On the Capacity of Multiple Input Multiple Output Broadcast Channels

Transcription

1 On the Capacity of Mutipe Input Mutipe Output Broadcast Channes Sriram Vishwanath, Nihar Jinda, and Andrea Godsmith Department of Eectrica Engineering, Stanford University, Stanford, CA 9430 Abstract We consider a muti-user mutipe input mutipe output (MIMO) Gaussian broadcast channe (BC), where the transmitter and receivers have mutipe antennas Since the MIMO broadcast channe is in genera a non-degraded broadcast channe, its capacity region remains an unsoved probem In this paper, we estabish a duaity between what is termed the dirty paper achievabe region (the Caire-Shamai achievabe region) for the MIMO broadcast channe and the capacity region of the MIMO mutipe-access channe (MAC), which is easy to compute Using this duaity, we greaty reduce the computationa compexity required for obtaining the dirty paper achievabe region for the MIMO BC The duaity aso enabes us to transate previousy known resuts for the MIMO MAC to the MIMO BC We aso show that the dirty paper achievabe region achieves the sum-rate capacity of the MIMO BC by estabishing that the maximum sum rate of this region equas an upper-bound on the sum rate of the MIMO BC I INTRODUCTION Mutipe input mutipe output (MIMO) systems have received a great dea of attention as a method to achieve very high data rates over wireess inks The capacity of singe-user MIMO Gaussian channes was first studied by Teatar [3] and Foschini [4] This work has aso been extended to the MIMO mutipe-access channe (MAC) [2, 3, 1] The capacity of MIMO broadcast channes (BC), however, is an open probem due to the ack of a genera theory on non-degraded broadcast channes In pioneering work by Caire and Shamai [], a set of achievabe rates (the achievabe region) for the MIMO broadcast channe was obtained by appying the dirty paper resut [8] at the transmitter (aso known as coding for non-causay known interference) It was aso shown in [, 6] that the sum rate MIMO BC capacity equas the maximum sum rate of this achievabe region for the two user broadcast channe with two transmit antennas ( ) and one receive antenna at each receiver ( ) However, computing this region is extremey compex and this approach does not appear to work for the more genera cass of channes which we consider In this paper, we consider a -user MIMO Gaussian BC in which receiver has receive antennas and the transmitter has transmit antennas The achievabe region for a genera MIMO BC requires an extension of the Caire-Shamai region to mutipe users and mutipe receive antennas, which was done by Yu and Cioffi in [7] We refer to this extension as the dirty paper region We estabish a duaity between the dirty paper region of the MIMO BC and the capacity region of the MIMO MAC In other words, we show that the dirty paper region is exacty equa to the capacity region of the dua MIMO MAC, with the transmitters having the same sum power constraint as the MIMO BC We estabish this duaity by showing that a rates achievabe in the dua MIMO MAC with power constraints whose sum equas the BC power constraint are aso achievabe in the MIMO BC, and vice versa This duaity is the mutipe-antenna extension of the previousy estabished duaity between the scaar Gaussian BC and MAC [1] Though we consider ony the constant channe case, this duaity can easiy be shown to hod for fading mutipe-antenna Gaussian BC s and MAC s, as it does in the scaar channe case Finding the fu capacity region of the MIMO BC is very difficut due to its non-degraded nature, but we are abe to show that the dirty paper region achieves the same sum rate as the actua MIMO BC capacity region through the use of the Sato upper bound on the sum-rate capacity of broadcast channes [9] The same upper bound is used in [] to find the sum rate capacity of the ( ) channe We upper bound the sum rate capacity of the MIMO BC by considering the capacity when the receivers perform joint signa detection (ie we consider a singe-user! antenna channe) when the noise at every antenna is correated with the noise at every other antenna except for those at the same receiver, and we anayticay show that this upper bound coincides exacty with the maximum sum rate in the dirty paper region Athough the optimaity of the dirty paper region has ony been shown for sum rate (and triviay for the corner points of the region), the fact that the dirty paper region is equa to the dua MIMO MAC capacity region together with the fact that the scaar Gaussian BC capacity region is equa to the dua MAC capacity region eads us to beieve that the dirty paper region may actuay be the capacity region of the MIMO BC However, this hypothesis has remains to be proved The remainder of this paper is organized as foows In Section II we describe the MIMO BC and the dua MIMO MAC In Section III we summarize some background information, incuding the achievabe dirty paper BC region, the MIMO MAC capacity region, and the duaity of the scaar MAC and BC We describe the MIMO MAC-BC duaity resut in Section IV and show that the dirty paper BC region achieves sum rate MIMO BC capacity in Section V We concude with Section VI II SYSTEM MODEL We use bodface to denote matrices and vectors #$ denotes the determinant and #& the inverse of a square matrix # For any genera matrix ', et ')( denote the conjugate transpose, and '* the trace + denotes the identity ,/ matrix and diag,4 denotes a diagona matrix with the entry equaing We consider a MIMO broadcast channe 67 with a singe - antenna transmitter and receivers with receive antennas, respectivey The transmitter sends independent information to each receiver The broadcast channe is the system on the eft in Fig 1 Let 8:9<;&=3> be the transmitted vector signa and et A@B9 ;CDE>= be the channe matrix of receiver F where A@ 102 G represents the channe gain from transmit antenna 0 to antenna of receiver F The white Gaussian noise at receiver F is represented by HI@J9K; C D > where HI@MLON QPG +! Let R7@A9K; C D > be the received signa at receiver F mathematicay represented as RVU R7W B8[Z H&U HIW where The received signa is KU AW The matrix represents the channe gains of a receivers The (1) /02/$ IEEE 1444

2 ` 8 R P o HU Z (U Z ]( _ <U RU \&U H^] A] R^] \V] H^W BW Z R7W \IW ( W Z ` Fig 1 System modes of the BC MIMO(eft) and the MAC MIMO (right) channes covariance matrix of the input signa is acbedgfih 8j8V(k The transmitter is subject to an average power constraint, which impies ambnpoq We assume the channe matrix is constant and is known perfecty at the transmitter and at a receivers Now consider the dua mutipe-access channe shown in the right haf of Fig 1 The dua channe is arrived at by converting the receivers in the BC into transmitters in the MAC and converting the -antenna transmitter into a -antenna receiver Notice that the channe gains of the dua 102 MAC are the same as that of the broadcast channe, ie A@ corresponds to the gain from transmit antenna 0 to antenna 0 of receiver F in the BC and to the gain from antenna of transmitter F to receive antenna in the MAC Let \I@:9e; C D > be the transmitted signa of transmitter F Let `r9:;=> be the received signa and _ 9:;s=3> the noise LtN QPG +! The received signa is mathemati- vector where _ cay represented as KU ( \UuZ \U ( \ 6 Zv ( \ Z _ Z _ where ( xw KU ( 6 ( zy In the dua MAC, each transmitter is subject to an individua power constraint of $ 6j, with V z (ie the sum of the MAC power constraints equas the BC power constraint) We aso assume perfect knowedge of the channe at the transmitters and the receiver in the dua MAC Lasty, we define the cooperative system to be the same as the broadcast channe, but with a receivers coordinating to perform joint detection If the receivers are aowed to cooperate, { the broadcast channe reduces to a singe-user }V mutipe-antenna system where R RVU RIW and ~8Zp (2) HU H^W of this system the cooperative capacity We ca the capacity 6j We use ƒ ƒ J, G / J(} and GˆŠ } 3 < to denote the capacity (regions) of the MIMO BC, MIMO MAC and cooperative system, respectivey III BACKGROUND To obtain our resuts, we use the achievabe region of the MIMO BC channe obtained in [,7] and resuts on the MIMO MAC capacity region [2,3,1] extensivey in this paper Hence, we summarize these resuts first, and then state resuts on the duaity of the scaar Gaussian BC and MAC [1] A Achievabe BC Region - The Dirty Paper Region An achievabe region for the MIMO BC was first obtained in [] In [7], the region was extended to the more genera mutipe-user, mutipe-antenna case using the foowing extension of the dirty paper resut [8] to the vector case: Lemma 1: [Yu, Cioffi] Consider a channe with RŒ AŒ 8^ŒVZA!ŒVZJH^Œ, where R^Œ is the received vector, 8VŒ the transmitted vector, Œ the vector Gaussian interference, and HŒ the vector white Gaussian noise If Œ and H^Œ are independent and non-causa knowedge of Œ is avaiabe at the transmitter but not at the receiver, then the capacity of the channe is the same as if Œ is not present In the MIMO BC, this resut can be appied at the transmitter when choosing codewords for different receivers The transmitter first picks a codeword for receiver 1 The transmitter then chooses a codeword for receiver 2 with fu (non-causa) knowedge of the codeword intended for receiver 1 Therefore receiver 2 does not see the codeword intended for receiver 1 as interference Simiary, the codeword for receiver 3 is chosen such that receiver 3 does not see the signas intended for receivers 1 and 2 as interference This process continues for a receivers Since the ordering of the users ceary matters in such a procedure, the foowing is an achievabe set of rates { +$Z: +$Z: Š Š 6 a 6š a ( ( 0 6j The dirty-paper region Eœ ž Ÿ 3! ž < is defined as the union of a such rates vectors over a covariance matrices a 67 a such that 1 a sz 6j2 6 a ab {oo and over a permutations 2 The transmitted signa is 8 8U^Z 6 ZB8IW and the input covariance matrices are of the form a fih 8^ªQ8^ª ( k B MIMO MAC Capacity Region The capacity region of a genera MIMO MAC was obtained in [2, 3, 1] We now describe this capacity region for the dua MIMO MAC as defined in Section II For any set of powers 66j, the capacity of the MIMO MAC is G i 4 67 «( &d 66j V -±j² ³µ! ^ Qºn» +$Z Qºn» ( s¼4½ ¾rÀ For *Ä ÅÆ 3Æ, we denote by GÅÆ 3Æ ( Ç È ³ÊÉË ² ³-±j² G i 4 $ 67Á*Â/à J( the foowing set 6j «( (3) 144

3 a Ö + It can be easiy shown that this region is the capacity region of a MAC when the transmitters have a sum power constraint instead of individua power constraints but are not aowed to cooperate C Duaity of the Scaar Gaussian MAC and BC Lasty, we state the duaity resut for scaar Gaussian MAC and BC channes [1] Theorem 1 (Jinda, Vishwanath, Godsmith) The capacity region of a Gaussian BC with power Í and channes Ì Í 6 Í is equa to the union of capacity regions of the dua MAC with powers $ 6j such that V ^ E ƒ «Í Ì Ç ³ÊÉ È ² ³ ² G i 4 6j m«í Ì The proof of this is obtained by showing that any set of rates achievabe in the BC is aso achievabe in the MAC, and vice versa One key point is that to achieve the same rate vector in the BC and MAC, the decoding order must in genera be reversed, ie if User 1 is decoded ast in the BC then User 1 is decoded first in the MAC In the next section, we wi derive a simiar resut that equates the dirty paper BC achievabe region with the union of MAC capacity regions for the MIMO channe we are considering IV DUALITY OF THE MAC AND DIRTY PAPER BC REGION In this section we show that the capacity region of the MIMO MAC with a tota power constraint of for the transmitters is the same as the dirty paper region of the dua MIMO BC with power constraint In other words, any rate vector that is achievabe in the dua MAC with power constraints 67 is in the dirty paper region of the BC with power constraint Conversey, any rate vector that is in the V dirty paper region of the BC is aso in the dua MIMO MAC region with the same tota power constraint Theorem 2: The dirty paper region of a MIMO BC channe with power constraint is equa to the the capacity region of the dua MIMO MAC with sum power constraint œ ž Ÿ 2 Šž J ÅÆ 3Æ 67 Proof: We prove this by showing that for every set of MAC covariance matrices Ï and the corresponding set of MAC rates, there exist BC covariance matrices aï 67 a using the same sum power as the MAC (ie V = V a ) such that the MAC rates are achievabe in the BC using the dirty paper coding method described in Section III-A We aso show that the inverse of this statement hods true, or that for every set of BC covariance matrices there exist MAC covariance matrices that achieve the same set of rates using the same sum power It is important to point out that we reverse the decoding/encoding order of the users in the dua MAC/BC channe In other words, if User 1 is decoded first in the MAC, then we must encode User 1 s signa ast in the BC to achieve the same rates using these transformations This competes the proof, provided we have the appropriate transformations that map the MAC covariances to the BC covariances and vice versa Next, we expain some terminoogy used in the transformations, foowed by the actua transformations ( (4) A Terminoogy First, we expain the terms effective channe and fipped channe A singe user MIMO system Ð with channe matrix, additive Gaussian noise with covariance, and additive independent Gaussian interference with covariance Ò is said to have an effective channe of ÓZÔÒ$ Õ3 The set of rates achievabe by Ð and a different system with channe matrix equa to the effective channe, additive white noise of unit variance, and no interference are the same Aso, the capacity and the ca- of a system Ð with effective channe matrix Ö pacity of system Ðc with effective channe matrix Ö (, termed the fipped channe, are the same [3] In other words, for every transmit covariance a in Ð, there exists a a in ÐÏ with ac aø such that the rate achieved by a in ÐÏ is equa to the rate achieved by a in Ð It can easiy be shown that ÔÙuÚ (a ÚMÙ rùüûcú ( meets this criterion where the SVD of Ö is ( Next, we describe the transformations B MAC to BC ansformation In this section we prove the existence of BC covariances that achieve the same rates as a set of MAC covariances and that use the same sum power Since the numbering of the users is arbitrary and because successive decoding at the receiver is known to be optima for the MAC, we assume that User 1 is decoded first, User 2 second, and so on at the receiver Let cþd +JZÓ Š ß a ß ( and àþd +JZ ß áâv ß( ß ß The rate achieved by User in the MAC is given by M ã +Z +$Z á ( /âv ( ( ã ä Z +Z áâv ã ä Zvà ( ã6 6 6 and use the prop- To simpify, we take the square root of à erty +/ZB[àA +/ZMàÏ We aso introduce Õ2 into the expression to get M +$Z:à Õ3 ( Õ3 eating à Õ2 ( Õ3 Õ2 ã! Õ3 2 ( ãg Õ3 Õ3 6à Õ3 as the effective channe of the system, we fip the channe and find Õ3 ã6 Õ3 such that M Õ3 {! Õ2 +$Zv Õ2 M6à Õ3 Õ2 ã6 Õ3 Õ2 ã! Õ3 à Õ3 ( Õ3 Now consider the rate of User in the BC assuming that the opposite encoding order is used (ie User 1 is encoded ast, 1446

4 à f ½ H ûs + User 2 second to ast,etc) B If we choose a as a +Z V Måa 6 ( +$Z Måa 6 ( äãz åa 6 ( ã äãz Õ3 Måa 6 ( Õ3 à Õ2 Õ3 {! Õ2 B By doing this for a users, we find à Õ3 () ceary we see M covariance matrices for the BC that achieve the same rate as in the MAC In Appendix A, we show that the transformations given in () satisfy the sum trace constraint, or that V = V a C BC to MAC transformation In this section we prove the existence of MAC covariances that achieve the same rates as a set of BC covariances and that use the same sum power For the dirty paper encoding at the BC, we assume that User K is encoded first, çæ second and so on in decreasing order Aong the same ines as the MAC- BC transformation, we treat Õ3!à Õ3 as the effective channe and à Õ3 a 6à Õ3 as the covariance matrix By fipping the effective channe, we obtain à Õ3 a 6à Õ3 and obtain the transformation { Õ3 Õ3 aã6à Õ3 Õ2 (6) As before, if we use the opposite decoding order in the MAC, this transformation ensures that the rates of a users in the BC and MAC are equivaent as is the tota power used in the BC and MAC < ) We now show that the bound can be tightened by introducing noise correation Since the capacity region of a genera BC depends ony on V SUM RATE CAPACITY OF BC CHANNELS In the previous sections we showed that the dirty paper region of the BC and the union of the dua MAC capacity regions are equivaent Now we show that the dirty paper broadcasting strategy is the capacity achieving strategy for the sum rate capacity of the MIMO BC To do this, we make use of the duaity of the dirty paper region and the dua MAC to show that the dirty paper region achieves an upper bound on the sum rate capacity of the MIMO BC We use the superscript sumrate to denote the maximum sum rate of the rate region under consideration In [9], Sato presents an upper bound on the capacity region of genera BCs This bound utiizes the capacity of the cooperative system as defined in Section II Since the cooperative system is the same as the BC, but with receiver coordination, the capacity of the cooperative system ( EˆŠ } 3 J ) is an upper bound on the BC sum rate capacity ( 7è ƒ Å6éVž 2Ÿ1 the margina transition probabiities (ie ê 1ë ìj ) and not on 1ë 672ë the entire joint distribution ê ìj, we can introduce correation between the noise vectors at different receivers of the BC without affecting the BC capacity region This correation does, however, affect the capacity of the cooperative system, which is sti an upper bound on the sum rate of the BC Therefore we retain f H ( + 0rí 6j, to maintain the same margina transition probabiities, but we et HVªŠHGî(}d - mð + 0{ñ for By searching over a feasibe positive definite noise covariance matrices, we get the foowing bound where öø ù úø è Å6éž ŸQ E Ä P J $o òïó ô» ˆŠ } 3 T + Õ2 J ( XÊü YÏý þø ÿ ø By using the fact that the capacity region of the dua MIMO MAC equas the capacity region of the dirty paper region of the MIMO BC (and therefore the maximum sum rate of the MAC and the dirty paper region are equivaent), we are abe to show that this bound is tight for the BC MIMO Theorem 3: The maximum sum rate in the dirty paper region of the MIMO BC equas the Sato upper bound on the sum rate capacity of broadcast channes and is therefore the actua sum rate capacity of the MIMO BC òcó ô È º» GˆŠ } 3 Õ3 < œ è Å6éž ŸQ ž Ÿ 3! ž E è Å6éž ŸQ Proof: We wi prove that the maximum sum rate in the dua MIMO MAC capacity region with a sum power constraint is equa to the Sato upper bound Due to the duaity of the MAC and BC, the maximum sum rate in the dirty paper region aso equas the Sato upper bound, which impies that the dirty paper region achieves the sum rate capacity of the MIMO BC We now show that the capacity of the cooperative system with worst case correated noise is equa to the maximum sum rate capacity of the MIMO MAC, or that òcó ô È ºn» ò -±4²^ ± Ç ò ³ÊÉ È ³ Š±4²^ ã +$Z J < Zv<aˆŠ } 3 <( ( V Note that the capacity of a cooperative system with channe and noise covariance can be rewritten as ò -±j²^ ò -±j²^ +$Z where Õ3 Õ3 Z:KaˆŠ } 3 J(å <aˆš } 3 ( Õ2 ( is a matrix such that Õ3 2( Õ3 Õ2 We obtain the fipped channe K( 2( from the effective 1447

5 o o È È Â ( channe Õ3 and a covariance matrix amˆš } 3 that ˆ } 2 ò È º such Õ3 < (7) +$Zv ( Õ3 ( Õ2 v where is the set À $oe è Ÿ1 {d òïó ô º» ò È º +$Zv ( Õ3 ( ò È º òïó ô È ºn» +$Zv ( Õ3 ( Note that Õ3 v Õ3 v (8) R Dirty Paper Region Sato Upper Bound ie, the order in which maximization and minimization is done can be interchanged We use the max-min version of the bound in the proof Equation (8) can be proved using Ky Fan s Theorem [11], which is an extension of von Neumann s Minimax Theorem [12] This theorem can aso be found in [13, Page 11] The conditions required for interchangeabiity are 1 The sets ½ and are compact and convex 2 The function +$Zv<( Õ3 2( Õ2 v is continuous and convex in and continuous and concave in These conditions can be easiy verified For a proof of the second condition and a proof of interchangeabiity of maximum and minimum for a case simiar to this, see [14] Note that the sum rate capacity of the MAC channe è Å6éVž 2Ÿ1 for given covariance matrices ã +Zv ( c In Appendix B, we show that a that ST! 6 6! for every 9# Here,! 6j definite matrices that satisfy V 67 ÅÆ 3Æ can be rewritten as (9) Õ3 9<½ can be found such Õ3 (! Õ3 (10) $! are symmetric positive (11) Substituting (10) in the expression for è 2Ÿ1 (8) we get è 2Ÿ1 ò% È º òïó ô º» +$Z: ( Õ3 ( Õ3 : ò% È º +$Zv ( Õ2 ( Õ3 : $! 6 ò% +$Zv ( 6! Sum Rate Capacity of MAC R 1 Fig 2 Dirty paper broadcast region - achievabiity and converse Since, from duaity, we have that the sum rate capacity for the MAC is ess than or equa to the cooperative capacity bound, we have the resut We prove Equations (10) and (11) in Appendix B The dirty paper BC region and the capacity regions of the dua MAC, aong with the Sato upper bound are iustrated for a two receiver BC in which the transmitter has two antennas and each receiver has a singe antenna ( 2s n n ) in Fig 2 The channe matrix is '& K A )( '&+* 6P and the power constraint is Notice that the dirty paper region is the union of the dua MAC capacity regions The boundary of the dirty paper region is a straight ine segment at the sum rate point, where the region touches the Sato upper bound This region is significanty different than the capacity region of a scaar BC In a scaar BC, the capacity region boundary does not contain ine segments Additionay, the sum rate point in a scaar BC is achieved by aocating a power to one user (the user with the argest channe gain) In the exampe MIMO channe, however, notice that the sum rate point is actuay achieved by aocating power to both users However, the shape of the dirty paper region is very simiar to the shape of the MIMO MAC capacity region when the transmitters have more than one antenna [2] There are two important impications of Theorems 2 and 3 First, it is easy to verify that the sum power constraint MAC capacity region GÅÆ 3Æ <(3 is convex due to a time-sharing argument, and hence the dirty paper region is aso convex Additionay, since the dua MAC capacity region can be posed as a convex optimization probem, the equivaence of the dirty paper region and the dua MAC aows us to use convex optimization techniques (such as the interior point method in [16]) to cacuate the entire dirty paper region and techniques such as iterative water-fiing [7] to cacuate the sum rate capacity of the MIMO BC - -, 0/ 1448

6 à ûs VI CONCLUSION In this paper, we estabished a duaity reationship between two seemingy different regions : the achievabe region of the MIMO BC obtained using the dirty paper coding strategy and the capacity region of the MIMO MAC This duaity makes the previousy intractabe probem of finding this dirty paper achievabe region much easier to sove Though the capacity region of the MIMO BC is unknown due to its non-degraded nature, we were abe to show that, for sum rate, the boundary of the dirty paper achievabe region and the MIMO BC capacity region are the same These resuts open up the possibiity that the dirty paper region is the actua capacity region of the BC MIMO and aso the possibiity that other instances of duaity exist in muti-termina networks APPENDIX A Proof of ace Constraint for Covariance ansformations In this section, we show that the transformations obtained in () satisfy the sum trace requirement Throughout this proof, we utiize the inearity of the trace and the property àï we compute a àïm when both à and àc are defined First, By adding V a a 7Z V Note that Hence V ß 7Z to both sides, we get V +Z ( 2 á}âv 6ൠà Õ2 Õ3 {! c6ã ã!7z a V ß7Z % V Assume the inductive hypothesis V ß /âv ß7Z V 3 3 +$Z 3 +$Z ( +Zv ( 2 Õ3 Õ3 ( ãm (12) ß ß á}âv ß( ß ß2 ß( ß ß2 By (12), we get For V ß á ß7Z, we have ß +sz ß ß á ß( ß ß2 A simiar method can be used to show that (6) aso satisfy the trace constraints B Proof of Equation (10) We prove (10) by recursive reduction We assume that the resut hods for a *æ user system and prove it for the user one Note that we can rewrite 9A½ as O Õ3 Õ3 ( '& + ( ( (13) where, using the notation of Section III-A, and ( h ( T + 1( aso must satisfy 6 66j ( 6 k ( + Õ3 ( &! 2 ( X ü Y Õ2 (14) where! and 2 are symmetric positive semidefinite matrices that satisfy!7z 2 (1) To prove this, we invoke the bock LDU factorization [10] of, which is given by where '& + à +3( & 4 ( & + à ( + ( (16) and are symmetric positive semidefinite matrices We pick to have the structure Õ3 Õ2 '& + æ ( B Õ3 B( 2 Õ2 2( Õ3 ( (17) 1449

7 Ò à Ö Ò C æ æ L L where +Üæp~( ~ Õ3 is the symmetric square root of + æ ( B and Õ3 is a bock upper trianguar square root of Note that this choice Õ3 of satisfies the requirement in (13) above Aso note that is positive definite and hence invertibe This is due to the recursive construction methodoogy empoyed in this proof and the fact that + Therefore, a Õ3 sufficient condition for to be fu rank, Õ3 and hence for Õ3 2( to be positive definite is that the spectra radius of Õ2 is ess than unity Let us suppose the optimum Õ3 is known Now, we construct Õ2 as foows 687:9 Let the singuar vaue decomposition (SVD) of à equa, 7 ;/ where ;/=< P diag where We choose Õ3 68>9 > to equa, where diag Z ; 2 By this construction, note that a singuar vaues of Õ2 are ess than unity Defining variabes Ö, Ò, and C as +uæ ( ~ Õ3 & Ö àö Õ3 2( ( & Ö Õ3 ( ( we can rewrite Equation (16) as foows D& + which equas +E( C+C & 4 ( C ( C ( & + àï( + ( & Ö zö Õ3 2( Õ3 ( Ò ( For the particuar choice of given above, we find that àmö Õ2, which impies Òi( Õ3 Defining Ö zö d! and Õ3 2( Õ3 df2 makes the expression for above and that given by Equation (14) identica Substituting (14) in (7) and caing h < 6 k, we obtain ˆ } 2 Õ3 J ( Õ2 (, ie, finding such that ã +$Z: ( ckz Õ3 Thus, our objective in (10) now simpifies to finding the optima Õ3 ( Õ3 2 $! $! for some positive semidefinite matrices :! }! 7Z 6 Z! 67! with Note that this is exacty the same objective as in (10) with %G instead of IHi6 W and instead of, and with users (Users ) instead of users Thus, the same procedure as in (13)-(18) can be appied to reduce the probem to a Óæ user one When reduced to a user (User J and User ) system, the optimum equas + Thus, one can run the agorithm backwards to generate the optima and thus, this recursive reduction estabishes the theorem Now, we show that the trace constraint is aso preserved in each step of this recursive reduction, ie that K K K Õ3 ( &! 32 ( &! 32 ( &! ( 2 cu 2 (! REFERENCES [1] N Jinda, S Vishwanath, and A Godsmith, On the Duaity of Mutipe- Access and Broadcast Channes, Aerton Conference on Commun, Contro, and Computing, Oct 3-, 2001 Aso in preparation for journa submission [2] W Yu, W Rhee, S Boyd, J Cioffi, Iterative Water-fiing for Vector Mutipe Access Channes, pp 322, Proc IEEE Int Symp Inf Theory, (ISIT), Washington DC, June 24-29, 2001 [3] E Teatar, Capacity of Muti-antenna Gaussian Channes, European ans on Teecomm ETT, 10(6):8-96, November 1999 [4] GJ Foschini and MJ Gans, On Limits of Wireess Communications in a Fading Environment when using Mutipe Antennas, Wireess Persona Communications, vo 6,311-33, 1998 [] G Caire and S Shamai, On Achievabe Rates in a Muti-antenna Broadcast Downink, 38th Annua Aerton Conference on Commun, Contro and Computing, Monticeo, IL, Oct 4-6, 2000 [6] G Caire and S Shamai, On the Achievabe Throughput of a Mutiantenna Gaussian Broadcast Channe, submitted to IEEE ans on Inform Theory, Juy 2001 [7] W Yu and J Cioffi, eis Precoding for the Broadcast Channe, Proc IEEE Gobecom, pp , Nov, 2001 [8] M Costa, Writing on Dirty Paper, IEEE ans Inf Theory, v 29, pp , May 1983 [9] H Sato, An Outer Bound on the Capacity Region of Broadcast Channe, IEEE ans Inf Theory, v 24, pp , May 1978 [10] RA Horn and CR Johnson, Matrix Anaysis, Cambridge University Press See Aso Topics in Matrix Anaysis, Cambridge University Press [11] K Fan, Minimax Theorems, Proc Nat Acad Sci 39, pp 42-47, 193 [12] A von Neumann, Zur Theorie der Geseshaftspiee, Math Ann 100, pp , 1928 [13] HW Kuhn and GP Szego, Differentia Games and Reated Topics, North Hoand Pubishing Company, 1971 [14] B Hassibi and BM Hochwad, How Much aining is Needed in Mutipe-Antenna Wireess Links, Technica Memorandum, Be Laboratories, Apri 2000 [1] S Verdu, Mutipe-access Channes with Memory With and Without Frame Synchronism, IEEE ans Inf Theory, vo 3, no 3, pp 60-19, May 1989 [16] L Vandenberghe, S Boyd and SP Wu Determinant maximization with inear matrix inequaity constraints, SIAM Journa on Matrix Anaysis and Appications, 19(2):499-33, 1998 Õ2 ML 140