Int. J. Communications, Network and System Sciences, 2010, 3, 1-18
1
doi:10.4236/ijcns.2010.31001 Published Online January 2010 (http://www.SciRP.org/journal/ijcns/).
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:
[email protected],
[email protected] 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 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. Keywords: MIMO Channels, Broadcast Systems, Capacity Region, Space-Time Coding
1. 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 [2–4]. 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 Copyright © 2010 SciRes.
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
IJCNS
2
M. KASSOUF
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 [16–18] 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.
Copyright © 2010 SciRes.
ET AL.
2. 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 d th component of a vector a is denoted by [a]d 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 n matrices { i }in1 is denoted by diag in1 [ i ]. The vertical stack of
n matrices with equal number of columns
i i1 n
in a
single matrix is denoted by stack [ i ] . The n -square identity matrix is denoted by Ι n . The n -dimensional vectors e i n for i 1, 2, , n are defined as n i 1
ein i,1 , i, 2 , , i, n , with the Kronecker symT
bol defined by i, j 1 if i j , and (i, j) 0
otherwise. For a scalar a , we have {a} max(0, a ) . Unless otherwise specified, the function log() denotes the base-2 logarithm, and the superscript ( k ) refers to the k th user in the system. We consider K orthogonally multiplexed users, each with power P k satisfying the constraint kK1 P k PT and affected by independent interference. Let DT
denote the total number of signal dimensions, with user k k occupying a sub-space of dimensionality D , where kK1 D k DT . Each user employs a different signal sub-space. This model corresponds to an orthogonally multiplexed MIMO broadcast channel (BC) without user cooperation. For user k , the propagation medium consists of L(t k ) 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 k we define the transmitted and received signal vectors s ( k ) (t ) [ s1( k ) (t ), s2( k ) (t ), , sN( kT) (t )]T and z ( k ) (t )
[ z1( k ) (t ), z 2( k ) (t ), , z N( k()k ) (t )]T , R
k
respectively, with NT and N R denoting the number of transmit and receive antennas. The continuous time channel model is specified by k Lt
z k t Fl k s k t l k i k t l 1
(1)
IJCNS
M. KASSOUF
ET AL.
3
Figure 1. System model for user k.
Figure 2. Examples of dimension allocation schemes for user k with N T = 4 transmit antennas and D (k) = 4 dimensions: a) ATA, b) OTA, c) POTA with N G(k) = 2 TOGs.
where l k and Fl k denote the propagation delay and
vector xjk n for transmit antenna j and the corre-
the N R k NT channel propagation matrix associated
sponding D k -dimensional signal space vector yjk n
k with cluster l (l {1, , Lt }) . The interference vector
are given by y jk ( n ) (
i (t ) [i1 k
k
t , i2 k , , iN k t ]T k R
is complex valued
k j
) T x jk ( n ) , with
j e Tt k N k Ι D k and t jk (t jk {1 , N G k }) indik
j
G
G
zero mean white Gaussian, with autocovariance matrix k k E [ i ( t ) ( i ( t )) † ] N o ( ) I N k where
cating the TOG to which antenna j belongs. The
denotes the Dirac’s delta function. Consider the system model from [26] illustrated in Figure 1. Assume a modulation process for user k that partitions the transmit antennas into groups called Transmit Orthogonal Groups (TOGs), each sharing a
sions (indexed by d {1, , D k } ) that can be used on
R
k
given subset of its D signal dimensions. Hence each TOG employs a different signal sub-space. Let N G k k denote the number of TOGs ( 1 N G NT ), and ni k
DG k D k matrix jk determines the signal dimen-
transmit antenna j by making [y jk (n)]d 0 if the d th signal dimension is not used on antenna j , or
[y jk (n)]d [xjk (n)]d for d {1, , DG k } otherwise.
The multi-dimensional space-time modulation format is determined by the matrices { jk }Nj T1 . With N G k 1
G
and N G k NT 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
Copyright © 2010 SciRes.
IJCNS
th
denote the number of transmit antennas in the i TOG, assumed to be adjacent. We assume an equal number of D k k signal dimensions per TOG, DG k and define NG k k k D N D . The D -dimensional complex input T
G
M. KASSOUF
4
k more general case ( N G NT ) 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
s j ( t ) k
n
T
( y j ( n )) ( t nT ) k
k
where
T
t 1 k t , 2 k , , D k k t denotes user k , s real orthogonal basis functions and 1 Τ 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 k
k d
t
k l
nT
k d
t
k l
sponds to OTA. Thus, increasing N G lowers complexity and increases parallelism. Let rm kl n and v mkl n denote the D k -dimenk
sional discrete time channel output and noise vectors of the l th time resolvable cluster received on the mth antenna. Hence, the vectors x k n vec({xjk n }Nj T1 ) and k
k
r k n vec({rm kl n }mN R1 }lLt 1 )
denote the input and
output of the discrete time channel defined by r k n H k x k n v k n k R
k
n vec({{vmkl n}mN1 }lL1 ) .Using we have y k n vec({y jk n }Nj 1 )
with noise vector v k diag Nj T1[jk ],
(2)
k
t
T
[ k ]T x k n . Assuming perfect multipath resolv-
nT dt
ability, the Lt k N R k D k D ( k ) discrete time channel
k , k l , l d , d n, n
k , k , l , l 1, , Lt k
ET AL.
k , d , d 1, , D
and n, n . Since different TOGs employ different signal subspaces, user k , s system is equivalent to N G k parallel MIMO systems each consisting of a TOG and the receiver. The overall Maximum Likelihood detection complexity for user k is the sum of his TOGs comk plexities, each being exponential in ni . By reducing each ni k the overall complexity decreases. It is minimum, and linear in NT when ni k 1 which corre-
matrix H k can be seen as the stack of N R k D k D ( k ) submatrices each associated with a time resolvable cluster. Therefore, we have k
H k stacklLt1 [(Fl k I D k ) [ k ]T ] k
stacklLt1 [(Fl I D k )] [ ]T k
k
or, equivalently,
H k C k I D k k
T
(3)
Figure 3. Space and time dispersive MIMO channel model for user k .
Copyright © 2010 SciRes.
IJCNS
M. KASSOUF k
with C k stacklLt1 [Fl k ] . Using the 3GPP spatial chan-
nel model [25] illustrated in Figure 3, let Sl k denote the number of propagation paths in cluster l , with path k s ( s 0, , Sl 1) characterized by the gain coefficient Gl;ks , angle of departure (AOD) T k;l;s and angle of arrival (AOA) R k;l; s .The total number of propagation paths for user k is
Lt l 1 k
Sl k . The spaces between
adjacent transmit and receive antennas are denoted by k d k k dT and d R . We use the notations l; s 2 R
ET AL.
5
tion for (2) is maximized by a Gaussian input distribution and it is given by [1] t
dT
sin(T k;l; s ) with
k
al ; s [1, e e
k
j NT 1 l ;s
k
k
jl ;s
, , e
k
k
j ( N R 1)l ;s
T
]
and
k
bl ; s [1,e
k
j l ; s
, ,
and from [25] we can write Fl k
]T
Gl;ks al;ks (bl;ks )† . Subsequently, we define the
Sl 1 s 0
Lt N R ni k
k
k
k propagation matrix Ci describing the
propagation between the i th TOG and the receiver of user k , such that C k [C1 k C2k CNkk ] . Let ri k G
denote the rank of Ci k and define rT k iNG1 ri k . (k )
k
k
k
We also use M i and Qi to denote the ni -square unitary and diagonal matrices associated with the eigenk k k k k value decomposition (Ci )† Ci (M i )† Qi Mi . Assuming a memoryless channel, we can drop the dependencies on the time index n for the remainder of this paper. Moreover, we consider orthogonally multiplexed broadcast MIMO channels with two users ( 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 {C k }2k 1 and the multi-dimensional space-time modulak tion formats { }k2 1 .
3. Capacity Region with Known Channel at the Transmitter Let
E[ x ( x ) ] k
k
k
†
denote the input covariance
matrix of user k constrained by Tr ( fixed
k
k
)P
k
. For a
, the input/output average mutual informa-
Copyright © 2010 SciRes.
k
variance matrices
satisfying Tr ( k ) P k .
From [32], we have that this capacity is obtained using a water-filling power allocation [31], and it is given by C
de-
noting the signal wavelength. The space signature vectors at the receiver and transmitter are given by
R
From [1,31], Shannon capacity is obtained by maximizing I (x k , r k ) over all positive semidefinite input co-
sin( R k;l; s ) and l;ks 2
1 k H k (H k )† |) N0 (4)
I (x k , r k ) log(| I L k N k D k
k
k
NG
1 N G k
k
ni
{log( )} k
i 1 l 1
k i ;l
bps / Hz
(5)
where i;1k i;2k i;kn k 0 denote the (real and i
non-negative) ordered eigenvalues of (Ci k )† Ci k , and the constant k satisfies N G P k
k
k
k
NG i 1
ni l 1 k
{ k (i ;l ) 1} k
k . The corresponding input signal x is zero
N0 D mean Gaussian with a block diagonal covariance matrix
diag k
N G i 1 k
[ i k ] where the ni k DG k -square input
covariance matrix of TOG i is given by
(M I k
k
i
i
DG k
)†
k
( N0 diaglni 1 [{ k i;kl
1
} ] I Dk ) (Mi k I D k ) G
G
that, using [33], reduces to
[(M ) k
k
i
†
i
k
( N 0 diaglni 1 [{ k (i;kl ) 1} ]) M i k ] I Dk
(6)
G
All rate pairs ( R 1 , R 2 ) such that R 1 I (x1 , r 1 ) and R 2 I (x 2 , r 2 ) are achievable, and the capacity 1 2 region is the closure of all such rate pairs ( R , R ) . We specify the power allocation between the two 1 P p users by PT , which is the fraction of power
allocated to user one. The fraction of power allocated P 2 . Using (5) and the notato user two is 1 p PT
tion C k ( p ) for C k , k 1, 2, the boundary of the capacity region is a parametric curve in p defined by
IJCNS
M. KASSOUF
6
C 1 p 1 1 1 NG ni 1 1 1 i 1 l 1 log i ;l N G 2 C p 2 2 2 2 NG ni 1 log i;l N 2 i 1 l 1 G
such that
N G i 1 2
2
ni l 1
1
NG i 1
1
ni l 1
{ i; l 1
{ ( i; l ) 1 } 2
2
1
1
bps / Hz
(7)
bps / Hz N G p PT 1
}
N0D 1
N G (1 p ) PT
and
2
N0D
2
with
ET AL.
tively. Considering a transmitter with NT 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 N G k 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 PT , we
p [0,1] . Using a time-sharing argument as in [31],
use p ; s to denote the power allocation that maximizes
we have that the capacity region is convex- , and also continuous since continuity is an underlying property of convexity [34]. Moreover, as p in-
the sum capacity. Define the maximum sum capacity 1 2 Cs max 0,1 (C 1 ( p ) C 2 ( p )) , and let (Cs , Cs )
creases in the interval [0, 1], P 1 increases and P 2 1 decreases, leading to an increase in C ( p ) and 2
1
decrease in C (p ) , respectively. Hence, C ( p ) is
monotonically increasing with p while C ( p ) is 2
monotonically decreasing with p . It follows that
C 2 ( p ) is a monotonically decreasing function of C 1 ( 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 PT . k For a fixed p (or P ), [32] shows that for a single
p
denote the point of the capacity region boundary that corresponds to p ; s . We have Cs Cs1 Cs 2 . Using the convexity property and considering the two-dimensional plane defined by the axes R 1 and R 2 , the point (Cs1 , Cs 2 ) corresponds to the intersection of the boundary of the capacity region (C 1 ( p ), C 2 ( p )) with
the
1
2
affine
when operating at capacity limits. We have ( Eb )T k P k
k
C ( p ) D
k
. The SNR per bit ( Eb / N 0 )T k referenced
at the transmitter satisfies N0 D
Copyright © 2010 SciRes.
Thus,
cated at the most right of the capacity region. Let ( Eb )T k denote the transmitted energy per bit
N G k 2 merging TOGs cannot decrease C k ( p ) . It
are respectively obtained when user k 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: 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-
2
(C 1 ( p ), C 2 ( p )) has a slope equal to −1, and is lo-
Pk
mum, intermediate and minimum values for C k ( p )
1
(Cs , Cs ) , is the point at which the support line to
user system C k ( p ) is maximized by ATA, and for follows that, for k = 1, 2 and for a given p , maxi-
R R Cs .
function
k
C ( p )( Eb / N 0 )T k
k
(8)
Next, we consider the capacity region and sum capacity at low and high SNR.
3.1. Capacity Region at Low SNR For user k at low SNR only the eigenmodes corresponding to the maximum eigenvalue ( k ) max k
k
{{ i;kj }nji1 }iNG1
are active. Let r ( k ) denote the numk
k
k
ber of elements of {{i ; j } ji1 }i G1 n
N
that are equal to
. Thus, using (8) and (5) we have (k )
IJCNS
M. KASSOUF
C ( p ) k
r ( k )
k
NG
k log(1
N G C ( p ) k
k
r
(k )
( Eb / N 0 )T ) . k
2
1
r ( k )
N G k C k ( p ) ( k ) r ( k )
(9)
k
log10 ( ( k ) ) dB ). Furthermore, (7) becomes
N 0 D ln2 1 C ( p ) leading to a 1 triangular capacity region (in the positive quadrant of the two-dimensional plane defined by the axes R 1 and R 2 ) bounded by the line R
2
1
.
ln2 N 0 D 2
1
PT
ln2 N 0 D 1
, 0) , which corresponds to
1
PT
ln2 N 0 D 1
.
2 D 1 1 then the support line with slope −1 1 D 2 lies on the boundary of the capacity region (11), thus, maximizing the sum capacity for any power allocation p . Hence, p ; s may take arbitrary value in [0, 1] and (Cs , Cs ) could be any point of the segment (11). The 2 PT 1 PT sum capacity is given by Cs . ln2 N 0 D 2 ln2 N 0 D 1 2
3.2. Capacity Region at High SNR At high SNR, all rT k DG k eigenmodes are active and the 1
channel capacity from (5) becomes C ( p ) k
k
NG i 1
k
rT
k
ri l 1 k
1
[ iNG1 k
log( i ;l ) with
k
ri l 1
1
d k
and
k
rT
N G k
N G P k
k
k
rT N 0 D k
k
k
(i;kl ) 1 ] . We define
log(i;kl )
ri l 1
k k
(11)
As PT / N 0 → 0, the segment (11) converges to the point (0, 0). We investigate the sum capacity at low SNR by considering three possible cases depending on the channel parameters: 2 D 1 1) If 1 2 > 1 , then the right most support line D
A k iNG1
NG i 1
k
k
k i ;l
ri l 1
1
yielding, C ( p ) k
A k N G k
rT
k
N G k
N G P k
log(
k
rT k N 0 D k
d ) . k
Equivalently, we have C ( p ) X k
k
rT
k
where
p ; s 0 with total transmit power allocated to user two
Using (8), we can write
X k
A k N G k
N G
Pk
k
N G k
R 1 R 2 Cs intersects the capacity region boundary 2 PT 1 2 ) , which corresponds to at (Cs , Cs ) (0, ln2 N 0 D 2
Copyright © 2010 SciRes.
PT
3) If
1
C 1 p r 1 N 1 P 1 p PT 1 log(1 1 1G p T1 ) ln2 N 0 D 1 r N 0 D NG (10) 2 C p N 2 1 p PT r 2 2 1 p PT 2 G ) 2 log(1 2 2 ln2 N 0 D 2 r N 0 D NG
2 D 1 1 2 PT 1 2 R ln2 N 0 D 2 D
2
and a sum capacity Cs
ln2 (which corresponds to −1.6 − 10 ( k )
Hence, we have p PT
2
p ; s 1 with total transmit power allocated to user one
showing that C ( p ) is linear in ( Eb / N 0 )T with 2 r ( k ) ( k ) slope . As C k ( p ) → 0, we have k 2 N G ( ln 2 ) k
at (Cs , Cs ) ( 1
N G k (ln2) 2 k ln2 C ( p ) ( k ) (k ) (k ) 2r k
Eb / N 0 T →
and a sum capacity Cs 2) If
k k NG C ( p )
k
7
2 D 1 1 then the right most support line 1 D 2 1 2 R R Cs intersects the capacity region boundary
Equivalently, we can write ( Eb / N 0 )T
ET AL.
log(1
rT
k
N G k
rT k N 0 D k d k
log(d ) . k
)
(12)
(13)
IJCNS
M. KASSOUF
8 [
k
( Eb / N 0 )T
k k k NG C p A
]
k
rT
2
d
k
N G k C k ( p )
[
k k k NG C p A
k
rT
2
]
N G k C k ( p )
rT k
(14)
k
As C ( p ) , ( Eb / N 0 )T
p;s
increases exponentially,
making ( Eb / N 0 )T k dB linear in C k ( p ) with slope N G k
N G k
3
. Furthermore, we prove the rT k rT k following theorem in the appendix Theorem 2: As PT / N 0 , the asymptotic capacity region with known channel at the transmitter becomes rec2 r P tangular, defined by the points (0, 0), (0, T 2 log( T )) N0 NG rT
P r P r P ( 1 log( T ), 0) and ( T 1 log( T ), T 2 log( T )) . N0 N 0 NG N0 NG NG 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: dC 2 ( p ) 1) 0 < p < 1 : Using (12), we can write dC 1 ( p ) 1
1
2
dC ( p )
d p
d p
dC 1 ( p )
dC 2 ( p ) dC 1 ( p )
The
p
2
, which yields d N0 D 1
1
2
d N0 D
2
N G 1
rT
N G 2 rT 2
1
p PT .
(15)
(1 p ) PT
for all p 0,1 . It follows that the support line at every point of the capacity region boundary with p 0,1 is unique and equal to the tangential line [34]. PT / N 0 ,
dC ( p ) dC 1 ( p )
we
N G1 rT 2
rT1 N G 2
have
from
(15)
that
p .The power allocation (1 p )
that maximizes the sum capacity can be obtained by 2 2 N dC ( ) solving [ 1 p ] 1 , yielding p ; s { G2 rT dC ( p ) p
p ;s
Copyright © 2010 SciRes.
N G rT1
NG
1
2
rT
2
[1 1
2
rT
2
N G
]1 .
(16)
Furthermore, from Theorem 2 we have that the capacity region boundary points corresponding to p 0,1 converge toward the upper right corner of the limiting rectangle. Therefore, we have 2 Cs r 1 P r P (Cs1 , Cs 2 ) ( T 1 log( T ), T 2 log( T )) and log(PT / N0 ) N0 N G N0 NG rT 1
(
NG 1
rT
2
) . It is seen that both Cs and p ; s depend
NG 2
on the users space-time modulation formats as well as the ranks of the TOGs propagation matrices without being dependent on the channel eigenvalues i;kl . 2) p 0 : From (15) the right derivative is such that dC ( p ) 2
[
dC 1 ( p )
] p 0 d 1 N 0 D 1 [d 2 N 0 D 2
N G 2 rT 2
PT ]1
The support line at (C 1 (0), C 2 (0)) is not unique and has a slope that varies in the interval [[
dC 2 ( p ) dC 1 ( p )
] p 0 , ] .
the point (C 1 (0), C 2 (0)) can have a slope equal to −1.
than 1, thus, making (C 1 ( p ), C 2 ( p )) differentiable
2
N G rT1
PT / N 0
The lower bound of this interval is larger than −1 for PT / N 0 sufficiently large and hence, no support line at
denominator of (15) becomes zero for 2 2 2 r d N 0 D which is strictly larger 1 T 2 N G PT
As
N 0 2 2 N 1 N 2 (d D d 1 D 1 )}{ G1 G2 }1 , and as PT rT rT
rT k
k
10log10 2
ET AL.
It follows that (C 1 (0), C 2 (0)) (for which p 0 ) cannot be an intersection of the capacity region boundary with the affine function R 1 R 2 Cs , yielding
p;s 0 . 3) p 1 : From (15) the left derivative is such that dC ( p ) 2
[
dC 1 ( p )
] p 1 (d 1 N 0 D 1
N G1 rT1
PT )[d 2 N 0 D 2 ]1
The support line at (C 1 (1), C 2 (1)) is not unique and it has a slope in (,[
dC 2 ( p ) dC 1 ( p )
] p 1 ] [0, ) that does
not include −1 if PT / N 0 is sufficiently large. Hence, no support line at the point (C 1 (1), C 2 (1)) can have a slope equal to −1. It follows that (C 1 (1), C 2 (1)) (for
IJCNS
M. KASSOUF
ET AL.
9
which p 1 ) cannot be an intersection of the capacity
region with unknown channel at the transmitter is given by
region boundary with the affine function R 1 R 2 Cs ,
I u1 p 1 1 p N G1 PT i;1j 1 1 iNG1 nji1 log(1 ) N 0 D 1 NT NG (19) 2 I u p 2 1 p N G PT m 2;l 2 2 1 NG nm ) 2 m 1 l 1 log(1 2 NT N 0 D NG
yielding p ; s 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 C1 and C 2 .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 k subject Tr ( k ) P k 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, 0 < R < . In [32] we considered input covariance matrices of the
diag k
form
N G i 1 k
S I
[ i k ] where
k
k
i
i
DG k
and Si is ni ni 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 i given by k
k
k i
k
N G P k
k
N T Dk
N G P k
I n k I D k i
G
k
N T D k
I n k D k . i
(17)
G
Using this uniform power allocation for each user in each TOG, the transmission rate (4) for user k k 1, 2 was shown to be [32] k
Iu
1
k k N G ni
k k 1 N G P
log(1 N N k G
i 1 j 1
0
N T Dk
i;kj ) bps / Hz (18)
which depends on p through P k and can be subsequently denoted as I u k ( p ) with p [0,1] . k
From [32], the capacity C ( p ) and the transmission rate I u k ( p ) present several common properties, such as continuity and convexity as well as similar asymptotic behaviour. Using (18), the boundary of the rate Copyright © 2010 SciRes.
Let ( Eb / N 0 )T k;u denote the SNR per bit referenced at the transmitter. When operating at rate I u k ( p ) , we have as in (8) P k N0 D k
I u ( p )( Eb / N 0 )T ;u . k
k
(20)
As for the case of known channel at the transmitter, I u1 ( p ) is monotonically increasing with p while I u 2 ( p ) is monotonically decreasing with p . Thus, I u 2 ( p ) is monotonically decreasing with I u1 ( p ) . Under similar transmit power constraint and for a given p , we have I u k ( p ) C k ( p ) , with equality achieved when the following necessary and sufficient conditions are satisfied: Theorem 3: The capacity region (C 1 ( p ), C 2 ( p ))
with transmitter channel knowledge is equal to the rate region ( I u1 ( p ), I u 2 ( p )) without transmitter channel knowledge if and only if for each user the TOG propagation matrices are full column rank (i.e. ri k ni k ) with all eigenmodes being active and with equal eigenvalues i;kl i k , such that (i ) 1
N G P k
k
k
N 0 NT D k
Constant
for all l 1, , ni k and all i 1, , N G k , k 1, 2. The proof of this theorem can be found in the appendix. In [32], we conjectured that the single-user information transmission rate I u k ( p ) 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 k k
diagiNG1 [ i k ] and
k
i
given in (17), the rate
IJCNS
M. KASSOUF
10
region ( I u1 ( p ), I u 2 ( p )) is largest, intermediate and smallest when the users employ ATA, POTA and OTA, respectively. Next, we investigate the asymptotic behaviour of the rate region ( I u1 ( p ), I u 2 ( p )) and the maximum sum rate at low and high SNR. For the remainder of this section, we use p ; s to denote the value of p that corresponds to the maximum sum rate. The associated point on the boundary of the rate region is denoted by (Iu1;s , Iu 2;s ) and the maximum sum rate by Iu;s Iu1;s Iu 2;s . We also prove Conjecture 1 in these extreme SNR regimes.
NG ni k k
N G k
i 1 j 1 k
N G
k
k ni
k G
By defining, B k N G k iN1 k
1 N G P k i ; j ]) N 0 NT D k k
k
ni j 1
k
k
i ; j
k
NG NT
NT
1
k
N G k
i;kl
k
NG i 1
ri k NT ),we have
ln2
k
( Eb / N 0 )T ;u
2
P k B k
NG i 1 k
Tr
N0 D k
(21)
)
k
( N G ) (ln2) 2 B
2
k
I u k ( p )
k
. Furthermore, if k
Eb / N 0 T ;u Eb / N 0 u ;min
B
k
I u k ( p ) 0
Tr [ ( C
Bk
) C †
k
k
R
k N ln2 G
B k
(which corresponds
k
k
Tr [( C i ) C i ] †
NG Tr[(C k )† C k ] , leading to NT
NT ln2
Tr[(C )† C ] k
]
k
NG i 1
k
,
we
have
Eb / N 0 u ;min k
which is independent of the space-time
Copyright © 2010 SciRes.
k
are equal and ri k ni k for all
yielding a triangular rate region (in the positive quadrant of the two-dimensional plane defined by the axes R 1 and R 2 ) and bounded by the line
then
k k to 1 .6 10 log 10 ( N G ) 10 log 10 ( B ) dB) . Since k
N
1 1 B p PT I u p 1 N G ln2 N 0 D 1 B 2 1 p PT 2 I u p 2 2 N G ln2 N 0 D
2
k
showing that
average eigenvalue for user k . From (21) we have
N G ln2
Thus, I u ( p ) is linear in ( Eb / N 0 )T ;u with slope ( N G k ) 2 (ln2) 2
k
k ( Eb / N 0 )T k . Subsequently, we refer to the ratio B as
. (22)
k
k
2B k
N G k
with a steeper slope than I u k ( p ) with respect to
1
B I u ( p ) 2
Bk
i 1, , N G . Thus, at low SNR, C k ( p ) grows
k k k
at
N G k
log(1
k k NG I u p
k
. Equality is achieved when all eigenva-
k
k
and, using (20), we have k
k
k k r k iNG1 i . Using ri k ni k (and hence NT
n lues {{i ;l }l i 1 }i G1
[(Ci ) Ci ] , we can write I u ( p )
Tr [( C ) † C ] k 2 NT N G (ln2) 2
I u k ( p ) is maximum, intermediate and minimum with ATA, POTA and OTA, respectively, proving Conjecture 1 at low SNR. k k k B iNG1 lni 1 One can also see that k NG
Bk
.
I u k ( p )
is decreasing with N G k , and we have that
k
k
k
†
Eb / N 0 u ;min
N G ln2
k
1 N G P k ]) log(1 [ k i; j i 1 j 1 N 0 NT D NG
1 k
log( [1
(which is equal to
At low SNR, (18) reduces to 1
modulation format. Hence, the slope of
NT
4.1. Rate Region at Low SNR
I u k ( p )
ET AL.
Since
Bk k
B 2 N G D 1 1
N G 2 B 1 D 2
R 1
B 2
PT
N G 2 ln 2 N 0 D 2
(23)
, it can be seen from (11) and (23) that k
NG 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 PT / N 0 0 . We distinguish the following cases: 2 2 B / N G D 1 1) If > 1 , then (23) has a steeper slope B 1 / N G1 D 2 IJCNS
M. KASSOUF
than −1. The right most support line with slope −1 intersects the rate region boundary at ( I u1; s , I u 2; s ) (0,
B
2
PT
B
Iu;s
2
yielding a maximum sum rate
)
N G ln2 N 0 D 2 2
PT
N G ln2 N 0 D 2 2
with power allocation p ; s 0 .
B / N G D 1 2
2
< 1 , then the right most support B 1 / N G1 D 2 line with slope −1 has a steeper slope than (23) and intersects the rate region boundary at ( I u1; s , I u 2; s )
2) If
B 1
(
PT
N G1 ln2 N 0 D 1 with p ; s 1 . B
2
, 0) ,
Iu;s
yielding
B 1
PT
N G1 ln2 N 0 D 1
the
/ NG 2
maximum B 1
1
1
B / NG
sum
rate
Iu;s
is
B
2
2
PT
N G ln2 N 0 D
2
PT
4.2. Rate Region at High SNR For large PT / N 0 , (18) becomes
k
k
NG i 1
ri j 1
I u
log( k
k
I u k ( p )
log(
N G k
k
N G k i;kj
k
k
k
k
k
T
k G
k G
k
0
(24)
k
10log10 2
NG k
[
k
Eb / N 0 T ;u Copyright © 2010 SciRes.
k k k N G I u p M
k
rT
2
I u ( p ) k
k
NG k
3
. Therefore, for large PT / N 0 rT k rT k 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: Theorem 4: As PT / N 0 , the rate region with uniform power allocation becomes rectangular, defined 1 2 r P by the points (0, 0), (0, rT 2 log( PT )), ( T 1 log( T ), 0) N0 N0 NG NG rT
PT r P ), T 2 log( T )) . 1 N N NG NG 0 0 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 2
log(
diag k
power allocation
k
NG i 1
[ i k ] with
k
i
given in (17) and for a given p , the necessary and sufficient conditions for the equality of C k ( p ) and I u k ( p )
k k at high SNR are that ri ni for all
i 1, , N G , k 1, 2 . Since these conditions hold for k
every p , they are also necessary and sufficient for the equality of the capacity region (C ( p ), C ( p )) and 1
2
the rate region ( Iu ( p ), Iu ( p )) at high SNR, yielding 1
2
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 k k TOG propagation matrices ri ni , for all i 1, , N G k and k 1, 2 .
The conditions of Theorem 5 are satisfied whenever
]
.
Finally, one can show that the conditions of Theorem 3 reduce to those of Theorem 5 for high values of PT / N 0 .
Using (20) and (24), we can write
linear in Iu p with slope
Eb / N0 T ;u dB
making
OTA is used, and if the channel propagation matrix has full column rank (rT k NT ) whenever ATA is used.
) we have
NT
MN Nr log NPD . p
1
1 N G P k , and with M k i ; j ) N 0 N T D k k
k
ri j 1
As I u ( p ) , ( Eb / N 0 )T ;u increases exponentially,
and (
. 1 1 N G ln2 N 0 D Comparison with Section 3 shows that at low SNR, the sum rate maximization is determined by the average eigenvalues B k / N G k when the channel is unknown at the transmitter, while being determined by the maximum eigenvalues k with known channel at the transmitter.
N G i 1
11
1
2
, the right most support D D 1 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 3) If
ET AL.
(25)
Consider now the effect of the space-time modulation k r k format. The term T k log( P / N 0 ) is dominant in NG (24), and hence the impact of the space-time modulation
IJCNS
rT k . Since N G k
format is determined through the ratio ri k min(ni k , Lt k N R k ) i k
rT
ET AL.
M. KASSOUF
12
takes the values rT;kA min( NT , Lt k N R k ) with
k
NG
k
rT ; P
ATA,
k
NG
k
rT ;O
POTA, and
k k
NG
m in (
n i
rT ; P
k
NG
k
NG
m in ( rT; P
k
k
NG
NG
NT
k
k
k
N G k
NG k
k
k
NG
k
NG i 1
k
1 k
NG
we
NG
k
k
k
1 N G
k
.
Also,
using
one can see that
. Thus,
rT
k
NG
is de-
NG
rT1
NG
which exists, thus, making
dI u ( p ) 1
2
(26)
p (1 p )
I u1 p , I u 2 p differenti-
able for all p 0,1 . The power allocation maximiz-
Copyright © 2010 SciRes.
which is identical to (16). Furthermore, using Theorem 4 it can be easily shown that 1 2 r P r P ( I u1; s , I u 2; s ) ( T 1 lo g( T ), T 2 log ( T )) , thus, N N NG NG 0 0 I u ;s
yielding
lo g PT / N
0
(
rT
1
N G1
rT
2
N G 2
)
as
[
dI u 2 p dI u p 1
] p 0 0 when PT / N 0 . The support
I 0 , I 0 is not unique and has a slope in 1 u
line at
2
u
the range [[
dI u ( p ) 1
] p 0 , ) which does not include
dI u ( p ) −1. As in Subsection 3.2 we can show in this case that p ; s 0 . dIu ( p ) 2
3) p 1 : From (26) we have [
dIu1 ( p )
when PT / N 0 . The support line at I u 2 1
] p 1
I 1 , 1 u
is not unique, with a slope in the set (,
dI u ( p ) 2
[
1
dI u ( p )
] p 1 ] [0, )
that does not include −1.
5. Numerical Results
d p 1
d p rT
(27)
Hence also in this case p ; s 1 .
dI u 2 ( p ) 2
NG
2
k
P 2 (1 p ) PT , we can write
dI u ( p )
r
]1
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 i;kl .
largest, intermediate and smallest when the users employ ATA, POTA and OTA, respectively. This proves Conjecture 1 at high SNR. We now investigate the maximum sum rate with uniform power allocation at high SNR by considering the following three cases: 1 1) 0 < p < 1 : Using (24) with P p PT and
1
1
2 T
have
follows easily that the rate region ( I u1 ( p ), I u 2 ( p )) is
N
N G rT1 2
[1
PT / N 0 . Similarly to the capacity region with known channel at the transmitter, both I u ; s and p ; s
creasing with increasing N G k , and so is I u k p . It
dI u 2 ( p )
r
1 G
using
. It follows that
k
NT
2
2 T
2) p 0 : From (26) the right derivative is such that
k
rT; O
1
, we have
k
k
Lt N R
)
k k
N G rT1
with
) 1 with
Similarly,
, Lt N R ) rT; A
L t N R
,
k R
N G
Lt N R
)
k
ni
)
.
N G
Lt N R
k
k
N
k
1 L N , NT NT
k G
)
N G
k t
p ; s
k
Lt N R
,
k
Lt N R
,
k
NG
k
k G
k
NT
k
k
N
k
NG
N G i 1
ni
m in (
k
min( k
ni
k
n i
k
NG i 1
Lt N R
,
k
rT ; P
k
NG
k
NG
ni
NG i 1
k
k
k
k
k
iNT1 min(
NT
OTA. Since min( rT ; P
for 3GPP channels [32],
ing the sum rate can be obtained by solving 2 dI u ( p ) [ 1 ] 1 yielding dI u ( p ) p p ;s
For numerical calculations, we assume equal number of NT k antenna elements per TOG for both users, nG k . NG
, NT 4 and Gl;ks 1 , 2 for all s {0, , Sl k 1}, l {1, , Lt k } and k 1, 2 .
We also assume dT d R k
IJCNS
M. KASSOUF
ET AL.
13
We consider the 3GPP spatial channel model of Figure 3 from the standardization document [25] with the following parameters for user k 1, 2 : k BS : The angle of the line-of-sight (LOS) direction be-
tween the base station and user k with respect to the antenna array normal at the transmitter. k MS : The angle of the LOS direction between user k and the base station with respect to the antenna array normal at the receiver. T;kl : The mean AOD of cluster l .
R;kl : The mean AOA of cluster l ˆT k;l; s : The offset AOD from T;kl of path s in cluster l . ˆR k;l; s : The offset AOA from R;kl of path s in cluster l . Hence, the AOD and AOA of path s in cluster l are given by T k; l; s B kS T;kl ˆT k; l; s and,
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 PT = 0.05, 5, 5102, 5104, 5106 . two employing ATA and N0 D(1)
R k; l; s M kS R;kl ˆR k; l; s , respectively. For numerical results, we fix the mean AODs and mean AOAs for all clusters and k 1, 2 .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 2o at the base station and RMS angle spread of 35o at the receiver with the following characterization: 1 1 15o . 20 o and MS User one: D 1 4, BS Furthermore, we assume Lt1 3 with the following clustering structure: S11 3 , mean AOD T1;1 4o and mean AOA
R1;1 38o . The offset AODs and AOAs in degrees are given by (ˆT1;1;1 , ˆR1;1;1 ) (0.2826, 4.9447), (ˆT1;1;2 , ˆR1;1;2 )
(1.3594, 23.7899)
and
ˆ
1 T ;1;3
1 , ˆR ;1;3
3.0389,
53.1816 . S 21 2 , me a n A O D T1;2 2o a n d me a n A O A
R1;2 10o . The offset AODs and AOAs in degrees are
1 1 given by ˆT1;2;1 , ˆR ;2;1 0.4984, 8.7224 and ˆT ;2;2 ,
R;2;2 4.3101, 75.4274 . ˆ1
1
S3 1 , mean AOD T;31 3o and mean AOA
R1;3 20o . The offset AOD and AOA in degrees are
1 1 1.0257,17.9492 . , ˆR ;3;1 given by ˆT ;3;1
2 2 User two: D 2 4, BS 10o and MS 5o . Fur2 thermore, we assume Lt 2 with the following clustering structure:
Copyright © 2010 SciRes.
Figure 5. Maximum sum rate for the two-user orthogonally multiplexed MIMO broadcast channel with user one employing POTA and user two employing ATA and PT = 0.05, 5, 5102 , 5104 , 5106 . The line p , s = 0.5 deN0 D(1) notes the power allocation (16), (27) that maximizes the sum rate at high SNR and “x” denote the maximum sum rate points.
S1 2 2 , mean AOD T;12 3o 2
and mean AOA
R;1 17 . The offset AODs and AOAs in degrees are o
ˆ 2 given by (ˆT 2;1;1 , R;1;1 ) (0.0894, 1.5649) and (ˆT 2;1;2 , 2 ˆ ) (1.7688,30.9538) . R ;1;2
IJCNS
M. KASSOUF ET AL.
14
S 2 2 2 , mean AOD T;22 4.5o and mean AOA
R;22 25o . The offset AODs and AOAs in degrees are 2 ˆ 2 given by (ˆT 2;2;1 , R ;2;1 ) ( 0.7431, 13.0045) and (ˆT ;2;2 , 2 ˆ ) (2.2961, 40.1824) .
R ;2;2
Effect of transmit power: The regions ( I u1 ( p ), I u 2 ( p )) and (C 1 ( p ), C 2 ( p )) are illustrated in Fi-
gure 4 for low, intermediate and high values of PT / N 0 when user one employs POTA and user two employs ATA. We consider single antenna receivers for both us1 2 ers, ( N R , N R ) (1,1) . Figure 4 illustrates the convergence of the capacity and rate regions with PT / N 0 toward a rectangle, and that the equality at high SNR of I u1 ( p ) and C 1 ( p ) is satisfied since ri1 ni1 2, i 1, 2 , following Theorem 5. However, the rate regions with and without channel knowledge at the transmitter are not equal at high SNR because user two em2 ploys ATA, and rT 2 while NT 4 . The corresponding maximum sum rate plots are presented in Figure 5 with respect to p , where “x” denotes the maximum sum capacity points. We see the convergence at high SNR of these points toward the line corresponding to (16), (27) given by p ; s 0.5 . Effect of space-time modulation: The regions ( I u1 ( p ), I u 2 ( p )) and (C 1 ( p ), C 2 ( p )) are illustrated in Figure 6 for users employing space-time modulation formats similar to those of Figure 2. For this example, the rate region ( I u1 ( p ), I u 2 ( p )) and capacity region (C 1 ( p ), C 2 ( p )) 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. Effect of the number of receive antennas: Figure 7 shows that increasing the number of receive antennas results in an expansion of the regions ( I u1 ( p ),
known channel at the transmitter is contained in the capacity 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 ( N R(1) , N R(2) ) = (1,1) antennas and different space-time modulation formats PT = 2. with N 0 D (1)
I u 2 ( p )) and (C 1 ( p ), C 2 ( p )) .
Effect of multipath propagation: Figure 8 considers two users employing POTA with single antenna receivers ( N R1 , N R 2 ) (1,1) . When the total number of multipath components increases from ( S 1 , S 2 ) (3,1) to ( S 1 , S 2 ) (6, 4) (where ( S 1 , S 2 ) (3,1) is obtained by considering the paths of the first time resolvable 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-
Copyright © 2010 SciRes.
Figure 7. Two-user orthogonally multiplexed MIMO broadcast channel with users employing ATA and different numbers of antenna elements with
PT = 40 . N 0 D (1 )
IJCNS
M. KASSOUF ET AL.
15
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 PT / N 0 . We distinguish the following cases: p 0 : Since no power is allocated to user one C (0) 0 ,
and
N G
PT
1
rT
2
2
log(
2
NG Figure 8. Two-user orthogonally multiplexed MIMO broadcast channel with users employing POTA with ( N R(1) , N R(2) ) = (1,1) and different numbers of propagation paths with
2
rT
(12)
2
N0 d D
2
C 2 0 X 2
yields
2 ) with X given in (13).
r C 1 (0) C 2 (0) , ) (0, T 2 ) log( PT / N 0 ) log( PT / N 0 ) NG 2
Hence,
PT =7. N 0 D (1)
(
p 1 : Similarly to the previous case, we can show r C 1 (1) C 2 (1) , ) ( T 1 , 0) . log( PT / N 0 ) log( PT / N 0 ) NG 1
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-
Copyright © 2010 SciRes.
that (
0 < p < 1 : Using (12) we have X k
rT k
N G k
log(
C ( p )
N G k
P k
rT k N 0 D k d rT
2
log( PT / N 0 )
1
)]0
