Energy efficient optimisation for large-scale multiple-antenna system with WPT

: In this study, an energy-efficient optimisation scheme for a large-scale multiple-antenna system with wireless power transfer (WPT) is presented. In the considered system, the user is charged by a base station with a large number of antennas via downlink WPT and then utilises the received power to carry out uplink data transmission. Novel antenna selection, time allocation and power allocation schemes are presented to optimise the energy efficiency of the overall system. In addition, the authors also consider channel state information cannot be perfectly obtained when designing the resource allocation schemes. The non-linear fractional programming-based algorithm is utilised to address the formulated problem. Their proposed schemes are validated by extensive simulations and it shows superior performance over the existing schemes.


Background
The high data rate wireless network has brought the explosive growth of intelligent mobile devices and applications, which has greatly enriched our daily life. However, frequent online activities and the high data rate transmission bring a higher requirement on the battery of the mobile terminals (MTs) while the development of battery technique is in a relevantly slow speed. Such a mismatch leads to a increasingly interest on how to prolong the lifetime of MTs, especially for those MTs that require continuous operations and have the difficulty to replace the battery.
One way of prolonging the lifetime of an MT is to provide energy supply whenever needed. However, in some cases, recharging and replacing the batteries of the MTs may be inconvenient or even impossible. In this context, energy harvesting techniques, can be leveraged into energy-constrained wireless networks, to prolong the system lifetime in a sustainable way. Conventional energy harvesting technologies mainly utilise solar, wind and vibration effects or other physical phenomena, which are location dependent and sometimes cannot be provided constantly. Therefore, scavenging energy from radio-frequency (RF) or electromagnetic signals offers an alternate way for energy supply [1]. The investigation on the simultaneously wireless information and power transfer (SWIPT), whereby the wireless nodes can harvest energy from RF signals, has received great interests recently and provides traditional energy-constrained networks with a promising solution for the convenient and permanent energy supply. As the RF signal is not location dependent and almost everywhere, the SWIPT has the potential to make a great contribution on prolonging the battery life time and improving the energy efficiency (EE) performance of the wireless system.
Meanwhile, in order to accommodate the high-speed transmission with limit amount of spectrum, the spectrum utilisation should be significantly improved. As one potential solution, employing large number of antennas at the base station (BS) to build up so-called massive multiple-input-multiple-output (massive MIMO) system, can make a great improvement on the spectrum efficiency comparing with the current MIMO system, a large number of extra antennas in the massive MIMO can focus transmit power into ever-smaller regions of space and correspondingly bring huge improvements in spectral efficiency (SE). However, employing a large number of antennas may bring some disadvantages on system complexity and additional energy consumption, which are caused by using a separate RF chain for each antenna [2]. To date, most of the energy-efficient algorithm development of the MIMO system typically focused on the transmit power minimisation, which is reasonable assuming the number of used RF chains is small. However, it is worthwhile to investigate whether some of the antennas can be switched off when the circuit power consumption can be comparable with or even dominates the transmit power [3], and spectrum and transmit power can be allocated accordingly to improve the EE performance in the a large-scale multiple-antenna system.

Related work
Recently, it can be found that the SWIPT system design has received increasing attention, and energy-efficient resource allocation is one of its major research topics. In [4], Zeng and Zhang present a throughput maximum scheme for the SWIPT system. In [5], Chen et al. focus on the problem of distributed power splitting for SWIPT in relay interference channels. Ma et al. [6] investigate the tradeoff between wireless power transfer (WPT) in downlink (DL) and information transfer in uplink (UL) in a SWIPT system. In Huang et al. [7], explore the average throughput performance by proposing novel energy beamforming schemes in a wireless-powered MIMO communication system. In [8], Chang et al. investigate the resource allocation scheme for a mobile cloud powered by the WPTwireless power transfer.
Meanwhile, the investigation of EE performance for a multipleantenna system has also received increasing attentions [2,[9][10][11][12]. In [2], Li et al. have proposed a transmit antenna selection scheme to improve the EE of a massive MIMO system. Considering a MIMO system, Hochwald et al. [9] have explored the mutual information optimisation problem and it can be found that increasing the number of antennas can lead to the SE improvement. On the other hand, though the use of MIMO can improve the system spectrum efficiency, the use of large number of antennas may lead to a significant decrease on the EE performance [2]. However, the corresponding energy consumption is also increased. Therefore, in terms of EE, the number of selected antennas should be decided in an optimised manner. In [10], with the assumption of imperfect channel state information (CSI) estimation, a rate adaptation and antenna selection scheme is introduced. The problem of the transmitting and receiving antenna selection is also investigated in [12] when channel estimation error exists. To improve the EE of the multiple-antenna system, different beamforming methods have been explored in [3,13] as well.
It can be observed that in most of the aforementioned works, CSI is assumed to be obtained perfectly at the BS or MTs. However, such an assumption may be too idealistic, since in practice, the CSI cannot be perfectly known due to the estimation error and/or feedback. Therefore, if the proposed algorithm cannot properly take imperfect CSI into consideration, system performance may be degraded. In [14], a robust beamforming algorithm for the multiple-antenna system with SWIPT is proposed under the assumption of imperfect CSI at the transmitter. In [15], Awad et al. investigate the resource allocation problem for an orthogonal frequency-division multiple access (OFDMA)-based network with diverse quality of service (QoS) requirements and assume imperfect CSI is available at the BS. In [16], the SE performance of an OFDMA multiple relays system is investigated with the assumption of imperfect CSI. So far, as we can observe, the system optimisation of imperfect CSI in WPT system is at its start phrase and typically under-investigated.

Contribution
Motivated by the previous works and existing problems, the main target is to design novel resource allocation algorithms to optimise the EE performance of a large-scale multiple-antenna point-topoint (P2P) system. In particular, we first present an antenna selection scheme to determine the optimal number of antennas to deliver wireless energy to the user and to receive data transmission from the MT. Moreover, with downlink WPT and uplink data transmission, we propose a time allocation scheme to optimise the WPT time duration as well as a power allocation scheme to optimise the transmit power consumption of the BS. In addition, as the CSI cannot be perfectly obtained in the practical system, the influence of imperfect CSI cannot be neglected. Thus, in this work, the impact of imperfect CSI is also considered when designing the algorithm. The contributions can be summarised as follows: • First, a novel resource allocation algorithm is proposed to improve the EE of the considered system with imperfect CSI. The proposed binary searching-based antenna selection scheme is to find the optimal number of used antennas at the BS. Moreover, energy beamforming scheme is also adopted to improve the system EE performance based on imperfect CSI.
• Second, the mutual information distributions with/without (w/wo) antenna selection of the considered system are derived. Accordingly, the expression of spectrum efficiency is derived with the consideration of channel estimation error. • Third, based on the above theoretical analysis, the time and power allocation schemes are proposed, where the time allocation for WPT and wireless information transfer (WIT), and the power allocation at the base station (BS) are jointly optimised. A nonlinear fractional programming-based scheme is then applied to solve the formulated problem.
• Through extensive simulation studies, the proposed schemes are illustrated and validated. The simulation results demonstrate the effectiveness and its superior performance of the proposed schemes.
The remainder of this paper is organised as follows. In Section 2, the system model and corresponding studies on mutual information distribution of the massive MIMO system are introduced. The optimisation problem and resource allocation algorithm are presented in Section 3. Performance evaluations are conducted in Section 4. Finally, we conclude this study in Section 5.

System model
As shown in Fig. 1, we consider a P2P massive MIMO system with bi-directional transmission. The BS has N ≫ 1 antennas and user is equipped with one antenna. In the considered system, the BS is able to provide energy via WPT in the downlink to the user during the scheduled time, and the user has no other energy supplements. The user can store the energy obtained from the BS and then utilise it transmit data to the BS in the uplink. Therefore, in the first time slot, through the WPT, the user can harvest the wireless power from the signal of the BS, and store it in a rechargeable battery. Then, in the second time slot, the user can utilise the collected energy to transmit the data to the BS. The entire transmission block is denoted as T, the first time slot is denoted as τ and the second time slot is represented with T − τ.
Furthermore, we assume the channel is quasi-static block fading, i.e. given transmission block T, the channel is constant and it can vary independently from one block to another. In each transmission block, the user can get the channel estimation via a channel estimator, and the estimated channel coefficient is H. The estimation error is △H and the CSI can be feedback to the BS. We denote σ e 2 as the variance of estimation error, i.e. △H ∼ CN(0, σ e 2 I e ), where I e is the identity matrix. Thus, denoting the channel coefficient as H, we have [15,16] To improve the efficiency of WPT, the energy beamforming is employed, where we assume that the RF chain architecture has no impact on the energy model. To maximise the system EE, an antenna selection scheme is presented at the BS, to select L antennas from N antennas, where 1 < L < N. Accordingly, the received power at the user can be given as follows: where P t is the transmit power of the BS and η is the conversion efficiency from harvested energy into electric energy stored by the user. Here, d is the distance-depend path loss between the BS and user. When L out of N antennas are selected, the estimated channel becomes Ĥ ∈ ℂ L × 1 . w is the energy beamforming matrix, and we consider the maximum ratio transmission [3] for the design of w as w = Ĥ / ∥ Ĥ ∥. Therefore, the obtained energy can be given as follows [17]: where Q = (σ e 2 /(1 + σ e 2 )) + ( ∥ H ∥ 2 /((1 + σ e 2 ) 2 )). During the time slot T − τ, the harvested energy can be used to transmit data from the user to the BS. At the BS, the received signal is given as [2] where x ∈ ℂ L × 1 is the information signal and n is additive Gaussian noise. n ∼ CN(0, σ n 2 ). E U /(T − τ) is the transmit power of the user.

Mutual information distribution under imperfect channel estimation
In a large-scale multiple-antenna system, the channel-hardening effect emerges as the number of antennas grows [9]. Thus, at first, we explore the mutual information distributions w/wo antenna selection with imperfect CSI. On the basis of the analytic results, we can then obtain the presentation of the expected throughput. We first derive Theorem 1 about the mutual information distribution of the considered system without antenna selection. Theorem 1: When the BS is with N ≫ 1 antennas, a numerical approximation of the mutual information in the uplink of the considered system with the imperfect CSI is given as where N represents the standard normal distribution and σ n 2 is the noise variance. Proof: The proof of Theorem 1 is similar to Appendix A of [18], so we omit here.□ Theorem 1 presents the distribution of mutual information when considering N antennas. On the basis of Theorem 1, we can obtain the expression of mutual information when L antennas are selected out of N in Theorem 2. Theorem 2: In the considered system, when L antennas are selected, the mutual information distribution is given as follows: where ρ is the signal-to-noise ratio with imperfect CSI and channel noise, i.e. ρ = (E U d/(T − τ))σ n 2 + (E U d/(T − τ))σ e 2 . ℱN is the folded normal distribution. Proof: The proof of Theorem 2 is similar to Appendix B of [18], so we omit here.□ As one can observe, if L = N, the expected value of the distribution is as same as that of the system without antenna selection, and the variance is approximately the same as well. Therefore, selecting a relatively big number of antennas in a massive MIMO system does not necessarily affect the channel hardening phenomenon. Thus, in each time block, the expected channel capacity under imperfect channel estimation is denoted by 3 Energy-efficient antenna selection and resource allocation

Problem formulation
After obtaining the mutual information of the considered system, the EE in [bits/J/Hz] is defined as U(P t , τ, L) denotes the total energy consumption in a time block T and it can be expressed as where P c is the circuit power consumption of the BS and the user. Thus, the power consumption P c after antenna selection is [2,19] P c ≃ P user + LP bs , where P user is the power consumption of the user and P bs denotes the power consumption related to the RF chain of antenna. Since the antennas of the BS need to be active for the whole time block T, the total power consumption in (9) can be rewritten as U(P t , τ, L) = (P user + LP bs )T + P t τ .
Therefore, the EE can be expressed as From (3), (7) and (12), the EE can be given as follows: (see (13)) To maximise the EE, the optimisation problem can be presented as follows: s.t.

C1: P t ≤ P bs, max
where P bs, max denotes the maximum of transmit power of BS and P user, max denotes the maximum of transmit power of the user. In (15), C1 is the BS transmit power constraint. C2 can ensure that QoS R min is satisfied. C3 is the WPT time allocation constraint and C4 is the transmit power constraint for the user. C5 is to ensure that the number of selected antennas is no bigger than N. We can substitute (3) into C4, and arrive at After combining C1 and (16), one can arrive τ ≤ P user, max T (ηdP bs, max Q + P user, max ) = τ max .

Proposed antenna selection scheme analysis
Before we propose to address the formulated resource allocation problem, we first determine the number of used antenna. A binary search-based scheme is applied here for antenna selection, and presented in Algorithm 1 (see Fig. 2). In this algorithm, we initialise three variables: the upper bound κ h , intermediate κ i and lower bound of the number of antennas κ l , respectively. We use κ i = (κ l + κ h )/2. At the beginning, we consider κ l = 1 and κ h = N. In each iteration, two values, i.e. Σ(κ i ) and Σ(κ i + 1) should be compared. Then, we can find which subset of the maximum value is located. If Σ(κ i ) < Σ(κ i + 1), κ i + 1 is assigned to κ l , and if Σ(κ i ) > Σ(κ i + 1), κ i is assigned to κ h . Consequently, by selecting the optimal number of antennas, the maximum EE can be obtained. The value of κ i is updated at the end Σ(P t , τ, L) = log 2 1 + 1 + ln(N /L) ((E U d/(T − τ))/(σ n 2 + (E U d/(T − τ))σ e 2 )L (T − τ) (P user + LP bs )T + P t τ .
of each iteration. When κ h − κ l = 1, the algorithm ends and optimal L can be found.

Proposed time and power allocation schemes
As we can see, the objective function in P 1 is the ratio of E[I] im (T − τ) to U(P t , τ, L), resulting in P 1 a non-linear fractional problem, which is generally not convex. However, these types of problems can be transformed into non-fractional format. According to Dinkelbach [20] and Zhang et al. [21], we can convert it into a subtractive form. First, given L antennas are selected, we consider q * as the global optimal solution of the EE, i.e.
where P * is the optimal solution for power allocation and τ * is the optimal solution for time allocation. Theorem 3 gives the necessary and sufficient condition for obtaining optimal q. Theorem 3: Optimal q can be reached if and only if (iff) [20] max P t , τ, L Accordingly, the problem P 1 can be transformed into a problem P 2 s.t.

C1
: P t ≤ P bs, max where We can see that Ω(P t , τ) is a concave function with respect to P t and τ. Therefore, the original problem can be transformed to a convex optimisation problem P 2 with constraints. Consequently, it can be solved in dual domain. The Lagrange dual function of P 2 is where α > 0, β > 0, μ > 0, φ > 0 are the Lagrange multipliers associated with the constraint in (21). Correspondingly, the dual problem is presented as follows: Then optimal P t * and τ * can be obtained via the Karush-Kuhn-Tucker condition and ∂ℒ(P t , τ, α, β, μ, φ) ∂τ where A, B, C, D, A 1 , B 1 , C 1 , D 1 are given as From (25), we can obtain Let As a result, we can obtain P t * as where a is iteration index, [x] + = max{0, x}, △α, △β, △μ, △φ are the step sizes. The proposed power and time allocation algorithm is summarised in Algorithm 2 (see Fig. 3) together with the proposed antenna selection scheme. If the convergence is not reached, the loop continues to find the optimal solution. During the loop, the dual variables are updated according to (30). The power allocation and time allocation solutions are obtained by (26) and (29). If E[I] im (T − τ′) − q[(P user + LP bs )T + P t ′τ′] ≤ ε where ε is a sufficiently small number, we can consider the convergence condition is reached and optimal solution can be obtained. As the output of Algorithm 2 (Fig. 3), the optimal solutions for power and time allocation and antenna selection can be achieved. To address the formulated problem P 1 , the problem has a linear time complexity where c is a constant. The time complexity of antenna selection algorithm is O (log 2 N). Therefore, the overall problem has a time complexisity O(c × log 2 N) where c is a constant.

Performance evaluation
In this section, extensive simulations have been conducted to evaluate the proposed algorithm. In Table 1, the key parameters based on the ones in [19] are given. In this table, we consider P DAC and P ADC denote the power consumption of the digital-to-analogue converter (DAC) and analogue-to-digital converter (ADC), respectively. P syn , P filt , P mix , P LNA , P IFA , and P filr , are the power consumption of the frequency synthesiser, transmit filter, the mixer, the low noise amplifier, the frequency amplifier, the receiver filter, respectively. Furthermore, we have P user = 2P syn + P LNA + P mix + P IFA + P filr + P ADC and P bs = P DAC + P mix + P filt .
In Fig. 4, the impact of imperfect CSI on the EE performance is illustrated. In this figure, we vary the variance of σ e 2 , which is the estimation error and also change the distance between the BS and user. In addition, the EE performance of the proposed scheme is compared with that of the system with perfect CSI, and we set σ e 2 = 0.3 and σ e 2 = 0.5. The simulation results show that the system with perfect CSI shows superior performance over the ones with imperfect CSI and the EE performance degrades when the value of σ e 2 increases. Moreover, when the distance becomes larger, the EE performance decreases as well due to the channel degradation. For example, when the distance is 50 m, EE of the system when σ e 2 = 0.5 is about four times lower comparing with the one of perfect CSI case. However, when the distance becomes 100 m, there is about only three times difference. From Fig. 4, the EE of the system with σ e 2 = 0.3 is higher than that of σ e 2 = 0.5, which evidences the significant impact that the imperfect CSI has on the EE.
In Fig. 5, the effectiveness of the proposed antenna selection and time allocation schemes are evaluated. We compare our proposed schemes with the one without antenna selection, e.g. the one that is modified from Chen et al. [3], to investigate the advantages of the presented antenna selection method. It can be observed from Fig. 5 that the system EE can be improved by selecting optimal number of antennas. The performance increases eight times higher with the proposed antenna selection scheme. Meanwhile, our proposed scheme is also compared with the one with equal time allocation, i.e. τ = 0.5. As one can see, the proposed scheme shows a superior performance, which evidences that proper design of time allocation is needed for a SWIPT system. Similar to the observation in Fig. 4, it is shown that the EE of the system decreases with the increase of the distance between the BS and user.
To examine the impact of number of selected antennas, we plot the EE performance in Fig. 6 by varying the number of antennas on the x-axis. The optimal time allocation scheme is considered in this case. In addition, we also change the value of transmit power P t . It shows that the EE of the considered system first increases and then decreases after reaching the maximum as the increase of the number of antennas, regardless of the value of transmit power level. Therefore, the number of used antenna needs to be optimised to enhance the system performance. Moreover, the results also reveal that the optimal antenna number is different for different values of transmit power. For example, the optimal L * = 45 for the case that when transmit power is 12 W. However, when transmit  power is 20 W, we can see L * = 35. Meanwhile, it can be found that when power allocation (transmit power is 15 W) is optimal, the EE performance is better comparing with the case with bigger transmit power, which confirms the advantages of using the power allocation algorithm. Fig. 7 demonstrates the influence of time allocation on the EE performance and presents the EE performance with different transmit powers P t . The proposed antenna selection scheme is considered here. The result shows that there is an optimal value of first time slot τ for certain transmit power to maximise the EE. In general, with the increase of time τ, the system EE first increases, then reaches its optimal value and finally decreases. Such phenomenon can be observed for the cases with different values of transmit power. In addition, it can be seen that the optimal EE is different when different transmit power is advocated, which further evidences the effectiveness of the proposed time and power allocation scheme. For example, when P t = 16 W, the optimal EE is higher than the one when P t = 12 W and P t = 18 W. Furthermore, the proposed scheme is also compared with the one without antenna selection [3] which is marked as 'traditional scheme'. As we can see, our proposed scheme with different transmit power outperforms the traditional scheme, which confirms the necessity of using antenna selection to improve the EE. Fig. 8 plots the EE by changing the transmit power P t and allocated time slot τ. The performance of the proposed scheme with optimal time allocation is compared with the one of traditional method with optimal time allocation, and the one of the proposed scheme with equal time allocation, e.g. τ = 1/2 T. By the comparison among these three curves, we can observe that with the increase of transmit power, the EE of the system first ascends and then descends. Similar to the results in Fig. 6, Fig. 8 shows that the transmit power has an optimal value, which confirms the advantages of power allocation scheme. Moreover, the proposed time allocation scheme can obtain additional EE gain when comparing with the equal time allocation scheme. We can also observe that the EE of our proposed scheme with antenna selection is the highest among all three, which validates the effectiveness of the presented schemes.
In Fig. 9, the impacts of number of selected antenna L and estimation error σ e 2 are illustrated. In this figure, the x-axis is the number of antennas, the y-axis is the estimation error and the z-axis is the EE. From Fig. 9, we can observe that the EE first increases with the increased number of antennas and then decreases after reaching its optimum. While we can also see that the increment of the estimation error can also deteriorate the performance of the EE, which is similar to Fig. 4. Finally, in Fig. 10, the convergence performance of our proposed Algorithm 2 (Fig. 3) is presented. We also plot the EE performance of the proposed power allocation and antenna selection schemes considering an equal time allocation scheme, and the performance of the presented time allocation and antenna selection schemes with a fixed power allocation. The results demonstrate that the proposed scheme is able to converge to a stable stage within a few iterations, which evidences the convergence of the proposed scheme. We can also see that the proposed scheme can outperform the other two schemes in terms of EE and the one with equal time allocation has the worst performance.

Conclusion
In this work, an energy-efficient optimisation scheme for a largescale multiple antennas P2P system with WPT is presented. Novel antenna selection, time allocation and power allocation schemes are presented to optimise the EE. We also take the imperfect CSI into account when designing the resource allocation schemes. Through extensive simulations, the effectivenesses of our proposed schemes are validated and the impact of optimisation variables on the system EE is also evaluated.