ω 1 I ( µ ) , and 16 also follows from the fact that, by d efinition, I ( µ ∗ ) = d L ( r ) . Consequently from (91) we have that − lim ρ →∞ P  N S D . ≥ ρ c r − ld ( r ) − 3 qT δ  log ρ < d L ( r ) , (93) and directly from th e definitio n of the co mplexity exponent, we have that c r − ld ( r ) ≥ c r − ld ( r ) − 3 q T δ . As the boun d h olds 15 In light of the fact that e vent V ′ p has zero measure, what the continuity of eigen v alues guarant ees is that we can construct a neighborhood of m atrice s around V ′ p which are full rank, and which have a non zero measure. W e also note that the matri ces V ′ p can be created recursi vely , starting from a singl e matrix V ′ n T . 16 Recal l that parameter t was previou sly introduced as a parameter that regul ates the near zero beha vior of the random vari able. for arbitrar ily small δ > 0 , it fo llows that c r − ld ( r ) = c r − ld ( r ) . Directly fr om [1, Th eorem 4] wh ich an alyzes the ML- based complexity expo nent c ml ( r ) , tog ether with the fact th at the ML-based sphere decoder, with or withou t MMSE prepro- cessing, shares the same u pper boun d c r − ld ( r ) as the MMSE- prepro cessed lattice d ecoder, g iv es that c ml ( r ) = c r − ld ( r ) , which in turns imp lies that c r − ld ( r ) = c ml ( r ) . This establishes T heorem 1 and Cor ollary 1a.  A P P E N D I X B P R O O F F O R C O R O L L A R I E S 1 B A N D 1 C Section A-A shows that c r − ld ( r ) can be o btained a s th e solution to th e co nstrained maxim ization problem c r − ld ( r ) , max µ T n T X j =1  r n T − (1 − µ j ) +  + s.t. I ( µ ) ≤ d L ( r ) , (94a) µ 1 ≥ · · · ≥ µ n T ≥ 0 . (94b) In so me cases thoug h, fur ther knowledge o f the error pe rfor- mance of the encoder and deco der, can result in an explicit characterizatio n of the com plexity exponen t. T ake f or instance the case of DMT optimal encoding [15], [16] a nd DMT optimal MMSE- prepro cessed lattice d ecoding [6], [8 ], where the co nstraint I ( µ ) ≤ d L ( r ) in (94a) reverts to the constraint P n T j =1 (1 − µ j ) + ≥ r (cf . [8]), which m ay b e recog nized to co rrespon d to the no -outage region (cf. [3]). In this case c r − ld ( r ) can then be explicitly o btained f rom th e o ptimization problem c r − ld ( r ) = max µ T n T X j =1  r n T − (1 − µ j ) +  + (95a) s.t. n T X j =1 (1 − µ j ) + ≥ r (95b) µ 1 ≥ .... ≥ µ n T ≥ 0 , (95c) which can b e solved in a straightfo rward manner to give that c r − ld ( r ) = T n T  r ( n T − ⌊ r ⌋ − 1) + ( n T ⌊ r ⌋ − r ( n T − 1)) +  , describing the up per b ound o n the comp lexity expo nent f or MMSE-prep rocessed lattice sphere decoding of DMT optimal full-rate code s, which fo r minimu m delay ( n T = T ) DM T optimal fu ll-rate codes takes the for m c r − ld ( r ) = r ( n T − ⌊ r ⌋ − 1) + ( n T ⌊ r ⌋ − r ( n T − 1)) + , (96) and which further simplifies to c r − ld ( r ) = r ( n T − r ) , for inte ger multiplexing gain s r = 0 , 1 , · · · , n T . In conjunction with the lo wer bound in Section A-B, under the conditions lay- ered cod es in Corollary 1 b, we have th at c r − ld ( r ) = c r − ld ( r ) , which proves Corollary 1b.  Moving on to the universal upper b ound , we can see from (71) that, regardless of the f ading statistics and the correspo nd- ing I ( µ ) , the expone nt c r − ld ( r ) is non- decreasing in d L ( r ) and is h ence maximized when d L ( r ) is itself maximized, i.e., it is maximize d in the p resence o f DMT optim al en coding and decodin g. Combined with the fact that the cor respond ing maximization pr oblem in (95) does n ot depen d on the fading distribution, other than the natu ral fact that its tail must vanish expon entially fast, results in the fact that, for any full- rate c ode and statistical cha racterization o f th e ch annel, the complexity o f MMSE- prepro cessed lattice SD is universally upper bou nded as (cf. [1]) T n T  r ( n T − ⌊ r ⌋ − 1) + ( n T ⌊ r ⌋ − r ( n T − 1)) +  . (97) This proves Corollary 1c.  A P P E N D I X C P R O O F F O R L E M M A 1 For R H r R r = M H r M r + α 2 r I (cf. ( 14)) 17 , it follows by the boun ded orthogo nality defect of LLL redu ced b ases that there is a constant K κ > 0 in depend ent of R r and ρ , for which (cf. [22] and the p roof in [3 0]) σ max ( ˜ R − 1 r ) ≤ K κ λ ( R r ) (98) where λ ( R r ) , min c ∈ Z κ \ 0 k R r c k (99) denotes the shortest vector in the lattice gener ated b y R r . As a result we have that σ min ( ˜ R r ) ≥ λ ( R r ) K κ . (100) Lookin g to lower bou nd σ min ( ˜ R r ) , we seek a bou nd o n λ ( R r ) . T owards this let r ′ = r − γ for some r ≥ γ > 0 , in whic h case fo r s bein g th e tran smitted sym bol vector, and for any ˆ s ∈ Z κ such that ˆ s 6 = s , it follows that k r − R r ′ ˆ s k = k ( r − R r ′ s ) + R r ′ ( s − ˆ s ) k ≤ k ( r − R r ′ s ) k + k R r ′ ( s − ˆ s ) k ( 101) and k R r ′ ( s − ˆ s ) k ≥ k r − R r ′ ˆ s k − k ( r − R r ′ s ) k = k r − R r ′ ˆ s k − k w k . (102 ) From (10 2) it is clear that to find a lower bou nd on λ ( R r ′ ) , we need to lower bou nd k r − R r ′ ˆ s k for all ˆ s ∈ Z κ and upper bound k w k . Let us, fo r now , assume that k w k 2 ≤ ρ b . T o lower bound k r − R r ′ ˆ s k , we dr aw from the equivalence o f MM SE prepro cessing an d the regularized m etric (cf. equation (45) in [8]), and r ewrite k r − R r ′ ˆ s k 2 = k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 − c, (103) where c , y H [ I − M H r ′ ( M H r ′ M r ′ + α 2 r ′ I ) − 1 M r ′ ] y ≥ 0 . W e now no te that for ˆ s = s then k y − M r ′ s k 2 + α 2 r ′ k s k 2 . ≤ ρ b , 17 Note the transition to the notation reflecti ng the dependence of R on r . and since the le ft han d side of (10 3) cann ot be negative, and further more g i ven th at c is independ ent of ˆ s , we co nclude that c . ≤ ρ b . W e will now proceed to lower bou nd k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 and then use (103) to lower boun d k r − R r ′ ˆ s k . T owards lower bo unding k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 we draw from Th eorem 1 in [8] and we let B b e the sphe rical region giv en by B , { d ∈ R κ | k d k 2 ≤ Γ 2 } where the r adius Γ > 0 is indepen dent of ρ an d is chosen so that d 1 + d 2 ∈ R f or any d 1 , d 2 ∈ B . The existence of the set B follo ws by the assum ption th at 0 is contained in the inter ior of R . Now let ν r ′ , min d ∈ ρ r ′ T κ B∩ Z κ : d 6 = 0 1 4 k M r ′ d k 2 , and for given γ > ζ > 0 choo se b > 0 suc h that 2 ζ T κ > b > 0 . This may clearly be done fo r ar bitrary ζ > 0 . W e will in the following temp orarily assum e that ν r ′ + ζ ≥ 1 and prove that, together with k w k 2 ≤ ρ b , the two con ditions are sufficient f or λ ( ˜ R r ′ ) . ≥ ρ ζ T κ to ho ld. In o rder to boun d the metric f or ˆ s ∈ Z κ where ˆ s 6 = s , we note that ν r ′ + ζ ≥ 1 implies th at ∀ d ∈ ρ ( r ′ + ζ ) T κ B ∩ Z κ , d 6 = 0 it is the case that 1 4 k M r ′ + ζ d k 2 ≥ 1 1 4    ρ 1 2 − ( r ′ + ζ ) T κ HGd    2 ( a ) ≥ 1 1 4    ρ 1 2 − r ′ T κ HGd    2 ≥ ρ 2 ζ T κ where ( a ) follows f rom the fact that M r = ρ 1 2 − rT κ HG . Consequently 1 4 k M r ′ d k 2 ≥ ρ 2 ζ T κ , ∀ d ∈ ρ ( r ′ + ζ ) T κ B ∩ Z κ , d 6 = 0 . (104) As R is b ounde d, an d as ζ > 0 , it holds th at R ⊂ 1 2 ρ ζ T κ B for all ρ ≥ ρ 1 , for a sufficiently large ρ 1 . This implies tha t s ∈ 1 2 ρ ( r ′ + ζ ) T κ B fo r ρ ≥ ρ 1 since s ∈ ρ r ′ T κ R . For s , d ∈ 1 2 ρ ( r ′ + ζ ) T κ B ∩ Z κ , th ere exists an ˆ s ∈ ρ ( r ′ + ζ ) T κ B ∩ Z κ , ˆ s 6 = s , such that ˆ s = d + s . Hen ce fo r a ny ˆ s ∈ ρ ( r ′ + ζ ) T κ B ∩ Z κ , we have from (104) that 1 4 k M r ′ ( ˆ s − s ) k 2 = 1 4 k M r ′ d k 2 ≥ ρ 2 ζ T κ . (105) As k w k 2 ≤ ρ b , it f ollows that 1 4 k M r ′ d k 2 ≥ k w k 2 for large ρ , and that k y − M r ′ ˆ s k 2 = k M r ′ ( s − ˆ s ) + w k 2 . ≥ ρ 2 ζ T κ . (106) Consequently k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 . ≥ ρ 2 ζ T κ . (107) On the other ha nd if ˆ s / ∈ ρ ( r ′ + ζ ) T κ B , the n by definition of B we have that α 2 r ′ k ˆ s k 2 ≥ 1 4 Γ 2 ρ 2 ζ T κ , and consequ ently that k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 ≥ 1 4 Γ 2 ρ 2 ζ T κ . (108) From (10 7) and (1 08) we then c onclud e that k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 . ≥ ρ 2 ζ T κ . (10 9) Giv en (107) and (109), for any ˆ s ∈ Z κ such that ˆ s 6 = s , it is the case that k y − M r ′ ˆ s k 2 + α 2 r ′ k ˆ s k 2 . ≥ ρ 2 ζ T κ , which comb ined with c . ≤ ρ b allows for (103) to give that k r − R r ′ ˆ s k 2 . ≥ ρ 2 ζ T κ . (110) Applying (99) and (1 02), we have λ ( R r ′ ) ≥ k r − R r ′ ˆ s k − k w k . ≥ ρ ζ T κ − ρ b 2 . = ρ ζ T κ (111) where the expo nential inequality follows f rom (110). Further- more we k now that λ ( R r ) = ρ − γ T κ λ ( R r ′ ) . ≥ ρ − ǫT κ (112) where ǫ = γ − ζ , r ≥ ǫ > 0 , and f rom (10 0) and (11 2) it follows that σ min ( ˜ R r ) . ≥ ρ − ǫT κ . W e now n ote that the above imp lies that fo r ν r ′ + ζ ≥ 1 and k w k 2 ≤ ρ b then σ min ( ˜ R r ) . ≥ ρ − ǫT κ , an d thus app lying the union bou nd yields P  σ min ( ˜ R r ) . < ρ − ǫT κ  = P  ( ν r ′ + ζ < 1 ) ∪ ( k w k 2 > ρ b )  ≤ P ( ν r ′ + ζ < 1) + P  k w k 2 > ρ b  . W e kn ow fro m the exponen tial tail o f th e Gaussian distri- bution th at P  k w k 2 > ρ b  . = ρ −∞ and fro m L emma 1 in [8] that P ( ν r ′ + ζ < 1 ) . ≤ ρ − d M L ( r ′ + ζ ) . Henc e P  σ min ( ˜ R r ) . < ρ − ǫT κ  . ≤ ρ − d M L ( r − ǫ ) for all r ≥ ǫ > 0 . The association with the singular values σ 1 ( ˜ R r,k ) ≤ · · · ≤ σ k ( ˜ R r,k ) is mad e using the interlac ing prop erty o f singular values of sub-matrices, which gives th at σ i ( ˜ R r,k ) ≥ σ i ( ˜ R r ) , i ≤ k = 1 , · · · , κ, (113) and for k = 1 , · · · , κ , that P  σ min ( ˜ R r,k ) . < ρ − ǫT κ  . ≤ ρ − d M L ( r − ǫ ) . Finally from the DMT op timality of the exact implementation of the regularized lattice deco der [6], [8], we h av e that P  σ min ( ˜ R r,k ) . < ρ − ǫT κ  . ≤ ρ − d L ( r − ǫ ) . This proves Lemma 1.  A P P E N D I X D P R O O F F O R L E M M A 2 For a search r adius that grows as ξ = √ z log ρ . = ρ 0 , we first prove that P  k w ′′ k 2 > ξ 2  . ≤ ρ − z ′ for z > z ′ > d L ( r ) . T owards establishin g the prope rties o f the equivalent n oise w ′′ (cf. (45)), we consider an equiv alent representatio n of the M MSE-prep rocessed lattice decoder an d let (cf. [31]) QR =  Q 1 Q 2  R =  M α r I  ∈ R ( n + κ ) × κ (114) be the th in QR factorization of th e mod ified chann el matr ix, where Q 1 = R − 1 M ∈ R n × κ , Q 2 = α r R − 1 ∈ R κ × κ and where R H R = M H M + α 2 r I . It then follows th at for F = Q H 1 , the M MSE-prep rocessed lattice decoder is eq uiv alent to lattice deco ding in the presen ce of channe l R and n oise w ′ = − α 2 r R − H s + R − H M H w = − α r Q H 2 s + Q H 1 w . (115) Consequently we calculate P  k w ′ k > ξ  ≤ P  k − α r Q H 2 s k + k Q H 1 w k > ξ  ( a ) = P  k − α r Q H  s 0  k + k Q H  w 0  k > ξ  ≤ P κ  k w k + s up s ∈ S κ r k − α r s k  > ξ ! ( b ) = P ( κ k w k + κK > ξ ) = P  κ k w k > ( z lo g ρ ) 1 2 − κK  ( c ) ≤ P  κ k w k > ( z 1 log ρ ) 1 2  = P  k w k 2 > z 1 κ 2 log ρ  ( d ) = P  k w k 2 > z 2 log ρ  . = ρ − z 2 (116) where ( a ) fo llows f rom the MMSE- prepr ocessed equ iv alent channel repr esentation (cf . 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