Distributed Energy Efficient Cross-layer Optimization for Multihop MIMO Cognitive Radio Networks with Primary User Rate Protection
Weiqiang Xu, Wenchu Yuan, Qingjiang Shi, Xiaodong Wang, Yake Zhang
aa r X i v : . [ c s . N I] O c t Distributed Energy Efficient Cross-layerOptimization for Multihop MIMO Cognitive RadioNetworks with Primary User Rate Protection
Weiqiang Xu,
Senior Member, IEEE , Wenchu Yuan,Qingjiang Shi,
Member, IEEE , Xiaodong Wang,
Fellow, IEEE , Yake Zhang
Abstract
Due to the unique physical-layer characteristics associated with MIMO and cognitive radio (CR), the networkperformance is tightly coupled with mechanisms at the physical, link, network, and transport layers. In this paper,we consider an energy-efficient cross-layer optimization problem in multihop MIMO CR networks. The objectiveis to balance the weighted network utility and weighted power consumption of SU sessions, with a minimum PUtransmission rate constraint and SU power consumption constraints. However, this problem is highly challenging dueto the nonconvex PU rate constraint. We propose a solution that features linearization-based alternative optimizationmethod and a heuristic primal recovery method. We further develop a distributed algorithm to jointly optimizecovariance matrix at each transmitting SU node, bandwidth allocation at each SU link, rate control at eachsession source and multihop/multi-path routing. Extensive simulation results demonstrate that the performanceof the proposed distributed algorithm is close to that of the centralized algorithm, and the proposed frameworkprovides an efficient way to significantly save power consumption, while achieving the network utility very closeto that achieved with full power consumption.
Index Terms
MIMO, cognitive radio, cross-layer optimization, alternative optimization, primal recovery.
Weiqiang Xu, Wenchu Yuan, Qingjiang Shi, and Yake Zhang are with School of Information Science & Technology, Zhejiang Sci-TechUniversity, Hangzhou, 310018, P. R. China. Email: [email protected], [email protected], [email protected], [email protected] Wang is with the Department of Electrical Engineering, Columbia University, New York, NY, 10027, USA. Email:[email protected].
I. I
NTRODUCTION
The cognitive radio (CR), with its built-in intelligence and cognitive capabilities, can flexibly adaptits transmission or reception parameters, and provides the means for unlicensed secondary users (SU) todynamically access the licensed spectrum held by primary users (PU) [1]. Thus, as a promising technologyto deal with the spectrum under-utilization problem, CR has quickly become the enabling technology forthe next-generation wireless communications, and will be adopted by many emerging applications, e.g.,smart grid communications, public safety, and medical applications [2].Research on effective spectrum sharing or spectrum allocation has been extensive. For multi-user single-hop CR networks (CRN), a number of approaches have been proposed. For example, [3] provided a surveyon dynamic resource allocation schemes with the interference temperature based spectrum-sharing model.For the multihop networking problem with CRs, there is a limited amount of work to date availablein the literature [4]–[12]. [4] proposed a mathematical formulation to modeling spectrum sharing, sub-band division, scheduling, and flow routing in multi-hop CRN. [5] developed a formal mathematicalmodel for a joint per-node based power control, scheduling, and flow routing problem in multi-hopCRN. These two joint formulation are the mixed-integer non-linear program (MINLP) problem. [6]proposed a framework of joint spectrum allocation and power control to utilize the open spectrum bands inCRN with both interference temperature constraints and spectrum dynamics. [7] investigated a multicastcommunication problem in a multihop CRN. A scheduling and routing approach was proposed to minimizethe network-wide resource utilization to support a set of multicast sessions. [8] addressed the stochastictraffic engineering problem in multihop CRN. The challenges induced by the random behaviors of theprimary users are addressed through the stochastic network utility maximization framework. [9] proposeda distributed routing algorithm to minimize the aggregate interference from the SUs to the PUs in CR meshnetworks. [10] studied a utility maximization framework by adapting SINR assignment and transmit powersubject to power budget constraints and additional interference temperature constraint. [11] investigatedthe problem of spectrum assignment and sharing to minimize the total delay of multiple concurrent flowsin multi-hop CRN. [12] provided a survey on the state-of-the-art of research on physical, medium access,and routing layer issues in the design of CRN, including the multihop scenario.In parallel to the development of CR, MIMO is a physical layer technology that can provide many types of benefits through multiple antennas and advanced signal processing.The potential network capacity gain with the use of MIMO depends on the coordinated mechanisms atthe physical, link, and network layers [13]. Thus, some works have addressed to exploit the benefits ofMIMO from a cross-layer prospective in single and multihop MIMO ad hoc networks. For example, [14]investigated the cross-layer optimization problem in multihop MIMO backhaul networks to maximize thefair throughput of the access points. [15] proposed the cross-layer algorithms for MIMO ad hoc networksto maximize the SINR. [16] considered the problem of jointly optimizing power and bandwidth allocationat each node and multihop multi-path routing in a MIMO ad hoc network, and developed a two-stepsolution to this cross-layer optimization problem.In CR networks, some efforts to explore the cognitive MIMO radio mainly focus on the physicallayer. The game theoretical approach and optimization-based approach have been applied to explore thepotential of MIMO CRN [17]–[25]. For instance, [25] proposed joint time scheduling and beamformingoptimization to minimize SU’s energy consumption while satisfying SU’s rate requirements and PUinterference constraints. However, these efforts mainly focused on the resource optimization for single-hopMIMO CRN. Few effort has addressed how to take advantage of the MIMO techniques in the context ofmultihop CRN through cross-layer design. [26] developed a tractable mathematical model for multihopMIMO CRN and jointly optimized channel assignment in CRN and degree-of-freedom allocation in MIMOto maximize the throughput. However, the centralized method is difficult to be implemented in the realisticmultihop MIMO CRN.In this paper, we propose a new formulation to address an energy efficient cross-layer optimizationproblem in a multihop MIMO CRN. Our objective is to optimize the weighted sum of network utilityand power consumption with a minimum PU transmission rate constraint and SU power consumptionconstraints. We jointly optimize covariance matrix at each transmitting SU node, bandwidth allocationat each SU link, rate control at each session source and multihop/multi-path routing. However, such ajoint formulation is usually highly nonconvex and difficult to solve due to the nonconvex MIMO linkcapacity constraint and the PU rate constraint. Although by exploiting the special structure we can easilydesign a distributed algorithm based on the popular dual-decomposition method, the dual-decompositionsolution is not feasible to the primal problem due to the nonconvexity nature of the problem. Hence, in this paper, we attempt to design a distributed algorithm that can yield a feasible suboptimal solution. Ouralgorithm includes two steps. The first step is to obtain a reasonable bandwidth allocation while in thesecond step we recover a feasible solution. Both steps are based on the dual-decomposition method, whichdecomposes the joint optimization problem into the network-transport layer subproblems and the physical-link layer subproblems. The network-transport layer subproblems in both steps are convex and can besolved efficiently and distributively. We mainly focus on the nonconvex physical-link layer subproblemin the first step and propose an iterative algorithm based on linearization of the nonconvex constraint andalternating optimization. We prove the algorithm can converge monotonically. The performance of ourproposed distributed optimization algorithm is examined by extensive simulations. By comparing with aperformance upper bound offered by a centralized algorithm, we find that the proposed algorithm canprovide a feasible solution with good performance, and the proposed formulation can provide an efficientmechanism to significantly save power consumption, while achieving the network utility very close to thatachieved at full power consumption. The key novelties and contributions of our work are summarized asfollows: • Note that for the MIMO-link capacity function Eq. (2) used in [16], the background noise is assumedto be not related to the allocated bandwidth. In this paper, we use a correlated but different MIMO-link capacity formula, where the background noise is related to the allocated bandwidth. Based on alinearization-based alternative optimization method and a heuristic primal recovery method, we designa new distributed algorithm that can find a feasible solution with good performance for the non-convexoptimization problem of multihop MIMO CRN, which is a generalization of the optimization problemin [16]. • Up to now, MIMO CRN design has mainly been focusing on utility (e.g., sum-rate) maximizationor the power minimization separately. However, achieving high transmission rate while consuminglow power consumption has recently become a main concern. In this paper, we extend traditionalMIMO CRN design towards a framework of utility-power trade-off with power related objective andconstraint. Specifically, we address the optimization objective of the weighted sum of network utilityand power consumption and consider non-full power consumption constraint. We quantify the utility-power trade-off with some surprising numerical results. Thus, our formulation provides an efficient way to design a green cross-layer optimization scheme for multihop MIMO CRN. • So far, many existing works proposed various algorithms for MIMO CRN based on the interferencepower constraints for protecting PU. In contrast to the conventional interference power constraintapproach, we protect the primary receiver by the direct constraint on the minimum PU transmissionrate subject to the SU interference. Although the non-convexity of the minimum PU rate constraint isintroduced, we apply the method of Taylor series expansion to approximate this non-convex constraintand propose an iterative algorithm to solve the non-convex optimization problem.The remainder of this paper is organized as follows. In Section II, we present the system modeland problem formulation. In Section III, we solve our problem through the dual decomposition and thesubgradient algorithm. In Section IV, we propose the linearization-based alternative optimization method.In Section V, we recover the primal solution. In Section VI, we provide the simulation results. Finally,we conclude the paper in Section VII.The upper-case and lower-case boldface letters denote matrices and vectors, respectively. The conjugatetranspose, Hermitian transpose, determinant are represented by ( · ) ∗ , ( · ) H , |·| , respectively. The trace isdenoted by Tr ( · ) . We let I denote the identity matrix, the dimension is determined by the context. Wedefine the operators ≥ and ≤ for vectors and matrices as componentwise, while A ≻ − B means that A − B is positive semidefinite. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a multihop MIMO CRN, where multi-antenna SUs share the same spectrum with multi-antenna PUs. We focus on the spectrum sharing model. In this model, from PU’s perspective, SU isallowed to transmit as long as the interference from SU does not degrade the quality of service (QoS)of PU to an unacceptable level. From SU’s perspective, SU should control its transmit power properlyin order to achieve a reasonably high transmission rate without causing too much interference to PU.The SUs form a multihop FDMA MIMO network, which is represented by a directed graph, denoted as G s = {N s , L s } , where N s = { , , . . . , n s , . . . , N s } and L s = { , , . . . , l s , . . . , L s } represent all SUs andall the possible SU MIMO links, respectively. The network is assumed to be always connected. Withinsuch a multihop MIMO CRN, the packets from a source node will reach a destination node throughmultihop/multipath. SU Node(Receiver)SU Node(Receiver) SU Node(Transmitter) p j H s l H PU Node(Transmitter)PU Node(Receiver) s p l j H PU BandwidthSU Bandwidth p k H PU Node(Transmitter)PU Node(Receiver) SU Node(Receiver)SU Node(Receiver)SU Node(Transmitter) p k B s m W PU BandwidthSU BandwidthGeolocationDatabase p s k m H s p l j H p s j l H p s j l H s l H p s k m H s p m k H s p m k H s m H s m H s m W p j B s l W s l W Fig. 1. An application scenario of multihop MIMO CRN operating over TVWS.
A. Application Scenario
Recently, a series of emerging wireless access standards incorporated CR technology over Digital TVwhite space (TVWS) [27] . Fig. 1 gives an application scenario of multihop MIMO CRN operatingover TVWS. The FCC ruling had opened up the possibility of designing wireless multihop networkswhere CR operates over TVWS [28]. The most recent FCC ruling requires that TVWS devices mustrely on a geolocation database to determine the spectrum availability [29] [30]. In such a database-assisted architecture, the primary licensed holders of TV spectrum provide the database with the up-to-date information including TV tower transmission parameters, TV receiver protection requirements, andetc.. Based on this information, the geolocation database implements a centralized spectrum allocationmechanism such that different SU nodes are assigned non-overlapping bandwidths. In this paper, weassume that the bandwidth of each SU node is given from the geolocation database. Our proposedbandwidth allocation scheme is to re-allocate/sub-divide the bandwidth of SU node among its all outgoinglinks. Note that its incoming and outgoing links is assigned different frequency bands, such that SU nodecan simultaneously transmit and receive signals, and cause interference among each other.
B. Channel Model
A MIMO SU n s ∈ N s includes possible multiple secondary outgoing links l s ∈ O ( n s ) ⊂ L s , where O ( n s ) denotes the set of links that are outgoing from node n s . Without loss of generality , we assumethat a MIMO SU n s only shares a portion or the entire of the bandwidth with a MIMO PU link j p .Let B j p denote the total bandwidth of PU link, W l s the overlapping bandwidth of PU link j p and SUlink l s . Furthermore, we assume that the transmit power of each PU link is distributed equally overfrequency band, i.e., the power allocation of the PU link for different frequency band is proportional tothe bandwidth. The transmission over the SU MIMO channel l s with T l s transmit and R l s receive antennascan be expressed as the following signal model: r l s = H l s t l s + s W l s B j p H j p l s t j p + n l s (1)where r l s ∈ C R ls × denotes the received signal vector of the secondary link l s , with R l s denoting thenumbers of receiving antennas. t l s ∈ C T ls × denotes the transmitted signal vector of link l s , with T l s denoting the numbers of transmitting antennas. H l s ∈ C R ls × T ls denotes the channel gain matrix fromthe transmitting node to the receiving node of link l s . H j p l s ∈ C R ls × T jp denotes the channel gain matrixfrom the transmitting node of the primary link j p to the receiving node of the secondary link l s , with T j p denoting the number of transmitting antenna of the primary link j p . t j p denotes the transmitted signalvector of the primary link j p . n l s denotes an additive white Gaussian noise vector of link l s . We assumethat the channel is block fading and H l s is known at the transmitting node of link l s . C. SU power constraint
Here, the transmit power constraint of each transmitting node n s of SU link is given by: X l s ∈O ( n s ) Tr ( Q l s ) α n s P n s ∀ n s (2)where Q l s = E { t l s t Hl s } denotes the transmitting covariance matrix of t l s at link l s , which is Hermitianand positive semi-definite (PSD). P n s denotes the maximum transmitting power at SU n s . < α n s ≤ isa chosen constant, and denotes the required power reduction with respect to the full power usage at SU n s . Note that from a mathematical perspective, accounting for α n s is straightforward. Note that it is readily seen that we can easily generalize to a scenario where a SU can coexist with multiple PUs, and our proposedscheme also applies to this scenario with minor modification.
D. PU Rate Constraint
In order to ensure the QoS of the PU network, we have the following rate requirement at PU link j p ,inspired by [31], R j p ≥ R min j p (3)where R min j p denotes the minimum transmission rate requirement of the primary link j p . R j p denotes theactual transmission rate of PU link j p .We assume that PU link is not fully interfered illustrated in the top-left of Fig. 1. So, R j p is composedof two parts : R j p = X l s ∈O ( n s ) W l s r l s j p + R NI j p (4)where W l s ≥ denotes the bandwidth assigned to SU link l s . The first term on the RHS is the achievablerate at the frequency band which suffers the interference from SU. Based on the classical capacity formula,we have r l s j p = log (cid:12)(cid:12)(cid:12) W l s N j p I + W ls B jp H j p Q ∗ j p H Hj p + H l s j p Q l s H Hl s j p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N j p W l s I + H l s j p Q l s H Hl s j p (cid:12)(cid:12)(cid:12) (5)The second term on the RHS R NI j p is the achievable rate at the frequency band which does not suffer theinterference from SU, R NI j p = B j p − X l s ∈ O ( n s ) W l s ∗ log (cid:12)(cid:12)(cid:12)(cid:12) I + 1 N j p B j p H j p Q ∗ j p H Hj p (cid:12)(cid:12)(cid:12)(cid:12) (6)where N j p is the noise power spectral density at PU j p . H j p ∈ C R jp × T jp denotes the channel matrix ofPU link j p . H l s j p ∈ C R jp × T ls denotes the channel matrix from the transmitting node of SU link l s to thetransmitting node of PU link j p . We assume based on the active support from the primary network, R min j p , B j p , H j p Q ∗ j p H Hj p and H l s j p are known at the transmitting nodes of link l s . E. Channel Capacity and Bandwidth Allocation
The capacity of a MIMO link l s is given by Φ l s ( W l s , Q sl ) , W l s log (cid:12)(cid:12)(cid:12) W l s N l s I + W ls B jp H j p l s Q ∗ j p H Hj p l s + H l s Q l s H Hl s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W l s N l s I + W ls B jp H j p l s Q ∗ j p H Hj p l s (cid:12)(cid:12)(cid:12) (7) Note that Eq. (4) is also suitable for the case where a MIMO SU link shares the full bandwidth with a MIMO PU link. where N l s is the noise power spectral density at SU l s . As seen in (7), the optimization of bandwidthallocation W l s and covariance matrix Q sl play an important role in improving the channel capacity. Noticethat Eq. (7) in this paper is different with the capacity function Eq. (2) of a MIMO link used in [16],where the background noise is assumed to be not related to the allocated bandwidth.Since the total outgoing bandwidth of node n s can not exceed its assigned bandwidth, we have X l s ∈O ( n s ) W l s B n s ∀ n s (8) B n s denotes the bandwidth assigned to node n s . F. Multi-commodity flow model
We use a multi-commodity flow model for the routing of data packets in the multihop wireless network.The source nodes send data packets to their intended destination nodes through multi-path and multi-hoprouting. We assume there are F sessions in the network. e f ≥ denotes the traffic demand of session f .Let x fl s > denote the transmission rate of session f over the link l s . Following the flow conservationlaw, we have X l s ∈O ( n s ) x fl s − X l s ∈I ( n s ) x fl s = a n s e f ∀ n s , ∀ f (9)where I ( n s ) denotes the sets of links that are incoming to node n s , a n s is defined as follow a n s = if n s = src ( f ) − if n s = dst ( f )0 otherwisewhere src ( f ) and dst ( f ) denote the source and destination node of session f , respectively.Obviously, the total traffic rate of all flows traversing a link cannot exceed the link’s capacity limitation.So we have F X f =1 x fl s Φ l s ( W l s , Q l s ) ∀ l s (10)Notice that (10) is a convex constraint due to the concavity of the channel capacity function (7) in ( W l s , Q l s ) . G. Problem Formulation
Our objective is to optimize the weighted sum of network utility and power consumption of SUcommunication sessions, with a minimum PU transmission rate constraint and SU power consumptionconstraints. We have the following formulation to jointly optimize the covariance matrix Q , [ Q , Q , · ·· , Q L s ] at the physical layer, bandwidth allocation W , [ W , W , · · · , W L s ] at the link layer, routing x , [ x , · · · , x L s , · · · , x F , · · · , x FL s ] at the network layer and rate control e , [ e , e , · · · , e f ] at the transportlayer: maximize F P f =1 U f ( e f ) − L s P l s =1 t l s Tr ( Q l s ) (11)subject to (2) , (3) , (8) , (9) and (10) where t l s , ∀ l s are weighting coefficients to specify the importance of power consumption. Remark 1 : We extend traditional MIMO CRN design towards a framework of utility-power trade-offwith power related objective and constraint. Different from the problem studied in [16], we address theoptimization of the weighted sum of network utility and power consumption by introducing the parameter t l s in (11), and consider non-full power consumption by introducing the parameter α n s in (2). We quantifythe utility-power trade-off with some surprising results through the simulation verification. Furthermore,the problem is more complex than that in [16] due to the non-convex constraint (3) and thus we need toseek a new approach to attack the problem.III. D UAL D ECOMPOSITION
In this section, we use the dual decomposition technique to decompose the problem (11) into twosubproblems. One is the network-transport layer subproblem, i.e., routing at the network layer and ratecontrol at the transport layer. The other is the physical-link layer subproblem, i.e., covariance matrix atthe physical layer and bandwidth allocation at the link layer. Both of them can be solved in a distributedfashion. If only (10) is dualized, the corresponding Lagrangian is L ( e , x , Q , W , u )= N s X n s =1 X l s ∈O ( n s ) { u l s Φ l s ( W l s , Q l s ) − t l s Tr ( Q l s ) } + F X f =1 ( U f ( e f ) − L X l s =1 u l s x fl s ) (12)where u , [ u , · · · , u l s , · · · , u L s ] , and u l s denotes the Lagrange multiplier (i.e., price) associated with theconstraint P Ff =1 x fl s Φ l s ( W l s , Q l s ) of link l s .The Lagrange dual function D ( u ) is given by D ( u ) , maximize L ( e , x , Q , W , u ) subject to (2) , (3) , (8) , (9) (13)The dual problem is minimize D ( u ) subject to u > (14)We notice that the Lagrange (12) is separable at each SU node or at each session. So, we decomposethe dual function (13) into two classes of subproblems, i.e., the physical-link layer subproblems at eachSU node and the network-transport layer subproblems at each session. Specifically, the network-transportlayer subproblem at each session is given by max e f , x f U f ( e f ) − L s X l s =1 u l s x fl s subject to X l s ∈O ( n s ) x fl s − X l s ∈I ( n s ) x fl s = a n s e f x fl s ≥ , e f ≥ (15)and the physical-link layer subproblem at each SU node is given by max Q ns , W ns X l s ∈O ( n s ) ( u l s Φ l s ( W l s , Q l s ) − t l s Tr ( Q l s )) subject to X l s ∈O ( n s ) W l s B n s X l s ∈O ( n s ) Tr ( Q l s ) α n s P n s R minj p R j p Q l s ≻ − , W l s ≥ ∀ l s ∈ O ( n s ) (16) NetworkNetwork source node f (cid:3) link s l (cid:3) f x , s l s u l ! " , s s n n Q W node s n , f f x s l u ! , s s l l W Q ! , s l s s u l n O , f f e x Computing Computing , s s n n Q W
Computing s l u Fig. 2. Local computation and information exchange
We denote x f , [ x fl s , ∀ l s ] , Q n s , [ Q l s , l s ∈ O ( n s )] , W n s , [ W l s , l s ∈ O ( n s )] .It is readily seen that the problem (15) is convex and can be globally solved at the source node of eachsession f . Although the problem (16) can be solved by each node n s , it is difficult to obtain the globalsolution due to the non-convexity of the constraint R minj p R j p . Hence in the next section we propose aniterative algorithm that converges monotonically.In the following, we formally solve the dual problem (14). Because the dual objective function is apiece-wise linear function which is non-differentiable, we can use the subgradient method to solve (14)[32]. We start with an initial u (0) . In the k -th iteration, after solving the problem (15) and the problem(16) for given u ( k ) , we update the dual variables u ( k +1) . The l s -th element u ( k +1) l s is updated as u ( k +1) l s = h u ( k ) l s − β ( k ) l s d ( k ) l s i + (17)where [ z ] + = max( z, , β ( k ) l s > is a positive scalar step size parameter. The subgradient d ( k ) l s can becalculated as d ( k ) l s = Φ l s ( W ∗ l s ( u ( k ) ) , Q ∗ l s ( u ( k ) )) − F X f =1 x f ∗ l s ( u ( k ) ) (18)where W ∗ l s and Q ∗ l s are a solution of the problem (16) at n s , and x f ∗ l s is a solution to (15) of session f for a given u ( k ) .It is worth pointing out that the dual variable u ( k ) l s can be interpreted as the “price” of link l s during the k -th iteration. This can help us to better understand the update of the dual variables u ( k ) l s . The subgradient d ( k ) l s indicates the usage of the link capacity during the k -th iteration. For link l s , when it is under-utilized,then d ( k ) l s > , and the price of link l s will reduce from (17). On the other hand, when link l s is over-utilized,then d kl s < , and the price of link l s will increase.We now give the following distributed algorithm, where each session source and each SU solve theirown problems with only local information. Algorithm 1: Dual decomposition algorithm
Initialization: u (0) , Q , W , x , s .Repeat until convergence: k ← k + 1 At each source node of session f , solve (cid:16) e ( k ) f , x ( k ) f (cid:17) in the problem (15).At each node n s , apply Algorithm 2 to solve (cid:16) W ( k ) l s , Q ( k ) l s (cid:17) , l s ∈ O ( n s ) in the problem (16) .At each node n s , update u ( k ) l s , l s ∈ O ( n s ) in (17). Remark 2 : Here, we discuss the distributed implementation of the proposed algorithm.For the network-transport layer subproblem, after receiving the dual variables, the source node src ( f ) of each session f locally solves the network-transport layer subproblem (15). Then the source nodehas updated the flow rate s f and routing information x f . Each intermediate node performs the routingaccording to the routing information x f in the source node.For the physical-link layer subproblem, each node n s locally solves the physical-link layer subproblem(16), and then updates W n s and Q n s .For the update of the dual variable u l s for link l s which is outgoing from node n s , from (17)-(18), wenotice that the computation of u l s only needs the local link capacity information Φ l s ( W l s , Q l s ) and thelocal traffic information P Ff =1 x fl s . Then link l s updates the dual variable. After that, node n s broadcaststhe dual information to its next hop neighbors. Meanwhile, it receives the dual information from otherneighbors and relays it to its next hop neighbors. Eventually, each source node will get all dual variablesof links that the session goes through.In Fig. 2, we give a graphic illustration of the local computation and the information exchange. IV. L
INEARIZATION - BASED A LTERNATING O PTIMIZATION
In this section, we focus on solving the problem (16) through linearization-based alternating optimiza-tion. First, it is seen that, fixing W n s , the problem (16) is a nonconvex problem with respect to Q n s dueto the primal user rate constraint (3). While fixing Q n s , the problem (16) is a convex problem with respectto W n s but is still not easy to handle. Here, we apply the method of Taylor series expansion twice tolinearize the primal user rate constraint at each step to get simple convex problem.Given ˜ Q l s , which satisfies the constraints of the problem (16), the rate function (5) can be linearizedas follows [31]: r l s j p = log (cid:12)(cid:12)(cid:12) W l s N j p I + W ls B jp H j p Q ∗ j p H Hj p + H l s j p Q l s H Hl s j p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W l s N j p I + H l s j p Q l s H Hl s j p (cid:12)(cid:12)(cid:12) ≃ log (cid:12)(cid:12)(cid:12)(cid:12) W l s N j p I + W l s B j p H j p Q ∗ j p H Hj p + H l s j p Q l s H Hl s j p (cid:12)(cid:12)(cid:12)(cid:12) − log (cid:12)(cid:12)(cid:12) W l s N j p I + H l s j p ˜Q l s H Hl s j p (cid:12)(cid:12)(cid:12) − Tr (cid:26)(cid:16) W l s N j p I + H l s j p ˜Q l s H Hl s j p (cid:17) − H l s j p Q l s H Hl s j p (cid:27) + Tr (cid:26)(cid:16) W l s N j p I + H l s j p ˜Q l s H Hl s j p (cid:17) − H l s j p ˜Q l s H Hl s j p (cid:27) , ˜ r l s j p Q (19)So the rate of PU link (4) can be approximated as R j p ≃ X l s ∈O ( n s ) W l s ˜ r l s j p Q + R NI j p , ˜ R j p Q (20)After fixing W n s to ˜W n s = [ ˜ W l s , l s ∈ O ( n s )] , using this linearized rate function ˜ R j p Q , the problem(16) is transformed into the problem (21), which is convex in Q n s . max Q ns X l s ∈O ( n s ) (cid:16) u l s Φ l s (cid:16) ˜ W l s , Q l s (cid:17) − t l s Tr ( Q l s ) (cid:17) subject to X l s ∈O ( n s ) Tr( Q l s ) ≤ α n s P n s R minj p ≤ ˜ R j p Q Q l s ≻ ∀ l s ∈ O ( n s ) (21)For given ˜W n s , it can be readily verified that Φ l s (cid:16) ˜ W l s , Q l s (cid:17) is concave with respect to Q l s . All constraintsof the problem (21) are linear with respect to Q n s . Thus, the problem (21) is a convex optimization problemwith respect to Q n s . Given ˜ W l s , which satisfies the constraints of the problem (16), W l s r l s j p can be linearized as follows: W l s r l s j p = W l s log (cid:12)(cid:12)(cid:12)(cid:12) N jp W ls I + WlsBjp H jp Q ∗ jp H Hjp + H lsjp ˜Q ls H Hlsjp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N jp W ls I + H lsjp ˜Q ls H Hlsjp (cid:12)(cid:12)(cid:12) ≃ W l s log (cid:12)(cid:12)(cid:12) N j p ˜ W l s I + ˜ W ls B jp H j p Q ∗ j p H Hj p + H l s j p ˜Q l s H Hl s j p (cid:12)(cid:12)(cid:12) +Tr { (cid:16) N j p ˜ W l s I + ˜ W ls B jp H j p Q ∗ j p H Hj p + H l s j p ˜Q l s H Hl s j p (cid:17) − ∗ (cid:18) N j p + H jpls Q ∗ jp H Hjpls B jp (cid:19) } (cid:16) W l s − ˜ W l s (cid:17) ˜ W l s − W l s log (cid:12)(cid:12)(cid:12) N j p W l s I + H l s j p ˜Q l s H Hl s j p (cid:12)(cid:12)(cid:12) (22)So the rate of PU link (4) can be approximated as R j p ≃ X l s ∈O ( n s ) ˜ r l s j p W + R NI j p , ˜ R j p W (23)After fixing Q n s to ˜Q l s , using this linearized rate function ˜ R j p W , the problem (16) is transformed intothe problem (24), which is convex in W n s . max W ns X l s ∈O ( n s ) (cid:16) u l s Φ l s (cid:16) W l s , ˜Q l s (cid:17) − t l s Tr (cid:16) ˜Q l s (cid:17)(cid:17) subject to X l s ∈O ( n s ) W l s ≤ B n s R minj p ≤ ˜ R j p W W l s ≥ ∀ l s ∈ O ( n s ) (24)For given ˜Q n s , it can be readily verified that Φ l s (cid:16) W l s , ˜Q l s (cid:17) is concave with respect to W l s . Allconstraints of the problem (24) are linear with respect to W n s . Thus, the problem (24) is a convexoptimization problem with respect to W n s .Let Q ∗ n s be its solution to the problem (21) and f ( W n s , Q n s ) denotes the objective value of the problem(21) or the problem (24). We have f ( W n s , Q ∗ n s ) ≥ f ( W n s , ˜ Q n s ) since ˜ Q n s is a feasible solution to theproblem (21). We then fix Q n s to Q ∗ n s in the problem (24) and solve for W n s in (24). Let W ∗ n s be itssolution to the problem (24). Then we have f ( W ∗ n s , Q ∗ n s ) ≥ f ( W n s , Q ∗ n s ) . By the above analysis, wepropose Algorithm 2 for solving problem (16). As Algorithm 2 iterates, the objective value of the problem(16) is nondecreasing. Algorithm 2: Linearization-based Alternating Optimization Algorithm Initialization: k = 0 , W (0) n s , Q (0) n s .Repeat until convergence: k ← k + 1 ;Linearize R j p at Q ( k ) n s s to get ˜ R j p Q ;Fix W n s = W ( k ) n s , solve (21) to get Q ( k +1) n s ;Linearize R j p at W ( k ) n s s to get ˜ R j p W ;Fix Q n s = Q ( k +1) n s , solve (24) to get W ( k +1) n s .V. R ECOVERY OF P RIMAL S OLUTION
Because of the non-convexity of the primal problem (11), the duality gap usually exists and the solutionto the dual problem D ( u ) is not always feasible for the primal problem (11). Hence, we still need togenerate a feasible primal solution with good performance. Our key idea is to fix W in the primalproblem. Although the resultant problem is still nonconvex, we can iteratively linearize the primal rateconstraint to get a sequence of convex problems that can be distributively solved by the dual-decompositionmethod. Such an iterative algorithm is called constrained concave-convex procedure (CCCP) [33] whichcan monotonically converge to a stationary point of the primal problem with fixed bandwidth allocation.To save the communication overhead and considering that the dual decomposition algorithm can quicklyconverge to the neighborhood of the optimal solution (generally oscillating around the optimal solution),we run a few iterations of the dual decomposition algorithm and perform a heuristic step, i.e., average W l s over iterations, ¯W ∗ = 1 n N X k = N − n +1 W ( k ) (25)where N represents the total number of iterations and n represents the last n iterations. Intuitively, weconsider that ¯W ∗ is the neighborhood of the optimal solution W . Hence, we fix W = ¯ W ∗ and linearizethe primal user rate function R j p at ˜Q to get a convex problem. Then we can use the dual-decompositionmethod to solve the resultant problem in a distributed fashion. Finally, we get a recovery solution. Thewhole algorithm is summarized in Algorithm 3. Note that, in Algorithm 3, ˜Q is initialized from the resultobtained by Algorithm 1 which is feasible to the primal problem. Hence, the recovery solution is alsofeasible for the primal problem [33]. Algorithm 3: Primal solution recovery algorithm
Implement Algorithm 1.Get ¯W ∗ in (25).Choose u (0) and initialize ˜ Q from the result obtained from Algorithm 1.Repeat until the required accuracy:Let k = 0 .Repeat until the required accuracy: k ← k + 1 At each source node of session f , compute (cid:16) s ( k ) f , x ( k ) f (cid:17) by solving the problem (15).At each node n s , compute Q ( k ) l s , l s ∈ O ( n s ) bysolving the problem (16) with ¯W ∗ .At each node n s , update u ( k ) l s , l s ∈ O ( n s ) in (17).Update ˜Q = Q ( k ) , linearize PU rate in (23), and update u (0) = u ( k ) . VI. SIMULATION RESULTSIn this section, we investigate the performance of the proposed distributed algorithm through simulations.The proportional fairness utility function is adopted, i.e., U f ( f ) = ln( e f ) for each session f . We randomlygenerate a SU network topology as shown in the Fig.3. In this network topology, there are 15 nodes, 56links , and three sessions: node 2 to node 6, node 8 to node 13, and node 15 to node 9. Each node inthe network is equipped with two transmitting antennas and two receiving antennas, and the allocatedbandwidth is 20. The maximum transmit power of each node is 100. The channel gain matrix from thetransmitting node of link l s to the receiving node of link l s is modeled as H l s = (200 /d l s ) . L l s ¯ H l s (26) We assume that there exists an SU link if the distance between two SU nodes is not larger than 300m.
100 200 300 400 500 600 700 800 90020030040050060070080090010001100 N1N2N3 N4N5 N6N7N8 N9N10N11N12 N13N14N15
Fig. 3. Topology of SU network with 15 nodes. where d l s is the distance from the transmitting node of link l s to the receiving node of link l s ;
10 log ( L l s ) is a real Gaussian random variable with zero mean and a standard deviation of 8 accounting for the largescale log-normal shadowing; finally, ¯ H l s is an 2-by-2 matrix containing random values drawn from thestandard normal distribution. Also, the channel gain matrix of PU link and the interference channel gainmatrix from the transmitting node of SU link l s to the receiving node of PU link j p are nearly similaras (26) with different distance. Specifically, we set the distance of PU link j p as d j p = 200 m , and thedistance from the transmitting node of SU link l s to the receiving node of PU link j p as d s l p j = 300 m .We assume that SU n s shares the full bandwidth with PU link j p . The transmitting covariance matrix Q ∗ j p of PU link j p is determined by the optimal value under the condition of interference from the SUlinks. We denote ρ j p = R min j p /R j p , where R j p denotes the maximal rate of PU j p achieved without anyinterferences from SUs. We set the default value of ρ j p for all PUs as 0.5.Through implementing the proposed distributed algorithm, we get the routing and flow rates of sessions1, 2 and 3, which are shown in Figs. 4,5 and 6, respectively. It is seen that flow routings for sessions 1, 2and 3 are multihop and multi-path. Moreover, it can be easily verified that the flow rates satisfy the flowconservation.Fig. 7 shows the convergence behavior of the linearization-based alternating optimization algorithmfor the physical-link layer subproblem with fixed u , while Fig. 8 shows the convergence behavior of therecovery algorithm with fixed bandwidth allocation. We can see that, the two algorithms can converge
100 200 300 400 500 600 700 800 90020030040050060070080090010001100 N1N2N3 N4N5 N6N7N8 N9N10N11N12 N13N14N15 N1N2N3 N4N5 N6N7N8 N9N10N11N12 N13N14N15 32.07239.52510.797 56.69248.77252.0245 7.924124.14825.781541.59734.46084.781 5.85342.0245 21.90993.757 61.7079 13.985813.5534 13.985819.40682.0245
Fig. 4. Routing and flow rates of session 1.
100 200 300 400 500 600 700 800 90020030040050060070080090010001100 N1N2N3 N4N5 N6N7N8 N9N10N11N12 N13N14N15 8.12052.33782.3378 5.6263 31.55189.660932.33119.66095.44862.517941.99210.6384 41.99231.551810.440210.63842.67190.00161210.153067.96412.52050.15381
Fig. 5. Routing and flow rates of session 2.
100 200 300 400 500 600 700 800 90020030040050060070080090010001100 N1N2N3 N4N5 N6N7N8 N9N10N11N12 N13N14N15 23.936717.18452.8272.827 40.8186 68.072938.12345.5246 22.70912.99790.001560739.057 22.709144.136143.645642.05655.0799
Fig. 6. Routing and flow rates of session 3 . O b j e c t v a l ue P h y − li n k l a y e r s ubp r ob l e m Fig. 7. Convergence behavior of linearization-based alternative algorithm for the physical-link layer subproblem. N e t w o r k U t ili t y Algorithm 3Centralized Algorithm
Fig. 8. Convergence of the outer loop in Algorithm 3 with n = 20 . N e t w o r k U t ili t y n=20n=50n=70Centralized AlgorithmEqual band width Allocation Fig. 9. Recovery of primal solution with n = 20 , , . (Compared to the centralized algorithm and the equal bandwidth allocationalgorithm) α N e t w o r k U t ili t y Utility(t=0)Utility(t=0.001)0.5 0.6 0.7 0.8 0.9 1010020030040050060070080090010001100120013001400150016001700180019002000 α P o w e r C on s u m p t i on Power Consumption(t=0)Power Consumption(t=0.001)
Fig. 10. Impact of α and t on network utility and power consumption. quickly and keep the objective function nondecreasing at each iteration. Particularly, the recovery algorithmcan achieve the most of performance in the first iteration. Hence, to save the communication overhead,we only need to implement once the dual decomposition method in the recovery step.To examine the performance of the bandwidth-allocation scheme, we run the dual-decompositionalgorithm 300 iterations in total. We set n = 20 , , and Let N = 200 + n + 5 M , M = 0 , , , . . . , .For each combination ( n, N ) , we carry out the recovery step and calculate the network utility. Fig. 9shows the simulation results. We can see that, for different SU n s , the proposed algorithm has verysimilar performance. Moreover, the algorithm is also robust to the choice of N , i.e., the total number ofiterations in the first step. Hence, in the other simulations, we set N = 250 and n = 20 .From Fig. 9, we can also see that the proposed bandwidth allocation scheme is better than the equalbandwidth allocation scheme. Moreover, the performance of the proposed distributed algorithm is veryclose to the performance of the centralized algorithm using Matlab solver fmincon . It is worth mentioningthat, although fmincon can only offer locally optimal solution, we find from simulations that, it cangenerate almost the same utility from different initialization. Hence, our distributed algorithm can find afeasible solution with good performance.We investigate the impact of two parameters α n s and t l s on the achieved network utility and the powerconsumption. The simulation result is shown Fig. 10, where α n s = α, ∀ n s is chosen from 0.5 to 1, and t l s = t, ∀ l s is chosen as 0 and 0.001, respectively. It is seen that there exist a good tradeoff between the network utility and the power consumption. Through choosing suitable α n s and t l s , we can significantlysave the power consumption, while achieving the network utility very close to that achieved at the fullpower consumption. For example, shown in Fig. 10, when α = 0 . and t = 0 . , we can achieve up to99% of the network utility achieved at full power consumption, while only expending about of fullpower. Thus, our formulation provides an efficient way to design a green cross-layer optimization schemefor multihop MIMO CRN. VII. C ONCLUSIONS
Both MIMO and CR are the enabling technologies for the next-generation wireless communications. Theperformance of multihop MIMO CRN is tightly coupled with mechanisms at the physical, link, network,and transport layers. In this paper, we have proposed a new formulation for green multihop MIMOCRN design. Our formulation balances the network utility and weighted total power consumption of SUcommunication sessions, with a minimum PU transmission rate constraint and SU power constraints.We have developed the distributed algorithms to tackle the highly nonconvex cross-layer optimizationproblem. Simulation results show that the proposed algorithm can provide a feasible solution with goodperformance, and the proposed formulation is power-efficient while maintaining the high network utility.R
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