Achievable Rate Regions for Orthogonally Multiplexed MIMO Broadcast Channels with Multi-Dimensional Modulation

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 antennas, 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 represent 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. Furthermore, 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.


Introduction
Multiple-Input Multiple-Output (MIMO) systems, employing 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][3][4].Furthermore, channel knowledge at the transmitter has been shown to increase capacity more significantly at low SNR [5,6].These favorable 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 information rate pairs.A larger capacity region may be obtained 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 nondegraded broadcast channels, thus, making the evaluation of their capacity regions very difficult.Superposition coding does not apply to non-degraded broadcast channels because users may employ different rates making successive decoding quite difficult if not impossible [11].However, this reference shows that a capacity region for broadcast channels can be achieved by using a coding technique, nicknamed dirty paper coding (DPC) [12], where the interference is non-causally known 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 systems with arbitrary number of users and multiple transmit and receive antennas has been carried out independently in [14] and [15].The sum rate optimality of DPC for Gaussian MIMO BCs has been investigated in [16][17][18] using the duality [19] between the DPC rate region 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 Division 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 Division Multiplexing (OFDM) transmission.
In this work, we consider a MIMO BC with orthogonally multiplexed non-cooperating users who employ space-time modulation.As in [23,24], we assume a non-fading space and time dispersive multipath environment.These schemes model the downlink of cellular communication systems with orthogonal user multiplexing.We consider a deterministic channel model since it provides an insight to the behaviour of the capacity region with respect to the number of antenna 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 users without mutual interference.When the channel is known at the transmitter, we consider 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 broadcast 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 Section 3. The case of unknown channel at the transmitter is considered in Section 4. Section 5 presents some illustrative numerical results.The conclusions follow in Section 6.

MIMO Broadcast Multipath Channels with Space-Time Modulation
In this paper, column vectors and matrices are represented by lower-case and upper-case bold letters.The component of a vector is denoted by [ ] Furthermore,  denotes the determinant of A, ⊗ denotes the matrix Kronecker product, and denotes the matrix product.We use the following superscripts: * for complex 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 The vertical stack of matrices with equal number of columns single matrix is denoted by .The -square identity matrix is denoted by .The -dimensional vectors , with the Kronecker symbol defined by 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.
We consider orthogonally multiplexed users, each with power and affected by independent interference.Let denote the total number of signal dimensions, with user occupying a sub-space of dimensionality where


. Each user employs a different signal 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 space only.For user we define the transmitted and received signal vectors respectively, with and denoting the number of transmit and receive antennas.
The continuous time channel model is specified by  , , }) is complex valued zero mean white Gaussian, with autocovariance matrix where     denotes the Dirac's delta function.
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 denote the number of transmit antennas in the TOG, assumed to be adjacent.We assume an equal number of signal dimensions per TOG, and define . The -dimensional complex input vector for transmit antenna and the corre- cating the TOG to which antenna belongs.The ) that can be used on transmit antenna by making The multi-dimensional space-time modulation format is determined by the matrices N  T we have the space-time modulation formats Aggregate Transmit Antenna (ATA) and Orthogonal 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 more general case corresponds to Partially Orthogonal Transmit 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.
The transmitted waveform through antenna j is given by where denotes user real orthogonal basis functions and is the symbol rate.We assume no inter-symbol interference and perfect synchronization at every receive antenna.Assuming perfect multipath time resolvability for each user as well as perfect orthogonality between users, we have the overall complexity decreases.It is minimum, and linear in when which corre-sponds to OTA.Thus, increasing lowers complexity and increases parallelism.Hence, the vectors denote the input and output of the discrete time channel defined by with noise vector .Using . Assuming perfect multipath resolvability, the can be seen as the stack of submatrices each associated with a time resolvable cluster.Therefore, we have or, equivalently, .Using the 3GPP spatial channel model [25] illustrated in Figure 3, let denote the number of propagation paths in cluster , with path characterized by the gain coefficient Copyright © 2010 SciRes.
noting the signal wavelength.The space signature vectors at the receiver and transmitter are given by and and from [25] we can write [1, , , . Subsequently, we define the propagation matrix describing the propagation between the TOG and the receiver of user , such that

C
and define .
We also use unitary and diagonal matrices associated with the eigenvalue decomposition . Assuming a memoryless channel, we can drop the dependencies on the time index for the remainder of this paper.Moreover, we consider orthogonally multiplexed broadcast MIMO channels with two users ( n K = 2) for simplicity.The results can be generalized to broadcast 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 and the multi-dimensional space-time modulation formats .

Capacity Region with Known Channel at the Transmitter
Let denote the input covariance matrix of user constrained by  , the input/output average mutual informa- tion for ( 2) is maximized by a Gaussian input distribution and it is given by [1] From [1,31], Shannon capacity is obtained by maximizing From [32], we have that this capacity is obtained using a water-filling power allocation [31], and it is given by where x is zero mean Gaussian with a block diagonal covariance matrix that, using [33], reduces to All rate pairs are achievable, and the capacity region is the closure of all such rate pairs We specify the power allocation between the two users by , which is the fraction of power allocated to user one.The fraction of power allocated to user two is . Using ( 5) and the nota- . 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 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 is maximized by ATA, and for follows that, for = 1, 2 and for a given k p  , maxi- mum, intermediate and minimum values for   ( ) are respectively obtained when user employs ATA, POTA and OTA, provided that the POTA system can be obtained from OTA by merging TOGs.A straightforward 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 dimension can be used on all the antennas.For POTA, a signal dimension is constrained to be used only on a subset of antennas, while for OTA it is constrained to be used only on one antenna.Thus, as 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 denote the point of the capacity region boundary that corresponds to ; p s  .We have . Using the convexity property and considering the two-dimensional plane defined by the axes C C corresponds to the intersection of the boundary of the capacity region ), ( ) with the affine function is the point at which the support line to denote the transmitted energy per bit when operating at capacity limits.We have Next, we consider the capacity region and sum capacity at low and high SNR.

For user
at low SNR only the eigenmodes corresponding to the maximum eigenvalue are active.Let denote the number of elements of {{ that are equal to k n i j j    .Thus, using ( 8) and ( 5) we have Copyright © 2010 SciRes.
Equivalently, we can write showing that   ( ) (which corresponds to −1.6 − 10 ).Furthermore, ( 7) becomes Hence, we have triangular capacity region (in the positive quadrant of the two-dimensional plane defined by the axes   1 R and As → 0, the segment (11) converges to the point We investigate the sum capacity at low SNR by considering three possible cases depending on the channel parameters: , then the right most support line intersects the capacity region boundary at , which corresponds to   with total transmit power allocated to user two and a sum capacity then the right most support line intersects the capacity region boundary at , which corresponds to   with total transmit power allocated to user one and a sum capacity C C ) could be any point of the segment (11).The sum capacity is given by

Capacity Region at High SNR
r D eigenmodes are active and the channel capacity from (5) becomes Equivalently, we have where Using (8), we can write Copyright © 2010 SciRes. As . Furthermore, we prove the following theorem in the appendix Theorem 2: As , the asymptotic capacity region with known channel at the transmitter becomes rectangular, defined by the points (0, 0), Hence, regardless of the space-time modulation format the capacity region of the orthogonally multiplexed MIMO broadcast channel converges to a rectangle, similar to that of orthogonal broadcast channels [7].Next we investigate the sum capacity at high SNR for the following cases: The denominator of (15) becomes zero for which is strictly larger than 1, thus, making , and as Furthermore, from Theorem 2 we have that the capacity region boundary points corresponding to verge toward the upper right corner of the limiting rectangle.Therefore, we have ) .It is seen that both s C and ; p s  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   The support line at is not unique and has a slope that varies in the interval It follows that cannot be an intersection of the capacity region boundary with the affine function 3) is unique and equal to the tangential line [34].

As
, we have from (15) that 0 The support line at )is not unique and it has a slope in

Rate Region with Unknown Channel at the Transmitter
In this section, we assume that the transmitter has no information about the channel matrices   .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 distribution on the channel propagation matrix, and considered the following optimality criterion: Definition 2 (Optimality Criterion 1) An input covariance matrix is said to be optimal in sense 1 if, as the transmitter channel knowledge converges to zero, it causes the fastest convergence to zero of the fraction of channels for which the input/output mutual information is below any specific value R, In [32] we considered input covariance matrices of the form where 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 x k of independent components, with input covariance matrix of TOG given by i Using this uniform power allocation for each user in each TOG, the transmission rate (4) for user was shown to be [32]  1, 2 which depends on p  through   k P and can be subsequently denoted as   ( ) From [32], the capacity   ( ) and the transmis- present several common properties, such as continuity and convexity as well as similar asymptotic behaviour.Using (18), the boundary of the rate region with unknown channel at the transmitter is given by E N u denote the SNR per bit referenced at the transmitter.When operating at rate   ( ) I  , we have as in ( 8) As for the case of known channel at the transmitter, ) with all eigenmodes being active and with equal eigenvalues The proof of this theorem can be found in the appendix.

i N
In [32], we conjectured that the single-user information transmission rate   ( ) is maximum, intermediate and minimum with ATA, POTA and OTA, respectively.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 .We also prove Conjecture 1 in these extreme SNR regimes.

At low SNR, (18) reduces to
and, using (20), we have Thus,   ( ) (which corresponds to .Since , we have which is independent of the space-time modulation format.Hence, the slope of   ( ) N , and we have that is maximum, intermediate and minimum with ATA, POTA and OTA, respectively, proving Conjecture 1 at low SNR.
One can also see that (and hence . Thus, at low SNR,   ( ) grows with a steeper slope than   ( ) . Subsequently, we refer to the ratio as average eigenvalue for user .k From (21) we have yielding a triangular rate region (in the positive quadrant of the two-dimensional plane defined by the axes ) and bounded by the line   , 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) reduces to the point (0, 0) as .We distinguish the following cases: ( , ) < , then the right most support line with slope −1 has a steeper slope than (23) and intersects the rate region boundary at ( , ) 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, ; p s   can take arbitrary value in [0, 1] and the maximum sum rate is Comparison with Section 3 shows that at low SNR, the sum rate maximization is determined by the average eigenvalues .

Rate Region at High SNR
For large ,

by
. Therefore, for large the rate of change of the SNR in dB with the transmission 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, 0), (0, log( )), Comparison with Theorem 2 shows that the limiting rectangle is the same with and without channel knowledge at the transmitter.From [32] and using the uniform power allocation (17) and for a given p  , the necessary and sufficient conditions for the equality of 2 .Since these conditions hold for every p  , they are also necessary and sufficient for the equality of the capacity region and the rate region ( ( ), ( )) at high SNR, yielding the following theorem.Theorem 5: Similar asymptotic capacity and rate regions are obtained with and without transmitter channel knowledge if and only if all users have full column rank TOG propagation matrices , for all The conditions of Theorem 5 are satisfied whenever OTA is used, and if the channel propagation matrix has full column rank   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 .
is dominant in (24), and hence the impact of the space-time modulation Copyright © 2010 SciRes.IJCNS format is determined through the ratio with POTA, and Similarly, using   differentiable for all .The power allocation maximiz-ing the sum rate can be obtained by solving which is identical to (16).Furthermore, using Theorem 4 it can be easily shown that Hence also in this case ; 1 p s    .

Numerical Results
For numerical calculations, we assume equal number of antenna elements per TOG for both users, We also assume We consider the 3GPP spatial channel model of Figure 3 from the standardization document [25] with the following parameters for user : , respectively.For numerical 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 presented 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 receiver with the following characterization:   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 expressions for such power allocation and the corresponding maximum sum rate at low and high SNR.At high SNR the power allocation that maximizes the sum capacity is determined by the users' space-time modulation format and ranks of the TOGs propagation matrices.However, when the channel is known (respectively unknown) 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 average) eigenvalue to signal space dimensionality; otherwise 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 antennas increases.Furthermore, these numerical results show that an increase in the number of multipath components 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
given in (13). Hence, ) ( 0 , ) in the two-dimensional plane defined by the axes   Consequently, the limiting capacity region is a rectangle with lower left corner (0, 0) and upper right corner

Proof of Theorem 3
The proof of this theorem consists first in deriving the necessary and sufficient conditions for the equality of   ( ) Proof: 1) Assume that     ( ) ( ) Due to the uniqueness of the input covariance matrix that maximizes the average mutual information [36], the covariance matrices associated with   ( ) From (30), , leading to full rank TOG propagation matrices.Therefore, i i and all eigenmodes in TOG i are active.Furthermore, (30) also leads to ( )  18) can be written as Using the arithmetic-geometric inequality [37], we have with equality achieved since . Hence, 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 to each TOG , for .Thus, the water-filling solution leading to the channel capacity results in a con- , leading to . Thus, we have which is similar to (32) and, hence, yielding

Figure 1 .
Figure 1.System model for user k.

Figure 2 .
Figure 2. Examples of dimension allocation schemes for user with transmit antennas and k 4 T N = 4 (k) D = dimensions: channel output and noise vectors of the time resolvable cluster received on the antenna.

Figure 3 .
Figure 3. Space and time dispersive MIMO channel model for user .k angle of departure (AOD)   equal to −1, and is located at the most right of the capacity region.Let of this interval is larger than −1 for sufficiently large and hence, no support line at ) can have a slope equal to −1.

− 1
if is sufficiently large.Hence, no support line at the point can have a slope equal to −1.It follows that

1 .
23) has a steeper slope than −The right most support line with slope −1 intersects the rate region boundary at the channel is unknown at the transmitter, while being determined by the maximum eigenvalues   k   with known channel at the transmitter.
Consider now the effect of the space-time modulation format.The term exists, thus, making        

− 1 .
on the users space-time modulation formats as well as the ranks of the TOGs propagation matrices without being dependent on the channel eigenvalues As in Subsection 3.2 we can show in this case that ) not unique, with a slope in the set ( ,  that does not include −1.


angle of the line-of-sight (LOS) direction between the base station and user with respect to the antenna array normal at the transmitter.k   k MS  : The angle of the LOS direction between user and the base station with respect to the antenna array normal at the receiver.of path s in cluster .l Hence, the AOD and AOA of path s in cluster are given by l

Figure 8 .
Figure 8. Two-user orthogonally multiplexed MIMO broadcast channel with users employing POTA with and different numbers of propagation paths with (1) (2) R R ( ) ( 1 N ,N = , 1 ) Similarly to the previous case, we can show that intersection point of the capacity region boundary with the function 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 if the corresponding TOGs propagation matrices have full column rank and is satisfied.Equation ( 1 p   : From (15) the left derivative is such that