This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.
Minimizing Transmit Power in a Virtual-cell
Downlink with Distributed Antennas
Boon Sim Thian1 , Sheng Zhou2 , Andrea Goldsmith1 , Zhisheng Niu2
1 Department
of Electrical Engineering,
Stanford University
Stanford, CA 94305
Email: {bsthian, andrea}@stanford.edu
2 Department
of Electronic Engineering
Tsinghua University
Beijing 100084, China
Email: zhouc@mails.tsinghua.edu.cn, niuzhs@tsinghua.edu.cn
Abstract—We consider the problem of allocating transmit
power in the downlink of a distributed wireless communication
system. We account for the power used in both channel estimation
and data transmission, with the objective of minimizing the
overall transmitted power while satisfying specified Quality of
Service (QoS) constraints to the mobile users. We consider
both single user and multi-user power control optimization; the
problem formulation for both cases lead to a nonconvex program.
We proposed solution strategies for both scenarios: For the single
user case, a simple intuitive solution, where power is allocated
to the antennas sequentially until the QoS constraint is satisfied,
is presented. For the multi-user case, we use successive convex
approximation (based on the single condensation method) to
find a provably convergent solution. We also demonstrate, via
numerical simulation, the convergence of the proposed multiuser power allocation strategy. Our numerical results indicate
that the proposed single and multi-user power allocation lead to
an overall savings of up to 45% when compared to the baseline
method of equal power allocation.
Index Terms—: distributed wireless communication systems,
imperfect channel state information, power control, convex optimization.
two distinct layers. Similar to DAS, the antennas are still
geographically distributed. Each of them is connected to a
processing node, typically via optic fiber. As shown in Figure
1, the processing nodes are also connected to one another, as
well as to a base station (central processor). Thus, in contrast
to a DAS, decisions can be made centrally by the base station
or in a distributed fashion, since each of the processing nodes
has a small computing facility so it can make its own decisions.
I. I NTRODUCTION
With increasing demands for data, multimedia and various
other types of services, future wireless communication systems
will be required to provide higher data rates to a larger number
of users. To meet the demands for higher reliability, higher
data rates and lower interference, different forms of spatial
diversity are used extensively. Distributed antenna systems
(DAS), with a number of remote antennas connected to a
base station, are receiving more attention from the research
community as a means to provide spatial diversity [1]-[3].
Distributed antennas were initially introduced to extend
coverage in environments hostile to radio propagation and
to cover dead spots in indoor wireless communications [4].
In addition to the aforementioned advantages, recent research
has demonstrated that DAS can reduce the overall transmit
power (and hence inter-cell interference) and increase system
capacity [5]-[8].
Distributed wireless communication systems (DWCS), an
extension of DAS, was first proposed by Zhou et. al in [9].
In this system, the antennas and processors are connected in
This work is supported in part by ONR under grant N000140910072,
DARPA’s ITMANET Program and China National Science Fund for Distinguished Young Scholars under grant 60925002.
Fig. 1.
Structure of a DWCS
In DWCS, the concept of physical cells is replaced by
virtual cells. By identifying a set of distributed antennas
dedicated to each mobile user, a virtual cell is determined.
Macro-diversity techniques and joint processing allow the
system to mitigate the effect of channel fading and inter-cell
interference. Recent studies have also shown that DWCS offer
the same advantages as DAS, along with capacity gain [10].
Previous research on DWCS focused on capacity analysis
[11] [12], antenna selection [13], transmission mode selection
[14] and antenna location design [15]. The objective of previous studies is primarily on capacity maximization under ideal
circumstances, such as delay-free perfect channel knowledge.
In this paper, we present a study of downlink power control
techniques in DWCS under practical constraints of imperfect
channel knowledge. Specifically, the goal is to minimize the
overall transmitted power while meeting a specified quality of
service (QoS) requirement for all mobile users. Minimizing
downlink power is an essential part of future wireless communication infrastructure design since energy consumption is
978-1-4244-9268-8/11/$26.00 ©2011 IEEE
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.
increasing rapidly due to exponential network growth. This is
especially true in countries with limited sources of clean and
reliable energy.
Both single user and multi-user scenarios are considered
in this paper. The problem formulation for both scenarios
leads to a nonconvex program. For the single user case, a
simple intuitive solution is presented; the solution is to allocate
power to the antennas successively, in descending order of the
path variance, until the specified QoS is met. For the multiuser case, no such simple solution exist. Instead, we utilize
successive convex approximation to the problem to find a
solution which converges to a point satisfying the KarushKuhn-Tucker (KKT) conditions of the original nonconvex
program.
The remainder of this paper is organized as follows. We
present our system model in Section II. In Section III, we
present the channel estimation strategy we utilize. The power
control problem for the single user and multi-user cases,
along with the solution strategies, are presented in Section
IV. Numerical results and discussions are presented in Section
V, and conclusions along with future research direction are
presented in Section VI.
Notation: In this paper, vectors and matrices are denoted
in bold. The symbols (·)T and || · || denote transposition and
Euclidean norm (l2 -norm), respectively.
where ci is the ith element of c and represents the channel
between the ith distributed antenna and the mobile receiver,
(1)
and Pi is the power used by the ith antenna during the first
phase (channel estimation phase). For ease of implementation,
linear estimation is considered; the received signal yCE,i is
multiplied by gi∗ . Hence, the estimated channel is
ĉi =
gi∗ yCE,i
=
gi∗
y = cT wx + n
(1)
where x ∈ X denotes the transmitted symbol; n ∼ CN (0, σn2 )
denotes the additive white
Gaussian
noise √
(AWGN)
T with
√
√
is the
variance σn2 ; c = h1 L1 , h2 L2 , . . . , hM LM
channel vector of dimension M × 1; hi ∼ CN (0, σh2 ) repre2
) represents path
sents Rayleigh fading; log Li ∼ N (μLi , σL
i
loss and lognormal shadowing; and w denotes the processing
vector.
+ ni
(3)
The variable gi∗ is chosen to minimize the mean squared
error (MSE) between ĉi and ci , which is the optimum criterion
if the noise ni is Gaussian. Hence,
gi∗ = arg min E|ĉi − ci |2
2
(1)
= arg min E ri
Pi ci + ni − ci ri
(4)
Differentiating the expression in (4) with respect to ri and
setting it to zero, we obtain
gi∗ =
(1)
Pi θ i
(5)
(1)
Pi θi + σn2
where θi = E|ci |2 . The corresponding MSE is
II. S YSTEM M ODEL
The channel model that we consider includes Rayleigh
fading, path loss and lognormal shadowing. We consider the
downlink scenario, where there are M distributed antennas at
the base station and either a single mobile user or N mobile
users. The received signal at a mobile user is expressed as
(1)
Pi ci
min E|ĉi − ci |2 ei =
θi σn2
(1)
Pi θ i +
σn2
(6)
The channel estimate can thus be expressed as
ĉi = ci + ei
(7)
where the channel
estimation error ei is zero-mean and has
θ σ2
variance (1)i n 2 . Hence, the power of the estimated channel
Pi θi +σn
is given by
E |ĉi |2 = E |gi∗ |2 |yCE,i |2
(1)
=
Pi θi2
(1)
Pi θi + σn2
(8)
III. C HANNEL E STIMATION
IV. P OWER C ONTROL FOR DWCS
We assume that the base station knows the statistical channel
state information (CSI), which is the mean and variance
(power) from each of the distributed antenna elements to
the mobile receiver. However, the exact channel gains are
not known and hence training sequences will have to be
transmitted in order to estimate the channel. There are several
ways to implement channel estimation; one approach is to preassign a training slot for each of the links.
During the channel estimation phase, training symbols are
transmitted sequentially along each of the links to facilitate
the estimation. The received signal at link i is
We first consider the single user scenario and jointly optimize the power (used by the antenna transmitters) over the
channel estimation and data transmission phases. The goal is
to minimize the overall transmitted power while meeting specified QoS constraints. We developed an algorithm to select the
antennas for transmission, and to find the optimum allocation
of power between the channel estimation and data transmission
phases. In a multi-user setting, the antennas are used to
transmit different messages to the different users (either via
code division multiple access (CDMA) schemes or frequency
division multiple access (FDMA) schemes). Successive convex
approximation, based on the single condensation method [16],
is used to find a solution to this problem which satisfies the
KKT conditions of the original problem.
yCE,i =
(1)
Pi ci + ni
(2)
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.
A. Single user
In the single user scenario, all the distributed antenna
elements cooperate to transmit a single message to the mobile
receiver. The processing vector w at the base station can be
expressed as
w=
T
(2)
(2)
(2)
P1 , P2 , . . . , PM
P2
(2)
Pi
M minimize
subject to
(9)
M (2)
Pi ĉi +
(2)
Pi ei x + n
2+
σn
The single user power allocation problem can be formulated
as
(1)
2
θi +σn
i
(2)
2
P
θi σn
i
i=1
(1)
2
P
θi +σn
i
(1)
minimize
subject to
γ=
2
M
i=1 E |ĉi |
M
2
σn + i=1 E [|ei |2 ]
(2)
=
σn2 +
M
(2)
2
Pi θi σn
i=1 P (1) θ +σ 2
i
n
i
≥ γ̄
(2)
M
(2)
Pi
(1)
Pi θi2 − θi σn2 γ̄
(1)
Pi θ i
+
P (1) θ 2 −θ σ 2 γ̄
(1)
Pi Pi θi2
i=1 P (1) θ +σ 2
i
n
i
(13)
σn2
≥ σn2 γ̄
(14)
To find an optimum power allocation which meets constraint
P (1) θ 2 −θj σ 2 γ̄
(14), we note that for θj > θi , we have P (2) P (1)jθj +σ2n >
(1)
(2)
P i + Pi
i
(1)
(2)
Pi + P i ≤ P T ,
i = 1, . . . , M
(1)
(2)
Pi , Pi ≥ 0, i = 1,
. . . , M
P̄SER ĉ P1 , P2 , e P1 , P2 ≤ P̄target
(11)
(1)
(2)
with variables Pi , Pi , i = 1, . . . , M . In (11), the first
constraint is a per antenna power constraint, and the third
constraint is the average symbol error rate (SER) constraint.
The average SER, P̄SER , depends on the decoding scheme
used at the mobile receiver; it is a nonlinear (and also
nonconvex) function of the estimated channel vector ĉ =
T
[ĉ1 , . . . , ĉM ] as well as the channel estimation error vecM
tor e = [e1 , . . . , eM ] . Both ĉ and e in turn depend on
T
(1)
(1)
P1 = P1 , . . . , PM
and P2 . The exact expression for
P̄SER is obtained by an M -fold integration of Q-function
expressions over lognormal and Rayleigh random variables.
Inclusion of the third constraint in (11) makes the problem
highly intractable, so a proxy measure for the average SER
constraint is used.
Since the value of the vector e is unknown to the mobile
receiver, it cannot be used for symbol decoding. These errors
will degrade the decoding error probability in a monotonic
manner; large a ei will increase the SER and vice versa. In
this sense, e is considered as pseudo interference. We define
the signal-to-channel- error-plus-noise-ratio (SCENR), γ, as
M
i = 1, . . . , M
i = 1, . . . , M
with variables Pi , Pi , i = 1, . . . , M . The single user
power allocation problem (13) is nonconvex in its original
form. We propose a method of finding the optimal solution via
solving a series of convex optimizaton problems as follows:
We write the third constraint as
i=1
M P
M
(10)
i=1
(1)
(2)
Pi + Pi
i
(1)
(2)
Pi + P i ≤ P T ,
(1)
(2)
Pi , Pi ≥ 0,
M Pi(2) Pi(1) θi2
i=1
th
is the power used by the i antenna elewhere
ment
during
the second phase (data transmission phase).
(1)
(2)
= (0, 0) if the ith antenna element is not
Pi , Pi
utilized. The received signal can therefore be expressed as
y=
−1
P̄target γ̄. The single user power allocation
γ ≥ P̄SER
problem can thus be reformulated as
(12)
The average SER can also be expressed as a function of γ,
i.e P̄SER
(γ); although no closed form expression exists, we
by numerical integration. For a given
can still obtain P̄SER
P̄target , we can reformulate the the third constraint of (11) as
P (1) θ 2 −θ σ 2 γ̄
n
i n
i n
as long as P (2) P (1)iθi +σ
> 0. This
P (2) P (1)iθi +σ
2
2
n
n
implies that allocating power to the terms with larger θi will
allow us to minimize power. Therefore, the goal is to allocate
power to the antenna with the largest θi in a way to maximize
2
γ̄
P (1) θ 2 −θi σn
. If the term exceeds the constraint
the term P (2) P (1)iθi +σ
2
n
after the power allocation, then it is not optimal. Power has to
be re-allocated in such a way that the term is just equal to the
constraint. Furthermore, if the constraint cannot be met, then
the same optimization is performed over the antenna with the
second largest θi and so on.
Hence, the power allocation problem can be decomposed
into a series of optimization problems, each parameterized by
θi . Assume θi is ordered in decreasing order, i.e θ1 ≥ θ2 . . . ≥
θM , then the first sub-optimization problem in the series is
maximize s
(1)
(2)
subject to P1 + P1 ≤ PT
(1)
(2)
P1 , P1 ≥ 0
(1)
(2) P θ 2 −θ σ 2 γ̄
P1 1 (1)1 1 2n
P1 θ1 +σn
(1)
(15)
≥s
(2)
with variables s, P1 , P1 . (15) is a geometric program and
we can convert it into a convex representation via a change
(1)
(2)
of variables: letting Q1 = log(P1 ), Q2 = log(P1 ) and
Q3 = log s, we reformulate the problem as
maximize Q3
subject to log (exp (Q1 ) + exp(Q2 )) ≤ PT
3
log k=1 bi exp aTi Q ≤ 0
T
(16)
with optimization variables Q = [Q1 , Q2 , Q3 ] ∈ R3 . In (16),
T
b1 = θi−1 σn2 γ̄, b2 = θi−1 , b3 = θi−2 σn2 , a1 = [−1, 1, 0] ,
T
T
a2 = [1, −1, 1] and a3 = [−1, −1, 1] . Denote the optimizer
of (16) to be Q∗ .
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.
If Q∗3 ≥ log σn2 γ̄, then the solution of the global
optimization problem (13) is obtained after we resolve (16) (and indirectly (15)) with the objective as
minimize log (exp (Q1 ) + exp(Q2 )) and an additional fourth
constraint Q∗3 = log σn2 γ̄. The optimizer of (15) is then taken
to be log Q∗ .
Otherwise, if Q∗3 < log σn2 γ̄, we continue and solve (16)
with parameter θ2 and so on. We summarize the solution
strategy in Algorithm 1 below.
Algorithm 1 Solution strategy for the single user power
control problem
2
1: Input: PT , γ̄, σn
, (θ1 , θ2 , . . . , θM )
2
2: α ← log σn γ̄
3: i ← 1
4: while α > 0 do
5:
Solve (16)
6:
if Q∗3 ≥ α then
7:
Replace the objective function of (16) by
include
a
minimize log (exp (Q1 ) + exp(Q2 )),
fourth constraint Q3 = α and re-solve it.
8:
end if
9:
Assign (log Q∗1 , log Q∗2 ) to be the transmission power
of the distributed antenna with θi .
10:
i←i+1
11:
α ← α − Q∗3
12: end while
B. Multiple users
In the multi-user scenario, all the distributed antenna elements cooperate to transmit different messages to all the
mobile users. The goal is to use minimal transmit power while
satisfying the QoS constraints, as measured by the SCENR,
from the base station to all the mobile users.
Suppose there are N mobile users and the channel vector
from the distributed antennas to the j th mobile user is denoted
T
by cj = [cj1 , cj2 , . . . , cjM ] , j = 1, . . . , N . Let E|cji |2 θji
T
and denote θj = [θj1 , θj2 , . . . , θjM ] to be the channel power
(second moment) vector from the distributed antennas to the
j th mobile user. The multi-user power allocation problem is
formulated as
minimize
subject to
M (1)
(2)
P i + Pi
i
(1)
(2)
Pi + P i ≤ P T ,
i = 1, . . . , M
(1)
(2)
0,
i
=
1, . . . , M
Pi , Pi ≥
(1)
2
2
M
Pi θji
−θji σn
γ̄
(2)
≥ σn2 γ̄,
(1)
i=1 Pi
P θ +σ 2
i
ji
subject to
6:
n
(1)
2
2
(2) Pi θji −θji σn γ̄
Pi
,
(1)
2
Pi θji +σn
(17) is
(1)
(2)
P i + Pi
i
(1)
(2)
i = 1, . . . , M
Pi + P i ≤ P T ,
(1)
(2)
,
P
≥
0,
i
= 1, . . . , M
P
i
i
(1)−1
(2)−1
+ dji2 tji Pi
i = 1, . . . , M
dji1 Pi
(1)−1 (2)−1
≤ 1, j = 1, . . . , N
+dji3 tji Pi
Pi
−1
M
≤ 1, j = 1, . . . , N
γ̄σn2
i=1 tji
Algorithm 2 Successive convex approximation method for the
multi-user power control problem
2
1: Input: PT , γ̄, σn
, (θ , θ , . . . , θ ), tolerance , initial fea 1 01 2 2 0 0 N
sible solution P , P , tj , ∀j,
2: k ← 1
3: for i = 1 to M , j = 1 to N do
5:
j = 1, . . . , N
M (18)
(1)
(2)
with variables Pi , Pi , tji , i = 1, . . . , M and j =
−1 2
−1
σn γ̄, dji2 = θji
and dji3 =
1, . . . , N . In (18), dji1 = θji
2 −2
σn θji , i = 1, . . . , M and j = 1, . . . , N . (18) is a signomial
programming problem because the last N constraints are
upper bound constraints on a ratio of posynomials. Directly
solving this problem is difficult, so we will use a series of
approximations to the last N constraints.
M
Let fji (tj ) =
i=1 tji , j = 1, . . . , N . The idea is to
find fˆji (tj ) such that fˆji (tj ) ≈ fji (tj ), ∀tj and that the
resulting optimization problem can be solved easily using
efficient interior point methods. The solutions of the series of
approximations will converge to a point satisfiying the KarushKuhn-Tucker (KKT) conditions of the original nonconvex
attributes
problem if fˆji (tj ) possesses the following
[17] [18]:
(i) fji (tj ) ≤ fˆji (tj ) , ∀tj , (ii) fji tkj = fˆji tkj where tkj is
the solution of the approximated
problem
in the preceding
iteration, and (iii) ∇fji tkj = ∇fˆji tkj .
The single condensation method method [16] [18] involves
approximating
fji (tj ) by a monomial, i.e fˆji (tj ) =
νji
M
tk
tji
. By choosing νji = f jitk , it can be shown
i=1 νji
ji ( j )
k
t
is
the
best
local
monomial
approximation to
that fˆji
j
fji tkj near tkj [18]. The successive approximation method for
solving the multi-user power control problem is summarized
in Algorithm 2 below.
4:
(17)
(1)
(2)
with variables Pi , Pi , i = 1, . . . , M . Similar to the
single user problem, (17) is also a nonconvex optimization
problem. It can reformulated as a signomial programming
problem [16] and we use successive convex approximations to
solve for the reformulated problem. Specifically, we introduce
auxiliary variables: Letting tji =
reformulated as
minimize
7:
8:
9:
10:
11:
12:
tk−1
k
νij
← f jitk−1
)
ji ( j
end for
Replace the last N constraints of (18) by fˆji (tj ) =
k
M tji νji
.
k
i=1 νji
Solve
the approximated problem to (18)
if ||(P1 )k − (P1 )k−1 || + ||(P2 )k − (P2 )k−1 || < then
Output (P1 )k and (P2 )k and terminate the loop.
else
k ← k + 1 and go to step 3
end if
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.
A. Performance of the single user power allocation strategy
In the first set of numerical simulations, we consider a
single user scenario and M = 12 distributed antennas. The
performance of the proposed power allocation strategy for
various values of the maximum allowable per antenna transmit
power PT , together with baseline method of equal power
allocation, is illustrated in Figure 2. The corresponding number
of antennas utilized for the proposed strategy is presented in
Figure 3.
In Figure 2, we see that when PT = 45, the total transmitted
power will increase rapidly with increasing SCENR, and when
PT = 60, the total transmitted power increases slowly. This is
an intuitive result because a larger PT implies that more power
can be allocated to the antenna with the largest θi , resulting in
more efficient use of power. Correspondingly, Figure 3 verifies
this by illustrating that as the required SCENR increases, the
number of antennas utilized increase quickly (implying that
antennas with smaller θi are used) for PT = 45 as compared to
PT = 60. As an extreme example, when the required SCENR
is 50, 11 antennas are used in the case when PT = 45 whereas
3 antennas are used when PT = 60.
An interesting observation is that in this scenario, by
allowing a 11.11% increase in PT from 45 to 50, one can
save approximately 100% in transmit power when the required
SCENR increases to 50 (and possibly much more for higher
SCENR).
800
PT = 45
700
PT = 50
P = 55
Total transmitted Power
T
600
P = 60
T
Equal power allocation
500
400
300
200
100
0
30
32
34
36
38
40
42
44
46
48
50
12
P = 45
T
PT = 50
10
Number of antennas utilized
V. N UMERICAL R ESULTS AND D ISCUSSIONS
In this section, we present simulation results for the proposed single user and multi-user power allocation solution
strategies. The mobile users are assumed to be randomly
located; the elements of the channel power (second moment) vector θ (in the multi-user case: channel power vectors θ1 , θ2 , . . . , θM ) are uniformly generated in the interval
(0, θmax ].
PT = 55
P = 60
T
8
6
4
2
0
30
32
34
36
38
40
42
44
46
48
50
SCENR
Fig. 3. Single user power allocation (M = 12 antennas): Number of antennas
utilized
B. Performance of the multi-user power allocation strategy
In the second set of numerical simulations, we consider
allocating power amongst M = 4 distributed antennas in
an N = 6 users scenario. We apply the successive convex
approximation to solve for the power allocation problem. Figof the power variables P1 =
ure 4 illustrates the convergence
(1)
(1)
(1)
(1)
T
(2)
(2)
(2)
(2)
T
P1 , P2 , P3 , P4
and P2 = P1 , P2 , P3 , P4
.
As seen in the figure, the variables converged in 7 iterations.
In addition, it is also observed that higher power is allocated
to P1 in the channel estimation phase (i.e between 10 to 20)
as compared to P2 in the data transmission phase.
In the multi-user scenario, it is not as straightforward to
allocate power as in the single user case where the antennas
are allocated power sequentially until the constraint is satisfied.
Here, all the antennas are utilized but different quantity of
power is allocated to each. Table I shows the total transmitted
power for the proposed power allocation strategy and compares it with the baseline method where power is allocated
equally until all constraints are satisfied. It is observed that
the proposed strategy leads to a savings of 44 − 45% in
total transmitted power, regardless of the required SCENR
constraints.
SCENR
Equal power allocation
Proposed power allocation
Percentage savings
5
64.9
36
44.53
10
134.4
75
44.19
15
204.6
114
44.28
20
275.6
153
44.48
25
346.4
192
44.57
SCENR
Equal power allocation
Proposed power allocation
Percentage savings
30
417.6
231.9
44.46
35
488.8
270.4
44.67
40
560
309
44.83
45
631.4
347.5
44.96
50
704
386
45.17
SCENR
Fig. 2. Single user power allocation (M = 12 antennas): Comparison of
the proposed power allocation strategy (with different PT ) and equal power
allocation method
TABLE I
M ULTI - USER POWER ALLOCATION (N = 6 USERS , M = 4 ANTENNAS ):
S AVINGS IN TOTAL TRANSMITTED POWER AS COMPARED TO EQUAL
POWER ALLOCATION
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE Globecom 2011 proceedings.
25
P(1)
1
P(1)
2
(1)
P3
Transmitted Power
20
(1)
P4
P(2)
1
P(2)
2
15
(2)
P3
(2)
P4
10
5
0
1
2
3
4
5
6
7
8
9
10
Iteration
Fig. 4. Multi-user power allocation (N = 6 users, M = 4 antennas):
Convergence of the power variables using successive convex approximation
VI. C ONCLUSIONS
We considered the power control problem in a downlink
distributed wireless communication system to minimize total
transmit power. We formulated the problem for both the singleuser and multi-user cases. While the problem formulations are
seemingly nonconvex, we provided a simple solution for the
single user problem and demonstrated, via mathematical argument, that it is the optimal power allocation. For the multi-user
case, we used an iterative method, based on successive convex
approximation, to find a convergent solution. We also demonstrated, via numerical simulation, that the proposed single
and multi-user power allocation strategies provided an overall
reduction in transmitted power of up to 45% as compared to
the baseline method of equal power allocation. The proposed
methods to date are based on centralized algorithms. Future
research directions include the development of distributed
algorithms so that each of the processing nodes (associated
with several antennas) can implement the algorithms in a
distributed fashion.
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