INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL ISSN 1841-9836, 10(1):112-122, February, 2015. Energy-Efficient Design for Relay-Aided MIMO-OFDM Cognitive Radio Networks B. Wu, J. Zuo, L. Zhao, C. Zou Bin Wu*, Jiakuo Zuo, Li Zhao, Cairong Zou School of Information Science and Engineering, Southeast University, Nanjing, 210096, China njwubin@seu.edu.cn, zuojiakuo85418@gmail.com zhaoli@seu.edu.cn, cairong@seu.edu.cn *Corresponding author:njwubin@seu.edu.cn Abstract: With the explosive growth of high-rate multimedia services and promptly boomed energy consumption in wireless networks, energy-efficient design is become more and more important. In this paper, we investigate energy-efficient design for relay-aided multiple-input multiple-output-orthogonal frequency division multiplex- ing (MIMO-OFDM) cognitive radio networks. We formulate an energy-efficient power allocation problem, which takes a form of nonlinear fractional programming. To solve the problem, we first make a joint concave approximation to the original problem which facilitates the optimal algorithm development. Then, we derive an equivalent parametric optimization problem of the approximated problem. Finally, an itera- tion energy-efficient power allocation algorithm is presented. Numerical results reveal that the proposed algorithm can improve energy efficiency over traditional capacity maximization method. Keywords: cognitive radio, power allocation, MIMO, relay, fractional programming. 1 Introduction Cognitive radio (CR) and multiple-input multiple-output-orthogonal frequency division mul- tiplexing (MIMO-OFDM) communications have been considered as a promising scheme to solve the spectrum scarcity problem and improve the quality of wireless communications [1]. Recently, to reduce the multi-path fading and improve the channel capacity, cooperative relaying technique is considered as a potent means to be adopted in the CR networks. Thus, the researches of relay aided MIMO CR networks are being received a growing attention in recent years [2, 3]. In [4], the relay selection and beamforming problem for the non-regenerative MIMO cognitive multi-relay network was considered and an optimal scheme was proposed via maximizing the capacity of the SUs by selecting the best cognitive MIMO relay. [5] studied a new paradigm for CR networks, which allowed the secondary users (SUs) to cooperatively relay the traffic for the primary users (PUs) while simultaneously transmitting their own traffic, and proposed a novel MIMO cooperative cognitive radio networks framework. [6] considered the power allocation problem for MIMO two-way CR sytem under a specturm sharing scenario, and presented an analytical expression of the optimal power allocation to each antenna of the treminals. [7] studied the power and channel allocation, and relay assignment for MIMO-OFDM based cooperative CR networks and proposed an optimal complexity algorithm and a sub optimal low complexity algorithm. [8] investigated subcarrier pairing and power allocation for MIMO-OFDM relay-aided CR networks and used environmental learning algorithm to mitigate the interference of the PUs. In the previous works, most researches intend to improve the throughput of relay aided CR systems. However, the energy efficiency (EE) has been considered more and more important in future wireless communication networks. The wireless devices and equipments consume about 9% of the total energy of information technology, the communication and information technology already contributes to about 2% of the global carbon dioxide emissions [9, 10]. Therefore green Copyright © 2006-2015 by CCC Publications Energy-Efficient Design for Relay-Aided MIMO-OFDM Cognitive Radio Networks 113 communication, which emphasizes on EE in wireless communication networks, is attracting more and more attention [11, 12]. A large amount of work has bee reported on energy-efficient design for CR networks [13-15]. For MIMO CR networks, [16] studies EE optimization problem of MIMO CR broadcast channels to improve the system throughput for unit energy consumption. In [17], the throughput and energy efficiency optimization under quality-of-service (QoS) constraints for MIMO CR systems are studied. In [18], a promising framework of spectrum sharing strategy selection based on EE is proposed for MIMO CR interference channels. In this paper, we focus on energy-efficient power allocation for relay-aided MIMO-OFDM CR networks. We formulate an optimization problem related to maximization of EE of the consider network under total power constraints of cognitive source node and cognitive relay node, and interference constraints of primary users. Since the original optimization problem is difficult to solve directly, we first get an approximated problem of the original problem, and then transform the approximated problem into an equivalent convex optimization problem. A new iterative energy-efficient power allocation scheme is presented at last. The rest of this paper is organized as follows: In Section 2, we introduce the system model and formulate an energy- efficient power allocation problem. In Section 3, the double-loop iterative method is presented. Finally, simulation results and Conclusions are presented in Section 4 and 5. The following notations are used in this paper, CM×N denotes M × N complex matrix, (·)H denotes the conjugate transpose, (·)+ means max (0, ·), the distribution of a circularly symmetric- complex-Gaussian vector with mean vector x and covariance matrix y is denoted by CN (x,y) , diag (·) returns a square matrix with the elements of (·) on the diagonal. SR nG SSN SRN SDN PU 1 PU L RD nG S P 1 n G H S P 1 n G S P L n G H S P L n G D P1 n G H D P 1 n G H D P L n G D P L n G R P 1 n G H R P L n G R P L n G HR P 1 n G Figure 1: Relay-aided MIMO-OFDM cognitive radio network 2 Signal Model and Problem Statement Consider a two-hop relay-aided cognitive radio (CR) network shown in Fig.1, there are a secondary source node (SSN), a secondary destination node (SDN), and a secondary relay node (SRN). The relay-aided CR network coexists with L licensed primary users (PUs). The SSN communicates with SDN through SRN, and they share the whole spectrum with PU. SSN, SDN, and SRN are equipped with MS antennas, each PU is equipped with MP (MP ≤ MS )antennas. The relay-aided CR network adopts OFDM modulation for transmission, and the total number of available subcarriers for CR network is N. Let GSR (n) ∈ CMS×MS and GRD (n) ∈ CMS×MS denote the channel matrices from SSN to SRN and SRN to SDN over the n-th subcarrier respectively. Let GSPl (n) ∈ CMp×MS , GRPl (n) ∈ CMp×MS and GDPl (n) ∈ CMp×MS denote the 114 B. Wu, J. Zuo, L. Zhao, C. Zou channel matrices from SSN to l-th PU, from SRN to l-PU, and from SDN to l-PU, respectively. The channel matrix from l-th PU to SSN, SDN, and SRN are GHSPl (n), G H RPl (n) and G H DPl (n). Assume there is no cooperation between CR network and PUs, environmental learning (EL) method [19] is performed to control the interference to the PUs. Via EL learning method, secondary nodes estimate the null space information of the channels between secondary nodes and PU. Assume the cognitive beamforming (CB) matrices at SSN, SRN, SDN for the n-th subcarrier are USPl (n) ∈ CMS×(MS−MP ) , URPl (n) ∈ CMS×(MS−MP ) and UDPl (n) ∈ CMS×(MS−MP ), respectively. These CB matrices satisfy UHSPl (n) GSPl (n) = 0, U H RPl (n) GRPl (n) = 0 and UHDPl (n) GDPl (n) = 0. However, the accurate CB matrices are difficult to be acquired, therefore in practical applications, the estimated CB matrices ŨSPl (n), ŨRPl (n) and ŨDPl (n) are used. In the first hop, the received signal at CRN in the n-th subcarrier is given by: yR (n) = GSR (n) ŨSPl (n) xS + G H RPl (n) w 1 P + zR (n) (1) where xS is the transmitted signal of SSN, w1P is the PU interference to SRN in the first hop. zR (n) ∼ CN ( 0,σ2RIMS ) is the additive white Gaussian noise (AWGN) at SRN. In the second hop, SRN first filters received signal yR (n) with Ũ H RPl (n), and then precodes the filtered signal by forwarding matrix B (n), finally precodes the resultant signal by ŨDPl (n). Therefore, the received signal at SDN in the n-th subcarrier is: yD (n) = GRD (n) ŨRPl (n) B (n) Ũ H RPl (n) yR (n) + G H DPl (n) w 2 P + zD (n) (2) where w2P is the PU interference to SDN in the second hop, zD (n) ∼ CN ( 0,σ2DIMS ) is additive white Gaussian noise (AWGN) at SDN in the n-th subcarrier. Finally, the CR-DN does receive CB by filtering yD (n) with Ũ H DPl (n), we have y = D2 (n) B (n) D1 (n) xS + D(n)2B (n) n1 (n) + n2 (n) (3) where D1 (n) = Ũ H DPl (n) GRD (n) ŨRPl (n) and D2 (n) = Ũ H RPl (n) GSR (n) ŨSPl (n), n1 (n) = ∆UHRPl (n) G H RPl (n) w 1 P +Ũ H RPl (n) zR (n), n2 (n) = ∆U H DPl (n) G H DPl (n) w 2 P +Ũ H DPl (n) zD (n), ∆UHRPl (n) = Ũ H RPl (n) − U H RPl (n), ∆U H DPl (n) = Ũ H DPl (n) − U H DPl (n) denote the first-order perturbations of the CB matrices due to imperfect environmental learning [8, 19]. Let the singular value decomposition of {Dk (n)}k=1,2 be Dk (n) = Uk (n) Λk (n) V H k (n) (k = 1,2), and define qm,n = √ pRm,n pSm,nλ SR m,n+σ 2 R , where pSm,n and p R m,n are the transmit power of SSN and SRN, λSRm,n is the eigenvalue of GSR (n). Thus, the forwarding matrix can be defined as B (n) = V H2 (n) Q (n) U1 (n). Multiplying y with U H 2,n at SDN , we have y = Λ2 (n) Σ (n) Λ1 (n) V 1 (n) xS + Λ2 (n) Σ (n) U H 1 (n) n1 (n) + U H 2 (n) n2 (n) (4) where Σn = diag (q1,n,q2,n, · · · ,qM,n). According to formula (4), the MIMO-OFDM channel between SSN and SDN can be decom- posed into N × MS parallel independent channels, therefore, the throughput of the Relay-aided MIMO-OFDM network is Ctp (p) = 1 2 M∑ m=1 N∑ n=1 log2 ( 1 + pSm,nαm,np R m,nβm,n 1 + pSm,nαm,n + p R m,nβm,n ) (5) where αm,n = λSRm,n σ2 R +ψ1 , βm,n = λRDm,n σ2 D +ψ2 , λRDm,n is the eigenvalue of GRD (n), ψ1 and ψ2 are constants and linear with 1 NEL ( NEL is the number of samples in EL stage), p = { pSm,n,p R m,n } is power Energy-Efficient Design for Relay-Aided MIMO-OFDM Cognitive Radio Networks 115 vector. The overall power consumption at SSN and SDN can be expressed respectively as follows: PSSN = τS M∑ m=1 N∑ n=1 pSm,n + P S c (6a) PSRN = τR M∑ m=1 N∑ n=1 pRm,n + P R c . (6b) where τS and PSc are the reciprocal of drain efficiency of power amplifier and circuit power at SSN. τR and PRc are the reciprocal of drain efficiency of power amplifier and circuit power at SDN. The EE of the cognitive relay network while selecting the l-th CRN for transmitting is defied as: ξEE (p) = 1 2 M∑ m=1 N∑ n=1 log2 ( 1 + pSm,nαm,np R m,nβm,n 1+pSm,nαm,n+p R m,nβm,n ) τS M∑ m=1 N∑ n=1 pSm,n + τ R M∑ m=1 N∑ n=1 pRm,n + P S c + P R c (7) Since we use the estimated CB matrices, the interferences to PUs cased by SSN and SRN are inevitably, the interferences cased by SSN and SRN to l-th PU are ISPl = µS σ2l M∑ m=1 N∑ n=1 pSm,n (8a) IRPl = µR σ2l M∑ m=1 N∑ n=1 pRm,n. (8b) where µS and µR are constants and linear with 1NEL , σ 2 l is the transmit power of the l-th PU signal. From (7), the objective of energy-efficient power allocation problem for the relay-aided MIMO-OFDM CR network can be expressed as: OP1 max pSm,n,p R m,n≥0 1 2 M∑ m=1 N∑ n=1 log2 ( 1 + pSm,nαm,np R m,nβm,n 1+pSm,nαm,n+p R m,nβm,n ) τS M∑ m=1 N∑ n=1 pSm,n + τ R M∑ m=1 N∑ n=1 pRm,n + P S c + P R c (9) subject to   C1 : M∑ m=1 N∑ n=1 pSm,n ≤ PSth C2 : M∑ m=1 N∑ n=1 pRm,n ≤ PRth C3 : µS σ2 l M∑ m=1 N∑ n=1 pSm,n ≤ Ith, l = 1,2, · · · ,L C4 : µR σ2 l M∑ m=1 N∑ n=1 pRm,n ≤ Ith, l = 1,2, · · · ,L. where PSth and P R th are the total power budgets of SSN and SRN respectively, Ith is the interference threshold of PUs. C1 and C2 are transmission power constraints of SSN and SDN, C3 and C4 are the interference constraints of the PUs. Duo to lack of convexity, it is difficult to solving OP1 directly. In the following, we make a joint concave approximation to OP1 and introduce a new equivalent optimization problem via nonlinear fractional programming (NFP) [21]. 116 B. Wu, J. Zuo, L. Zhao, C. Zou 3 Energy Efficient Power Allocation Algorithm To make OP1 more tractable, the throughput Ctp can be approximated at the high signal- to-noise ratio (SNR) as C̃tp (p) = 1 2 M∑ m=1 N∑ n=1 log2 ( 1 + pSm,nαm,np R m,nβm,n pSm,nαm,n + p R m,nβm,n ) (10) Note: As in [20], C̃tp (p) is joint concave with pSm,n and p R m,n. Thus, we can also get the approximation of EE as: ξ̃EE (p) = C̃tp (p) Ptotal (p) (11) where Ptotal (p) = PSSN + PSRN. Substitute ξ̃EE into OP1, we get the approximated optimization problem OP2 max pSm,n,p R m,n≥0 ξ̃EE (p) (12) subject to C1 ∼ C4 For notational simplicity, we define ℵ as the set of feasible solution of OP2, and let p ={ pSm,n,p R m,n } be variable vector. Define the maximum EE ρ∗ of network as follows: ρ∗ = C̃tp (p∗) Ptotal (p∗) = max p∈ℵ C̃tp (p) Ptotal (p) (13) where p∗ is the optimal solution of OP2. Introducing a new parametric optimization problem OP3 OP3 max pSm,n,p R m,n≥0 { C̃tp (p) − ρPtotal (p) } (14) subject to C1 ∼ C4 where ρ is non-negative parameter. Since C̃tp (p) is joint concave with pSm,n and p R m,n, for a given ρ, OP3 is a convex optimization problem. Next, introduce a theorem based on NFP [21]: Theorem The optimal solution achieve the maximum EE if and only if max p∈ℵ { C̃tp (p) − ρ∗Ptotal (p) } = C̃tp (p∗) − ρ∗Ptotal (p∗) = 0 (15) with C̃tp (p) ≥ 0, Ptotal (p) > 0. Proof: Similar proof can be found in [21]. The Theorem implies that for fractional OP2, there is an equivalent problem whose objective function is in subtractive form, e.g. C̃tp (p) − ρ∗Ptotal (p) . Therefore, solving OP2 is equivalent to solve problem OP3 for a given ρ and then update ρ until the Theorem is satisfied. Energy-Efficient Design for Relay-Aided MIMO-OFDM Cognitive Radio Networks 117 An alternative method solving OP3 is through deriving the Lagrange dual [22] of the opti- mization problem OP3. The Lagrange function of OP3 is defined as bellow: Lag ( p,θ1,θ2,{χl} L l=1 ,{ηl} L l=1 ) = 1 2 M∑ m=1 N∑ n=1 log2 ( 1 + pSm,nαm,np R m,nβm,n pSm,nαm,n+p R m,nβm,n ) −ρ [ τS M∑ m=1 N∑ n=1 pSm,n + τ R M∑ m=1 N∑ n=1 pRm,n + P S C + P R C ] −θ1 ( M∑ m=1 N∑ n=1 pSm,n − PSth ) − θ2 ( M∑ m=1 N∑ n=1 pRm,n − PRth ) − L∑ l=1 χl ( µS σ2 l M∑ m=1 N∑ n=1 pSm,n − Ith ) − L∑ l=1 ηl ( µR σ2 l MS∑ m=1 N∑ n=1 pRm,n − Ith ) (16) where θ1, θ2 , χl and ηl are the Lagrange multipliers. Therefore, the Lagrange dual function of the primal problem OP3 can be written as: Dual ( θ1,θ2,{χl} L l=1 ,{ηl} L l=1 ) = max p≥0 Lag ( p,θ1,θ2,{χl} L l=1 ,{ηl} L l=1 ) (17) The corresponding Lagrangian dual problem of OP3 can be expressed as: min θ1,θ2,χl,ηl≥0 Dual ( θ1,θ2,{χl} L l=1 ,{ηl} L l=1 ) (18) The problem (17) is convex, according to the Karush-Kuhn-Tucker condition: ∂Lag ∂pSm,n = 0 and ∂Lag ∂pRm,n = 0, then we have: 1 2 ln 2 αm,nβ 2 m,n ( pRm,n )2( pSm,nαm,n + p R m,nβm,n )( pSm,nαm,n + p R m,nβm,n + p S m,np R m,n ) = ρτS + θ1 + µS L∑ l=1 χl σ2l (19) 1 2 ln 2 α2m,nβm,n ( pSm,n )2( pSm,nαm,n + p R m,nβm,n )( pSm,nαm,n + p R m,nβm,n + p S m,np R m,n ) = ρτR + θ2 + µR L∑ l=1 ηl σ2l (20) Solving the above two equations, we get the optimal power allocation solutions as: pSm,n = 1 √ αm,nx1 (√ x1 αm,n + √ x2 βm,n )(1−(√ x1 αm,n + √ x2 βm,n )2)+ (21) pRm,n = 1√ βm,nx2 (√ x1 αm,n + √ x2 βm,n )(1−(√ x1 αm,n + √ x2 βm,n )2)+ (22) where [x]+ = max (0,x), x1 = ρτS + θ1 + µS L∑ l=1 χl σ2 l and x2 = ρτR + θ2 + µR L∑ l=1 ηl σ2 l . Note: (21) and (22) show that pSm,n and p R m,n are either both positive or both zero, this implies that if power allocated to the n-th subcarrier in the first hop is zero, then no power is allocated to its corresponding subcarrier in the second hop, which meets the intuition very well. 118 B. Wu, J. Zuo, L. Zhao, C. Zou Table 1: Algorithm: approximated energy-efficient power allocation 1 Initialization: initial ρ, ϖθ1, ϖ θ 2, ϖ χ l and ϖ η l , the maximum tolerance δ 2 repeat 3 repeat 4 update pSm,n and p R m,n according to (21) and (22) 5 update ϖθ1, ϖ θ 2, ϖ χ l and ϖ η l according to (23) 6 until ϖθ1, ϖ θ 2, ϖ χ l and ϖ η l converge 7 Update ρ = ξ̃EE (p) via (11) 8 until ∣∣∣C̃tp (p) − ρPtotal (p)∣∣∣ ≤ δ The optimal dual variables can be obtained from the dual problem (18) using the subgradient method[23]. The dual variables could be updated as: θ1 = ( θ1 + ϖ θ 1 ( PSth − M∑ m=1 N∑ n=1 pSm,n ))+ (23a) θ2 = ( θ2 − ϖθ2 ( PRth − M∑ m=1 N∑ n=1 pRm,n ))+ (23b) χl = ( χl − ϖ χ l ( Ith − µS σ2l M∑ m=1 N∑ n=1 pSm,n ))+ (23c) ηl = ( ηl − ϖηl ( Ith − µR σ2l M∑ m=1 N∑ n=1 pRm,n ))+ (23d) where ϖθ1, ϖ θ 2, ϖ χ l and ϖ η l are the step length. According to the aforementioned analysis, we propose a two loop iterative algorithm to solve the approximated energy-efficient power allocation problem OP2, which is termed as AEE-PA and tabulated as in Table 1. Note: [23] shows that the subgradient algorithm can converge to the optimal solution of convex optimization problems within a small range. Therefore, the inner loop can converge to the optimal solution of the dual problem (18) with in a small range. Since OP3 is convex optimization problem, the duality gap for OP3 is zero, the inner loop also converges to the optimal solution of OP3 within a small range. The detailed proves of the convergence of the outer loop, i.e. NFP can be found in [21]. 4 Performance Simulations We perform numerical simulations to evaluate the present some numerical experiments to evaluate the performance of our proposed scheme. Without loss of generality, the channel gains are assumed to be Rayleigh fading with an average power gain of 1dB, and set the parame- ters N = 10, L = 2 , MS = 4, MP = 2, σ2R = σ 2 D = 10 −6W , σ2l = 1W , τ S = τR = 1, PSc = P R c = 10 −2W , PSth = P R th = Pmax. Since ψ1, ψ2, µS and µR are linear with 1 NEL , for Energy-Efficient Design for Relay-Aided MIMO-OFDM Cognitive Radio Networks 119 simplicity, let , , and are equals with . All the results have been averaged over 500 iterations. We compare the proposed algorithm with the traditional throughput maximum problem. Change the objective function ξEE (p) in OP1 with the throughput Ctp (p) in (5) and change the approx- imated objective function ξ̃EE (p) in OP2 with the approximated throughput C̃tp (p) in (10), then we formulate the traditional throughput maximum problem. Since C̃tp (p) is joint con- cave in with pSm,n and p R m,n, the throughput maximum problem is convex problem which can be solved by many standard convex optimization algorithms [22]. We name the method to solve the throughput maximum problem as TM-PA. In the following, we compare the proposed algorithm with the TM-PA scheme. Since the proposed AEE-PA consists of two loops, we only consider the affect of the number of outer loop iterations tO and set the number of inner iterations large enough to guarantee that the inner loop can find the optimal solution of OP3. Fig.2 shows the EE versus the outer loop iterations tO for different total power budget under Ith = 1W , NEL = 500 . It can be observed in Fig.2 that AEE-PA converges to the optimal value within eleven iterations for all considered value of total power budgets. The maximum EE can be improved when there are more total power budgets. Fig.3 depicts the EE versus total power budget Pmax for different interference thresholds under NEL = 500. As shown in Fig.3, the EE of the both algorithms increases with the increasing of the total power budget, however the proposed AEE-PA has a higher EE than the non energy efficiency scheme TM-PA. The EE versus interference threshold Ith for different total power budgets under NEL = 500 is evaluated in Fig.4. It is shown in Fig.4 that the EE of the both algorithms grows with the growth of the interference threshold. This is because that the lower the interference threshold is, the more the CR network suffers outage. We also evaluate the impact of NEL (the number of samples in EL stage) on the proposed algorithm. In Fig.5, EE versus interference threshold Ith for different NEL under Pmax = 0.5W is depicted. Obviously, the algorithm has a better performance with lager NEL than small NEL. This is because smaller NEL performs poor learning and yields large interference to PUs. 1 2 3 4 5 6 7 8 9 10 11 7 8 9 10 11 12 13 t O E n e rg y E ff ic ie n cy ( b it/ s/ Jo u le ) P max =1W P max =0.8W P max =0.6W Figure 2: Energy efficiency versus the outer iterations for different total power budget 120 B. Wu, J. Zuo, L. Zhao, C. Zou 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 P max (W) E n e rg y e ff ic e n cy ( b it/ s/ Jo u le ) AEE−PA I th =1W TM−PA I th =1W AEE−PA I th =0.01W TM−PA I th =0.01W Figure 3: Energy-efficiency versus total power budget for different interference threshold 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2 3 4 5 6 7 8 I th (W) E n e rg y e ff ic e n cy (b it/ s/ Jo u le ) AEE−PA P max =1W TM−PA P max =1W AEE−PA P max =0.5W TM−PA P max =0.5W Figure 4: Energy efficiency versus interference threshold for different total power budget 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 3.5 4 4.5 5 5.5 I th (W) E n e rg y e ff ic e n cy (b it/ s/ Jo u le ) N EL =1000W N EL =500W N EL =100W Figure 5: Energy efficiency versus interference threshold for different NEL Energy-Efficient Design for Relay-Aided MIMO-OFDM Cognitive Radio Networks 121 5 Conclusions In this paper, we investigated the power allocation for relay-aided MIMO-OFDM cognitive radio networks from energy efficiency perspective. 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