Energy Efficient Resource Allocation for MIMO SWIPT Broadcast Channels

Transcription

1 1 Energy Efficient Resource Allocation for MIMO SWIPT Broadcast Channels Jie Tang 1, Daniel K. C. So 2, Arman Shojaeifard and Kai-Kit Wong 1 South China University of Technology, eejtang@scut.edu.cn 2 University of Manchester, d.so@manchester.ac.uk University College London, {a.shojaeifard, kai-kit.wong}@ucl.ac.uk. Astract In this paper, we address the energy efficiency (EE) optimization prolem for SWIPT multiple-input multiple-output roadcast channel (MIMO-BC) with time-switching (TS) receiver design. Our aim is to imize the EE of the system whilst satisfying certain constraints in terms of imum transmit power and minimum harvested energy per user. The coupling of the optimization variales, namely, transmit covariance matrices and TS ratios, leads to a EE prolem which is non-convex, and hence very difficult to solve directly. Hence, we transform the original imization prolem with multiple constraints into a min- prolem with a single constraint and multiple auxiliary variales. We propose a dual inner/outer layer resource allocation framework to tackle the prolem. For the inner-layer, we invoke an extended SWIPT-ased BC-multiple access channel (MAC) duality approach and provide an iterative resource allocation scheme under fixed auxiliary variales for solving the dual MAC prolem. A su-gradient searching scheme is then proposed for the outer-layer in order to otain the optimal auxiliary variales. Numerical results confirm the effectiveness of the proposed algorithms and illustrate that significant performance gain in terms of EE can e achieved y adopting the proposed extended BC-MAC duality-ased algorithm. I. INTRODUCTION Energy harvesting (EH) is considered a prominent solution for prolonging the lifetime of power-constrained wireless devices [1]. In addition to well-recognized renewale energy sources such as iomass, wind, and solar, wireless power transfer (WPT) has emerged as a new enaler for EH. With WPT, the transmitter can transfer energy to the receivers via amient radio frequency (RF) electromagnetic waves [2]. The integration of RF-ased EH capaility in communication systems opens up the possiility for simultaneous wireless information and power transfer (SWIPT). This topic has attracted great attention in oth academia and industry recently []. Most research works on SWIPT systems aim to imize the rate or the harvested energy, or otherwise achieve a certain rate-energy alance. Nevertheless, the standalone imization of the system throughput would inherently constitute to the highest network power consumption. This trend goes against gloal commitments for tackling the socalled capacity crunch in a sustainale and economically viale manner. On the other hand, the sole goal of imizing the harvested energy may degrade the delivered information, and in turn quality of service (QoS). An alternative strategy is therefore to consider energy-efficiency (EE), which is nowadays considered a fundamental performance metric in the design and deployment of wireless networks [4]. In addition to the many works on the EE optimization prolem for conventional communication setups, the imization of EE has een recently considered in the context of SWIPT systems [5] [7]. In contrast to previous literature on EE for SWIPT, such as multi-carrier OFDMA systems [8], MISO systems ased on a fixed precoder such as ZF [6] or MMSE [7], in this paper, we address the EE optimization prolem for SWIPTased MIMO-roadcast channel (BC) where TS technique is employed. By coupling the optimization variales in terms of transmit covariance matrices and TS ratios, the EE imization prolem under consideration ecomes non-convex. Hence, it is very difficult to otain the system EE solution using direct methods. Hence, to tackle this prolem, we transform the original optimization prolem with multiple constraints into a min- prolem with a single constraint and multiple auxiliary variales. In order to tackle the transformed prolem, we propose a dual inner/outer layer resource allocation framework. By invoking the conventional BC-multiple access channel (MAC) duality principle [9], we formulate a dual SWIPT-ased MIMO-MAC EE optimization prolem with fixed auxiliary variales and accordingly provide an iterative resource allocation algorithm ased on the Dinkelach method [10]. A su-gradient approach is then proposed in order to reach the optimal auxiliary variales in the outer-layer. II. SYSTEM MODEL AND PROBLEM FORMULATION The system consists of a BS with N T transmit antennas and K users k {1, 2,..., K} each with N R receive antennas. We denote the channel matrix from the BS to the k th user as H k C Nr Nt. Channel state information (CSI) is assumed to e perfectly known at the corresponding transmitter and receivers. Different from the conventional MIMO downlink system, each transmission lock in the SWIPT-ased MIMO- BC system is divided into two orthogonal time slots, one for ID and the other for EH, per illustrated in in Fig. 1. In particular, the TS-ased receiver periodically switches its operations etween ID and EH. Hence, the received signal from the BS to the k th user efore TS is written as y k = H k x + n k, (1) where n k C Nr 1 is the independent zero mean Gaussian noise with each entry CN (0, σ 2 ), x is the transmitted signal

2 2 Fig. 1. A downlink SWIPT-ased MIMO-BC system with TS-ased receivers. on the downlink. In addition, x = x 1 + x x K where x k C Nt 1, is the signal transmitted to the k th user. In this paper, receivers consisting of a harvesting energy unit and a conventional signal processing unit for concurrent EH and ID is under consideration. Let α k with 0 α k 1 denote the percentage of transmission time allocated to the ID time slot for user k. Thus, 1 α k corresponds to the percentage of transmission time allocated to the EH time slot for user k. Hence, the harvested energy at the receiver of user k can e written as E k = (1 α k )η k tr(g k Q), (2) where η k is a constant that accounts for the loss in the energy transducer for converting the harvested energy to electrical energy to e stored, G k = H H k H k is the channel covariance matrix, Q denotes the total transmit covariance matrix at the ) is the corresponding transmit covariance matrix, Q k 0 i.e., Q k is a positive semidefinite matrix. On the other hand, the total capacity of the MIMO-BC SWIPT system can e expressed as follows BS, Q = Q k, Q k = E(x k x H k C BC = α k Rk, () where Rk denotes the rate achieved y the user k in the downlink. Further, note that dirty paper coding (DPC) is the capacity achieving scheme for Gaussian MIMO-BC [11]. With DPC, the information for different users is encoded in a sequential fashion. It should e noted that the transmit covariance matrices remain the same during each transmission lock. This implies that the same transmit covariance matrices are shared in ID and EH models. Without loss of generality, with an encoding order (1,, K), i.e., the codeword of user 1 is encoded first, the data rate Rk for the kth user can e written as [9] Rk = W log I N r N r + 1 σ H 2 k ( i=k Q i)h H k I Nr N r + 1 σ H 2 k ( i=k+1 Q i)h H k. (4) For the SWIPT-ased MIMO-BC system considered in this work, the power consumption model should e extended considering EH devices. In general, small amounts of energy is consumed y the RF EH devices. On the other hand, the system power consumption is intuitively compensated y the harvested energy. It may then e apparent that enaling EH can improve the EE of a wireless communication system. Thus, as in [8], we take the harvested power into consideration in the formulation of the system power consumption model. Specifically, the total power consumption is given y P = ζp T + P C E k, (5) where E k represents the harvested power at all the receivers. Note that the minus sign denotes that a portion of the power radiated in the RF from the transmitter can e harvested y the K receivers. Recall that P C is the total circuit power required for supporting reliale communication P C = P BS ant N T + P sta + KP R C, (6) where Pant BS N T denotes the power consumption proportional to the numer of transmit antennas, P sta is the constant signal processing circuit power consumption in the transmitters (due to filters, frequency synthesizer, etc., independent of the power radiated y the transmitter), and PC R denotes the total circuit power consumption in the K receivers. The EE for SWIPT-ased MIMO-BC can e defined as the total numer of delivered its per unit energy. Hence, we define EE in a SWIPT-ased MIMO-BC as λ EE C BC P = α krk ζp T + P C E, (7) k where C BC is the total capacity achieved y all users and P T = tr(q k) is the total transmission power. Given the expression of the system sum rate and power consumption, we can proceed with the prolem formulation. The ojective of this paper is to imize the EE in SWIPTased MIMO-BC whilst meeting two constraints in terms of transmission power and harvested energy. By invoking the linear power model in (5), the optimization prolem can e formulated as α krk Q k 0,α k ζp T + P C (1 α k)η k tr(g k Q), (8) (1 α k )η k tr(g k Q) E k,min, k K, (9) tr(q k) P, (10) 0 α k 1, k K, (11) where P and E k,min are the imum total transmit power constraint at the BS and the minimum harvested energy constraint for user k (1, 2,, K), respectively. Note that (11) corresponds to the inherent constraints in terms of TS ratios. It is easy to see that the coupling of optimization variales leads to the prolem (8)-(11) eing non-convex and challenging to solve directly. Therefore, in the following sections, we develop resource allocation schemes for SWIPTased MIMO-BC to solve the aove optimization prolem.

3 III. EQUIVALENCE AND EXTENDED BC-MAC DUALITY In our SWIPT-ased MIMO-BC setting, the EE optimization prolem in (8)-(11) has not only a sum power constraint ut also multiple minimum harvested energy constraints. The imposed multiple constraints complicates the formulation of an efficient solvale dual prolem. In order to overcome the these challenges, we first transform this multi-constrained EE imization prolem into its equivalent prolem that has a single constraint with multiple auxiliary variales. Thus, we develop a duality etween a SWIPT-ased MIMO-BC and a SWIPT-ased dual MIMO-MAC in the case where the multiple auxiliary variales are fixed. Proposition 1: Prolem (8)-(11) shares the same optimal solution with the following equivalent prolem min χ,µ k Q k,α k χ( + α krk ζp T + P C (1 α k)η k tr(g k Q) tr(q k ) P ) (12) µ k (E k,min (1 α k )η k tr(g k Q)) 0, (1) where χ and µ k are the auxiliary dual variales for the imum power constraint and the k th minimum harvested energy constraint, respectively. However, it is still very difficult to directly find an efficiently solvale dual prolem for (12)-(1). Thus, in the following, we first investigate the EE imization prolem considering fixed auxiliary dual variales χ and µ k. The prolem in (12)- (1) can hence e reduced to α krk Q k,α k ζp T + P C (1 α k)η k tr(g k Q) (14) χ( tr(q k) P ) + µ k (E k,min (1 α k )η k tr(g k Q)) 0. (15) The solution of the aove prolem is unfortunately nontrivial given the ojective function is non-concave even under fixed auxiliary dual variales χ and µ k. Thus, we exploit an extended SWIPT-ased BC-MAC duality principle ased on results from [9] and [12]. Consequently, the weighted sum rate imization prolem in SWIPT-ased MIMO-BC under constraints in (1) can e formulated as Q k 0,0 α k 1 α k Rk, (16) χ tr(q k) µ k (1 α k )η k tr(g k Q) P all, (17) where P all := χp µ ke k,min. Since χ and µ k are fixed, P all is a constant (16)-(17). Hence, y extending the general BC-MAC duality principle from [9] to our SWIPTased MIMO-BC scenario, we have the following SWIPTased dual MAC prolem corresponding to the original SWIPT-ased BC prolem in (16)-(17). Proposition 2: The SWIPT-ased dual MAC prolem of (16)-(17) is given y Q m k 0,0 α k 1 α k Rk m (18) tr(q m k ) P all, (19) where Q m k is the transmit signal covariance matrix of the kth user, and Rk m is the rate achieved y the kth user of the dual MAC defined as Rk m = W log N + HH k ( i 1 Qm i )H k N + H H k ( i 1 1 Q m i )H k, (20) with the noise covariance at the BS denoted with N = χi K µ k(1 α k )η k [tr(g k Q)]). Proposition 2 indicates that the capacity region of a SWIPTased MIMO-BC with power constraint P all is equal to the union of capacity regions of the SWIPT-ased dual-mac with power constraints such that tr(qm k ) = P all. However, Proposition 2 descries the rate region for SWIPT-ased MIMO-BC system and its duality relationship with SWIPTased MIMO-MAC, meaning the EE aspect is still an open question. Hence, in order to tackle the prolem in (14)-(15), we develop the following proposition (EE aspect) ased on the results in Proposition 2. Proposition : The solution of the SWIPT-ased dual MAC EE imization prolem, namely, Q m k 0,0 α k 1 α krk m ζtr(q m k ) + P (21) C tr(q m k ) P all, (22) is an upper-ound of the solution to the prolem in (14)-(15). Consequently, instead of directly tackling the EE imization prolem in (14)-(15), in this work, we have provided a dual-mac upper ound solution of (21)-(22). To solve the optimization prolem in (21)-(22), it is generally helpful to relate it to a concave program y separating numerator and denominator with the help of parameter β, this is what is known as the Dinkelach method [10]. In the following sections, we propose an iterative resource allocation algorithm ased on the Dinkelach method to otain the upperound solution to the prolem in (14)-(15). IV. ITERATIVE RESOURCE ALLOCATION SCHEME BASED ON DINKELBACH METHOD Recall that the optimization prolem in (21)-(22) elongs to a family of non-linear fractional programming prolems which are non-convex and difficult to solve directly. Nevertheless, y invoking the theory of non-linear fractional programming in [10], we can use the Dinkelach method to solve this nonconvex non-linear fractional programming prolem. Specifically, ased on Dinkelach method [10], here, we propose an

4 4 1) Initialize β = 0, and δ as the imum tolerance; 2) REPEAT ) For a given β, otain an intermediate resource allocation policy {Q, α} y solving the prolem in (2)-(24); 4) IF U R (Q, α) βu T (Q, α) δ 5) Convergence = TRUE; 6) RETURN {Q, α } = {Q, α} and β = U R(Q,α) U T (Q,α) ; 7) ELSE 8) Set β = U R(Q,α) U T (Q,α) and n = n + 1, Convergence = FALSE; 9) END IF 10) UNTIL Convergence = TRUE. TABLE I PROPOSED DINKELBACH METHOD-BASED SOLUTION iterative algorithm for solving (21)-(22) with an equivalent ojective function such that the otained solution satisfies the conditions stated in Proposition 4. The proposed algorithm is summarized in Tale I. It can e oserved from Tale I that the key step for the proposed iterative algorithm concerns the solution to the following optimization prolem for a given parameter β in each iteration (i.e., step ), Q m k 0,0 α k 1 α k Rk m β(ζ tr(q m k ) + P C ) (2) tr(q m k ) P all. (24) To solve this prolem, we define f(q m 1,, Q m K, α 1,, α K ) = k log N + HH k Q m k H k, where k = α k α k+1, and thus the optimization prolem in (2)-(24) can e reformulated as β(ζ Q m k 0,0 α k 1 f(qm 1,, Q m K, α 1,, α K ) tr(q m k ) + P C ) tr(q m k ) P all. (25) The corresponding Lagrangian function can e expressed as L(Q m 1,, Q m K, α 1,, α K, τ) := f(q m k, α k ) β(ζ tr(q m k ) + P C ) τ[ tr(q m k ) P all ], (26) where τ 0 is the Lagrangian multipliers associated with the imum power constraint. The dual ojective function of (2) is written as g(τ) = Q m k 0,0 α k 1 L(Qm 1, Q m K, α 1,, α K, τ), (27) and the dual prolem is given y min τ g(τ) τ 0. (28) In this work, an iterative approach is used here in order to achieve the optimum Q m k and α k. In particular, we update Q m k through the gradient of the Lagrangian function (26) with respect to Q m k and α k as follows Q m k L := f(qm 1,, Q m K) Q m k (n 1) βζτi Nr N r, (29) αk L := f(α 1,, α K ), (0) α k (n 1) Q m k (n) = [Q m k (n 1) + t Q m k L] +, (1) α k (n) = α k (n 1) + t αk L, (2) where t represents the step size, and the notation [A] + is defined as [A] + := i [q i] + v i vi H, with q i and v i denote the i th eigenvalue and the corresponding eigenvector of A respectively. Therefore, we can compute the gradient in (29) and (0) as follows f(q m ) Q m k f(α) α k = = H k (I Nt N t + 1 K σ 2 H H k Q m kh k ) 1 H H k, i=1 i i tr[µ i η i G i ( N + H H k Q m k H k ) 1 )] + Rk m. () After we otain the optimum Q m k and α k, our next task is to find out the optimal τ. Given that the Lagrangian function g(τ) is a convex function with respect to τ, we can achieve the optimal τ through a one-dimensional searching approach. Nevertheless, it is not guaranteed that g(τ) is differentiale, and thus the gradient approach may not availale in this case. On the other hand, the su-gradient approach can e applied to search the optimal solution where τ is updated in accordance with the su-gradient direction as P all tr(qm k ). Upon convergence of the transmit covariance matrices Q m k, k = 1, 2,, K and the TS ratios α k, k = 1, 2,, K, the current consumption power is saved in order to compare with P all. In particular, the value of τ should e increased if K tr(qm k ) P all, and decrease otherwise. This procedure is continued until convergence, i.e., τ min τ ε. V. SOLUTION TO THE SWIPT-BASED EE MAXIMIZATION PROBLEM Here, we provide a complete solution to the EE optimization prolem in (12)-(1), namely an extended BC-MAC dualityased EE imization algorithm. Under fixed χ and µ, the prolem can e reformulated as follows x(χ, µ) = α krk Q k,α k ζpt m + P C (1 α k)η k E[tr(G k Q)] (4) χ( tr(q k) P ) + µ k (E k,min (1 α k )η k E[tr(G k Q)]) 0, (5) In addition, the prolem (12)-(1) is equivalent as follows min χ,µ x(χ, µ) (6)

5 t = 0.01 t = 0.1 EE (/Hz/J) 2 EE (/Hz/J) Proposed Dinkelach-ased method Full-search-ased optimal approach Numer of iterations Numer of Iterations Fig. 2. scheme. Convergence ehavior of the proposed Dinkelach method-ased Fig.. Convergence ehavior of the proposed extended BC-MAC dualityased EE imization algorithm in terms of EE. χ 0 and µ k 0, k K. (7) By applying the BC-MAC duality in Section III together with the proposed Dinkelach method-ased iterative resource allocation scheme in Section IV, one can achieve x(χ, η). We then apply the BC-MAC covariance mapping approach from [12] to otain the corresponding BC transmit covariance matrices Q k, k = 1,, K. Once we have otained the solution for a given χ and µ, we can update χ and µ through a su-gradient approach. It should e noted that with a constant step size, the su-gradient approach will converge to a point that is very close to the optimal value [1], i.e., lim n χn χ < ɛ, and lim n µn k µ k < ɛ, k = 1,, K, (8) where χ and µ k are the optimal values, and χn and µ n k are the values of χ and µ k at the n th iteration of the sugradient approach, respectively. This result indicates that the su-gradient approach determines an ɛ-suoptimal point in a finite numer of iterations. VI. SIMULATION RESULTS In our simulation, the BS employs N t = 4 transmit antennas, each user is equipped with N R = 2 receive antennas, and the total numer of users is set to K = 4. The path-loss is calculated using log 10 d with distance d, and the radius of the cell is set to 500 m. The drain efficiency of the power amplifier ζ is set to 8% in our simulation whilst the energy harvesting efficiency is set to η = 50%. The power udget for each BS is considered to e 46 dbm and the circuit power P C is 40 dbm. The minimum harvesting energy E k is set to 10% of the imum transmit power. In the first simulation, the performance of the proposed extended BC-MAC duality-ased EE imization algorithm is studied. The convergence ehavior of the proposed Dinkelach method-ased scheme is first evaluated y illustrating how the EE performance ehaves with the numer of iterations. As shown in Fig. 2, the proposed Dinkelach method-ased iterative resource allocation scheme converges to the optimal value. In addition, the convergence ehavior of the proposed upper ound su-gradient resource allocation algorithm is also studied. Fig. plots the EE versus the numer of iterations for step sizes 0.1 and As can e seen from the figure, the proposed extended BC-MAC duality-ased EE imization algorithm converges to a stale value, and the step size affects the accuracy and convergence speed of the algorithm. In the next simulation, the proposed extended BC-MAC duality-ased EE imization algorithm under different imum transmit power allowance is evaluated and presented in Fig. 4. To show the EE gain achieved y TS-ased SWIPT system, we compare our proposed scheme with the scheme that imize the EE without EH [14] and the scheme that aims for imizing the system sum rate [15]. It is oserved that the EE achieved y our proposed extended BC-MAC duality-ased EE imization algorithm is monotonically non-decreasing with respect to the imum transmit power allowance P. Particularly, the EE is increasing dramatically with an increasing imum transmit power allowance P in the higher transmit power constraint region, i.e., P > 25 dbm. This is ecause in the higher transmit power constraint region, a alance etween the system EE and the total power consumption can e achieved. On the other hand, all the algorithms achieve similar performance in terms of the system EE criterion in the lower transmit power constraint region, i.e., 5 < P < 15 dbm. Besides, due to the fact that the received power of the desired signal may not e sufficiently large for delivering information and energy harvesting at the same time, the system with TS-ased energy harvesting receivers achieves a small performance gain compared to the system without energy harvesting receivers. Nevertheless, in the region of higher transmit power, the proposed extended BC-MAC duality-ased EE imization algorithm outperforms the other two schemes sustantially. In particular, there is aout a 5% gain can e achieved y our proposed extended BC-MAC duality-ased EE imization algorithm compared to the scheme that without energy harvesting receivers [14].

6 6 EE (/Hz/J) EE imization with BC-MAC duality EE imization without EH [14] SE imization without EH [15] Transmit Power Constraint (dbm) Fig. 4. The performance of the proposed extended BC-MAC duality-ased EE imization algorithm. EE (/Hz/J) E k 0.1P T E k 0.05P T E k 0.01P T Transmit Power Constraint (dbm) Fig. 5. Energy efficiency versus the imum transmit power allowance for the proposed extended BC-MAC duality-ased EE imization algorithm. Furthermore, due to the fact that the increasing sum rate of the system cannot offset the consumption of the transmit power, the sum rate imization scheme without energy harvesting [15] achieves a very low EE. Finally, we investigate the system EE versus the imum transmit power allowance for the proposed extended BC-MAC duality-ased EE imization algorithm with different level of minimum required harvested energy. As shown in Fig. 5, the increasing level of minimum required harvested energy will not always lead to an increasing system EE. Furthermore, jointly considering the results in Fig. 4 and Fig. 5, we can conclude that there exists an optimal minimum required harvested energy value for the EE optimization prolem. Hence, the performance of EE can e further improved if the TS ratios and the minimum required harvested energy are jointly considered, and that would e investigated in our future works. VII. CONCLUSIONS In this paper, we addressed the EE optimization prolem for SWIPT-ased MIMO-BC with TS receiver. The corresponding EE imization prolem from the coupling of the optimization variales, namely the transmit covariance matrices and TS ratios, is non-convex. Hence, to tackle the prolem, we transform the original imization prolem with multiple constraints into a min- prolem with single constraint and multiple auxiliary variales. For the min- prolem with single constraint, a dual-layer resource allocation strategy is proposed. We incorporate an extended SWIPT-ased BC- MAC duality principle in order to simplify the inner-layer prolem, and accordingly provide an iterative resource allocation algorithm for solving the dual MAC prolem with fixed auxiliary variales. A su-gradient-ased searching scheme is then proposed to otain the optimal auxiliary variales in the outer-layer. Numerical results validate the effectiveness of the proposed algorithms and show that significant performance gain in terms of EE can e achieved y our proposed extended BC-MAC duality-ased EE imization algorithm. REFERENCES [1] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, Wireless networks with RF energy harvesting: A contemporary survey, Commun. Surveys Tuts., vol. 17, no. 2, pp , 2nd Quart [2] I. Krikidis, S. Timotheou, S. Nikolaou, G. Zheng, D. W. K. Ng, and R. Schoer, Simultaneous wireless information and power transfer in modern communication systems, IEEE Commun. Mag., vol. 52, no. 11, pp , Nov [] Z. Xiang and M. 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