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A Bio-Inspired Swarming Algorithm for Decentralized Access in Cognitive Radio Paolo Di Lorenzo, Student Member, IEEE, and Sergio Barbarossa, Member, IEEE

Abstract—The goal of this paper is to propose a bio-inspired radio access mechanism for cognitive networks mimicking the behavior of a flock of birds swarming in search for food in a cohesive fashion without colliding with each other. The equivalence between swarming and radio resource allocation is established by modeling the interference distribution in the resource domain, e.g., frequency and time, as the spatial distribution of food, while the position of the single bird represents the radio resource chosen by each radio node. The swarming mechanism is enforced by letting every node allocate its resources (power/bits) in the time-frequency regions where the interference is minimum (the food density is maximum), avoiding collisions with other nodes (birds), yet limiting the spread in the time-frequency domain (i.e., maintaining the swarm cohesion). The solution is given as the distributed minimization of a functional, borrowed from social foraging swarming models, containing the average interference plus repulsion and attraction terms that help to avoid conflicts and maintain cohesiveness, respectively. Index Terms—Bio-inspired networks, cognitive radio, distributed resource allocation, social foraging swarms.

I. INTRODUCTION HE opportunistic access of the radio spectrum by unlicensed users is a problem that is attracting a large interest in the research community as well as in the industry sector, as a way to improve the efficiency thanks to a dynamic radio resource allocation as opposed to conventional rigid spectrum access [1], [2]. A broad survey on cognitive radios is [3], whereas a more specific survey on dynamic spectrum access methods is given in [4]. The basic idea in cognitive networks is to have a hierarchical structure where unlicensed users, also known as secondary users (SUs), are allowed to use temporally unoccupied communication resources, like frequency bands, time slots or user codes, under the constraint of not interfering (or producing a tolerable interference) towards licensed, or primary, users. A key step in cognitive networks is the ability of the opportunistic users to sense the resource domains, either time slots or frequency subchannels, and then use the unoccupied resources until possible and release them as soon as primary users access them. Besides cognitive radios, another interesting area

T

Manuscript received January 24, 2011; revised May 20, 2011; accepted August 19, 2011. Date of publication August 30, 2011; date of current version November 16, 2011. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yao-Win Peter Hong. This work was supported by FREEDOM Project, Nr. ICT-248891. The material in this paper was presented, in part, at the 2010 IEEE Workshop on Signal Processing Advances for Wireless Communications,June 2010. The authors are with the Department of Information, Electronics and Telecommunications, University of Rome “Sapienza,” 00184 Rome, Italy (e-mail: [email protected]; [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.2011.2166549

of application of dynamic radio access is femtocell networks, where a potential massive deployment of femto-access points can determine an intolerable interference towards macrocell station users. In this case, the high number of femto-access points demands for decentralized radio access strategies, aided with proper channel sensing. Dynamic access based on sensing has been studied in a series of works; see, e.g., [5]–[11] and [14]–[21]. In all of these works, it has been emphasized how the sensing and access strategies should be designed jointly to optimize the system performance. In particular, the authors of [10] show how to maximize the throughput of a secondary user considering the identification of spectral opportunities, the sensing strategy and the access strategy jointly. The primary user activity is modeled as a discrete-time Markov chain and the effect of channel estimation errors is taken into account. Discrete-time Markov models assume some kind of synchronization between primary and secondary users. In situations where this synchronization cannot be taken for granted, as in WLAN for instance, a continuoustime Markov process is assumed to model the primary user activity, as in [9]. The proposed methods do not prevent collisions, but try to maximize throughput under a collision constraint. In [12], it is shown that, when the collision constraint is tight, the strategy can be implemented with a simple memoryless policy with periodic channel sensing. The previous works concentrate on the decisions about accessing the available channels or not. In [14], it was shown how to determine the decision thresholds in order to maximize the opportunistic throughput, in a multicarrier setup, for a given set of rates over the available subcarriers. In [16], it was then shown in the same multicarrier framework, that a substantial performance improvement can be obtained by choosing decision thresholds and power allocation jointly, rather than separately. Most works concentrate on the access from a single secondary user into a system partially occupied by primary users, whereas relatively fewer papers address the uncoordinated access from multiple secondary users. The multiuser case was specifically addressed in [13]. Game-theoretic approaches have also been proposed, as a way to derive decentralized access strategies. In particular, in [17], the multiuser access problem was formulated as a game whose players are the secondary nodes, who aim at maximizing their rate under the constraint of inducing no interference at all, or only limited interference, to the primary users. The joint optimization of detection thresholds and power allocation in a multiple secondary user scenario was addressed in [21]. In [17] and [21], every cognitive user was assumed to have perfect knowledge of the aggregated interference from primary users. In practice, this is a rather idealistic assumption. A Bayesian game-theoretic approach was alternatively considered in [18]–[20]. In this paper, we follow a rather alternative path and, inspired by biological models, we formulate the search for radio

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resources, i.e., time and frequency slots, as the search for food by a flock of birds swarming in a cooperative manner, but without any centralized control. The equivalence between the two problems is the following. The interference distribution in the time-frequency plane takes the role of the food spatial distribution: The birds (radio nodes) fly (allocate their resources) over the regions (time-frequency domain) where there is more food (less interference). During the flight, the birds move (choose their time-frequency slots) in a coordinated way, even in the absence of any central control, in order to avoid collisions (conflicts over common radio resources), yet maintaining the swarm cohesion (i.e., avoiding unnecessary spread in the occupancy of the time-frequency plane). Inspired by the swarm models proposed by Gazi and Passino in [22] and [23], properly modified according to our goals, we show how the decentralized resource allocation can be formulated as the minimization of a proper functional including the interference distribution over the radio resource plane, plus the combination of a repulsion and an attraction term, introduced to avoid conflicts over common resources while preventing, at the same time, an excessive spread in the resource domain. The basic contributions of this paper are the following: 1) we propose the application of swarming mechanisms to radio resource allocation in cognitive radios; 2) we provide upper and lower bounds on the spread of the swarm, as a function of the swarm connectivity; 3) we propose fast versions of the swarming algorithm, useful for our application, and we apply such procedures to the dynamic resource allocation in the frequency domain; and 4) we apply the proposed procedure to the case where the primary users in a cognitive radio are modeled as statistically independent homogeneous continuous-time Markov processes. The paper is organized as follows. In Section II, we recall briefly some basic concepts from algebraic graph theory that will be used throughout the paper. Section III describes the swarm model and formulates the search of available time/frequency slots as the distributed minimization of a global potential function. In Section IV, we analyze the cohesiveness of the swarm in case of local interactions among the nodes, providing closed form expressions for the upper and lower bounds of the swarm size. This analysis is useful to capture the effect of the network topology and of the swarm parameters on the spread over the resource domain. In Section V, we study in more detail the swarming mechanism in a one-dimensional domain, e.g., the frequency domain. We provide first a local stability analysis useful to show that the introduction of the attraction and unbounded repulsion terms in the functional to be minimized does not affect the stability of the system. Then, we propose fast swarming methods based on a proper selection of the descent direction of a scaled gradient optimization. The first method is an approximation of a Newton based optimization and improves the convergence speed of the algorithm; the second method adapts the swarming speed with respect to the interference power perceived by the swarm. Numerical examples show the main algorithm’s features. In Section VI, we describe the application of the proposed model to the distributed resource allocation problem on a time-frequency plane. We consider both a static interference scenario, where the interference activity is assumed to be known and constant along the duration of the swarming algorithm, and a dynamic interference scenario. In this latter case, the interference activity over each frequency

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subchannel is modeled as a continuous-time Markov chain. We provide also a numerical validation of our theoretical findings about the spread of the swarm, as a function of the main system parameters. Finally, in Section VIII we draw some conclusions. II. BRIEF REVIEW OF ALGEBRAIC GRAPH THEORY In this section, we briefly recall some basic concepts from algebraic graph theory to be used throughout the paper. We model the interaction among the individuals (agents) of a swarm as an , where denotes undirected graph the set of nodes and is the edge set. The structure adjacency of the graph is described by the symmetric matrix , whose entries are either positive or zero, if there is or not a link between nodes and , respectively. The set of neighbors of a node is , defined as (1) communicates with node if is a neighbor of (or ). Denoting by the degree of node , the degree matrix is a diagonal matrix with entries that are the row sums of the adjacency matrix . The graph Laplacian is defined as and we denote by its ordered eigenvalues. The matrix always has, by construction, a right eigenvector composed of all ones, associated to the null eigenvalue . Properties: Let be an undirected graph of order with a symmetric nonnegative adjacency matrix . Then, the following statements hold true. 1) is a positive semidefinite matrix that satisfies the following sum-of-squares (SOS) property Node

(2) connected components iff 2) The graph has . In particular, is connected iff 3) Let be a connected graph, then for any such that , we have1

.

(3) is known as the algebraic connectivity of The quantity the graph and is a measure of performance/speed of consensus algorithms [26]. If the symmetric graph modeling the interaction among the nodes is connected, the multiplicity of the null eigenvalue of the Laplacian is one. In particular, we consider -dimensional graph Laplacians defined by (4) where denotes Kronecker product. This multidimensional Laplacian satisfies the following property:

(5) 1We recall that, for a connected graph, the nullspace of L has dimension 1 1 indicates a vector and it is spanned by the vector 1. Hence, a vector x lying in a subspace orthogonal to the nullspace of L .

?

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where and . Furthermore, the spectrum of the multidimensional Laplacian matrix is such that (6) where taining the eigenvalues of .

is the vector con-

III. SWARM MODEL Let us consider now a set of secondary users whose goal is to allocate resources (bits/power) dynamically in a domain typically occupied by primary users, with the aim of minimizing the interference towards the primary users. A typical example is the one where the resource space is the time-frequency domain and the goal of every secondary user is to find out a time slot and/or a frequency subchannel, temporally unoccupied by other users. The problem arises when the number of secondary users is very high and there is no central authority assigning the resources on demand. In such a case, it is necessary to devise a decentralized mechanism to assign resources under the constraint of keeping the interference towards primary and secondary users as low as possible. To have a notation as general as possible, we denote by the dimension of the resource domain and the single re, source selected by node is described by a vector whose entries denote, for example, a frequency subchannel and a time slot. In the presence of many SUs, the problem is how to avoid conflicts between them, without requiring a centralized coordination node. To avoid conflicts, we propose an iterative algorithm where, at each iteration, every node broadcasts the vector that is planning to occupy to its nearest neighbors. The interaction among the SU nodes is modeled as an undirected . We assume that there is a link (edge) begraph tween two nodes if the distance between them is less than a prescribed value (the coverage radius), dictated by the node’s transmit power and the radio channel characteristics. We formulate the resource allocation problem as the distributed minimization of a global function, defined as follows. Each node allocates its resources in the position represented by , whose entries may be time slots the vector and/or subchannel frequencies. The function to be minimized is

(7) whose first term represents the overall interference power over the optimization domain (e.g., the time-frequency plane) perceived by node , while the second and third term are two penalty terms taking into account, respectively, the spread of resources and the collisions, in the resource domain. Neglecting for a moment the second term on the right-hand side (RHS) of (7), the minimization of (7) leads every node to find a position such that the overall interference power, as perceived by the swarm, is minimum. This is a way to let the SUs to fill the gaps in the time-frequency domain. However, such a solution would not prevent two different SUs to choose the same position, thus conflicting with each other. Actually, if the

had a common single minsum of interference functions imum, every node would tend to occupy the same value, thus inducing an overall conflict among SUs. To avoid this situation, the second term on the RHS of (7) contains a repulsion function that is maximum when ). Hence, the two nodes occupy the same position (i.e., purpose of the repulsion term is to avoid collisions. However, the repulsion term alone might lead to an excessive dispersion of the slots occupied by the whole set of nodes, in the resource domain. This is also undesirable, because it may imply the occupation of an unjustified large region in the spectral domain or of distant time slots in the time domain, thus running into a nonstationarity problem. To avoid excessive dispersion, we introduce that is an attraction function minimum when the vectors are all close to each other. Hence, in summary, the attraction and repulsion terms in (7) are chosen so that the overall system tends to remain cohesive, without creating conflicts. Furthermore, there is a unique distance at which the attraction and repulsion forces balance: the so called equilibrium distance in the biological literature. This distance, in our case, is related to the bandwidth of the frequency slot or the duration of the time interval. Function (7) is reminiscent of the social foraging function, introduced by Gazi and Passino in [23] and subsequently generalized in [24]. The objective of [23] and [24] was to model the behavior of a swarm of birds searching for food, while moving collectively as a swarm and yet avoiding collisions. In their case, the three terms represented, respectively, the spatial distribution of food, the attraction and repulsion forces among the birds. In [23], the interaction among birds was modeled as a fully connected . The model was then generalized in graph, i.e., [24] to deal with an asymmetric graph. In our work, the coeffitake into account the existence of a radio link between cients two nodes and then they depend on radio channel characteristics, more specifically, on the physical distance between the nodes and . In our setup, the coefficients may assume any real nonnegative value, but because of the radio channel reciprocity, they . This simplifies our satisfy the symmetry condition analysis with respect to [24], yet providing more general results than [23]. In particular, both [23] and [24] concentrated on the swarm cohesiveness, or stability. The main goal of their analysis was then to provide an upper bound for the spatial spread of the swarm. Conversely, in our application, we are concerned with two main issues, collisions and spread: We want to limit the spread in the resource domain and avoid, or limit as much as possible, collisions, especially between nearby nodes. For this reason, the goal of our analysis is to provide both a lower and an upper bound of the swarm, to be able to assess the swarm properties in terms of overall occupancy and collision. Interestingly, thanks to the symmetry of our graph, we will provide simple expressions for the lower and upper bounds which yield physical insight about the role played by the graph connectivity. Before starting our analysis, it is worth to point out that the function (7), while mathematically similar to the function studied in [23] and [24], it is functionally different. In fact, in [23] and [24], the vectors indicate a spatial position and the are also related to the spatial positions of nodes coefficients and . Conversely, in our case, the vectors indicate the

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resource domain, i.e., time/frequency, while the coefficients are related to the spatial domain, as they depend on the spatial distance between the nodes. In this work, we consider, for simplicity, time-invariant coefficients. Since in our application the exchange of information among nodes occurs over wireless channels, in a parallel are paper we have analyzed the case where the coefficients random, in order to incorporate channel fading phenomena [25]. Our goal in this work is the distributed minimization of (7). A possible way to achieve the solution in decentralized form is to use a simple gradient based optimization, so that every node , and then it updates starts with an initial guess, let us say in time according to the its own resource allocation vector following dynamical system:

(8)

The trajectory of the center is given by

(12) where the equality of the second term of the RHS of (12) to zero and from the follows from the symmetry condition fact that is an odd function. The above equation means that the center of the swarm moves until the agents reach a position where the average gradient is zero. The fact that is the average gradient to determine the motion of the swarm center, rather than the individual gradients, is appealing in our context because the averaging operation reduces the effect of undesired zero-mean fluctuations due to observation noise or to errors in the estimate of the gradient. A. Profiles With Bounded Gradient

, with

denoting a vector function defined as (9)

where and are the derivatives of and with respect to , respectively. In this paper we consider a constant attraction term (10) and unbounded repulsion

We will now analyze the cohesiveness of the swarm under some assumptions on the attraction/repulsion functions and on the interference profile. To this end, we define the displacement vector between the position of node and the center of the swarm as . Deriving bounds on the magnitude of the vector is useful to quantify the size of the swarm and then, ultimately, the spread in the resource allocation domain. The assumptions needed for our derivations are the following. Assumption A.1: The interference profile functions and there exists a constant such that

(11)

(13)

These choices are instrumental to endow the system with the desired behavior and they are simple enough to allow for mathematical tractability. The constant attraction term in (10) determines an intensity of the attraction force in (9) that is directly proportional to the distance between resources. Unbounded repulsion is appealing in our intended application as it prevents collisions among nodes and ensures the existence of a lower bound on the swarm size, as we will see in the following section. and are chosen The parameters of the functions so that, at large distances (in the resource domain), the attraction term dominates, while at short distances is the repulsion term to dominate, and there is a unique distance where attraction and repulsion balance. In our setting, this equilibrium distance is chosen proportional to the bandwidth of the frequency slot, in the frequency domain, or to the duration of the elementary time and . slot and can be adjusted acting on the parameters It is important to remark, about the updating rule (8), that each individual in the swarm has to estimate only local parameters: the gradient of the interference level, evaluated only on its intended running position , and the balance of attraction and repulsion forces with its immediate neighbors.

This assumption is quite general and it only requires the gradient of the profile to be bounded. This hypothesis is indeed very reasonable in the context of interest. Under Assumption A.1

IV. COHESION ANALYSIS Before studying the stability of the swarm (8), it is useful to . analyze the motion of the swarm center:

(14) Assumption A.2: Given the initialization vector , the set is compact. In our application, the resource allocation domain, either a frequency band or a time interval (or both), is always a compact set. The incorporation of the frequency and/or time interval limits in our problem can be done either imposing box constraints on our optimization or by adding a barrier to the interference profiles , for all , i.e., a positive continuous term that starts from the boundary of the resource domain and goes to infinity linearly, with constant derivative , in order to keep satisfying Assumption A.2.2 Under this choice, Assumption A.1 holds true. We follow this second approach. We are now able to state the main theorem on swarm behavior: Theorem 1: Given the swarming algorithm described in (8), under Assumptions A.1 and A.2, the state converges to the largest invariant subset of the set 2To preserve C continuity it is also necessary to ensure a smooth transition from the original interference profile to the modified profile incorporating the barrier.

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and the modulus of the displacement vector is upper and lower bounded as follows:

From (21), the time derivative of the Lyapunov function along the system trajectory can be rewritten as

(15) where (16) and (17) where denotes the trace of . Proof: The evolution of the time derivative of the global potential function (7) along the trajectory described by (8)3 is

(18)

(22)

This means that, while moving along the trajectory given by (8), the potential function is always nonincreasing and it stops ) only if . If decreasing (i.e., the set defined as is compact, then using LaSalle’s invariance principle, we can conclude that, as , the state converges to the largest invariant subset of the set defined as

Exploiting the features of linear attraction and unbounded re, as expressed in (10) and pulsion of the coupling function (11), we obtain

(23)

(19) Let us consider now the displacement vector derivative of the distance is given by

. The time

The state vector can always be decomposed into the motion of the center of the swarm plus the displacement vector , as follows: (24)

(20) Defining a cumulative Lyapunov function as , with , and taking its time derivative along the system trajectory (20), we can write

where the disagreement vector satisfies , where is the vector of the canonical basis, with all entries equal to zero, except the th component, equal to one. The vector belongs to an -dimensional subspace (the disagreement eigenspace of ) that is orthogonal to the nullspace of . As a consequence of (5) and (24), expression (23) can be recast as

(25) Applying (3) and considering the expression of obtain an upper bound

in (6), we

(21) 3In the following, we drop the dependency on time t, to avoid an excessive overcrowding of the formulas, and we introduce the notation := ( ;...; ).

r

r

r

(26)

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This result proves the swarm cohesiveness. Lower Bound For the lower bound case, (31) assumes the form (35) . Also in this case, the RHS of (35) is a with parabola, with two real roots having opposite sign, which we and , with . Only is feasible in this indicate as case. In this case, the evolution of the system occurs in the space above the dashed parabola depicted in Fig. 1. At convergence, it must be (36)

Fig. 1. Upper and lower bounds of the potential function time derivative.

where and a lower bound

(37)

(27) of the time derivative of the Lyapunov function. Exploiting the Cauchy–Bunyakovsky–Schwarz (CBS) inequality and the Assumption A.1, (26) and (27) can be further bounded as (28) (29)

, the Now, considering final expressions for the lower and upper bounds of the time derivative of the global Lyapunov function take the form (30) (31) Let us consider now the two bounds in more detail. Upper Bound For the upper bound case, (30) assumes the form (32) with . The RHS of (32) is a polynomial in , having two real roots, with opposite sign, which we denote, respectively, as and , with . Clearly, only the positive root is feasible, as is certainly nonnegative. In, the evolution troducing the 2D plane having axes and of the system occurs below the solid parabola shown in Fig. 1. Clearly, at convergence, i.e., when , it must be (33)

This proves that the vectors cannot become all zero, i.e., the vectors cannot collapse all to the same value. This is a result of the repulsion force. Combining both upper and lower bounds, the evolution of the system occurs in the dashed area sketched in Fig. 1 and then, at convergence, (15) must hold true. We can easily check that, . This concludes the proof of the theorem. indeed, Remark: Inequality (15) implies that the modulus of cannot be zero, avoiding the overall collapse on the swarm center, but does not prevent some pairs of nodes to end up with the same resource allocation vector. This does not happen if the graph is fully connected, as in such a case the repulsion between nodes with the same allocation vector would go to infinity. However, in a sparse graph, two nodes and with no direct link between them (i.e., with ), may end up with the same resource vector. Actually, in our intended application, where the coefficients depend on the distance between the nodes, it may happen that two nodes get the same resource vector only if they are not neighbors. But this is indeed a positive behavior as it gives rise to what is typically known as spatial reuse of frequency slots. In Section V we will exhibit some numerical results showing that the proposed approach is intrinsically able to provide a spatial reuse of frequencies. Theorem 1 has been derived under the assumption of a bounded norm of the profile gradient. In the following, we will provide closed form expressions for the lower and upper bounds in the case of an unbounded gradient, e.g., in the case in which the profile is quadratic. B. Quadratic Profile We consider now a quadratic interference profile given by (38) where

, and

. Its gradient at a point

is given by (39) Theorem 2: Given the swarming algorithm described in (8), under the assumption of a quadratic profile as in (38), we have

where (34)

(40)

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where the lower and upper bounds are, respectively, given by (41) . with This proves swarm cohesiveness and prevents the individuals to collapse on the swarm center. Moreover, if the network is fully connected, the swarm converges to the only stable equilibrium of the system, given by (42) Proof: Under the hypothesis of quadratic profiles we have (43) As a consequence, the upper bound (26) can be rewritten as

(44)

A numerical validation of our theoretical findings will be provided in the Sections V and VI. In the next section, we consider in more detail the situation where the resource domain is monodimensional (1D), like the frequency axis, for example. In the ensuing section we will then provide some examples of application for 2D allocation domain, e.g., time-frequency domain. V. SWARMING IN THE FREQUENCY DOMAIN In this section we focus our attention on the swarming over a 1D domain, for example the frequency axis. First, we provide a local stability analysis useful to show that the introduction of the attraction and unbounded repulsion terms in the functional to be minimized does not affect the stability of the system. Moreover, considering the swarm discrete-time model, we provide an upper bound for the step size, depending on simple system parameters, that assures the local convergence to an equilibrium point. Then, we propose fast swarming methods based on a proper selection of the descent direction of a scaled gradient optimization. The first method is an approximation of the Newtonbased optimization and improves the convergence speed of the algorithm; the second method adapts the swarming speed with respect to the interference power perceived by the swarm. Finally, we show some numerical results to assess the performance of the resource allocation technique based on swarming. A. Local Stability Analysis In this section, we analyze the behavior of the swarm in proxwhere the gradient of the potenimity of a solution point tial function is equal to zero. To study the local stability of the system around a solution point, we consider the secondorder Taylor series expansion of the scalar-valued function around , given by

Similarly, the lower bound (27) becomes

(45)

(49)

Arguing as in Theorem 1, the evolution of the system in the plane occurs in the region comprised between the two parabolas described by (44) and (45), both centered on the axis . In particular, at convergence, i.e., when , we must have (46) where

where is the Hessian matrix, computed in , whose en. If is positive definite, the tries are potential function is locally approximated by a positive definite quadratic form, in the neighborhood of . This guarantees the local stability of the system. In the one-dimensional case, and . Considering a nonlinear coupling function characterized by linear attraction and unbounded repulsion as given in (10) and (11), the potential function assumes the form

(47) with . If the network is fully connected, (44) and (45) hold with strict equality and the swarm converges to the unique stable equilibrium point of the dynamical system. The final convergence value can be obtained substituting, in one of the bounds in (47), and , thus achieving (48) This concludes the proof of the theorem.

(50) The entries

of the Hessian matrix are then

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(51) According to Gershgorin theorem, is a positive definite matrix with all the eigenvalues greater than zero if

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represents the iteration matrix of where the discrete time algorithm evaluated at the equilibrium point. In order to assure the convergence of the swarming algorithm, the discrete time mapping must be a contraction having fixed point such that . In this way, letting , the discrete time procedure can be rewritten as (57)

(52) In our case, this gives

and the error vector converges to zero if the spectral radius of the iteration matrix is less than one in modulus. and the step size is This condition holds true if chosen such that . From the analysis carried out in Section V-A, the positive definiteness of the Hesis assured by the convexity of the interfersian matrix ence profiles evaluated at . Under this condition, the step size must be chosen in order to satisfy the previous bound. The upper limit may be difficult to evaluate in a distributed manner. Nevertheless, using again Gershgorin’s theorem, the maximum can be upper bounded as eigenvalue of

(53) As a consequence, the convexity of the interference profiles , evaluated in the system equilibrium point , guarantees the local stability. This is an important result as it shows that the introduction of the attraction and unbounded repulsion terms in the functional to be minimized does not affect the system stability. B. Discrete-Time Implementation

(58)

The swarm evolution has been described, up to now, in continuous time. In practice, the exchange of information between nodes of the networks requires a discrete-time implementation. This requires the time discretization of (8), which yields

where is the maximum convexity of the interference profile, is determined by the repulsion constant and is the maximum degree of the network connectivity. Hence, the convergence of the discrete time algorithm is ensured by choosing the step size in the interval

(59) (54) where the step size must be sufficiently small to ensure convergence. In the following, we will provide some upper bounds on , in order to guarantee convergence, at least in the neighborhood of the solution points. In the neighborhood of the equilibrium point , the continuous-time dynamical system can be approximated as (55) The discretization of this vector differential equation leads to the following difference equation:

(56)

whose upper bound depends now on global parameters that can be exchanged through the network. C. Fast Swarming Algorithms One of the main drawbacks of gradient-based methods is their speed of convergence, which is known to be low. Clearly, a distributed technique is amenable for resource allocation only if it guarantees convergence in a few iterations. In this section, we modify the basic swarming algorithm (8) in order to increase its convergence speed. In general, the minimization of the functional in (7) can be achieved through a general scaled-gradient method given by (60) where is some positive definite matrix representing a scaling along the direction of steepest descent. The introduction of is useful to increase the speed of the algorithm or to enforce particular behaviors on the swarm individuals. The

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evolution of the time derivative of the global potential function (7) along this modified system trajectory (60) is given by

PU’s activation. The discrete-time recursion of node can then be expressed as

(61) means inverse and transpose of a matrix. This where means that, moving along the trajectory given by (8), the potential function is always nonincreasing and it stops de) only if . Under assumption A.2, creasing (i.e., the set is compact, then using the LaSalle’s invariance principle we can conclude that, as , the state converges to the largest invariant subset of the set . defined as Typically, is chosen equal to the inverse of the Hessian in order to implement a Newton recursion having matrix an improved convergence speed with respect to the normal steepest descent. The Newton’s method approximates at each iteration by a quadratic function, as in (49), and then it moves towards the minimum of that quadratic function. However, the matrix is a full matrix, with elements given by (51). This implies that the computation of requires a centralized mechanism. One suboptimal, but parallelizable, solution consists in approximating the Hessian matrix by retaining only its diagonal entries. This simplifies the inversion of the Hessian matrix and allows parallel computation. We consider then the trajectory (60), where the diagonal scaling matrix has entries given by

(62)

The th element is computable at the correspondent node having access to the second derivative of the profile at the local point and to the positions of its neighbors. For general non convex profiles, the value of can be negative. To ensure the positive definiteness of the matrix and to preserve the descent direction of the algorithm, we can add a scaled identity matrix, with the scale adjusted at each iteration, to the Hessian’s approximation. An alternative solution to improve the convergence speed of the swarming algorithm uses a scaling matrix whose diagonal elements are functions of the power perceived at each node . The goal is to accelerate the motion of the resources perceiving a high interference and, at the same time, to slow down the resources that are allocating on idle subbands. This adaptive feature can be implemented using a variable step size depending on a monotonically increasing function of the perceived interference power. The values of this function are lower and upper bounded by positive values ensuring that is positive definite. Examples include linear, the matrix quadratic, logarithmic functions, etc. The diagonal entries of the scaling matrix can then be expressed as (63) This solution improves the reaction time needed by the algorithm to perform a resource allocation on idle bands in case of a

(64) is given by (62) or (63). In the following secwhere tions, we will illustrate how these solutions outperform the convergence speed of the gradient-based algorithm. D. Numerical Examples In this section we provide some numerical results to assess the performance of the proposed algorithms. Example 1: Validation of the Theoretical Results for Profiles With Bounded Gradient: In this example we show some numerical results supporting Theorem 1. We consider the one-dimensional evolution of ten agents constituting the swarm in the presence of a bounded profile composed of the superposition of several Gaussian functions. Each agent interacts with its neighbors according to a connected topology and updates its intended position according to (8), implemented in discrete-time. To provide a validation of our theoretical findings, we report, on the top side of Fig. 2, the behavior of the swarm size parameter and its theoretical bounds in (16) and (17), versus the node’s covering radius that determines the network connectivity. The results have been averaged over 100 independent realizations. The attraction and repulsion parameters used in this simulation are and . As we can see, the effect of an increment of connectivity slightly reduces the swarm size and remains always inside the theoretthe swarm parameter ical bound interval. The increment of the network connectivity implies tighter bounds that converge on constant values if the network is fully connected. A further example is given on the bottom side of Fig. 2, where we show the behavior of the swarm size parameter and its theoretical bounds in (16) and (17), versus the swarm attraction constant . As expected, an increment of the attraction force, while keeping constant the repulsion, decreases the swarm size. Also in this case, we can see how the theoretical bounds are always satisfied. Example 2: Collision Avoidance and Spatial Reuse of Frequency: One of the main features of the proposed swarming technique is the capability to prevent collisions between nearby nodes while, at the same time, allowing for spatial reuse of the frequency channels from secondary nodes far away from each other. The coupling coefficients in (8) depend on the distance between the SUs and reflect the network topology, as dictated by the coverage radius of each node. Interestingly, as will be shown next, the proposed swarming algorithm leads naturally to spatial reuse of frequencies, as the network topology becomes more and more sparse. To quantify the spatial reuse of frequencies, we introduce the reuse factor , defined as the ratio between the number of resources (frequency subchannels) necessary to guarantee one slot for each node and the number of channels really

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Fig. 3. Network topology (top) and allocation example (bottom). Fig. 2. Swarm size parameter versus the node covering radius (top) and the swarm attraction parameter (bottom).

allocated by the algorithm. As an example of channel allocation, in Fig. 3 we consider a network composed of 100 nodes (top side), where each node senses the interference spectrum shown on the bottom side. The swarm parameters are and . Every node starts from a random initial position and then it updates its intended position according to (8), implemented in discrete-time. In the application at hand, there is an intrinsic quantization of the frequency resources given by the subchannel bandwidth. In our implementation, we let the system evolve according to (54) until successive differences in allocation become smaller than the bandwidth of a frequency subchannel. At that point, the evolution stops and every SU is allowed to transmit over the selected channel. The final choice is indicated by assigning a different shape to different subchannels, as shown in the topology plot reported in Fig. 3 (top). In our experiment, the number of available channels with low interference is eight, hence much smaller than the number of users. Interestingly, from Fig. 3 (top), we can observe that the nodes that have picked up the same channel are never neighbor of each other. This is indeed one of the most interesting features of the proposed algorithm. This means that the algorithm is capable

of implementing a decentralized mechanism for spatial reuse of frequencies. In this case, the network frequency reuse parameter is . Clearly, the reuse factor depends on the sparsity of the graph describing the network topology. To quantify the effect of the coverage radius of each node on the reuse factor, in Fig. 4, we report the average behavior of , averaged over 100 independent initializations, versus the covering radius of each node, considering three different values of the repulsion constant . In this simulation, we consider nodes, , the same interference profile as in Fig. 3 (bottom) and 15 available channels on the idle band in the middle of the spectrum. As expected, the behavior of is monotonically decreasing and it reaches the unit value when the covering radius is such that the network is fully connected. Furthermore, Fig. 4 shows also that, by reducing the value of , the repulsion force is weaker and this facilitates the reuse of frequency slots. Hence, the parameter has to be chosen as a tradeoff between reuse factor and number of collisions. We have checked numerically that in all simulations, choosing appropriately the swarm parameters, the final channel allocation never determines collisions among spatial neighbors. Example 3: Convergence Speed: One of the main issues for distributed resource allocation algorithms is convergence speed.

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Fig. 4. Frequency reuse parameter versus coverage radius.

Fig. 6. Dynamic resource allocation by swarming: Reaction time to PU’s activations.

Fig. 5. Normalized system potential function versus time index, for different coverage radii (top), and for different descent directions of the algorithm (bottom): gradient descent (dashed) and Newton approximation (solid).

In this section we show some numerical examples to evaluate the convergence time of the proposed allocation algorithm and its modified version given in (64). The first example assumes the same settings as in Fig. 3. To assess convergence time, on the top side of Fig. 5, we report the average behavior of the evolution of the system potential function (7), normalized with

respect to the maximum and the minimum value, averaged over 500 independent realizations, versus the iteration index. Three different coverage radii are considered to evaluate the impact of network topology on the convergence speed. The attraction and repulsion parameters of the swarm for this simulation are and , and the step size is equal to . As expected, the convergence rate increases as the connectivity increases. The second example compares the convergence speed of the gradient based swarming algorithm in (8) and of the approximated Newton method with scaling coefficients given by (62). On the bottom side of Fig. 5, we report the average behavior of the evolution of the system potential function, e.g., (7), normalized with respect to the maximum and the minimum value, averaged over 500 independent realizations, versus the iteration index. The evolution of the gradient-based algorithm is given by the dashed curve while the approximated Newton version is depicted by the solid curve. In this simulation we consider the same interference profile and network topology depicted in Fig. 3. The parameters are the same of the previous simulation and the step size for the approximated Newton version is equal to . The step sizes were empirically decided because slightly greater values determine instability of the algorithm in both cases. From Fig. 5, we can notice how the approximated Newton-scaled version greatly outperforms the gradient

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based algorithm. This means that the convergence time of the swarming algorithm can be considerably improved if every node is able to evaluate the second order derivative of the system potential function given by (64). Example 4: Dynamic Response of the Swarm to a Predator (Interferer): Natural swarms are adaptive systems whose individuals cooperate in order to improve their food search capabilities and to increase their robustness against predators’ attacks. We show next that the proposed resource allocation increases, as a by-product, the network robustness against the intrusion of a primary user (predator). We consider again the network topology depicted in Fig. 3, plus the inclusion of two PUs that start emitting, at different times, thus causing a dynamic change of the occupied spectrum. Our goal is to test the dynamic response of the network to this changing environment. Playing again with the swarm analogy, PUs take now the role of predators whose positions must be avoided by the swarm individuals. To give an example of the reaction time needed by the algorithm to react to the PU’s intrusion and adjust the resource allocation consequently, in Fig. 6 we show the behavior of the average interference perceived by the swarm versus the time index. The two peaks at the iterations 67 and 123 correspond to the two PU’s activation times. The low power value represents the noise level. In particular, we compare the results of the gradient-based algorithm in (8), on the top side, and its scaled version in (63), on the bottom side, that adapts the convergence speed with respect to the perceived interference. The attraction and repulsion paand ; the rameters used in this simulation are step size is equal to for both algorithms. We can notice how the adaptive scaled version needs only a small number of iterations to leave the PU’s regions, thus outperforming the gradient-based version. This positive behavior is given by the adaptation of the algorithm with respect to the perceived interference, determining that resources allocating on high interference regions move faster due to the increment of the profile gradient and the cohesion force. These examples show that the cohesion force represents an intrinsic robustness factor of the algorithm. In fact, resources allocating over high interference bands might measure a flat spectrum, thus resulting in limited capabilities to move out of (flat) occupied bands, if the only cause of change is spectrum gradient. However, increasing the cohesion force, the agents allocating over the low interference band tend to form cohesive blocks that exert an attraction towards the agents trapped by mistake over the flat regions of the spectrum occupied by the primary users. This is an example of cooperation gain. VI. SWARMING IN THE TIME-FREQUENCY PLANE The proposed swarming procedure can be extended to the two-dimensional domain, representing, for instance, the timereferred to node frequency plane. In this case, the vector has two entries representing the position of the frequency subchannel and the time slot that node intends to occupy. Before occupying the slot, each secondary node interacts with its neighbors exchanging information on the intended slot. Every node then updates its intended position according to (8), implemented in discrete-time. In the application at hand, there is an intrinsic quantization of the time-frequency resources given by the subchannel bandwidth and by the duration of the time slot. In our

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implementation, we let the system evolve according to (8) until successive differences in allocation become smaller than the discretization step in the time-frequency domain. At that point, the evolution stops and every SU is allowed to transmit over the selected resource, i.e., a pair of frequency subchannel/time slot. A major difference with respect to the frequency domain is that the allocation over successive time slots requires the knowledge of the primary users’ activities through time. Of course, this noncausal knowledge of the future is not available. However, if we have a statistical model of the interference activity, we may derive a resource allocation mechanism, based on our statistical model. In this section we provide some examples of swarming in the time-frequency plane. We consider first the ideal case of known interference profile, which can represent a limit case of a static environment. Then, we will consider the more realistic case in which the PU’s activity is modeled as a Markov chain. A. Swarming in a Static Interference Environment In the first example we provide numerical support to Theorem 2. We consider the two-dimensional evolution of 15 agents constituting the swarm, in the presence of a quadratic at the center of profile having a global minimum the plane. On the top side of Fig. 7, we report the behavior of the swarm size parameter and its theoretical bounds in (47), versus the magnitude of the quadratic profile. The results have been averaged over 100 independent realizations. The swarm parameters are and . As we can see, the effect of an increment of the profile magnitude reduces the swarm size and the swarm parameter remains always inside the theoretical bound interval. The increment of the network connectivity implies tighter bounds on the final swarm size and the presence of a unique equilibrium point in case of a fully connected network. This behavior is shown on the bottom side of Fig. 7, where we repeat the previous simulation increasing the coverage radius of each node and, as a consequence, the connectivity of the network graph. A second example is reported on the top side of Fig. 8. The boxes in Fig. 8 represent the power allocation of primary users, in the time-frequency domain, supposed to be static and known, within the convergence time of the swarm. These are the regions that do not have to be occupied by the SUs. The initial guesses of the SUs are represented as squares, scattered randomly across the time-frequency plane. The evolution of the resource allocation is depicted by the dotted curves and the final allocations are given by dots. It is evident how the secondary users avoid the positions occupied by the primary users, tend to keep the spread in the time-frequency plane as small as possible and, at the same time, they avoid collisions with each other. The previous examples refer to the idealistic case where the interference is assumed to be static and known, at least within the convergence time of the swarming algorithm. Next section generalizes the approach to the case where the interference is known only in a statistical sense. B. Swarming in the Presence of Dynamic Markov Interference In this section, we model the interference activity, over each frequency subchannel, as a two-state continuous-time homogeneous Markovian chain. The two (ON/OFF) states refer to the cases where the interferer is transmitting or is idle. This model gives a map of the expected interference power over the time-

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where [see (67) at the bottom of the page]. Let us suppose that, on each channel, it is known, through preliminary estimation, and the transition rates from idle to idle the average power and from active to active . Let us denote by the expected power on channel , at time , conditioned to the knowledge of the channel status at time 0. Suppose now that at time 0, the channel is sensed as idle. The probability vector that, at , the channel will be either idle or busy is then: time (68) Conversely, if, at time 0, the channel is sensed as active, the probability vector at time is (69) Hence, the expected power, at time pressed as

Fig. 7. Swarm size parameter versus the magnitude of the quadratic profile A .

frequency plane, conditioned to the measurement performed in the initial time slot. This expected profile is then used by the swarm algorithm to allocate resources according to the assumed statistical model. Let us denote by (65) the transition matrix of the interferer over the th frequency subchannel. At each time , each subchannel can be either idle or and the probabilities that active. Let us denote by subchannel , at time , is idle or active, respectively. Let us also introduce the probability vector . Given the vector at time 0, the vector probability at time is (66)

on channel

can be ex-

idle at

(70)

active at

(71)

Expressions (70) and (71) model the time evolution of the expected interference power over each subchannel. This model allows the construction of a map of the expected interference distribution, conditioned to the estimation performed in the first slot. This profile is then used by the swarm to dynamically allocate power over the time-frequency plane. As an example, the bottom side of Fig. 8 shows the evolution of the swarm in the time-frequency plane. The grey level represents the expected interference profile. At time 0, there are two disjoint subbands with a high level of interference and three interference-free regions. The evolution of the (expected) interference in time is given by (70), in case the channel at time 0 is sensed as idle, or by (71), in case the channel is sensed as busy. The initial guesses of the SUs are represented as squares, scattered randomly across the time-frequency plane. The evolution of the resource allocation is depicted by the dotted curves and the final allocations are given by dots. This figure shows that the swarm tends to keep the spread in the time-frequency plane as limited as possible while avoiding, at the same time, collisions. Interestingly, the swarm tends to stay as close as possible, to time 0, where the prediction is better, under the constraint of avoiding collisions. VII. CONCLUSION In this paper, we have proposed a distributed resource allocation strategy for the access of opportunistic users in cognitive networks based on a swarm model mimicking the foraging behavior of a swarm, where the interference distribution over the radio resource domain plays the role of spatial distribution of food. The swarm mechanism includes an attraction force, useful

(67)

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techniques outperform the basic swarming algorithm. Several directions for future research are open, incorporating more realistic channel models, and taking into account non ideal communications among the sensors. REFERENCES

Fig. 8. Examples of time-frequency allocation.

to minimize the spread over the resource domain, and a repulsion force, useful to avoid collisions among radio nodes. In the proposed model, each agent is supposed to listen only to nearby nodes, in a narrow band spectral interval, over consecutive time slots. This is useful to limit the complexity of the secondary users’ equipments and, interestingly enough, it produces, as a by-product, an intrinsic capability of the system to provide spatial reuse of frequency, through a purely decentralized mechanism. We have derived closed form expressions, validated by numerical results, for the upper and lower bounds of the final spread as a function of the main system parameters and of the network topology. Specific attention has been devoted to the analysis of the proposed swarming mechanism in the frequency domain, where a local stability analysis has shown that the introduction of the attraction and unbounded repulsion terms in the functional to be minimized does not affect the stability of the system. Furthermore, we have introduced fast swarming methods, useful to improve the convergence rate and the algorithm’s reaction time in dynamic environments. Simulation results confirm that these

[1] J. Mitola, “Cognitive radio for flexible mobile multimedia communications,” Mobile Netw. Appl., vol. 6, no. 5, pp. 435–441, 2001. [2] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [3] I. Akyildiz, W. Lee, M. Vuran, and S. Mohanty, “NeXt generation/ dynamic spectrum access/cognitive radio wireless networks: A survey,” Comput. Netw., vol. 50, no. 13, pp. 2127–2159, Sep. 2006. [4] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,” IEEE Signal. Process. Mag., vol. 24, no. 3, pp. 79–89, May 2007. [5] Q. Zhao, L. Tong, and A. Swami, “Decentralized cognitive MAC for dynamic spectrum access,” in Proc. 1st IEEE Int. Symp. New Frontiers Dyn. Spectrum Access Netw., Baltimore, MD, Nov. 2005, pp. 224–232. [6] Q. Zhao, L. Tong, A. Swami, and Y. Chen, “Decentralized cognitive MAC for opportunistic spectrum access in ad hoc networks: A POMDP framework,” IEEE J. Sel. Areas Commun., vol. 25, no. 3, pp. 589–600, Apr. 2007. [7] Q. Zhao and A. Swami, “A decision-theoretic framework for opportunistic spectrum access,” IEEE Wireless Commun. Mag. (Special Issue Cogn. Wireless Netw.), vol. 14, no. 4, pp. 14–20, Aug. 2007. [8] S. Geirhofer, L. Tong, and B. Sadler, “Cognitive medium access: Constraining interference based on experimental models,” IEEE J. Sel. Areas Commun. (Special Issue on Cogn. Radio: Theory Appl.), vol. 26, no. 1, pp. 95–105, Jan. 2008. [9] S. Geirhofer, L. Tong, and B. Sadler, “Opportunistic spectrum access via periodic channel sensing,” IEEE Trans. Signal Process., vol. 56, no. 2, pp. 785–796, Feb. 2008. [10] Y. Chen, Q. Zhao, and A. Swami, “Joint design and separation principle for opportunistic spectrum access in the presence of sensing errors,” IEEE Trans. Inf. Theory, vol. 54, no. 5, pp. 2053–2071, May 2008. [11] Q. Zhao, B. Krishnamachari, and K. Liu, “On myopic sensing for multichannel opportunistic access: Structure, optimality, and performance,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5431–5440, Dec. 2008. [12] X. Li, Q. Zhao, X. Guan, and L. Tong, “Optimal cognitive access of Markovian channels under tight collision constraints,” in Proc. 2010 IEEE Int. Conf. Commun. (ICC), Cape Town, South Africa, Jul. 2010, pp. 1–5. [13] S. Chen and L. Tong, “Multiuser cognitive access of continuous time Markov channels: Maximum throughput and effective bandwidth regions,” in Proc. Inf. Theory Appl. Workshop (ITA) 2010, San Diego, CA, Jan.–Feb. 2010, pp. 1–10. [14] Z. Quan, S. Cui, H. V. Poor, and A. H. Sayed, “Optimal multiband joint detection for spectrum sensing in cognitive radio networks,” IEEE Trans. Signal Process., vol. 57, pp. 1128–1140, Mar. 2009. [15] J. Unnikrishnan and V. V. Veeravalli, “Algorithms for dynamic spectrum access with learning for cognitive radio,” IEEE Trans. Signal Process., vol. 58, no. 2, pp. 750–760, Feb. 2010. [16] S. Barbarossa, S. Sardellitti, and G. Scutari, “Joint optimization of detection thresholds and power allocation for opportunistic access in multicarrier cognitive radio networks,” in Proc. CAMSAP 2009, Aruba, Dec. 12–14, 2009, pp. 404–407. [17] G. Scutari, D. P. Palomar, and S. Barbarossa, “Cognitive MIMO radio,” IEEE Signal Process. Mag., vol. 25, pp. 46–59, Nov. 2008. [18] V. Krishnamurthy, “Decentralized spectrum access amongst cognitive radios—An interacting multivariate global game-theoretic approach,” IEEE Trans. Signal Process., vol. 57, no. 10, pp. 3999–4013, Oct. 2009. [19] J. Huang and V. Krishnamurthy, “Transmission control in cognitive radio as a Markovian dynamic game: Structural result on randomized threshold policies,” IEEE Trans. Commun., vol. 58, no. 1, pp. 301–310, Jan. 2010. [20] M. Maskery, V. Krishnamurthy, and Q. Zhao, “Cecentralized dynamic spectrum access for cognitive radios: Cooperative design of a non-cooperative game,” IEEE Trans. Commun., vol. 57, no. 2, pp. 459–469, Feb. 2009. [21] S. Barbarossa, S. Sardellitti, and G. Scutari, “Joint optimization of detection thresholds and power allocation in multiuser wideband cognitive radios,” presented at the COGIS 2009, Paris, France, Nov. 16–18, 2009.

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Paolo Di Lorenzo (S’08) received the B.S. and M.S. degrees in electrical engineering from University of Rome ‘‘Sapienza’’, Italy, in 2005 and 2008, respectively. He is currently working towards the Ph.D. degree in the Department of Information, Electronics and Telecommunications, University of Rome, ‘‘Sapienza’’. During 2010, he held a visiting research appointment at the Department of Electrical Engineering, University of California at Los Angeles (UCLA). His primary research interests are in bio-inspired signal processing, synthetic aperture radar systems, distributed algorithms

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for communication and sensor networks, adaptive filtering and stochastic optimization. Mr. Di Lorenzo received the 2010 IEEE Best Student Paper Award at SPAWC’10 and the 2011 EURASIP Best student paper award at EUSIPCO 2011 for two works in the area of bio-inspired signal processing for communications.

Sergio Barbarossa (S’84-M’88) received the M.Sc. degree in 1984 and the Ph.D. degree in electrical engineering in 1988, both from the University of Rome ‘‘La Sapienza’’, Rome, Italy. He is currently a Full Professor at the University of Rome “La Sapienza”. He has held positions as a Research Engineer at Selenia SpA from 1984 to 1986 and at the Environmental Institute of Michigan (1988), and he was a visiting professor at the University of Virginia in 1995 and 1997 and the University of Minnesota in 1999, and he has taught short graduated courses at the Polytechnic University of Catalonia in 2001 and 2009. He has been principal investigator in several international projects. He has been the scientific lead of the European project WINSOC, on wireless sensor networks. He is currently the scientific lead of the European Project FREEDOM, on femtocell networks, he is a Principal Investigator on the European Project SIMTISYS, on the monitoring of maritime traffic from constellations of satellites. He is also involved in the design of the radar system to be used for the landing module of ExoMars, a spacecraft to be launched by the European Space Agency to land on Mars. He is the author of a research monograph titled Multiantenna Wireless Communication Systems (Artech House, 2005). His current research interests lie in the areas of cognitive radios, sensor networks, cooperative communications, and distributed decision. Dr. Barbarossa has been a member of the IEEE Signal Processing for Communications Technical Committee from 1998 to 2004, and he served as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1998 to 2001 and from 2004 to 2006. He has been a plenary speaker at SAM 2008 and SPAWC 2009. He has co-edited a special issue of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS on Optimization of MIMO Transceivers for Realistic Communication Networks: Challenges and Opportunities and a special issue of the EURASIP Journal of Applied Signal Processing on MIMO Communications and Signal Processing. He received the 2000 IEEE Best Paper Award from the IEEE Signal Processing Society and the 2010 EURASIP Technical Achievements Award.