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Sensor Selection and Precoding Strategies for
Wireless Sensor NetworksAlessandro Nordio, Member, IEEE, Alberto Tarable, Member, IEEE, Fabrizio Dabbene, Senior Member, IEEE, and
Roberto Tempo, Fellow, IEEE
AbstractSensor selection has recently received a growing in-
terest in the literature, motivated by the worldwide deployment
of wireless sensor networks and by the increase in the number of
available applications. In our framework, sensors take remote mea-
surements of a quantity of interest and communicate their observa-
tions through a noisy, multiantenna wireless communication link.
In this context, we propose a scheme for optimally selecting out
of sensor nodes on the basis of the amount of information they
convey to a common receiver/actuator. Moreover, a suitable linear
precoder is employed and optimized at each transmitter with the
aim of maximizing the mutual information between the observed
variable and the signal received by the actuator. The sensor se-lection problem is known to be combinatorial, and several com-
putable relaxations are available in the literature. In this paper,
the optimality conditions are formally expressed in an informa-
tion-theoretic context, and both semi-definite-programming relax-
ations and greedy schemes, leading to computable techniques for
large values of and , are presented. Moreover, specific results
for the cases of high and low signal-to-noise ratio on the wireless
channel are derived. Numerical simulations show that knowledge
of the channel state at the transmitter mayleadto an increase of the
achievable mutual information and determine a different choice of
sensors, thus pointing out that our approach significantly improves
upon selection schemes that neglect the characteristics of the com-
munication layer.
Index TermsChannel precoding, convex relaxation, green net-works, mutual information, sensor selection, wireless sensor net-
works.
I. INTRODUCTION
W IRELESS SENSOR NETWORKs (WSNs) and wire-less sensor-and-actuator networks (WSANs) havefound applications in many fields, including traffic control,
weather forecast, pollution control, environmental monitoring
and surveillance, see for instance the surveys [1], [2]. In
WSNs/WSANs, the sensors measure a given physical variable
and transmit their observations to a processing center, some-times coinciding with the network gateway, which processes
the received information.
Manuscriptreceived June27, 2014; revisedFebruary 20, 2015; accepted May
04, 2015. Date of publication June 01, 2015; date of current version July 09,
2015. The associate editor coordinating the review of this manuscript and ap-proving it f or publication was Prof. Y.-W. Peter Ho ng.
The authors are with IEIIT-CNR (Institute of Electronics, Telecommu-nications and Information Engineering of the Italian National Research
Council), Torino 10129, Italy (e-mail: [email protected]; [email protected]; fabrizio.dabbene@polito .it; roberto [email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2015.2439239
The technology growth of recent years has allowed for a re-
duction in the production and deployment cost of sensors. As
a consequence, the number of deployed sensors is often larger
than needed for processing. On the other hand, the operational
cost of sensors is usually still rather large, due to battery con-
sumption, especially in the signal transmission phase. To cope
with this issue, at fixed time intervals, the processing center
needs to perform a sensor selection, with the aim of reducing
the number of transmitting sensors, while keeping the others in
sleep mode. This green-networkstrategy allows power saving
and increases the lifetime of sensor batteries.
Motivated by these issues, the literature considering the
problem of optimally selecting a subset of sensors from a set of
possible choices has received increasing interest. The interested
reader is referred to [3] for a survey on different schemes
currently used to select sensors, based on different kinds of
metrics (coverage, target-tracking and localization schemes,
single/multiple mission assignments). Clearly, whenever the
number of sensors is large, optimal selection becomes a compu-
tationally difficult combinatorial problem. Several approaches
have been proposed to obtain suitablerelaxations with the goal
of making the problem computationally feasible, even for a
moderate-to-large number of sensors.Forexample, in [4] the sensor scheduling problemis reformu-
lated as a nonlinear deterministic optimal control problem that
could be in principle attacked via a tree-search approach, but this
solution is generally impractical for a relatively large number of
sensors. The work [5] proposes a convex relaxation technique
to reduce the combinatorial problem to a convex programming
problem, which can be easily solved in polynomial time. The
most prominent advantages of this approach are its immediate
applicability, due to the powerful existing convex optimization
algorithms, and the possibility of handling various performance
criteria with energy and topology constraints. In [6], the case of
multistep measurements is considered, and the process state isestimated by means of a Kalman filter with the goal of mini-
mizing an objective function related to the Kalman filter error
covariance matrix.
An information-theoretic approach to sensor selection and lo-
calization in sensor networks for target tracking is studied in [7].
There, the problem of choosing a sensor from a set of candidates
is formulated, with the goal of minimizing the expected condi-
tional entropy of the posterior target location distribution. In this
case, both the prior target location distribution and the locations
and observation models of the candidate sensors are assumed to
be known. In [8], [9], it is shown that the information gain is a
submodular function. This property allows to prove guaranteed
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lower bounds on the performance of a greedy selection scheme,
in which the sensor providing the larger performance gain is
added to the selected ones iteratively. In particular, the perfor-
mance of a greedy algorithm is proved to be within a fraction
of from the optimal one. In [10] the so-called budgeted
case is considered, in which a positive integer cost is associated
to each sensor. The paper presents a general algorithm for max-
imizing non-decreasing submodular functions under a budget
constraint with general (additive) costs, which still guarantees
the same optimality gap of . A greedy selection algo-
rithm is also proposed in [11], whose complexity scales linearly
with the total number of sensors.
It should be remarked that the approaches previously dis-
cussed assume that all sensors send their observations to
a centralized processing center. However, all these works
completely neglect the influence of the wireless channel char-
acteristics, which can largely affect the quality of the received
signal. This consideration is at the basis of the present work, in
which we demonstrate how optimal sensor selection strongly
benefits from the knowledge of the channel characteristics, andhow a careful transmitter design should take into account the
statistics of the observations.
In order to carry out this task, we study a WSN composed of
sensors, transmitting signals to a common receiver through
multiple antennas. We explicitly consider the influence of the
communication channel characteristicson the sensor selection
problem and we assume that sensors communicate by using
orthogonal channels, so that they do not interfere with each
other. Among the total of received signals, the receiver has
to choose , ( ) according to some optimality criterion.
To formally define this criterion, we follow an information-the-
oretic approach. In particular, we adopt as objective function of
the wireless sensor selection problem the mutual information
between the measured variable and the set of selected signals.
Such mutual information is maximized by employing at each
node an optimal linear precoder. This scenario is also consid-
ered in our previous work [12]. However, in [12], the sensors
do not employ precoders to optimize the amount of information
conveyed to the receiver/actuator. The addition of suitably op-
timized precoders proves to be crucial in the considered frame-
work.
Signal precoding for multiantenna systems is a widely inves-
tigated technique, which has also been implemented in many
commercial applications (see [13] and references therein).
The idea is to process the information to be transmitted byexploiting the knowledge of the communication channel. In
particular, linear precoding techniques are the most commonly
used, thanks to their low complexity and their optimality from
the information-theoretic point of view. In our scenario, sensor
precoding is akin to relay precoding in amplify-and-forward
relay communication systems, for which optimal precoding
is derived in [14]. There, however, a single-relay network is
considered, while in our case multiple sensor nodes perform
linear precoding. In [15], joint precoder optimization in a
network of multiple nodes transmitting correlated signals is
considered. However, in our scenario, sensor node inputs are
affected by independent noise samples, and this makes the joint
precoder optimization of [15] not applicable. The main noveltyof the precoders proposed in this paper is that they take into
specific account the available information on the remote sensor
characteristics.
In summary, the contributions of our work are:
the formulation of the sensor selection problem in pres-
ence of a multiple-input multiple-output (MIMO) wire-
less channel with channel state information available at the
transmitters (CSIT);
introduction and optimization of precoders taking into spe-
cific account sensor characteristics, by using an informa-
tion-theoretical approach;
relaxation of the optimization problem to a suboptimal
semidefinite program (SDP) [5] and approximation
through a greedy algorithm;
analysis of the cases of high and low signal-to-noise ratio
(SNR) on the wireless links;
numerical assessment of the impact of sensor selection pa-
rameters and of the influence of CSIT on the system per-
formance.
The rest of the paper is structured as follows. Section II de-
scribes in detail the considered scenario. In Section III, wederive locally optimal linear precoders for sensor nodes.
In Section IV, we formulate the problem of optimal sensor
selection for our scenario. In Section V, we describe sub-
optimal solutions based on relaxation and greedy search. In
Section VI, we consider the limit regimes for high and low
received signal-to-noise ratio (SNR) on the wireless channel.
In Section VII, we compare the proposed sensor selection
algorithms by means of numerical simulations and we also
assess the impact of the precoders on the performance. Finally,
in Section VIII, we draw some conclusions.
II. SYSTEM MODEL
We consider the scenario where we are interested in esti-
mating an unknown vector1 from the observations
performed by sensing devices. This approach is similar to
the one discussed in [5] and subsequent works. However, dif-
ferently from these works, we here consider the more realistic
situation where the sensing devices act remotely, and their ob-
servations are to be sent to a common receiver through a wire-
less channel.
Each sensing device performs a different observation of the
quantity , and obtains linear measurements corrupted by
additive noise, which can be written as
where is a vector of size and is the -th observation
matrix of size (different for each sensor). The mea-
surementnoise vectors aremodeled as independent Gaussian
random variables with zero mean and covariance matrices ,
. Similarly, we assume that the quantity is a
Gaussian random vector with zero mean and covariance matrix
.
As summarized in Fig. 1, in thesetup considered in this paper,
the sensors have to transmit their observations through a wire-
less channel, and are hence equipped with transmit antennas.
1
Throughout the paper, uppercase and lo wercase bo ldface letters denote m a-trices and vectors, respectively. The identity matrix is denoted by , and
denotes the determinant of the matrix .
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Fig. 1. Wireless sensor.
To optimize transmission, the observation is first precoded
by the matrix and then sent to the antennas. In other words,
the transmitted signal is given by where the pre-
coders are matrices. Due to limited sensor energy,
the average power irradiated by the transmitter of the -th sensor
is assumed to be bounded by , i.e.,
(1)
The sensing devices transmit their (precoded) signals to a
common processing center, which consists of a wireless re-
ceiver equipped with antennas. Signals are assumed to be
transmitted over MIMO orthogonal channels, so that they do
not interfere with each other. The signal received by the pro-
cessing center from the -th sensor is then given by
(2)
where the matrix represents the -th MIMO channel
and represents additive noise, assumed independent of ,
and modeled as a complex Gaussian random vector with zero
mean and covariance matrix . More specifically, the -th
entry of the matrix is the gain of the link connecting the
-th transmit antenna to the -th receive antenna. In the case
allsensors transmit, the received signals can be arranged in the
vector , of size , given by
..
.
..
.
..
.
(3)
We define the matrices ,
, and . Note that and
are block-diagonal of size and
respectively, and is of size . Then, we can rewrite
(3) more compactly as where
and collect the measure-
ment noise and the thermal noise, respectively. The equivalent
noise at the receiver is represented by the random vector
, which has covariance matrix
(4)
and is the covariance matrix of .
III. PRECODEROPTIMIZATION
The precoders represent free variables of the system. Usu-
ally, they are designed in order to maximize a suitable cost
function. In our case, we are interested in the amount of infor-
mation about the unknown that the sensors convey towards
the receiver, under the transmit power constraints in (1). In-
deed, by maximizing the information transfer, we allow the re-ceiver to form better estimates of . The information-theoretical
concept of mutual information [16] between two random vari-
ables provides us with the suitable quantitative measure. There-
fore, our aim is to find the optimal precoding matrix maxi-
mizing the mutual information . Such optimal precoder
can be obtained by solving the following constrained maximiza-
tion problem
(5)
where denotes the mutual information between and
, conditioned on the knowledge of and . For Gaussian
random variables is defined as
(6)
where and are
the covariance matrices of and , respectively. Also, the
symbol in (6) denotes conditioning. It follows that the mutual
information can be written as
(7)
where in the third line of (7) we used the property
(Sylvesters theorem) and in the last
equality we defined and . Then the
term in (7) can be rewritten as follows
(8)
where in the first equality we used the expression for given
in (4), and in thesecond equality we applied the matrix inversion
lemma [17] (i.e., ,
where and ). Moreover, in (8) for
the sake of simplicity we defined . The
last equality of (8) comes from the fact that
. By joining the
results in (7) and (8) the mutual information can be rewritten as
(9)
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By definition the covariance matrix is block-diag-
onal with blocks , is also block-diagonal with
blocks and where
. Then, is also block-diagonal with
blocks
(10)
As for the transmit power constraint in (1), we have
(11)
Then, the maximization problem in (5) can be rewritten as
(12)
where . We observe that the solution
of the above problem involves a joint optimization of all pre-
coders. In general, the optimal precoder for the -th sensor, ,
is function not only of and , but rather depends on the
whole matrices and . This follows from the fact that the
matrix does not have a block-diagonal structure and the
cost function in (12) cannot be written as a sum of terms, each
depending on the parameters of a single sensor. Thus, the com-
putation of the optimal precoder, , requires the knowledge of
the matrices and at every sensor. This assumption is how-
ever unrealistic in our case since it would require an intensive
exchange of information among sensors.
We then resort to a locally optimal solution: instead of
finding the optimal precoder , we compute the precoder
where is obtained by solving the
local maximization problem
(13)
for . In (13) we recalled the definition of
given in (10). We moreover observe that the dependence on
can be found only in the term . Then, solving the
above problem is equivalent to maximize over the mutual
information at each sensor, and the solution, , does
not depend on and for . Therefore no exchange of
information among sensors is required.
Proposition 1: The locally optimal precoder that maxi-
mizes the problem in (13) is given by
(14)
where
the matrices , , and are obtained from
the eigenvalue decomposition of the Hermitian matrices
and , i.e., and
;
the eigenvalues are ordered in decreasing order, i.e,
where and
where ;
is an rectangular diagonal matrix whose diag-
onal elements are given by
(15)
, where and
. Moreover
, a nd t he p arameter is c hosen so that
.
The expression of the locally optimal precoder in (14) has
the following intuitive explanation: the purpose of the unitarymatrices and is to rotate the channel and observation
matrices, respectively, in order to allow the precoder to act
directly over the system eigenvalues. The term
normalizes to unity the average power of the signal , and
the diagonal matrix shares the available power, , among
the available channel modes. The proof of (14) is given in
Appendix A. Expressions for the locally optimal precoder in
the high- and low-SNR regimes are derived in Section VI.
IV. SENSORSELECTION PROBLEM
As discussed in Section I, we are interested in selecting a
suitable subset of sensors of cardinality maximizingan appropriate performance metric, i.e, in our case, mutual in-
formation. In practice, to estimate the quantity , the processing
center makes use of sensors only, out of the available, at
the advantage of a lower complexity and energy saving. Clearly,
better estimates can be obtained if the amount of information
contained in the selected signals is larger.
Mathematically, the sensor selection problem can be formal-
ized by introducing a selection matrix , which
left-multiplies the vector . This matrix is formed by
sub-blocks of size each, and the -th block of ,
, is defined as
if sensor is the -th selected sensorotherwise
for and . Also has the following
properties: i) for each , there exists one and only
one index such that , and ii) for each ,
there is at most one index such that .
In particular, i) guarantees that exactly sensors are selected,
while ii) ensures that sensors are selected no more than once.
With this notation defined, the optimal sensor selection can be
formulated as the following problem.
Problem 1 (Wireless sensor selection): Based on the knowl-
edge of sensor characteristics (i.e., measurement matrices
and measurement noise covariance matrices ,
), and of the wireless communication channels
(i.e., transmission matrices , and thermal
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noise variance ), the problem of choosing the selection
matrix that maximizes the mutual information , can
be formulated as:
(16)
where denotes the received signal in the case the optimal
precoders have been designed according to (14), i.e.,
where and
. The mutual information obtained by an optimal choice of
signals received from sensors is thus given by . In the
next section we show how this problem can be reformulated in
terms of the notation introduced so far.
A. Problem Reformulation
Since and are Gaussianrandom variables wefirst rewrite
the mutual information as
(17)
where and are,
respectively, the covariancematricesof and , and where
denotes the covariance matrix of . Thus (17) rewrites
(18)
It is important to observe that, by construction, the covari-
ance matrix has a block-diagonal structure with blocks of
size . Such a property is due to the fact that both
and are block-diagonal matrices. Therefore we can write
whence it follows that the mutual
information (18) can be rewritten as
(19)
where in we have applied Sylvesters theorem, and in
we defined , we exploited the fact that the matrix
is idempotent (i.e., ) and that, because of the block-di-
agonal structure of , . Note also that is
a block-diagonal matrix with blocks .
Specifically the -th block of , , can be written aswhere , , are Boolean variables. It
turns out that the sensor selection problem can be reformulated
by introducing the Boolean selection vector de-
fined as where means that the -th
sensor has been selected. It follows that the matrix is
block-diagonal and its -th block is given by where
is the -th diagonal block of and is given by
Thus where
(20)
Finally, by substituting (20) in (19), we obtain that the mutual
information can be expressed as
(21)
This observations allow us to state the next proposition, which
provides a useful reformulation of the optimal sensor selectionproblem in terms of the Boolean selection vector .
Proposition 2 (Optimal wireless sensor selection): The op-
timal selection vector that maximizes the mutual information
is given by the solution of the following optimization
problem
(22)
where is defined in (20), and denotes the -norm,
equal to the number of nonzero elements in .Remark 1: Note that the problem formulated in Proposition
2 can be solved based solely on the knowledge of the quantities
, , , , , and thus can be solved prior to per-
forming the actual measurements, thus allowing to switch off
the sensors that are not selected. This will permit an
optimization of the energy resources.
In the next Section, we show relaxations and greedy algo-
rithms aiming at reducing the complexity of the sensor selection
problem in Proposition 2.
V. SENSORSELECTION ALGORITHMS
The solution of the optimal wireless sensor selection problem
formulated in Proposition 2 amounts at solving a hard non-convex optimization problem, due to its intrinsic combinatorial
nature. In principle, a solution would require the evaluation of
the performance index for each of the possible selections
of active sensors. Specific branch-and-bound techniques, in the
spirit of [18], [19], can be devised for the numerical solution
of this problem, but this approach is clearly not practical un-
less and are relatively small. In the other cases, different
approximations or relaxations are possible, as discussed in the
next subsections.
A. Concave Relaxation
A computable continuous relaxation of the optimizationproblem (22) can be immediately obtained by relaxing the
requirement that the selection vector is to be binary. In fact,
the cost function is concave for
. This approach, analogous of that introduced in [5],
leads to the following semidefinite program (SDP)
(23)
Note that, in the above maximization problem, the -norm is
replaced by the -norm. This is a standard technique employed
in optimization to derive convex relaxations for combinatorial
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problems, and has been applied, e.g., in the context of com-
pressive sensing in [20]. The complexity of this SDP algorithm
scales as , and hence it can be applied also to rather large
networks. In general, the optimal solution of this SDP relax-
ation will take fractional values, so that some kind of sorting
and rounding is necessary to obtain the desired solution. The
simplest approach consists in selecting the elements of
with the largest values. A more sophisticated technique, whichleads to better solutions, consists in applying an iterative proce-
dure, where each iteration solves areweighted -problem. This
procedure has been suggested in [21] to enhance the sparsity of
the solution of a -minimization problem, and is summarized
in Algorithm 1.
Algorithm 1:Reweighted SDP Relaxation
1) 1) Initialization: ,
2) 2) Solve the weighted SDP relaxation:
3) 3) Update the weights:
4) IF OR t he n umber o f elements o f larger t han
is equal to then EXIT; ELSE set , GOTO 2.
In Algorithm 1 at step the weights are updated to give
more importance to the smallest values of , in order to force
their values towards zero. For a formal analysis of the algorithm
the reader is referred to [21]. The parameter at step 3) is
introduced in order to provide stability and to ensure that a zero-
valued component in does not strictly prohibit a nonzero
component at step . In particular, in [21] it is suggested to
select slightly smaller than the expected nonzero magnitudes
of the components of , but the algorithm is rather robust to this
choice.
B. Greedy Algorithm
In this section, we study the properties of a greedy selec-
tion scheme providing an approximated solution of the problem
described in Proposition 2. The greedy algorithm consists inchoosing the sensors one at a time, until sensors are finally
selected. At the -th step ( ), the selected sensor
is the one maximizing the objective function when combined
with the previously chosen . Formally, we first note that
the objective function in (22) can be rewritten as the following
set-function2}
(24)
where is a set of sensors. Thegreedy sensor selection algo-
rithm is described in Algorithm 2.2A set-function is defined as a function whose input is a given set .
Algorithm 2:Greedy Sensor Selection
1) 1) Initialization: , ,
2) 2) Select greedily the next sensor:
(25)
3) 3) Update the selected measurement set:
(26)
4) 4) Set , IF GOTO 2.
The use of Algorithm 2 in the context of sensor selection has
been originally proposed in [11], [22], where its suboptimality
properties are discussed in details, based on the concept of sub-
modularity. Submodularity plays for discrete functions the same
role as convexity for continuous functions, see e.g. the survey
[23], and has been leveraged in various problems in the contexts
of optimal sensor placement [24] and leader selection [25].Definition 1 (Submodular function): For a given finite set ,
a set-function , where denotes the power set, is
said to be submodularif for any , and for any
, it holds
(27)
In other words, a submodular function satisfies the so-called
diminishing incrementsproperty. The submodular function is
monotone if , .
The importance of the submodularity stems from the results
proven in [26], where the authors study the problem of maxi-
mizing a monotone submodular function under a simple cardi-nality constraint, that is
(28)
They prove that the solution of the greedy procedure applied to
problem (28) is not less than times the optimal
one. Since the mutual information in (21) is a submodular func-
tion [23], an immediate application of Definition 1 provides the
following lemma.
Lemma 1 (Suboptimality of the greedy algorithm): Let
be the mutual information obtained by solving Algorithm 2 and
let be the mutual information achieved by the optimal solution
of (22). Then,
It should be remarked that in [26], [27] it is also shown that
the performance of the greedy algorithm is the best practi-
cally achievable, unless . Indeed, they show that no
polynomial-time algorithm can provide a better bound than
. This consideration motivates our choice of adopting
the greedy algorithm in the solution of the numerical examples
in Section VII. On the other hand, the bound does
not hold for more general constraints than the cardinality one
considered here, and specifically tailored algorithms have to be
devised, see e.g. the works [9], [10] that consider budget-type
constraints. In this latter case, the use of the SDP relaxation,
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where such constraints can be included in the problem almost
for free, can become competitive.
VI. HIGH- A NDLOW-SNR REGIMES
In this section, we consider two particular scenarios for
which the sensor selection problem considerably simpli-
fies, i.e., the cases of high and low SNR on the wirelesschannel. More precisely, the high-SNR regime is defined
by the condition , while the low-SNR regime cor-
responds to . For the whole section, let the
eigenvalue decompositions of and be, respec-
tively, and
where , and
.
A. High-SNR Regime
We start from the high-SNR regime and assume that
, . This condition can be
obtained by choosing , and supposing thatthe channel matrices are all full rank: the latter assumption
is physically common both in the case when the receiver is in
line of sight, provided that it is not exceedingly far from the
transmitter, and when the receiver is not in line of sight, in
presence of a rich-scattering propagation environment. With
such hypotheses, we can prove that optimal sensor selection be-
comes independent of the channel matrices , ,
and boils down to the same problem of, e.g., [5]. As a direct
consequence, in the high-SNR regime it is inessential to pre-
code the transmitted signals.
Proposition 3(High-SNR regime): If ,
, then, for , the mutual information
does not depend anymore on the channel realizations. More pre-cisely, the optimal sensor selection problem defined in (22) can
be rewritten as
(29)
Proof: Under the hypotheses of the theorem, from (15) we
see that, for , the optimal precoder for the -th sensor
satisfies
(30)
where is a constant needed to satisfy . By
substituting the above values into (14), we obtain
(31)
where is the diagonal matrix of the largest eigenvalues
of . In turn, this can be substituted into (10), where
, yielding .
Now, suppose that has rank . As a consequence,
we have that, for
(32)
where has rank . It follows that
. Thus, for , (9) be-
comes
(33)
which does not depend on the channel matrices .
Next, we derive the expression of the optimal sensor
selection problem in the high-SNR regime. We first ob-
serve that for any matrix and any nonzero constant ,
. Thus the term
in (20) can be rewritten as
where is given in (10) and
. By substituting this result into (20) we obtain,
for
(34)
Substituting the above expression into (22), we obtain (29).
The fact that, in the high-SNR regime, the optimal sensor se-
lection does not depend on channel realizations has a great prac-
tical importance. Indeed, the problem (29) can be solved once
at the beginning and can be used whenever the channels are ingood conditions, without the need for changing the selection ac-
cording to channel time variations. However, notice that the re-
ceiver must know the channel realizations for the selected nodes
in order to properly exploit the received signals.
Notice also that any precoder that makes full rank will
give the same mutual information as in (33) for . Even
without precoding, i.e., with a choice like , the re-
sulting mutual information would be the same as in the case of
optimal precoding. Seeing the overall channel from to as a
cascade of the measurement channel from to and the com-
munication channel from to , at high SNR saturates
to , and the sensor selection problem becomes equiva-
lent to the one treated in [5].
B. Low-SNR Regime
In the next proposition, we consider instead the low-SNR
regime and we show that, in such case, the cost function to be
maximized is a linear combination of the contributions of each
sensor.
Proposition (Low-SNR regime): If , the optimal
sensor selection problem in (22) reduces to
(35)
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Proof: It is shown in Appendix B that, for large , all
defined in (15) eventually reach zero but the first. Thus, for
, we obtain
otherwise (36)
In words, beamforming in the direction of the best parallel
channel is the optimal precoding strategy in the low-SNRregime, as it is usual for waterfilling power allocation.
Regarding the mutual information given in (19), we observe
that, for , . Under this condition (19)
becomes
(37)
where we have used the fact that for ,
, and . The trace
in (37) can be written as
(38)
where we have used the expression of the optimal precoder in
(14) and the value of in (36). Substituting (38) into (37) and
noticing that under the operator the constant terms in-
dependent of can be removed, we find that the optimal sensor
selection can be written as in (35).
The locally optimal precoder can be computed by observing
that the problem in (35) can be easily solved by a greedy algo-
rithm that takes the largest values of
. Unlike the high-SNR case, however, the solu-tion does depend on the channel matrices and thus must be
recomputed every time the channel realizations change. More-
over, the value of can be interpreted as the total useful power
reaching the receiver from the -th sensor.
VII. NUMERICAL RESULTS
In this section, through computer simulations, we assess the
performance of the different sensor selection algorithms and we
evaluate the impact of the precoder on the achievable mutual
information. In our tests, sensors are chosen out of a total set
of . The variable under control has length . We
assume , except otherwise stated. Each sensor performs
scalar measurements and transmits through
antennas. The receiver is equipped with antennas. The
Fig. 2. (Top) Achievable mutual information versus SNR for the optimal
sensor selection algorithm and its counterparts designed for high-SNR and
low-SNR regimes. (Bottom) Sets of sensors selected by the optimal algorithm,for different SNR (circle markers), and by its low- and high-SNR counterparts
(triangle markers).
Fig. 3. Comparison among sensor selection algorithms in terms of achievable
mutual information versus SNR, with and without precoder. Average over 100channel realizations.
measurement noise vectors have covariance matrices
, . In each simulation, the matrices ,
, are kept fixed and their (real) entries are drawn from
a normal distribution.
The entries of the matrices , , are i.i.d. com-
plex circularly Gaussian with zero-mean and unit-variance, as it
is the case when the sensors transmit through Rayleigh-fading
channels.
To assess the benefit of precoding, we compare the case where
is chosen to be equal to the locally optimal precoder
defined in (13) (or its counterparts for high or low SNR) with
the no-precoder case, i.e., . In all cases, the averagetransmitted power of sensor is set to ,
. The signal-to-noise ratio of every wireless channel is
therefore .
In a first test, we concentrate on the benefits of the optimal
sensor selection of Proposition 2. To this end, we set
and we keep fixed the set of s and the corresponding locally
optimal precoder for the whole simulation.
The results are depicted in Fig. 2, where we also show the
mutual information achieved by the high-SNR and low-SNR
approximations of optimal selection, as defined in Propositions
3 and 4, respectively. In the lower part of Fig. 2, we show the
sets of sensors selected by each algorithm as a function of SNR.
As it can be seen, the performance of the algorithm adapted
to the low-SNR regime and described in Proposition 4 merges
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Fig. 4. Outage probability plotted vs. SNR for 5.5 bit threshold mutual information (a), and plotted vs. the threshold mutual information for (b).In both figures and .
with the performance of the optimal selection scheme for SNRs
not larger than 0 dB, while it exhibits bumps for medium SNR
values dueto the fact the low-SNR selection rule is suboptimal.
For high SNR, the low-SNR algorithm reaches an asymptoticvalue of about 6.5 bits, which is more than 1 bit lower than the
maximum achievable mutual information. The high-SNR algo-
rithm of Proposition 3 merges with the optimal selection for
, while it loses up to 1 bit of mutual informa-
tion formedium SNR values. There are barely perceptible slope
changes in the optimal selection curve in correspondence with
changes in the set of selected sensors. It can be observed that,
for , the optimal selection significantly differs
from the choice made neglecting the effect of the transmission
channel. For instance, for SNR around 10 dB the optimal choice
corresponds to sensors 2, 17 and 20, while the choice made ne-
glecting the channel would be 1, 8 and 13, corresponding to a
loss of about 0.4 bits.In a second test shown in Fig. 3 we compare the mutual in-
formation achieved by the optimal sensor selection of Proposi-
tion 2, the reweighted convex relaxation of Algorithm 1 and the
greedy sensor selection of Algorithm 2. We also evaluate the
benefit of introducing the locally optimal precoder, by consid-
ering the cases with and without signal precoding. To this end,
we set and we average over 100 Rayleigh-fading channel
realizations. We here consider the case where the entries of
are correlated. To this end in this test the covariance matrix of
is kept fixed and drawn from a Wishart distribution. As it can
be seen, the precoder yields a gain of almost 2 dB at low SNR,
while for sufficiently high SNR the gain in terms of mutual in-
formation provided by the optimal precoders tends to be negli-gible. In both scenarios, the two approximated algorithms per-
form almost as well as the optimal sensor selection for low SNR,
while they become slightly suboptimal for larger SNR, losing
about 0.2 bits w.r.t. optimal performance. Among the two sub-
optimal solutions, concave relaxation has a slightly better per-
formance for medium SNR values than greedy sensor selection,
although the gain is always lower than 0.1 bits. Considering that
concave relaxation still exhibits a larger complexity than greedy
sensor selection, our conclusion is that the latter represents the
better trade-off between algorithm complexity and achievable
mutual information.
Inthe next two tests, we investigate more in detail the per-
formance of the greedy algorithm and in particular we derive
its outage probability which is defined as the probability that
the achieved mutual information is below a given threshold. We
compare in terms of outage probability the cases where locally
optimal precoders and no precoders are used. In addition, we
consider a suboptimal precoder which neglects the informationabout the sensors matrices. This precoder maximizes
by assuming that the signals , , are uncorre-
lated. The expression of this suboptimal precoder is given by
where the -th diagonal element of is
given by where is a constant allowing
for power normalization and is the -th eigenvalue of . In
Fig. 4(a), we show the outage probability versus SNR for a mu-
tual information threshold equal to 5.5 bit, while in Fig. 4(b) we
depict the outage probability versus the threshold on the mutual
information for . In both figures, we observed
10000 channel realizations per point and we selected
sensors out of a total of .As it can be seen from Fig. 4(a), the choice provides a
significant gain with respect to the case in terms of both
SNR and decay slope. In general, the decay slope increases with
due to the diversity effect provided by the channel matrices,
which are assumed to be independent. For all considered values
of , the locally optimal precoder can save up to 3 dB of SNR
w.r.t. the case where no precoder is employed, at an outage prob-
ability of . Instead, the suboptimal precoder loses 12 dB
against the no-precoder case. For there is a further gain
of about 45 dB with respect to the case , at the price
of a small increase in the complexity of the greedy algorithm.
These results clearly show the impact of the precoder on the per-
formance of sensor selection and that a careful precoder designcannot neglect the information on the sensor matrices.
In Fig. 4(b), the outage probability provided by the greedy se-
lection algorithm for and is depicted
as a function of the threshold mutual information. In this case,
an increase of from 2 to 3 yields a gain of about 1.5 bits at
an outage probability of , while a further increase of to 4
provides an additional gain of more than 1 bit, for all considered
precoders. Notice that for precoder optimization entails
a gain of almost 1 bit.
VIII. CONCLUSIONS
Inthis paper, we studied the sensor selection problem in the
case where the observations are transmitted through noisy wire-
less MIMO channels. At each transmitter we employed and op-
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timized linear precoders in order to maximize the information
conveyed to the processing center. The optimality conditions
have been derived in an information-theoretic framework. The
numerical simulations clearly demonstrate how optimal sensor
selection strongly benefits from the knowledge of the channel
characteristics, and how precoder design should take into ac-
count the structure of the sensor matrices. This testifies for the
importance of the aggregate framework proposed in this paper,which considers simultaneously the sensor and the channel char-
acteristics. Further studies will consider the problem of remote
Kalman filtering in the same wireless context. Finally, the pos-
sibility of introducing additional constraints in the choice of the
sensors can be considered, as in [28].
APPENDIX A
PROOF OFPROPOSITION 1
In order to solve (13) we first consider the eigen-
value decompositions and
, where and are unitary
matrices. We also assume that the eigenvalues are orderedin decreasing order, i.e, where
and where
. By following the same steps as in [14] we
define the arbitrary matrix such that is given by
(39)
By using (39) in (10) the matrix is given by
and the
constraint in (13) reduces to
. Then, by substituting these results in (13)
and after some algebra, we can show that the problem in (13)
is equivalent to finding the matrix solving the followingproblem [14]
(40)
For the case of a square matrix , it has been shown in [14]
that the optimal is diagonal. Even though in our case is
rectangular, the same result applies. In particular the diagonal
elements of , are given by
where and .
This result can be obtained by following the same steps as in
[14, Section III] and generalizing for the case of a rectangular
. For a diagonal (40) takes the form
(41)
The value of the variables solving (41) can be obtained by
applying the Karush-Kuhn-Tucker conditions, as done in [14],
and is given by (15).
APPENDIX B
PRECODEROPTIMIZATION IN THE LOW-SNR REGIME
To prove that for the result in (36) holds we first re-
call the solution for given in (15). Also, we remind that
and appearing in (15) are the
eigenvalues of and , respectively, sorted in de-
creasing order. We start by noticing that for any value of ,if
(42)
After some elementary manipulations, such condition can be
written as where , .
Notice that, since and
, then also . From
what we have said, it follows that there are exactly nonzero
precoder gains if and only if
. Now suppose that and that , i.e., only the
first precoder gain is nonzero. As a consequence, we have:
(43)
where the second equality comes from the power normalization.
Solving for gives
(44)
It is easy to verify that is a continuous, strictly
increasing function of with
and . Because of that, and since
, there will be a value of , call it , for which
. It follows that, for all , only the first
precoder gain is nonzero.
REFERENCES
[1] I. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, A surveyon sensor networks,IEEE Commun. Mag., vol. 40,no. 8, pp.102114,2002.
[2] J. Yick, B. Mukherjee, and D. Ghosal, Wireless sensor networksurvey,Comput. Netw., vol. 52, no. 12, pp. 22922330, 2008.
[3] H. Rowaihy, S. Eswaran, M. Johnson, D. Verma, A. Bar-Noy, T.Brown, and T. L. Porta1, A survey of sensor selection schemes inwireless sensor networks,SPIE, vol. 6562, p. 65621A, 2007.
[4] L. Meier, J. Peschon, and R. Dressler, Optimal control of measure-ment subsystems, IEEE Trans. Autom. Control, vol. 12, no. 5, pp.
528536, 1967.[5] S. Joshiand S. Boyd, Sensorselection viaconvexoptimization,IEEE
Trans. Signal Process., vol. 57, no. 2, pp. 451462, 2009.[6] Y. Mo, R. Ambrosino, and B. Sinopoli, Sensor selection strategies
for state estimation in energy constrained wireless sensor networks,Automatica, vol. 47, pp. 13301338, 2011.
[7] H. Wang, K. Yao,and D. Estrin, Information-theoretic approaches forsensor selection and placement in sensor networks for target localiza-tion and tracking,J. Commun. Netw., vol. 7, no. 4, pp. 438449, Dec.2005.
[8] A. Krause and C. Guestrin, Near-optimal value of information ingraphical models, presented at the Conf. Uncertainty Artif. Intell.(UAI), Edinburgh, Scotland, U.K., Jul. 2629, 2005.
[9] A. Krause and C. Guestrin, Optimizing sensing: From water to theweb,IEEE C omput. Mag., vol. 42, no. 8, pp. 3845, 2009.
[10] M. Sviridenko, A note on maximizing a submodular set function sub-ject to knap sack constraint, Oper. Res. Lett., vol. 32, pp. 4143, 2004.
[11] M. Shamaiah, S. Banerjee, and H. Vikalo, Greedy sensor selectionunder channel uncertainty, IEEE Wireless Commun. Lett., vol. 1, no.4, pp. 376379, 2012.
7/23/2019 07114341
11/11
NORDIOet al.: SENSOR SELECTION AND PRECODING STRATEGIES FOR WIRELESS SENSOR NETWORKS 4421
[12] A. Nordio, A. Tarable, and F. Dabbene, Optimal sensor selectionstrategies in the presence of wireless communication links, presentedat the 19th World Congr. Int. Federation Autom. Control (IFAC), CapeTown, South Africa, Aug. 2429, 2014.
[13] M. Vu and A. Paulraj, MIMO Wireless Linear Precoding, IEEESignal Process. Mag., vol. 24, no. 5, pp. 86105, Sep. 2007.
[14] X. Tang and Y. Hua, Optimal design of non-regenerative MIMOwireless relays, IEEE Trans. Wireless Commun., vol. 6, no. 4, pp.13981407, 2007.
[15] J. Fang, H. Li, Z. Chen, and Y. Gong, Joint precoder design for dis-tributed transmission of correlated sources in sensor networks,IEEETrans. Wireless Commun., vol. 12, no. 6, pp. 29182929, Jun. 2013.
[16] D. MacKay, Information Theory, Inference and Learning Algo-rithms. New York, NY, USA: Cambridge Univ. Press, 2002.
[17] R.Hornand C.Johnson, Topics in Matrix Analysis. Cambridge, U.K.:Cambridge Univ. Press, 1991.
[18] E. Lawler and D. Wood, Branch-and-bound methods: A survey,Oper. Res., vol. 14, pp. 699719, 1966.
[19] W. Welch, Branch-and-bound search for experimental designs basedon D-optimality and other criteria,Technometrics, vol. 24, no. 1, pp.4148, 1982.
[20] D. Donoho, Compressed sensing,IEEE Trans. Inf. Theory, vol. 52,no. 4, pp. 12891306, 2006.
[21] E. Cands, M. Wakin, and S. Boyd, Enhancing sparsity by reweightedminimization, J. Fourier Anal. Appl., vol. 14, pp. 877905, 2008.
[22] M. Shamaiah, S. Banerjee, and H. Vikalo, Greedy sensor selection:
Leveraging submodularity, presented at the IEEE Conf. DecisionControl, Atlanta, GA, USA, Dec. 1517, 2010.
[23] A. Krause and D. Golovin, Submodular function maximization, inTractability: Practical Approaches to Hard Problems (to appear).Cambridge, U.K.: Cambridge Univ. Press, Feb. 2014.
[24] A. Krause, R. Rajagopal, A. Gupta, andC. Guestrin, Simultaneous op-timization of sensor placements and balanced schedules, IEEE Trans.
Autom. Control, vol. 56, no. 10, pp. 23902405, Oct. 2011.[25] A. Clark, L. Bushnell, and R. Poovendran, A supermodular opti-
mization framework for leader selection under link noise in linearmulti-agent systems,IEEE Trans. Autom. Control, vol. 59, no. 2, pp.283297, 2014.
[26] G. Nemhauser, L. Wolsey, and M. Fisher, An analysis of the approxi-mations for maximizing submodular set functions, Math. Programm.,vol. 14, pp. 265294, 1978.
[27] A. Krause, A. Singh, and C. Guestrin, Near-optimal sensor place-ments in Gaussian processes: Theory, efficient algorithms and empir-
ical studies,J. Mach. Learn. Res., vol. 9, pp. 235284, 2007.[28] Y. Wang, M. Sznaier, and F. Dabbene, A convex optimization ap-
proach to wo rst-case optimal sensor selection, presented at the I EEEConf. Decision Control, Firenze, Italy, Dec. 1013, 2013.
Alessandro Nordio (S00M03) is a Researcher
with the Institute of Electronics, Computer andTelecommunication Engineering of the Italian
National Research Council. In 2002 he receivedthe Ph.D. in Telecommunications from Ecole
Polytechnique Federale de Lausanne, Lausanne,
Switzerland. From 1999 to 2002, he performedactive research with the Department of Mobile
Communications at Eurecom Institute, Sophia
Antipolis (France). From 2002 to 2009 he was apost-doc researcher with the Electronic Departmentof Politecnico di Torino, Italy. His research interests are in the field of signal
processing, wireless sensor networks, theory of random matrices, and crowd-
sourcing systems.
Alberto Tarable (S00M02) received the Laureadegree (summa cum laude) in 1998 and the Ph.D.
degree in Electronic Engineering in February 2002,
both fro m Politecnico di Torino. Fro m 2002 to 201 2,he worked as a researcher in the Department of
Electronics and Telecommunications of Politecnicodi Torino. From 2012, he holds a research position in
the Institute of Electronics, Computer and Telecom-munication Engineering of the Italian National
Research Council. His research interests includeMIMO systems and space-time coding, anytime
coding and coding schemes for relay channels.
Fabrizio Dabbene (M02SM09) is currently
Senior Researcher at CNR-IEIIT, Torino, Italy.He received the Ph.D. in Systems and Computer
Engineering in 1999 from Politecnico di Torino.His research interests include probabilistic and
randomized methods for systems and control, robust
control and identification of complex systems,convex optimization and modeling of environmental
systems. On these topics, he has published morethan 80 research papers, which include 30 articles
published in international journals a monographand an edited book. Dr. Dabbene has been an Associate Editor for the IEEE
TRANSACTIONS ON AUTOMATIC CONTROL (20082012) and of Automatica(20082014), Program Chair for the CACSD Symposium of the 2010 IEEE
MSC, Chair of the IEEE Technical Committee on CACSD (20102013) and of
the IFAC Technical Committee on Robust Control (2011-present), member ofCEB (20022008) and IPC member of various IEEE conferences. He is elected
member of the IEEE-CSS Board of Governors for the years 20142016, andserves as IEEE-CSS Vice-President for Publication Activities for the year 2015.
Roberto Tempo (F00) is currently a Director of
Research at CNR-IEIIT, Torino, Italy. He has held
visiting positions at Chinese Academy of Sciences,Beijing, China; Kyoto University, Kyoto, Japan; The
University of Tokyo, Tokyo, Japan; University ofIllinois at Urbana-Champaign, USA; and Columbia
University, New York, USA. He is a co-author ofthe book Randomized Algorithms for Analysis and
Control of Uncertain Systems (Springer-Verlag,
London) 2013. His research activities are focusedon the analysis and design of complex systems with
uncertainty, and various applications within information technology.Dr. Tempo is a Fellow of the IFAC. He is a recipient of the IFAC Out-
standing Paper Prize Award for a paper published in Automatica and of theDistinguished Member Award from the IEEE Control Systems Society. He is
a Corresponding Member of the Academy of Sciences, Institute of Bologna,Italy, Class Engineering Sciences. In 2010 he has served the IEEE Control
Systems Society as President. Since 2015, he is serving as Editor-in-Chief
of Automatica. He has been Editor for Technical Notes and Correspondenceof the IEEE TRANSACTIONS ON AUTOMATIC CONTROL in 20052009 and a
Senior Editor of the same journal in 20112014. He was General Co-Chair forthe IEEE Conference on Decision and Control, Florence, Italy, in 2013.
