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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 5, MARCH 1, 2013
A Game-Theoretic Approach for Energy-Efficient
Contention-Based Synchronization in
OFDMA Systems
Giacomo Bacci, Member, IEEE, Luca Sanguinetti, Member, IEEE, Marco Luise, Fellow, IEEE, and
H. Vincent Poor, Fellow, IEEE
Abstract—The purpose of this paper is to provide a novel energyefficient perspective to the problem of contention-based synchronization in orthogonal frequency-division multiple-access communication systems. This is achieved by modeling the terminals and
their corresponding receivers at the base station as economic and
rational agents that engage in a noncooperative game. In the proposed game, each one trades off its available resources (transmit
power and detection strategy) so as to selfishly maximize its own
revenue (in terms of probability of correct detection) while saving
as much energy as possible and satisfying quality-of-service requirements given in terms of probability of false alarm and timing
estimation accuracy. The existence and uniqueness of the equilibrium of the game are studied. In particular, a necessary and sufficient condition on the system parameters is given for the equilibrium to exist. An iterative and distributed algorithm based on
best-response dynamics (at the transmit side) and a practical parameter estimation (at the receive side) are proposed to achieve the
equilibrium point. Numerical results are used to highlight the effectiveness of the proposed solution and to make comparisons with
existing alternatives in terms of power consumption, synchronization time, and estimation accuracy.
Index Terms—Best-response dynamic, contention-based synchronization, energy efficiency, game-theory, generalized Nash
equilibrium, GLRT, IEEE 802.16, LTE-Advanced, OFDMA,
power allocation.
I. INTRODUCTION
A. Motivation
T
HE demand for high data rates in wireless communications has led to a strong interest in orthogonal
frequency-division multiple-access (OFDMA) technologies
Manuscript received June 11, 2012; revised August 27, 2012 and November
13, 2012; accepted November 13, 2012. Date of publication December 03,
2012; date of current version February 12, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Dr.
Samson Lasaulce. The research leading to these results has received funding
from the People Programme (Marie Curie Actions) of the European Union’s
Seventh Framework Programme (FP7/2007-2013) under REA grant agreement
n. PIOF-GA-2011-302520 GRAND-CRU Game-theoretic Resource Allocation
for wireless Networks based on Distributed and Cooperative Relaying Units.
This work was presented in part at the Fiftieth Annual Allerton Conference
on Communications, Control, and Computing, Monticello, IL, USA, October
2012.
G. Bacci is with the Department of Information Engineering, University of
Pisa, 56126 Pisa, Italy, and also with the Department of Electrical Engineering,
Princeton University, Princeton, NJ 08544 USA (e-mail: giacomo.bacci@iet.
unipi.it; [email protected]).
L. Sanguinetti and M. Luise are with the Department of Information Engineering, University of Pisa, 56126 Pisa, Italy (e-mail: luca.sanguinetti@iet.
unipi.it; [email protected]).
H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TSP.2012.2231679
due to their advantages in terms of dynamic channel allocation,
spectral efficiency and robustness to multipath distortions. For
these reasons, OFDMA-based technologies have been adopted
by the WiMAX alliance [1] and the Third-Generation Partnership Project (3GPP) [2] for efficient packet-based mobile
broadband communications.
To maintain orthogonality among subcarriers of different
users and avoid the occurrence of multiple access interference
(MAI), uplink signals arriving at the base station (BS) should
be aligned to the local time and frequency references [3]. For
this purpose, the WiMAX alliance specifies a network entry
procedure called initial ranging (IR) by which subscribers can
achieve uplink synchronization and power control [4]. A similar
procedure called random access (RA) is specified by the 3GPP
for the (advanced) long-term evolution (LTE) standard [5]. In
their basic forms, the IR and RA functions are contention-based
synchronization procedures developing through the following
steps. Each terminal that intends to establish a communication
link with the BS (also known as eNodeB) achieves downlink
synchronization using a common control channel. The synchronization parameters estimated in the downlink are used by each
terminal as references in the subsequent uplink phase, during
which each terminal notifies its entry request by transmitting
a randomly chosen code over a specified set of subcarriers.1
As a result of different positions within the radio coverage
area, synchronization signals transmitted by different terminals
arrive at the BS with different time delays and power levels.
After identifying the active codes and their corresponding
timing and power information, the BS broadcasts a response
packet indicating which codes have been detected and giving
instructions for timing and power adjustments. If a terminal
does not receive any response message, after a specified time
it performs another synchronization attempt using a higher
power level. To minimize the probability of collision, the time
between two successive attempts is typically randomized on
the basis of some specified contention resolution methods. For
instance, the approach adopted in WiMAX relies on the binary
exponential backoff (BEB) algorithm [4].
From the above discussion, it follows that code identification as well as multiuser timing and power estimation are the
main tasks of the BS in contention-based synchronization processes. These problems have received great attention in the last
1In WiMAX, the entry request of each terminal is generated from a binary
phase-shift keying ranging code and is transmitted over groups of adjacent subcarriers arbitrarily positioned within the available spectrum. In LTE, it is obtained by applying different cyclic shifts to a Zadoff-Chu sequence [5] and it is
transmitted over a single set of adjacent subcarriers.
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BACCI et al.: A GAME-THEORETIC APPROACH FOR ENERGY-EFFICIENT CONTENTION-BASED SYNCHRONIZATION IN OFDMA
few years and some solutions are currently available (see for
example [6]–[11]). All these works are focused on the detection
and estimation problems mentioned above, under the assumption that the terminals deterministically increase their transmit
powers upon successful synchronization. In this work, we look
at the problem from a different perspective by focusing on the
joint detection/estimation and power allocation issues. In particular, we derive a low-complexity and scalable procedure that
allows each terminal and the BS to locally choose the transmit
power and the detection strategy so as to obtain a good tradeoff
between detection capabilities and power consumption. This is
achieved by modeling the terminals and the BS as economic and
rational agents that engage in a noncooperative game [12], [13]
in which each one trades off its available resources (transmit
power and detection strategy). The main goal of each terminal
is to selfishly maximize its own revenue in terms of probability
of being detected while saving as much energy as possible and
satisfying quality of service (QoS) requirements given in terms
of probability of false alarm and accuracy of its timing estimate.
B. Prior Work
Game theory provides an analytical framework to study
complex interactions among rational entities, and has attracted a significant interest by the wireless communication
and signal processing communities in the last few years (see
[14]–[16] and references therein). In particular, there exists
a substantial literature on power control techniques based on
noncooperative game theory [12]. Most of them are focused
on the data transmission phase in wireless communications.
Among the different formulations available in literature, particularly significant is the energy-efficient approach (see [17]
and references therein), which is specifically tailored for a
battery-powered mobile population as it aims at capturing the
tradeoff between achieving a satisfactory error-rate and saving
as much energy as possible. This framework, originally developed for voice traffic [18], has been extended to code-division
multiple-access (CDMA) systems [19], [20], multicarrier
networks [21], ultrawideband systems [22], and ad-hoc networks [23]. Energy-efficient resource allocation schemes for
multicell OFDMA-based networks based on noncooperative
game theory have been recently investigated in [24]–[27]. As
mentioned before, the focus of the aforementioned works is on
the data transmission phase, in which the terminals are already
synchronized to the BS references. In this work, we investigate
the network association phase, in which the terminals notify
the BS of their entries through the procedure summarized in
Section I.A. To the best of our knowledge, energy efficiency
during this stage has been investigated only in [28], [29]
for a CDMA wireless network in a flat-fading scenario, and
extended in [30] to include a frequency-selective channel. In
this work, we use a similar approach to improve the energy
efficiency of contention-based synchronization procedures for
wideband OFDMA single-cell systems. Although we adopt a
similar optimization criterion, notable differences with respect
to [28]–[30] can be highlighted: i) the detection strategy is
included in the optimization problem, and not just the detection
threshold given a particular code detection technique; ii) the
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parameter estimation accuracy is included in the game, given
the particular requirements of OFDMA synchronization; and
iii) in [28]–[30], simulations are conducted using the average
performance of the detectors, including average auto- and
cross-correlation properties of the spreading codes and perfect
channel state information (CSI); in this paper, the simulation
platform contains more physical-layer details, and all the results are obtained using random channel realizations (based on
frequency-selective models), system parameters specified by
existing standards (including spreading codes), and estimated
parameters fed back by the BS (thus including the impact of
imperfect CSI and estimation errors on the resource allocation
performance).
C. Contributions
The main contributions of this paper are as follows.
• A novel perspective on contention-based synchronization
in OFDMA-based systems is provided. This is achieved
by modeling the network elements as economic and rational agents that trade off transmit powers and detection
strategies so as to selfishly maximize the probability of
correct synchronization using an energy-efficient approach
and satisfying given QoS requirements.
• A game-theoretic framework is used: i) to identify optimization criteria for system design; ii) to derive a low-complexity and scalable procedure that allows each terminal to
regulate its power in a distributed manner on the basis of its
estimated signal-to-interference-plus-noise ratio (SINR) at
the BS; and iii) to derive a low-complexity detection/estimation scheme allowing the BS to detect the selected codes
and to estimate the associated synchronization parameters,
without any prior information on the terminals currently in
the network association stage.
• Numerical results are used to highlight the effectiveness of
the proposed solution with respect to existing alternatives
based on a deterministic increase of the transmit power
(with or without contention resolution methods). It turns
out that the proposed solution allows each terminal to be
detected with a limited amount of power in a shorter time
interval while providing better timing and power estimation accuracy.
D. Organization
The remainder of the paper is structured as follows.2
Section II describes the system model, whereas Section III
formulates the problem and uses the analytical tools of game
theory to investigate its solution. Section IV presents an iterative and distributed algorithm to reach the equilibrium point,
whereas Section V reports some simulation results and comparisons with alternatives. Conclusions are drawn in Section VI.
2The following notation is used throughout the paper. Matrices and vectors
and
are the
identity matrix and the
are denoted by boldface letters.
all-zero vector, whereas
denotes an
diagonal matrix with entries
along its main diagonal. We use
and
for expectation, transposition and Hermitian transposition, resp.,
for the Euclidean norm of the enclosed vector, and
to round to the
nearest integer towards zero.
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II. SYSTEM AND SIGNAL MODEL
A. System Model
We consider an OFDMA-based transmission technology emsubcarriers with frequency spacing
. To avoid
ploying
aliasing problems at the receiver,
null subcarriers are placed
at both edges of the signal spectrum. The available
subcarriers are split into a synchronization subchannel, used for
the network association, and several data subchannels, used for
data transfer. The former is employed by synchronization terminals (STs) that must complete their synchronization procedure,
while the latter are assigned to data terminals (DTs) that entered
the network at an earlier stage and have already achieved synchronization. Following [4], the synchronization subchannel is
further divided into subbands and a given subband comprises
a set of adjacent subcarriers, which is called a tile. We denote
the index of the th subcarrier within the th tile
by
and call
the set collecting the subcarrier indices of the th tile. The only constraint in the selection of the
is that the tiles must be disjoint, which amounts to
indices
saying that
for
. We denote by the
number of simultaneously active STs and assume that each ST
notifies its entry request by transmitting a synchronization code
of length
, which is randomly chosen from a specified set
. For simplicity, we assume that different STs choose different
denoting the set collecting the
codes, with
indices of the selected codes. This assumption is reasonable in
practical systems, since is usually much smaller than the code
set cardinality .
We assume that DTs have been successfully synchronized to
the BS and do not generate interference over the received synchronization channel. On the other hand, the signals transmitted
by the STs are not aligned, both in frequency and time domains.
Let us denote the carrier frequency offset (CFO) (normalized to
the subcarrier spacing
) and the timing error (normalized to
the sampling period
) of the th ST by and ,
respectively. As explained in [31], CFOs are only induced by
Doppler shifts and/or downlink estimation errors. We consider
low mobility applications and assume that frequency estimation
errors in the downlink are within 2% of
as specified in [4].
Under such hypotheses, the impact of CFOs on the synchronization process can reasonably be neglected. On the contrary,
timing errors cannot be neglected, as they are related to the different (unknown) positions
occupied by the users within
the cell, and their maximum value corresponds to the round trip
propagation delay
for a user located at the cell
boundary [31], where is the cell radius and is the speed of
light. A simple way to counteract the effects of timing errors
would be the use of a sufficiently long cyclic prefix (CP) composed by
sampling intervals, with
being the maximum expected path delay. In this case, timing
errors would not produce any interblock interference (IBI), and
would appear only as phase shifts at the output of the receiver
discrete Fourier transform (DFT) unit. Unfortunately, this solution is not suited for the data section of the frame as it leads to an
intolerably large overhead. This implies that a successful synchronization procedure calls not only for identifying the transmitted codes actually buried in the received signal, but also for
providing accurate timing estimates during the synchronization
process in order to avoid IBI during the subsequent data transmissions.
B. Signal Model
To model the received signal, we assume that the channel response experienced by the th ST over the th tile is nearly flat
over each tile and independent across tiles (with an amplitude
that depends on both large-scale and small-scale fading effects).
The validity and impact of this assumption on the system performance will be discussed in detail in Section V.B by means of
numerical results obtained with standard channel models. Denoting by
the vector collecting the DFT outputs over the
th tile
, we may write
(1)
is ST ’s transmit power, and
denotes its
where
channel frequency response within the th tile. In addition,
is additive white Gaussian noise with zero mean and
covariance matrix
, the vector
(2)
accounts
for
the
phase
shifts
within
a
tile, and
, with
being the code randomly
chosen by the th ST.
For simplicity, in this paper we focus on single-user
strategies that operate individually for each
. More
, we assume the BS to
precisely, for each
decide in favor of one of the two following hypotheses:3
the code
is not present in the observation vector
is present in
. Although suboptimal, this approach has the advantage of
allowing a simple formulation of the detection problem as a
composite binary hypothesis test:
(3)
(4)
, and
accounts for
where
the contribution of MAI plus thermal noise. To statistically describe the distribution of
, we assume that
and
in (4) are
unknown deterministic parameters, and we model the entries of
as independent Gaussian random variables with zero
means and (unknown) powers
(5)
3As better detailed in Section IV, during contention-based synchronization,
detectors, one for each possible
, since it does
the BS must allocate
not know a-priori which codes are selected by the STs. Note that the code
constitutes the only temporary means of identification between the th ST and
the BS, that will be also used by the BS (on a broadcast downlink channel) to
provide feedback to the th ST (e.g., outcome of the test, and estimated parameters, such as the timing offset).
BACCI et al.: A GAME-THEORETIC APPROACH FOR ENERGY-EFFICIENT CONTENTION-BASED SYNCHRONIZATION IN OFDMA
where
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is terminal ’s received power at the BS, and
(6)
is ST ’s average channel power gain across the tiles. For later
convenience, the probability density functions of
under the
hypotheses
and
are defined as:
(7)
(8)
III. ENERGY-EFFICIENT SYSTEM DESIGN
A. Problem Formulation
As mentioned in the introduction, the goal of this work is to
make each ST locally and selfishly choose its transmit power so
as to maximize its own utility, paired with an optimal detection
strategy at the BS. Although OFDMA-based 4G candidate technologies, such as IEEE 802.16m [32] and 3GPP LTE [5] standards, allow for cooperation among terminals in the network,
the contention-based synchronization is the very first active operation performed by a mobile terminal (after listening to broadcast channels on the downlink) to get successful association to
the network. In this phase, it is thus reasonable to assume selfish
behavior by the STs, which aim at achieving a good tradeoff between detection capabilities and power consumption. In other
words, the utility function to be maximized must be defined so
as to guarantee an energy efficient management of the available
resources. This tradeoff can be quantified by defining the utility
function of the th ST as the ratio of the probability of correct code detection
to the transmitted energy per OFDMA
symbol
, with
being the duration
of the cyclically extended OFDMA block. This yields
(9)
where
is the vector containing the transmit
is the detection strategy adopted at
powers of all STs, and
the BS to detect
(the code selected by the th ST). We have
explicitly introduced the dependence of
on in
but also with
affects
(9) to emphasize that not only
the detection probability of ST . Thus, the maximization of
with respect to
cannot be performed by means of
a unilateral optimization, but rather requires a multidimensional
one.
The physical interpretation of
, which has units of
J , can be captured using the finite-state machine of Fig. 1,
where the two possible states of a terminal in the network are
sketched: the synchronization state, during which a terminal acts
as an ST, and the data transmission state, in which it acts as a
Fig. 1. Finite-state machine describing the terminal operational status.
DT. The probability that the synchronization procedure is successfully completed is
, and thus the average time
spent by an ST in the first state is
. As a conserepresents the expected energy consumed
quence,
by the th ST when transmitting at power .4
To proceed further, we observe that the detection strategy
may lead to a wrong detection of the th code in the two following cases: i) the code
is detected even though it is not
present in the observation vector ; or, ii) the code
is detected with an inaccurate timing estimation when it is actually
present. The former situation gives rise to a false alarm event,
and it can be handled imposing an upper bound
on the
. On the other hand,
probability of its occurrence
the latter case may be particularly harmful in OFDMA systems,
as it may give rise to a significant IBI during the data transmission phase. A possible solution to circumvent this situation is to
make the th ST regulate its power
while satisfying a constraint on the mean-square-error (MSE) of its timing estimation
error, defined as
, where
is the
timing offset estimate of
at the BS.
Based on the above considerations, we can formulate our optimization problem in terms of optimal transmit power (at the
ST side) and of optimal detection strategy
(at the BS side)
as follows:5
(10)
STs attempting to achieve successful
for all
synchronization, where:
is ST ’s transmit power
denoting the maximum transmit power (to be set
set, with
large enough to allow STs located at the cell boundaries to correctly achieve successful synchronization);
is the single-user
and
are the network QoS
detection strategy set; and
requirements in terms of maximum probability of false alarm
and maximum MSE of the timing estimation error, respectively.
Note that the problem in (10) cannot be solved in a centralized fashion by the BS as it actually does not know which STs
are transmitting (this is indeed one of the outcomes of the synchronization process). This means that each ST must solve
(10) in a decentralized way in order to compute its optimal .
Furthermore, (10) cannot be decoupled into subproblems, as
depends not only on , but also on all other STs’
4Although other metrics (based for instance on a global-performance criterion and possibly coordinated by the service provider) can be used, we find the
proposed metric a physically sound one to properly capture the description of
the problem and to measure how efficient a resource allocation strategy is.
5Strictly speaking, the solution to this problem is not necessarily unique, as
operator provides a whole set of values as its output. The equality
the
stands for all possible values that provide the maximum of the utility function.
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transmit powers
.
Otherwise stated, any action taken by a user affects the performance of all others. This makes the problem naturally fall within
the scope of noncooperative game theory [12].
In this respect, the problem in (10) can be restated as a noncooperative game with complete information [12] defined as
in which:
is the player
set, where player corresponds to the pair composed by the
ST using code , and its receiver at the BS;
denotes the strategy set for which the constraints in (10) are satisfied; and
is player ’s payoff function defined as in (9). By
restating (10) as game , in the next subsections we will use the
analytical tools of game theory to investigate its solution.
In other words, meeting the constraint on the MSE means
restricting ST ’s power strategy set to the subset
(19)
which depends on the opponents’ strategies
. Since not only
the utility
but also the strategy set
depends on
, the
game belongs to the category of generalized Nash games [13].
For such games, the most widely used solution is the generalized
Nash equilibrium (GNE) [12], [13], defined as a set of strategies
such that no player can unilaterally improve its own utility. Formally, a vector
, with
, is a
GNE of if for any
B. Optimal Detection Strategy (at the BS Side)
(20)
To satisfy the constraint on the MSE in (10), let us focus
on the properties of a generic timing estimator . In general,
takes the form [33]
(11)
where
and
are the bias and the variance of the timing estimate
, respectively. The bias term
can be generally measured through extensive simulations, whereas finding
is usually a challenging task. To circumvent this problem, we
assume that
is well approximated by its Cramér-Rao
[33] and rewrite (11) as
bound (CRB)
(12)
from which it follows that the inequality constraint in (10) can
be reasonably replaced by
(13)
In Appendix A, we show that
(14)
where
is the SINR of the th ST at the BS, given by
(15)
and for all strategies
, with
. In this work, we focus only on
pure (i.e., deterministic) strategies6 and study the existence and
uniqueness of the corresponding GNEs.
Using the considerations above and taking into account that
the strategy
only impacts on the numerator of (9), the optimization problem in (10) can be reformulated as
for all transmit powers
such that
(21)
from which the optimal detection strategy is easily identified as
the solution of the inner maximization. Under the assumption of
single-user detection introduced in Section II, the utility function in (9) is independent of the others’ strategies with
.
can be found
This means that the optimal detection strategy
using standard optimization techniques. In particular, the solution can be obtained using the Neyman-Pearson theorem and is
given by [33]
(22)
where
and
are defined as in (7) and
is such that
(8), and the threshold
(23)
Letting
(16)
the constraint
in (13) is satisfied provided that
and
As follows from (7) and (8), computing
in (22) requires knowledge of the unknown pa, and . Then, we approximate the likelihood
rameters
ratio test (LRT) in (22) with the generalized LRT (GLRT) given
by (see Appendix B) [33]
(17)
(24)
with
given by
(18)
6This choice is motivated by the fact that there exist no mixed (i.e., statistical)
strategies that are more efficient (in the Pareto sense [12]) than any pure (i.e.,
degenerated mixed) one, assuming such exists.
BACCI et al.: A GAME-THEORETIC APPROACH FOR ENERGY-EFFICIENT CONTENTION-BASED SYNCHRONIZATION IN OFDMA
where
with
and
(25)
is the maximum likelihood (ML) estimate of
in which
obtained as
(26)
with being a tentative value for
we obtain
. From (24), following [34]
(27)
where
(28)
is the incomplete beta function [35]. The above result indicates
that, for a given pair
, the probability of false alarm is
a function of the threshold
only. The latter must be chosen
is satisfied.
such that the identity
Note that, unlike [28]–[30], the probability of false alarm (27)
does not depend on the transmit power . As better detailed in
Section IV, this is of significant relevance for practical implementation purposes as the BS can dispense with any prior information on the active codes.
C. Optimal Transmit Power (at the ST Side)
Substituting the above results into (21) leads to the problem
(29)
on
for nowhere we have omitted the dependence of
tational simplicity. To solve (29), we first need to evaluate the
probability of correct detection
. For this purpose, we
recall from Section II that the channel coefficients
can
be modeled as independent complex Gaussian random variables
with zero means and variances given by
. Under these assumptions, we have that [34]
(30)
where
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being the solution of
(33)
The game admits a unique pure-strategy GNE if and only if
in (32) is such that
(34)
Proof: The proof of existence and uniqueness of
in (29) follows the same steps described in [28] and
proceeds according to the following line of reasoning. First, the
probability of correct code detection (30) is proven to be eventually a concave function of . Then, the existence is proven as in
[13] and [36]–[39], whereas the arguments illustrated in [40] are
used to prove the uniqueness. Next, we observe that the detection strategy based on the GLRT (22) maximizes
for a given probability of false alarm
irrespectively of all
transmit powers. This easily follows from (21) where it is shown
that each player can always choose
. Hence, all the
properties of the solution of (29) apply to
as well. This means the optimal action at the GNE is given by
with
(selected by the th ST)
given by (24) (selected by the detector for code at the
and
BS).
Although related to a different application scenario, the
results shown in Proposition 1 have a mathematical structure
which is similar to those obtained by other relevant works in
the field of game-theoretic energy efficiency (see for example
[19], [20], [22], and [41]). Unlike previous works, Proposition
1 establishes a necessary and sufficient condition for the GNE
to exist. This condition is not present in the aforementioned
works and stems from the constrained formulation (10), which
makes the game a generalized one (such as that investigated
in [28]). As seen, the condition (34) is not always met, as it
depends on the number of STs, the tile size , and the QoS
and
(through ). It is worth observing
constraints
that, if (34) is not fulfilled, it may happen that the set of power
strategies
of some users
may be empty. In
this case, such users will not be able to reach the equilibrium
with a resulting degradation of the system performance (see
Section V.B for some numerical examples). On the other hand,
when (34) is fulfilled the following result holds true.
Corollary 1: If the condition (34) is satisfied, at the GNE the
th ST would end up transmitting at
is defined as
(31)
(35)
with
being the SINR of the th ST, defined as in (15). From
(16), it follows that there exists a one-to-one correspondence
between
and
for a given vector
. This means that the
can be equivalently
probability of correct detection
replaced with
, which is a function of rather than of
. To proceed further, we introduce the following proposition.
Proposition 1: Define
(36)
(32)
Proof: The above result can be derived using the mathematical arguments developed in [28].
It is worth observing that neither (35) nor (36) give a practical method to enforce that each ST noncooperatively (i.e., distributely) reaches the GNE. This easily follows by observing
that (35) requires knowledge of
whereas (36) depends
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on and . All these parameters are unknown at the STs and
cannot be directly estimated. To overcome this problem, in the
next section we show how to exploit the GNE analysis provided
so far to derive a practical power control algorithm.
IV. IMPLEMENTATION
To derive a practical algorithm to reach the GNE of in a distributed fashion, we start by assuming that the STs with indices
have already chosen their optimal transmit powers. This
amounts to saying that
. Then, from (15) we have
(37)
from which using (35) yields
(38)
for a given
requires
We see that the computation of
knowledge of
only, but this information is not available at
the th ST. A possible solution is performing the estimation at
back to the ST through a
the BS and feeding the estimate of
reliable reverse link. An estimate of
is obtained as [11] (see
Appendix B for further details):
(39)
Since the BS does not have any a-priori knowledge of which
codes are currently active, it must estimate the received SINR
for all possible codes
. The estimated SINRs
are
then fed back to the STs together with the corresponding code
indices. The STs finding their code information in the response
message will update the transmit power according to (38) after
replacing
with .
Based on the above considerations, an iterative and distributed algorithm can be derived to let each active user reach
the GNE of the proposed game . In the sequel, this scheme is
referred to as best-response synchronization algorithm (BRSA),
as it is based on best-response dynamics [40] operating as follows.
a. Initialization of the game:
a1) each active ST
computes the
SINR level
at the equilibrium using (32), in
, and are assumed
which
to be common knowledge across the network;
a2) each active ST
initializes the
to a predetermined value;
transmit power
a3) each active ST
sets
; and
a4) the BS sets the optimal detection threshold
based on
by numerically inverting (27).
b. GLRT algorithm: at each step of the algorithm, for each
code
, the BS
;
b1) applies (26) to obtain
b2) decides on
or
according to (24);
b3) estimates the received
using (39);
b4) feeds back the results of the GLRT test and the
SINR
on the return broadcast channel.
c. Best-response algorithm: at each step of the algorithm,
each ST
c1) receives the SINR
estimated by the BS, and
the result of the code detection test (24) based on
current estimated parameters;
c2) if the GLRT for code
is verified and
,
exits the game, otherwise goes to the next step;
c3) adjusts the transmit power according to
(40)
c4) updates
.
Note that both the GLRT algorithm (performed by the BS for
each available code
) and the best-response power update (implemented by each active ST) follow directly from the
solution of the optimization problem (10). A close inspection
of steps c1-c4 reveals that the BRSA requires each ST to know
only its estimated SINR and the GLRT outcome. This means
that it can be performed in a fully decentralized manner, in
which the number of STs and the transmit power levels change
across time.
It is worth emphasizing that the convergence of BRSA to the
GNE is ensured for any initial power by the theoretical analysis
presented in [40] as long as condition (34) is fulfilled. This result
holds true because the relationship (35) describing the best-reponse dynamics possesses the properties of a standard function
[40]. Observe that, although in general the properties of uniqueness of the game equilibrium and of the convergence of iterative
algorithms approaching it are not necessarily related, in this specific formulation the property of the standard function ensures
both GNE uniqueness and BRSA convergence.
Note that, in step c1, the ST can extract the relevant details in the downlink broadcast channel using , that serves as
a temporary identifier between it and the BS. When the th ST
exits the game or, equivalently, when hypothesis
is verified,
the BS will broadcast the estimated timing offset and the estimated received power. The ST will make use of both to adjust its
time scale and transmit power in order to access the subsequent
data transmission phase with the minimum impact in terms of
IBI [11]. The exit condition on
in step c2 is introduced to
guarantee a sufficient accuracy of the timing estimation as indicated by the constraint
, whereas
in step c3 is considered to make sure that the constraints on the maximum transmit
powers are met at every step of the BRSA.
V. NUMERICAL RESULTS
The performance of the BRSA is now investigated by means
of numerical results. The simulation parameters are reported
below and are chosen in compliance with the IEEE 802.16
family of standards [4]. All numerical results are obtained by
averaging 20,000 independent network realizations.
A. Simulation Parameters
The DFT size is
, and the sampling period is
ns, corresponding to a subcarrier spacing of
kHz. There are
unmodulated subcarriers for both the
releft and the right guard bands. Out of the
maining subcarriers,
are reserved for synchronization. The
tiles composed of a set of adjacent subcarriers
are randomly positioned within the available bandwidth so as to
BACCI et al.: A GAME-THEORETIC APPROACH FOR ENERGY-EFFICIENT CONTENTION-BASED SYNCHRONIZATION IN OFDMA
effectively exploit the frequency diversity offered by the multipath channel. We assume that a single OFDMA synchronization
symbol is present within a frame, and that the frame duration is
equal to
ms. The synchronization codes are chosen from
a set composed of
binary phase shift keying signafor
) generated
tures (i.e.,
as specified in [4]. We consider a cell radius
3 km, corresponding to a maximum normalized propagation delay
.
Path gains are modeled as statistically independent and
circularly symmetric Gaussian random variables with zero
means and power delay profile specified by the 6-tap ITU modified vehicular-A channel model [42]. In these circumstances,
the maximum channel impulse response has a time span of
samples, and the coherence bandwidth of the
kHz. The average
channel is approximately given by
channel power is normalized with respect to a distance
,
and a path loss exponent
is used.
The CP length during the data transmission phase
is chosen equal to
. This means
that the maximum timing estimation error that does
not to give rise to IBI during the frame data section
is
.
Hence, the constraint on the maximum MSE translates into
. Extensive
simulations (not reported here due to space limitations) indicate
that the estimate
obtained as in (26) is Gaussian-distributed
with a bias
, where
is the delay spread
of the channel [43]. Although the channel parameter
is generally unknown, it can be roughly upper bounded by
. Therefore, combining this result with
(13),
. In addition, the
.
maximum probability of false alarm is set to
The DTs are supposed to be perfectly aligned with the BS references, whereas the number of STs is fixed at
. Without
loss of generality, we concentrate on the first ST and assess
the performance of the BRSA when its distance
from the
BS is kept constant, while all other STs with indices
are located at random distances
uniformly distributed in the
range between
and . The maximum normalized transmit
powers
are assumed to be the same for all STs and equal to
dB. On the other hand, two different scenarios are envisaged for the initial normalized powers
:
i) the minimum power (MP) scenario (shown using upper triis fixed to
dB by all
angular markers), in which
STs; and ii) the single-user (SU) scenario (reported with circular
markers), in which
is chosen equal to
, rep. The latter case
resenting the optimal power level when
minimizes the convergence time of the best-response algorithm
[44] and it is used in the sequel as a benchmark for the minimum
synchronization time required by the BRSA. Note that, in the
SU scenario, knowledge of the average channel gain
is required at the th ST. This information can be easily obtained in
time division duplexing systems, whereas it calls for additional
complexity in systems operating according to a frequency division duplexing mode.
B. Network Design Criteria
In this subsection, we exploit the theoretical analysis provided in Section III-C to derive some criteria for the network
1265
Fig. 2. Average normalized power consumption as a function of
and
).
Fig. 3. Average synchronization time as a function of
and
).
(
(
system design, and make sue of the simulation results to illustrate the major findings. We begin by assessing the impact of
and on the performance of the BRSA. For this purpose,
in Fig. 2 we assume
and report the average normalized power expenditure
, required by ST 1 from the time
it accesses the network until it successfully completes the synchronization procedure, as a function of the number of tiles .
Similarly, Fig. 3 shows the average time
needed to complete a successful synchronization, obtained by scaling the average number of steps required by the BRSA by .
The results of Figs. 2 and 3 indicate that, in both the MP
and the SU scenarios, the best performance is achieved for
. This is motivated by the following reason.
On one hand, the larger the tile size , the larger the processing gain yielded by the spreading code, as is apparent
from (15). On the other hand, the smaller , the more accurate
the matching between the selectivity of the channel in the
frequency domain and the flat-fading assumption adopted in
1266
Fig. 4. Optimal SINR at the GNE as a function of
and
).
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 5, MARCH 1, 2013
Fig. 5. Average MSE of the timing estimate as a function of
and
).
(
Section II to derive the GLRT. The best choice to meet these
two conflicting requirements is to choose
according to the
coherence bandwidth of the channel. In the network under
investigation, this amounts to setting
and
, which is in accordance with the results of Figs. 2 and
is not an integer, the optimal choice is to
3. If
select the closest acceptable value for . In the case the code
length is a prime number, as considered for LTE-compliant
and
systems [5], the best approach is to set
, thus partially sacrificing some synchronization
subcarriers, with a tolerable loss on the code cross-correlation
performance [11].
The results of Figs. 2 and 3 show a large performance degradation for
. This can be explained using the results
of Section III.C. For the reader’s convenience, Fig. 4 reports
the numerical values of the optimal SINR
at the GNE (star
markers) as a function of . The SINR levels and are also
shown, using dashed and dash-dotted curves, respectively. By
inverting (34), it follows that the maximum number of STs that
can be simultaneously accommodated in the cell to guarantee
the existence and uniqueness of the GNE of is
(41)
the maxIn the investigated network, we have that for
should be smaller than 5. Since we
imum number of STs
assume
, it follows that the condition (34) is not met for
. This means that resources are not sufficient to accomSTs while meeting the QoS requirements.
modate all the
This makes the power consumption as well as the average synchronization time increase, although with a graceful behavior
for
. The sudden increase in the resources consumed
by the pair
can be again interpreted using the
theoretical analysis of Section III-C, in a twofold manner. On
one side,
when
. On the other side,
for
. Since at the GNE
lies on the global maximum
of the unconstrained utility function (i.e., that without the constraint on the MSE) if and only if
(as follows from the
(
proof of Proposition 1, not reported for the sake of brevity), it
is always preferable to have
to ensure a better energy-efficient exploitation of the available resources.
To complete the simulation analysis, we assess the accuracy
of the timing estimator in (26) once the synchronization process
is successfully completed. For this purpose, Fig. 5 illustrates the
MSE of the timing estimate for different values of . From
the simulation results, we can see that the QoS requirement
is largely satisfied by all considered values of .
In particular, we note that the best estimation accuracy is again
obtained for
and
for both MP and SU scenarios.
C. Performance Assessment
Based on the results of Section V-B, in all subsequent simulations we choose
and
, from which, using
(18) and recalling that
and
, we get
dB. From (27), we have that
when
. Moreover, the optimal SINR turns out to be
dB, from which, using (30) and (31), we
.
get
The BRSA is compared with two alternative solutions relying on a deterministic increase of the transmit power. The
first one is referred to as the deterministic synchronization algorithm (DSA), whereas the second one is called the BEB-DSA
[4]. Both differ from the BRSA in the transmit power update
(step c3) only, as follows: if the selected code is not successfully detected, the th ST updates its transmit power according
to
(42)
where
is a design parameter common to all STs. For the
DSA,
, whereas for BEB-DSA
is an integer number
, with
being the
randomly chosen in the interval
current value of ST ’s backoff window. The parameter
of
BEB-DSA is referred to as the backoff counter and indicates
how many OFDMA synchronization symbols the th ST must
BACCI et al.: A GAME-THEORETIC APPROACH FOR ENERGY-EFFICIENT CONTENTION-BASED SYNCHRONIZATION IN OFDMA
Fig. 6. Average normalized power consumption as a function of the normalized
(
and
).
distance
Fig. 7. Average synchronization time as a function of the normalized distance
(
and
).
wait before re-transmitting using
. During the first
synchronization attempt, the backoff window size is set to
and it is doubled after each unsuccessful attempt.
After successive unsuccessful attempts,
is reset to its
. In all subsequent simulations, we assume
initial value
dB,
, and
. Similarly to the BRSA, we
assume that the maximum normalized transmit powers for both
schemes are the same for all STs and such that
dB. The initial powers are chosen equal to
,
with
dB.
and
as functions of
,
Figs. 6 and 7 show
i.e., the normalized distance from the BS of the ST of interest,
using lower triangular and square markers to report the results
of the DSA and BEB-DSA, respectively. As seen, the BRSA
provides better results in terms of both energy efficiency and
. As expected, the
fairness for all investigated values of
SU initialization requires the minimum average time for a successful synchronization, whereas the MP initialization guarantees the minimum amount of power expenditure. The BEB-DSA
1267
requires roughly the same power consumption as the BRSA,
but at the expense of an increased average time per acquisition.
The DSA is faster than the BEB-DSA, but requires a significant
amount of battery drain. On the contrary, BRSA maintains the
average acquisition time limited up to the maximum of about
two frames for any BS-terminal distance in the SU initialization case, and less than five frames in the MP initialization case
for the far users, and requires a limited amount of power expenditure (around 5 dB above the noise power for the far users in
both initialization schemes). Therefore, the BRSA outperforms
the DSA both in terms of power expenditure and of time needed
for successful synchronization, at the expense of an increased
amount of feedback information from the BS. Currently, IEEE
802.16m [32] and 3GPP LTE [5] standards provide feedback for
every detected code (in terms of corrections to be applied both to
timing offsets and to transmit powers). In the proposed method,
we only require the BS to broadcast the outcome of the GLRT
(1 bit), and the received SINR for each code in . Since the
number of codes
is usually on the order of tens to hundreds,
the amount of extra resources needed on the downlink channel is
limited, also considering that preliminary experimental results
(not reported for the sake of brevity) show that introducing a
1-byte SINR quantization does not significantly affect the numerical performance.
We now observe that the analysis presented in Section III is
valid under the hypothesis of a sufficiently large power constraint
that allows far users to reach the optimal SINR
even in the presence of a bad channel realization. To properly
set this parameter, consider an SU scenario in which the single
ST is placed at
. To meet , we just need the channel
power gain
(averaged across the
tiles as in (6)) to be
higher than
. Since the channel powers are normalized with respect to
, in the considered scenario it is
sufficient that
be higher than
dB. Since the average
value of
is
dB, the probability of not
meeting
is negligible. When considering
, it might
happen (with a low probability considering that channel power
gains are averaged across
tiles than span across a bandwidth
), that some near STs
show
, thereby preventing ST from meeting . However,
this situation usually occurs only for a short time interval since
“strong” users are synchronized very fast, and the new incoming
users (randomly placed in the cell) are likely not to be “strong”
again. This allows the “weak” ST to eventually achieve
and enter the network (although after a longer time interval). To
support this observation, note that throughout our experiments
channel realizations
(obtained by averaging at least
per simulation point) we never encountered a situation in which
an ST was not able to achieve . Finally, observe that the design of
plays a more important role for BEB-DSA and
DSA rather than BRSA. While increasing
does not substantially affect the performance of BRSA, a large degradation
is observed for both BEB-DSA and DSA. This is because, for
a given incremental power step, increasing
forces STs to
remain in the collision phase for a longer time interval before
resetting their powers to .
For the sake of completeness, in Figs. 8 and 9 we compare
the estimation accuracy of the investigated solutions in terms of
and the normalized variance of the (unbiased) power
1268
Fig. 8. Average MSE of the timing estimate as a function of the normalized
(
and
).
distance
Fig. 9. Average normalized variance of the power estimate as a function of the
(
and
normalized distance
).
estimate, given by
, as func. The normalized CRB of the received power, detions of
fined as
with
given in Appendix A, is
also reported for comparison. As before, the BRSA represents
the best allocation scheme, whereas the DSA and BEB-DSA exhibit a significant loss for values of
larger than 0.6.
Finally, Figs. 10 and 11 show
and
for the invesof STs ranges from 2 to
tigated solutions when the number
. The user of interest is placed at distance
.
Comparisons are similar to the ones shown above and lead to
similar conclusions: the proposed BRSA is particularly suitable
for a bursty-traffic scenario, with messages composed of a small
number of packets, and in which users are required to be synchronized in a limited amount of time. This makes it particularly suited for applications in which a refinement of the synchronization parameters is periodically required. Nowadays, the
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 5, MARCH 1, 2013
Fig. 10. Average normalized power consumption as a function of number
of STs (
and
).
Fig. 11. Average synchronization time as a function of number
and
).
of STs (
common procedure adopted by current standards for synchronization refinement is known as periodic ranging [5], [32]. The
latter is a process by which any mobile terminal periodically adjusts its transmission parameters in order to stay synchronized
with the BS even during its idle states, thereby reducing the
transmission latency but at the same time the battery life. Since
the proposed solution enables a fast initial synchronization with
low power consumption (especially when far terminals are considered), it can be used for reducing the frequency of periodic
ranging procedures, thereby further increasing the energy efficiency of the mobile terminals.
VI. CONCLUSION
In this work, we have proposed a game-theoretic approach
to derive an energy-efficient solution for contention-based
synchronization in OFDMA-based systems. A noncooperative
game has been formulated in which each terminal and the
BACCI et al.: A GAME-THEORETIC APPROACH FOR ENERGY-EFFICIENT CONTENTION-BASED SYNCHRONIZATION IN OFDMA
base station are allowed to locally and selfishly choose the
transmit power as well as the detection strategy to maximize
the probability of correct terminal synchronization while saving
as much energy as possible. This is achieved by controlling
the performance of the network in terms of probability of false
alarm and estimation accuracy of the timing errors. We have
shown that the proposed game admits a unique generalized
Nash equilibrium under some mild conditions on the system
parameters.
An iterative algorithm based on best-response dynamics has
been derived to let each user achieve the equilibrium point in a
distributed manner, using the unknown parameters estimated at
the BS (through a low-complexity method that follows from the
resource allocation optimization problem). Numerical results
based on realistic network parameters and widely agreed-upon
channel models have shown that the proposed solution outperforms deterministic-increase approaches (both with and
without contention resolution methods) in terms of average
transmit power and synchronization time as well as in terms of
accuracy of the parameter estimation. This is achieved at the
expense of an increased amount of feedback information from
the BS. Further work is needed to assess the impact of quantized
levels for both the signal-to-interference-plus-noise ratios and
the transmit powers, with the former particularly important to
significantly reduce the amount of feedback required across the
network.
It is worth observing that the approach developed in this work
can also be expedient to optimize the performance and the efficiency of other radio networks. As an example, in [45] a similar formulation for the utility function is used to investigate the
tradeoff between detection capabilities and power consumption
for a distributed radar sensor network.
1269
with
(47)
and
From (46), it follows that
(48)
and defining
[33]
Using (47), recalling that
, we get (14). From
(49)
where
(50)
is the gradient of
with respect to
(50) into (49) yields
. Substituting (46) and
(51)
APPENDIX B
Maximizing
in (7) with respect to
produces
(52)
from which it follows that
(53)
APPENDIX A
In this Appendix, we compute the CRBs for the
joint estimation of
and
. Let us define
the set of unknown paand
denoting the real and the
rameters, with
imaginary parts of vector
, respectively. Using the above
definitions and the pdfs given in (7)–(8), the Fisher information
matrix is found to be [33]
(43)
in (8) with respect
Looking for the maximum of
to while keeping
and
fixed yields (26), where
takes the form (25). Maximizing
with respect to
using
as in (26) leads to
(54)
with
. Substituting this result back into
and maximizing with respect to
yields
(55)
where
from which we get
(44)
(56)
and
(45)
The inverse of
is equal to
Substituting (53) and (56) into (22) yields
(57)
(46)
which can be equivalently rewritten as in (24) with
. Using the invariance property of the ML estimator,
it follows that the estimate of the received power
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 5, MARCH 1, 2013
can be obtained as
using (25) and (54), as
or, equivalently,
(58)
It is worth observing that, if the timing offset is perfectly estimated (i.e.,
), then
(59)
and
(60)
From (59) and (60), it follows that
and
are biased estimates of
and , respectively. Using (59), we can see that
takes the form
.
an unbiased estimate of
Recalling (55),
(61)
is found to
From the above results, an unbiased estimate of
, from which, using (61), we obtain
be
(62)
Combining (61) and (62) with (15), the estimated SINR
(39) easily follows.
in
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Giacomo Bacci (S’07–M’09) received the B.E. and
the M.E. degrees in telecommunications engineering
and the Ph.D. degree in information engineering from
the University of Pisa, Pisa, Italy, in 2002, 2004, and
2008, respectively.
Since 2005, he has been with the Department of
Information Engineering at the University of Pisa,
where he is currently a Postdoctoral Research Fellow.
In 2006–2007, he was a visiting student research collaborator at the Department of Electrical Engineering
at Princeton University, Princeton, NJ, USA. In 2008,
he joined Wiser srl, Livorno, Italy, as a software engineer. From May 2012 to
April 2013, he was also enrolled as a Visiting Postdoctoral Research Associate
at the Department of Electrical Engineering, Princeton University. His research
interests are in the areas of digital communications, signal processing, and estimation theory. His current research topics focus on resource allocation for multiple-access and relay-based wireless networks, time delay estimation for satellite positioning systems and wireless communications, and channel coding for
IMT-advanced technologies.
Dr. Bacci is the recipient of the FP7 Marie Curie International Outgoing Fellowships for career development (IOF) 2011 GRAND-CRU Game-Theoretic
Resource Allocation for wireless Networks Based on Distributed and Cooperative Relaying Units.
Luca Sanguinetti (S’04–M’06) received the Laurea
Telecommunications Engineer degree (cum laude)
and the Ph.D. degree in information engineering
from the University of Pisa, Pisa, Italy, in 2002 and
2005, respectively.
Since 2005 he has been with the Department
of Information Engineering of the University of
Pisa. In 2004, he was a visiting Ph.D. student at
the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. During the period June 2007
to June 2008, he was a Postdoctoral Associate in
the Department of Electrical Engineering at Princeton University, Princeton,
NJ, USA. During the period June 2010 to September 2010, he was selected
1271
for a research assistantship at the Technische Universitat München. He is
currently an Assistant Professor at the Department of Information Engineering
of the University of Pisa. His expertise and general interests span the areas of
communications and signal processing, estimation, and detection theory.
Dr. Sanguinetti is currently serving as an Associate Editor for the IEEE
TRANSACTIONS ON WIRELESS COMMUNICATIONS.
Marco Luise (M’85–SM’94–F’12) received the
M.E. and Ph.D. degrees in electronic engineering
from the University of Pisa, Pisa, Italy, in 1984 and
1989, respectively.
He was formerly a Research Fellow of the European Space Agency (ESA) at ESTEC Noordwijk,
The Netherlands, a Researcher of the Italian National Research Council (CNR), at the Centro Studio
Metodi Dispositivi Radiotrasmissioni (CSMDR),
Pisa, Italy, and an Associate Professor at the Department of Information Engineering of the University
of Pisa, where he is currently a Full Professor of telecommunications. His main
research interests lie in the broad area of communication theory, with particular
emphasis on wireless communications, mobile and satellite communications,
positioning systems, and software-defined radios.
Prof. Luise served as an Editor of the IEEE TRANSACTIONS ON
COMMUNICATIONS and of the European Transactions on Telecommunications. He was the founder and first Co-Editor-in-Chief of the International
Journal of Navigation and Observation and is currently an Associate Editor of
the Journal of Communications and Networks. Recently, he was the General
Chairman of EUSIPCO 2006, the General Co-Chair of European Wireless
2010, and will be the General Co-Chair of the IEEE International Conference
on Acoustics, Speech, and Signal Processing (ICASSP) 2014.
H. Vincent Poor (S’72–M’77–SM’82–F’87) received the Ph.D. degree in electrical engineering and
computer science (EECS) from Princeton University,
Princeton, NJ, USA, in 1977.
From 1977 until 1990, he was on the faculty of the
University of Illinois at Urbana-Champaign. Since
1990, he has been on the faculty at Princeton, where
he is the Michael Henry Strater University Professor
of Electrical Engineering and Dean of the School of
Engineering and Applied Science. His research interests are in the areas of stochastic analysis, statistical
signal processing, and information theory, and their applications in wireless networks and related fields such as social networks and smart grid. Among his
publications in these areas are the recent books Classical, Semi-classical and
Quantum Noise (Springer, 2012) and Smart Grid Communications and Networking (Cambridge Univ. Press, 2012).
Dr. Poor is a member of the National Academy of Engineering and the
National Academy of Sciences, a Fellow of the American Academy of Arts and
Sciences, and an International Fellow of the Royal Academy of Engineering
(U.K.). He is also a Fellow of the Institute of Mathematical Statistics, the
Acoustical Society of America, and other organizations. In 1990, he served as
President of the IEEE Information Theory Society, and from 2004 to 2007, he
served as the Editor-in-Chief of the IEEE TRANSACTIONS ON INFORMATION
THEORY. He received a Guggenheim Fellowship in 2002 and the IEEE
Education Medal in 2005. Recent recognition of his work includes the 2010
IET Ambrose Fleming Medal, the 2011 IEEE Eric E. Sumner Award, the
2011 Society Award of the IEEE Signal Processing Society, and honorary
doctorates from Aalborg University, the Hong Kong University of Science and
Technology, and the University of Edinburgh.