Robust Resource Allocation for Full-Duplex Cognitive Radio Systems

Transcription

1 Robust Resource Allocation for Full-Duplex Cognitive Radio Systems Invited Paper) Yan Sun, Derric Wing wan Ng, Niola Zlatanov, and Robert Schober Institute for Digital Communications, Friedrich-Alexander-University Erlangen-Nürnberg FAU), Germany School of Electrical Engineering and Telecommunications, The University of New South Wales, Australia Department of Electrical and Computer Systems Engineering, Monash University, Australia Abstract In this paper, we investigate resource allocation algorithm design for full-duplex FD) cognitive radio systems. The secondary networ employs a FD base station for serving multiple half-duplex downlin and uplin users simultaneously. We study the resource allocation design for minimizing the maximum interference leaage to primary users while providing quality of service for secondary users. The imperfectness of the channel state information of the primary users is taen into account for robust resource allocation algorithm design. The algorithm design is formulated as a non-convex optimization problem and solved optimally by applying semidefinite programming SDP) relaxation. Simulation results not only show the significant in interference leaage compared to baseline schemes, but also confirm the robustness of the proposed algorithm. I. INTRODUCTION Bandwidth has become a scarce resource in wireless systems due to the tremendous demand for ubiquitous and high data rate communication. Recently, cognitive radio CR) has emerged as a promising paradigm to improve spectrum efficiency. In particular, CR technology allows a secondary networ to share the spectrum of a primary networ without severely degrading the quality of service QoS) of the primary networ. The authors of [1] proposed an optimal beamforming and power control algorithm to guarantee communication security in multiuser CR networs. In [2], distributed beamforming and rate allocation for multiple secondary users were considered for maximization of the minimum data rate achieved by secondary users. However, the spectral resource is still underutilized in [1], [2]. Specifically, since the secondary networ operates in the traditional halfduplex HD) mode, orthogonal radio resources are used for uplin UL) and downlin DL) transmission which limits the spectral efficiency. Full-duplex FD) wireless communication has recently attracted significant research interest due to its potential to double the spectral efficiency by performing simultaneous DL and UL transmission using the same frequency [3] [5]. Therefore, it is expected that the spectral efficiency of existing wireless communication systems can be further improved by employing an FD base station BS) in CR networs. However, the simultaneous UL and DL transmission may lead to excessive interference leaage to the primary networ and degrade the quality of communication. Therefore, different resource allocation designs for FD-CR networs were proposed to overcome this challenge. For example, the authors of [6] studied the rate region of a secondary single-antenna user served by a secondary FD BS while guaranteeing the primary user s QoS. In [7], a suboptimal resource allocation algorithm was proposed for the maximization of the sum throughput of secondary Robert Schober is also with the University of British Columbia, Canada. FD users. However, [6], [7] assumed that the channel state information CSI) of the lin between the secondary networ and the primary networ is perfectly nown at the secondary FD BS which is a highly idealistic assumption. In fact, the perfect CSI of the primary users may not be available at the secondary FD BS since they do not directly interact with the secondary networ. Besides, the obective of the resource allocation algorithms in [6], [7] was to improve the performance of the secondary networ from the secondary networ s point of view. However, in FD-CR systems, interference leaage is more serious than in traditional HD-CR systems due to the simultaneous secondary DL and UL transmission. Therefore, in FD-CR systems, a careful design of the resource allocation is necessary. Motivated by the aforementioned observations, we formulate an optimization problem to minimize the maximum interference leaage caused by the secondary FD networ to the primary networ while guaranteeing the QoS of all secondary users. The imperfectness of the CSI of the interference leaage channels is taen into account in the proposed problem formulation to facilitate a robust resource allocation. II. SYSTEM MODEL In this section, we present the considered FD-CR wireless communication system model. A. Notation We use boldface capital and lower case letters to denote matrices and vectors, respectively. A H, TrA), andrana) denote the Hermitian transpose, trace, and ran of matrix A, respectively; A 0 and A 0 indicate that A is a positive semidefinite and a positive definite matrix, respectively; I N is the N N identity matrix; C N M denotes the set of all N M matrices with complex entries; H N denotes the set of all N N Hermitian matrices; and denote the absolute value of a complex scalar and the Euclidean vector norm, respectively; E{ } denotes statistical expectation; diagx 1,,x ) denotes a diagonal matrix with diagonal elements {x 1,,x } and diagx) returns a diagonal matrix having the main diagonal elements of X on its main diagonal. R ) extracts the real part of a complex-valued input; the circularly symmetric complex Gaussian distribution with mean μ and variance σ 2 is denoted by CNμ, σ 2 );and stands for distributed as. B. Cognitive Radio System Model The considered CR system comprises one secondary FD BS, secondary DL users, J secondary UL users, one primary transmitter, and R primary receivers. The secondary FD BS is equipped with N T > 1 antennas for facilitating simultaneous

2 Fig. 1. A CR system where a secondary FD BS, 1 secondary HD downlin user, and J 1secondary HD uplin user share the same spectrum with R 2primary HD receivers. DL transmission and UL reception in the secondary networ in the same frequency band. The + J secondary users, the primary transmitter, and the secondary receivers are singleantenna HD devices that share the same spectrum, cf. Figure 1. The number of antennas at the secondary FD BS is assumed to be larger than the number of secondary UL users to facilitate reliable UL signal detection, i.e., N T J. The secondary FD BS provides wireless service to the secondary users applying multiuser multiple-input multiple-output MU-MIMO) techniques. The primary transmitter provides conventional broadcast services to the primary receivers. In this paper, we focus on slow frequency flat fading channels. In each scheduling time slot, the secondary FD BS transmits independent signal streams simultaneously at the same frequency to the secondary DL users. In particular, the information signal to secondary DL user {1,...,} can be expressed as x w d DL, where ddl C and w C NT 1 are the information bearing signal for DL user and the corresponding beamforming vector, respectively. Without loss of generality, we assume E{ d DL 2 } 1, {1,...,}. Therefore, the received signal at secondary DL user {1,...,}, the secondary FD BS, and primary receiver r {1,...,R} are given by y DL h H x + y UL y PU r h H x m + m }{{} multiuser interference P g d UL l H r x + + H SI P f, d UL + n DL, 1) } {{ } co-channel interference + n UL, and 2) x }{{} self-interference P e,r d UL + n PU r, 3) respectively. The DL channel between the secondary FD BS and secondary DL user is denoted by h C NT 1 and f, C represents the channel between secondary UL user and secondary DL user. Variables d UL, E{ d UL 2 } 1,and P are the data and transmit power sent from secondary UL user to the secondary FD BS, respectively. Vector g C NT 1 denotes the channel between secondary UL user and the secondary FD BS. Matrix H SI C NT NT denotes the selfinterference SI) channel of the secondary FD BS. The SI is caused by the signal leaage from DL transmission to UL reception in the secondary networ. Vector l r C NT 1 denotes the channel between the secondary FD BS and primary receiver r. Scalar e,r C denotes the channel between secondary UL user and primary receiver r. Variables h, f,, g, H SI, l r, and e,r capture the oint effect of path loss and small scale fading. n UL CN0,σUL 2 I N T ) and n DL CN0,σn 2 ) are the equivalent noises at the secondary FD BS and secondary DL user, which capture the oint effect of the received interference from the primary transmitter and thermal noise. n PU r CN0,σPU 2 r ) represent the additive white Gaussian noise AWGN) at primary receiver r. In1), the term J P f, d UL denotes the aggregated co-channel interference CCI) caused by the UL users to DL user. In2), the term H SI x represents the SI. III. RESOURCE ALLOCATION PROBLEM FORMULATION In this section, we formulate the resource allocation design as a non-convex optimization problem, after introducing the adopted performance metrics and the CSI assumed for resource allocation. For the sae of notational simplicity, we define the following variables: H h h H, {1,...,}, G g g H, {1,...,J}, andv v v H, {1,...,J}. A. Performance Metrics The receive signal-to-interference-plus-noise ratio SINR) at secondary DL user is given by h H w 2. 4) h H w m 2 + J P f, 2 + σn 2 m On the other hand, the receive SINR of secondary UL user at the secondary FD BS is given by Γ UL P g H v 2, 5) P n gn Hv 2 + I SI +σul 2 v 2 n where v C NT 1 is the receive beamforming vector for decoding the information received from secondary UL user. Besides, we define I SI Tr ρv diag H SIw w H HH SI)), where 0 <ρ 1 is a constant modelling the noisiness of the SI cancellation at the secondary FD BS [8, Eq. 4)]. In this paper, we adopt zero-forcing receive beamforming ZF-BF) [9] as it approaches the performance of optimal minimum mean square error beamforming MMSE-BF) when the noise term is not dominating [9] or the number of antennas is sufficiently large [10]. Besides, ZF-BF facilitates the design of a computational efficient resource allocation algorithm. B. Channel State Information In this paper, we assume that the CSI of all secondary users is perfectly nown at the secondary BS because of frequent channel estimation. However, for the secondary networ-toprimary networ channels, the perfect CSI assumption may not hold since the primary receivers do not interact directly with the secondary networ. Hence, the CSI of the lin between the secondary FD BS and primary receiver r {1,...,R}, i.e., l r, and the lin between the secondary UL user {1,...,J} and primary receiver r, i.e., e,r, are modeled as l r ˆl } r +Δl r, Ω DLr {l r C NT 1 : Δl r ε DLr }, 6) e,r ê,r +Δe,r, Ω UL,r {e,r C : Δe,r ε UL,r, 7) respectively, where ê,r and ˆl r are the CSI estimates, and Δe,r and Δl r denote the unnown CSI estimation errors. The continuous sets Ω UL,r and Ω DLr contain all possible channel uncertainties, and ε UL,r and ε DLr denote the bounded magnitude of Ω UL,r and Ω DLr, respectively.

3 C. Optimization Problem Formulation The system obective is to minimize the maximum interference leaage from the secondary networ to the primary receivers. The optimal power allocation and beamformer design are obtained by solving the following optimization problem: minimize max l H w,p Δe,r Ω UL,r,Δl r Ω DL r, r w 2 + P e,r 2 r {1,...,R} s.t. C1:,, C2: Γ UL Γ UL,, C3: w 2 PDL max, C4: 0 P PUL max,. 8) Constants > 0 and Γ UL > 0 in constraints C1 and C2 in 8) are the minimum required SINR for secondary DL users {1,...,} and secondary UL users {1,...,J}, respectively. Constants PDL max max > 0 and PUL > 0 in constraints C3 and C4 in 8) are the maximum transmit power allowance for the secondary FD BS and secondary UL users {1,...,J}, respectively. The problem in 8) is a non-convex problem due to the non-convex constraints C1 and C2. Besides, the obective function of 8) involves infinitely many functions due to the continuity of the CSI uncertainty sets. IV. SOLUTION OF THE OPTIMIZATION PROBLEM To solve the non-convex problem in 8) efficiently, we first reformulate the problem in an equivalent form and then transform the non-convex constraints into equivalent linear matrix inequality LMI) constraints. Finally, the problem is solved by semidefinite programming SDP) relaxation. To facilitate the SDP relaxation, we define W w w H and rewrite the problem in the following equivalent form: minimize τ W H N T,P,τ s.t. C1: TrH W ) C2: P TrV G ) C3: I DL + σ 2 n,, Γ UL I UL + σul 2 TrV ),, req TrW ) PDL max, C4: 0 P PUL max,, l H r W l r + P e,r 2 τ, C5: max Δe,r Ω UL,r, Δl r Ω DL r, r {1,...,R} C6: W 0,, C7: RanW ) 1,, 9) where W 0, W H NT, and RanW ) 1 in 9) are imposed to guarantee that W w w H holds after optimization. Furthermore, we use I DL m TrH W m )+ J P f, 2 and I UL Tr ρv diag H SIW HSI)) H + J n P n TrG n V ).τis an auxiliary optimization variable and 9) is the epigraph representation of 8). Constraint C5 involves an infinite number of inequality constraints, as the estimation error variables Δe,r and Δl r are involved. Here, we introduce a scalar slac variable δ r to handle the coupled estimation error variables in constraint C5. In particular, constraint C5 can be equivalently represented by C5a: l H r W l r δ r, l r Ω DLr, r, 10) C5b: δ r τ P e,r 2, e,r Ω UL,r,, r. 11) Now, we introduce a lemma which allows us to transform constraint C5a into an LMI. Lemma 1 S-Procedure [11]): Let a function f m x),m {1, 2}, x C N 1, be defined as f m x) x H A m x +2R{b H mx} + c m, 12) where A m H N, b m C N 1,andc m R 1 1. Then, the implication f 1 x) 0 f 2 x) 0 holds if and only if there exists a variable δ 0 such that δ A1 b 1 b H 1 c 1 A2 b 2 b H 0, 13) 2 c 2 provided that there exists a point ˆx such that f ˆx) < 0. By applying 6), constraint C5a can be equivalently expressed as C5a: 0 Δl H r W Δl r +2R{ˆl H r W Δl r } +ˆl H r W ˆlr δ r. By exploiting Lemma 1, we obtain the following implications: Δl H r Δl r ε 2 DL r 0 C5a holds if and only if there exists a variable α r 0 such that ) C5a: R C5ar W,α r,δ [ r ] αr I NT 0 0 α r ε 2 B DL r + δ H l r W B lr 0,, r, 14) r holds, where B lr I NT ˆlr. Similarly, by applying Lemma 1 to constraint C5b, we obtain an equivalent constraint C5b: R C5br P,β r,δ r,τ ) βr I NT P Pê r ê H r P β r ε 2 UL r δ r + τ ê H 0, r, 15) r Pê r where β r 0, P diag ) P 1,...,P J, and êr ] T. [ê1,r,...,ê J,r Next, we relax the non-convex constraint C7: RanW ) 1 by removing it from the problem formulation such that the considered problem becomes a convex SDP: minimize τ W H N T,P,τ,δ r,α r,β r s.t. C1, C2, C3, C4, C6, C8: ) δ r,α r,β r 0, r, C5a: R C5ar W,α r,δ r 0, r, C5b: R C5br P,β r,δ r,τ ) 0, r. 16) The relaxed convex problem in 16) can be solved efficiently by standard convex program solvers such as CVX [12]. Besides, if the solution obtained for a relaxed SDP problem is a ran-one matrix, i.e., RanW )1for W 0,, thenitisalso the optimal solution of the original problem. Next, we reveal the tightness of the SDP relaxation in the following theorem. Theorem 1: Assuming the considered problem is feasible, for > 0, we can always obtain or construct an optimal ranone matrix W. Proof: Please refer to the Appendix. V. RESULTS In this section, we investigate the performance of the proposed resource allocation scheme through simulations. The most important simulation parameters are specified in Table I. There are 3 secondary DL users, J 5secondary UL users, and R 2 primary receivers in the system. We assume that the primary transmitter is 100 meters away from the secondary FD BS. The secondary users and primary receivers are randomly and uniformly distributed between the reference

4 TABLE I SYSTEM PARAMETERS. Carrier center frequency 1.9 GHz System bandwidth 200 Hz Path loss exponent 3.6 SI cancellation 80 db Secondary DL user equivalent noise power, σn 2 90 dbm Secondary FD BS equivalent noise power, σul 2 90 dbm Secondary FD BS antenna gain 10 dbi Max. transmit power at the secondary FD BS, PDL max 30 dbm Max. transmit power at the secondary UL users, PUL max 10 dbm Max. transmit power at the primary transmitter 30 dbm Average maximum interference leaage dbm) , N T Baseline scheme 1, N T Baseline scheme 2, N T, N T Baseline scheme 1, N T Baseline scheme 2, N T Interference leaage Baseline scheme 1 Interference leaage Baseline scheme Minimum required DL SINR db) Fig. 2. Average maximum interference leaage dbm) versus the minimum required DL SINR db), req, for different resource allocation schemes. distance of 5 meters and the maximum service distance of 50 meters of the corresponding secondary FD BS and primary transmitter, respectively. The small scale fading of the secondary DL channels, secondary UL channels, CCI channels, and secondary networ-to-primary networ channels are modeled as independent and identically distributed Rayleigh fading. The multipath fading coefficients of the SI channel are generated as independent and identically distributed Rician random variables with Rician factor 5 db. To facilitate the presentation, we define the maximum normalized estimation error of the secondary FD BS-to-primary receiver channel and the secondary UL user-toprimary receiver channel as ε2 DLr l r κ 2 2 DL r and ε 2 UL,r e,r κ 2 2 UL,r, respectively. Besides, we assume that all channels have the same maximum normalized estimation error, i.e., κ 2 DL m κ 2 UL,m κ 2 est. Furthermore, we assume that all secondary DL users and all secondary UL users require the same minimum SINRs, respectively, i.e., req and ΓUL Γ UL req. In Figure 2, we investigate the average maximum interference leaage versus the minimum required secondary DL SINR, req, for a minimum required secondary UL SINR of Γ UL req 6 db, a maximum normalized channel estimation error of κ 2 est 5%, and different numbers of antennas at the secondary FD BS. It can be observed that the average maximum interference leaage caused by the secondary networ depends only wealy due to the proposed robust optimization. Besides, Figure 2 also indicates that the interference leaage can be significantly reduced by increasing the number of secondary BS antennas. This is due to the fact that the extra degrees of freedom DoF) offered by the additional antennas facilitate a more accurate DL beam-steering. For comparison, we consider two baseline resource allocation schemes. For baseline scheme 1, we perform ZF DL transmission for the secondary networ where the direction of on req Average maximum interference leaage dbm) Interference leaage Interference leaage Baseline scheme -96 Baseline scheme 1 Baseline scheme Maximum normalized channel estimation error, κ est %) Fig. 3. Average maximum interference leaage dbm) versus the maximum normalized channel estimation error, κ 2 est,forn T 9. beamformer w for secondary DL user is fixed and lies in the null space of the other secondary DL user channels. Then, we ointly optimize P and the power of w subect to constraints C1-C4 as in 8) via SDP relaxation. For baseline scheme 2, we consider a secondary networ with an HD BS equipped with N T antennas. We set log 2 1+ )1/2log 2 1+ HD ) and log 2 1+Γ UL )1/2log 2 1+Γ UL HD ) for a fair comparison. Thus, the required SINRs for the secondary DL and UL users served by the secondary HD BS are HD 1+ ) 2 1 and Γ UL HD 1+Γ UL ) 2 1, respectively. Besides, the power consumption of DL and UL transmission for the secondary HD networ is divided by two as DL and UL transmission do not overlap. Then, we optimize w and P to minimize the maximum interference leaage to the primary users for the optimal MMSE receiver at the secondary HD BS [9]. It can be observed from Figure 2 that the average maximum interference leaage of the baseline schemes is higher than that of the proposed FD-CR system. In particular, the average maximum interference leaage increases with req for baseline scheme 1 due to the fixed beamforming design. Besides, the average maximum interference leaage of baseline scheme 2 is insensitive to req since the w and P are optimized for the considered system setting. In Figure 3, we study the average maximum interference leaage versus the maximum normalized channel estimation error, κ 2 est, for a minimum required secondary DL SINR of req 10dB and a minimum required secondary UL SINR of Γ UL req 5dB. As can be observed, the average maximum interference leaage increases with increasing κ 2 est. In fact, with increasing imperfectness of the CSI, it is more difficult for the secondary FD BS to perform accurate DL beam-steering. In particular, more DoF are utilized to reduce interference leaage as the channel uncertainty increases which leads to a higher maximum interference leaage. Besides, as more DoF are consumed for interference leaage, there are fewer DoF available to suppress the SI which degrades the UL reception in the secondary networ. Thus, the secondary UL users are forced to transmit with a higher power to satisfy the UL QoS requirements which in turn results in a larger interference leaage to the primary networ. Furthermore, we note that the baseline schemes cause significantly higher interference leaages compared to the proposed scheme due to their inefficient resource allocation.

5 VI. CONCLUSIONS In this paper, we studied the robust resource allocation design for CR secondary networs employing an FD BS for serving multiple secondary HD DL and UL users simultaneously. The algorithm design was formulated as a non-convex optimization problem with the obective to minimize the maximum interference leaage to the primary networ while taing into account the QoS requirements of all secondary users. The imperfectness of the CSI of the secondary networto-primary networ channels was taen into account for robust resource allocation algorithm design. The proposed non-convex problem was solved optimally by SDP relaxation. Simulation results unveiled a significant in interference leaage compared to baseline schemes. Besides, we showed that the proposed scheme is indeed robust with respect to imperfect CSI. APPENDIX -PROOF OF THEOREM 1 We first solve the convex optimization problem in 16) and obtain the optimal solution P, W, and the optimal auxiliary variables which are collected in Ξ {τ,δr,α r,β r }. If RanW ) 1,, then the globally optimal solution of problem 16) is achieved. Otherwise, we substitute P and Ξ into the following auxiliary problem: minimize TrW ) W H N T s.t. C1, C2, C3, C4, C5a, C5b, C6, C8. 17) Since the problem in 17) has the same feasible set as problem 16), problem 17) is also feasible. Now, we claim that for a given P and Ξ in 17), the solution W of 17) is a ran-one matrix. First, the problem in 17) is ointly convex with respect to the optimization variables and satisfies the Slater s constraint qualification. Therefore, strong duality holds and solving the dual problem is equivalent to solving the primal problem [11]. For obtaining the dual problem, we first need the Lagrangian function of the primal problem in 16) which is given by L λ TrH W )+ θ TrρV diagw H H SI H SI)) R ) +1+μ) TrW ) TrR C5ar W,α r,θ r DC5ar ) r1 TrW Y )+Δ. 18) Here, Δ denotes the collection of terms that only involve variables that are independent of W. λ, θ,andμ are the Lagrange multipliers associated with constraints C1, C2, and C3, respectively. Matrix D C5ar C NT+1) NT+1) is the Lagrange multiplier matrix for constraints C5a. MatrixY C NT NT is the Lagrange multiplier matrix for the positive semidefinite constraint C6 on W. For notational simplicity, we define Ψ as the set of scalar Lagrange multipliers and Φ as the set of matrix Lagrange multipliers. Thus, the dual problem for the problem in 17) is given by maximize Ψ 0,Φ 0 minimize W H N T ) L W, Ψ, Φ. 19) Then, we reveal the structure of the optimal W of 17) by studying the arush-uhn-tucer T) conditions. The T conditions for the optimal W are given by: Y, D C5a r 0, λ,θ,μ 0, 20) Y W 0, 21) W L 0, 22) where Y, D, λ C5a r, θ,andμ are the optimal Lagrange multipliers for dual problem 19), W L denotes the gradient of Lagrangian function L with respect to matrix W. The T condition in 22) can be expressed as R 1+μ )I NT + θ ρv diagh H SI H SI)+ B lr D B H C5a r l r r1 Y+λ H. 23) Hence, 23) implies Y Π λ H, 24) where Π 1 + μ )I NT + J θ ρv diagh H SI H SI) + R r1 B l r D C5a r B H l r. Premultiplying both sides of 24) by W, and utilizing 21), we have W Π λ W H. By applying basic inequalities for the ran of matrices, the following relation holds: Ran W ) a) Ran WΠ ) Ran λ WH ) b) { min Ran λ W ) ) }, Ran H c) Ran ) H, 25) where a) is due to Π 0, b) is due to the basic result RanAB) min { RanA), RanB) },andc) is due to the fact that min{a, b} a. SinceRan ) H 1, by utilizing 25), the ran of W is given by RanW ) Ran ) H 1. 26) We note that W 0 for ΓDL > 0. Thus, RanW ) 1. 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