Optimal Large-MIMO Data Detection with Transmit Impairments

Transcription

1 Opima Large-MIMO Daa Deecion wih Transmi Impairmens Ramina Ghods, Chares Jeon, Arian Maeki, and Chrisoph Suder Absrac Rea-word ransceiver designs for muipe-inpu muipe-oupu (MIMO wireess communicaion sysems are affeced by a number of hardware impairmens ha aready appear a he ransmi side, such as ampifier non-ineariies, quanizaion arifacs, and phase noise Whie such ransmiside impairmens are rouiney ignored in he daa-deecion ieraure, hey ofen imi reiabe communicaion in pracica sysems In his paper, we presen a nove daa-deecion agorihm, referred o as arge-mimo approximae message passing wih ransmi impairmens (shor LAMA-I, which akes ino accoun a broad range of ransmi-side impairmens in wireess sysems wih a arge number of ransmi and receive anennas We provide condiions in he arge-sysem imi for which LAMA-I achieves he error-rae performance of he individuayopima (IO daa deecor We furhermore demonsrae ha LAMA-I achieves near-io performance a ow compuaiona compexiy in reaisic, finie dimensiona arge-mimo sysems I INTRODUCTION Pracica ransceiver impemenaions for wireess communicaion sysems suffer from a number of radio-frequency (RF hardware impairmens ha aready occur a he ransmi side, incuding (bu no imied o ampifier non-ineariies, quanizaion arifacs, and phase noise [1] [11] This paper deas wih opima daa deecion in he presence of such impairmens for arge (mui-user muipe-inpu muipeoupu (MIMO wireess sysems wih a arge number of anenna eemens a (possiby boh ends of he wireess ink [12], [13] In paricuar, we consider he probem of esimaing he M T -dimensiona ransmi daa vecor s O MT, where O is a finie conseaion se (eg, QAM or PSK, observed from he foowing (impaired MIMO inpu-oupu reaion [1], [2]: y = H(s + e + n (1 Here, he vecor y C MR corresponds o he received signa, he marix H C MR MT represens he MIMO channe, he vecor e C MT modes ransmi impairmens, and he vecor R Ghods, C Jeon, and C Suder are wih he Schoo of Eecrica and Compuer Engineering, Corne Universiy, Ihaca, NY; e-mai: rghods@cscorneedu, jeon@cscorneedu, suder@corneedu A Maeki is wih Deparmen of Saisics a Coumbia Universiy, New York Ciy, NY; e-mai: arian@sacoumbiaedu The work of R Ghods, C Jeon, and C Suder was suppored in par by Xiinx Inc, and by he US Naiona Science Foundaion under grans ECCS and CCF The work of A Maeki was suppored by he US Naiona Science Foundaion under gran CCF n C MR corresponds o receive noise; he number of receive and ransmi anennas is denoed by M R and M T, respecivey A Conribuions We buid upon our previous resus in [14] and deveop a nove, compuaionay efficien daa deecion agorihm for he mode (1, referred o as LAMA-I (shor for arge-mimo approximae message passing wih ransmi impairmens We provide condiions for which LAMA-I achieves he error-rae performance of he individuay opima (IO daa-deecor, which soves he foowing opimizaion probem: ŝ IO = arg min s O P( s s (2 In words, LAMA-I aims a minimizing he per-user symboerror probabiiy [15], [16] Assuming p(s = M T i=1 p(s i and iid circuary-symmeric compex Gaussian noise wih variance N 0 per compex enry of he noise vecor n, we define he effecive ransmi signa x C MT as x = s+e wih he ransmi-impairmen disribuion p(x s = M T =1 p(x s Besides user-wise independence, we do no impose any condiions on he saisics of he ransmi impairmens his aows us o mode a broad range of ransmi-side impairmens, incuding hardware non-ideaiies ha exhibi saisica dependence beween impairmens and he daa symbos, as we as deerminisic effecs (eg, non-ineariies Our opimaiy condiions are derived via he sae-evouion (SE framework [15], [16] of approximae message passing (AMP [17] [19] and for he asympoic seing, ie, he socaed arge-sysem imi Specificay, we fix he sysem raio β = M T /M R and e M T, and assume ha he enries of H are iid circuary-symmeric compex Gaussian wih variance 1/M R per compex enry To demonsrae he efficacy of LAMA-I in pracice, we provide error-rae simuaion resus in finie-dimensiona arge-mimo sysems Figure 1 iusraes he performance of LAMA-I in a and arge-mimo sysem (we use he noaion M R M T wih QPSK ransmission, and ransmi impairmens modeed as iid circuary-symmeric compex Gaussian noise [1] We observe significan symbo error-rae (SER improvemens compared o ha of reguar LAMA, which achieves given cerain condiions on he MIMO sysem are me he error-rae performance of he individuayopima (IO daa deecor in absence of ransmi impairmens (see [14], [20] for he deais We emphasize ha LAMA-I enais viruay no compexiy increase (compared o reguar

2 symbo error rae (SER LAMA, no EVM LAMA, EVM LAMA-I, EVM LAMA whiening, EVM IO, EVM average received SNR [db] (a 128 BS anennas and 8 users symbo error rae (SER LAMA, no EVM LAMA, EVM LAMA-I, EVM LAMA whiening, EVM IO, EVM average received SNR [db] (b 128 BS anennas and 128 users Fig 1 Symbo error-rae of LAMA-I in arge-mimo sysems wih QPSK and srong Gaussian ransmi noise (EVM = 10 db LAMA-I enabes significan performance improvemens compared o conveniona LAMA and requires ower compexiy han reguar LAMA wih noise whiening LAMA and achieves he same SER performance of whieningbased approaches, which require prohibiive compuaiona compexiy in arge MIMO sysems B Reevan Prior Ar Channe capaciy expressions for he ransmi-impaired MIMO sysem mode (1 have firs been derived in [1] A corresponding asympoic anaysis has been provided receny in [21], which uses he repica mehod [22] o obain capaciy expressions for arge MIMO sysems The resus in [1], [21] buid upon on he so-caed Gaussian ransmi-noise mode, which assumes ha he ransmi impairmens in e can be modeed as iid addiive Gaussian noise ha is independen of he ransmi signa s Whie he accuracy of his mode for a paricuar RF impemenaion in a MIMO sysem using orhogona frequency-division muipexing (OFDM has been confirmed via rea-word measuremens [1], i may no be accurae for oher RF ransceiver designs and/or moduaion schemes LAMA-I, as proposed in his paper, enabes us o sudy he fundamena performance of more genera ransmi impairmens (which may, for exampe, exhibi saisica dependence wih he ransmi signa and even incude deerminisic non-ineariies, which is in sark conras o he commony used ransmi-noise mode in [1] [11], [21] For he we-esabished Gaussian ransmi-noise mode, we wi show in Secion IV ha he sae-evouion equaions of LAMA-I coincide o he couped fixed poin equaions in [21], which reveas ha LAMA-I is a pracica agorihm ha deivers he same performance as prediced by repica-based capaciy expressions in he arge-sysem imi Daa deecion agorihms in he presence of ransmi impairmens were sudied in [1] The proposed mehods rey on he Gaussian ransmi-noise mode, which enabes one o whien he impaired sysem mode (1 by muipying he received vecor y wih a so-caed whiening marix W = N 0 Q 1 2, where Q = N T HH H + N 0 I M is he covariance marix of he effecive ransmi and receive noise n + He, and N T denoes he variance of he enries of he ransmi-noise vecor e By appying he whiening fier W o he received vecor in (1, we obain he foowing saisicayequivaen, whiened inpu-oupu reaion [1], [2]: ỹ = Hs + ñ, (3 where ỹ = Wy, H = WH, and ñ = W(n + He Opima (as we as subopima daa deecion can hen be performed by considering he whiened sysem mode in (3 Whie such a whiening approach enabes opima daa deecion in conveniona, sma-scae MIMO sysems (see [1] for he deais under he Gaussian ransmi-noise mode, compuaion of he whiening marix W quicky resus in prohibiive compuaiona compexiy in arge-mimo sysems consising of hundreds of receive anennas a siuaion ha arises in massive MIMO [12], [13], [23], an emerging echnoogy for 5G wireess sysems LAMA-I avoids compuaion of he whiening marix W aogeher, which resus in (ofen significany reduced compuaiona compexiy Furhermore, he generaiy of our sysem mode enabes LAMA-I o be resiien o a broader range of ransmi-side impairmens C Noaion Lowercase and uppercase bodface eers designae coumn vecors and marices, respecivey For a marix A, we define is conjugae ranspose o be A H The enry on he k-h row and -h coumn is A k,, and he k-h enry of a vecor a is a k The M M ideniy marix is denoed by I M and he M N a-zeros marix by 0 M N We denoe he averaging operaor by a = 1 N N k=1 a k Muivariae compex-vaued Gaussian probabiiy densiy funcions (PDFs are denoed by CN (m, K, where m represens he mean

3 vecor and K he covariance marix; E X [ ] and Var X [ ] denoe expecaion and variance wih respec o he PDF of he random variabe (RV X, respecivey We use P(X = x o denoe he probabiiy of he RV X being x D Paper Ouine The res of he paper is organized as foows Secion II deais he LAMA-I agorihm aong wih he sae-evouion framework Secion III provides condiions for which LAMA-I achieves he performance of he IO daa deecor Secion IV anayzes he specia case of Gaussian ransmi-noise We concude in Secion V II LAMA-I: LARGE MIMO APPROXIMATE MESSAGE PASSING WITH TRANSMIT IMPAIRMENTS Large MIMO is beieved o be one of he key echnoogies for 5G wireess sysems [24] The main idea is o equip he base saion (BS wih hundreds of anennas whie serving a ens of users simuaneousy and wihin he same frequency band One of he key chaenges in pracica arge MIMO sysems is he high compuaiona compexiy associaed wih daa deecion [25] We nex inroduce LAMA-I, a nove ow-compexiy daa deecion agorihm for arge-mimo sysems ha akes ino accoun ransmi-side impairmens We derive he associaed compex sae-evouion (cse framework, which wi be used in Secions III and IV o esabish condiions for which LAMA-I achieves he errorrae performance of he IO daa deecor for he impaired sysem mode (1 A Summary of he LAMA-I Agorihm In he remainder of he paper, we consider a compexvaued daa vecor s C MT, whose enries are chosen from a discree conseaion O, eg, phase shif keying (PSK or quadraure ampiude moduaion (QAM We furher assume iid priors p(s = M T =1 p(s wih p(s = p a δ(s a, (4 where p a corresponds o he (known prior probabiiy of he conseaion poin a O In he case of uniformy disribued conseaion poins, we have p a = O 1, where O is he cardinaiy of he se O We define he effecive ransmi signa x = s + e, which is disribued as p(x = M T =1 p(x wih p(x = p(x s p(s ds, (5 C where p(x s modes he ransmi-side impairmens We can now rewrie he inpu-oupu reaion (1 as y = Hx + n (6 The key idea behind LAMA-I is o perform daa deecion in wo seps We firs use message passing on he facor graph for he disribuion p(s, x y, H in order o obain he margina disribuion p(s y, H Once he message passing agorihm (a Impaired MIMO sysem wih LAMA-I as he daa deecor (b Equivaen decouped MIMO sysem Fig 2 In he arge-anenna imi, LAMA-I decoupes he impaired MIMO sysem ino M T parae and independen AWGN channes, which aows us o perform impairmen-aware MAP daa deecion, independeny per user converged, we assume ha i converged o he margina disribuion, which aows us o perform maximum a-poseriori (MAP deecion on p(s y, H o obain esimaes ŝ for he ransmi daa signas independeny for every user Since he facor graph for p(s, x y, H is dense, ie, for every enry in he receive vecor y we have a facor ha is conneced o every ransmi symbo x, an exac message passing agorihm is compuaionay expensive However, by expoiing he biparie srucure of he graph and he high dimensionaiy of he probem (ie, boh M T and M R are arge, he enire agorihm can be simpified 1 In paricuar, we simpify our message-passing agorihm using compex Bayesian AMP (cb- AMP as proposed in [14], [20] for he MIMO sysem mode (6 cb-amp cacuaes an esimae for he effecive ransmi signa ˆx, The MAP esimae can hen be cacuaed from ˆx independeny for every user The resuing wo-sep procedure of LAMA-I is iusraed in Fig 2 As iusraed in Fig 2(a, we firs use cb-amp o compue he Gaussian oupu z and he effecive noise variance σ 2 a ieraion Since he Gaussian oupu of cb-amp can be modeed as z = x + w wih w N (0, σ2, being independen from x, s and e, in he arge-sysem imi (see [16] for he deais, he MIMO sysem is effecivey decouped ino a se of M T parae and independen addiive whie Gaussian noise (AWGN channes Fig 2(b shows he equivaen decouped sysem Since he effecive ransmi 1 We refer o [26] for more deais on hese caims

4 signas are defined as x = s + e, = 1,, M T, we have z = s + e + w, (7 which aows us o compue he MAP esimae for each daa symbo independeny using ŝ = arg max p(s z (8 s O Here, he probabiiy p(s z is obained from Bayes rue p(s z p(z s p(s and from p(z s = p(x = z w s p(wdw (9 C The resuing LAMA-I agorihm is summarized as foows Agorihm 1 (LAMA-I Iniiaize ˆx 1 = E X[X], r 1 = y, and τ 1 = β Var X [X]/N 0 wih X p(x as defined in (5 1 Run cb-amp for max ieraions by compuing he foowing seps for = 1, 2,, max : z = ˆx + H H r ˆx +1 = F(z, N 0 (1 + τ τ +1 = β N 0 G(z, N 0 (1 + τ r +1 = y Hˆx +1 + τ τ r The scaar funcions F(z, σ2 and G(z, σ2 operae eemen-wise on vecors, and correspond o he poserior mean and variance, respecivey, defined as F(z, σ 2 = x f(x z, σ 2 dx (10 C G(z, σ 2 = x 2 f(x z, σ 2 dx F(z, σ 2 2 (11 C Here, he message poserior disribuion is f(x z, σ2 1 = Z p(z x, σ 2 p(x, where p(z x, σ 2 CN (x, σ 2 and Z is a normaizaion consan 2 Compue he MAP esimae using (8 for = max wih he poserior PDF p(s z max as defined in (9 and p(w max N (0, σ 2 max The effecive noise variance σ 2 max is esimaed using he posuaed oupu variance N 0 (1 + τ max from cb-amp (see [20, Def 3] B Sae Evouion for LAMA-I Viruay a exising heoreica resus are incapabe of providing performance guaranees for he success of messagepassing on dense graphs In our specific appicaion, however, he srucure of he facor graph of p(s, x y, H enabes us o sudy he associaed heoreica properies in he arge-sysem imi As shown in [20], he effecive noise variance σ 2 can be cacuaed anayicay for every ieraion = 1, 2,, max, using he compex sae evouion (cse recursion In Secion III, we wi use he cse framework o derive opimaiy condiions for which LAMA-I achieves he error-rae performance of he IO daa deecor in (2 The cse for cb-amp is deaied in he foowing heorem Theorem 1 ([20, Thm 3] Suppose ha p(x M T =1 p(x and he enries of H are iid circuary-symmeric compex Gaussian wih variance 1/M R Le n CN (0 MR 1, N 0 I MR and F be a pseudo-lipschiz funcion as defined in [27, Sec 11, Eq 15] Fix he sysem raio β = M T /M R and e M T Then, he effecive noise variance σ 2 +1 of cb-amp a ieraion is given by he foowing cse recursion: σ 2 +1 = N 0 + βψ(σ 2 (12 Here, he mean-squared error (MSE funcion Ψ is defined by Ψ(σ 2 = E X,Z [ F ( X + σ Z, σ 2 X 2 ] (13 wih X p(x, Z CN (0, 1, and F is defined in (10 The cse recursion is iniiaized by σ 2 1 = N 0 + β Var X [X] The cse in Theorem 1 racks he effecive noise variance σ 2 for every ieraion, which enabes us o compue he poserior disribuion (9 required in Sep 2 of Agorihm 1 Remark 1 The poserior mean funcion F and consequeny he MSE funcion Ψ(σ 2 in (13 depend on he effecive ransmi signa prior p(x in (5, which is a funcion of he daa-vecor prior p(s and he condiiona probabiiy p(x s ha modes he ransmi-side impairmens III OPTIMALITY OF LAMA-I We now anayze he opimaiy of LAMA-I for he impaired sysem mode (1 A Opimaiy Quesions The cse framework enabes us o characerize he performance of LAMA-I in he arge-sysem imi In his secion, we use his framework o sudy he opimaiy of LAMA-I In paricuar, we address he foowing wo opimaiy quesions: (i We derived LAMA-I using a message-passing agorihm However, here exiss a broader cass of agorihms o accompish he same ask More specificay, he version of LAMA-I ha uses sum-produc message passing empoys he poserior mean funcion F as defined in (10 One can poeniay change F (or even pick differen funcions a differen ieraions and come up wih esimaes ˆx,, and perform daa deecion by using MAP deecion on o hese new esimaes Such aernaive daa-deecion agorihms can si be anayzed hrough he sae evouion framework The firs opimaiy quesion we can ask is wheher we can improve he performance of LAMA-I by choosing funcions differen o hose we inroduced in (10? As we wi show in Secion III-B, he poserior mean funcions we used in (10 are indeed opima (ii We aso ask ourseves wheher he opima LAMA-I, ie, LAMA-I ha ses F o a poserior mean, achieves he same error-rae performance as he IO daa deecor (2? In wha foows, we wi answer boh of hese quesions in he arge-sysem imi

5 B Opimaiy of Poserior Mean for LAMA-I Consider he foowing generaizaion of LAMA-I, where he poserior mean funcion is repaced wih a genera pseudo- Lipschiz funcion F [16] ha depends on he ieraion sep, ie, where we use ˆx +1 = F (z, N 0 (1 + τ The firs opimaiy quesion we woud ike o address is wheher here exiss a choice for he funcions F 1, F 2,, such ha he resuing daa-deecion agorihm achieves ower probabiiy of error The foowing heorem esabishes he fac ha i is impossibe o improve upon he choice of LAMA-I, where we use he poserior mean Theorem 2 Le he assumpions made in Theorem 1 hod for F 1,, F max Suppose ha we run LAMA-I for max ieraions and hen, perform eemen-wise daa deecion Le ŝ be he esimae we obain for s We denoe he deecion error probabiiy as P F1,,F max (ŝ s o emphasize on he dependence of his probabiiy on he funcions empoyed a every ieraion The choice of F 1,, F max ha minimizes P F1,,F max (ŝ s is he poserior mean empoyed in (10 A deaied version of he proof for his heorem is given in [26] For he sake of breviy, we ony skech he main seps of he proof Since F 1,, F max are pseudo-lipschiz according o Theorem 1, we know ha z max can be modeed as z max = s + e + w max, where w max is Gaussian The effec of F 1,, F max is summarized by he variance of w max I is sraighforward o prove ha he smaer he variance of w max is, he smaer he error probabiiy P F1,,F max (ŝ s wi be Hence, we shoud use a sequence of funcions F 1,, F max ha minimize he variance of w max We can use inducion o esabish ha he poserior mean eads o he minimum variance In he as ieraion max, if he variance of w max 1 is fixed, hen i is sraighforward o prove ha we shoud use he poserior mean in he as ieraion o minimize he variance of w max By empoying inducion and by foowing he same ine of argumenaion, we can show ha F 1,, F max mus a be he poserior mean We now use he cse framework in Theorem 1 o esabish condiions for which LAMA-I is opima We consider he case where he number of ieraions max for which, as expained in [20, Sec IV], he cse recursion (12 converges o he foowing fixed-poin equaion: σ 2 = N 0 + βψ(σ 2 (14 This equaion can in genera have one or more fixed poins If i has more han one fixed poin, hen LAMA-I may converge o differen fixed poins, depending on is iniiaizaion [28] As he firs sep oward proving ha LAMA-I is opima, we derive condiions under which he fixed poin equaion (14 has a unique souion To esabish such condiions, we firs define he foowing quaniies (aso see [14, Defs 1-4] Definiion 1 For a given ransmi daa-vecor prior p(s and ransmi-impairmen disribuion p(x s, we define he exac recovery hreshod (ERT β max and he minimum recovery hreshod (MRT β min as { (Ψ(σ 2 1 } { (dψ(σ 2 1 } β max = min σ 2 >0 σ 2, β min = min σ 2 >0 dσ 2 The minimum criica noise N0 min (β is defined as { } N0 min (β = min σ 2 βψ(σ 2 : β dψ(σ2 σ 2 >0 dσ 2 = 1, and he maximum guaraneed noise N0 max (β is defined as { } N0 max (β = max σ 2 βψ(σ 2 : β dψ(σ2 σ 2 >0 dσ 2 = 1 Using Definiion 1, he foowing heorem esabishes severa regimes in which he fixed poin of LAMA-I is unique Lemma 3 (Opimaiy Condiions of LAMA-I Le he assumpions made in Theorem 1 hod and e max Fix p(s and p(x s If he variance of he receive noise N 0 and sysem raio β are in one of he foowing hree regimes: 1 β ( 0, β min] and N 0 R + 2 β ( β min, β max and N 0 [ 0, N0 min (β (N0 max (β, 3 β [β max, and N 0 (N0 max (β, hen LAMA-I soves he opima probem The proof foows from [14, Tabe II] Noe ha for LAMA-I, he quaniies in Definiion 1 do no ony depend on he daa-vecor prior p(s, bu aso on he ransmi-impairmen disribuion p(x s (cf Remark 1 C LAMA-I vs Individuay Opima (IO Daa Deecion We now show ha in he arge-sysem imi, LAMA-I achieves he error-rae performance of he IO daa deecor (2, if he fixed-poin equaion (14 has a unique fixed poin As wi be cear from our argumens, even in cases where LAMA-I does no have a unique fixed poin, one of is fixed poins corresponds o he souion of IO I is, however, difficu o find a suiabe agorihm iniiaizaion ha woud cause our mehod o converge o he opima fixed poin The core of our opimaiy anaysis is he resu on he performance of IO daa deecion based on he repica anaysis presened in [29] The repica anaysis for IO daa deecion makes he foowing assumpion abou ŝ IO Definiion 2 The IO souion is said o saisfy hard-sof assumpion, if and ony if here exis a funcion D : R O, whose se of disconinuiies has Lebesgue measure zero and ŝ IO = D(E(s y, H For some popuar conseaion ses, we can prove ha he hard-sof assumpion is in fac rue For exampe, for equiprobabe BPSK conseaion poins, we have E(s y, H = P(s = 1 y, H P(s = 1 y, H,

6 and hence, ŝ IO = sign(e(s y, H The nex heorem esabishes condiions for which LAMA-I achieves he performance of he IO daa deecor Theorem 4 Suppose ha he IO souion saisfies he hardsof assumpion Furhermore, assume ha he assumpions underying he repica symmery in [29] are correc Then, under a he condiions of Lemma 3 and in he argesysem imi, he error probabiiy of LAMA-I is he same as probabiiy of error of he IO daa deecor For he sake of breviy, we ony presen a proof skech; see [26] for he proof deais From he hard-sof assumpion we reaize ha in order o characerize he probabiiy of error of he IO daa deecor, we have o characerize he join disribuion of (s, E(s y, H Noe ha in [29] he imiing disribuion of (x, E(x y, H is cacuaed A simiar approach wi work for our probem oo However, we have o sighy modify he probem and make i coser o he one in [29] As he firs sep, we firs derive he imiing disribuion of (s, x, E(s y, H Noe ha he join disribuion of (s, x is known Furhermore, s x y form a Markov chain Hence, s x E(x y, H is a Markov chain, and condiioned on x, he wo quaniies s and E(x y, H are independen This impies ha in order o characerize he disribuion of (s, x, E(s y, H, we ony need o characerize he disribuion of (x, E(s y, H Furhermore, we have E(s y, H = E(s x dp(x y, H Define L(x = E(s x Our origina probem of characerizing he imiing disribuion of (s, E(s y, H is simpified o characerizing he imiing disribuion of (x, E(L(x y, H This aer probem can be soved by he repica mehod as expained in [22] The fina resu is he foowing: he join disribuion of (s, x, E(s y, H converges o (S, X, E(S X+ σz, where S p(s, X S = s p(x s, Z N(0, 1 and is independen of boh S and X, and finay σ saisfies he fixed poin equaion σ 2 = N 0 + βψ( σ 2 (15 Noe ha his is he same fixed poin equaion as he one we have for LAMA-I (14 Hence, whenever (15 has a unique fixed poin, he repica anaysis and LAMA-I wi necessariy ead o he same souion So far, we have shown ha he effecive noise eve is he same for LAMA-I and IO I is sraighforward o show ha since he effecive noise eves are he same, he error probabiiy of boh schemes is he same For he deais, refer o our journa paper [26] IV LAMA-I FOR THE GAUSSIAN TRANSMIT-NOISE MODEL Theorem 2, Lemma 3, and Theorem 4 as given above hod for genera ransmi-impairmen disribuions p(x s We now focus on he we-esabished Gaussian ransminoise mode [1], [3] In paricuar, we sar by providing he remaining LAMA-I agorihm deais and hen, derive more specific condiions for which LAMA-I is opima We furhermore provide simuaion resus for finie-dimensiona sysems A Agorihm Deais We assume e CN (0, N T I MT, where e is independen from s and n, and N T is he ransmi-noise power The foowing emma provides he remaining deais for Agorihm 1 wih his mode The proof is given in Appendix A Lemma 5 Assume he MIMO sysem in (1 wih e CN (0, N T I MT being independen of s and n For Sep 1 of Agorihm 1, he probabiiy disribuion p(x is given by p(x = p a 1 πn T exp ( 1NT x a 2 The poserior mean F and variance G funcion corresponds o F(z, σ 2 = N T N T + σ 2 z σ 2 + N T + σ 2 w a a, G(z, σ 2 = N Tσ 2 N T + σ 2 + N T z w + σ2 2 a a N T + σ 2 F(z, σ 2, respecivey, wih w a = p a exp p a exp ( z a 2 N T+σ ( 2 z a 2 N T+σ 2 (16 For Sep 2, he MAP esimaor (8 is given by ( z max a 2 ŝ = arg min N T + N 0 (1 + τ max og p a (17 For he Gaussian ransmi-noise mode, we see ha LAMA-I ony requires a few sube modificaions o he funcions F and G compared o reguar LAMA [20, Ag 1], which ignores ransmi-side impairmens Hence, making LAMA robus o he Gaussian ransmi-noise impairmens comes a viruay no expense in erms of compexiy, bu resus in ofen significan performance improvemens (cf Secion IV-C B Opimaiy Condiions The opimaiy condiions in Lemma 3, which depend on he sysem raio β, receive noise variance N 0, as we as he signa prior and he ransmi-impairmen mode, can be obained via he fixed-poin equaion in (14 I can be shown ha for he Gaussian ransmi noise mode, he fixed-poin equaion (14 is equivaen o he couped fixed poin equaions derived in [21, Eqs 48 and 49], which have been used o characerize he capaciy of he impaired sysem (1 Whie he resus in [21] have been obained via he repica mehod [22], LAMA-I provides a pracica agorihm ha achieves he same performance in he arge-sysem imi

7 The foowing emma provides a condiion for which LAMA-I is opima Our condiion is independen of he receive noise variance N 0 and he ransmi-noise power N T The proof is given in Appendix B Lemma 6 Le he assumpions in Theorem 1 hod and suppose ha he IO souion saisfies he hard-sof assumpion Define βm min = min NT β min (N T Furhermore, assume he Gaussian ransmi-noise mode If β βm min, hen LAMA-I is opima This emma impies ha here is a hreshod βm min on he sysem raio β ha enabes LAMA-I o achieve he same error-rae performance as he IO daa deecor in he argesysem imi Noe ha his condiion is independen of he receive and ransmi noise eves N 0 and N T, respecivey C Simuaion Resus We now demonsrae he efficacy of LAMA-I for he Gaussian ransmi-noise mode in more reaisic, finie-dimensiona arge-mimo sysems We define he average receive signao-noise-raio (SNR as SNR = E [ Hs 2] E [ n 2 ] = β E s N 0, where E s = E [ s 2] We aso define he so-caed errorvecor magniude (EVM as EVM = E [ e 2] E [ s 2 ] = N T E s Figures 1(a and 1(b iusrae he symbo error rae (SER simuaion resus for arge MIMO sysems wih QPSK moduaion, Gaussian ransmi-noise, and wo anenna configuraions, ie, and In boh figures, he soid bue ine corresponds o he performance of reguar LAMA [14], [20] in absence of ransmi noise (ie, EVM = db As shown by he dashed red ine, reguar LAMA experiences a significan performance oss in he presence of ransmi noise wih EVM = 10 db In conras, LAMA-I (indicaed wih he dash-doed magena ine yieds significan performance improvemens (he maximum number of LAMA-I ieraions for and was max = 10 and max = 15, respecivey The soid green ine shows he opima arge-sysem imi performance As i can be seen, LAMA-I cosey approaches he opimum SER performance for finie-dimensiona sysems Figures 1(a and 1(b furhermore compare LAMA-I o reguar LAMA operaing on he whiened sysem (3 shown by he doed back ine Whie boh approaches achieve nearopima performance, he whiening-based approach enais prohibiive compexiy, mainy caused by he inverse marix square roo In addiion, he whiening-based approach is designed specificay for he Gaussian ransmi-noise mode; in conrary, LAMA-I is appicabe o a broader range of rea-word ransmi-side impairmens V CONCLUSION We have inroduced LAMA-I, a nove, compuaionay efficien daa deecion agorihm suiabe for arge-mimo sysems ha are affeced by a broad range of ransmi-side impairmens We have deveoped condiions in he argesysem imi for which LAMA-I achieves he error rae performance of he individuay opima (IO daa deecor For he specia case of he Gaussian ransmi-noise mode and for pracica anenna configuraions, we have demonsraed ha LAMA-I enabes significan performance improvemens compared o impairmen-agnosic agorihms a viruay no overhead in erms of compuaiona compexiy As a consequence, LAMA-I is a pracica daa-deecion agorihm ha renders pracica arge-mimo sysems more resiien o user equipmen ha suffers from srong ransmi-side impairmens APPENDIX A DERIVATION OF F AND G FOR GAUSSIAN TRANSMIT NOISE We firs derive p(x as used in Sep 1 of Agorihm 1 From (4 and he Gaussian ransmi-noise mode e CN (0, N T I MT, which assumes independence from s, we can wrie effecive ransmi signa prior (5 as foows: ( 1 p(x = exp 1 s x 2 p a δ(s ads C πn T N T = 1 p a exp ( 1NT x a 2 πn T Wih his resu, we can wrie he message poserior disribuion f(x ˆx, σ 2 defined in Sep 1 of Agorihm 1 as: f(x z, σ 2 1 = Zπ 2 N T σ 2 p a exp ( z a 2 N T + σ 2 ( exp N T + σ 2 N T σ 2 x N Tz + σ2 a 2 N T + σ 2 Here he normaizaion consan Z is chosen so ha C f(x ˆx, σ 2 dx = 1, which can be compued as: Z = 1 p a ( π(n T + σ 2 exp z a 2 N T + σ 2 Therefore, he poserior mean F(z, σ2 in (10 is given by: F(z, σ 2 = x f(x z, σ 2 dx = N T z w + σ2 a a C N T + σ 2, wih he shorhand noaion (16 The message variance G(z, σ2 defined in (11 can be derived simiary For Sep 2 of Agorihm 1, he effecive noise w max is disribued as p(w max CN (0, N 0 (1 + τ max wih saisica independence from he ransmi-noise mode e CN (0, N T, which yieds p(w max + e CN (0, N 0 (1 + τ max + N T Wih his resu and reaion (7, we have p(z max s CN (s, N 0 (1 + τ max + N T, which ogeher wih (4, yieds he foowing poserior disribuion:

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