Energy efficiency resource allocation for amplify and forward relaying system with energy harvesting

Jianfang Xin; Guangjun Liang; Lu Li; Xueli Ni; Lingling Xia; Youhong Feng; Jianfang Xin; Guangjun Liang; Lu Li; Xueli Ni; Lingling Xia; Youhong Feng

doi:10.48130/wpt-0025-0021

With the vigorous advocacy of green communication, research on energy harvesting technology in mobile communication resource allocation has become increasingly urgent. This paper studies the resource allocation problem of multi-relay energy harvesting systems. To improve the diversity gain of the relay system, an improved three-phase energy harvesting relay protocol is proposed. Complete with established non-convex optimization models and transform them into quasi-convex problems through convex optimization methods. Then, by constructing a nonlinear programming model with optimal energy efficiency, parameters such as energy harvesting time slot allocation, subcarrier pairing, and power allocation are jointly optimized. Finally, the Dinkelbach method and Hungarian algorithm were used to iteratively obtain the optimal solution. The Monte Carlo simulation experiment verified the accuracy and fast convergence of the algorithm.

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Introduction

With the widespread recognition of the concept of green communication, radio frequency energy harvesting (RF-EH) technology has gradually become a research hotspot^[1,2]. A hybrid spectrum access mechanism integrating NOMA-based cooperative transmission and beamforming technology was proposed, with the outage probability of secondary users theoretically derived for the proposed mechanism^[3]. A multi-input single-output (MISO) downlink communication system architecture based on Rate Splitting Multiple Access (RSMA) technology is proposed^[4]. RSMA achieves optimal performance by decoding the information intended for the target user while treating the information of other users as noise. The selection of secure relays for device-to-device (D2D) communication in cellular networks was investigated^[5]. By defining the closed form of the relay selection area in the D2D link and its interruption probability, a safe relay selection scheme using social relations as a safety factor for relay-assisted D2D communication systems is proposed.

Buffer-aided relays utilizing finite-size energy buffers to accumulate energy harvested from source signals were studied^[6]. A relay selection scheme that considers the states of data buffer and energy buffer at the same time is proposed, and its performance is theoretically analyzed through the Markov chain model. Two energy and spectrum-efficient transmission strategies were proposed^[7]. For the proposed WPT-NOMA scheme, considering the hybrid continuous interference cancellation (SIC) decoding sequence, it is proven that WPT-NOMA can avoid the error threshold of power failure probability and achieve full diversity gain. Unlike WPT-NOMA, BAC-NOMA has an error threshold for the probability of power failure, and the asymptotic behavior of the error threshold can be proved theoretically. Methods for improving performance indicators, including spectrum efficiency and system capacity, were studied^[8]. A scenario to extend D2D communication technology to vehicular infrastructure was proposed. The performance of the TLE frequency band for vehicular traffic was analyzed and network simulation was performed.

A nonlinear recursive model was proposed for energy harvesting communication networks, enabling effective simulation of the collected energy transmission process in real-world environments^[9]. A new media access control (MAC) protocol was introduced^[10]. Devices in a WLAN with 802.11 IEEE Distributed Coordination Function (DCF) can use the remaining energy to selectively execute energy transmission requests, which can minimize the network throughput. The energy efficiency (EE) maximization problem in RF-EH heterogeneous cellular networks (HetNets) was investigated^[11]. Considering the fairness of users when allocating resources, the problem of maximizing network EE is reduced to a hybrid combination and non-convex optimization problem. The impacts of energy efficiency, delay, and random data arrival on system performance were discussed^[12]. A resource allocation model for wireless mobile edge computing (MEC) scenarios was established and an online computing offloading method proposed based on random channels and Lyapunov theory.

This paper investigates an RF-EH multi-relay cooperative communication network. By adopting the improved hybrid time switching (HTC) protocol, the base station transfers energy to the source node and relays during the energy transmission phase. In the data transmission stage, the source node first sends signals to both relays and the base station simultaneously in the first time slot. In the second time slot, all relays forward the received signals to the base station, while the source node retransmits its first-slot signals. This retransmission mechanism is designed to enhance system reliability by leveraging spatial diversity.

A joint optimization model is formulated to maximize the average EE under practical constraints, considering optimal power allocation, subcarrier pairing, and time slot assignment. The problem is a mixed-integer nonlinear programming (MINLP) issue with high computational complexity, intractable for conventional solvers. To address this, a nonlinear programming model of the objective function is first established, which is linearized via the Dinkelbach method. The Hungarian algorithm is then employed to iteratively tackle the joint resource allocation problem. Finally, an iterative algorithm combining dual decomposition and subgradient methods is used to derive the optimal EE solution. Part of this work was previously presented at the EAI BROADNETS 2020 conference^[13].

The main contributions of this paper are as follows：

(1) A three-phase energy harvesting cooperative communication protocol is proposed, allowing the source node to transmit data twice. This enables relays and the source to harvest sufficient energy for collaboration, ultimately achieving higher system diversity gain.

(2) An optimal EE-oriented joint resource allocation model is established, integrating subcarrier pairing, power allocation, and time slot assignment in a complex scenario while prioritizing energy efficiency maximization.

(3) For the NP-hard problem, an iterative optimization algorithm based on the Dinkelbach method is developed to reduce computational complexity. The Hungarian algorithm ensures rapid convergence to the optimal joint resource allocation solution.

Materials and methods

An HTC protocol was presented for a three-node relay scenario^[14]. The source node and the relay node harvest energy from the HAP in the downlink and then use the collected energy to assist the source node's information transmission. A closed-form expression of the delay-constrained average throughput under the Rayleigh channel condition of the protocol is proposed, and an asymptotic analysis of the approximate throughput under the multi-relay HTC scheme is given. An energy-efficient hybrid energy harvesting system was developed, which harvests energy from solar, vibration, and radio frequency (RF) sources. The distribution problem of RF energy harvesting was further explored^[15]. A resource allocation strategy for drone-assisted edge computing in a wireless power communication network was studied^[16]. By jointly optimizing factors such as power and time, the optimal system performance was achieved.

The resource allocation problem in a wireless power communication network (WPCN) with a hybrid access point (HAP), and two energy harvesting users was investigated^[17]. Using the HTC-based user cooperation protocol, the user close to the HAP uses the collected energy and allocated time to provide relay services for another user. Considering user quality of service (QoS) requirements and fairness, the total throughput of WPCN is maximized. The optimal configuration of parameters for the NOMA-based IoT relay system was explored^[18]. The key indicators, such as EH time and NOMA coefficients are mainly studied. A wireless communication system with a relay was considered^[19]. Aiming at maximizing the network throughput, a new optimization problem is proposed. Solving the optimization problem can jointly optimize the ratio of transmission energy and signal power over time. A hybrid relay scheme was proposed, realized by a self-sustaining intelligent reflector (IRS) in a WPCN^[20]. For TS and power allocation (PS) schemes, the problem of maximizing the sum is proposed by jointly optimizing the phase shift of IRS and network resource allocation. Relay selection schemes in energy harvesting wireless sensor networks were discussed^[21]. To take into account the requirements of transmission delay and energy efficiency, a relay selection scheme based on reinforcement learning is proposed. Considering that there may be eavesdroppers in the communication process between two users, it is necessary to carry out secret communication between users. A secure two-way relay scheme based on time-division broadcasting was proposed^[22]. Under the constraints of average peak transmit power and data buffer, a safe total rate maximization problem is proposed. Different model predictive control was studied to optimize HEV nonlinear energy management to achieve better fuel economy^[23]. By treating MPC as a non-linear programming problem, using sequential secondary programming to obtain the descending direction of the control variable, the current control input is obtained. The control strategy for edge nodes with caching capabilities was studied, a dynamic programming model based on truncation concepts was established, and an asymptotically optimal algorithm was proposed^[24]. The resource allocation problem in a reconfigurable smart surface-assisted wireless power transfer system was studied^[25]. To deal with the beamforming optimization model, a scheme based on alternating direction multipliers was proposed to perform the beamforming optimization algorithm.

Energy efficiency optimization algorithm

Convex optimization preprocessing

The P1 is a MINLP problem that needs the Dinkelbach method to solve it. The Dinkelbach algorithm is an effective algorithm for solving fractional programming problems. The algorithm solves the fractional programming problem by converting it into a series of parameterized sub-problems. The core of the algorithm is to find the optimal solution in an iterative way. The Dinkelbach algorithm can ensure that the optimal solution of the problem is reached within a finite number of steps, thanks to its unique parameter update mechanism and convergence properties.

By scaling Eq. (4), P1 can be transformed into a quasi-convex form^[26].

$ {\phi _{i,j}} \in \left[ {0,1} \right],\forall i,j $

(20)

$ \begin{aligned} P2:\;\;\;\;\;\;\;\;\;& \mathop {\max }\limits_{{\tau _I},P_i^{S{\text{1}}},P_{i,j}^{S{\text{2}}},P_{i,j,k}^R,{\phi _{i,j}}} \dfrac{{{R_{total}}({\mathbf{\Phi }},{\mathbf{P}})}}{{{P_{total}}({\mathbf{\Phi }},{\mathbf{P}})}} \\ &s.t. {\phi _{i,j}} \in \left[ {0,1} \right],\forall i,j \\ &\qquad(19B) - (19E) \end{aligned} $

(21)

Lemma 1. The relaxation problem P2 has the same optimal solution as the original problem P1.

Proof of Lemma 1. It is not difficult to find that the value range of the constraint ϕ_i,j $ \in $ {0, 1] of the original problem P1 is scaled from 0 or 1 to 0 to 1 in the ϕ_i,j $ \in $ [0, 1]. The slack problem P2 is a slack version of the original problem P1. Therefore, the optimal solution $ \left( {{{\mathbf{\Phi }}^*},{{\mathbf{P}}^*}} \right) $ of the relaxation problem P2 is naturally the upper bound of the original problem P1. Further, the Lagrange multiplier method used to solve the relaxation problem P2, the $ {{\mathbf{\Phi }}^*}\left( {\alpha ,\beta } \right) $ that satisfies the binary constraint is continuously iterated to convergence in the sub-gradient algorithm, and finally the $ \left( {{{\mathbf{\Phi }}^*},{{\mathbf{P}}^*}} \right) $ that meets all the constraints in the original problem P1 is also given. The corresponding lower bound. Therefore, the relaxation problem P2 has the same optimal solution as the original problem P1.

Dinkelbach method and its outer loop algorithm

Adopting the Dinkelbach method

$ {\eta ^*} = \mathop {\max }\limits_{{\mathbf{\Phi }},{\mathbf{P}}} \dfrac{{{R_{total}}({\mathbf{\Phi }},{\mathbf{P}})}}{{{P_{total}}({\mathbf{\Phi }},{\mathbf{P}})}} = \dfrac{{{R_{total}}({{\mathbf{\Phi }}^*},{{\mathbf{P}}^*})}}{{{P_{total}}({{\mathbf{\Phi }}^*},{{\mathbf{P}}^*})}},\forall \{ {\mathbf{\Phi }},{\mathbf{P}}\} \in {\mathbf{T}} $

(22)

Define function $ F(\eta ) $

$ F(\eta ) = \mathop {\max }\limits_{{\mathbf{\Phi }},{\mathbf{P}}} \left[ {{R_{total}}({\mathbf{\Phi }},{\mathbf{P}}) - \eta {P_{total}}({\mathbf{\Phi }},{\mathbf{P}})} \right] $

(23)

Proposition 1. According to Eqs (22), (23), the sufficient and necessary conditions for the optimization problem P2 to obtain the optimal solution $ \left\{ {{\eta ^*},{{\mathbf{\Phi }}^*},{{\mathbf{P}}^*}} \right\} $ is:

$ \begin{aligned} F({\eta ^*}) & = \mathop {\max }\limits_{{\mathbf{\Phi }},{\mathbf{P}}} \left[ {{R_{total}}({\mathbf{\Phi }},{\mathbf{P}}) - {\eta ^*}{P_{total}}({\mathbf{\Phi }},{\mathbf{P}})} \right] \\ & = {R_{total}}({{\mathbf{\Phi }}^*},{{\mathbf{P}}^*}) - {\eta ^*}{P_{total}}({{\mathbf{\Phi }}^*},{{\mathbf{P}}^*}) = 0,\forall \{ {\mathbf{\Phi }},{\mathbf{P}}\} \in {\text{T}} \end{aligned} $

(24)

Proof of Proposition 1. First, the optimization problem P2 is reduced to a fractional programming problem, and its optimal energy efficiency value is $ {\eta ^*} = \frac{{{R_{total}}({{\mathbf{\Phi }}^*},{{\mathbf{P}}^*})}}{{{P_{total}}({{\mathbf{\Phi }}^*},{{\mathbf{P}}^*})}} $. The denominator polynomial $ {P_{total}}({\mathbf{\Phi }},{\mathbf{P}}) = {P_C} + \varsigma \sum\limits_{m = 1}^N {\sum\limits_{n = 1}^N {{P_{i,j}}} } $, where the reciprocal of the loop current $ {P_C} $ and the power conversion efficiency $ \varsigma $ is constant, and the denominator polynomial is the convex function of P_i,j. Finally, examine the domain of the variable {ϕ_i,j, P_i,j} satisfying 0 ≤ ϕ_i,j ≤ 1, P_i,j ≥ 0, which is a convex set.

In summary, the denominator polynomial $ {P_{total}}({\mathbf{\Phi }},{\mathbf{P}}) $ of the optimization problem P2 is a convex function, the numerator polynomial $ {R_{total}}({\mathbf{\Phi }},{\mathbf{P}}) $ is a concave function, and its domain {ϕ_i,j, P_i,j} is a convex set, which satisfies the conditions of Dinkelbach's fractional programming^[27].

According to Eq. (22), the Dinkelbach method can achieve optimal energy efficiency $ {\eta ^*} $ under the given user power allocation $ {\mathbf{P}} $ and subcarrier allocation $ {\mathbf{\Phi }} $.

Lemma 2. For $ \forall \{ \Phi ',{{\text{P}}^\prime }\} \in {\text{T}} $ and $ \eta ' = {\eta _E}(\Phi ',{{\text{P}}^\prime }) $, there is $ F(\eta ') \geqslant 0 $, and if and only when $ \eta ' = {\eta ^*} $ is $ F(\eta ') = 0 $, that is:

$ F(\eta ') = \mathop {\max }\limits_{{\mathbf{t}},{\mathbf{P}}} \left[ {{R_{total}}(\Phi ,{\text{P}}) - \eta '{P_{total}}(\Phi ,{\text{P}})} \right] \geqslant {R_{total}}(\Phi ',{{\text{P}}^\prime }) - \eta '{P_{total}}(\Phi ',{{\text{P}}^\prime }) = 0 $

(25)

Proposition 2. The sequence {η_n} generated by Algorithm 1 is a convergent sequence, and the convergence value η^lmt is reachable.

Table 1. Energy efficiency optimization iterative algorithm.

1. Initialize.

2. Set the iteration termination times $ N_{outer}^{max} $, iteration termination accuracy $ {\varpi }_{outer} $, and the initial iteration value $ {\eta }_{0}=0 $ and $ n=0 $.

3. Loop body:

4. Update iteration index $ n=n+1 $.

5. Solving the Optimization Problem $ F({\eta }_{n-1})=\underset{\mathbf{\Phi },\mathbf{P}}{\max }\left[{R}_{total}(\mathbf{\Phi },\boldsymbol{P})-{\eta }_{n-1}{\mathbf{P}}_{total}(\mathbf{\Phi },\mathbf{P})\right] $, gain $ \left\{{\mathbf{\Phi }}^{*},{\mathbf{P}}^{*}\right\} $.

6. Calculate the energy efficiency under the current iterative index using $ \left\{{\mathbf{\Phi }}^{*},{\mathbf{P}}^{*}\right\} $, $ {\eta }_{n}={R}_{total}({\mathbf{\Phi }}^{*},{\boldsymbol{P}}^{\text{*}})/{\mathbf{P}}_{total}({\mathbf{\Phi }}^{*},{\mathbf{P}}^{*}) $.

7. End Condition: $ \left| {\eta }_{n}-{\eta }_{n-1}\right| < {\varpi }_{outer} $ or $ n > N_{outer}^{max} $.

8. Output optimal solution $ \left\{{\eta }^{*},{\mathbf{\Phi }}^{*},{\mathbf{P}}^{*}\right\} $.

Proof of Proposition 2. We first discuss the monotonicity of the sequence {η_n}. In order not to lose the generality, arbitrarily choose the n-th iteration, and solve the energy efficiency value of the$ n $-th iteration by Eq. (23), denoted as η_n, and the energy efficiency value of the n+1-th iteration as η_n+1, with:

$ {\eta _{n + 1}} = \dfrac{{{R_{total}}({\Phi _n},{{\text{P}}_n})}}{{{P_{total}}({\Phi _n},{{\text{P}}_n})}} $

(26)

It is assumed that η_n and η_n+1have not yet reached the convergence value η^lmt of the energy efficiency sequence, namely η_n,η_n+1 ≠ η^lmt. According to Lemma 2, F(η_n) > 0, according to Eq. (25), expands as follows

$ \begin{aligned} F({\eta _n}) & = {R_{total}}({\Phi _n},{{\text{P}}_n}) - {\eta _n}{P_{total}}({\Phi _n},{{\text{P}}_n}) = {\eta _{n + 1}}{P_{total}}({\Phi _n},{{\text{P}}_n}) \\ & - {\eta _n}{P_{total}}({\Phi _n},{{\text{P}}_n}) = {P_{total}}({\Phi _n},{{\text{P}}_n})({\eta _{n + 1}} - {\eta _n}) \gt 0 \end{aligned} $

(27)

The total power consumption $ {P_{total}}({\Phi _n},{{\text{P}}_n}) $ is non-negative, so η_n+1 > η_n.

The total power consumption of the system is $ {P_{total}}({\Phi _n},{{\text{P}}_n}) = {P_C} + \varsigma \sum\limits_{m = 1}^N {\sum\limits_{n = 1}^N {{{\text{P}}_{i,j}}} } $. Assuming that the given energy efficiency value can obtain the maximum average transmission rate R_max of the system, ignoring the second part of $ {P_{total}}({\Phi _n},{{\text{P}}_n}) $, the upper bound of the system energy efficiency value η^ub meets:

$ \dfrac{{{R_{max}}}}{{{P_C}}} = {\eta ^{ub}} $

(28)

Therefore, the non-negative sequence {ηⁿ} increases monotonically, and there is an upper bound η^ub, so the sequence {ηⁿ} must converge, and the convergence value is η^lmt, which satisfies the following formula:

$ \begin{aligned} F({\eta ^{lmt}}) & = \mathop {\lim }\limits_{n \to \infty } F({\eta _n}) = \mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{{\mathbf{t}},{\mathbf{P}}} \left[ {{R_{total}}(\Phi ,{\text{P}}) - {\eta _n}{P_{total}}(\Phi ,{\text{P}})} \right] \\ &= \mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{{\mathbf{t}},{\mathbf{P}}} \left[ {{\eta _{n + 1}}{P_{total}}({\Phi _n},{{\text{P}}_n}) - {\eta _n}{P_{total}}(\Phi ,{\text{P}})} \right] \\ & = \mathop {\lim }\limits_{n \to \infty } ({\eta _{n + 1}} - {\eta _n}){P_{total}}({\Phi _n},{{\text{P}}_n}) = 0 \end{aligned} $

(29)

In summary, the sequence {η_n} converges to η^lmt, and Proposition 2 is proved. The P2 is converted into other forms and solved by using a progressive optimization method in the following content.

Energy efficiency inner loop optimization algorithm

This section presents a solution method for the inner loop optimization problem $ F({\eta _{i - 1}}) $, which is expressed as:

$ P3:\mathop {\max }\limits_{ {\mathbf{\Phi }},{\mathbf{P}}} F(\eta ) = \left[ {{R_{total}}({\mathbf{\Phi }},{\mathbf{P}}) - \eta {P_{total}}({\mathbf{\Phi }},{\mathbf{P}})} \right] $

(30)

Equivalent channel gain and optimization problem conversion

The total power consumption defined on the subcarriers SP_i,j of the first and the second time slot is $ P_{\left( {i,j} \right)}^{TS1} $ and $ P_{\left( {i,j} \right)}^{TS2} $, respectively, so we have:

$ {P_{(i,j)}} = P_{(i,j)}^{TS1} + P_{(i,j)}^{TS2} $

(31)

The first time slot source S of information transmission sends data, and the second time slot K relay of information transmission simultaneously forwards data. In the improved transmission protocol of this article, the second time slot source S uses the collected excess energy to send data again. To increase the diversity gain, the power relationship is as follows:

$ P_{(i,j)}^{TS1} = P_i^{S1} $

(32)

$ P_{(i,j)}^{TS2} = P_{i,j}^{S2} + \sum\limits_{k = 1}^K {P_{i,j,k}^R} ,\;\;\;\;\;\;\;P_{i,j}^{S2} \geq 0,P_{i,j,k}^R \geq 0 $

(33)

To obtain the power allocation based on Eq. (33), it is necessary to simplify P_(i,j) in Eq. (31) and obtain the equivalent channel gain of the system, which will help the subcarrier pairing in the following content. The definition is as follows:

$ P_{(i,j)}^{TS1} = {\varepsilon _{(i,j)}}{P_{(i,j)}},{\varepsilon _{(i,j)}} \in (0,1] $

(34)

$ P_{(i,j)}^{TS2} = (1 - {\varepsilon _{(i,j)}}){P_{(i,j)}} $

(35)

where, ε_{(i, j)} represents the ratio of $ P_{(i,j)}^{TS1} $ and P_(i,j), and 1 − ε_{(i, j)} represents the ratio of $ P_{(i,j)}^{TS2} $ and P_{(i, j)}. When ε_{(i, j)} = 1 indicates that the source node S transmits data in the first time slot at full power P_{(i, j)}.

Since Eqs (34) and (35) are both functions of P_(i,j), we have:

$ SN{R_{(i,j)}} \approx {A_{(i,j)}}{P_{(i,j)}} $

(36)

where, SNR_{(i, j)} is the signal-to-interference and noise ratio of the subcarrier to SP_{(i, j)}. A_{(i, j)} represents the equivalent channel gain of the subcarrier to SP_{(i, j)}. A_{(i, j)} can be used to further optimize the simplified end-to-end transmission rate expression. A_{(i, j)} can be expressed as:

$ {A_{(i,j)}} = \dfrac{1}{{{N_0}}}[{\left| {g_{j,2}^{SB}} \right|^2} + {\varepsilon _{(i,j)}}({\left| {g_{i,1}^{SB}} \right|^2} - {\left| {g_{j,2}^{SB}} \right|^2}) + \sum\limits_{k = 1}^K {\dfrac{{{\varepsilon _{(i,j)}}(1 - {\varepsilon _{(i,j)}}){{\left| {g_{i,k}^{SR}} \right|}^2}{{\left| {g_{j,k}^{RB}} \right|}^2}}}{{{\varepsilon _{(i,j)}}{{\left| {g_{i,k}^{SR}} \right|}^2} + (1 - {\varepsilon _{(i,j)}}){{\left| {g_{j,k}^{RB}} \right|}^2}}}} ] $

(37)

According to Eq. (37) and the golden section method, the optimal solution of ε_{(i, j)} can be obtained.

$ \varepsilon _{(i.j)}^* = \arg \max {A_{(i,j)}} $

(38)

Power allocation and subcarrier pairing

Rewriting the expression of $ SNR_{(i,j)}^{TS2} $ in Eq. (11):

$ SNR_{(i,j)}^{TS2} = \dfrac{{P_i^{S1}}}{{{N_0}}}\cdot\dfrac{{{{\left( {\displaystyle\sum\limits_{k = 1}^K {\dfrac{{\left| {g_{i,k}^{SR}g_{j,k}^{RB}} \right|\sqrt {P_{i,j,k}^R} }}{{\sqrt {P_i^{S1}{{\left| {g_{i,k}^{SR}} \right|}^2} + {N_0}} }} + \left| {g_{j,2}^{SB}} \right|\sqrt {\dfrac{{P_{i,j}^{S2}}}{{P_i^{S1}}}} } } \right)}^2}}}{{1 + \displaystyle\sum\limits_{k = 1}^K {\dfrac{{{{\left| {g_{j,k}^{RB}} \right|}^2}P_{i,j,k}^R}}{{P_i^{S1}{{\left| {g_{i,k}^{SR}} \right|}^2} + {N_0}}}} }} $

(39)

$ P_{i,j,k}^R $ and $ P_{i,j}^{S2} $ represent the optimized power of the relay node, represents the optimized power of the source node can be expressed as:

$ P_{i,j,k}^R = \dfrac{{{{\left| {g_{i,k}^{SR}g_{j,k}^{RB}} \right|}^2}\theta _{i,j}^k}}{{\left( {\dfrac{{{{\left| {g_{j,k}^{RB}} \right|}^2}}}{{P_i^{S1}}} + \displaystyle\sum\limits_{k = 1}^K {\dfrac{{{{\left| {g_{i,k}^{SR}g_{j,k}^{RB}} \right|}^2}\theta _{i,j}^k}}{{{{(1 + \theta _{i,j}^k{{\left| {g_{j,k}^{RB}} \right|}^2}P_{i,j}^{S2})}^2}}}} } \right){{\left( {\dfrac{1}{{\sqrt {P_{i,j}^{S2}} }} + \theta _{i,j}^k{{\left| {g_{j,k}^{RB}} \right|}^2}\sqrt {P_{i,j}^{S2}} } \right)}^2}}} $

(40)

$ P_{i,j}^{S2} = P_{\left( {i,j} \right)}^{TS2}{\left| {g_{j,2}^{SB}} \right|^2}/\left( {{{\left| {g_{j,k}^{RB}} \right|}^2} + \sum\limits_{k = 1}^K {\dfrac{{{{\left| {g_{i,k}^{SR}g_{j,k}^{RB}} \right|}^2}\theta _{i,j}^kP_i^{s1}}}{{{{(1 + \theta _{i,j}^k{{\left| {g_{j,k}^{RB}} \right|}^2}P_{\left( {i,j} \right)}^{TS2})}^2}}}} } \right) $

(41)

where:

$ \theta _{i,j}^k = 1/(P_i^{S2}{\left| {h_{i,2}^{S{R_k}}} \right|^2} + {N_0}),k = 1,2,...,K $

(42)

Energy-efficient inner resource allocation algorithm

Taking Eq. (37) into Eq. (30), the optimization problem P3 can be further transformed into:

$ \begin{gathered} P4:\mathop {\max }\limits_{{\tau _I},P_i^{S{\text{1}}},P_{i,j}^{S{\text{2}}},P_{i,j,k}^R,{\phi _{i,j}}} \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {\frac{{{\tau _I}}}{2}} } {\phi _{i,j}}\log (1 + {A_{(i,j)}}{P_{(i,j)}}) - \eta \left( {{P_C} + \varsigma \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{P_{\left( {i,j} \right)}}} } } \right) \\ \;\;\;\qquad\qquad s.t. (5)(16) - (18)(20) \\ \end{gathered} $

(43)

Under the premise of obtaining the known equivalent channel gain A_i,j, the $ {\phi _{i,j}} $log₂(1+A_i,jP_i,j) is a convex function of {$ {\phi _{i,j}}$, P_i,j}^[27]. Equation (43) meets the Slater condition and can be solved by the dual method. Although the derivation of the equivalent channel gain A_i,j is an approximate process, the problem P4 may not be the optimal solution to the original problem P3. But when the number of sub-carrier pairs increases, the double gap in the optimization problem can be ignored. Then, the dual method is used to solve, and the progressive optimization method is obtained^[28]. When P_(i,j) ≥ 0 is satisfied, using the KKT condition, we can get:

$ P_{(i,j)}^{} = {\left( {\dfrac{{{\tau _I}}}{{2\mu }} - \dfrac{1}{{{A_{(i,j)}}}}} \right)^ + } $

(44)

$ {\tau _I} = \sqrt {4(\mu - \lambda )\eta {P_A}\sum\limits_{i = 1}^N {(h_{i,1}^{AS} + \sum\limits_{k = 1}^K {h_{i,1}^{A{R_k}}} )} {\text{ /}}\sum\limits_i {\sum\limits_j {{\phi _{i,j}}\log \left( {1 + {A_{\left( {i,j} \right)}}{P_{\left( {i,j} \right)}}} \right)} } } $

(45)

$ L_{\left( {i,j} \right)}^*{\text{ = }}\dfrac{{{\tau _I}}}{2}\log {\left( {\dfrac{{{\tau _I}{A_{\left( {i,j} \right)}}}}{{2\mu }}} \right)^ + } - \mu {\left( {\dfrac{{{\tau _I}}}{{2\mu }} - \dfrac{1}{{{A_{\left( {i,j} \right)}}}}} \right)^ + } $

(46)

where, $ \mu $ and $ \lambda $, are Lagrange dual variables. $ L_{\left( {i,j} \right)}^* $ is a N × N subcarrier pair matrix, which can be solved using the standard Hungarian algorithm. The Hungarian algorithm is a combinatorial optimization algorithm used to solve the assignment problem. The algorithm has a time complexity of O(N³).

Algorithm 1 shows the energy-efficiency internal resource allocation algorithm for solving $F({\eta _{n - 1}}) = \mathop {\max }\limits_{{\mathbf{\Phi }},{\mathbf{P}}} [ {R_{total}}({\mathbf{\Phi }},{\mathbf{P}}) - $ $ {\eta _{n - 1}}{P_{total}}({\mathbf{\Phi }},{\mathbf{P}}) ] $.

In the joint subcarrier pairing proposed in this section, the calculation complexity of the time and power allocation strategy is O(N³ + N²K). Because in actual scenarios, N is much larger than K, the complexity of the calculation mainly depends on the Hungarian algorithm, and its complexity is O(N³).

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Energy efficiency resource allocation for amplify and forward relaying system with energy harvesting

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