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Downlink Optimization of Indoor Wireless
Networks Using Multiple Antenna Systems
Bongyong Song, Rene L. Cruz and Bhaskar D. Rao
Department of Electrical and Computer Engineering
University of California, San Diego, La Jolla, CA 92023-0407
Abstract - We compare the performance of two
multiple antenna systems to be used in quality of
service (QoS) supported indoor wireless networks.
While a conventional array antenna system (AAS)
has collocated, closely spaced antenna elements,
a distributed antenna system (DAS) has largely
spaced antennas over the entire area of radio
coverage. To support multimedia applications
requiring high bandwidth and on time delivery,
we propose a set of highly spectrum efficient radio
resource management algorithms. We focus on
the optimization of downlink since many kinds of
Internet traffic show the downlink dominance in
their traffic asymmetry. To maximize the downlink
throughput, we present a new transmit beamforming
algorithm which can be equally applied to both DAS
and AAS. The beamforming algorithm is integrated
with a link scheduling algorithm that exploits
the space division multiplexing (SDM) capability
of multiple antenna systems to meet the QoS
requirements of all terminals. Numerical examples
conducted for a line of sight (LOS) environment
demonstrate that a network with DAS outperforms
one with AAS in terms of signal coverage and
provides 40 - 160% higher capacity.
Index Terms - Multiple Antenna Systems, Beamforming, Power Control, Link Scheduling, Duality
I. I NTRODUCTION
With the recent dramatic growth in wireless communications together with the Internet, technologies in wireless communications and networking are being advanced
with the goal of delivering multimedia applications and
services, at anytime, anywhere and on any devices.
This rapid growth of the untethered multimedia demand
facilitates the migration of current wireless networks into
broadband networks. Since many kinds of Internet traffic
This research was supported in part by the National Science
Foundation Grant No. ANI-0123421.
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show the downlink dominance in their traffic asymmetry,
maximizing downlink capacity is particularly important.
CDMA/HDR [1] is designed to provide the downlink
capacity of up to 2.4Mbps and high speed downlink
packet access (HSDPA) mode of UMTS aims at over
10Mbps downlink capacity. The latest wireless local
area network (WLAN) standards (802.11a/g and HIPERLAN/2) support up to 54Mbps.
High demand for capacity can be handled by various
techniques in communication systems. Utilizing multiple
antennas is one of the most promising physical layer
approaches to increasing system capacity. A conventional
array antenna system (AAS) has an array of antenna
elements closely spaced altogether and an array processor computes a beam pattern that increases signal
power and reduces interference at the receiver. Due to
the interference suppression capability of an array, space
division multiplexing (SDM) in downlink and space
division multiple access (SDMA) in uplink are attainable. Beamforming at a transmitter is in general more
difficult than beamforming at a receiver since a transmit
beamforming affects the performance of all receivers
via interference whereas beamforming at a receiver can
be executed independently. However, higher capacity
demand for downlink necessitates more efficient transmit
beamforming at an access point or a base station. The
zero-forcing approach proposed by Gerlach and Paulraj
[2] maximized the individual SNRs while placing nulls
to all directions causing interference. Although computationally efficient, zero-forcing can result in reduced
signal power at the receiver and, limits the order of SDM
to the number of antenna elements. Rashid-Farrokhi, et
al. [3] found an iterative joint beamforming and power
control solution for reaching specified SINR levels at
each receiver with the minimum total transmit power for
arbitrary number of users. Another iterative algorithm
developed by Montalbano and Slock [4] finds a solution
that maximizes the minimum signal to interference ratio
(SIR) of all users. The convergence of the algorithm
is provided in [5]. Recently, Koutsopoulos, et al. [6],
proposed a set of iterative heuristics attempting to amIEEE INFOCOM 2004
plify the signal and suppress the interference signals. A
more analytic approach by Wong, et al. [7] addressed the
problem of maximizing the total downlink throughput
in multiple-input and multiple-output (MIMO) context.
They provided a closed-form beamforming solution that
maximizes the lower bound of the total downlink capacity. They computed the array weight of each link
independently and did not jointly optimize the beam
patterns of different users. In this paper, we propose
an iterative algorithm for maximizing total downlink
capacity that jointly optimizes the array weights of all
users.
Another domain of multiple antenna systems is a
distributed antenna system (DAS). In a DAS, multiple
antennas belonging to a single access point/base station
are dispersed across the entire coverage area. Those
antennas are connected to a central tranceiver either
via coaxial cables or optical fibers. This approach was
originally proposed to provide a uniformly good signal
coverage. Saleh et al. [8] equally distributed downlink
power to all antennas and obtained dramatic reduction
in signal attenuation and multipath delay spread. The
virtual cellular network approach proposed by Kim
and Linnartz [9] increases spectral reuse efficiency by
communicating with a user via the antenna port that
best serves the user. In addition to the reduced signal
attenuation, DAS provides another degree of freedom
for fully utilizing the spatial resources. A large spatial
distribution leads to fully uncorrelated channels among
different antennas and the maximum degree of spatial
diversity is facilitated which is called macrodiverstiy.
When applied to CDMA networks where fast power control is incorporated, macrodiversity reception contributes
to a large reduction in the dynamic range of transmitters
and ultimately increases the network capacity [10].
In this paper, we consider a more advanced utilization of a DAS. We apply different complex weights to
different antennas. The allocation of downlink power
(magnitude) and the phase angles are determined such
that the signal is coherently summed up at the receiver
and destructively suppressed at interfering locations.
Different complex weights are used to different users
for space division multiplexing. By doing this we can
treat a DAS as a single spatially spread antenna array
and apply the same array processing principles being
used in conventional arrays. We equalize the different
propagation times in cables/fibers if necessary. Notice
that the delay equalization is not always necessary since
its purpose is to maintain the difference in propagation
delay sufficiently small compared to the modulation
symbol duration. Hence, in some narrowband context
such as a OFDM sub-carrier, the signals from different
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antennas can be coherently combined without delay
equalization.
Other physical layer strategies to achieve higher capacity are power control and link adaptation. Transmit
power level, spreading factor, modulation and coding
schemes are to be chosen so as to maximize the network capacity. Our proposed radio resource management
policy encompasses these capabilities.
To fully support multimedia communications, guaranteeing quality of service (QoS) is as important as
providing high capacity. An efficient radio resource
management mechanism aims at maximizing the radio
capacity while maintaining the QoS requirements of
users. Although the end-to-end QoS can be enhanced
using various technologies across all protocol layers,
we focus on the QoS support in the wireless hop of
a connection. Medium access control (MAC) in most
advanced communication systems provide the means
of supporting QoS by introducing priority and reservation schemes. In particular, 802.11x wireless local
area network (WLAN) standards define a contention
free period (CFP) and contention period (CP) in a radio
superframe [11]. Time-bounded services are periodically
polled during the CFP, and the remaining best-effort type
services are served during the CP. Hence, the goal of
a radio resource management policy can be interpreted
as minimizing the length of a CFP for a given set of
minimum data rate requirements. In our multiple antenna
context, this can be accelerated by fully utilizing the
SDM capability of an array. In this paper, we discuss
a link scheduling algorithm that jointly considers the
beamforming capability to minimize the length of a CFP.
This paper is organized as follows. In Section II, the
system model for a multiple antenna indoor wireless
network is introduced. Section III formulates the joint
beamforming, power control and scheduling problem
and derives optimal and suboptimal policies. Numerical
examples are then presented in Section IV. Finally,
Section V concludes with some closing remarks.
II. S YSTEM M ODEL
Consider an indoor wireless network composed of an
access point and N wireless terminals. We consider the
downlink of this network amounting to N individual
links. A single radio channel/bandwidth is shared by
multiple users and the channel access is coordinated
by an access point to support applications requiring
QoS. The aggregate QoS requirement from applications
running on terminal l is expressed by the aggregate minimum required downlink bandwidth Cl . Then the network
rate requirement vector is defined as C [C1 , . . . , CN ]T
where the superscript T denotes transpose operation.
IEEE INFOCOM 2004
2
:
s1 (t)
s l (t )
D
P1
.
.
.
Pl
.
.
.
s N (t)
1
Transmit
Array
Processing
PN
.
.
.
link
k
.
.
.
:
Q
D
:
l
Q
N
(a)
Fig. 1. Transmission Schematic at an Access Point. Array processor
computes an array weight for each link. Delay elements compensate
for different propagation delays in cables/fibers if necessary.
A. System Architecture
A multiple transmit antenna system of size Q is
used at an access point to enhance signal quality at
terminals and to facilitate the space division multiplexing
(SDM) of signals to different users. (In case of uplink,
it improves the signals received by an access point and
promotes the space division multiple access (SDMA).)
Fig. 1 illustrates the transmission schematic at an access point. N signals, s1 (t), . . . , sN (t), having powers
P1 , . . . , PN , respectively, are fed into the transmit array
processor. The transmitter is subject to a peak power
constraint P max , i.e.,
Pl ≤ P max and Pl ≥ 0 for all l.
(1)
l=1
The network power vector defined as P [P1 , . . . , PN ]T
represents the power control status of a network.
The transmit array processor maintains N transmit
weight vectors used for N different links. The weight
vector assigned to link l is denoted as vl and its l2 -norm
is always kept to be unity, i.e., vl 2 = 1. We construct
the network transmit weight matrix V = [v1 , . . . , vN ]
where the l-th column represents the transmit weight
vector applied to the signal designated to terminal l. We
consider two types of multiple antenna systems :
1) Array Antenna System (AAS)
2) Distributed Antenna Systems (DAS)
Fig. 2 contrasts the configurations of these two systems.
In an array antenna system (AAS, part (a)), Q antennas are collocated and separated only by half carrierwavelength or so. In a distributed antenna system (DAS,
part (b)), however, Q antennas are largely separated over
the indoor coverage. We assume the cabling in DAS
doesn’t cause any signal loss, which is reasonable in
the case of RF transport over optical fiber. We also
assume that the propagation time in the cable caused by
different cable length is perfectly compensated by delay
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:
.
.
.
link
N
1
:
Q
D
link
1
:
xl (t )
j
.
.
.
:
(b)
Fig. 2. Two multiple antenna configurations. (a) Array antenna
system (AAS) (b) Distributed antenna system (DAS).
compensation elements. Then, without loss of generality,
we can assume zero propagation delay in the cable for
both AAS and DAS. Also, both AAS and DAS are
subject to the power constraint given by (1).
B. Transmission Mode Scheduling Policy
A set T = {P, V} precisely describes the network
resource management status and we call it a transmission mode. A transmission mode scheduling policy, in
short a policy, is defined as the way we construct a
transmission frame by scheduling several transmission
modes. A transmission frame comprises a QoS period
and a non-QoS period. QoS period is dedicated to
supporting the minimum data rate requirements C. NonQoS period is used for different purposes - delivery of
best-effort traffic, provision of extra capacity, control
activities e.g. measurement of channel and/or delivery of
uplink traffic in case of time division duplexing (TDD)
networks. An access point computes M transmission
modes, T1 , . . . , TM , and use Tl for λl fraction of time
(1 ≤ l ≤ M ). This frame format is depicted in
Fig. 3. An optimal policy minimizes the length of the
QoS period and maximizes the length of the non-QoS
period for a given C. Since there are infinitely many
possible transmission modes, M can be an arbitrarily
large number. But it will shown later that no more than
N transmission modes are required in QoS period in an
optimal policy, i.e., M ≤ N . We denote the time fraction
used for the non-QoS period as λ0 . It is obvious then that
M
l=0 λl = 1. We associate an imaginary transmission
mode T0 = {0, V} with the non-QoS period. Having
P = 0 in T0 doesn’t actually mean that the access point
is not transmitting any signal during the non-QoS period.
Rather it means no power is being used to satisfy the
requirement C. A policy π is represented as
π = {T, λ}
(2)
where T = {T0 , . . . , TM }, λ = [λ0 , . . . , λM ]T and M is
integer.
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Non-QoS period
QoS period
T1
T2
1
2
...
DAS, a LOS channel is given by
2π
2π
√
√
hl = [ gl1 e−j λ dl1 , . . . , glQ e−j λ dlQ ]
TM
M
Fig. 3.
A transmission Frame comprising a QoS period and a
non-QoS period. (Ti : i-th transmission mode, λi : fraction of time
assigned to Ti )
C. Signal Representation
yl (t) =
where glj is the link gain from the j -th antenna element
to the terminal l and dlj is the distance between them.
We assume that all the signals are zero-mean, wide
sense stationary and signals from different sources are
independent. Based on this, the total received signal
power at terminal l is obtained as
Pyl
Assuming a flat-fading channel environment, the channel between an access point and terminal l is described
by a 1 × Q row vector hl where the j -th element of
hl represents the complex gain from the j -th transmit
antenna to the antenna at the receiver l. We further
assume a quasi-static channel where the variation of
channel is much slower than a frame duration. An
example is an office environment where the mobility of
both terminals and of reflectors is very small. We can
then safely assume that the channel is constant for the
duration of a frame. In the rest of this section, we’ll
focus on a single transmission mode in a frame and we’ll
suppress the transmission mode index and the frame
index unless otherwise stated.
Using the complex envelope notation, the composite
signal received at terminal l is given by
N Pi hl vi si (t) + nl (t)
(3)
i=1
=
Pl hl vl sl (t) +
Pi hl vi si (t) + nl (t)
i=l
where nl (t) is the ambient noise at terminal l. If the
channel provides a set of independent Rayleigh fading,
the elements of hl are circularly symmetric complex
Gaussian for both AAS and DAS. In a line of sight
(LOS) channel environment, another extreme scenario
having no fading at all, the channel vector in AAS is
given by
√
hl = gl aTQ (θl )
(4)
where aQ (θ) represents the array manifold vector describing the relative phase delay between Q array elements for the incoming signal at an angle θ, and θl
is the angle to the terminal l at an access point. If
λc 2
)
we further assume a free space pathloss, gl = ( 4πd
l
for some wavelength λc and the distance dl . The array
manifold vector for a standard uniform linear array
(ULA) with Q elements, for instance, is given by
Q−1
[. . . ej(n− 2 )π cos(θ) . . .]T , 0 ≤ n ≤ Q − 1. Here, we
modelled a signal as a narrowband plane wave. For a
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(5)
= E{| yl (t) |2 }
= Pl vlH hH
l hl vl +
(6)
2
Pi viH hH
l hl vi + σnl ,
i=l
where the superscript H denotes the conjugate transpose
operation. Hence, the signal to interference and noise
ratio (SINR) of link l can be written as
Γl =
=
Pl vlH hH
l hl vl
H
H
2
i=l Pi vi hl hl vi + σnl
Pl Gll
,
2
i=l Pi Gli + σnl
(7)
(8)
where Gij = |hi vj |2 . We refer to Gij as the effective link
gain since it reflects the multi-antenna gain as well as the
channel gain. Obviously, this quantity can vary from a
transmission mode to another as we apply different array
weights.
D. Link Adaptation
We assume that the data rate of a link is a monotonically non-decreasing function of the corresponding
SINR. We call this function a link adaptation function.
Many advanced communication systems adaptively adjust the data rate of a link as the link quality varies
[12],[13]. Though real world systems only provide a discrete set of link rates, we assume continuous link adaptation functions for the sake of mathematical tractability.
We consider two link adaptation functions commonly
used in the literature - linear and logarithmic. A linear
link adaptation mechanism adjusts the transmission rate
in proportion to the SINR [14], [15], [16]. In this model
which is appropriate for CDMA systems where data
rate is primarily adjusted by a spreading factor, the
transmission rate for link l is give by
Xl = W Γl
Pl Gll
,
2
i=l Pi Gli + σnl
= W
(9)
where W is a constant.
In a logarithmic model [17], [18], [19], the data rate
Xl of link l is given by
Xl = W log2 (1 + kΓl )
(10)
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Xl
Xl
Xl
Xl
W 'F l
policy is optimal. Then we consider a logarithmic link
adaptation model which in general requires simultaneous
scheduling of multiple links. The maximum capacity
beamformer algorithm designed for efficient concurrent
transmissions is presented at the end of this section.
W log 2 (1 k Fl )
Fl
Fl
(a)
(b)
A. Problem Formulation : Primal Problem
Fig. 4. Two link adaptation functions. (a) The data rate is a Linear
function of SINR. (b) The data rate is a Logarithmic function of
SINR.
where W is a communication bandwidth and k is a
positive constant indicating the consistent gap to the
Shannon capacity limit. Under the Gaussian interference
assumption, this becomes a Shannon capacity formula
for k = 1. Since k = 0.16 well describes the performance of the uncoded M-QAM [18] we can assume
0.16 ≤ k ≤ 1 for a practical communication systems.
These link adaptation functions are illustrated in Fig. 4.
We define the (guaranteed) average rate of link l as
Xlavg ∞
λm Xl (m)
An optimal policy π ∗ is defined as a policy that
minimizes the lengthof QoS period and, equivalently,
∞
minimizes q(λ) m=1 λm = 1 − λ0 for a given
C. Since an optimal policy depends on C and channel
conditions, an access point may need to recompute a
policy whenever either of them changes. The primal
problem for any given C and channel condition is then
mathematically stated as
min
π
subject to 0 ≤
λm Pl (m)
N
Pl (m) ≤ P max for all m
(12)
m=0
where Pl (m) is the transmit power for link l in transmission mode m. Again, we do not include the power
consumed during the non-QoS period. The reason we
are interested in these quantities will become clear in
the subsequent section. Finally, we define the network
average rate vector and the network average power
avg T
] and Pavg =
vector as Xavg = [X1avg , . . . , XN
avg
avg T
[P1 , . . . , PN ] , respectively.
Xlavg ≥ Cl for all l.
AN
O PTIMUM P OLICY
Our goal in this paper is to find an optimal policy
that minimizes the length of the QoS period for a
given set of minimum rate requirements. We present a
duality approach for solving this optimization problem.
We first find an optimal policy for a linear link adaptation
model using this approach and show that a “one-by-one”
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(14)
The constraints in (14) represent the minimum rate
requirements from all terminals and (13) states the peak
power constraint of an access point. We denote the
minimum of this optimization problem for a given C
as Q(C).
can easily show that, in an optimal policy,
We
N
max for any transmission mode m = 0.
P
l=1 l (m) = P
max for
To see this, suppose we have N
l=1 Pl (m) < P
some m = 0. It can be easily seen from (8) that for
α > 1, Γl (αP, V) > Γl (P, V) for all l, Pl = 0. This
implies that we can obtain a higher SINR and thus a
higher data rate for each active link by further assigning
the residual power. This allows reducing the length of
transmission mode m without decreasing the amount of
data delivered in any link. This contradicts the optimality
of the policy. Thus the constraint (13) can be replaced
by
h(P(m)) N
Pl (m) =
l=1
III. F INDING
(13)
l=1
(11)
where Xl (m) is the data rate of link l in transmission
mode m. From the definition of T0 , we can see this
quantity only adds up data bits delivered during the QoS
period of a frame. Even though we can assign more
capacity to users during the non-QoS period we do not
include them for the average rate computation. Similarly,
we define the average power consumed by an access
point for link l as
∞
λm
m=1
m=0
Plavg
∞
q(λ) =
P max if m = 0
if m = 0
0
(15)
avg
From (12), we can see N
= P max q(λ) in an
l=1 Pl
optimal policy. This observation introduces an equivalent
cost function to minimize given by
h(P
avg
)=
N
Plavg .
(16)
l=1
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Then the new optimization problem is to minimize the
total average power consumed during the QoS period and
is stated as
min h(P
π
avg
) subject to (14) and (15).
(17)
Note this optimization involves considering every possible policy. Using a duality approach below, we reduce
the problem to an optimization problem over a single
transmission mode.
B. Duality Approach
Let the value of the optimal cost in the problem (17)
as a function of C be denoted by H(C). If for a given
value of C, no schedule of network power vectors and
network transmit weight matrices exists satisfying (17),
define H(C) = +∞. It can be easily shown using a time
sharing argument that H(C) is a convex function of C.
Define a set of dual variables β = [β1 , . . . , βL ]T for
each link. Define a potential function V according to
V (P, C, X) = h(P) +
βl [Cl − Xl ].
(18)
l∈L
The dual object function is defined as
g(β) = min{V (Pavg , C, Xavg )} subject to (15). (19)
π
Note the absence of the minimum rate constraints (14)
in the definition of g(β). For any non-negative vector β ,
using (14), note that
H(C) ≥ g(β).
(20)
Thus, for any non-negative vector β , the minimum value
of the objective function in the optimization problem (17)
is lower bounded by g(β). This observation leads to the
dual optimization problem,
max{g(β) : β ≥ 0}.
(21)
A geometric interpretation of the inequality (20) can be
obtained by considering the surface of H(·) in (N + 1)dimensional Euclidean space. In particular, H(C) is
lower bounded by considering a supporting hyperplane
that is normal to the vector (β, −1). Since H(·) is convex, there exists a supporting hyperplane which passes
through the point (C, H(C)). This is simply the classical
argument used to prove that there is no duality gap, [20]
i.e.,
P, C and X, it follows that g(β) can be computed by
an optimization over a single transmission mode, i.e.,
g(β) = min{V (P, C, X) : h(P) = P max or 0} (23)
P,V
where X = X(P, V) is defined as in (9) in case
of linear link adaptation and (10) in case of logarithmic link adaptation. Any pair (P, V) that minimizes
(23) represents a point on a hyperplane in (N + 1)dimensional Euclidean space. Since a hyperplane in
(N + 1)-dimensional Euclidean space is determined by
(N + 1) linearly independent points contained within it,
an optimal policy can be constructed that consists of at
most (N + 1) transmission modes. Since T0 representing
the non-QoS period is always included in an optimal
policy, considering a policy with M ≤ N transmission
modes for QoS period is sufficient for optimality. It will
become clear later that this is true even when λ0 = 0.
C. Linear Link Adaptation
After including the linear link adaptation (9) into a
dual objective function (23), we can decouple power
control and beamforming, as we show below.
Theorem 1: Under linear link adaptation assumption,
an optimal policy can always be constructed using a oneby-one scheme where an access point transmits to one
terminal at a time at its peak power.
Proof Let P∗ and V∗ be a minimizer of (23). Consider
the case that P∗ = 0. We have
g(β) = V (P∗ , C, X(P∗ , V∗ ))
= min V (P, C, X(P, V∗ ))
P
where the second equality is ensured from the fact
that finding P∗∗ achieving a smaller value than
g(β) contradicts the optimality of P∗ and V∗ . Since
h(P) = P max
problem is to maximize
, an equivalent
∗ ) over P. Define Pext S(P) β
X
(P,
V
l
l
l
l
[0, . . . , P max , . . . , 0]T where the l-th element is P max
and the others are zero. It can be easily seen that
S(P) =
= g(β ).
=
φl βl W l
(22)
Computation of g(β) involves optimizing over all
schedules of network power vectors and network transmit weight matrices over every transmission mode. However, since the potential function V (P, C, X) is linear in
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Pl G∗ll
∗
2
i=l Pi Gli + σnl
βl W l
H(C) = max{g(β) : β ≥ 0}
∗
=
P max G∗ll
σn2 l
φl S(Pext
l )
l
≤
l
φl max S(Pext
l ),
l
≤ max S(Pext
l ),
l
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σ2
l
where φl = PinGl ∗ +σ2 P Pmax
and G∗ij = |hi vj∗ |2 .
nl
li
i=l
Hence the maximum is always achievable using one of
extreme power vectors. It follows that
ext
∗
g(β) = min{V (Pext
l , C, X(Pl , V )) : 1 ≤ l ≤ N }
l
ext
= min{V (Pext
l , C, X(Pl , V)) : 1 ≤ l ≤ N }
l,V
where the second step is ensured since finding V∗∗ for
achieving smaller value than g(β) again
any of Pext
l
contradicts the optimality of P∗ and V∗ . Since this
result holds for all β including β ∗ , H(C) = g(β ∗ ) is
achievable using only extreme power vectors.
It is obvious from (18) that, for any given β and P,
the optimal beamformer is given by
∗
V (P, β) = arg max
V
N
βl Xl (P, V).
(24)
l=1
Hence an optimal transmit beamformer must maximize
the weighted sum capacity and the weight is defined by
β ∗ . In a one-by-one policy, this narrows down to finding
Vl∗ for each extreme power vector Pext
satisfying
l
Vl∗ = arg max βl∗ Xl (Pext
l , V).
V
(25)
Notice that Xl is maximized when Γl is maximized and,
in a one-by-one policy, Γl is affected by vl only. Thus
the optimal transmit weight for link l is given by
vl∗ = arg max
v
v=1
hH
P max |hl v|2
l
.
=
σn2 l
hl (26)
Since we assume that the channel remains unchanged
during the entire frame period, a single network transmit
weight matrix
∗
V∗ = [v1∗ , . . . , vN
]
is optimal for all extreme power vectors, i.e., for all
transmission modes. In addition, this matrix can be used
without modification as long as the channel remains
unchanged. This maximum SNR beamformer is equally
applicable to both AAS and DAS.
It is interesting that the dependence of V on β found
in (25) is removed. Hence the dual objective function is
further simplified to
ext
∗
g(β) = min {V (Pext
l , C, X(Pl , V ))}.
1≤l≤N
(27)
Having reduced the complicated scheduling problem
(19) to a simpler problem (27), we revisit the primal
problem with the knowledge obtained from the dual
analysis. Although we can complete the problem in
the dual domain as in [14], the primal domain approach eliminates the complexity for finding optimal
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
dual variables β ∗ . It is easy to see that in any optimal policy, the average rate on each link l, Xlavg ,
is exactly equal to Cl . Otherwise, we could decrease
Q(C) further without violating any rate constraints thus
contradicting the optimality of the solution. Thus all
inequality constraints in (14) are active, i.e., Aλ = C
∗
ext
∗
where A = [X(0, V∗ ), X(Pext
1 , V ), . . . , X(PN , V )]
T
and λ = [λ0 , . . . , λN ] . This set of linear equations
determine an unique λ∗ and thus Q(C) = q(λ∗ ) if
q(λ∗ ) ≤ 1 otherwise C is infeasible.
D. Logarithmic Link Adaptation
When a logarithmic link adaptation is used, Theorem
1 doesn’t hold and solving (23) and finding optimal dual
variables β ∗ satisfying (22) are very hard. Due to these
difficulties, we go back to the primal domain and present
a heuristic that attempts to maximize the capacity while
satisfying all QoS requirements. The heuristic is based
on the implication obtained in the dual analysis. Our
algorithm considers only a finite set of power vectors
which are chosen to represent all possible link schedules.
And, for each power vector, we compute a network
transmit weight matrix that maximizes the total downlink
capacity. This is equivalent to using β = 1 in equation
(24) where 1 [1, . . . , 1]T . Finally, we select best N
pairs (transmission modes) by solving a linear program
that minimizes the length of QoS period subject to N
minimum rate constraints.
We first describe our transmit beamforming algorithm
that maximizes the sum capacity for a given P, i.e.,
∗
V (P, 1) = arg max
N
V
log(1 + kΓl ).
(28)
l=1
To the best of our knowledge, a transmit beamforming
solution that maximizes the sum capacity is not found
yet. We propose an iterative beamforming algorithm
maximizing the sum of downlink throughput by jointly
optimizing the array weights of all users. Our algorithm
updates the weight vector of one link at a time. When we
update vl , we freeze all other weight vectors {vj }j=l .
Let the summation in r.h.s. of (28) as a function of
V be denoted as J(V). Since, for a given set {vj }j=l ,
J(V) is a function of only vl , we denote this function
as Jl (v). At any iteration i, vl is updated such that
(i)
vl = arg max
Jl (v)
v
(29)
v=1
(i)
(i)
for a given set of N − 1 weight vectors {v1 , . . . , vl−1 ,
(i−1)
(i−1)
vl+1 , . . . , vN }.
The entire procedure of our algorithm is as follows :
IEEE INFOCOM 2004
Choose V(0) ;
i←1;
While not converged do
For l = 1 : N
(i)
vl ← arg max Jl (v) ;
(i)
(i)
(i) (i−1)
(i−1)
Vl ← [v1 . . . vl vl+1 . . . vN ] ;
End
(i)
V(i) ← VN ;
i←i+1 ;
End
(i)
(i)
and corresponding transmission modes.
We remark that the complexity of our link scheduling algorithm is exponential. We are currently making
effort to create simple criteria to identify inefficient
link schedules based on channel conditions and the
array size. Using heuristics to eliminate many inefficient
link schedules from consideration, we will be able to
significantly improve the scalability of our algorithm.
E. Maximum Capacity Beamformer
(i)
Matrix Vl−1 is used when vl is computed. (V0 is
(i−1)
and hence V(i−1) .)
defined to be equivalent to VN
Notice that a single update of matrix V(i) is completed
(i)
(i)
after updating all N columns of it, i.e., v1 , . . . , vN .
An algorithm for solving (29) is described in the
subsequent section.
This beamforming algorithm is designed to maximize
the downlink sum capacity for a given network power
vector P. Then how to find a set of optimal power
vectors? To maintain the complexity of our algorithm
manageably low, we only consider scheduling power
vectors. A set of scheduling power vectors represent
all possible link combinations in link scheduling with
a equal power allocation. Let N̂ be the number of
scheduled (active) links in a transmission mode. Clearly,
we have L = 2N − 1 possible link scheduling vectors
max
and the l-th element of a link scheduling vector is P
N̂
if link l is active, otherwise zero. This link scheduling
power vector approach is justified from the result in
[19] - in a network incorporating link adaptation and
scheduling, the additional capacity gain due to power
control is insignificant. Let Psch
s , 1 ≤ s ≤ L denote the
scheduling power vectors and Vs∗ be the corresponding
optimal network transmit weight matrices. Their pairs
∗
{Psch
s , Vs } constitutes a set of candidate transmission
modes. We then choose best N transmission modes for
satisfying C by solving a linear program :
minimize
L
λs
(30)
Now we describe an algorithm for solving (29). To
update the weight of link l at iteration i, we need to
(i)
(i)
(i) (i−1)
(i−1)
compute Vl = [v1 . . . vl vl+1 . . . vN ] from the
(i)
(i)
(i)
(i−1)
(i−1)
. . . v N ].
current matrix Vl−1 = [v1 . . . vl−1 vl
The following explanation holds for all links throughout
the iteration. Assume that an update in a single weight vl
changes the individual SINRs by a small amount. This
assumption is valid when the current network transmit
weight matrix is close to the maximizer of J(V). We
can then approximate the function log(1 + kΓj ) by
be the SINR of
an affine function Lj (Γj ). Let Γold
j
link j corresponding to the network transmit weight
(i)
matrix Vl−1 . The affine function passes through the
k
1
old
point (Γold
j , log(1 + kΓj )) with a slope ln 2 1+kΓold , i.e.,
j
Lj (Γj ) = aj + bj Γl ,
k
1
old
with aj = log(1 + kΓold
.
j ) − bj Γj and bj = ln 2 1+kΓold
j
By applying this approximation to all links, we obtain
Jl (v) ≈ a0 +
(32)
(i)
where a0 = j aj . Apparently, vl has to be chosen
such that the weighted sum of SINR presented in the
second term of r.h.s is maximized. The dependence of
the SINR of each link j = l on vl can be identified by
the following manipulation of (7),
Γj
=
Pj vjH hH
j hj vj
2
Pi viH hH
j hj vi + σnj
ωj
ϕj + Pl vlH hH
j hj vl
i=j
(31)
=
Λ ≥ 0,
∗
sch
∗
where  = [X(Psch
1 , V1 ), . . . , X(PL , VL )] and Λ
T
[λ1 , . . . , λL ] . Here we are assigning time fraction
=
to
each transmission mode to satisfy all minimum rate
requirements with smallest length of the QoS period.
If the solution is greater than 1, we declare C violates
the capacity limit of this algorithm. If less than or equal
to 1, a policy is constructed by collecting non-zero λs ’s
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
bj Γj ,
j=1
s=1
subject to ÂΛ = C and
N
=
ωj Pl H H
ωj
ϕj vl hj hj vl
−
ϕj
ϕj + Pl vlH hH
j hj vl
=
vlH ωϕj Pj l hH
ωj
j hj vl
− H
ϕj
vl (ϕj IQ + Pl hH
j hj )vl
=
v H A j vl
ωj
− lH
ϕj
vl B j vl
(33)
IEEE INFOCOM 2004
H H
2
where ϕj =
=
i=j,l Pi vi hj hj vi + σnj , ωj
H
H
Pj vj hj hj vj , IQ represents an Q × Q identity matrix,
H
Aj = ωϕj Pj l hH
j hj and Bj = ϕj IQ +Pl hj hj . We isolated
the term containing vl from the denominator for the
y
1
= x1 − x(x+y)
for
second step, applied the identity x+y
the third step. Two nonnegative scalars ϕj and ωj , a
symmetric positive semi-definite matrix Aj and a symmetric positive definite matrix Bj are only determined
by transmit weight vectors other than vl .
We don’t need above manipulation of SINR for link l
and it is simply expressed as
Γl = vlH Al vl
(34)
where Al = Pi vHPhlH hl vi +σ2 hH
l hl is again a symnl
i
l
i=l
metric positive semi-definite matrix independent of vl .
(i)
Using (33) and (34), the weight vl that maximizes (32)
is written as
(i)
vl = arg max
{bl vH Al v −
v
bj
j=l
v=1
v H Aj v
}. (35)
v H Bj v
Although the standard method for solving this optimization problem is the Newton method, we propose a set of
non-iterative heuristics to avoid Newton iterations. If we
(i−1)
(i)
to vl is small then the
assume the change from vl
problem (35) is well approximated to
(i)
H
vl = max
v
{b
A
−
bj Aj }v
(36)
l
l
v
bj
.
(i−1)
B j vl
The maximizer of this funcv
tion is simply an eigenvector of matrix bl Al − j=l bj Aj
associated with the largest eigenvalue. We denote the
(i1)
maximizer as vl .
A lower bound of the function in (35) can be obtained
by replacing each denominator vlH Bj vl by its lower
bound λmin (Bj ), the smallest eigenvalue of Bj . We
can maximize this lower bound by solving (36) after
replacing bj by λminbj(Bj ) , and we denote this solution
(i−1) H
l
(i2)
as vl . Similarly, we can maximize an upper bound of
the function in (35) by solving (36) after replacing bj
(i3)
by λmaxbj(Bj ) , and we denote this solution as vl . We
(i)
finally determine vl
using
(i)
(i)
vl = arg max{Jl (v) : v ∈ Svl },
v
(i)
In this section, we compare the performance of a conventional array antenna system with that of a proposed
distributed antenna system. The first example examines
the signal strength distributions over the entire indoor
coverage area when the maximum SNR beamformers are
being used. It will become clear that the latter provides
stronger SNR and more stable coverage under the same
peak power constraint. The second example models a
multi-user indoor network and finds optimal policies
for a linear link adaptation system. The lengths of the
minimum required QoS period for two multi-antenna
systems are compared. The third example considers a
network with a logarithmic link adaptation mechanism.
We first explore the performance of the proposed maximum capacity beamformer algorithm. Then we examine
the lengths of QoS period for AAS and DAS. The
superiority of the link scheduling policy to the one-byone policy will be also illustrated.
The performance of each system will be highly dependent on the communication channel environment. With
DAS, almost everywhere in the area may have a clean
line of sight (LOS) to at least one antenna. It is more
likely to have blockage with AAS. But we assume a LOS
multiple-input and single-output (MISO) channel given
by (4) and (5) for both DAS and AAS. We further assume
that the free-space pathloss governs the propagation law
in all examples in this paper.
j=l
v=1
where bj =
IV. N UMERICAL E XAMPLES
(i1)
(i2)
(i3)
(i−1)
(37)
where Svl = {vl , vl , vl , vl
}. The inclusion
(i−1)
of vl
in the candidate set ensures the sequence
{J(V(i) )}∞
i=0 to be monotonic nondecreasing, and hence
our algorithm converges.
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
A. Network Coverage - SNR Performance
We consider an access point that provides the wireless
coverage to a circular area with a radius 25 meters. We
consider a standard uniform array composed of Q = 6
omnidirectional antennas as the conventional AAS. We
place six omni-antennas equally spaced on the circular
wall for the DAS set up. Antenna locations are depicted
in Fig. 5. The antenna array for an AAS is located at
the double-circled point on the right. Both multi-antenna
systems are subject to peak power constraint of P max =1
Watt.
For any given terminal at position (xl , yl ), the array
processor of each multiple antenna system computes the
transmit array weight vl∗ maximizing the SNR using
(26). Then we can compute the effective link gain Gll
and the corresponding signal strength at that position. At
each sample positions spaced by 1 meter, we compute
and compare the signal strength of two systems. The
dotted area in Fig. 5 is where the DAS provides higher
signal strength than that of the AAS. Clearly, most of
the coverage area belongs to the DAS-superior region
and the AAS yields stronger signal strength only in the
IEEE INFOCOM 2004
DAS−superior
region
AAS−superior
region
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
Fig. 5. Network coverage comparison of the AAS and DAS. DAS
provides stronger signal in the dotted area and AAS is superior only
in the blank area. The DAS-superior region gradually expands to the
dashed lines as the propagation exponent increases.
proximity of the array. Note that we are assuming a LOS
environment where the propagation exponent is 2, i.e.,
signal attenuation in proportion to the distance square.
We can easily show that, as the propagation exponent
goes to the infinity, the boundary of the DAS-superior
region and the AAS-superior region gets closer to the
two dashed lines drawn in Fig. 5. This is simply because
the array gain provided by an AAS is overwhelmed by
the distance based propagation law and can’t play any
role. Hence we can expect that the DAS-superior region
will further extend in a practical indoor environment
where the propagation exponent ranges from 2 to 5.
We can more accurately compare the coverage of
those two systems by contrasting the distributions of
the effective link gain presented by AAS and DAS.
By collecting the link gains at all sample locations we
portray the cumulative distribution function (CDF) in
Fig. 6. We plot the link gain of system with a single
omnidirectional antenna in part (a) as a reference. Since
the array gain in AAS is 7.8 dB (=Q=6) regardless of
the terminal position, the CDF of AAS (b) is simply
a 7.8 dB shifted version of (a). The CDF of DAS in
(c) is further shifted to the right and shows more rapid
accumulation. This means that the DAS supplies not only
stronger signal coverage on average but also more stable
coverage, i.e., uniformly good coverage. The mean and
the standard deviation of the effective link gain in DAS
are -64.7dB and 3.2dB, respectively, whereas they are
-67.9dB and 4.8dB in AAS.
The LOS channel considered in this example is not
random at all. The issue of how the network coverage
performance changes when the random fading is present
in the channel should also be examined. We leave this
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
−80
−60
−40
eff. link gain (dB)
(a)
0
−80
−60
−40
eff. link gain (dB)
(b)
0
−80
−60
−40
eff. link gain (dB)
(c)
Fig. 6. CDF comparison of effective link gains provided by AAS
and DAS. (a) CDF of link gain with a single omnidirectional antenna.
(b) CDF of effective link gain in AAS. This is simply a 7.8 dB (=6)
right shifted version of (a). (c) CDF of effective link gain in DAS.
Further shifted to the right and more concentrated.
for future work.
B. Linear Link Adaptation - One-by-one Policy
Now we examine a network performance serving
multiple users. In this example, we assume a linear link
adaptation model given by (9) with a slope W = 14.4×
106 . An optimal policy is the one-by-one policy as shown
in the previous section. The same antenna configurations
and peak power constraint as in the previous example are
assumed. N =10 users each of them requiring a minimum
data rate of 384kbps (C = C0 × [1, . . . , 1]T , C0 = 384 ×
103 ) are randomly scattered over the network coverage in
each simulation. Terminals suffer from identical ambient
noise of -66dBm. We computed the minimum required
portion of QoS period Q(C) for 100 simulations.
In the network with AAS, the QoS requirements of
all users are satisfied in 99 simulations out of 100. In
those successful simulations, we have spent 66% of the
frame to support the minimum data rates required by
users (the mean of Q(C) is 0.66). The standard deviation
of Q(C) is found to be 0.14. In the network with
DAS, the QoS requirements of all users are satisfied in
all 100 simulations. (no outage) Furthermore, the mean
and the standard deviation of Q(C) are 0.25 and 0.03,
respectively. The reduction in the mean suggests that we
can accommodate 16 more 384kbps users using DAS
without increasing the length of QoS period in a frame.
This implies a 160% capacity increase. Also, a very
small standard deviation indicates that we can provide a
very reliable service even in a mobile environment where
terminals keep changing their locations.
IEEE INFOCOM 2004
Initial Beam Patterns
Converged Beam Patterns
0
User 1
1
10
User 3
1
10
0
10
10
0
0
10
10
−2
−2
10
10
−1
−1
10
10
0
50
100
θ (degree)
150
0
50
100
θ (degree)
150
−2
−2
10
10
0
0
10
10
0
−2
−2
10
10
0
50
100
θ (degree)
150
0
0
50
100
θ (degree)
150
120
60
θ (degree)
(a)
180
0
120
60
θ (degree)
(b)
180
Fig. 8. A transmit beamforming for N =5 users with a standard
ULA of size Q=4. Users are at angles 0.3063π, 0.3580π, 0.4877π,
0.6609π, and 0.9382π. (a) Beam pattern of user 1. (b) Beam pattern
of user 3.
0
10
10
−2
−2
10
10
0
50
100
θ (degree)
150
0
50
100
θ (degree)
150
Fig. 7. A transmit beamforming for N =3 users with a standard
. Left three beam
ULA of size Q=4. Users are at angles π4 , π2 and 3π
4
patterns are initial beamsteering patterns. Converged beam patterns
are depicted on the right of each initial beam pattern.
We remark that the superiority of DAS is even more
distinctive when the users show heterogeneity in their
QoS requirements. We executed 200 simulations with
C = C0 × [3, 3, 1, 1, 1, 1]T and no outage happened in
DAS whereas AAS experienced twelve. This superior
outage performance attributes to the uniformly good
coverage provided by DAS investigated above.
C. Logarithmic Link Adaptation
We finally analyze a network with a logarithmic
link adaptation with parameter k =0.398 (=-4dB). The
channel bandwidth W is assumed to be 10MHz. We
first look at the performance of our maximum capacity
beamformer algorithm. Consider a standard ULA with
size Q=4. Three equidistance terminals are located at
angles π4 , π2 and 3π
4 with 20dB SNR. The maximum SNR
beamforming vectors given by (26) are used as initial
values. This initial conditions are depicted on the left
half of Fig. 7 and solid lines represent signal directions
and dashed lines denote interfering directions. Clearly
three initial beams are interfering each other. The three
beam patterns on right half of Fig. 7 are outcomes of
our maximizing capacity beamforming algorithm. These
beams are not interfering each other any more.
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
Our beamforming algorithm is applicable even when
the number of interfering directions exceeds the array’s
null placing capability. We randomly generated five angles (0.3063π , 0.3580π , 0.4877π , 0.6609π , and 0.9382π )
and placed five equidistance users (SNR = 20dB) at those
angles. Note that user 1 and user 2 are very close in
their angles. We started our algorithm with beamsteering
vectors and obtained beam patterns depicted in Fig. 8.
Only beam patterns of user 1 (part (a)) and user 3 (part
(b)) are shown. The beam pattern for user 1 nulls out
interference to user 3, 4 and 5. But it doesn’t place a null
to the direction of user 2 to keep the signal strength high.
A similar result was found in the beam pattern of user 2.
The beam patten of user 3 eliminates interference to user
4 and 5 and places a null in between user 1 and 2. This
is a result of an effort to avoid an undesirable situation
where one of them suffers too much interference alone.
Similar results were found in the beam patterns of user
4 and 5.
Since it is difficult to visualize beam patterns of a
DAS, we indirectly examine them by looking at the
network level performance in the following examples.
N users are now randomly generated across the circular
indoor coverage. They are all subject to an identical
ambient noise -86dBm. We consider 3 policies for both
AAS and DAS :
I. one-by-one policy with maximum SNR beamformer
II. link scheduling (based on scheduling power vectors) policy with maximum SNR beamformer
III. link scheduling policy with the proposed
maximum capacity beamformer.
A set of minimum rate requirements C = C1 ×[1, 1, 1]T ,
C1 =667kbps is required when N =3, and C = C2 ×
IEEE INFOCOM 2004
TABLE I
C OMPARISON OF F RACTION OF TIME USED FOR Q O S PERIOD
Policy
I
II
III
N =3 users
AAS
DAS
0.5480
0.4128
0.4784
0.3865
0.4322
0.2851
N =5 users
AAS
DAS
0.5533
0.4184
0.4470
0.3823
0.4149
0.2977
[1, 1, 1, 1, 1]T , C2 =400kbps when N =5. The minimum
fraction of a QoS period Q(C) is averaged over 100 simulations and summarized in Table I. Comparing policies,
policy II always requires shorter QoS period than policy
I irrespective of number of users and antenna scheme.
This indicates that, even though beamformers do not care
about the interference in a maximum SNR beamforming,
link scheduling itself is taking advantage of spatial multiplexing capability to some extent. Policy III is always
superior to policy II because beamformers together with
link scheduling are jointly exploiting the space division
multiplexing capability. Comparing AAS and DAS, DAS
always outperforms AAS regardless of number of users
and type of policy used. If we look at the best policy
× 100)
(policy III), DAS provides 51.6% ( 0.4322−0.2851
0.2851
more capacity than AAS when N =3, and 39.4% more
capacity when N =5. Obviously, distributed antennas are
taking advantage of spatial resources more efficiently.
V. C ONCLUSION
In this paper, we proposed a radio resource management policy that maximizes the downlink capacity while
satisfying QoS requirements from all users. This includes
array processing, power control, link adaptation and link
scheduling. We found that we can better utilize the
spatial resources by using a distributed antenna system
rather than a conventional array antenna system. The
benefits are stronger and uniformly good signal coverage
and increased radio capacity. These are obtained at the
expense of increased hardware complexity - extended
cabling, optical modulator and detector cost, etc. The
selection of an antenna scheme should be based on a
tradeoff between capacity and cost. The proposed maximum capacity beamformer algorithm further exploits
the spatial dimension and is equally applicable to both
DAS and AAS. It presents a significant capacity increase
compared to a simple maximum SNR beamformer. Link
scheduling plays an important role in supporting QoS
requirements. Numerical examples demonstrate a dramatic capacity gain when these schemes are combined
all together.
0-7803-8356-7/04/$20.00 (C) 2004 IEEE
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IEEE INFOCOM 2004