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Degrees of Freedom in Multi-user Spatial Multiplex
Systems with Multiple Antennas
Wei YuElectrical and Computer Engineering Dept., University of Toronto
10 King’s College Road, Toronto, Ontario M5S 3G4, Canadaemail: [email protected]
Wonjong RheeASSIA, Inc., Atherton, CA 94027 USA
email: [email protected]
Abstract
This paper characterizes the structure of the optimal spatial multiplex scheme and provides a theoretical upperbound on the maximum number of active users in a multiuser multi-antenna wireless fading environment. Thesum capacities of the multiuser uplink and downlink channels are used as the performance criterion, and a unifiedapproach based on a duality between the multiple-access channel and the broadcast channel is provided. The sum-capacity achieving transmission strategy in a complex random i.i.d. fading environment, where the base-station hasn antennas and the K remote users have m antennas each, is shown to involve up to n2 data streams in total witheach user transmitting or receiving up to m2 data streams. In particular, the total number of users that are allowedto transmit/receive simultaneously in the uplink/downlink channel is shown to be theoretically upper bounded byn2. This gives a dimension counting interpretation for multiuser diversity. Multiple antennas at the base-stationincreases the total number of dimensions thus allowing more users to transmit and receive at the same time. Bycontrast, multiple antennas at the remote terminal allow a single user to occupy multiple dimensions, which increasesits transmission rate, but also has the potential effect of precluding simultaneous transmission by other users.
Keywords
Channel Capacity, Multiple-access Channel, Broadcast Channel, Multiuser communications, Spatial Diversity,Multiuser Diversity, Spatial Multiplex, Multiple-Input Multiple-Output (MIMO) Systems
Submitted to IEEE Transactions on Communications on March 10, 2004. This work was supported by Natural Science
and Engineering Research Council (NSERC), the Canada Research Chair program, Bell Canada University Laboratory and
Communication and Information Technology Ontario (CITO). Conference versions of this work have appeared in ICC’01 and
Globecom’03. Correspondence should be addressed to Wei Yu.
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I. Introduction
While the use of multiple antennas has been increasingly recognized as an effective means to pro-
vide both transmitter and receiver diversities for spectral efficient wireless applications, efforts to
quantify their precise benefit have been mainly limited to single-user systems. The emergence of
multiuser transmission techniques , however, opens up another dimension. When multiple spatial
dimensions are available at the base-station, the system capacity can be significantly increased by
allowing multiple users to access the base-station simultaneously. The design of such a multiuser
multi-input multi-output (MIMO) system demands not only sophisticated multiuser detection and
multiuser precoding techniques at the signal processing level, but also the implementation of multi-
user scheduling algorithms at the system level. The focus of this paper is on the optimal structure
of multiuser scheduling and power control in a multi-antenna wireless system.
Multiple antennas enhance the performance of wireless systems by creating multiple dimensions
in the spatial domain. Spatial dimensions can be used in two different ways. Multiple spatial
dimensions can carry duplicate copies of the same information. Because not all paths are likely to
experience deep fades at the same time, the overall information transmission reliability is likely to
improve. This is known as spatial diversity. On the other hand, multiple spatial dimensions may
also carry independent information in multiple data streams, thus improving the overall data rate.
This is known as spatial multiplex. In general, there is a tradeoff between diversity and multiplex
[1].
The concepts of diversity and spatial multiplex are also applicable to multiuser systems. Consider
a cellular system with one base-station and many remote terminals geographically scattered in a
single cell. Diversity in this multiuser system may be understood as follows: since not all remote
terminals are likely to experience deep fades at the same time, the total throughput of the multiuser
system can be made resilient to channel fading by intelligently selecting a best subset of users. Thus,
diversity in a multiuser system occurs not only across the antennas within each user, but also across
the set of all users. This type of diversity is referred to as multiuser diversity. Multiuser diversity
was first introduced by Knopp and Humblet [2] and has since been explored by various authors [3]
[4]. Note that multiuser diversity differs from single-user spatial diversity in one aspect. In a single-
user system, spatial diversity refers to the ability for the multiple antennas to transmit or receive
the same information across several paths, while in a multiuser system, independent information is
transmitted and received by different users. Thus, a system that utilizes multiuser diversity may
also operate in a spatial multiplex mode whenever more than one user are active at the same time.
The notion of multiuser diversity brings up the following set of questions. In a spatial multiplex
system with multiple antennas at the base-station, how many users should be active at any given
time? How should the optimal set of users be chosen? What are the appropriate power level and
data rate for each user? These questions should provide useful insights to the design of both the
3
Base-Station
User 1
User 2
User N
Fig. 1. Multiuser Multi-Antenna Wireless Environment
physical (PHY) layer and the medium-access-control (MAC) layer in a wireless network (e.g. [5].)
The purpose of this paper is to develop a rigorous theory to tackle these questions. Toward this
end, this paper characterizes the sum capacities of multiuser uplink and downlink channels assuming
perfect channel side information at both the transmitter and the receiver, and explores the structure
of the optimal power allocation strategy in the multiuser setting. The optimal power allocation
strategy not only gives a solution to the optimal multiuser scheduling problem, but also sheds light
on the interaction between spatial diversity and multiuser diversity.
The optimal power control strategy for single-antenna single-user fading channel is the well-known
water-filling strategy. Under the assumption of flat fading and perfect channel side information at
both the transmitter and the receiver, the ergodic capacity of a single-user single-antenna fading
channel is achieved with “water-filling-in-time” [6], where the transmit power is adapted to the
channel fading state so that more bits are transmitted when the channel is good. In a multiuser
environment, because of the presence of multiuser interference, the optimal power control strategy
needs to consider not only the fading state of the transmission channel but also that of the interfering
channels. For example, in the uplink direction, with a single transmit antenna at each remote user
and a single receive antenna at the base-station, Knopp and Humblet [2] showed that to maximize
the sum ergodic capacity, only the single user who has the best fading state should transmit at any
given time instant. A similar result exists in the downlink. In fact, even in the non-fading case,
the optimal downlink transmission scheme involves the single active user with the best channel.
Thus, with a single antenna at the base-station, the optimal uplink and downlink strategies are
exactly time-division multiple access (TDMA) strategies in which each user would wait until his
channel realization becomes the best among all users before it could transmit. TDMA is a key
feature that underlines Tse’s work on the Qualcomm HDR system [3]. As pointed out in [3], the
sum capacity maximizing strategy by itself does not provide fairness among the users. Thus, a fair
4
TDMA scheduler needs to be design to ensure equal access by all users.
The results in this paper are generalizations of above ideas to systems with multiple antennas
at the base-station and remote users. Intuitively, with n antennas at the base-station, n − 1 extra
spatial dimensions are created. Thus, one might expect that the optimal transmission strategy
should involve up to n users active at the same time. Despite its intuitive appeal, this unfortunately
is not true. For example, in the uplink direction, if there is no fading, the capacity maximizing
strategy involves all users transmitting at the same time. In fact, the sum capacity grows without
bound as the number of users increases to infinity [7]. In a fading channel, because of the channel
fluctuation, it makes sense for a user to conserve power at a time instant so that it can be more
efficiently used later. However, even in this situation, as the result of this paper shows, the maximum
number of users involved in any time instant may well be above the number of base-station antennas.
A similar situation exists for the downlink where a counter-example, in which the sum-capacity
achieving strategy involves more users than the number of transmit antennas, was first pointed out
by Caire and Shamai [8] for non-fading channels. One of the main objectives of this paper is to find
a rigorous way to quantify the interaction between spatial diversity and multiuser diversity. As the
main theorem of the paper shows, while the maximum number of active users may be larger than
the number of base-station antennas n, it is theoretically upper bounded by n2. In fact, n2 can be
considered as the theoretical maximum number of degrees of freedom in the system. In both uplink
and downlink channel, a remote user equipped with m antennas can utilize up to m2 degrees of
freedom, with the total number of degrees of freedom summed over all users bounded by n2.
The above dimension-counting argument is applicable to both uplink and downlink channels.
A key feature of the approach contained in this paper is a unified analysis for both uplink and
downlink channels developed using a duality between the multiple-access channel and the broadcast
channel. As first pointed out in [9] [10], the capacity region of a multiple-access channel under a
sum power constraint is exactly the same as the best achievable rate region of a dual broadcast
channel. Later, as shown in [11], uplink-downlink duality is equivalent to Lagrangian duality in
convex optimization. This optimization viewpoint provides a crucial link between capacity and the
optimal power allocation that allows a unified theoretical analysis to be developed for both uplink
and downlink channels.
The rest of the paper is organized as follows. In section II, the cellular multiuser channel model is
established. The optimal transmission problem is formulated for both uplink and downlink channels,
and the main result is stated. Section III derives the structure of the optimal uplink transmission
strategy and proves the main result of the paper for the uplink channel. Section IV gives an outline
of the recent information theoretical results on the downlink channel, provides an optimization
viewpoint for uplink-downlink duality, and establishes a similar result for the downlink channel.
Section V discusses the interaction between spatial diversity and multiuser diversity. Section VI
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presents simulation results. Section VII contains concluding remarks.
II. Multi-User Multi-Antenna Fading Channel
A. Uplink
Consider a cellular wireless environment with a base-station equipped with n antennas and K
remote users each equipped with m antennas. The remote-to-base-station transmission can be
modeled as follows:
y(i) =K∑
k=1
Hk(i)xk(i) + n(i), (1)
where k is the user index and i is the time index. The input signals xk(i) are m-dimensional vectors.
The received signal y(i) is an n dimensional vector. n(i) is the i.i.d. Gaussian noise. The user k’s
channel at time instant i is represented by Hk(i), which is an n × m matrix. This paper assumes
an i.i.d. fading model with instantaneous channel state information available to all the transmitters
and the receiver. The availability of channel state information is crucial for power control and in
practice it must be estimated at the receiver and transmitted back to the transmitter by a reliable
feedback mechanism. For simplicity, the entries of Hk are assumed to be independent and identically
distributed complex Gaussian random variables. This corresponds to a Rayleigh channel model with
rich scatterers.
The capacity of a K-user multiple access channel is a K-dimensional convex region whose boundary
points characterize the trade-off among data rates for various users. In this paper, the focus will be
on the single boundary point that maximizes the K-user rate sum. The characterization of other
boundary points involves the maximization of a weighted average of all users’ data rates, which, while
numerically possible [12] [13], does not appear to have a closed-form solution. The single maximum
rate-sum point is arguably the most effective single figure of merit for the multiple access channel.
It has an efficient numerical characterization [14] and has been the subject of many previous studies
[2] [15].
The capacity for the multiple access channel is achieved with superposition coding and successive
decoding. The relevant mutual information corresponding to the sum capacity (assuming perfect
channel side information at both the transmitters and the receiver) can be expressed using the chain
rule as:
I(X1, X2, · · · , XK; Y |H) = I(X1; Y |H) + I(X2; Y |X1, H) + · · ·+ I(XK; Y |X1, · · · , XK−1, H). (2)
Ergodic capacity is the channel capacity in the traditional Shannon sense. In this case, channel
coding is done over a block length sufficiently large to cover all fading states. The mutual information
can then be averaged over all fading states, and the ergodic capacity is expressed as an expectation:
C(u) = maxEH1,··· ,HKI(X1, · · · , XK; Y |H1, · · · , HK). (3)
6
Here, the expectation is over the joint channel distribution. The mutual information is now a
random variable, which depends on the channel in two ways. First, the explicit computation of
I(X1, · · · , XK; Y ) depends on the channel. Secondly, because of the perfect transmitter side infor-
mation assumption, the input distribution for (X1, · · · , XK) is also a function of (H1, · · · , HK). The
maximization is over all such input distributions, which are called power allocation policies.
Gaussian signaling is optimal in the i.i.d. fading multiple access channel, so the optimal transmit
signal is a zero-mean Gaussian process. Given the instantaneous channel realization (H1, · · · , HK),
each transmitter sets its power spectral density subject to its total power constraint.
Let Sk(H1, · · · , HK) be the m × m signal covariance matrix for user k at the given channel
realization, i.e.
Sk(H1, · · · , HK) = E[XkX∗k |H1, · · · , HK], (4)
where (·)∗ denotes complex conjugate, and the expectation is over the transmitted codebook. An
uplink power allocation policy for user k is a mapping
Pk : (H1, · · · , HK) 7→ Sk(H1, · · · , HK). (5)
The average power constraint Pk for user k is satisfied when
EH1··· ,HK[tr(Sk(H1, · · · , HK))] ≤ Pk, (6)
where the expectation is over the joint channel distribution and “tr” denotes the matrix trace
operator. The optimal sum capacity point is the solution to the following optimization problem,
maxS1,··· ,SK
EH1,··· ,HKI(X1, · · · , XK; Y |H1, · · · , HK), (7)
subject to the average power (or trace) constraints (P1, · · · , PK) on (S1, · · · , SK).
B. Downlink
The power control problem for downlink transmission can likewise be defined as follows. In the
base-station-to-remote-terminal direction
y′k(i) = H ′
k(i)x′(i) + n′
k(i), k = 1 · · ·K, (8)
where x′(i) is an n-dimensional vector representing the transmitted signal from the base-station,
y′k(i) is an m-dimensional vector representing the received signal for kth user, and n′
k(i) is the i.i.d.
Gaussian noise. Again, H ′k(i) is an m × n matrix, and it denotes the channel for user k at time
instant i. Note that the base-station jointly encodes independent information for all users, as all
transmitting antennas are co-located. Remote terminals are scattered geographically, so they cannot
coordinate.
7
Given the instantaneous channel realization (H1, · · · , HK), a downlink power allocation policy is
a mapping
P ′ : (H1, · · · , HK) 7→ S ′(H1, · · · , HK), (9)
where S ′(H1, · · · , HK) is the transmit covariance matrix defined as:
S ′(H1, · · · , HK) = E[X ′X ′∗|H1, · · · , HK]. (10)
The sum-rate optimum downlink transmission strategy is a power control policy that maximizes the
sum capacity of the broadcast channel subject an average power constraint P ′,
EH1··· ,HK[tr(S ′(H1, · · · , HK))] ≤ P ′. (11)
The sum capacity of the broadcast channel is not a simple mutual information expression. However,
it turns out that via a so-called uplink-downlink duality, the broadcast channel problem can be
transformed into a multiple access channel problem, and both uplink and downlink problems have
the same solution.
C. Main Result
Intuitively, as each user experiences a different channel fading state, the sum-rate maximizing
strategy should involve only a subset of good channels in each time instant. This is true not only in
the downlink where the base-station is free to allocate power to the best users in each time instant,
but also in the uplink direction because each user has the freedom to choose the best time instants
to transmit. The objective of this paper is to give a rigorous analysis of the structure of the optimal
power control algorithm and to provide a theoretical upper bound on the maximum number of active
users.
The main result of the paper is the following: Consider a multiuser multi-antenna fading channel
with n antennas at the base-station, K remote users each equipped with m receive antennas and
with i.i.d. Gaussian fading coefficients and i.i.d. Gaussian noise. With probability one, the sum-
capacity achieving transmission scheme in both uplink and downlink directions involves up to n2
data streams in total, with each remote terminal using up to m2 data streams. In particular, at
most n2 users can be active at any given time.
Two simplifying assumptions are made in deriving the above result, First, perfect and instan-
taneous channel state information is assumed to be available at both the base-station and remote
terminals. Second, the sum rate is used as the criterion in the analysis. Although a real system
design has to take into account channel estimation error and the performance tradeoff among the
users, these two assumptions allow an amenable framework under which theoretical analysis can be
carried out. The analysis also provides useful guidance to practical system design.
8
III. Optimal Uplink Power Control
A. Simultaneous Water-filling
We start the development by considering first the uplink direction. Let ν be a random variable
representing the channel fading state with a probability density function ρ(ν). Denote the channel
fading distribution as Hk(ν), and the power allocation strategy as Sk(ν). The ergodic sum capacity
maximization problem (7) can be written down as follows:
maxSk(ν)
∫
ν
log
∣∣∣∣∣K∑
k=1
Hk(ν)Sk(ν)H∗k(ν) + I
∣∣∣∣∣ dρ(ν) (12)
s.t
∫
ν
tr(Sk(ν))dρ(ν) ≤ Pk, k = 1, · · · , K (13)
Sk(ν) ≥ 0, (14)
where | · | denotes matrix determinant, the noise covariance matrix is assumed to be an identity
matrix, and the entropy expression for Gaussian random vectors H(X) = log ((2πe)n|E[XX∗]|) is
used. Here, Sk(ν) ≥ 0 is taken to mean that Sk(ν) is a positive semi-definite matrix. Equations (13)
and (14) need to be satisfied for all k = 1 · · ·K and for all fading states ν.
Observe that the objective function in (13) is concave and the constraints are linear. Thus, it
belongs to the class of convex optimization problems for which numerical solution can be efficiently
computed. The goal of this section is to characterize the form of the optimal solution by further
exploiting the problem structure.
In a single-user channel (i.e. K = 1) with constant fading, it is well-known that the optimal
transmit covariance matrix S is the water-filling covariance over the set of singular values of the
channel H. With fading, the optimal covariance S(ν) must water-fill in both time and space. A
set of channel singular values need to be computed in each time instant, and the same water-
filling level must be set over both time and space. The main result of this section is based on
a natural generalization of these ideas to the multiuser setting. The key is to recognize that the
optimal solution must satisfy the water-filling condition over both space and time for every user
simultaneously.
Consider the optimization problem (13). Clearly the global optimal solution {Sk(ν)}Kk=1 must
be such that each Sk(ν) is a single-user “water-filling” power allocation against the noise and the
combined interference from all other users, as otherwise, Sk(ν) can be optimized to improve the
global objective. This must be true for Sk(ν) for each k = 1, · · · , K. Conversely, if the single-user
water-filling condition is satisfied for every user k = 1, · · · , K, because the optimization problem
is concave over the positive semidefinite matrices Sk(ν), the local optimum is also guaranteed to
be globally optimal. Thus, the set of Sk(ν) is the solution to (13) if and only if each Sk(ν) is the
single-user water-filling covariance against combined interference and noise.
9
The above discussion can be made more rigorous by writing down the Karush-Kuhn-Tucker (KKT)
condition for the optimization problem. Associate dual variables λk to each power constraint and
Φk(ν) to each positivity constraint, where λk is a scalar, and Φk(ν) are n×n matrices. We have the
following theorem.
Theorem 1: A power control strategy Sk(ν) maximizes the sum ergodic capacity for a fading
multiple access channel (1) with perfect side information at all the transmitters and at the receiver
if and only if each Sk(ν) is the single-user water-filling power allocation against the combined noise
and interference from all other users. More formally, the set of Sk(ν) must satisfy the following set
of equations at each fading state ν and for each user k:
λkI = H∗k(ν)
(K∑
j=1
Hj(ν)Sj(ν)H∗j (ν) + I
)−1
Hk(ν) + Φk(ν)
∫
ν
tr(Sk(ν))dρ(ν) ≤ Pk (15)
tr(Φk(ν)Sk(ν)) = 0 (16)
Φk(ν), Sk(ν), λk ≥ 0, (17)
where I is the n × n identify matrix.
The proof for Theorem 1 is a straightforward generalization of a similar result in [14] where the
non-fading case is treated. The key is to recognize that the KKT conditions naturally separate
into K groups of single-user water-filling conditions, one corresponding to each user. The only
modification from the single-user case is that the interference from all other users is now regarded
as additional noise.
Note that the power control strategy is a function of the channel fading state, but the water-filling
level λk is a function of the fading distribution only, which can be pre-computed. Given a fixed λk,
the power allocation at each time instant can be readily derived. The fixed λk itself is computed
from the statistics of the channel fading states to ensure that the total power constraint is satisfied.
In fact, as shown in [14], this simultaneous water-filling condition leads to an iterative water-filling
procedure to compute the optimal Sk(ν). Starting from any initial power allocation, each step of
the iterative procedure consists of a single-user water-filling for Sk(ν) with all other users’ signals
regarded as noise. The iterative procedure goes through all users. The set of covariance matrices
Sk(ν) eventually converges to the optimum solution to (13).
We are now ready to derive the main result of the paper by focusing on the set of KKT conditions
(15)-(17).
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B. Single-Antenna Case
When there is only one antenna for each transmitter and for the receiver, [2] showed that the sum-
rate maximizing power control strategy is a TDMA-like strategy where a single-user with the best
channel transmit at every moment. This result will be re-derived here using the simultaneous water-
filling interpretation, thus setting the stage for subsequent development where multiple antennas are
introduced.
In the single antenna case, the KKT condition simplifies to the following:
λk =h2
k(ν)∑Kj=1 h2
j(ν)sj(ν) + σ2+ φk(ν) (18)
∫
ν
tr(Sk(ν))dρ(ν) ≤ Pk (19)
φk(ν)sk(ν) = 0 (20)
φk(ν), sk(ν), λk ≥ 0. (21)
Without the φk(ν) term, equation (18) is the familiar water-filling condition, with water level equal
to λ−1k . The slack variable φk(ν) is used to account for the possibility that a fading state may be
sufficiently weak so that no power is allocated for that state. In that case, a positive slack variable
φk(ν) is used to make up the difference. Note that the slack variable can only be non-zero when
sk(ν) = 0. This is also reflected in the matrix case as (16).
Now, for two users k and l to both transmit at a fading state ν, they must both satisfy the
single-user water-filling condition:
λk =h2
k(ν)∑Kj=1 h2
j(ν)sj(ν) + σ2(22)
λl =h2
l (ν)∑Kj=1 h2
j(ν)sj(ν) + σ2(23)
The denominator for the two conditions are the same, so if both users transmit, then
h2k(ν)
λk=
h2l (ν)
λl. (24)
In other words, the fading state ν may not be shared by the two users unless the channel gains differ
by exactly the factor λk/λl. Since the channel fading state is assumed to be i.i.d. complex Gaussian
distributed, such event has zero probability. Therefore, we have proved the following:
Lemma 1 (Knopp and Humblet [2]) : In a single-antenna multiple access fading channel with i.i.d.
Gaussian fading statistics, assuming perfect side information at the transmitters and the receiver,
with probability one, the sum capacity is achieved with a power control strategy that allows only
one user to transmit at a time.
11
This same conclusion was reached earlier by Cheng and Verdu [16] in the context of multiple
access channel with intersymbol interference. The power allocation problem for the fading channel
is identical to the loading problem for the ISI channel if the fading statistics is assumed to be i.i.d.,
and if the ISI channel is equipped with guard periods which ensure the orthogonality of subchannels.
C. Multiple-Antenna Case
The intuition for Knopp and Humblet’s result is the following. A single-antenna receiver is limited
by the single degree of freedom it has. To achieve the sum capacity, only one user can transmit at
a time. With multiple antennas however, multiple dimensions may be available. So, the optimal
power control strategy may involve more than one user transmitting at the same time. Nevertheless,
the maximum number of simultaneous users is still related to the number of antennas. This intuition
is made precise in the following theorem:
Theorem 2: In a multiple-antenna multiple-access fading channel with i.i.d. complex Gaussian
channel matrices, assuming perfect channel side information at all the transmitters and at the
receiver, the optimal power control strategy that achieves the maximum sum ergodic capacity has
the following property: with m antennas for each user and n antennas at the base-station, at any
time instant, the rank of transmit signal rk for each users must satisfy∑
k r2k ≤ n2. In particular, a
maximum of n2 users can transmit simultaneously.
Theorem 2 is applicable when H has complex entries. When H has real entries, the results need
to be modified as follows: The rank bound becomes∑
k rk(rk + 1)/2 ≤ n(n + 1)/2. The maximum
number of simultaneous users becomes n(n + 1)/2. This is also the case for Theorem 4 presented
later in the paper.
Proof of Theorem 2: First, consider the case in which each transmitter has one antenna and the
receiver has n antennas, so that the channel matrix Hk(ν) is an n × 1 vector and the transmitter
covariance is just a scalar sk(ν), and the slack variable is also a scalar φk(ν):
λk = H∗k(ν)
(K∑
j=1
sj(ν)Hj(ν)H∗j (ν) + I
)−1
Hk(ν) + φk(ν).
The claim is that at any fading state ν, only a maximum of n2 users can have sk(ν) > 0 and
φk(ν) = 0. The rest of the users must have sk(ν) = 0 and φk(ν) > 0. As in the single-antenna case,
the key is to recognize that the matrix inversion in the expression is common to all users. Recall
that λk is determined by the channel fading distribution, so it can be considered fixed. We first ask
whether there exists a positive definite symmetric matrix M such that H∗kMHk = λk for more than
n2 Hk’s. The following Lemma answers this question.
12
Lemma 2: Fixing positive λ1, · · · , λK, let H1, · · · , HK be n× 1 random vectors whose entries are
i.i.d. complex Gaussian variables. If K > n2, then with probability one, there does not exist a
positive definite symmetric matrix M such that H∗kMHk = λk, ∀k = 1 · · ·K.
Proof: Let Hk = [hk1 hk2 · · · hkm]T . Denote the (i, j) entry of M by Mij. M is a positive semi-
definite symmetric matrix with complex entries. Because of the symmetry, Mij = Mji, so there are12n(n + 1) independent complex variables in M . However, the diagonal entries are real. So, there
are effectively n2 real independent variables in M . To have H∗kMHk = λk, we need
∑
ij
hkihkjmij = λk, (25)
for all k = 1 · · ·K. Because Hk are i.i.d. Gaussian, with probability 1, these K equations are linearly
independent. So a solution to (25) exists only if K ≤ n2. 2
Lemma 2 shows that the number of users that can transmit simultaneously is n2 or fewer. It does
not guarantee that exactly n2 users will transmit because the existence of a matrix M satisfying
(25) does not guarantee that such an M can be synthesized by sk(ν) as in (25).
Next we turn our attention to the case where remote transmitters are equipped with multiple
antennas. In this case, the water-filling level is now an m × m identity matrix, and the transmitter
power spectrum and the slack variables are both m × m positive semidefinite matrices. The water-
filling condition is:
λkI = H∗k(ν)
(K∑
j=1
Hj(ν)Sj(ν)H∗j (ν) + I
)−1
Hk(ν) + Φk(ν), (26)
for k = 1 · · ·K. Parallel to the previous development, we ask: Does there exist a positive semidefinite
matrix M that satisfies λkI = H∗k(ν)MHk(ν) + Φk(ν)? The idea is again to count the number of
independent equations and the number of unknowns. To satisfy the matrix equation, we need to
satisfy one equation for each matrix entry. By symmetry, there are 12m(m+1) independent complex
entries for each k. However, as the diagonal entries of the matrix equations are automatically real,
the total number of independent real equations is Km2.
The number of unknown variables is counted as follows. Again, the matrix M introduces n2
degrees of freedom. The number of unknowns introduced by the slack variable Φk(ν) depends on its
rank. Φk(ν) are m×m matrices. An m×m complex symmetric matrix with real diagonal entries has
at most m2 degrees of freedom. However, if the matrix is restricted to rank r, the number of degrees
of freedom decreases to m2 − (m − r)2. To see this, recall that a positive semidefinite symmetric
matrix can be represented by its Cholesky factorization as LL∗, where L is triangular. The number
of degrees of freedom can be thought of as the number of independent variables in the triangular
matrix L. The diagonal terms are always real. The off-diagonal terms are complex. Thus, the total
13
number of independent entries is:
2
(m∑
i=1
i
)− m = 2
m(m + 1)
2− m = m2. (27)
If an m × m matrix is of rank r, its Cholesky factor is an m × r triangular matrix. Again, the
(m − r) diagonal entries are always real, and the off-diagonal terms may be complex. So, the
number of degrees of freedom is:
2
(m∑
i=r
i
)− (m − r) = m2 − (m − r)2. (28)
Now, the slack variables need to satisfy the complementary slackness condition
tr(Φk(ν)Sk(ν)) = 0. (29)
So, if the transmit signal Sk(ν) is of rank rk, the rank of Φk(ν) is at most m − rk. Therefore, each
Φk(ν) introduces at most m2 − r2k extra degrees of freedom. The total number of unknown variables
is then n2 coming from the matrix M plus m2 − r2k coming from each of Φk(ν).
The matrix equations involve the channel realization Hk(ν), which are Gaussian random matrices.
So, with probability one, these equations are independent. Thus, for a solution to exist, there need
to be at least as many unknown variables as there are equations, so:
n2 +K∑
k=1
(m2 − r2
k
)≥ Km2, (30)
from which the condition∑
k r2k ≤ n2 follows.
At any time instant, a user transmits with positive power if the rank of its transmitted signal is
at least 1. Therefore, in a multiple access scenario with n receive antennas, a total of n2 users can
transmit at the same time. The power control strategy can be thought of as choosing the “best”
set of n2 users, when transmitting together (using power determined by the fading state and their
respective water levels), provides the highest sum capacity. This concludes the proof. 2
Theorem 2 establishes an upper bound on the number of simultaneous users that can transmit si-
multaneously in a multiple access channel. Although the theorem does not guarantee that the bound
is tight, simulation results indicate that the maximum number of simultaneous users is somewhere
between n and n2.
IV. Optimal Downlink Power Control
A. Gaussian Vector Broadcast Channel
We now proceed to derive a counterpart of the previous result for the downlink multi-antenna
channel. The downlink solution relies on recent results on the sum capacity of Gaussian vector
14
broadcast channels [17] [18] [19] [10], and on a duality of the uplink and downlink channels [9] [10]
[11]. This section provides a brief overview of relevant information theoretical results first, then
proceeds to show that the exact same dimension counting argument is applicable to the downlink
multi-antenna channel as well.
Consider first the non-fading broadcast channel
y′k = H ′
kx′ + n′
k, k = 1 · · ·K. (31)
The transmitter is equipped with n antennas, so x′ is an n×1 vector. The remote users are equipped
with m antennas each, so y′k are m×1 vectors. H ′
1, · · ·H ′K are m×n matrices. The transmit signal is
subject to a power constraint E[X′TX′] ≤ P . Independent information is to be transmitted to each
user. The capacity region refers to the set of rate tuples (R1, · · · , RK) simultaneously achievable by
users 1 to K with arbitrarily small probabilities of error. We focus on the sum rate R1 + · · ·+ RK .
The broadcast channel capacity is a long-standing problem in network information theory. The
capacity region for a broadcast channel is known only in several special cases (such as the degraded
broadcast channel.) The Gaussian vector broadcast channel considered in this paper where the
transmitter and receivers are equipped with multiple antennas is not necessarily degraded, and the
capacity region is still not known completely. However, several recent work [17] [18] [19] [10] has
successfully solved the sum capacity case.
There are two simultaneous and independent approaches to solving the sum capacity problem.
A key ingredient in both solutions is a connection between the broadcast channel and channels
with side information, first published in [17]. In a classic result known as “writing on dirty paper”,
Costa [20] showed that if a Gaussian channel is corrupted by an interference signal s that is known
non-causally to the transmitter but not to the receiver, i.e.
y = x + s + n, (32)
the capacity of the channel is the same as if s does not exist. Thus, in a broadcast channel, if we let
x′ = x1 + · · ·+xK, where xk is intended for the kth user, the capacity of the channel from xk to yk is
as large as if multiuser interference x1, · · · ,xk−1 can all be pre-subtracted. This precoding strategy
turns out to be optimal for achieving the sum capacity in a Gaussian broadcast channel. This was
proved for the 2-antenna case by Caire and Shamai [17], and has since been generalized by several
authors [18] [19] [10] using two different approaches. In the rest of this section, the decision-feedback
approach [18] is outlined first. Then, a new derivation for the duality approach (which differs from
the original approaches given in [19] and [10]) is presented.
B. Precoding
The approach in [18] is based on the observation that multiuser interference pre-subtraction at
the transmitter is similar to a decision-feedback equalizer (DFE) at the receiver. The interfer-
15
ence cancelation operation can in effect be “moved” to the transmitter. However, although the
decision-feedback structure achieves the capacity of a Gaussian vector channel, it also requires coor-
dination among the receivers because the DFE structure has a feedforward matrix that operates on
all of y′1, · · · ,y′
K . Clearly, such coordination is not possible in a broadcast channel. However, pre-
cisely because y′1, · · ·y′
K cannot coordinate, they are also ignorant of the noise correlation between
n′1, · · · ,n′
K. Thus, the sum capacity of the broadcast channel must be bounded by the cooperative
capacity with the worst possible noise correlation, i.e.
C(d) ≤ minSnn
I(X′;Y′1, · · ·Y′
K), (33)
where Snn is the covariance matrix for n = [n′T1 · · ·n′T
K]T , and the minimization is over all Snn whose
kth block diagonal term is the covariance matrix of n′k. This outer bound is due to Sato [21]. Sato’s
outer bound can be evaluated explicitly by writing down the Karush-Kuhn-Tucker (KKT) condition
for the minimization problem:
S−1nn − (HSxxH
T + Snn)−1 =
Ψ1 0. . .
0 ΨK
, (34)
where H = [H ′1T · · ·H ′
KT ]T and Sxx is the transmit covariance for x′ and Ψk is the Lagrangian dual
variables corresponding to the diagonal constraints. Interestingly and surprisingly, S−1nn −(HSxxH
T +
Snn)−1 also corresponds to the feedforward matrix of the decision-feedback equalizer. So, if the
noise covariance happens to be the worst-case noise, the feedforward matrix of the decision-feedback
equalizer would be diagonal. Thus, after moving the feedback operation to the transmitter, the
entire equalizer de-couples into independent receivers for each user, and no coordination is needed
whatsoever. Consequently, Sato’s outer bound is achievable. Now, this achievable rate may be
further maximized over all Sxx subject to the power constraint. Therefore, the sum capacity of a
Gaussian vector broadcast channel is:
maxSxx
minSnn
1
2log
|HSxxHT + Snn|
|Snn|(35)
subject to S(i)nn = I, i = 1, · · · , K,
tr(Sxx) ≤ P,
Sxx, Snn ≥ 0,
where S(i)nn refers to the ith block-diagonal term of Snn.
C. Uplink-Downlink Duality
The sum capacity of a Gaussian vector broadcast channel can be solved using a completely different
method. In [9], it was observed that the achievable region of a broadcast channel using the precoding
16
technique is exactly the same as the capacity region of a dual multiple access channel with the channel
matrix transposed and a sum power constraint applied to all inputs. This uplink-downlink duality
is closely related to convex duality. Based on the duality, [19] and [10] showed that the sum capacity
of the broadcast channel is precisely the sum capacity of the dual multiple access channel. The
proof also relies on Sato’s outer bound. Duality is essential in our characterization of the structure
of the optimal transmission strategy in the broadcast channel. In the rest of the section, we give a
derivation of duality that is different from that of [9] and [10].
To simplify matters, assume that H is square and invertible. (The derivation can be generalized
to arbitrary channels. See [22] and [11] for details.) The starting point of the new derivation is
the minimax optimization problem (35). The objective function of the problem is concave in Sxx
and convex in Snn, so its KKT conditions completely characterize the saddle points. The KKT
conditions are:
HT (HSxxHT + Snn)−1H = λI (36)
S−1nn − (HSxxH
T + Snn)−1 = Ψ (37)
where λ is the dual variable associated with the power constraint and Ψ = diag(Ψ1, · · · , ΨK) is the
dual variable associated with the diagonal constraint. Multiple (37) by HT on the left and H on the
right, substitute in (36). We obtain:
HT S−1nn H = HTΨH + λI. (38)
The above is equivalent to the following condition:
H(HTΨH + λI)−1HT = Snn, (39)
where the transmit covariance matrix is a diagonal matrix Ψ, and the dual variable associated with
the constraint has 1’s on the diagonal (as the worst noise Snn has 1’s on the diagonal.) This condition
is precisely the KKT condition for a sum-power multiple access channel. After a proper scaling of
the power constraint, Ψ, λ and Snn, it can be shown that the KKT condition of the following
optimization problem is exactly (39):
maxD
1
2log |HTDH + I| (40)
s.t. D is diagonal
tr(D) ≤ P,
D ≥ 0,
(where D = Ψ/λ.) Thus, the solution to the minimax problem (35) is just the solution to a
maximization problem (40). Since D is a diagonal matrix, the maximization problem above solves
17
the sum capacity of a multiple-access channel. This establishes the duality of a multiple access
channel and a broadcast channel. Note that this multiple access channel differs from that considered
in the previous section in that the power constraint is applied across all users, rather than to each
user individually.
Uplink-downlink duality gives us a convenient method to compute the optimal power allocation
for a downlink channel. From (36) and (37), it is not difficult to derive an expression for Sxx based
on the dual variables:
Sxx = (λI)−1 − (HT ΨH + λI)−1. (41)
In fact, the worst-case noise can also be found easily via (39). Thus, the basic procedure for obtaining
the optimal power allocation reduces to the following. First, find a primal-dual solution (D, λ) to
the multiple-access channel (40). Set Ψ = λD. Then, the optimal transmit covariance matrix Sxx
may be found via (41).
D. Optimum Downlink Transmission Strategy
We are now ready to provide counterparts of Theorem 1 and Theorem 2 for the downlink channel.
Again, let ν be a random variable representing the fading state. The following two statements first
establish the optimal power allocation for the broadcast fading channel, then provide a theoretical
upper bound for the maximum number of active users for the downlink using a dimension-counting
argument:
Theorem 3: The power control strategy S ′(ν) that maximizes the sum ergodic capacity for a fading
broadcast channel (8) with perfect side information at the transmitter and at all the receivers can
be found as follows:
S ′(ν) = (λI)−1 − (H∗λD(ν)H + λI)−1, (42)
where D(ν) is a diagonal matrix with diagonal terms D1(ν), · · ·DK(ν) that satisfy the optimality
condition for the dual multiple-access channel. The optimality condition for the dual channel is:
λI = Hk(ν)
(K∑
j=1
H∗j (ν)Dj(ν)Hj(ν) + I
)−1
H∗k(ν) + Φk(ν)
∫
ν
K∑
j=1
tr(Dj(ν))dρ(ν) ≤ P (43)
tr(Φk(ν)Dk(ν)) = 0 (44)
Φk(ν), Dk(ν), λk ≥ 0. (45)
Further, when the optimal precoder is used, λDk(ν) is precisely the optimal receiver matrix at each
remote user.
18
Theorem 4: In a multiple-antenna broadcast fading channel with i.i.d. complex Gaussian channel
matrices, the optimal transmission strategy with perfect channel side information at all transmitters
and at the receiver that achieves the maximum sum ergodic capacity has the following property:
With m antennas for each user and n antennas at the base-station, at any time instant, the rank of
receiver matrix rk for each users must satisfy∑
k r2k ≤ n2. In particular, a maximum of n2 receivers
can be active simultaneously.
The proof of Theorem 3 is a straightforward generalization of the development in the previous
section. As the previous section shows, the dual of a broadcast channel is a multiple-access channel
with sum power constraint across all users. In a fading environment, the same duality holds. The
power constraint is now across all users and across all fading states. Consequently, the only difference
between the optimality condition for a broadcast fading channel and that of a multiple access fading
channel is that the same water-filling level λ is now fixed for all users and for all fading states at the
same time.
The proof of Theorem 4 also essentially follows along the same line as the proof of Theorem 2. As
the water-filling level λ is fixed by the fading distribution, the same dimension counting argument
can be made for the dual multiple-access channel. The analysis of the number of variables versus the
number of equations reveals that the rank of Dk must satisfy∑
k r2k ≤ n2. Now, the decision-feedback
equalization interpretation of the precoding strategy reveals that λDk is precisely the receiver matrix
at the remote user. Thus, the rank bound for the dual multiple-access channel translates directly
to the rank bound for the receiver matrix. In particular, at most n2 receivers can be active at any
given time. This establishes Theorem 4.
V. Spatial Diversity vs Multiuser Diversity
In both the multiple-access fading channel and the broadcast fading channel, the number of
degrees of freedom created by multiple antennas is bounded by a quadratic function of the number
of antennas. With n antennas at the base-station, the maximum number of degrees of freedom at the
base-station is theoretically upper bounded by n2. These many degrees of freedom are to be divided
among the remote users. Each remote user, with m antennas, can potentially use up to m2 degrees
of freedom. Thus, the number of base-station antennas has the effect of allowing more users to
transmit simultaneously, while the number of antennas at the remote users has the opposite effect.
The remote antennas theoretically have the potential to crowd out receiver dimensions and thus
prevent other users from transmitting at the same time. Such crowding-out potentially increases
system sum capacity at the expense of delay and unfairness. It should be noted that such crowding-
out, although theoretically possible, rarely happens in an i.i.d. Gaussian channel. As was pointed
out in [23], the remote users are most likely to employ a beamforming strategy which effectively
reduces each remote user to a single-dimension transmission or receiving terminal.
19
When the base-station is equipped with a single antenna only, Theorem 2 and 4 reduce immediately
to the result in [2] for the uplink channel and to the well-known degraded broadcast channel for the
downlink. The main contribution of this paper is a generalization of these results to the situation
where the base-station is equipped with multiple antennas. In this case, there appears to be an
interesting interplay between spatial diversity and multiuser diversity. In a single-user vector channel,
the maximum number of usable dimensions is the rank of the channel matrix. The rank is the number
of independent data streams that a vector channel can process. In a multiuser systems however, the
maximum number of independent users that the system can accommodate is bounded by the square
of the rank. The difference is that in a multiuser environment, no signal coordination among the
users is possible, so spatial dimensions for each user overlap. Consequently, more users are cramped
into the limited number of dimensions. As the simulation results in the next section show, the
optimal number of active users is typically four times the number of base-station antennas in a high
SNR-regime.
It should be noted that the maximum number of users derived in this paper is a theoretical upper
bound only. In several recent results [24] [25], independent simulations also showed that a suboptimal
scheme with only as many active users as the number of base-station antennas might lose only a
small fraction of sum capacity. Thus, it appears that the sum capacity may not be very sensitive to
the number of active users as long as at least n dimensions are used. This is true particularly when
n goes to infinity. In an earlier result, [15] showed that in the asymptotic regime where the number
of users and the number of receive antennas both go to infinity (while keeping their ratio fixed), a
single-user water-filling power allocation strategy is essentially optimal. We expect a similar result
to hold for the downlink.
VI. Simulation Results
Theorem 2 and 4 provide a theoretical upper bound on the total number of active users in a multi-
user multi-antenna wireless channel. Simulation results suggest that this upper bound is tight when
the number of antennas at the base-station is small, but becomes loose as the number of antennas is
large. Fig. 2 and Fig. 3 show simulation results on a Rayleigh fading channels with multiple remote
users each with a single-antenna and a base-station with varying number of transmit antennas. The
optimization problem for the uplink is solved using the iterative water-filling algorithm [14]. The
corresponding downlink problem is solved using a combination of iterative water-filling and a nu-
merical approach called dual decomposition [26]. The number of active users versus the number of
transmit antennas over many realizations of the channel is plotted in Fig. 2. It is seen that although
the theoretical upper bound on the number of active users goes up quadratically with the number of
antennas, the actual number of active users goes up roughly linearly. Note that the optimal number
of users is significantly larger than the number of base-station antennas. In particular, the square
20
2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
Number of Base−Station Antennas
Num
ber
of A
ctiv
e U
sers
Fig. 2. The number of active users vs the number of base-station antennas over random realizations of the channel.
bound is achieved when the number of base-station antennas is small. For example, with 2 antennas
or 4 antennas at the base-station, the number of active users can be as high as 4 or 15, respectively.
Fig. 3 shows a histogram of the number of active users over 500 random realizations of the channel
for the cases of 2 to 16 base-station antennas. There is substantial variation among different real-
izations of the channel. However, we see that when the number of base-station antennas is large,
the number of active users is almost always several times higher than the number of antennas.
VII. Concluding Remarks
When the base-station in a cellular system is equipped with a single antenna, the sum rate
maximizing transmission strategy consists of only a single active user. Essentially, only the user with
the best channel is being served at any given time. The main point of this paper is to show that
with multiple antennas at the base-station, the sum-rate maximizing strategy consists of multiple
simultaneous active users. Theorems 2 and 4 make a precise dimension counting argument that
gives a theoretical upper bound on the maximum number of active users. There is an interesting
interplay between spatial diversity and multiuser diversity. In a random propagation environment,
a downlink broadcast channel with n transmit antennas has up to n2 spatial dimensions. These
dimensions are divided among the remote terminals. A remote terminal with m receive antennas
can utilize up to m2 dimensions. The dimensions are additive, and the total number of dimensions
21
0 1 2 3 4 5 6 7 80
50
100
150
200
250
300
Number of active users: n=2
Dis
trib
utio
n
0 5 10 15 20 25 300
20
40
60
80
100
120
140
Number of active users: n=4
Dis
trib
utio
n
0 10 20 30 40 50 600
20
40
60
80
100
120
140
160
Number of active users: n=8
Dis
trib
utio
n
0 10 20 30 40 50 60 70 80 900
20
40
60
80
100
120
Number of active users: n=16
Dis
trib
utio
n
Fig. 3. Histograms of the number of active users over 500 random realizations of the channel. Top-left, top-right,bottom-left and bottom-right figures correspond to the cases of 2, 4, 8 and 16 base-station antennas respectively.
is upper bounded by n2. This same result applies to both uplink and downlink fading channels.
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