Power efficiency optimisation of wireless‐powered full‐duplex relay systems
2018; Institution of Engineering and Technology; Volume: 12; Issue: 5 Linguagem: Inglês
10.1049/iet-com.2017.0384
ISSN1751-8636
Autores Tópico(s)Advanced MIMO Systems Optimization
ResumoIET CommunicationsVolume 12, Issue 5 p. 603-611 Research ArticleFree Access Power efficiency optimisation of wireless-powered full-duplex relay systems Ruirui Chen, Corresponding Author Ruirui Chen rrchen@stu.xidian.edu.cn State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorHailin Zhang, Hailin Zhang State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this author Ruirui Chen, Corresponding Author Ruirui Chen rrchen@stu.xidian.edu.cn State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this authorHailin Zhang, Hailin Zhang State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an, People's Republic of ChinaSearch for more papers by this author First published: 08 March 2018 https://doi.org/10.1049/iet-com.2017.0384Citations: 8AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, the authors study the power efficiency optimisation in wireless-powered full-duplex relay (WFR) systems, where the entire transmission process can be partitioned into wireless power transfer phase (WP) and full-duplex information transmission phase (FP). For WFR systems with same source transmit power, where the source transmit powers of WP and FP are same, they first prove that the power efficiency is a strictly concave function over the time-switching factor, and an optimal time allocation scheme is proposed to maximise the power efficiency. Then, for the given time-switching protocol, the optimal power allocation strategy is obtained by analysing the derivative of the power efficiency with respect to the transmit power. Finally, they propose the joint power and time allocation scheme. For WFR systems with different source transmit power, where the source transmit powers of WP and FP are different, the optimal time allocation scheme and optimal power allocation strategy are derived by studying the derivative of the power efficiency. Furthermore, the joint power and time allocation scheme is proposed to maximise the power efficiency. Simulation results are presented to validate the authors' proposed schemes for the WFR systems. 1 Introduction Wireless power transfer has been envisioned as a promising technique to provide perpetual power supply for energy-constrained systems [1, 2], which are generally battery powered and have a limited operational lifetime. Since radio-frequency (RF) signal can carry both information and energy, time-switching and power-splitting protocols were proposed in [3] to achieve simultaneous wireless information and power transfer. Due to the easy implementation of half-duplex transmission, many wireless-powered relay systems, which can use wireless power transfer to charge the relay, employed the half-duplex protocol to complete the information transmission from source to destination [4–6]. Through simultaneous transmission and reception using the same frequency band at the same time, full-duplex relay can significantly improve the spectrum efficiency as compared with half-duplex relay [7–11]. Motivated by the advantages of wireless power transfer and full-duplex transmission, some recent works have been devoted to the performance analysis of wireless-powered full-duplex relay (WFR) systems [12–17]. The time-switching protocol based WFR systems were investigated in [13]. The authors of [14] proposed a two-phase protocol for amplify-and-forward (AF)-based WFR systems. Particularly, the energy harvested from the self-interference link was taken into account [14]. The authors in [15] studied the decode-and-forward (DF)-based WFR systems, where there are two relays and the inter-relay interference serves as an additional source of energy to the relay. The throughput of the two-hop WFR systems with multi-antenna relay was analysed in [16]. By jointly optimising the power-splitting factor, energy consumption proportion, and transmit beamformer of the relay, the authors of [17] maximised the throughput of the power-splitting protocol-based WFR systems. Due to the increasing energy cost and awareness of environmental protection, power efficiency has been an important performance metric in wireless communications [18–22], especially for WFR systems. However, to the best of our knowledge, there exists few researches about power efficiency analysis of WFR systems. Therefore, in this paper, our objective is to maximise the power efficiency of WFR systems by analysing time and power allocations. For the WFR systems with same source transmit power, since the power efficiency is a strictly concave function over time-switching factor, a time allocation scheme is first proposed to maximise the power efficiency. Then, through deriving the derivative of the power efficiency with respect to transmit power, we give the power allocation strategy to obtain the optimal transmit power under the given time-switching factor. Finally, the joint power and time allocation scheme is obtained. For the WFR systems with different source transmit power, we propose the optimal time allocation scheme and optimal power allocation strategy by analysing the derivative of the power efficiency. Moreover, the joint power and time allocation scheme is obtained to maximise the power efficiency. The remaining of this paper is organised as follows. Section 2 describes the system model. In Section 3, for the WFR systems with same source transmit power, we first propose the optimal time allocation scheme. Then, for given time-switching protocol, an optimal power allocation scheme is presented. Finally, a joint power and time allocation scheme is derived to maximise the power efficiency of WFR systems. In Section 4, for the WFR systems with different source transmit power, by analysing the derivative of the power efficiency, we obtain the optimal time allocation scheme and optimal power allocation strategy. Then, the joint power and time allocation scheme is proposed to maximise the power efficiency. Section 5 conducts the simulation results to demonstrate the performance of our proposed resource allocation schemes for WFR systems. The paper concludes with Section 6. 2 System model As shown in Fig. 1, the WFR system model, which consists of one source S, one relay R, and one destination D, is considered in this paper. R is equipped with two separate antennas. One antenna is used for transmission and the other antenna is utilised for reception. S and D both have a single antenna. The power gains of the channels from S to R and from R to D are denoted by and , respectively, where and are the channel responses corresponding to the links from S to R and from R to D, respectively. There does not exist the direct link from S to D. All channel power gains follow the stationary block fading model, where they remain unchanged during the frame duration T, but vary independently across different time frames. The additive noise is the independent and circularly symmetric complex white Gaussian noise with zero mean and variance . Figure 1Open in figure viewerPowerPoint Wireless-powered full-duplex relay system model R employs decode-and-forward protocol to assist the information transmission from S to D. In addition, we assume that R has no external power supply, and is powered through wireless power transfer from S [13]. The time-switching protocol is applied to R. Therefore, the entire transmission process can be partitioned into wireless power transfer phase (WP) and full-duplex information transmission phase (FP). As depicted in Fig. 2, fraction of the frame duration T is used for energy harvesting at R. The remaining time is devoted for full-duplex information transmission from S to D. The source transmit powers of WP and FP are denoted by and , respectively. Figure 2Open in figure viewerPowerPoint Time-switching protocol for wireless-powered full-duplex relay systems During the first phase, the energy harvested by R can be expressed as follows: (1) where denotes the energy conversion efficiency. The energy harvested from the noise is small and thus ignored. As in [5], we assume that the energy harvested by R during the first phase is stored in a supercapacitor, and then fully consumed by R to complete the information transmission in the second phase. Thus, the relay transmit power, denoted by , can be derived as follows: (2) In the second phase, with the full-duplex DF protocol, the signal-to-noise ratio (SNR) of the system can be written as [10] (3) where and denote the channel-to-noise ratios at R and D, respectively; is defined as the cancellation coefficient to characterise the effect of self-interference on full-duplex information transmission. When approaches 0, the self-interference has large interference on full-duplex link from S to D. When approaches 1, the self-interference has little interference on full-duplex information transmission. Then, the spectrum efficiency of WFR systems can be expressed as follows [16]: (4) Due to the limitation of power resource and the requirement of green communications, power efficiency has been an important performance metric in wireless communications [18–22], especially for WFR systems. Similarly to [21, 22], we can model the power efficiency of WFR systems as (5) where is the reciprocal of the power amplifier efficiency which varies in ; denotes the constant circuit power consumption, which is independent of the information transmission. 3 WFR systems with same source transmit power In this section, we will study the WFR systems with same source transmit power. Furthermore, it is assumed that . To maximise the power efficiency, we first propose the optimal time allocation scheme. Then, an optimal power allocation scheme is obtained for given time-switching protocol. Finally, we propose a joint power and time allocation scheme. 3.1 Optimal time allocation scheme Under the assumption of , the power efficiency optimisation problem can be formulated as (6a) (6b) The SNR of WFR systems is determined by the minimum value between the received SNRs at R and D. When the received SNR at R equals the received SNR at D, i.e. , we can derive the switching threshold, which is defined as follows: (7) Thus, problem can be solved by two cases: (i) and (ii) . For case (i), problem can be simplified as (8a) (8b) Taking the derivative of with respect to , we can obtain (9) Because , is less than zero in the region . Power efficiency is linear with and monotonically decreases as increases in . When is equal to , power efficiency will reach the maximum value in the range of . Therefore, the optimal time-switching factor is for problem . For case (ii), we can convert problem to problem , which can be expressed as follows: (10a) (10b) Then, we can obtain the derivative of with respect to as follows: (11) Note that when . Taking the second-order derivative of with respect to , we can have (12) Because in , the objective function of problem is a strictly concave function of in . Furthermore, for problem , is linear with in . Thus, the objective function of problem is a strictly concave function over in . Note that is the optimal time-switching factor of problem and also belongs to . The objective function of problem is a strictly concave function of in . Therefore, we can draw a conclusion that the optimal time-switching factor of problem belongs to the region . The optimal solution of problem is the optimal solution of problem . Let (11) be equal to zero, we can obtain the solution . Since , is larger than zero. If , and is the optimal time-switching factor . Otherwise, and is . Thus, the following time allocation scheme is proposed to obtain (see Fig. 3). Figure 3Open in figure viewerPowerPoint Time allocation scheme In this paper, the main computational complexity of the proposed scheme is , i.e. linear with the iteration number N. For the optimal time allocation scheme, N is a variable associated with the selection of initial point and maximum tolerance . 3.2 Optimal power allocation scheme Now, we will give the optimal power allocation scheme to maximise the power efficiency. For any , the optimal time-switching factor belongs to the region . Thus, we only analyse the case where . The optimisation problem can be formulated as follows: (13a) (13b) where denotes the maximum transmit power. Then, taking the derivative of with respect to , we can obtain (14) Because the denominator of (14) is larger than zero, judging the sign of (14) can be determined by just evaluating the sign of the numerator. For simplicity, we define the numerator of (14) as (15) Taking the derivative of with respect to , we can have (16) Clearly, in the region , is less than zero. Thus, is a monotonically decreasing function of in the region . Note that . If there exists the value that satisfies , will be a monotonically increasing function of in the region , and then will monotonically decrease as increases in the range of . is the optimal solution . Otherwise, will be the optimal solution because is a monotonically increasing function of in the whole range of . Therefore, the following power allocation scheme is proposed to obtain the optimal transmit power (see Fig. 4). Figure 4Open in figure viewerPowerPoint Power allocation scheme The main computational complexity of the proposed power allocation scheme can be denoted by . The iteration number is a constant . 3.3 Joint power and time allocation scheme We first attempt to obtain an optimal joint power and time allocation scheme. The optimisation problem can be formulated as follows: (17a) (17b) (17c) If the objective function in (17a) is concave on the space spanned by , the optimal joint power and time allocation scheme can be obtained. The Hessian matrix of , denoted by , can be written as follows: (18) If and , in (17a) will be concave [23]. We can derive as (19) where can be expressed as (20) Since the item and the denominator of are both larger than zero, the sign of can be determined by just judging the sign of . Then, we derive the first- and second-order derivatives of with respect to as follows: (21) and (22) Clearly, is larger than zero under the constraints of problem . Based on and , we can draw a conclusion that will be larger than zero for relatively large transmit power, which can be defined as . Correspondingly, will be larger than zero when is larger than . Therefore, the objective function of problem will not be concave when the transmit power is relatively large. We can not obtain the closed-form expression for the exact . Furthermore, the expression of is extremely complicated to determine the sign. In other words, it is difficult to judge whether the objective function of is concave. Based on the above analysis, it is hard to obtain the optimal joint power and time allocation scheme. Consequently, the following joint power and time allocation scheme is proposed to find the approximate optimal solution of (see Fig. 5). Figure 5Open in figure viewerPowerPoint Joint power and time allocation scheme Note that the main computational complexity of the proposed joint power and time allocation scheme is , which consists of two parts: the outer iteration number and the inner iteration number (i.e. the iteration number of time allocation scheme). Generally, the computational complexity of joint power and time allocation scheme will be larger than those of the proposed time and power allocation schemes. 4 WFR systems with different source transmit power In this section, to further improve the power efficiency, we will consider a more general WFR systems, where the source transmit powers of the two phases can be different (i.e. ). Similarly to the previous section, the optimal time allocation scheme is first obtained. Then, we propose an optimal power allocation scheme under the total transmit power constraint. Finally, a joint power and time allocation scheme is obtained to maximise the power efficiency. 4.1 Optimal time allocation scheme For the WFR systems with , we can formulate the power efficiency optimisation problem as follows: (23a) (23b) We can obtain the switching threshold (24) when the received SNR at R is equal to the received SNR at D, i.e. . Then, we can solve the problem by analysing two cases: (i) and (ii) . For case (i), problem can be expressed as (25a) (25b) We can obtain the derivative of with respect to as follows: (26) Obviously, is less than zero in the region . Therefore, power efficiency will monotonically decrease as increases in . For problem , power efficiency will reach the maximum value when . For case (ii), problem can be converted to the problem . (27a) (27b) Then, taking the derivative of with respect to , we can obtain (28) Note that the denominator of (28) is larger than zero, the sign of (28) only depends on the sign of the numerator. The numerator of (28) can be defined as follows: (29) Taking the derivative of with respect to , we can obtain (30) Obviously, is less than zero in the region . is larger than zero. If there does not exist the value that satisfies , is the optimal time-switching factor of problem because will monotonically increase as increases in . Otherwise, will be a monotonically increasing function of in the region , and then will monotonically decrease as increases in the range of . is the optimal solution . The following time allocation scheme is proposed to obtain (see Fig. 6). Figure 6Open in figure viewerPowerPoint Time allocation scheme We can study the main computational complexity of the proposed time allocation scheme by . The iteration number is a constant . 4.2 Optimal power allocation scheme under total transmit power constraint For the WFR systems with same source transmit power, the total transmit power is in one frame duration T. Thus, for the WFR systems with different source transmit power, we will study the optimal power allocation scheme under the total transmit power constraint . Then, the power efficiency optimisation problem can be formulated as (31a) (31b) (31c) Based on (31b), we can obtain (32) Problem can be converted to problem by using (32). (33a) (33b) The switching threshold is (34) Thus, problem can be solved by analysing two cases: (i) and (ii) . For case (i), based on (34), it can be derived as follows: (35) Moreover, the objective function of problem is (36) Obviously, is a monotonically increasing function of for . As for case (ii), based on (34), we can obtain . The objective function of problem is (37) Note that will monotonically decrease as increases for the case where . Thus, the optimal is (38) Based on (38), we can convert problem to the following problem: (39a) (39b) Then, taking the derivative of with respect to , we can obtain (40) Since the denominator of (40) is larger than zero, we can determine the sign of (40) by judging the sign of the numerator. The numerator of (40) can be defined as (41) Taking the derivative of with respect to , we can have (42) Note that in and . If there does not exist the value which satisfies , will be the optimal solution because monotonically increases as increases in . Otherwise, is the optimal solution. This is because will be a monotonically increasing function of in , and then will monotonically decrease as increases in . The following power allocation scheme is proposed to obtain the optimal powers , and (see Fig. 7). Figure 7Open in figure viewerPowerPoint Power allocation scheme The main computational complexity of the proposed power allocation scheme is . The iteration number is a constant . 4.3 Joint power and time allocation scheme For the WFR systems with different source transmit power, an optimal joint power and time allocation scheme is first attempted to study. We can formulate the optimisation problem as follows: (43a) (43b) (43c) (43d) Based on (38) and (43b), the problem can be converted to the following problem: (44a) (44b) (44c) The optimal joint power and time allocation scheme can be obtained when the objective function in (44a) is concave on the space spanned by . We can derive the Hessian matrix of , denoted by , which can be expressed as follows: (45) Based on convex optimisation [23], when and , in (44a) will be concave. can be derived as (46) where can be expressed as (47) Because the denominator of and the item are both larger than zero, we can determine the sign of by judging the sign of . The first- and second-order derivatives of with respect to can be expressed as (48) (49) Note that is larger than zero. Based on and , we can draw a conclusion that will be larger than zero for relatively large total transmit power, which can be denoted by . For , will be larger than zero. The objective function of problem is not concave if the total transmit power is relatively large. The closed-form expression for the exact can not be obtained. Moreover, the expression of is difficult to determine the sign. That is, it is difficult to determine whether the objective function of is concave. Based on the above analysis, it is hard to obtain the optimal joint power and time allocation scheme. Therefore, the following joint power and time allocation scheme is presented to obtain the approximate optimal solution () (see Fig. 8). Figure 8Open in figure viewerPowerPoint Joint power and time allocation scheme We can analyse the main computational complexity of the proposed joint power and time allocation scheme by , which contains two parts: the outer iteration number and the inner iteration number (i.e. the iteration number of power allocation scheme). In general, the computational complexity of joint power and time allocation scheme will be larger than those of the proposed time and power allocation schemes. 5 Simulation results In this section, simulation results are presented to evaluate the proposed schemes for the two WFR systems. Throughout simulations, the channel-to-noise ratios are set to be and . We set the energy conversion efficiency as , the cancellation coefficient as , and the reciprocal of the power amplifier efficiency as . The circuit power consumption and maximum transmit power are and , respectively. 5.1 WFR systems with same source transmit power For the WFR systems with same source transmit power, we can obtain the switching threshold based on (7). Fig. 9a shows the power efficiency versus the time-switching factor. We set the transmit powers as 10, 12 and 14 dBm, respectively. The vertical dash line represents the switching threshold . The pentagram symbol denotes the maximum power efficiency corresponding to the optimal time-switching factor . We can see that the power efficiency will monotonically increase as the time-switching factor increases in , and then power efficiency will be a monotonically decreasing function of the time-switching factor in the region . Note that the optimal time-switching factor is not larger than the switching threshold . This is because the SNR of WFR systems is determined by the minimum value between the received SNRs at R and D. Figure 9Open in figure viewerPowerPoint Performance of proposed time and power allocation schemes for WFR systems with same source transmit power (a) Power efficiency under various time-switching factor, (b) Power efficiency versus the transmit power In Fig. 9b, we plot the power efficiency under various transmit power . The time-switching factors are , and , respectively. The pentagram symbols of these three curves correspond to the maximum power efficiency with the different optimal transmit power . We can find that the power efficiency will be a monotonically increasing function of in the region , and then power efficiency will monotonically decrease as increases in the range of . Since the switching threshold is , the power efficiency with is the largest among the three cases. Fig. 10 evaluates the proposed joint power and time allocation scheme for WFR systems with same source transmit power. The pentagram symbol is the point of the maximum power efficiency. As shown in Fig. 10, the power efficiency will monotonically increase and decrease as the transmit power increases in the regions and , respectively. However, the power efficiency is not concave on the space spanned by . The joint power and time allocation scheme can be used to find the approximate optimal solution . Figure 10Open in figure viewerPowerPoint Power efficiency of WFR systems with same source transmit power 5.2 WFR systems with different source transmit power In Fig. 11a, we plot the power efficiency under various time-switching factor. The transmit powers are set as , and , respectively. The pentagram symbol represents the maximum power efficiency corresponding to the optimal time-switching factor . It can be observed that the power efficiency will be a monotonically increasing function of the time-switching factor in the region , and then power efficiency will monotonically decrease as the time-switching factor increases in . Fig. 11b depicts the variation of power efficiency with the total transmit power . We set the time-switching factor as , and , respectively. The pentagram symbols of these three curves correspond to the maximum power efficiency with the different optimal total transmit power . We can see that the power efficiency will monotonically increase as increases in the range of , and then power efficiency will be a monotonically decreasing function of in the region . Figure 11Open in figure viewerPowerPoint Proposed time and power allocation schemes for WFR systems with different source transmit power (a) Variation of power efficiency with time-switching factor, (b) Power efficiency under various total transmit power In Fig. 12, we evaluate the proposed joint power and time allocation scheme for WFR systems with different source transmit power. The pentagram symbol corresponds to the maximum power efficiency. From Fig. 12, it can be observed that the power efficiency will monotonically increase and decrease as the total transmit power increases in the regions and , respectively. Furthermore, the approximate optimal solution can be obtained by utilising the proposed joint power and time allocation scheme. Fig. 13 compares the power efficiency of the two WFR systems. From Fig. 13, we can see that the power efficiency of WFR systems with different source transmit power is larger than that of WFR systems with same source transmit power. This is because the power allocation between the first and second phases improves the power efficiency of the WFR systems with different source transmit power. Figure 12Open in figure viewerPowerPoint Power efficiency of WFR systems with different source transmit power Figure 13Open in figure viewerPowerPoint Power efficiency comparison of the two WFR systems 6 Conclusions For the WFR systems with same source transmit power, the optimal time allocation scheme was first proposed to maximise the power efficiency. Then, we gave the optimal power allocation strategy to obtain the optimal transmit power under the given time-switching factor. Finally, a joint power and time allocation scheme was presented to maximise the power efficiency. As for the WFR systems with different source transmit power, we obtained the optimal time allocation scheme and optimal power allocation strategy by analysing the derivative of the power efficiency. Then, the joint power and time allocation scheme was proposed to maximise the power efficiency of WFR systems. 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