Cross-Layer Optimization of MIMO-Based Mesh Networks Under Orthogonal Channels

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1 Cross-Layer Optimization of MIMO-Based Mesh Networks Under Orthogona Channes Jia Liu, Tae Yoon Park, Y. Thomas Hou,YiShi, and Hanif D. Sherai Bradey Department of Eectrica and Computer Engineering Grado Department of Industria and Systems Engineering Virginia Poytechnic Institute and State University, Backsburg, VA 406 Abstract MIMO-based systems have great potentia to improve network capacity for wireess mesh networks WMNs. Due to unique physica ayer characteristics associated with MIMO systems, network performance is tighty couped with mechanisms at physica ayer and ink ayer. So far, research on MIMO-based WMNs is sti in its infancy and itte resuts are avaiabe in this important area. In this paper, we consider the probem of jointy optimizing power and bandwidth aocation at each node and mutihop/mutipath routing in a MIMO-based WMN where inks operate in orthogona channes. To sove this probem, we deveop a mathematica soution procedure, which combines Lagrangian dua decomposition, gradient projection, and cutting-pane methods. We provide theoretica insights in deriving gradient projection and cutting pane methods. We aso use simuations to verify the efficacy of our agorithm. I. INTRODUCTION Since Winters s [], Teatar s [] and Foschini s [3] pioneering works predicting the potentia of high spectra efficiency provided by mutipe antenna systems, the ast decade has witnessed the soar of research activity on Mutipe-Input Mutipe-Output MIMO technoogies. Without costs of extra spectrum, MIMO technoogy, which expoits the rich scattering characteristic of wireess channes, is abe to increase channe capacities substantiay than conventiona communication systems. However, compared to the research on the capacity of singe-user MIMO, for which many resuts are avaiabe see [4] and [5] and references therein, the capacity issue of mutiuser MIMO systems is much ess studied and many fundamenta probems remain unsoved [5]. With the emergence of wireess mesh networks WMNs, which are mutiuser and mutihop in nature, the need to extend the MIMO communication concept from singe-user systems to mutiuser systems has become increasingy compeing. In a WMN, however, appying MIMO technique becomes far more compicated. Power contro and power aocation at each ink as we as mutihop/mutipath routing across the network a interact with one and another, and cross-ayer optimization is not ony desirabe, but aso necessary. In this paper, we study the probem of cross-ayer optimization on mutihop/muti-path routing, power contro, power aocation, and bandwidth aocation CRPBA for MIMO-based mesh network where inks operate in orthogona channes. Specificay, we consider how to support a set of user communication sessions by jointy optimizing power contro, power aocation, bandwidth aocation, and fow routing such that some network utiity e.g., proportiona fairness is maximized. This probem, to the best of the authors knowedge, has not been studied thus far. A. Main Contribution The main contribution of this paper are the foowing: We deveoped a mathematica soution procedure to sove CRPBA by combining Lagrangian decomposition, gradient projection, and cutting-pane agorithms. For the chaenging ink ayer subprobem, we deveop a rigorous gradient projection method as opposed to the heuristic one in [6]. 3 Our proposed cutting-pane method can not ony sove the Lagrangian dua, but aso easiy recover the optima prima feasibe soutions, thus circumventing a major difficuty of the popuar subgradient-based approaches for soving Lagrangian dua probems. B. Paper Organization The remainder of this paper is organized as foows. In Section II, we review reated work. In Section III, we discuss the network mode and probem formuation. Section IV introduces the key components in our soution procedure, incuding gradient projection, cutting-pane agorithm, and the recovering of optima prima feasibe soutions. Numerica resuts are presented in Section VI. Section VII concudes this paper. II. RELATED WORK Research on appying MIMO to WMN is sti in its infancy and resuts remain extremey imited. In this section, we provide a synopsis of reated work on the MIMO research evoution from singe-hop ad hoc networks to mesh networks. For singe-hop MIMO ad hoc networks, in [6], Ye and Bum introduced a gradient projection method to find a suboptima soution for the nonconvex optimization probem for singehop ad hoc networks. However, the way they handed gradient projection is based on heuristic: in soving the constrained Lagrangian dua probem in projection, they simpy set the first derivative to zero to get the soution. Such a method does not work for genera constrained optimization probem. For mutihop WMNs, Hu and Zhang [7] examined the probem of joint medium access contro and routing, and in particuar,

2 considered the optima hop distance to minimize the endto-end deay. However, power contro and power aocation were not considered in this work. In [8], the authors designed different routing protocos for WMN to expore the tradeoff between mutipexing gain and diversity gain [9]. However, this work is argey based on simuation observations. III. NETWORK MODEL We first introduce notation for matrices, vectors, and compex scaars in this paper. We use bodface to denote matrices and vectors. For a matrix A, A denotes the conjugate transpose. Tr{A} denotes the trace of A. Diag [ ] A... A n represents the bock diagona matrix with matrices A,...,A n on its main diagona. We et I denote the identity matrix with dimension determined from context. A 0 represents that A is Hermitian and positive semidefinite PSD. and 0 denote vectors whose eements are a ones and zeros, respectivey, and their dimensions are determined from context. v m represents the m th entry of vector v. For a rea vector v and a rea matrix A, v 0 and A 0 mean that a entries in v and A are nonnegative, respectivey. We et e i be the unit coumn vector where the i th entry is and a other entries are 0. The dimension of e i is determined from context as we. The operator, represents vector or matrix inner product operation. A. Link Capacity Mode In this paper, it is assumed that the system has perfect channe knowedge, that is, the transmitters have perfect channe state information CSI. Let the matrix H C nr nt represent the wireess channe gain matrix from the transmitting node to the receiving node of ink, where n t and n r are the numbers of transmitting and receiving antenna eements at each node, respectivey. Athough wireess channes in reaity are timevarying, we consider a constant channe mode in this paper, i.e., H s coherence time is arger than the transmission period we consider. This simpification is of much interest for the insight it provides and its appication in finding the ergodic capacity for bock-wise fading channes [5]. The received compex base-band signa vector for MIMO ink with n t transmitting antennas and n r receiving antennas in a Gaussian channe is given by r = ρ H t + n, where r and t represent the received and transmitted signa vectors, n is the normaized additive white Gaussian noise vector, ρ j captures path-oss effect. By adopting the pathoss mode with path-oss exponent being equa to α, ρ can be computed as [0] ρ = GtGrλ 4π /N 0 WD α, where D denotes the ength of ink, N 0 represents the power spectra density of white Gaussian noise, and W denotes the communication bandwidth, G t and G r are transmit and receive antenna gains, respectivey, which are assumed to be in this paper, λ is the waveength of the transmitted signa. Let matrix Q represent the covariance matrix of a zeromean Gaussian input symbo vector t at ink, i.e., Q = { E }. The definition of Q impies that it is Hermitian t t and Q 0. Physicay, Q represents the power aocation in different antenna eements in ink s transmitter and correations between each of these eements. In this paper, we use the compex matrix Q [ ] Q Q... Q L C n t n t L to denote the coection of a input covariance matrices. The ink capacity of a MIMO ink in an AWGN channe can be written as Φ W, Q W og det I + ρ H Q H, where W represents the communication bandwidth of ink. It can be readiy verified that Φ Q is a monotone increasing concave function in W and Q. B. Data Routing and Network Fows One of the most chaenging aspects of a WMN is that its connectivity is highy dependent on the transmission power at each node. As a resut, the network topoogy is not fixed. Our approach to hande this eusive network topoogy is to denote the network by a directed graph consisting of N nodes and L possibe MIMO inks. By saying that a ink is possibe we mean the distance between its transmitting node and receiving node is ess than or equa to the transmission range by using the maximum transmission power. We assume that the graph is aways connected. The network topoogy can be represented by a node-arc incidence matrix NAIM [] A R N L, whose entry a n associating with node n and ink is defined as if n is the transmitting node of ink a n = if n is the receiving node of ink 0 otherwise. We define O n and I n as the sets of inks that are outgoing from and incoming to node n, respectivey. We use a muticommodity fow mode for the routing of data packets across the WMN. Suppose that there are F sessions in tota in the network, representing F different commodities. The source and destination nodes of session f, f F, are denoted as srcf and dstf, respectivey. For each session, we define a source-destination vector s f R N, whose entries, other than at the positions of srcf and dstf, are a zeros. In addition, from the fow conservation aw, we must have s f srcf = s f dstf. Without oss of generaity, we ets f srcf 0 and simpy denote it by a scaar s f. Therefore, we can further write the source-destination vector of session f as s f = s f [ ] T, 3 where the dots represent zeros, and and are in the positions of srcf and dstf, respectivey. Using the notation = x,y to represent the component-wise equaity of a vector except at the x th and the y th entries, we have s f = srcf,dstf 0. Using matrix S [ s s... s F ] R N F to Note that for the source-destination vector of a fow f, does not necessariy appear before as in 3, which is ony for iustrative purpose.

3 denote the coection of a source-sink vectors s f, we further have On Se f = srcf,dstf 0, f F, 4, Se f =0, f F, 5 Se f srcf = s f, f F, 6 where e f is the f th unit coumn vector. On each ink, weetx f 0 be the amount of fow of session f. We define x f R L as the fow vector for session f. At each node n, components of the fow vector and source-destination vector for the same session satisfy the fow conservation aw: x f x f =s f n, n N, f F. In With NAIM, the fow conservation aw across the whoe network can be compacty written as Ax f = s f, f F. We use matrix X [ x x... x ] F R L F to denote the coection of fow vectors x f. With X and S, the fow conservation aw can be further compacty written as AX = S. 7 C. Probem Formuation The goa of this paper is to design an agorithm that performs the cross-ayer optimization on mutihop/mutipath routing, power contro, power aocation, and bandwidth aocation CRPBA for a MIMO-based WMN. We consider an FDMA MIMO-based WMN, where each node has been assigned non-overapping possiby reused frequency bands for its incoming and outgoing inks so that each node can simutaneousy transmit and receive, and cause no interference to other nodes. How to perform channe assignments is a huge research topic by itsef, and there are a vast amount of iterature that discuss channe assignment probems. Thus in this paper, we focus on how to jointy optimize routing in the network ayer and power contro/aocation as we as bandwidth aocation in the ink ayer. We adopt the we-known proportiona fairness utiity function, i.e., ns f for fow f []. We wish to perform the crossayer optimization such that the sum of a utiities of fows is maximized. Since the network fows in a ink cannot exceed the ink s capacity imit, we have F f= xf Φ W, Q, for L. Using matrix-vector notations, it can be further compacty written as, X T e Φ W, Q, L. 8 In the ink ayer, since the tota transmit power of each node is subject to a maximum power constraint, we have On Tr{Q } P max, n n N, where P max n represents the maximum transmit power of node n. Aso, the sum of bandwidths of a the outgoing inks for a node n cannot exceed the assigned bandwidth for node n, i.e., On W B n, n N, where B n is the assigned transmission [ band for node ] n. We use the matrix W = T W W... W L R L to denote the coection of the bands for inks from to L. Couping the MIMO ink capacity mode in Section III-A and the network fow mode in Section III-B, we have the probem formuation for CRPBA as in 9. CRPBA :Maximize F f= ns f subject to AX = S X 0, X T e Φ W, Q Se f = srcf,dstf 0 f, Se f =0 f 9 Se f srcf = s f f On Tr{Q } P max n n Q 0 On W B n n Variabes: S, X, Q, W IV. SOLUTION PROCEDURE It can be observed that the CRPBA possesses a specia structure: the network ayer variabes and the ink ayer variabes are couped through the ink capacity constraints, X T e Φ W, Q. Thus, we can expoit this specia structure using Lagrangian dua decomposition to sove CRPBA efficienty. In [3], the authors used a simiar decomposition technique to sove simutaneous routing and resource aocation probems. However, their routing setting was very different to our work and their ink ayer was not MIMObased. Due to the spatia dimension resuted from MIMO, the ink ayer subprobem in this paper is competey different and substantiay more chaenging. Generay, given a noninear programming probem, severa different Lagrangian dua probems can be constructed depending on which constraints are associated with Lagrangian dua variabes [4]. For CRPBA, we associate Lagrangian mutipiers u i to the ink capacity couping constraints, X T e Φ W, Q. Hence, the Lagrangian can be written as [4] Θu = sup {LS, X, Q, W, u S, X, Q, W Γ}, S,X,Q,W where LS, X, Q, W, u = n s f + u Φ W, Q f, X T e 0 and Γ is defined as AX = S X 0 Se f = srcf,dstf 0 f, Se Γ S, X, Q, W f =0 f Se f srcf = s f f On Tr{Q } P max n n Q 0 On W B n n The Lagrangian dua probem of CRPBA can thus be written as [4]: : Minimize Θu subject to u 0.

4 It is easy to recognize that, for any given Lagrangian mutipier u, the Lagrangian in 0 can be separated into two terms: Θu =Θ net u+θ ink u, where Θ net and Θ ink are two subprobems respectivey corresponding to network ayer and ink ayer: net :Θ net u Maximize f n s f u, X T e subject to AX = S X 0 Se f = srcf,dstf 0 f, Se f =0 f Se f srcf = s f f Variabes: S, X ink :Θ ink u Maximize u Φ W, Q subject to On Tr{Q } P max n n On W B n n Q 0 Variabes: Q, W The CRPBA Lagrangian dua probem can be thus transformed into the foowing master dua probem: M : Minimize Θ net u+θ ink u subject to u 0 Now, the task of soving the decomposed Lagrangian dua probem bois down to how to evauate the subprobems net and ink, and how to hande the master probem. Note that in the network ayer subprobem net,the objective function is concave and a constraints are affine. Therefore, net is readiy sovabe by using many poynomia time convex programming methods. However, soving ink is not trivia because the objective function and constraints invove many compex matrices variabes, even though it can be shown that ink is a convex probem. In the foowing subsections, we wi discuss each techniques we use to sove the ink ayer subprobem and the master probem in detai. A. Modified Gradient Projection Method MGP In this paper, we propose a modified gradient projection MGP method to sove the ink subprobem. Gradient projection, originay proposed by Rosen [5], is a cassica noninear programming method aiming at soving constrained optimization probems. But its forma convergence proof has not been estabished unti very recenty [4]. The framework of MGP is shown in Agorithm. Due to the compexity of the objective function, we cannot afford the uxury of performing an exact ine search which requires the expense of excessive objective function evauations. Therefore, we adopt the Armijo rue inexact ine search method [4], which sti enjoys provabe convergence. The basic idea of Armijo rue is that at each step of the ine search, we sacrifice accuracy for efficiency as ong as we have sufficient improvement. According to Armijo rue, we choose s k =and α k = β m k Agorithm Modified Gradient Projection Method Initiaization: Choose the initia conditions W 0 =[W 0,W 0,...,W 0 L ]T, Q 0 =[Q 0, Q0,...,Q0 L ]T.Letk =0. Main Loop:. Cacuate the gradients G k W = W Θ ink u, W k, Q k and = Q Θ ink u, W k, Q k,for =,,...,L. G k Q. Choose an appropriate step size s k.letw k Q k = Q k + s k G k = W k + s k G k W, Q,for =,,...,L. k k 3. Let [ W n, Q n ] T be the projection of [W n k Ω + n, whereω + n {W, Q On,W 0, Q 0, On W B n, On Tr{Q } P max}. n 4. Choose appropriate step size α k.letw k+ W k, Q k+ stop; ese go to step. = Q k + α k, Q k n ] T onto = W k k +α k W Q k, =,,...,L. <ɛ,for =,,...,L,then Q k 5. k = k +. If the maximum absoute vaue of the eements in Q k Q k <ɛand W k W k the same as in [6], where m k is the first non-negative that satisfies Θ ink Q k+ Θ ink Q k σβ m k L Tr = [ Q Θ ink Q k Qk ] Q k, where 0 <β<and 0 <σ<are fixed scaars. It is evident that the gradient G W W Θ ink = u og deti + ρ H Q H.Byusingtheformua X n deta + BXC = [ CA + BXC B ] T [6], [6], where matrices A C p p, B C p m, X C m n, C C n p, and A + BXC is invertibe, we are abe to derive the gradient G Q Q Θ ink as foows [7]: G Q = W u ρ H I + ρ H Q H H. n Noting that G Q are Hermitian, we have that Q k is Hermitian as we. Then, for a node n having O n outgoing inks, the projection probem becomes how to simutaneousy project the O n W -scaars and O n Q-covariance matrices onto Ω + n {W, Q On,W 0, Q 0, On W B n, On Tr{Q } P max}. n We construct a bock diagona matrix D n as foows: [ ] Wn 0 D n = C On nt+ On nt+ 0 Q n where W n Diag [ W : On ] C On On, and Q n Diag [ Q : On ] C On nt On nt. Moreover, we introduce two more matrices E n and E n as foows: [ ] E n I On 0 = C On nt+ On nt+, 0 0 [ ] E n 0 0 = C On nt+ On nt+. 0 I On nt It is easy to recognize that if D n Ω + n, we have TrE n D n = On W B n, TrE n D n =

5 On Tr Q P n max, and D n 0. In our projection, given a bock diagona matrix D n,wewishtofindamatrix D n Ω + n such that Dn minimizes D n D n F, where F denotes Frobenius norm. For more convenient agebraic manipuations, we instead study the foowing equivaent optimization probem: Minimize D n D n F subject to TrE n D n B n TrE n D n P max n D n 0 Notice that the probem is a convex minimization probem and we can sove this minimization probem by soving its Lagrangian dua. Associating Hermitian matrix X to the constraint Dn 0, ν to the constraint TrE n D n B n, and µ to the constraint TrE n D n P max, n we can write the Lagrangian as gx,ν,µ = min / D n D n F TrX Dn + ν D n ] B n + µ D n ] P max n Tr[E n D n { Tr[E n }.3 Since D n becomes unconstrained after removing its positive semidefinite constraint correspondingy, adding a penaty term to the Lagrangian, we can compute the minimizer of 3 by simpy setting the derivative of 3 to zero. Thus, we have D n = D n + X νe n µe n. Substituting D n back into 3, and after some agebraic simpifications, we can rewrite the Lagrangian dua probem as Maximize D n νe n µe n + X F νb n µp max n + D n subject to X 0,ν 0,µ 0. 4 Eq. 4 beongs the cass of so-caed matrix nearness probems, which are not easy to sove in genera see [8], [9] and references therein. However, based on the specia structure in E n and E n, we are abe to design a poynomia time agorithm to sove 4 Due to space imitation, we ony give the pseudo-codes in Agorithm and Agorithm 3 and refer readers to [7] for more detais. Agorithm Projection onto Ω + n. Construct a bock diagona matrix D n. Perform eigenvaue decomposition D n = U nλ nu n, separate the eigenvaues in two groups corresponding to W n and Q n, and sort them in non-increasing order within each group, respectivey.. For each group of eigenvaues, ca Agorithm 3 to find the optima dua variabe ν and µ. 3 Compute D n = U nλ n ν E n µ E n +U. V. CUTTING-PLANE METHOD FOR SOLVING Compared to the popuar subgradient-based approaches for soving Lagrangian dua probems, the attractive feature of the cutting-pane method is its speed of convergence and its simpicity in recovering optima prima feasibe soutions. As Agorithm 3 Search the Optima Dua Variabe Initiation: Introduce λ 0 = and λ K =. LetÎ =0. Let endpoint objective vaue ψî λ 0 =0, φ = ψî λ 0,andµ = λ 0. Main Loop:. If Î>K, return µ ; ese et µ Î =Î j= λ j P /Î.. If µ [λ Î Î+,λ Î ] R +, then et µ = µ and return Î µ. 3. Compute ψîλî+.ifψîλî+ <φ, then return µ ; ese et µ = λî+, φ = ψîλî+, Î = Î +and continue. opposed to the cumbersomeness of subgradient method, in cutting-pane method, prima optima feasibe soutions can be exacty computed by averaging a the prima soutions may or may not be prima feasibe using the dua variabes as weights [7]. We briefy introduce the basic idea of cutting-pane method as foows. Letting z = Θu, the inequaity z f n s f + u Φ W, Q, X T e must hod for a S, X, Q, W Γ. Thus, the dua probem is equivaent to Minimize z subject to z f n s f + u Φ W, Q, X T e 5 u 0, where S, X, Q, W Γ. Athough 5 is a inear program with infinite constraints not known expicity, we can consider the foowing approximating probem: Minimize z subject to z f n u u 0, s j f + Φ W j, Q j, X jt e 6 where the points S j, X j, Q j, W j Γ, j =,...,k. The probem in 6 is a inear program with a finite number of constraints and can be soved efficienty. Let z k, u k be an optima soution to the approximating probem, which we refer to as the master program. Ifthe soution is feasibe to 5, then it is an optima soution to the Lagrangian dua probem. To check the feasibiity, we consider the foowing subprobem: Maximize f n s f + uk Φ W, Q, X T e subject to S, X, Q, W Γ 7 Suppose that S k, X k, Q k, W k is an optima soution to the subprobem 7 and Θ u k is the corresponding optima objective vaue. If z k Θ u k, then u k is an optima soution to the Lagrangian dua probem. Otherwise, for u = u k, the inequaity constraint in 5 is not satisfied for S j, X j, Q j, W j. Thus, we can add the constraint z f n s k f + u Φ W k, Q k, X kt e 8

6 to 6, and re-sove the master inear program. Obviousy, z k, u k vioates 8 and wi be cut off by 8. The cutting pane agorithm is summarized in Agorithm 4. Agorithm 4 Cutting Pane Agorithm for Soving Initiaization: Find a point S 0, X 0, Q 0, W 0 Γ. Letk =. Main Loop:. Sove the master program in 6. Let z k, u k be an optima soution.. Sove the subprobem in 7. Let S k, X k, Q k, W k be an optima point, and et Θ u k be the corresponding optima objective vaue. 3. If z k Θu k, then stop with u k as the optima dua soution. Otherwise, add the constraint 8 to the master program, repace k by k +, and go to step. VI. NUMERICAL RESULTS We present some numerica resuts through simuations to provide further insights on soving CRPBA. We use a 5- node network exampe, as shown in Fig. a, to show the convergence process of the cutting-pane agorithm for soving. Each node in the network is equipped with two antennas and assigned a unit transmit bandwidth. In this exampe, there are three fows transmitting across the network: N4 to N, N6 to N0, and N5 to N4, respectivey. The convergence process is iustrated in Fig. b. It is seen that the cutting-pane method ony takes 7 iterations to converge N0 N3 N8 N4 N7 N3 N5 N N4 N N N N9 a Network topoogy. N5 N6 Network Utiity ogb/s/hz Lagrangian Dua UB Lagrangian Dua LB Number of Iterations b Convergence process. Fig.. A simpe exampe iustrating the need of scheduing to improve performance. VII. CONCLUSION In this paper, we investigated the probem of cross-ayer optimization of routing, power contro, power aocation, and bandwidth aocation for MIMO-based mesh networks. We deveoped a mathematica soution procedure, which combines Lagrangian decomposition, gradient projection, and cutting-pane methods. We provided the theoretica insights of our proposed agorithms and conducted simuations to verify their efficacy. Our resuts show that the nice decouped structure and the high efficiency of our proposed agorithm make it an attractive method for optimizing the performance of MIMO-based mesh networks. ACKNOWLEDGEMENT The work of Y.T. Hou, J. Liu, and Y. Shi has been supported in part by NSF Grant CNS The work of H.D. Sherai has been supported in part by NSF Grant REFERENCES [] J. Winters, On the capacity of radio communication systems with diversity in a Rayeigh fading environment, IEEE J. Seect. Areas Commun., vo. 5, no. 5, pp , June 987. [] I. E. Teatar, Capacity of muti-antenna Gaussian channes, European Trans. Teecomm., vo. 0, no. 6, pp , Nov [3] G. J. Foschini and M. J. Gans, On imits of wireess communications in a fading environment when using mutipe antennas, Wireess Persona Commun., vo. 6, pp , Mar [4] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, From theory to practice: An overview of MIMO space-time coded wireess systems, IEEE J. Seect. 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