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4306 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
Distributed Beamforming for Relay Networks
Based on Second-Order Statistics of the
Channel State Information
Veria Havary-Nassab, Student Member, IEEE, Shahram Shahbazpanahi, Member, IEEE,
Ali Grami, Senior Member, IEEE, and Zhi-Quan Luo, Fellow, IEEE
Abstract—In this paper, the problem of distributed beam-
forming is considered for a wireless network which consists of a
transmitter, a receiver, and
relay nodes. For such a network,
assuming that the second-order statistics of the channel coeffi-
cients are available, we study two different beamforming design
approaches. As the first approach, we design the beamformer
through minimization of the total transmit power subject to the
receiver quality of service constraint. We show that this approach
yields a closed-form solution. In the second approach, the beam-
forming weights are obtained through maximizing the receiver
signal-to-noise ratio (SNR) subject to two different types of power
constraints, namely the total transmit power constraint and
individual relay power constraints. We show that the total power
constraint leads to a closed-form solution while the individual
relay power constraints result in a quadratic programming opti-
mization problem. The later optimization problem does not have a
closed-form solution. However, it is shown that using semidefinite
relaxation, this problem can be turned into a convex feasibility
semidefinite programming (SDP), and therefore, can be efficiently
solved using interior point methods. Furthermore, we develop a
simplified, thus suboptimal, technique which is computationally
more efficient than the SDP approach. In fact, the simplified
algorithm provides the beamforming weight vector in a closed
form. Our numerical examples show that as the uncertainty in
the channel state information is increased, satisfying the quality
of service constraint becomes harder, i.e., it takes more power
to satisfy these constraints. Also our simulation results show
that when compared to the SDP-based method, our simplified
technique suffers a 2-dB loss in SNR for low to moderate values
of transmit power.
Index Terms—Convex feasibility problem, distributed beam-
forming, distributed signal processing, relay networks, semidefi-
nite programming.
Manuscript received October 5, 2007; revised April 14, 2008. First published
May 23, 3008; last published August 13, 2008 (projected). The associate ed-
itor coordinating the review of this manuscript and approving it for publication
was Prof. Andreas Jakobsson. This paper has been published in part in the Pro-
ceedings of IEEE International Conference on Acoustics, Speech, and Signal
Processing (ICASSP), Las Vegas, NV, March 30–April 4, 2008. The research of
V. Havary-Nassab, S. Shahbazpanahi, and A. Grami is supported by the Natural
Sciences and Engineering Research Council (NSERC) of Canada. The research
of Z.-Q. Luo is supported by the National Science Foundation, grant number
DMS-0610037.
V. Havary-Nassab, S. Shahbazpanahi, and A. Grami are with the Faculty of
Engineering and Applied Science, University of Ontario Institute of Technology,
Oshawa, ON L1H 7K4, Canada (e-mail: veria.havarynassab@uoit.ca; shahram.
shahbazpanahi@uoit.ca; ali.grami@uoit.ca).
Z.-Q. Luo is with the Department of Electrical and Computer Engineering,
University of Minnesota, Minneapolis, MN 55455 USA (e-mail: luozq@ece.
umn.edu).
Digital Object Identifier 10.1109/TSP.2008.925945
I. I
NTRODUCTION
T
HE explosive growth of research in wireless communi-
cations has been inspired by the demand for developing
affordable bandwidth-efficient technologies to provide users
with wireless access anywhere anytime. To develop such tech-
nologies, various means of diversity, including time, frequency,
code, and space have to be exploited. These types of diversity
have been well studied in the literature. Recently another
type of diversity, namely multiuser cooperation diversity, has
attracted attentions in the research community [1]–[3]. Com-
munication based on user cooperation, often called cooperative
communications, exploits the spatial diversity of multiuser sys-
tems without the need for using multiple antennas at each user
[4]. In cooperative communications, users relay each other’s
messages thereby providing multiple paths from the source to
the destination.
Emerging wireless technologies, such as sensor and relay
networks, have found applications in cooperative communi-
cations. In fact, users of a wireless network can cooperate
by relaying each other’s messages thus improving the com-
munications reliability. However, the limited communication
resources, such as battery lifetime of the devices and the
scarce bandwidth, challenge the design of such cooperative
communication schemes. Therefore, while ensuring that each
user receives a certain quality of service (QoS), one is often
confronted with the challenge that communication resources
are subject to stringent constraints.
Various cooperative communication schemes have been pre-
sented in the literature. A three-node network is considered in
[5], where one of the nodes relays the messages of another node
towards the third one. For such a network, different cooperative
protocols are then developed and the outage and ergodic capac-
ities are analyzed. This analysis was later extended in [6] to the
case of a relay network where the relay nodes as well as the
receiving and transmitting nodes were equipped with multiple
antennas. The common assumption used in [5] and [6] is that the
perfect
instantaneous channel state information (CSI) is avail-
able at the receiver as well as at the relaying nodes.
Other examples of cooperative communication schemes
are amplify-and-forward [3], coded-cooperation [7], and
compress-and-forward [8]. Of all these schemes, the am-
plify-and-forward approach, due to its simplicity, is of partic-
ular interest. Recently, the amplify-and-forward approach has
been extended to develop space-time coding strategies for relay
networks, thereby opening a new research avenue called dis-
tributed space-time coding [9]–[15]. While the aforementioned
1053-587X/$25.00 © 2008 IEEE
HAVARY-NASSAB et al.: DISTRIBUTED BEAMFORMING FOR RELAY NETWORKS ON SECOND-ORDER STATISTICS OF THE CSI 4307
Fig. 1. Relay network.
cooperative approaches assume different levels of CSI avail-
ability in the network, they all share the common assumption
that the relay nodes operate at their maximum allowable power.
For different relaying strategies, the problem of power allo-
cation between the source and the relay node(s) has been well
studied in the literature [4]. In [16], the problem of optimal
power allocation is considered in the context of coherent com-
bining the relay signals under the aggregate relay power con-
straint. This approach assumes that the relays have the perfect
knowledge of both their receive and transmit
instantaneous CSI.
In [17] and [18], a distributed beamforming strategy has been
developed for the case where the relaying nodes cooperate to
build a beam towards the receiver under individual relay power
constraints. To do so, each relay multiplies its received signal
by a complex weight and retransmits it. In this scheme, the am-
plitude and the phase of the transmitted signals are properly
adjusted such that they are constructively added up at the re-
ceiver. While assuming that the power of each individual relay
is limited, it is assumed in [17] that each relay knows the in-
stantaneous CSI for both backward (transmitter to the relay)
and forward (relay to the receiver) links. Using such an assump-
tion, the network beamforming approach is simplified to a dis-
tributed power control method. In fact, each relay matches the
phase of its weight vector to the total phase of the backward and
forward links. Therefore, only the amplitudes of the complex
weights remain to be determined. These amplitudes are then ob-
tained through maximizing the signal-to-noise ratio (SNR) at
the receiver while guaranteeing that the individual relay powers
meet the corresponding constraints. Interestingly enough, such
a maximization results in relay powers that are not necessarily
at their maximum allowable values. The relaying schemes de-
veloped in [16] and [17] are based on the availability of instan-
taneous CSI, and therefore, they do not allow any uncertainty in
the channel modeling.
In this paper, we consider the problem of distributed beam-
forming under the assumption that the second-order statistics
of the channel coefficients are available. Such an assumption al-
lows us to consider uncertainty in the channel modeling through
introducing the covariance matrices of the channel coefficients.
Based on this assumption, we develop two distributed beam-
forming algorithms. As the first approach, we aim to minimize
the total transmit power required in the relay network subject to
a constraint which guarantees that the receiver QoS (measured
by the receiver SNR) remains above a predefined threshold.
We show that this approach results in a closed-form solution
for the beamforming weights. In the second approach, our goal
is to maximize the receiver SNR subject to two different types
of power constraints: aggregate power constraint as well as
individual relay power constraints. We show that in the case
of constrained aggregate power, the beamforming problem has
a closed-form solution. We also show that in the case of indi-
vidual relay power constraints, the beamforming problem can
be approximately written as a semidefinite programming (SDP)
problem which can be efficiently solved using interior point
methods. Furthermore, to avoid the computational complexity
of SDP, we present a simplified (but suboptimal) technique
which provides the beamforming weight vector in a closed
form.
The remainder of the paper is organized as follows. In
Section II, we present the data model. The power-minimiza-
tion-based beamforming technique is developed in Section III.
Section IV presents the SNR-maximization-based beamforming
algorithms. Simulation results are provided in Section V, and
concluding remarks are given in Section VI.
II. S
YSTEM MODEL
Consider a wireless network which consists of a transmitter,
a receiver, and
relay nodes, as shown in Fig. 1. We assume that
due to the poor quality of the channel between the transmitter
and receiver, there is no direct link between them. As a result,
the transmitter deploys the relay nodes to communicate with the
receiver. Each relay has a single antenna for both transmission
and reception. Assuming a flat fading scenario, let
denote
the channel coefficient from the transmitter to the
th relay and
represent the channel coefficient from the th relay to the
receiver. We also assume that the second-order statistics of the
channel coefficients
and are known. In fact, we
model
and as random variables with known second-order
statistics.
We herein study a two-step amplify-and-forward (AF) pro-
tocol. During the first step, the transmitter broadcasts to the re-
lays the signal
, where is the information symbol and
4308 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
is the transmit power. We assume that , where
represents the statistical expectation, and denotes the
amplitude of a complex number. The signal
received at the
th relay is given by
(1)
where
is the noise at the th relay whose variance is known
to be
.
During the second step, the
th relay transmits the signal
which can be expressed as
(2)
where
is the complex beamforming weight used by the th
relay. At the destination, the received signal can be written as
(3)
where
is the received signal and is the receiver noise whose
variance is known to be
. Using (1) and (2), we can rewrite
(3) as
(4)
Our goal is to obtain the weight coefficients
such
that the SNR at the receiver is either maximized subject to some
power constraint(s) or kept above a certain threshold while min-
imizing the total transmit power.
III. P
OWER MINIMIZATION
In this section, we aim to find the beamforming weights
such that the total relay transmit power is mini-
mized while maintaining the receiver QoS at a certain level,
i.e., the receiver SNR is required to be larger than a certain
predefined threshold
. Mathematically, we solve the
following optimization problem:
subject to (5)
where SNR is defined as the ratio of the signal power
to
the noise power
. The total relay transmit power can be
obtained as
(6)
where
represents Hermitian transpose and the following
definitions are used:
Here, denotes the transpose operator, represents
a diagonal matrix whose diagonal entries are the entries of the
vector
, and is the identity matrix.
Using (4) and assuming that the relay noises
, the re-
ceiver noise
, and the channel coefficients are all in-
dependent from each other, the total noise power
can then be
obtained as
(7)
where
represents complex conjugate and the following def-
initions are used:
Also, using (4), the signal component power can be ob-
tained as
(8)
where
is the correlation matrix of the vector
and represents the ele-
ment-wise Schur–Hadamard product, that is,
(9)
Using (6), (7), and (8), the optimization problem in (5) can be
written as
subject to (10)
or, equivalently, as
subject to (11)
where we have changed the optimization variable to
.
HAVARY-NASSAB et al.: DISTRIBUTED BEAMFORMING FOR RELAY NETWORKS ON SECOND-ORDER STATISTICS OF THE CSI 4309
It is worth mentioning that if is chosen such that is
negative definite, then the optimization problem in (11) becomes
infeasible.
One can easily show that the inequality constraint in (11) is
satisfied with equality at the optimum, for otherwise, the optimal
could be scaled down to satisfy the constraint with equality,
thereby decreasing the objective function and contradicting op-
timality. Therefore, we can rewrite (11) as
subject to (12)
The Lagrange multiplier function can now be defined as
(13)
Using the following definition for differentiation of
with respect to :
where and denote the real and imaginary parts, we obtain
that
(14)
Equating
to zero, we obtain that
(15)
It follows from (15) that
should be chosen as one of the eigen-
vectors of the matrix
and is the
corresponding eigenvalue. Multiplying both sides of (15) with
yields
(16)
where in the last equality we have used the constraint in (12).
It follows from (16) that minimizing
amounts to mini-
mizing
(or, equivalently, maximizing ). This means that
has to be selected as the largest eigenvalue of
. As a result, the solution to (5) is given by
(17)
where
(18)
Here
represents the normalized principal eigen-
vector of a matrix, and
is a scalar which is
chosen to satisfy the equality constrain in (12), i.e.,
.
Eventually, the optimum beamforming weight vector can be
written as
(19)
The minimum total relay transmit power for any feasible
is
given by
(20)
where
denotes the principal eigenvalue of a matrix.
IV. SNR M
AXIMIZATION
In this section, we consider a different approach to obtain the
beamforming weight vector. Our goal is to maximize the re-
ceiver SNR subject to two different types of relay power con-
straints. We first study the case where the total relay transmit
power is constrained, and then investigate the scenario where
the individual relay transmit powers are limited.
A. Total Power Constraint
In this subsection, we aim to maximize the SNR subject to
a constraint on the total transmit power. That is, we solve the
following optimization problem:
subject to (21)
where
is the maximum allowable total transmit power.
Using (6), (7), and (8), the optimization problem (21) can be
rewritten as
subject to (22)
To solve (22), let us write the weight vector
as
(23)
where
satisfies . The optimization problem (22)
can be rewritten as
subject to and (24)
where the following definitions are used:
4310 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
As the objective function in (24) is monotonically increasing
in
, for any value of , this objective function is maximized
for
. Hence, the optimization problem in (24) can be
simplified as
subject to (25)
or, equivalently, as
subject to (26)
It is well known [23] that the objective function in (26) is glob-
ally maximized when
is chosen as the principal generalized
eigenvector of
, or, equivalently, as the prin-
cipal eigenvector of the matrix
. It is easy
to show that such a global maximizer of the objective function
in (26) can be normalized to satisfy the unit-norm constraint in
(26). Therefore, the solution to (26) is given by
(27)
As a result, the beamforming weight vector can be written as
(28)
and the maximum achievable SNR can be expressed as
(29)
B. Individual Power Constraint
In this subsection, we consider a different type of power con-
straint. More specifically, we consider the case where each relay
node is restricted in its transmit power. Such a case is of partic-
ular interest when the relay nodes are restricted in their battery
lifetimes. In this case, we aim to solve the following optimiza-
tion problem:
subject to for (30)
or, equivalently
subject to for (31)
where
is the maximum allowable transmit power of the th
relay, and
is the th diagonal entry of the matrix . Using
the definition
, the optimization problem in (31) can
be written as
subject to for
and rank (32)
or, equivalently, as
subject to
and for
and rank (33)
where
represents the trace of a matrix and means
that
is constrained to be a symmetric positive semidefinite
matrix. The optimization problem in (33) is not convex and may
thus not be amenable to a computationally efficient solution. Let
us ignore the rank constraint in (33). That is, using a semidefinite
relaxation, we aim to solve the following optimization problem:
subject to
and for
and (34)
Due to the relaxation, the matrix
obtained by solving the
optimization problem in (34) will not be of rank one in general.
If
happens to be rank one, then its principal eigenvector
yields the optimal solution to the original problem.
Note that the optimization problem in (34) is quasi-convex. In
fact, for any value of
, the feasible set in (34) is convex. Let
be the maximum value of obtained by solving the optimization
problem (34). If, for any given
, the convex feasibility problem
[19]
find
such that
and for
and (35)
is feasible, then we have
. Conversely, if the convex
feasibility optimization problem (35) is not feasible, then we
conclude
. Therefore, we can check whether the op-
timal value
of the quasi-convex optimization problem in
(34) is smaller than or greater than a given value
by solving
the convex feasibility problem (35).
Based on this observation, we can use a simple algorithm to
solve the quasi-convex optimization problem (34) using bisec-
tion technique, solving a convex feasibility problem at each step.
We assume that the problem is feasible, and start with an interval
known to contain the optimal value . We then solve
the convex feasibility problem at its midpoint
,
to determine whether the optimal value is larger or smaller than
. We update the interval accordingly to obtain a new interval.
That is, if
is feasible, then we set , otherwise, we choose
and solve the convex feasibility problem in (35) again.
HAVARY-NASSAB et al.: DISTRIBUTED BEAMFORMING FOR RELAY NETWORKS ON SECOND-ORDER STATISTICS OF THE CSI 4311
Remark 1: In order for any bisection method to achieve a
global optimum, it is required that the feasible values of the
search parameter constitute a connected set, otherwise the al-
gorithm can lead to a local optimal. In our problem, feasible
values of
are the same as the set (denoted by ) of the achiev-
able objective values of the optimization problem in (32) when
the rank constraint is relaxed. The set
is certainly connected
because the objective function in (32) is a continuous function
which maps any connected set (in this case the convex feasible
region of (35)) to another connected set.
Remark 2: To choose the initial value for
and , one can se-
lect
and where is
chosen. In fact, one can easily show that the maximum achiev-
able SNR with the total power constraint
is
larger than or equal to the maximum SNR achieved by solving
(31).
Remark 3: To solve the convex feasibility problem (35),
one can use the well-studied interior-point-based methods. For
example, the SeDuMi [20] is an interior point-method-based
package which produces a feasibility certificate if the problem
is feasible.
Remark 4: Once the maximum feasible value for
is ob-
tained, one can replace it into (34). This turns (34) into a convex
problem which can be solved efficiently using interior-point-
based methods.
Remark 5: It is worth mentioning that our problem formu-
lation is applicable to both random and deterministic channel
cases. In the case of random channels, our beamforming
methods may not be optimal for individual channel realiza-
tions, rather our techniques are designed to be optimal in a
statistical sense. Naturally our algorithms may perform poorly
when they are applied for a specific channel realization. In
order to design beamforming techniques for a specific channel
realization, one needs to know the channel coefficients pre-
cisely. In this case, our formulation is still applicable, however,
if the channel coefficients deviate slightly from their nominal
values by unknown random fluctuations, the performance of
our beamforming algorithms can become degraded drastically.
In such a channel modeling, the maximum SNR can be very
low due to lack of coherent combining of relay signals at
the receiver. This is a well-known phenomenon and has been
studied in the recent literature where robust beamforming has
been of primary concern (see, for example, [23] and references
therein). To compensate the lack of coherence in relay signals,
one of the two following approaches can be taken: one can
model the channel deviations from their nominal values into
the correlation matrices and design a beamforming technique
which is statistically optimal. Obviously, the receiver SNR will
be smaller than that for the known channel case. Alternatively,
one can use a robust technique which guarantees the worst-case
performance for all channel coefficients that belong to an
uncertainty set. Naturally the “size” of the uncertainty set deter-
mines the SNR loss compared to the known channel (coherent
combining) case. Recently, worst-case optimization-based
beamforming has been the focus of several studies, see [23] and
[25]. The results developed in [23] can be straightforwardly
applied to the beamforming techniques developed in this paper
to provide robustness against unknown mismatch between
the presumed and the actual channel coefficients, thereby
protecting the performance against the lack of perfect coherent
combining.
Remark 6: In semidefinite relaxation, the solution may not
be rank-1 in general simply because the feasible set of the op-
timization problem (34) is a subset of that of the optimization
problem (33). Interestingly, in our extensive simulation results,
we never encountered a case where the solution to the SDP
problem had a rank higher than one. For the cases where the
SDP problem has a solution with rank higher than one, several
randomization techniques have been proposed in the literature
which use the solution to the SDP problem to provide a good ap-
proximation to the rank-1 problem [21]. The basic idea in ran-
domization is to use
to generate a set of candidate weight
vectors
and then select the best solution among these can-
didates. One such randomization technique, eigendecomposes
as and chooses , where is
a vector of zero-mean, unit-variance complex circularly sym-
metric uncorrelated Gaussian random variables. That is
’s
are samples from the complex Gaussian distribution
.
For cases where the SDP problem has a solution with rank
higher than one, it is possible to establish a bound for perfor-
mance of the randomization technique. To show this, consider
the problem
subject to (36)
where
is a matrix with all zero entries except for the th
diagonal element which is equal to
. The SDP relaxation
can be written as
subject to (37)
Using bisection, we can solve the SDP relaxation in polynomial
time yielding an optimal
and a satisfying
(38)
Clearly,
is an upper bound for the optimal value of (36).
Now consider the nonconvex quadratic optimization problem
subject to (39)
Its SDP relaxation can be written as
subject to (40)
By the definition of
, it follows that is a global
optimal solution for (40). Let us sample from the complex
Gaussian distribution
. By the result of [22], we
can generate in randomized polynomial time an approximate
solution
satisfying
4312 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
where is a constant. In light of (38), we fur-
ther obtain
implying
where the last step follows from the positive semidefiniteness of
. Rearranging the terms, we obtain
implying that is a -optimal solution of (36). In other words,
the SDP relaxation approach provides a
ap-
proximation to the nonconvex fractional quadratic optimization
problem (36).
As was shown above, designing the beamformer based on the
SNR maximization with individual relay power constraints re-
quires an iterative procedure where, at each step, a convex feasi-
bility problem is solved. We now turn (31) into an optimization
problem which can be solved easily without significant compu-
tational complexity. To do so, we ignore
in the numerator
of the objective function in (31) and aim to solve the following
optimization problem:
subject to (41)
In fact, the objective function in (41) is an upper bound to the
objective function in (31).
The objective function in (41) is globally maximized for
(42)
where
is the normalized principal eigenvector of the matrix
and can be any scalar parameter. If we choose
(43)
where
denotes the th entry of and
(44)
then, the global maximizer in (42) becomes the solution to (41)
as well.
V. S
IMULATION RESULTS
In our numerical examples, we consider a network with 20
relay nodes
. The channel coefficients and are
assumed to be independent from each other for any
and .We
also assume that the channel coefficient
can be written as
(45)
where
is the mean of and is a zero-mean random vari-
able. We assume that
and are independent for .
For any
, we choose and
, where is a uniform random variable randomly
chosen from the interval [0,
] and is a parameter which
determines the level of uncertainty in the channel coefficient
.
Note that as
,if is increased, the variance of
the random component
is increased while the mean is de-
creased. This, in turn, means that the level of the uncertainty in
the channel coefficient
is increased.
Similarly, we model the channel coefficient
as
(46)
where
is the mean of and is a zero-mean random vari-
able. We assume that
and are independent for .
For any
, we choose and
, where is a uniform random variable chosen from
the interval [0,
] and is a parameter which determines the
level of uncertainty in the channel coefficient
.
Based on this channel modeling, we can write the
entry
of the matrices
and , respectively, as
where is the Kronecker function. It is worth mentioning that
the distributions of
and do not play a role, for our algo-
rithms use only the second-order statistics of
and and not
their distributions. Also, we consider asymptotic regimes where
the correlation matrices
and are exactly known, and there-
fore, we do not need to generate the channel coefficients. Alter-
natively, one can obtain the sample estimates of these correlation
matrices from a finite number of samples of the channel coeffi-
cients which are generated randomly. In this case, the mismatch
between the true and the sample correlation matrices may de-
grade the performance of our beamforming techniques. To cope
with such performance degradation, one has to resort to robust
techniques proposed in [23], where positive and negative diag-
onal loading techniques are used to compensate for the lack of
the precise knowledge of the correlation matrices.
Throughout our numerical examples, the transmit power
is assumed to be the same as receiver noise power which is 0
dBW.
A. Power Minimization
Fig. 2 shows the minimum total relay transmit power,
versus the SNR threshold for 5 dB and for
different values of
. Fig. 3 illustrates versus for
5 dB and for different values of . In these figures,
the transmit powers have been plotted only for those values of
that are feasible.
As can be seen from these figures, when the uncertainty in
and coefficients (measured, respectively, by and )is
increased, it becomes exceedingly difficult to guarantee that the
SNR is above a certain threshold
. That is, as (or ) is in-
creased, it takes more power to ensure that the SNR is above a
certain (feasible)
. Also, as (or ) is increased, the max-
imum feasible value of
is decreased.
As can be seen from Figs. 2 and 3, the transmit power in-
creases drastically near some limiting
. This limiting value of
is the one which makes the optimization problem infeasible.
HAVARY-NASSAB et al.: DISTRIBUTED BEAMFORMING FOR RELAY NETWORKS ON SECOND-ORDER STATISTICS OF THE CSI 4313
Fig. 2. Minimum total relay transmit power versus SNR threshold , for dif-
ferent values of
and for 5dB.
Fig. 3. Minimum total relay transmit power versus SNR threshold
, for dif-
ferent values of
and for 5dB.
B. SNR Maximization
In Fig. 4, we have plotted the maximum achievable SNRs,
given as in (29), versus the maximum allowable total transmit
power
for 5 dB and for different values of .In
Fig. 5, we have shown the maximum achievable SNRs versus
for 5 dB and for different values of . As can
be seen from these figures, for any given
, the maximum
achievable SNR is decreased as the uncertainty in the
(or in
the
) coefficients is increased.
In the next numerical example, we consider the case where
the individual relay nodes are limited in their transmit powers.
We assume that the relay nodes are divided into two groups.
The relay nodes in each group have the same maximum al-
lowable transmit power, while the maximum allowable transmit
power for one group is twice that for the other group, that is,
.
We use the SDP-based technique proposed in Section IV-B to
Fig. 4. Maximum achievable SNR versus the maximum allowable total
transmit power
for different values of and for
0dB.
Fig. 5. Maximum achievable SNR versus the maximum allowable total
transmit power
for different values of and for 0dB.
obtain the optimum value for matrix , say . We have inves-
tigated the solution to SDP problem for different values of
and , for different maximum allowable transmit powers, and
for different values of
and . In our intensive simulation
examples, we have observed that the matrix
is always rank
one, and therefore, no randomization technique is required. As
a result, the optimum value for the vector
is the same as the
principal eigenvector of
. Fig. 6 shows the maximum achiev-
able SNRs, when the individual relay nodes have the aforemen-
tioned power constraints, versus the total relay transmit power
, for 5 dB and for different values of
. Fig. 7 illustrates the maximum achievable SNRs versus
for 5 dB and for different values of . For this ex-
ample, we have also plotted the performance of the simplified
technique in Figs. 8 and 9. As can be seen from Figs. 6–9, for any
given , the maximum achievable SNR of both the SDP-based
4314 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
Fig. 6. Maximum achievable SNR, with individual relay power limits
, versus the transmit power for different values of
and for 0dB.
Fig. 7. Maximum achievable SNR, with individual relay power limits
, versus the transmit power for different values of
and for
0dB.
technique and the simplified method is decreased when the un-
certainty in
(or in ) coefficients is increased. In Fig. 10, we
compare the performance of the techniques developed in this
paper for SNR maximization for
5 dB. As can be
seen from this figure, in this example, the maximum achievable
SNR under constrained total transmit power and that under con-
strained individual replay powers are very close to each other.
It can also be seen that when the individual relay powers are
constrained, the simplified method suffers a 2-dB loss in SNR
as compared to the SDP-based technique for low to moderate
values of
. For large values of , the simplified method has
a maximum SNR close to that of the SDP-based approach.
Fig. 8. Maximum achievable SNR of the simplified technique, with individual
relay power limits
, versus the transmit power for
different values of
and for 0dB.
Fig. 9. Maximum achievable SNR of the simplified technique, with individual
relay power limits
, versus the transmit power for
different values of
and for 0dB.
VI. CONCLUSIONS
In this paper, we studied the problem of distributed beam-
forming in a network which consists of a transmitter, a receiver
and
relay nodes. Assuming that the second-order statistics
of the channel coefficients are available, we considered two
different approaches to beamforming design. As the first ap-
proach, we designed the beamformer through minimization of
the total transmit power subject to a constraint which guarantees
the receiver quality of service. We showed that this approach
yields a closed-form solution. In the second approach, we
obtained the beamforming weights through maximizing the re-
ceiver SNR subject to two different types of power constraints,
namely total transmit power constraint and individual relay
power constraints. We herein have shown that the total power
constraint leads to a closed-form solution while the individual
HAVARY-NASSAB et al.: DISTRIBUTED BEAMFORMING FOR RELAY NETWORKS ON SECOND-ORDER STATISTICS OF THE CSI 4315
Fig. 10. Maximum achievable SNR versus the total transmit power for
different methods.
relay power constraints result in a quadratic programming
optimization problem. The later optimization problem does
not have a closed-form solution. However, it is shown that
using semidefinite relaxation, it can be turned into a convex
feasibility semidefinite programming, and therefore, can be
efficiently solved using interior point methods. Furthermore,
we presented a simplified (but suboptimal) technique which can
be used to avoid the computational complexity of semidefinite
programming. Our simplified algorithm provides the beam-
forming weight vector in a closed form. Simulation results show
that when compared to the semidefinite programming-based
method, our simplified technique suffers a 2-dB loss in SNR
for low to moderate values of transmit power.
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Veria Havary-Nassab (S’08) was born in Sanandaj,
Kurdistan, Iran. He received the B.Sc. and M.Sc. de-
grees in electrical engineering from Sharif University
of Technology, Tehran, Iran, in 2000 and 2003, re-
spectively. He is currently working towards the Ph.D.
degree at the Department of Electrical and Computer
Engineering, University of Toronto, ON, Canada.
From September 2001 to January 2002, he was a
Research Assistant at the Electronic Research Center
of Sharif University of Technology. From February
2002 to January 2007, he worked as an Electrical En-
gineer in Iran’s industry sector. Since February 2007, he has been a Research
Associate at the Faculty of Engineering and Applied Science, University of On-
tario Institute of Technology, Oshawa, ON, Canada. His research interests in-
clude signal processing for communications, MIMO communications, cooper-
ative communications, and cognitive radio.
4316 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
Shahram Shahbazpanahi (M’02) was born in
Sanandaj, Kurdistan, Iran. He received the B.Sc.,
M.Sc., and Ph.D. degrees in electrical engineering
from Sharif University of Technology, Tehran, Iran,
in 1992, 1994, and 2001, respectively.
From September 1994 to September 1996, he was
a Faculty Member with the Department of Electrical
Engineering, Razi University, Kermanshah, Iran.
From July 2001 to March 2003, he was a Post-
doctoral Fellow with the Department of Electrical
and Computer Engineering, McMaster University,
Hamilton, ON, Canada. From April 2003 to September 2004, he was a Visiting
Researcher with the Department of Communication Systems, University of
Duisburg-Essen, Duisburg, Germany. From September 2004 to April 2005,
he was a Lecturer and Adjunct Professor with the Department of Electrical
and Computer Engineering, McMaster University. Since July 2005, he has
been with the Faculty of Engineering and Applied Science, University of
Ontario Institute of Technology, Oshawa, ON, Canada, where he holds an
Assistant Professor position. His research interests include statistical and array
signal processing, space–time adaptive processing, detection and estimation,
smart antennas, spread-spectrum techniques, MIMO communications, DSP
programming, and hardware/real-time software design for telecommunication
systems.
Dr. Shahbazpanahi is currently serving as Associate Editor for the IEEE
T
RANSACTIONS ON
SIGNAL
PROCESSING and IEEE S
IGNAL PROCESSING
LETTERS. He is also a member of the Sensor Array and Multichannel (SAM)
Technical Committee of the IEEE Signal Processing Society.
Ali Grami (M’86–SM’06) received the B.Sc. degree
from the University of Manitoba, Winnipeg, Canada,
in 1978, the M.Eng. degree from McGill University,
Montreal, Canada, in 1980, and the Ph.D. degree
from the University of Toronto, Toronto, Canada, in
1986, all in electrical engineering.
Upon his graduation in 1986, he joined Nortel net-
works, Montreal, where he contributed to the defini-
tion and the development of the first North American
digital cellular mobile standard. In 1989, he joined
Telesat Canada, Ottawa, where he was the lead re-
searcher and principal designer of Canada’s Anik-F2 Ka-band system—the first
broadband access satellite system in North America. He taught at the Univer-
sity of Ottawa, Ottawa, and Concordia University, Montreal, while he was with
the industry. Since 2003, he has been with the University of Ontario Institute
of Technology, Oshawa, ON, Canada, where he is currently an Associate Pro-
fessor. His research interests include satellite and wireless communications.
Zhi-Quan Luo (F’07) received the B.Sc. degree
in applied mathematics from Peking University,
Beijing, China, in 1984 and the Ph.D. degree from
the Massachusetts Institute of Technology (MIT),
Cambridge, in 1989.
Subsequently, he was selected by a joint com-
mittee of the American Mathematical Society and
the Society of Industrial and Applied Mathematics
to pursue Ph.D. studies in the United States. After
a one-year intensive training in mathematics and
English at the Nankai Institute of Mathematics,
Tianjin, China, he entered the Operations Research Center and the Department
of Electrical Engineering and Computer Science at the Massachusetts Institute
(MIT). From 1989 to 2003, he held a faculty position with the Department of
Electrical and Computer Engineering, McMaster University, Hamilton, ON,
Canada, where he eventually became the department head and held a Canada
Research Chair in Information Processing. Since April 2003, he has been with
the Department of Electrical and Computer Engineering at the University of
Minnesota (Twin Cities) as a Full Professor and holds an endowed ADC Chair
in digital technology. His research interests lie in the union of optimization
algorithms, data communication and signal processing.
Prof. Luo serves on the IEEE Signal Processing Society Technical Commit-
tees on Signal Processing Theory and Methods (SPTM), and on the Signal Pro-
cessing for Communications (SPCOM). He is a corecipient of the 2004 IEEE
Signal Processing Society’s Best Paper Award and has held editorial positions
for several international journals, including the
Journal of Optimization Theory
and Applications, the SIAM Journal on Optimization, the Mathematics of Com-
putation, and the IEEE T
RANSACTIONS ON
SIGNAL PROCESSING. He currently
serves on the editorial boards for a number of international journals, including
Mathematical Programming and Mathematics of Operations Research.
