Linear precoding via conic optimization for fixed MIMO receivers

Transcription

1 Lnear precodng va conc optmzaton for fxed MIMO recevers Am Wesel, Yonna C Eldar, and Shlomo Shama (Shtz) Department of Electrcal Engneerng Technon - Israel Insttute of Technology June 8, 004 Abstract We consder the problem of desgnng lnear precoders for fxed multple nput multple output (MIMO) recevers Two dfferent desgn crtera are consdered In the frst, we mnmze the transmtted power subject to sgnal to nterference plus nose rato (SINR) constrants In the second, we maxmze the worst case SINR subject to a power constrant We show that both problems can be solved usng standard conc optmzaton packages In addton, we develop condtons for the optmal precoder for both of these problems, and propose two smple fxed pont teratons to fnd the solutons whch satsfy these condtons The relaton to the well known downlnk uplnk dualty n the context of jont downlnk beamformng and power control s also explored Our precoder desgn s general, and as a specal case t solves the beamformng problem In contrast to most of the exstng precoders, t s not lmted to full rank systems Smulaton results n a multuser system show that the resultng precoders can sgnfcantly outperform exstng lnear precoders INTRODUCTION Multple nput multple output (MIMO) systems arse n many modern communcaton channels, such as multple user communcaton 3], and/or multple antennas channels 4] It s well known that the use of multple antennas promses substantal capacty gans when compared to tradtonal sngle antenna systems In order to explot these gans, the system must deal wth the dstorton caused by the channel and/or the nterference The conventonal way to deal wth these dstortons s recever optmzaton Recently, the quest for better performance wth lower complexty led researchers to also optmze the transmtter 5 0], e-mals: amw@txtechnonacl, yonna@eetechnonacl and sshlomo@eetechnonacl Presented n part n, ] The work was supported n part by the NewCom project

2 and even to jontly optmze the transmtter and recever 7] Ths, as well as new results and algorthms n convex optmzaton theory 8], have sgnfcantly mproved state of the art communcaton systems H Ch, H Rx, y w b T H Tx H Ch, H Rx, y w Fxed! H Ch,M H Rx,M w M y M Fgure : Block dagram of a precoder for a fxed MIMO recever In ths paper, we explore the desgn of a centralzed precoder gven fxed lnear MIMO transmtter, channel and recever We defne a precoder as a lnear transformaton on the transmtted symbols If the precoded symbols are sent as s to the channel, then the precoder s the transmtter tself However, n general, the precoded symbols may be transformed agan before the channel We refer to ths transformaton as the transmtter, and we assume that t s a fxed desgn parameter An example of a fxed transmtter s the multuser spreadng transmtter, whch spreads the symbols usng standardzed sgnatures whch cannot be altered The output of the transmtter s then sent over a fxed MIMO channel (or channels), and s receved usng fxed lnear recevers (See Fg ) Naturally, a system whch jontly optmzes the precoder and recevers wll outperform a system wth fxed recevers However, ths problem s nterestng as t allows mprovng the system wthout modfyng the recevers Snce optmzaton technques are usually costly n terms of complexty, by optmzng only the precoder, we allow for smple low complexty recevers An appealng applcaton s the downlnk channel of a cellular system The base staton, whch s less restrcted n complexty terms, s optmzed and allows better performance wthout changng the low complexty moble hand sets For example, one of our nterestng results s that n a symmetrc channels, usng an optmzed precoder acheves the performance of mnmum mean squared error (MMSE) recevers whle usng smple matched flter (MF) recevers In our development, we restrct ourselves to desgn when there s perfect knowledge of the channel

3 state nformaton (CSI) at the transmtter In scenaros when there s no CSI at the transmtter, the common approach s to resort to space tme codng or spatal multplexng However, n many stuatons the transmtter has ths knowledge In tme dvson duplex (TDD) t can be estmated, whereas n frequency dvson duplex (FDD) t can be fedback from the recever Therefore, followng pror work, we assume that the CSI s perfect and errorless One of the frst results on optmzng a precoder for a fxed MIMO lnear model s due to 5] n the context of multuser transmsson In that work, two precoders were derved The precoders appled a lnear transformaton on the transmtted symbols pror to the spreadng The frst precoder was desgned to mnmze the mean squared error (MSE) between the receved vector and the symbols vector It was then scaled to ft an average power constrant on the output of the transmtter The second was desgned to mnmze the MSE subject to an average power constrant It was observed that the frst precoder decorrelates the channel at the transmtter sde, and asymptotcally outperforms the second Therefore, t s sometmes referred to as the transmt zero forcng (ZF) precoder One of the man drawbacks of the transmt ZF precoder s ts degraded performance n low sgnal to nose rato (SNR) Ths phenomena s well known n MIMO recepton and multuser detecton 3] Invertng the channel decreases the output SNR Therefore, n low SNR, the preferable recever s the conventonal MF whch maxmzes the SNR Ths motvated the desgn of transmt MF precoders 7] and transmt rakes 9 ] Clearly, just as n the recever case, n hgh SNR, the nterference domnates the nose, and the performance of these precoders degrades Recently, transmt scaled MMSE precoders were derved n 7 0], and tred to compensate for the performance n the dfferent SNR regons It s mportant to note that the name transmt MMSE or Transmt Wener s borrowed from the termnology of recever desgn These recevers do not mnmze the MSE On the contrary, as we explaned, the transmt ZF mnmzes t 5] Ths stems from the fact that the choce of the precoder does not effect the nose snce t s placed n the transmtter sde Other precoders usng dfferent knds of optmzaton crtera were also derved We only provde a bref non comprehensve revew of the general schemes and do not elaborate on all of the precoders Varants of the prevous precoders were dscussed n 7] Lnear precoders based on an approxmate maxmum lkelhood approach and maxmum asymptotc multuser effcency wth dfferent power constrants were derved n 8] A lnear precodng technque based on a decomposton approach was proposed n 9] A lnear precoder desgn for non lnear maxmum lkelhood (ML) recevers was dscussed n 30] Moreover, non lnear precoders have also been derved Among them are the non lnear Tomlnson Harashma MIMO 3

4 precoder 6 0, 3], the well known Drty Paper precoder 3], and the vector perturbaton precoder 33] Another non lnear precoder whch optmzes the transmtted symbol s vector tself was derved n 34] The problem of precoder desgn s hghly related to other problems n the lterature As stated, we nvestgate the desgn of lnear precoders for fxed lnear recevers A related problem s the problem of jontly optmzng the precoder/transmtter and the recever, whch has been treated, eg, n 9, 7,35,36] The desgn of the optmal sgnatures (whch can be consdered as a lnear precodng scheme) for matched MMSE recevers was dscussed n 37, 38], whereas sgnature desgn for matched decson feedback recevers was explored n 39] One of the nterestng propertes of these jont desgns s that maxmzng SINR s related to mnmzng MSE 5] Thus, although dfferent crtera have been explored, most of the research was dedcated to varants of the MMSE crteron Another related problem s the rank one transmt beamformng desgn and optmal power control, 40 4] Ths problem s well nvestgated n the lterature The man dfference between precodng and transmt beamformng s that n beamformng t s assumed there s no transmtter, e, the precoder tself s the transmtter Ths smplfes the transmtted power constrants At frst glance t seems that the precodng problem can be solved by addressng the transmt beamformng problem and then compensatng for the fxed transmtter Unfortunately, ths s not possble when the transmtter s rank defcent and cannot be nverted In ths aspect, our problem s more general Unlke the prevous references regardng precodng whch usually dealt wth the MSE crteron and ts varants, the beamformng communty has successfully managed to optmze SINR based crtera, whch are more related to practcal performance measures, such as bt error rate (BER) and capacty Ths problem s mathematcally more dffcult than MSE optmzaton It was solved usng an nterestng dualty between downlnk and uplnk beamformng 43 45] The uplnk beamformng problem has been solved before n 46, 47] Usng the dualty, the downlnk SINR problem can be handled as well 4,48,49] Recently, a non lnear Tomlnson Harashma verson of these papers was presented n 50] In ths paper, we ntegrate the deas above n the context of MIMO precodng for fxed recevers As explaned before, the desgn of most of the prevous precoders s based on mnmzng varants of the common MSE crteron Ths crteron s usually computatonally attractve and performs qute well However, as far as the applcatons are concerned, the nterestng and relevant crtera are BER and capacty It s well known that these are ntmately assocated wth maxmzng SINR 3] Unlke jont optmzaton, optmzng the system to mnmze MSE does not necessarly maxmze SINR when the recever s fxed Thus, followng On the other hand, most of the above references deal wth beamformng for rank r > and are therefore more general than beamformng 4

5 the transmt beamformng approach, we focus on SINR based crtera, and, n partcular, try to optmze the worst SINR We consder two desgn strateges The frst maxmzes the worst SINR subject to an average power constrant The second mnmzes the requred average power subject to a constrant on the worst SINR We prove that the proposed precoders have the attractve property of equal performance among all the sub channels Our precoders desgn s based on the powerful framework of convex optmzaton theory 8], whch allows effcent numercal solutons usng standard optmzaton packages 5] A bref revew of such programs and ther standard forms s provded n Secton 3 The frst contrbuton of ths paper s the castng of the precoder desgn problems as standard conc optmzaton packages The power optmzaton can be formulated as a Second Order Cone Program (SOCP) 5], or a sem defnte program (SDP) 53] (otherwse known as a lnear matrx nequaltes (LMI) program) Ths procedure has already been addressed n 4] n the context of beamformng We generalze t for the case of precodng Moreover, we show that the SINR optmzaton can also by formulated as a standard conc program known as the generalzed egenvalue problem (GEVP) 54] Another contrbuton of ths paper s the dervaton of the optmalty condtons for both of the desgn problems By analyzng the Karush-Kuhn-Tucker (KKT) condtons for conc programs we present smple condtons for optmalty These provde more nsght to the problem We derve a smple expresson for the structure of the optmal precoder as a functon of the dual varables The condtons can be used to verfy whether a proposed soluton s optmal For example, usng these condtons t s easy to show that the so called scaled MMSE precoder proposed n 7 9] does not necessarly maxmze the worst SINR, except n the case of a symmetrc channel Another use for these condtons s as a stoppng crtera n prevous teratve optmzaton algorthms Probably the most mportant use of the optmalty condtons s n dervng new desgn algorthms Usng the condtons, we provde a smple fxed pont teraton whch s guaranteed to converge to the soluton of the power optmzaton As a specal case, ths smple teraton can solve the well known beamformng problem Ths allows a smple soluton to the problem wthout the need for specal optmzaton packages A smlar fxed pont teraton s derved for the SINR optmzaton problem wthout a convergence proof In comparson to the downlnk-uplnk dualty based solutons, whch consst of nner and outer teratons and egenvalues optmzatons, our fxed pont teratons are consderably more appealng In addton, followng 4], we derve an alternatve approach for satsfyng the optmalty condtons n the power optmzaton through a dual SDP/LMI program 5

6 One of the advantages of our proposed algorthms s ther robustness to the rank of the effectve channels Most of the prevous precoders assume a full rank effectve channel For example, one cannot decorrelate the channel n 5] f the channel s rank defcent, as s the case when the number of users s greater than the spreadng factor (or the number of transmt antennas) Our desgn algorthms, both the conc solutons and the fxed pont teratons, are ndfferent to the rank of the channel, and are therefore applcable to such scenaros as well In addton, followng 37,38] whch addressed ths problem n the context of optmal sequences desgn for MMSE recevers, we provde an upper bound for the maxmal feasble SINRs n these cases An nterestng result of our precoders s ther performance n symmetrc systems As already mentoned, n ths case, our precoders admts a smple closed form expresson that has already been derved n 7 9] through dfferent consderatons and for general channels Usng our optmalty condtons, t s easy to show that these precoders maxmze the worst SINR n symmetrc channels A realstc example of such systems s a code dvson multple access (CDMA) scheme, usng pseudonose (PN) sequences as sgnatures Moreover, we analytcally show that the achevable SINRs usng these precoders wth Matched Flter (MF) recevers s dentcal to those obtaned by usng MMSE recevers wth no precoders Ths result s nterestng, as t allows for each user to use a smple recever that does not requre nether the knowledge of all the other sgnatures nor a matrx nverson It s mportant to note that ths feature does not extend to non symmetrc channels The paper s organzed as follows We begn n Secton by ntroducng the problem formulaton A bref revew of conc optmzaton s provded n Secton 3 The power optmzaton problem s explored n Secton 4, n whch we dscuss ts feasblty, and provde standard conc optmzaton solutons In order to mprove our desgn algorthms and n order to gan more nsght nto the problem, we then provde optmalty condtons, and suggest a smple fxed pont teraton for fndng the varables that satsfy them Next, n Secton 5 we follow the same steps whle addressng the SINR optmzaton problem A few specal cases for whch a closed form soluton exsts are explored n Secton 6 In Secton 7, we llustrate the use of the aforementoned precoders n the context of multple user communcaton systems The followng notaton s used Boldface upper case letters denote matrces, boldface lower case letters denote column vectors, and standard lower case letters denote scalars The superscrpts ( ) T, ( ) H, ( ) and ( ) denote the transpose, the Hermtan, the matrx nverse operators, and the Moore Penrose pseudonverse, respectvely X],j denotes the (th,jth) element of the matrx X By dag {x } we denote a dagonal matrx wth x beng the (th,th) element, by vec (X) we denote stackng the elements of X n one long column 6

7 vector, by e we denote a zeros vector wth a one at the th element, by we denote an all ones vector, and by I we denote the dentty matrx of approprate sze Tr { },, and denote the trace operator, the Kronecker product, the absolute value operator, the standard Eucldean norm, and the nduced matrx norm, respectvely Fnally, X 0 denotes that the matrx X s a Hermtan postve sem defnte matrx, and N { } denotes the Null space operator PROBLEM FORMULATION Consder a general, block orented, MIMO communcaton system wth a centralzed transmtter At each tme nstant, a block of symbols s modulated and transmtted over a channel The possbly dstorted output s then processed at the recever n a lnear fashon, as depcted n Fg Denotng by y the length L output of the th recever, for =,, M, we have that: y = H Rx, H Ch, H Txb + H Rx, w, () y M H Rx,M H Ch,M H Rx,M w M where the matrces H Rx, and H Ch, denote the recever and channel assocated wth the th user, the matrx H Tx s the centralzed transmtter, b s the length K = M L vector of ndependent, and unt varance transmtted symbols, and w are the nose vectors Ths system model s qute general The nose vectors may be correlated or even dentcal to each other, and the channels are completely arbtrary The only restrcton s that the transmtter s centralzed and has access to all of the K transmt components Let us present two specfc examples for whch the followng problem formulaton holds: CDMA system - Consder a multple user downlnk system usng CDMA At each tme perod, the base staton transmts K symbols usng an N K sgnature matrx H Tx = S wth columns s For smplcty, we assume deal channels H Ch, = I and conventonal matched flter recevers H Rx, = s H where s denotes the th column of S Denotng by y the output of the th user s recever, we have that y = s H Sb + s H w, () y K s H K s H K w K whch s clearly a specal case of () 7

8 Beamformng - Consder a multuser system ncorporatng multple antennas At each tme perod, K symbols are modulated, wthout any transformaton at the transmtter H Tx = I, through K transmt antennas to K users, each usng one receve antenna The dfferent pathes between the transmt antennas and the th user are represented n the length K vectors H Ch, = h H Denotng by y the output of the th user s recever, we have that y = h H b + w, (3) y K h H K w K whch s agan a specal case of () In the sequel, we wll assume that the transmtter H Tx, the channels H Ch, and the recevers H Rx, are fxed, and cannot be altered due to budget restrctons, standardzaton, or physcal problems Gven ths fxed structure, we wll try to mprove the performance of such systems by ntroducng a lnear precoder The precoder, denoted by T, lnearly transforms the orgnal symbol vector pror to the transmsson, so that the outputs of the recever are now gven by y = H Rx, H Ch, H TxTb + H Rx, w (4) y M H Rx,M H Ch,M H Rx,M w M For ease of representaton, we wll use the followng notaton: y = HTb + w, (5) where H = H RxCh H Tx and y = y, H RxCh = H Rx, H Ch,, w = H Rx, w (6) y M H Rx,M H Ch,M H Rx,M w M Our goal s to mprove the system performance by optmally desgnng the precoder The system performance s usually quantfed by ts qualty of servce (QoS) and the resources t uses The most common QoS metrcs are BER and capacty, both of whch are hghly related to the output SINRs, and n partcular 8

9 to the worst SINR In our model, the output SINR of the th sub channel s defned as: SINR = { HT], } E b { ]} = E y HT], b j HT],, =,, K, (7) HT],j + σ where σ = E{ w } > 0 Another range of crtera deal wth the use of system resources, eg, peak to average rato, or maxmal transmtted power The most common resource measure s average transmtted power, whch s defned as: P = E { H Tx Tb } = Tr { T H H H TxH Tx T } (8) It s easy to see that the SINRs metrcs and average power metrc conflct One cannot maxmze the SINRs whle also mnmzng the power, and vce versa Dependng on the applcaton, the desgner must decde whch crtera s strcter We therefore consder one of the followng two complementary strateges The frst optmzaton strategy seeks to mnmze the average transmtted power subject to QoS constrants Ths crteron s nterestng from a system level perspectve Gven the requred QoS, the system tres to satsfy t wth mnmum transmtted power 6], 4]: mn T Tr { T H H H Tx H TxT } P(γ o ) = HT] st mn, γ j HT],j o, +σ (9) where γ o > 0 s the gven worst SINR constrant The second strategy s maxmzng the mnmal SINR subject to a power constrant 48], 5] Ths problem formulaton s nterestng when the power constrant s a strct system restrcton whch cannot be relaxed In ths case, the problem can be formulated as HT] max T mn, S(P o ) = j HT],j +σ st Tr { T H H H Tx H TxT } P o, (0) where P o > 0 s the gven power constrant Note that although we are optmzng the mnmum SINR n both problems, n the sequel, we wll show that at the optmal soluton of both problems all users wll attan equal SINRs In other words, the above desgn crtera both promse farness among all the sub streams Ths s an mportant property n MIMO communcaton systems In systems where some streams demand dfferent QoS, eg, systems wth voce 9

10 and data streams, the desgner can replace each SINR n the optmzatons wth SINR /ρ where ρ are constant weghts that denote the mportance of the sub streams Ths wll ensure weghted farness among the streams One of the man observatons of our work s that both optmzaton problems (9) and (0) can be solved usng standard conc optmzaton algorthms Therefore, n the followng secton, we revew these algorthms 3 REVIEW OF CONIC OPTIMIZATION In recent years, there has been consderable progress and development of effcent algorthms for solvng a varety of optmzaton problems In order to use these algorthms, one must reformulate the problem nto a standard form whch the algorthms are capable of dealng wth In ths secton, we wll brefly revew the three formulatons whch we use n the paper: SOCP, SDP and GEVP programmng The most wdely researched feld n optmzaton s convex optmzaton A convex program s a program wth a convex objectve functon and convex constrants It s well known that n such programs a local mnmum s also a global mnmum Thus, the global mnmum can be found by any Hll Clmbng or Gradent Descent algorthm The most common convex program s probably the Lnear Program (LP) 8], e, an optmzaton wth a lnear objecton functon and lnear (affne) constrants Recent advances n convex optmzaton generalze the results and algorthms of LPs to more complcated convex programs Specal attenton s gven to conc programs, e, LPs wth generalzed nequaltes The two standard conc programs are SOCP and SDP optmzaton The standard form of an SOCP s 5]: SOCP : mn x st f H x ch x + d A H x + b K 0, =,, N, () where the optmzaton varable s the vector x of length n and f, A, b, c and d for =,, N are the data parameters of approprate szes The notaton K denotes the followng generalzed nequalty: z z K 0 z z () 0

11 The standard form of an SDP s 53]: mn x f H x SDP : st A(x) 0, (3) where A(x) = A 0 + n = x A s an Hermtan matrx that depends affnely on x The data parameters are the Hermtan matrces A for = 0,, n The notaton denotes the postve sem defnte generalzed nequalty A smple case of an SDP s an SOCP Ths stems from the property that any SOC nequalty can be wrtten as an LMI 8]: ch x + d A H x + b K 0 ch x + d x H A + b H A H x + b ( ) c H x + d I 0 (4) A common optmzaton package desgned to solve SOCP and SDP s SEDUMI 5] Although most of the research n the feld of optmzaton concerns convex programs, due to ther mportance, some cases of non convex problems have also been nvestgated Among them s the generalzed egenvalue mnmzaton program (GEVP) 54], whch s not convex but can stll be effcently solved Its standard form s GEVP : mn β,x β st βb(x) A(x) 0; B(x) 0; C(x) 0, (5) where A(x) = A 0 + n = x A, B(x) = B 0 + n = x B and C(x) = C 0 + n = x C are Hermtan matrces that depend affnely on x The data parameters are the Hermtan matrces A, B and C for = 0,, n The name of the GEVP arses from ts resemblance to the well known problem of mnmzng the maxmal generalzed egenvalue of the pencl A, B], e, mnmzng the largest β such that Av = βbv It s easy to show that ths problem can be expressed as mn β β st βb A 0, (6) whch s of course a smple SDP The GEVP generalzes ths program to the case where A and B also depend on the optmzaton varables

12 4 POWER OPTIMIZATION In ths secton, we consder the power optmzaton subject to SINR constrants, e, the P problem of (9) We begn n Secton 4 by dscussng ts feasblty, and then provde a few alternatve approaches for ts soluton In partcular, n Secton 4, we derve a soluton to the problem whch s based on standard SOCP or SDP optmzaton packages Next, n Secton 43, we develop optmalty condtons for ths problem, and use them to derve two alternatve solutons For completeness, n Secton 44, we dscuss the downlnk uplnk dualty n the context of the power optmzaton 4 FEASIBILITY The frst mportant property of any optmzaton problem s ts feasblty (admssblty), e, whether a soluton exsts In other words, we need to verfy whether for a gven γ o there exsts a T such that mn j HT], γ o (7) HT],j + σ It s easy to see that the SINRs are bounded by the sgnal to nterference ratos (SIR): j HT], HT],j + σ HT], j HT],j, =,, K, (8) wth equalty f and only f σ = 0 for =,, K, e, n a zero nose envronment By scalng T to at for large enough a > 0, the dfference between the SIRs and the SINRs can be made nsgnfcant Therefore, for the sake of examnng the feasblty, the nterestng case s the zero nose envronment A condton for the feasblty n ths case s provded n the followng lemma Lemma There exsts a T such that (7) holds only f γ o (9) K rank(h) Proof In order to prove the lemma we must upper bound the mnmal SIR: mn HT], j = mn HT],j ξ = mn ξ, (0)

13 where ξ = HT],, and we have used the monotoncty of f(a) = HTT H H H ] n a Due to monotoncty,, we can bound f(a) by boundng ts argument Thus, we now develop a bound on the mnmum ξ Let HT have a sngular values decomposton (SVD) HT = UΛV H, where U and V are untary K r matrces, Λ s an r r dagonal matrx, and r = rank (HT) Then, a ξ = u H Λv u H, =,, K, () Λ u where u and v are the th columns of U H and V H, respectvely For every =,, K, we can bound () by applyng the Cauchy-Schwarz nequalty to the vectors Λu and v u H Λv ( v H v ) ( u H Λ u ), =,, K () Snce v H v = ] (HT) HT, we conclude that, ξ ] (HT) HT, =,, K (3), Thus, the mnmum ξ s bounded by mn ξ K K ξ K = K = (HT) HT ], = { } K Tr (HT) rank (HT) rank (H) HT = K K (4) Substtutng (4) nto (0) yelds the requred condton on γ o If the effectve channel H s full rank, then the lemma results n γ o, e, any SIR s feasble Ths s easly verfed as the condton n (7) can be satsfed by choosng T = ah for large enough a > 0 Ths choce of precoder nverts the channel and elmnates all nterference Unfortunately, when the effectve channel s rank defcent, the nterference cannot be elmnated, and there s an upper bound on the maxmal SIRs A smlar lemma was provded n 37] n the context of optmal sgnature desgn usng MMSE recevers (whch s a specal case of a MIMO system) There, t was shown that the above condton s necessary and suffcent for feasblty usng MMSE recevers In our case, the recevers are fxed, and therefore the condton s only necessary In general, we cannot always attan the bound when the recever s fxed Two smple examples for channels n whch the bound cannot be acheved are a dagonal H wth K rank (H) dagonal zeros, or a channel H wth two dentcal rows In both of these examples, t s easy to see that, no matter what the precoder s, we wll not attan the bound 3

14 Nonetheless, expermentng wth arbtrary channels shows that n almost all practcal channels the bound can be acheved even for a fxed sub optmal recever For example, consder a rank K channel H, wth the normalzed null vector u N { H H} Except for the case n whch u = 0 for some =,, K, the bound can always be attaned by choosng T = H dag {/u } Q, (5) where Q],j =, = j; K, j (6) Ths s easly shown by consderng the followng chan: HT = HH dag {/u } Q = dag {/u } Q, (7) where we have used HH = I uu H and the fact that N {Q} Substtutng the above HT nto the SIRs yelds the maxmal SIRs n rank K channels: HT], j = K HT] K,j, =,, K (8) 4 CONIC OPTIMIZATION SOLUTION We now show that the P problem of (9) can be represented as a standard conc optmzaton program Thus, usng off the shelf optmzaton packages, we can numercally verfy ts feasblty, and fnd ts optmal soluton In order to use the standard forms of the conc programs, we must cast our problem constrants usng the standard notatons descrbed n Secton 3 Usng a slack varable, the program can be rewrtten as P(γ o ) : mn T,Po st P o HT], j HT],j +σ Tr { T H H H Tx H TxT } P o γ o, =,, K; (9) The argument T of the P program s optmal up to a dagonal phase scalng on the rght, e, f T s 4

15 optmal, then Tdag { e jφ }, where φ for =,, K are arbtrary phases, s also optmal Ths s easy to verfy, as the phases do not change the objectve nor the constrants Therefore, we can restrct ourselves to precoders n whch HT], 0 for =,, K, e, each has a non negatve real part, and a zero magnary part Takng ths nto account, we now recast the SINR constrants n standard form Rearrangng the constrants and usng matrx notatons, the constrants yeld ( + γ o ) HT], HTT H H H], + σ, =,, K, (30) or ( + γ o ) HT], T H H H e σ, =,, K (3) Snce HT], 0 for =,, K, we can take the square root of HT], resultng n + HT] γ, o T H H H e σ, =,, K, (3) whch can be wrtten as the SOCs + γ o HT], T H H H e σ K 0, =,, K (33) The power constrant Tr { T H H H TxH Tx T } P o, (34) can be reformulated usng the vec( ) operator as vec(h Tx T) P o, whch s equvalent to the SOC Po vec(h Tx T) K 0 (35) 5

16 Usng (33) and (35), and denotng p = P o, the program (9) can be cast n the standard SOCP form 5]: P(γ o ) : mn T,p st p + γ o HT], T H H H e K 0, =,, K; (36) p σ vec(h Tx T) K 0 Thus t can be effcently solved usng any standard SOCP package 5] Such a solver can also numercally determne the feasblty of the optmzaton problem A smlar approach was taken n 4] n the context of transmt beamformng As explaned n Secton 3, each SOC constrant can be replaced wth an SDP constrant usng (4) Thus the problem can also be expressed as a standard SDP: P(γ o ) : mn T,p p st A (T) 0, =,, K; C (T) 0, (37) where A (T) = + γo HT], TH H H e σ e H HT σ ] + γ o HT], I, =,, K, (38) and C (T) = p vec H (H Tx T) vec(h Tx T) pi (39) However, solvng SOCPs va SDP s not very effcent Interor pont methods that solve SOCP drectly have a much better worst case complexty than ther SDP counterparts 5] It s mportant to note that the above formulatons are general and do not depend on the rank of the channel Thus, these solutons are also approprate for rank defcent channels 6

17 43 OPTIMALITY CONDITIONS In ths secton, we wll derve the KKT optmalty condtons for the power optmzaton As explaned n the prevous secton, the problem can be numercally solved usng standard optmzaton packages Nonetheless, the condtons provde more nsght nto the soluton A smple structure based on the Lagrange dual varables s derved for the optmal soluton Gven ths structure, we propose two alternatve methods for fndng these varables In Secton 43, we derve a smple fxed pont teraton whch converges to these varables The computatonal complexty of ths approach s lower than that of the conc soluton Moreover, ths soluton does not requre any external conc package whch s not always avalable Alternatvely, n Secton 43, we propose a dual SDP program, whose optmal arguments are the necessary varables The man results are summarzed n the followng theorem: Theorem Consder the power optmzaton program P(γ o ) of (9) Defne the dual varables λ > 0 for = K, and denote Λ = dag {λ } If there exst λ > 0 such that γ o = Λ H(H H ΛH+H H Tx H Tx) H H Λ ],, =,, K, (40) holds, then the program s strctly feasble Moreover, f (40) hold, then the optmal T s of the form T = ( H H ΛH + H H TxH Tx ) H H Λ dag {δ }, (4) where δ are the postve weghts that allocate the power between the users: δ = F],j = ] γo I F λ j σj, =,, K; (4) + γ j o,j ( Λ H H H ΛH + H H ) ] TxH Tx H H Λ,, j =,, K (43),j Ths structure of T s unque wthn the range of H H Tx At ths optmal soluton, all the constrants are actve, e, there are equal SINRs for all the subchannels The optmal objectve value s P o = λ σ (44) Proof The proof conssts of two parts Frst, we show that f (40) holds then the problem s strctly feasble Next, assumng t s strctly feasble, we wll use the KKT optmalty condtons to show that the proposed 7

18 soluton s necessary and suffcent We begn by provng that f (40) holds, then the proposed soluton n (4)-(43) s feasble Frst, let us ] prove that ths soluton exsts, e, that the matrx γo +γ o I F n (4) s nvertble and that the argument of the squared root s non negatve The matrx s nvertble because that the maxmal egenvalue of F s less than γo +γ o : eg max (F) F (45) ( = max Λ H H H ΛH + H H ) ] TxH Tx H H Λ,j (46) j ( = max Λ H H H ΛH + H H ) TxH Tx H H ΛH ( H H ΛH + H H ) ] TxH Tx H H Λ (47) = max = = { ( Λ H H H ΛH + H H ) ] TxH Tx H H Λ ( Λ H H H ΛH + H H ) ( TxH Tx H H Tx H Tx H H ΛH + H H ) TxH Tx H H Λ { ( max Λ H H H ΛH + H H ) ] TxH Tx H H Λ H ( H H ΛH + H H ) ] TxH Tx H H Λ γ o + γ o,,,, }, ], } (48) (49) (50) (5) where the nequalty n (45) stems from the fact that any nduced matrx norm upper bounds the maxmal egenvalue of the matrx The equalty n (46) s the defnton of the row sum nduced matrx norm The nequalty n (49) stems from neglectng the non negatve terms n (48), and the equalty n (50) s due to (40) We stll need to prove that the nequalty s strct, but ths can be proven as follows Assume that the nequalty s not strct, e, there exsts an such that the second element n (48) s zero, e, ( H Tx H H ΛH + H H Tx H Tx) H H ( ] Λ e = 0, and therefore Λ H RxCh H Tx H H ΛH + H H Tx H Tx) H H Λ 0 But, snce γ o > 0, ths a contradcton to (40), and therefore the nequalty n (49) must be strct We now show that the arguments of the squared roots n (4) are non negatve Usng a seres expanson for the matrx nverson yelds 55], = δ = γo + γ o I F ] λ σ = + γ o γ o j= ] j γo F + γ o λ σ (5) δ K λ K σ K λ K σ K The elements of γ o +γ o F are nonnegatve Therefore, the elements of the sum wll also be non negatve, and we can take the element wse squared roots and solve for δ for =,, K 8

19 Thus, we have shown that the soluton n (4)-(43) exsts Pluggng ths soluton nto the SINR constrants satsfes all the constrants wth equalty Therefore, the problem s feasble Moreover, snce σ > 0 for =,, K, we can always scale the soluton T by c >, and satsfy the constrants wth nequaltes, e, the problem s strctly feasble In the next part of the proof, we wll show that f (40) holds, then the soluton n (4)-(43) s necessary and suffcent for optmalty The power optmzaton problem can be wrtten as follows: P(γ o ) : mn T Tr { T H H H Tx H TxT } st TH H H e ) ( + HT] γo, 0, =,, K σ (53) The above program s not wrtten n convex form (n order to wrte t n convex form, conc nequaltes must be used) In general, the KKT condtons are not suffcent for optmalty n non convex programs However, n Appendx A, we show that n ths specal case, f the program s strctly feasble, then ts KKT condtons are necessary and suffcent for optmalty The Lagrangan assocated wth program (53) s L = Tr { T H H H TxH Tx T } + λ TH H H e = Tr { T H H H ΛH + H H ] } TxH Tx T + λ σ σ ( + ) HT], γ o λ ( + γ o ) HT],, (54) where λ 0 are the Lagrange dual varables As we have shown n the frst part of the proof, f (40) holds, the problem s strctly feasble Therefore, ts prmal and dual varables are optmal f and only f the followng condtons are satsfed: Feasblty: The varable T s feasble ( + γ o ) HT], TH H H e σ, =,, K, (55) and the dual varables are dual feasble, e, λ 0 for =,, K Complementary Slackness: For each =,, K, ether λ = 0 or ( + γ o ) HT], = TH H H e σ (56) 9

20 3 Zero dervatve: The dervatve of L wth respect to T s zero, resultng n H H ΛH + H H TxH Tx ] T = H H Λ dag {( + γ o ) Λ HT ], } (57) At the optmal soluton all the constrants are actve, e, (56) holds wth equalty for =,, K As proof, note that f one constrant does not hold wth an equalty, then we can always scale the row n T assocated wth t, and arrve wth a feasble soluton that results n a lower objectve value, whch s a contradcton Another mportant property of the optmal soluton, s that all the dual varables are strctly postve As proof, assume the contrary, e, there exsts an such that λ = 0 Then, multplyng (57) by T H on the left and examnng the th dagonal element, we have e H TH H H ΛH + H H Tx H Tx] Te = 0, whch holds f and only f Λ HTe = 0 and H Tx Te = 0, n whch case the th SINR s clearly zero But, snce γ o > 0, ths contradcts the SINR constrants In general, the T that satsfes (57) s not unque Nonetheless, expressng T as T = T + T where T = PT, T = (I P) T, and P = H Tx H Tx, we can fnd T whch s unque wthn the range of H H Tx Usng HT = HT, we arrve wth the followng necessary and suffcent condtons: T = H H ΛH + H H TxH Tx ] H H Λ dag {δ } ; (58) ( + γ o ) HT ], = TH HH e σ, =,, K, (59) ) ] where δ = ( + Λ HT γo, for =,, K As already explaned, f (40) holds then the soluton n (4)-(43) satsfes (59) In addton, t has the structure of (58) and s therefore suffcent Moreover, t s easy to show that ths structure s also necessary (wthn the range of H H Tx ) Pluggng T from (58) nto (59) yelds ( + γ o ) λ F], δ = j λ F],j δ j + σ, =, K, (60) 0

21 where F s the matrx defned by (43) Rewrtng n matrx form, we have ] γo I F + γ o δ = λ σ (6) δ K λ K σ K Snce HT], 0 for =,, K, the unque soluton to ths set of equatons n gven by (4)-(43) Fnally, the optmal objectve value n (44) can be easly found usng (8) and (57) Theorem provdes a smple strategy for desgnng the precoder Gven a feasble γ o, all one has to do s fnd λ > 0 whch satsfy (40) Once these are found, T can be derved through (4)-(43) As we wll show n Secton 6, n some specal cases, these varables can be derved n closed form Otherwse, we now propose two alternatve methods for fndng these varables In Secton 43, we present a smple fxed pont teraton, and n Secton 43 we propose the use a SDP dual program 43 FIXED POINT ITERATION FOR FINDING λ The structure of (40) motvates a fxed pont teraton for fndng λ By rearrangng (40), we arrve wth the followng smple teraton: λ (n+) = γ o + γ ( { } o H H H dag λ (n) Clearly, the optmal λ satsfy ths fxed pont teraton wll converge from any λ (0) to a set λ (n) ] H + H H Tx Tx) H H H,, =, K (6) As we now show, f P(γ o ) s feasble, then the above > 0 that satsfes (40) The convergence proof s based on the standard functon approach ntroduced n 56], whch can be summarzed as follows Consder the fxed pont teraton λ (n+) = f (Λ (n) ), =, K, (63) where Λ (n) = dag { λ (n)} If the functons f (Λ) obey the followng propertes: Postvty f (Λ) > 0 for all ; Monotoncty If λ λ for all, then f (Λ) f (Λ ) for all ; Scalablty If α >, then αf (Λ) > f (αλ) for all,

22 b H Ch, H Rx, y w b W H H Tx H Ch, H Rx, y w b M M H Ch,M H Rx,M w M y M Fgure : Block dagram of a downlnk (broadcast) system The matrces m for m =,, M, are dagonal matrces wth the δ s assocated wth b m then the teraton has a fxed pont, t s unque, and for any ntal Λ (0), the teraton wll converge to t In Appendx B, we show that f the problem s feasble, and H ( ] H H Tx H Tx) H H < for =,, K, then the functons n (6) satsfy these exact propertes Thus, the teraton wll converge, 43 DUAL PROGRAM FOR FINDING λ Alternatvely, the dual varables can be found through a dual program The dual program s a concave program that optmzes the dual varables The detals of ts dervaton are provded n Appendx C The resultng program s: max λ 0 P D (γ o ) : st λ σ γ o +γ o H H ΛH + H H Tx H Tx ] λ H H e e H H 0, =,, K (64) Ths s a smple SDP/LMI program, whch can be effcently solved by any standard SDP/LMI optmzaton package Moreover, t has only K optmzaton varables, n comparson to K optmzaton varables n the orgnal program, and therefore has a lower computatonal complexty A smlar result was obtaned n 4] n the context of beamformng

23 b Λ H H Rx, H H Ch, y b Λ H H Rx, H H Ch, H H Tx w W y b M Λ M H H Rx,M H H Ch,M y M Fgure 3: Block dagram of a uplnk (multple access) system The matrces Λ m for m =,, M, are dagonal matrces wth the λ s assocated wth b m The vector w s the vrtual uplnk nose vector 44 INTERPRETATION USING DOWNLINK UPLINK DUALITY In ths secton, we provde an alternatve soluton for the power optmzaton problem based on the well known downlnk uplnk dualty 43, 44] As explaned n the prevous sectons, the power optmzaton can be solved effcently wthout the use of dualty However, prevous attempts for solvng the downlnk beamformng problem, whch s a specal case of precodng (where H Tx = I), are based on ths approach Therefore, for completeness, we now revew ths method and generalze t to the case of precodng, e, arbtrary H Tx Moreover, the dualty s nterestng from an engneerng pont on vew, as t provdes an nterestng physcal nterpretaton for the soluton Recently, an nterestng dualty was found between downlnk beamformng and another problem called uplnk beamformng It s usually referred to as downlnk (broadcast) - uplnk (multple access) dualty, snce one problem s usually found n the broadcast channel of a downlnk system, and the other s found n the multple access channel of an uplnk system Fortunately, the uplnk beamformng problem s easer to solve Usng the dualty, the downlnk soluton can be derved through the uplnk soluton For smplcty, n the sequel, we restrct ourselves to full rank channels (as dd all the works n ths context) Mathematcally, the dualty can be stated as follows: 3

24 Theorem Consder the downlnk program P: mn T Tr { T H H H Tx H TxT } P(γ o ) = HT] st, γ o, =,, K, j HT],j +σ (65) and the uplnk program: mn λ >0 P(γ o ) = st σ λ λ WHH ], j λ j WH H ],j +WH H Tx H TxW H ], γ o, =,, K (66) If the optmal arguments and objectve value of P are W, λ, and P o, then the optmal objectve value of P s also P o, and ts optmal argument s T = W H Λ dag {δ }, where δ are defned n Theorem Proof It s easy to see that each constrant n P deals wth one row of W and that the objectve s not a functon of W at all Therefore, t s clear that each row of W wll be chosen to maxmze the SINR assocated wth t Thus, for fxed λ, the optmal recever W s the well known MMSE matrx 3]: W = H ( H H dag {λ } H + H H TxH Tx ), (67) whch s unque up to a dagonal matrx multplcaton on the left In addton, smlarly to the downlnk problem, all the constrants of the uplnk problem are actve (otherwse, one can always decrease the λ assocated wth the passve constrant and decrease the objectve) Thus, at the optmum: λ WH H ], j λ j WH H ],j + WH H Tx H Tx W H], = γ o, =,, K (68) Pluggng n the optmal W and smplfyng the terms results n λ H ( H H dag {λ } H + H H TxH Tx ) H H ], = γ o + γ o, =,, K (69) Thus, the optmal λ s of P satsfy (40) Due to (67), the precoder T = W H Λ dag {δ } satsfes also (4) Therefore, accordng to Theorem, ths precoder s optmal for P Ths uplnk downlnk dualty was developed n 43] for the specal case of H Tx = I In Theorem we generalze ths result to arbtrary H Tx The mportance of ths theorem s n ts nterestng nterpretaton of the optmal soluton and to the dualty between the downlnk and the uplnk problem For example, t 4

25 provdes a physcal nterpretaton to the postve dual varables λ > 0 as the vrtual power allocaton In order to vsualze ths dualty, we provde block dagrams of the two dual systems n Fgs and 3 Moreover, prevous attempts for solvng the P problem are based on ths dualty As we have shown n the prevous secton, the problem can be solved wthout the dualty usng the optmalty condtons However, for completeness, we now present the dualty based approach as well Ths approach confronts the P problem by addressng the P problem frst and then adjustng the soluton based on Theorem Fortunately, there s an ntutve teratve soluton to program P The problem can be solved by teratvely solvng for each of the parameters, whle keepng the others fxed 4, 46]: P(γ o ) repeat for to K λ w 3 w arg {max HHH e w mn λ 4 Λ arg σ λ st 5 untl convergence j λ j w H H e j + H Tx w λ w H HH e j λ j w H H e j + H Tx w γ o, =,, K } where w H s the th row of W Each optmzaton n lne 3 tres to maxmze an SINR usng a receve weght vector Ther optmal solutons are gven by the columns of (67) The optmzaton n lne 4 can be carred out usng the followng fxed pont teraton 56]: j λ γ λ j w H H e j + HTx w o w HHH e, =,, K (70) In 46] t was shown that the above algorthm usng (70) always converges to the optmal soluton However, the algorthm s qute complcated, and conssts of an outer teraton wth two nner teratons Therefore, t s not as appealng as our smple fxed pont teraton n (6) 5 SINR OPTIMIZATION We now consder the problem of maxmzng the worst SINR subject to a power constrant, e, the S program of (0) As before, we begn by examnng ts feasblty Fortunately, t s easy to verfy that the S program s always feasble, as we can always scale T so that t would satsfy the power constrant In Secton 5, we dscuss the connecton between the power optmzaton and the SIR optmzaton and explan how ths connecton can be used to solve the SIR optmzaton Then, we follow the steps we took before n the 5

26 context of the power optmzaton, and repeat them n the context of the SINR optmzaton In Secton 5, we formulate the SIR problem as a standard GEVP conc program, n Secton 53 we provde a fxed pont teraton, and n Secton 54 we dscuss ts downlnk uplnk dualty 5 CONNECTION WITH POWER OPTIMIZATION The most nterestng property of the SINR optmzaton program s ts relaton to the power optmzaton program In order to mathematcally defne ths relaton, we ntroduce the followng theorem: Theorem 3 The power optmzaton problem of (9) and the SINR optmzaton problem of (0) are nverse problems: γ o = S(P(γ o )); (7) P o = P(S(P o )) (7) In addton, the optmal objectve value of each program s contnuous, and strctly monotonc ncreasng n ts nput argument: γ o > γ o P(γ o ) > P( γ o ); (73) P o > P o S(P o ) > S( P o ) (74) Proof We begn by provng (7) by contradcton Assume the contrary, e P and T are the optmal value and argument of P(γ), and γ γ and T are the optmal value and argument of S(P ) If γ < γ, then ths s a contradcton for the optmalty of T for S(P ), snce T s feasble for t, and provdes a larger objectve value γ Otherwse, f γ > γ, then ths s a contradcton for the optmalty of T for P(γ), snce γ > γ, and we can always fnd c < such that c T wll stll be feasble, but wll result n a smaller objectve Next, we prove (73) by contradcton Assume the contrary, e, P and T are optmal for γ, and P P and T are optmal for γ < γ We can always multply T by c < so that t wll stll acheve the SINRs constrants of γ, wth an effectve power constrant c P < P P Ths contradcts the assumpton that T was optmal for γ The contnuty can be verfed usng smlar arguments to those n Lemma of 57] The proofs of (7) and (74) are smlar and are therefore omtted Usng the propertes n the Theorem 3, we can solve S(P o ) for a gven P o by teratvely solvng P(γ o ) for dfferent γ o s Due to the nverson property, f P o = P(γ o ), then ts soluton wll be optmal also for 6

27 S(P o ) The strct monotoncty and contnuty guarantees that a smple one dmensonal bsecton search wll effcently fnd the requred γ o Ths procedure s summarzed n the followng algorthm (see also 47]): S(P o ) γ max MaxSINR γ mn MnSINR 3 repeat 4 γ o (γ mn + γ max ) / 5 Po P(γ o ) 6 f P o P o 7 then γ mn γ o 8 else γ max γ o 9 untl P o = P o 0 return γ o where MnSINR and MaxSINR defne a range of relevant SINRs for a specfc applcaton, and where we have used the conventon that = P(γ o ) f t s nfeasble Theoretcally, ths means that the SINR optmzaton problem can be solved through the prevous results concernng the power optmzaton Nonetheless, due to ts mportance and n order to obtan more effcent numercal soluton, we now provde drect solutons for the SINR optmzaton through conc optmzaton, va the optmalty condtons, and through the downlnk uplnk dualty 5 CONIC OPTIMIZATION SOLUTION The SINR optmzaton can be cast as a standard conc program Usng a slack varable, the problem can be rewrtten as S(P o ) : max T,γo st γ o HT], j HT],j +σ Tr { T H H H Tx H TxT } P o γ o, =,, K; (75) At frst glance, (75) seems smlar to (9) However, t turns out to be consderably more complcated Ths s because the SINR matrx nequaltes n (38) are lnear n β = + γ o or n T, but not n both smultaneously Thus, when β s an optmzaton varable and not a parameter, these constrants are no 7

28 longer LMIs In fact, the sets whch they defne are not convex Nonetheless, we can stll express them usng generalzed matrx nequaltes as n (37) and (38) If we rewrte the A (T) s n (38) and separate out the terms whch are lnear, we have A (T) = βa (T) A (T), (76) where A (T) and A (T) are matrces that depend affnely on T: A (T) = A (T) = HT], 0 0 HT], I 0 e H HT TH H H e 0 σ ; (77) σ ] (78) Usng (76) we can express S n the standard GEVP form: S(P o ) : mn T,β β st βa (T) A (T), =,, K; A (T) 0, =,, K; C(T) 0 (79) It can be solved usng approprate software, eg, the GEVP command n the LMI toolbox 58] As before, t s mportant to note that the GEVP formulaton s general and does not depend on the rank of the channel Thus, ths solutons s also approprate for rank defcent channels 53 A FIXED POINT ITERATION FOR FINDING λ The SINR optmzaton problem can also be solved usng the condtons n Theorem As explaned n Theorem 3, S and P are nverse problems Thus, the optmal soluton of the SINR optmzaton s also optmal for an nverse power optmzaton problem, and therefore must satsfy ts optmalty condtons as well All we need to do n order to solve the SINR optmzaton s fnd λ > 0 that satsfy (40) and (44) Unfortunately, n ths case, γ o s an optmzaton varable and not a parameter and has to be found as well The exact defnton of such sets s quas convex 8] 8

29 Ths can be overcome by adjustng the fxed pont teraton n (6): λ = ( { H H H dag λ (n) } H + H H Tx Tx) H H H ],, =,, K, (80) and then normalzng the result so that t wll satsfy (44): λ (n+) = P o λ j σ j λ j, =, K (8) If ths teraton converges to a fxed pont λ (n) > 0 then t wll satsfy (40) and (44) We do not have a convergence proof for ths teraton However, numerous numercal smulatons wth arbtrary ntal ponts and parameters show a rapd convergence rate 54 INTERPRETATION USING DOWNLINK UPLINK DUALITY Followng the success of the uplnk downlnk dualty n the power optmzaton, the dualty was recently used to confront the SINR optmzaton 48, 49] The uplnk downlnk dualty n the case of the SINR optmzaton can be stated as follows: Theorem 4 Recall the downlnk program S: S(P o ) = max T,γo st γ o HT], j HT],j +σ Tr { T H H H Tx H TxT } P o γ o, =,, K; (8) and consder the followng uplnk program: S(P o ) = max W,λ 0,γ o st γ o λ WHH ], j λ j WH H ],j +WH H Tx H TxW H ], γ o, =,, K; (83) σ λ P o If the optmal arguments and objectve value of S are W, λ, and γ o, then the optmal objectve value of S s also γ o, and ts optmal argument s T = W H Λ dag {δ }, where δ are non negatve weghts Proof The proof s smlar to the other proofs n ths paper, and therefore we only provde a sketch of t As before, the problems P and S are nverse problems But n Theorem we have shown that P and P 9