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Int. J. Communications, Network and System Sciences, 2010, 3, 1-18 doi:10.4236/ijcns.2010.31001 blished Online January 2010 (http://www.SciRP.org/journal/ijcns/). Copyright © 2010 SciRes. IJCNS 1 Pu Achievable Rate Regions for Orthogonally Multiplexed MIMO Broadcast Channels with Multi-Dimensional Modulation Marthe KASSOUF, Harry LEIB Department of Electrical and Computer Engineering, McGill University, Montreal, Canada Email: marthe.kassouf@mail.mcgill.ca, harry.leib@mcgill.ca Received July 23, 2009; revised September 12, 2009; accepted November 27, 2009 Abstract In this work, we consider a multi-antenna channel with orthogonally multiplexed non-cooperative users, and present its achievable information rate regions with and without channel knowledge at the transmitter. With an informed transmitter, we maximize the rate for each user. With an uninformed transmitter, we consider the optimal power allocation that causes the fastest convergence to zero of the fraction of channels whose mutual information is less than any given rate as the transmitter channel knowledge converges to zero. We assume a deterministic space and time dispersive multipath channel with multiple transmit and receive an- tennas, generating an orthogonally multiplexed Multiple-Input Multiple-Output (MIMO) broadcast system. Under limited transmit power; we consider different user specific space-time modulation formats that repre- sent assignments of signal dimensions to transmit antennas. For the two-user orthogonally multiplexed MIMO broadcast channels, the achievable rate regions, with and without transmitter channel knowledge, evolve from a triangular region at low SNR to a rectangular region at high SNR. We also investigate the maximum sum rate for these regions and derive the associated power allocations at low and high SNR. Fur- thermore, we present numerical results for a two-user system that illustrate the effects of channel knowledge at the transmitter, the multi-dimensional space-time modulation format and features of the multipath channel. Keywords: MIMO Channels, Broadcast Systems, Capacity Region, Space-Time Coding 1. Introduction Multiple-Input Multiple-Output (MIMO) systems, em- ploying multiple antennas at the transmitter and receiver, have been shown to yield significant capacity gains for single-user channels [1]. A gain in the capacity of MIMO channels is also observed when increasing the number of multipath components [2–4]. Furthermore, channel know- ledge at the transmitter has been shown to increase ca- pacity more significantly at low SNR [5,6]. These fa- vorable features trigerred a considerable interest in the application of MIMO technology to multi-user systems as well. The capacity region of the two-user scalar orthogonal broadcast channel (BC) is shown in [7] to be a rectangle generated by the set of jointly achievable mutual infor- mation rate pairs. A larger capacity region may be ob- tained by allowing multi-user data superposition instead of simple time sharing [8]. Assuming perfect channel state information (CSI) at transmitter and receiver, the optimality of Code Division Multiple Access (CDMA) with successive decoding has been established in [9,10] for flat and frequency selective fading channels. MIMO broadcast channels (BCs) belong to the class of non- degraded broadcast channels, thus, making the evalua- tion of their capacity regions very difficult. Superposi- tion coding does not apply to non-degraded broadcast channels because users may employ different rates mak- ing successive decoding quite difficult if not impossible [11]. However, this reference shows that a capacity re- gion for broadcast channels can be achieved by using a coding technique, nicknamed dirty paper coding (DPC) [12], where the interference is non -causally know n to the transmitter and unknown to the receiver. The optimality of DPC in terms of maximizing the sum rate was proved in [13] for a constant two-user BC with single-antenna receivers, and known channel at the transmitter as well as all receivers. Generalizations of results from [13] to M. KASSOUF ET AL. 2 systems with arbitrary number of users and multiple transmit and receive antennas has been carried out inde- pendently in [14] and [15]. The sum rate optimality of DPC for Gaussian MIMO BCs has been investigated in [16–18] using the duality [19] between the DPC rate re- gion of the MIMO BC and the capacity region of a Gaussian MIMO MAC with similar power constraint. In [20] it was shown that the DPC rate region is in fact the MIMO BC capacity region. Scaling laws of the sum rate for block fading Rayleigh MIMO BCs with large number of users are considered in [21] using DPC, Time Divi- sion Multiple Access (TDMA) and beamforming. The rate balancing problem (i.e. the selection of the capacity region boundary point that satisfies given constraints on the ratios between the users’ rates) is considered in [22], which also provides optimal and suboptimal algorithms for MIMO BCs employing Orthogonal Frequency Divi- sion Multiplexing (OFDM) transmission. In this work, we consider a MIMO BC with ortho- gonally multiplexed non-cooperating users who employ space-time modulation. As in [23,24], we assume a non-fading space and time dispersive multipath envi- ronment. These schemes model the downlink of cellular communication systems with orthogonal user multiplex- ing. We consider a deterministic channel model since it provides an insight to the behaviour of the capacity re- gion with respect to the numb er of anten na and multipath components, and often serves as a first step towards the study of fading channels. We investigate the achievable rate region of such orthogonally multiplexed broadcast schemes with multi-dimensional space-time modulation, where a transmitter attempts simultaneously to transfer information to several u sers without mutual interference. When the channel is known at the transmitter, we con- sider the optimal power allocation that maximizes the rate for each user. We also consider the power allocation for each user that causes the fastest convergence to zero of the fraction of channels whose mutual information is less than any given rate, as the transmitter channel knowledge goes to zero. For both cases, we investigate the maximum sum rate. Considering a two-user broad- cast system, we investigate the asymptotic behaviour of the achievable rate regions at low and high SNR, and provide the optimum power allocations that correspond to the maximum sum rate. Illustrative numerical results are provided for users having different propagation channels, using different multi-dimensional space-time modulation schemes and employing different number of antennas. This paper is structured as follows. Section 2 presents the system model. The capacity region with known channel at the transmitter is investigated in Sec- tion 3. The case of unknown channel at the transmitter is considered in Section 4. Section 5 presents some illustra- tive numerical results. The conclusions follow in Section 6. 2. MIMO Broadcast Multipath Channels with Space-Time Modulation In this paper, column vectors and matrices are repre- sented by lower-case and upper-case bold letters. The component of a vector is denoted by [] Fur- thermore, th dad a denotes the determinant of A, ⊗ denotes the matrix Kronecker product, anddenotes the matrix product. We use the following superscripts: ∗ for com- plex conjugate, T for matrix transpose, and for Hermitian conjugate. The vec() operator denotes the stack in a single column vector of matrix columns or a set of column vectors. The direct sum of matrices † n 1 n ii {} n is denoted by ] . The vertical stack of matrices with equal number of columns 1[ n ii diag 1 n ii n in a single matrix is denoted by . The -square identity matrix is denoted by . The -dimensional vectors 1[ i ] n i stack n Ιn in e for 1, ,2,in are defined as ,1 ,, 2, ,, T inii i n e, with the Kronecker sym- bol defined by ,1ij if , and ij(,ij) 0 otherwise. For a scalar , we have . Unless otherwise specified, the function denotes the base-2 logarithm, and the superscript () refers to the user in the system. a{} lo max( () 0, ) aa g k th k We consider orthogonally multiplexed users, each with power K k P satisfying the constraint 1 k K kP T D and affected by independent interference. Let denote the total number of signal dimensions, with user occupying a sub-space of dimensionality T P k k D, where 1 k k D KD T . Each user employs a different sig- nal sub-space. This model corresponds to an orthogonally multiplexed MIMO broadcast channel (BC) without user cooperation. For user , the propagation medium consists of time resolvable multipath clusters following the 3GPP space and time dispersive channel model [25]. The signal paths of same cluster have equal propagation delays and are resolved in spac e only. For use r we define the transmitted and received signal vectors k ] T ()k t L () T N k s() () () [ kk ts() 12 (),(), , k tst () k s t and () ()t k z () [(zt 1k), () 2(), () () , ()] k R kk N T z tzt , respectively, with and T N k R N denoting the number of transmit and receive ante nnas. The continuous time channel model is specified by 1 k t L kkkk ll l tt zFs i k t (1) Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. Copyright © 2010 SciRes. IJCNS 3 Figure 1. System model for user k. Figure 2. Examples of dimension allocation schemes for user with transmit antennas and k4 T N= 4 (k) D = dimensions: a) ATA, b) OTA, c) POTA with TOGs. 2 (k) G N= where k l and k l F denote the propagation delay and the k R T NN channel propagation matrix associated with cluster . The interference vector ,, }) k t ll L({1 ,k R kk N i 12 , ,ti()[ kk ti] T ti is complex valued zero mean white Gaussian, with autocovariance matrix ()k † ))][()(( R kk ii o NN IEt t where denote s the Di rac ’s delta functi on. Consider the system model from [26] illustrated in Figure 1. Assume a modulation process for user that partitions the transmit antennas into groups called Transmit Orthogonal Groups (TOGs), each sharing a given subset of its k k D signal dimensions. Hence each TOG employs a different signal sub-space. Let k G N denote the number of TOGs ( 1k G NN T ), and k i n denote the number o f transmit anten nas in the TOG, assumed to be adjacent. We assume an equal number of signal dimensions per TOG, th i k k Gk G D DN and define . The -dimensional complex input vector k TG DND k k G D k jnx for transmit antenna and the corre- sponding j k D-dimensional signal space vector k jny are given by () ()( kk T jj nn yx ) k j , with kk j T tN e k GG D Ι k j and ({ j 1, kk jj G tt N }) k indi- cating the TOG to which antenna belongs. The k j k G Dk D matrix determines the signal dimen- sions (indexed by {1, ,} k dD j ) that can be used on transmit antenna by making [( k j )] d n0 y if the signal dimension is not used on antenna , or th dj [( )][ k( )] k j djd nn yx {1, ,}D k G for d otherwise. The multi-dimensional space-time modulation format is determined by the matrices 1 } T 1 k G N { kN j j . With and k G NNT we have the space-time modulation formats Aggregate Transmit Antenna (ATA) and Or- thogonal Transmit Antenna (OTA), respectively. The space-time coding system [27] and the Alamouti transmit diversity scheme with two transmit antennas [28] are examples of ATA. The orthogonal transmit diversity technique of IS 2000 [29,30 ] is an example of OTA. The M. KASSOUF ET AL. 4 more general case corresponds to Par- tially Orthogonal Tr ansmit Antenna (POTA), that can be viewed as a combination of ATA and OTA. Examples of ATA, OTA and POTA are illustrated in Figures 2(a), 2(b) and 2(c), respectively. ( k GT NN ) The transmitted waveform through antenna j is given by ()(( ))() T kkk jnj s tn y 12 ,,, k T kkkk D tt t tnT dt where denotes user , ks real orthogonal basis functions and is the symbol rate. We assume no inter-symbol interference and perfect synchronization at every receive antenna. Assuming per- fect multipath time resolvability for each user as well as perfect orthogonality between users, we have 1Τ ,,, , kkk k dld l tnTt nT kk ll ddnn ,, ,1,,k t kk llL , ,1,, k dd D and . Since different TOGs employ different sig- nal subspaces, user system is equivalent to ,nn , ks k G N parallel MIMO systems each consisting of a TOG and the receiver. The overall Maximum Likelihood detection complexity for user is the sum of his TOGs com- plexities, each being exponential in . By reducing each k k i n k i n the overall complexity decreases. It is mini- mum, and linear in when T N 1 k i n which corre- sponds to OTA. Thus, increasing lowers complex- ity and increases parallelism. k G N Let k ml n r and k ml n v denote the k D th -dimen- sional discrete time channel output and noise vectors o f the time re solvabl e cluster received on the antenna. th l m Hence, the vectors 1 } ) T k N jj nn xx 11 } ) kk t RL l ({ k vec and denote the input and output of the discrete time channel defined by ({ } kk N ml m nvecnrr kkkk nrHxvn ml n n (2) with noise vector .Using 11 ({{}} ) kk t R kk NL m l n vec vv 1[ T k N] k j j diag , we have 1 } ) T N jj n({ kk vecn yy [] kk Tnx. Assuming perfect multipath resolv- ability, the kkk tR DLN ()k D discrete time channel matrix k H can be seen as the stack of kk ND R submatrices each associated with a time resolvable cluster. T herefore, we have ()k D 1 1 [([] ] [([ ] k t k t kk LT ll kk LT ll stack stack HFI k ) )] k k D D FI T or, equivalently, k kk k D HCI (3) Figure 3. Space and time dispersive MIMO channel model for user . k Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 5 ] k F with . Using the 3GPP spatial chan- nel model [25] illustrated in Figure 3, let denote the number of propagation paths in cluster , with path characterized by the gain coeffi- cient 1[ k t k L ll stack C (0,, 1) k l S k l S l ss ; k ls G, angle of departure (AOD) ;;s k Tl and angle of arrival (AOA) ;; k R ls k k .The total number of propagation paths for user is . The spaces between adjacent transmit and receive antennas are denoted by and 1 k tk L ll S T d R d. We use the notations ;2 k ls C opyright © 2010 SciRes. IJCNS k R d ;; ) k sin( R ls and ; 2sin( k T Tl d ;; k ) s ls with de- noting the signal wavelength. The space signature vec- tors at the receiver and transmitter are given by and and from [25] we can write ; k ls a jN e ;; (1) [1,, , kk R ls ls jjN ee ; 1] k TlsT ] k Tb ; k ls ;k ls j [1, e , , k l F . Subsequently, we define the propagation matrix describing the propagation between the TOG and the receiver of user , such that 1† 0;; ; () k Skkk slslsls G ab kk k tR i n k l LN k i C th i 12 kkk CCC[] k k G N C. Let k i r denote the rank of k i C and define . We also use kk ii rr () 1 k G N T k i M and k i Q to denote the k i n-square unitary and diagonal matrices associated with the eigen- value decomposition k i M †† i M()( kk ii CC ) k Q k i . Assuming a memoryless channel, we can drop the de- pendencies on the time index for the remainder of this paper. Moreover, we consider orthogonally multi- plexed broadcast MIMO channels with two users ( n K = 2) for simplicity. The results can be generalized to broad- cast systems with an arbitrary number of users. In the next section, we investigate the capacity region assuming that the transmitter and both receivers have perfect knowledge of the channel propagation matrices 21 {} k k C and the multi-dimensional space-time modula- tion formats . 21 {} k k 3. Capacity Region with Known Channel at the Transmitter Let denote the input covariance matrix of user constrained by † [() kkk E xx k ] () k Tr P k . For a fixed k , the input/output average mutual informa- tion for (2) is maximized by a Gaussian input distribution and it is given by [1] † 0 1 (,)log(|( )|) kkk tR kkkk k LN D IN xrIHH (4) From [1,31], Shannon capacity is obtained by maximiz- ing (, kk Ixr) over all positive semidefinite input co- variance matrices k satisfying () kk Tr P . From [32], we have that this capacity is obtained using a water-filling power allocation [31], and it is given by ; 11 1{log()}/Hz kk Gi Nn kkk il k il G Cb N ps (5) where ;1 ;2;0 k i kk k ii in denote the (real and n-negative) ordered eigenvalues of no † () k i CC k i , and the constant k satisfies k N 11 { kk Gi n il 1} ; () k il 0 kk G k NP ND . The corresponding input signal k ] x is zero mean Gaussian with a block diagonal covariance matrix 1[ k G kk N ii diag where the kk iG nD -square input covariance matrix of TOG is given by i † 1 01 ; () ([{ }])( k G k i kk GG kk ii D kk k n lil i DD Ndiag ) MI IMI that, using [33], reduces to † 1 01 ; [( ) ([{()}])] k i k G kk ii kk k n lili D Ndiag M MI (6) All rate pairs 12 (,RR) such that 11 (,)RIxr 1 and 22 (,)xr 2 RI are achievable, and the capacity region is the closure of all such rate pairs 12 (, )RR . We specify the power allocation between the two users by 1 p T P P , which is the fraction of power allocated to user one. The fraction of power allocated to user two is 2 1p T P P . Using (5) and the nota- tion () k p C for ,1,2 k Ck, the boundary of the capacity region is a parametric curve in p defined by M. KASSOUF ET AL. 6 11 22 1 11 11 ; 1 2 22 11 ; 2 1log/H 1log/ H Gi Gi p Nn il il G p Nn il il G C bps z N C bps z N (7) such that 111 1 11 11 ;1 0 {} Gi GpT Nn il il N P ND and 222 22 1 11 ;2 0 (1 ) {()} Gi Gp Nn il il NP ND T with [0,1] p . Using a time-sharing argument as in [31], we have that the capacity region is convex- , and also continuous since continuity is an underlying property of convexity [34]. Moreover, as p in- creases in the interval [0, 1], 1 P increases and 2 P decreases, leading to an increase in 1( p)C and decrease in 2() p C , respectively. Hence, 1( p)C is monotonically increasing with p while 2() p C is monotonically decreasing with p . It follows that 2() p C is a monotonically decreasing function of 1() p C . In order to assess the transmission performance of a multi-user system using a comparison of capacity regions, we introduce the following definition: Definition 1: A capacity or rate region is said to be larger (respectively, smaller) than another region if the former contains (respectively, is contained in) the latter for the same power . T P For a fixed p (or ), [32] shows that for a single user system k P () k p C is maximized by ATA, and for 2 k G N merging TOGs cannot decrease () k p C . It follows that, for = 1, 2 and for a given kp , maxi- mum, intermediate and minimum values for () k p C are respectively obtained when user employs ATA, POTA and OTA, provided that the POTA system can be obtained from OTA by merging TOGs. A straightfor- ward application of this result to the boundary points of the capacity region (7) of the orthogonally multiplexed MIMO BC leads to the following theorem: k Theorem 1: With channel known at the transmitter, the capacity region is largest, intermediate and smallest when the users employ ATA, POTA and OTA, respec- tively. Considering a transmitter with transmit antennas, ATA represents a transmission strategy where there are no constraints on the assignment of a signal dimension to the transmit antennas. As such, with ATA a signal di- mension can be used on all the antennas. For POTA, a signal dimension is constrained to be used only on a subset of antenn as, while for OTA it is constrained to be used only on one antenna. Thus, as T N k G N increases the assignment of a signal dimension to transmit antennas is more constrained, yielding a decrease or no change in the capacity. We now investigate the sum capacity. Given , we use T P ; p s to denote the power allocation that maximizes the sum capacity. Define the maximum sum capacity 12 0,1(( )( )) p sp CCC max p , and let 12 s (, s CC) denote the point of the capacity region boundary that corresponds to ; p s . We have 12 s s s CC . Using the convexity property and considering the two-dimen- sional plane defined by the axes C 1 R and 2 R, the point 12 (, ) ss CC corresponds to the intersection of the boundary of the capacity region 12 (( 12 ),() pp CC) with the affine function s RRC. Thus, 1 (, ss CC 2 ), is the point at which the support line to 1 (( 2 ),()) pp CC has a slope equal to −1, and is lo- cated at the most right of the capacity region. Let () k bT E denote the transmitted energy per bit when operating at capacity limits. We have () k bT E () k k p CD k P k . The SNR per bit referenced at the transmitter satisfies 0 (/) bT EN 0 0 ()(/ ) k kk p bT k PCEN ND (8) Next, we consider the capacity region and sum capac- ity at low and high SNR. 3.1. Capacity Region at Low SNR For user at low SNR only the eigenmodes corre- sponding to the maximum eigenvalue k () max k ;11 {{}} k k iG kN n ij ji ()k are active. Let denote the num- ber of elements of {{ that are equal to ()k r 1 k G N i ;1 } } k i kn ij j . Thus, usin g ( 8) and (5) we h a ve Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 7 () 0 () () ()log(1(/ )) kk k Gp kk pb k k G NC r CE r N k T N . Equivalently, we can write () () 0 () () 2 () ()() 21 (/) () (ln2) ln2 () 2 kk p G k NC r k bT kk Gp k k k k G p kk k EN NC r NC r (9) showing that () k p C is linear in 0 (/) k b EN T with slope () () 2 (ln2) kk 2 k G r N . As () k p C → 0, we have 0 / bT ENk→() ln2 k (which corresponds to −1.6 − 10 ). Furthermore, (7) becomes () 10 log() d k B 1 1 11 1 1111 00 2 2 22 2 222 00 log(1) ln2 11 log(1) ln2 p GpT pT G p GpTp G C NP P r NrNDND C NP r Nr NDND 2 T P (10) Hence, we have 11 0 1 ln2 () p Tp ND PC leading to a triangular capacity region (in the positive quadrant of the two-dimensional plane defined by the axes 1 R and 2 R) bounded by the line 212 21 12 2 0 ln2 T P D RR DN D (11) As → 0, the segment (11) converges to the point (0, 0). 0 / T PN We investigate the sum capacity at low SNR by con- sidering three possible cases depending on the channel parameters: 1) If 21 12 >1 D D , then the right most support line 12 s RR C intersects the capacity region boundary at 2 12 (, ) ss CC ;0 ps 2 0 (0,) ln2 T P ND , which corresponds to with total transmit power allocated to user two and a sum capacity 2 2 0 ln2 T s P CND . 2) If 21 121 D D then the right most support line 12 s RR C intersects the capacity region boundary at 1 12 (, ) ss CC ;1 ps 1 0 ( ,0) ln2 T P ND , which corresponds to with total transmit power allocated to user one and a sum capacity 1 1 0 ln2 T s P CND . 3) If 21 121 D D then the support line with slope −1 lies on the boundary of the capacity region (11), thus, maximizing the sum capacity for any power allocation p . Hence, ;ps may take arbitrary value in [0, 1] and 12 (, ss CC) could be any point of the segment (11). The sum capacity is given by 21 21 00 ln2 ln2 TT s PP C. ND ND 3.2. Capacity Region at High SNR At high SNR, all kk TG rD eigenmodes are active and the channel capacity from (5) becomes 1 () k pk G CN 11; log( ) kk Gi kk Nr il il with 0 kk kG kk T NP rND 1 11; 1[( kk Gik Nr ilil k T r )] . We define 1 k G kN i A and 1; log( ) k ik r lil 1 11; k Gi kk r ilil 1k N k T dr yielding, 0 ( )log() kk k k kk G T pkkkk GG T NP r A Cd NN rND . Equivalently, we have 0 ( )log(1) k kk kk G T pkkk GT N rP CX NrNDd k (12) where log( ) k k kk T kk GG r A X NN d . (13) Using (8), we can write C opyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 8 [][] 022 (/) () () kk kkkk pp GG kk TT NC ANCA k rr k bT kk kk Gp G kk TT d EN NC NC rr p (14) As 0 (),(/ ) kk p b CE T N increases exponentially, making 0 (/)dB k bT EN linear in () k p C with slope 10 10log 23 kk G k TT NN rr G k . Furthermore, we prove the following theorem in the appendix Theorem 2: As , the asymptotic capacity region with known channel at the transmitter becomes rec- tangular, defined by the points (0, 0), 0 / T PN 2 20 (0,log( )) TT G rP N N 1 10 (log(),0) TT G rP N N and 1 10 (log(), TT G rP N N 2 20 log( )) TT G rP N N. Hence, regardless of the space-time modulation format the capacity region of the orthogonally multiplexed MIMO broadcast channel converges to a rectangle, simi- lar to that of ortho gonal broadcast channels [7]. Next we investigate the sum capacity at high SNR for the follow- ing cases: 1) 0< <1 p : Using (12), we can write 2 1 () () p p dC dC 2 () p 1 () p p p dC d d dC , which yields 1 11 201 12 22 02 () () (1 ) G pT pT pG p T T N dND P dC r dC N dND P r . (15) The denominator of (15) becomes zero for 22 2 0 2 1T p GT rd ND NP which is strictly larger than 1, thus, making 12 ((),() p CC 12 22 111 0 12 ()}{ GG TTT NN dD dD Prr } N , and as 0 / T PN 22 11 1 ;2121 [1 ] GG T ps TGTG NN rr rN rN T . (16) Furthermore, from Theorem 2 we have that the capacity region boundary points corresponding to 0, 1 p con- verge toward the upper right corner of the limiting rec- tangle. Therefore, we have 1 12 10 (,)(log( ), TT ss G rP CC N N 2 20 log( )) TT G rP N Nand 0 log( /) s T C P N 12 12 ( TT GG rr NN ) . It is seen that both s Cand ; p s depend on the users space-time modulation formats as well as the ranks of the TOGs propagation matrices without be- ing dependent on the channel eigen values ; k il . 2) 0 p : From (15) the right derivative is such that 22 11221 00 0 12 () [] [ () p pG T pT dC N dND dNDP dC r ] The support line at 12 ((0), (0))CC is not unique and has a slope that varies in the interval 2 0 1 () [[], ] () p p p dC dC . The lower bound of this interval is larger than −1 for sufficiently large and hence, no support line at the point 0 / T PN 1 ((0),CC 2 (0)) can have a slope equal to −1. It follows that 12 ((0), (0))CC (for which 0 p ) cannot be an intersection of the capacity region boundary with the affine function 12 s RRC, yielding ;0 ps . 3) 1 p : From (15) the left derivative is such that ) p differentiable for all . It follows that the support line at every point of the capacity region boundary with 0,1 p 10, p is unique and equal to the tangential line [34]. As , we have from (15) that 0 / T PN 2() () 1 11 p G 21 112 2 1 10 0 11 () []()[ () p pG T pT dC N dNDP dND dC r ] The support line at 12 ((1), (1)CC )is not unique and it has a slope in 2 1 1 () (,[]][0,) () p p p dC dC 0 / T PN 12 ((1), (1))CC that does not include −1 if is sufficiently large. Hence, no support line at the point can have a slope equal to −1. It follows that 1 ((1),CC 2 (1) 2 2 (1 ) p T p pT N rG r N dC dC .The power allocation that maximizes the sum capacity can be obtained by solving ; 2 1 () [] () pps p p dC dC 1 , yielding 2 ;2 {G ps T N r ) (for Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 9 which 1 p ) cannot be an intersection of the capacity region boundary with the affine function 12 s RR C, yielding ;ps 1 . 4. Rate Region with Unknown Channel at the Transmitter In this section, we assume that the transmitter has no information about the channel matrices 1 C and 2 C.Without channel knowledge at the transmitter, [1, 35] advocate to uniformly distribute the transmit power among all antennas. In [32], we represented the lack of channel knowledge at the transmitter in a single user system by an uninformative a-prior probability distribu- tion on the channel propagation matrix, and considered the following optimality criterion: Definition 2 (Optimality Criterion 1) An input covari- ance matrix k subject () k k PTr is said to be optimal in sense 1 if, as the transmitter channel knowledge converges to zero, it causes the fastest con- vergence to zero of the fraction of channels for which the input/output mutual information is below any specific value R, . 0<R < dia In [32] we considered input covariance matrices of the form where 1[ k G kk N ii g ] Ik G kk ii D S and k i S k i n is Hermitian positive semidefinite, similarly to the water-filling matrix (6), and have shown that Optimality Criterion 1 is satisfied using a zero mean Gaussian input vector k i n xk of independent components, with input covariance matrix of TOG given by i I kk ii GG kk II kk TT kk GG kk nD nD NP ND k i NP ND . (17) Using this uniform power allocation for each user in each TOG, the transmission rate (4) for user was shown to be [32] 1, 2kk ; 11 kk Gi Nn ij G 0 11 log(1)bps/ Hz T kk kk G uij kk NP IN NND (18) which depends on p through k P and can be subse- quently denoted as () p k u I with [0,1] p . From [32], the capacity () k p C and the transmis- sion rate ( k u I) p present several common properties, such as continuity and convexity as well as similar as- ymptotic behaviour. Using (18), the boundary of the rate region with unknown channel at the transm itter is given by 11 22 1 11 ; 11 11 0 2 22 ; 11 22 0 1log(1 ) 1 1log(1 ) Gi Gm up pGT ij Nn ij T G up pGT mlNn ml T G I NP N NND I NP N NND (19) Let 0; (/) k bT EN u denote the SNR per bit referenced at the transmitter. When operating at rate () k up I , we have as in (8) 0; 0 ()(/ ) k k upb T k PIEN ND k u . (20) As for the case of known channel at the transmitter, 1() up I is monotonically increasing with p while 2() up I is monotonically decreasing with p . Thus, 2() up I is monotonically decreasing with 1 I() up . Under similar transmit power constraint and for a given p , we have () () kk up p IC , with equality achieved when the following necessary and sufficient conditions are satisfied: Theorem 3: The capacity region 12 ((), () pp CC ) with transmitter channel knowledge is equal to the rate region 12 ((),()) upu p II without transmitter channel knowledge if and only if for each user the TOG propaga- tion matrices are full column rank (i.e. kk rn ii ) with all eigenmodes being active and with equal eigenvalues ; k il k i , such that 1 0 () Constan kk kG ik T NP NND t for all 1,,k ln i and all 1,,,1, 2. kk G The proof of this theorem can be found in the appendix. iN In [32], we conjectured that the single-user informa- tion transmission rate () k up I is maximum, intermedi- ate and minimum with ATA, POTA and OTA, respec- tively. Since this statement holds for any p , it can be easily extended to the orthogonally multiplexed MIMO BC as follows: Conjecture 1: Using the uniform power allocation for each user with input covariance matrix k 1[ k Gk N ii diag ] and k i given in (17), the rate C opyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 10 region 12 ((),()) upu p II is largest, intermediate and smallest when the users employ ATA, POTA and OTA, respectively. Next, we investigate the asymptotic behaviour of the rate region 1 ((), upu II ;ps 2 ()) p and the maximum sum rate at low and high SNR. For the remainder of this sec- tion, we use to denote the value of p that cor- re spo nds to th e max imu m su m rate. The associated point on the boundary of the rate region is denoted by and th e maximum sum rat e by 12 ;; (, us u II ) s 12 ;;us us III ;us . We also prove Conjecture 1 in these extreme SNR re- gimes. 4.1. Rate Region at Low SNR At low SNR, (18) reduces to ; 11 0 ; 11 0 11 ( )log([1]) 11 log(1 []) kk Gi kk Gi kk Nn kk G up ij kk ij GT kk Nn k G ij kk ij GT NP IN NN NP N NND D . By defining, ; 11 1 kk k Gi G kk ij kk NN nG Gi ji TT N BN T NN r † [( )] kk ii CC, we can write 0 1 () log(1) kk k up k G PB INND k (21) and, using (20), we have 0; 22 21 (/) () () (ln2)ln2 () 2 kk up G NI k bTukk up kk k GG up k EN BI NN I B k B . (22) Thus, () k up I is linear in 0; (/) k bT EN u with slope 22 (ln2) k B2 () k NG . Furthermore, if () k up I 0 then 00 ;min ln2 // k u k NG k B ENEN ; k bb Tu .6 10log ()] kk CC (which corresponds to . Since , we have 10 1 1() 10log G N † 1 [ [() k Gk N ii Tr CC 0()dB) kk B † ] k i Tr † [( )] k kkk G T N BTr N CC , leading to ;min k 0 / bu EN † ln2 [( )] T kk N CCTr which is independent of the space-time modulation format. Hence, the slope of () k up I (which is equal to † 2 2[( (ln2) k T G Tr N N C)] kk C at 0;min /k bu EN is decreasing with k G N, and we have that () k up I is maximum, intermediate and minimum with ATA, POTA and OTA, respectiv ely, proving Conjecture 1 at low SNR. One can also see that 11 kk Gi k k G B N Nn il ; k il 1 k G k kNi i TT r NN . Using k i rk i n (and hence 1 k Gk N ii r T N ),we have k G B N k k showing that ln2 ln2 k G k N Bk . Equality is achieved when all eigenva- lues ;1 {{} k kn il l 1 }k iG N i are equal and i n k i rk for all 1, ,k G iN. Thus, at low SNR, () k p C grows with a steeper slope than () up k I with respect to 0 (/) k bT EN . Subsequently, we refer to the ratio k k G B N as average eigenvalue for user . k From (21) we have 1 1 2 2 22 ln2 ln2 up G 11 0 0 1 pT p T up G B INN P B ID P D NN yielding a triangular rate region (in the positive quadrant of the two-dimensional plane defined by the axes 1 R and 2 R) and bounded by the line 1 212 21 21 22 0 2 GT NP BD B R NBD ND 2 ln GG R N (23) Since k k k G B N , it can be seen from (11) and (23) that the capacity region bounded by (11) contains the one bounded by (23). Similar to (11) the segment (23) re- duces to the point (0, 0) as . We distinguish the following cases: 0 / T PN 0 1) If 22 1 112 /1 / G G BN D BND >, then (23) has a steeper slope Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 11 than −1. The right most support line with slope −1 in- tersects the rate region boundary at 12 ;; (,) us us II 2 2 0 (0, ) ln2 T G P B NND 2 yielding a maximum sum rate 2 ;2 0 ln2 T us G P B INND 2 with power allocation ;0 ps . 2) If 22 1 112 /1 / G G BN D BND <, then the right most support line with slope −1 has a steeper slope than (23) and in- tersects the rate region boundary at 12 ;; (,) us us II 1 11 0 ( ln2 T G P B NND ,0) , yielding 1 11 0 ln2 T G P B NND ;us I with ;1 ps . 3) If 22 1 2 // G BN BN DD 1 1 G , the right most support line with slope −1 lies on the boundary of the triangular rate region (23), thus, maximizing the sum rate at every point. Hence, ;ps can take arbitrary value in [0, 1] and the maximum sum rate is 2 0 ln2 T G P B NN ;22 us ID 1 ln2 T P B NND 11 0G Comparison with Section 3 shows that at low SNR, the sum rate maximization is determined by the average eigenvalues . / k G BN k when the channel is unknown at the transmitter, while being determined by the maximum eigenvalues k with known channel at the transmitter. 4.2. Rate Region at High SNR For large, (18) becomes 0 / T PN 1 () up N k k G I 11 lo g( k Gi r ij N ; 0 1) kkk k NG ij k NP ND T , and with k M ; 11 lo g( k Gi r ij ) k kk Gij N T N N we have 0 log k kk kT up kk GG r M INN N k P D . (24) Using (20) and (24), we can write [] 0; 2 /() kkk up G k T NIM r k bTu k up EN I . (25) As 0; (),(/) k upb T IE k u N increases exponentially, making 0; /dB k bTu EN linear in k up I with slope 10 10log23 kk G k TT NN rr G k by . Therefore, for large the rate of change of the SNR in dB with the transmis- sion rate remains unchanged with and without channel knowledge at the transmitter. By using a proof similar to that of Theorem 2 with equation (24) instead of (12), we can easily prove the following theorem: 0 / T PN Theorem 4: As , the rate region with uniform power allocation becomes rectangular, defined the points 0 / T PN 2 20 (0,0), (0,log()), TT G rP N N 1 10 (log( ),0) TT G rP N N and 12 12 00 (log(), log()) TTT T GG rPrP NN NN . Comparison with Theorem 2 shows that the limiting rectangle is the same with and without channel knowl- edge at the transmitter. From [32] and using the uniform power allocation 1[ k G kk N ii diag ] with k i given in (17) and for a given p , the necessary and suf- ficient conditions for the equality of ( k) p C and () k up I at high SNR are that kk i n i rfor all , 1, k Gk1, ,iN2. Since these conditions hold for every p , they are also necessary and sufficient for the equality of the capacity region and the rate region 1 ((CC 2 ), ( p )) p 12 ((), ()) upu p II at high SNR, yielding the following theorem. Theorem 5: Similar asymptotic capacity and rate re- gions are obtained with and without transmitter channel knowledge if and only if all users have full column rank TOG propagation matrices , for all k ii rnk1,i ,k G N and 1, 2k . The conditions of Theorem 5 are satisfied whenever OTA is used, and if the channel propagation matrix has full column rank ( k T rN) T whenever ATA is used. Finally, one can show that the conditions of Theorem 3 reduce to those of Theorem 5 for high values of . 0 / T PN Consider now the effect of the space-time modulation format. The term 0 log(/ ) k k T k G rPN N is dominant in (24), and hence the impact of the space-time modulation C opyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 12 format is determined through the ratio k T k G r N . Since min( ,) kkkk iitR rnLN i for 3GPP channels [32], k T k G r N takes the values ;min( ,) kk TATtR rNLNk with ATA, ;1 k G k TP N kkk GGG rnLN NNN min( ,) kk k itR i with POTA, and ;11 min(,)1 T kkk TO NtR i TTT rLN NNN with OTA. Since min( ,) kkkk itR i kk GG G nLN n NN N k , we have ; k TP r 1 k G k NiT i kk k GGG nN NN N . Similarly, using min(, k i k G n N ) kk kk tR tR kk GG LN LN NN we have ; k TP r 1 k G kk kk NtR itR kk GG LN LN NN . It follows that ; k TP r ; min(,) kk k T tR TA kk GG NLN r NN . Also, using 1 min ( ,) kkk itR kk k GG G nLN NN N one can see that ;; 111 k G kk TO N i kk T GG rr N NN TP . Thus, k T k G r Nis de- creasing with increasing , and so is k G N k up I . It follows easily that the rate region 1 (( ) up 2 ),() u p II is largest, intermediate and smallest when the users employ ATA, POTA and OTA, respectively. This proves Con- jecture 1 at high SNR. We now investigate the maximum sum rate with uni- form power allocation at high SNR by considering the following three cases: 1) 0 p1 <<: Using (24) with 1 P p T P and 2(1 ) p T P P, we can write 22 2 21 ( u p T 11 1 ()) () () (1 ) upp p up up p G p GT d N r Nr dI dI d dI dI (26) which exists, thus, making 1 , upu II 2 p differenti- able for all . The power allocation maximiz- ing the sum rate can be obtained by solving 0, 1 p ; 2 1 () [] () pps up up dI dI 1 yielding 22 11 1 ;2121 [1 ] GG T ps TGTG NN rr rN rN T (27) which is identical to (16). Furthermore, using Theorem 4 it can be easily shown that 12 12 ;; 12 00 (,)(log( ),log( )) TTTT us us GG rPr P II NN NN , thus, yielding 12 ; 12 0 () log / us TT TGG Irr PN NN as . Similarly to the capacity region with known channel at the transmitter, both and 0 / T PN ;us I;ps depend on the users space-time modulation formats as well as the ranks of the TOGs propagation matrices without being dependent on the channel eigenvalues ; k il . 2) 0 p : From (26) the right derivative is such that 2 0 1 [p up up ]0 dI dI /PN when . The support line at 0T 12 0, 0 u II uis not unique and has a slope in the range 2 0 1 () ] , () p up up dI dI [[ which does not include −1. As in Subsection 3.2 we can show in this case that ) ;ps 0 . 3) 1 p : From (26) we have 2 1 1 () [] when . The support line at () p up up dI dI 0/ T PN 11, u I 2 u I1 is not unique, with a slope in the set (, 2 1 dI dI 1] () [] () p up up [0,) that does not include −1. Hence also in this case ;1 ps . 5. Numerical Results For numerical calculations, we assume equal number of antenna elements per TOG for both users, kT Gk G N nN . We also assume , 4 2 k TR T dd N and ;1 k ls G , for all {0,,1}, {1,,} kk lt s SlL and 1, 2k . Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 13 We consider the 3GPP spatial channel model of Figure 3 from the standardization document [25] with the follow- ing parameters for use r : 1, 2k k B S : The angle of the line-of-sight (LOS) direction be- tween the base station and user with respect to the antenna array normal at the transmitter. k k M S : The angle of the LOS direction between user and the base station with respect to the antenna array normal at the receiver. k ; k Tl : The mean AOD of cluster . l ; k R l : The mean AOA of cluster l ;; ˆk Tls : The offset AOD from ; k Tl of path s in cluster . l ;; ˆk R ls : The offset AOA fr om ; k R l of path s in cluster . l Hence, the AOD and AOA of path s in cluster are given by l ;;; ;; ˆ kkkk Tls BS TTls l and, ;;; ;; ˆ kkkk R lsMS RlRls , respectively. For numeri- cal results, we fix the mean AODs and mean AOAs for all clusters and .The values of the offset AODs and AOAs are chosen from the simulation model pre- sented in [25]. We consider a macrocell environment with a root mean square (RMS) angle spread of at the base station and RMS angle spread of at the re- ceiver with the following characterization: 1, 2k 2o 35o User one: 11 4, 20o BS D and 115o MS . Furthermore, we assume 13 t L with the following clustering structure: 1 13S, mean AOD 1 ;1 4o T and mean AOA 1 ;1 38o R . The offset AODs and AOAs in degrees are given by 11 ;1;1 ;1;1 ˆˆ , )(0.2826,4.9447), TR ( 11 ;1;2 ;1;2 ˆˆ (, TR ) and (1.3594, 23.7899) 11 ;1;3 ;1;3 ˆˆ , TR 3.0389, . 53.1816 1 22S, mean AOD 1 ;2 2o T and mean AOA 1 ;2 10o R . The offset AODs and AOAs in degrees are given by and 11 ;2;1 ;2;1 ˆˆ ,0.4984, 8.7224 TR 1 ;2;2 ˆ, T . 1 ;2;2 ˆR 4.3101,75.4274 1 31S, mean AOD 1 ;3 3o T and mean AOA 1 ;3 20o R . The offset AOD and AOA in degrees are given by . 11 ;3;1 ;3;1 ˆˆ , 1.0257,17.9492 TR User two: 22 4, 10o BS D and 25o MS . Fur- thermore, we assume with the following clus- tering structure: 22 t L Figure 4. The capacity and uniform power allocation rate regions for two-user orthogonally multiplexed MIMO broadcast channel with user one employing POTA and user two employing ATA and 24 (1) 0.05,5, 510, 510, 510 T 0 P= ND 6 . Figure 5. Maximum sum rate for the two-user orthogonally multiplexed MIMO broadcast channel with user one em- ploying POTA and user two employing ATA and 24 (1)0.05, 5, 510, 510, 510 T 0 P= ND 6=0.5. The line ,ps de- notes the power allocation (16), (27) that maximizes the sum rate at high SNR and “x” denote the maximum sum rate points. 2 12S , mean AOD 2 ;1 3o T and mean AOA 2 ;1 17o R . The offset AODs and AOAs in degrees are given by 22 ;1;1 ;1;1 ˆˆ (,)( 0.0894,1.5649) TR and 2 ;1;2 ˆ (, T 2 ;1;2 ˆ)( R 1.7688,30.9538) . C opyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 14 2 22S, mean AOD 2 ;2 4.5o T and mean AOA 2 ;2 25o R ( . The offset AODs and AOAs in degrees are given by and 22 ;2;1 ;2;1 ˆˆ ,)( 0.7431,13.0045) TR 2 ;2;2 ˆT (, 2 ;2;2 ˆ) (2.2 R 961,40.1824). Effect of transmit power: The regions 1 (() up I, 2()) up I and 12 ((), () p CC ) p ) are illustrated in Fi- gure 4 for low, intermediate and high values of 0T when user one employs POTA and user two employs ATA. We consider single antenna receivers for both us- ers, . Figure 4 illustrates the conver- gence of the capacity and rate regions with 0T to- ward a rectangle, and that the equality at high SNR of /PN /N 12 (, )(1,1 RR NN P 1 u I() p and 1() p C is satisfied since 11 ii rn , following Theorem 5. However, the rate re- gions with and without channel knowledge at the trans- mitter are not equal at high SNR because user two em- ploys ATA, and while . The corre- sponding maximum sum rate plots are presented in Fig- ure 5 with respect to 2, i1,2 2 T r24 T N p , where “x” denotes the maxi- mum sum capacity points. We see the convergence at high SNR of these points toward the line corresponding to (16), (27) given by ;ps 0.5 . Effect of space-time modulation: The regions 12 ((), ()) upu p II and 12 ((), () p CC ) p are illus- trated in Figure 6 for users employing space-time modula- tion formats similar to those of Figure 2. For this example, the rate region 1 (( up )) p 2 ), ()) u p II and capacity region for OTA coincide. From Figure 6 we see that the largest capacity and rate regions are obtained when both users employ ATA, while POTA yields a smaller region, and the smallest regions are obtained with OTA. These results reinforce Theorem 1 and Conjecture 1. 12 ((), ( p CC Effect of the number of receive antennas: Figure 7 shows that increasing the number of receive antennas results in an expansion of the regions 1) up I((, and ((. 2()) up I 12 ), () pp CC ) Effect of multipath propagation: Figure 8 considers two users employing POTA with single antenna receiv- ers 12 (, )(1,1 RR NN ) ) . When the total number of multi- path components increases from to 12 (, )(3,1SS 12 (, )(6,4)SS (where 12 )(3,1)SS (, is ob- tained by considering the paths of the first time resolv- able cluster for user one and the first path for user two), an expansion of the rate regions is observed, with and without channel knowledge at the transmitter. In all figures, we see that the rate region with un- known channel at the transmitter is contained in the ca- pacity region for channel knowledge at the transmitter. 6. Conclusions This paper considered orthogonally multiplexed MIMO broadcast systems with multi-dimensional space-time modulation over a deterministic multipath additive Gaussian channel. We showed that the largest capacity region is achieved when each user employs all his signal dimensions on all transmit antennas (which corresponds to ATA space-time modulation format). The capacity Figure 6. Two-user orthogonally multiplexed MIMO broadcast channel with users employing antennas and different space-time modulation formats with (1) (2) RR ()(1N,N =,1) (1) 2 T 0 P= ND . Figure 7. Two-user orthogonally multiplexed MIMO broadcast channel with users employing ATA and different numbers of antenna elements with (1) 40 T 0 P= ND . Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 15 Figure 8. Two-user orthogonally multiplexed MIMO broadcast channel with users employing POTA with and different numbers of propagation paths with (1) (2) RR ()(1N,N =,1) (1) 7 T 0 P= ND . region with informed transmitter and the rate region with uninformed transmitter using a uniform power allocation are triangular at low SNR and become rectangular at high SNR. At high SNR these regions become the same if and only if all users have full column rank TOG propagation matrices. We also investigated the power allocation among users that maximizes the sum rate, and provided explicit ex- pressions for such power allocation and the correspond- ing maximum sum rate at low and high SNR. At high SNR the power allocation that maximizes the sum capac- ity is determined by the users’ space-time modulation format and ranks of the TOGs propagation matrices. However, when the channel is known (respectively un- known) at the transmitter the sum rate is maximized at low SNR by an arbitrary power allocation between users if they have equal ratios of maximum (respectively av- erage) eigenvalue to signal space dimensionality; other- wise it is maximized by allocating the total transmit power to one user only. Numerical results for a two-user system using some examples from the 3GPP spatial channel model show that the capacity region with an informed transmitter and the rate region with an uninformed transmitter using a uniform power allocation expand when the number of transmit antennas per TOG or the number of receive an- tennas increases. Furthermore, these numerical results show that an increase in the number of multipath com- ponents leads to a rate region expansion with known and unknown channel at the transmitter. In this paper we assumed that users do not share signal dimensions, resulting in an orthogonal multiplexed sys- tem without interference between users. Future work will explore systems where some dimensions are shared between users, and hence interference plays a major role. Appendix Proof of Theorem 2 Fix p and take T PWe distinguish the fol- lowing cases: 0 /N . 0 p : Since no power is allocated to user one 1(0) 0C , and (12) yields 22 0CX 2 2log( GT N Nr 22 22 0 ) G TT rP NdD 2 with X given in (13). Hence, 2 2 )(0,) 12 00 (0) (0) (, log( /)log( / T TT G r CC PN PNN ) 1 p : Similarly to the previous case, we can show that 1 12 1 00 (1) (1) (,)(,0) / )log(/ ) T TT G r CC PN PNN log( . 1 p 0<< : Using (12) we have () k p C 0 log( ), leading to k kk kG T kkkk GT N rP XNrNDd 1 0 () [( , log( /) p T C PN 212 01 12 0 () )]( ,) log( /)p pTT TGG Crr PN NN << . Thus, the rate of change of both 1() p C and 2() p C with in dB is independent of 0 / T PN p . In the limit, all the points 01 ))] p p << 12 [((),( p CC converge toward the intersec- tion point of the capacity region boundary with the func- tion 1 2 2 21 G T GT N r1 R R Nr (28) in the two-dimensional plane defined by the axes 1 R and 2 R. The capacity region boundary is characterized by three points 21 21 00 (0,log( )),(log( ),0) TTTT GG rPrP NN NN and 12 12 00 (log( ),log( )) TTT T GG rPrP NN NN . Since, by definition, the capacity region is the closure of all the set of achievable rate pairs, it follows that its boundary contains all the points 2 1 20 (,log( )) T G rP RN N T with 1 1 10 0log( ) TT G rP RN N , C opyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 16 and 12 10 (log( ),) TT G rP R N N with 2 2 20 0lo TT G rP RN N g() . Consequently, the limiting capacity region is a rectangle with lower left corner (0, 0) and upper right corner 12 12 00 (log( ),log( )) TTT T GG rPrP NN NN . Proof of Theorem 3 The proof of this theorem consists first in deriving the necessary and sufficient conditions for the equality of () k up I and () k p C for a given p . The necessary and sufficient conditions for equality of the regions 1 (( up 2 ()) p ), u II and 12 ((), )CC ( p) p follow by varying the power allocation p in the interval [0, 1]. Assuming a fixed power allocation p , we have Lemma 1: () () kk p up CI () k i n if and only if the cor- responding TOGs propagation matrices have full column rank and k i r ; k ili k for all {1, ,} k i ln such that 1 0 () kk kG ik T NP NND = Constant ,1, G ii N k. (29) Proof: 1) Assume that () () kk p up IC . Due to the uniqueness of the input covariance matrix that maxi- mizes the average mutual information [36], the covari- ance matrices associated with () k p C and () k up I must be equal. Hence, (6) and (7) yield 1 01; [] k i k i kk kk nG lil kn T NP Ndiag ND I. (30) From (30), 1 ; 0 T , leading to full rank TOG propagation matrices. Therefore, ii and all eigenmodes in TOG i are active. Fur- thermore, (30) also leads to () kk kk G il k NP l NND k() ()k rn 1 ; 0 () kk kk G il k T NP NND = Constant ,1, k i ll n. It follows that ; k ili k , and the water-filling con- stant becomes 1 0 (), 1, kk kk G iG k T NP ii N NND making 1 0 () kk kG ik T NP NND equal for all TOGs. 2) Consider the channel of user with k k G N TOGs, k ii rnk and ; k il i k for all 1,, i k nl such that (29) is satisfied. Equation (18) can be written as 1 1 11 0 1 1 ()log[(())] 1log( ) kk Gi T k G kk Nn kk N G T up i kk il T G N kk ii k i G NP N IN NN n N D . Using the arithmetic-geometric inequality [37], we have 1 11 0 1 11 ( )log[(())] 1log( ) kk Gi k G kk Nn kk G T up i kk il TT G N kk ii k i G NP N INN NN n N D with equality achieved since 1 0 1 () kk kG ik T NP NND is constant for all 1, ,k G iN. Hence, 1 10 1 1 ()log[(( ))] 1log( ) k G k G kk N kkk G T up ii kk i T GT N kk ii k i G NP N In N NN n N ND .(32) If the channel is known at the transmitter, (29) shows that there exists a water-filling power allocation where all eigenmodes are active, and all TOGs contribute to the overall channel capacity with power kk i T nP N allocated to each TOG , for .Thus, the water-filling solution leading to the channel capacity results in a con- stant i 1, ,k G iN 1 0 () kk G k T NP NND kk pi , leading to 11 1 ( )log() kk Gi Nn kk pp k il G CN k i with 1 11 0 (()) kk Gi kk kk NnG ilpik NP ND . Thus, we have 1 1 0 1(()) k G kk kkk NG piii k TT NP n NNND and k (31) Copyright © 2010 SciRes. IJCNS M. KASSOUF ET AL. 17 () () () () ()()1 10 () () () 1 1 ()log[(() )] 1log( ) k G k G Nkk kkk G T pii kk i T G N kk ii k i G NP N Cn N N n N T N ND (33) which is similar to (32) and, hence, yielding () () kk p up CI . This proves Lemma 1. Now we return to the proof of Theorem 3. 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