Energy Rate Maximization with Sum-Rate Constraint for SWIPT in Multiple-Access Channels

Abstract

This paper considers simultaneous wireless information and power transfer (SWIPT) systems in the two-user Gaussian multiple access channel (G-MAC). In SWIPT systems, for a transmit signal each transmitter consists of an information-carrying signal and energy-carrying signal. By controlling a different set of the power for the information transmission and power for the energy transmission under a total power constraint, the information sum-rate and energy transmission rate can be achieved. As the information carrying-to-transmit power ratio at transmitters and the information sum-rate increases, however, the energy transmission rate decreases. In other words, there is a fundamental trade-off between the information sum-rate and the energy transmission rate according to the power-splitting ratio at each transmitter. Motivated by this, this paper proposes an optimal power-splitting scheme that maximizes the energy transmission rate subject to a minimum sum-rate constraint. In particular, a closed-form expression of the power-splitting coefficient is presented for the two-user G-MAC under a minimum sum-rate constraint. Numerical results show that the energy rate of the proposed optimal power-splitting scheme is greater than that of the fixed power-splitting scheme.

1. Introduction

Radio frequency (RF) signals can be used for information and energy transmission simultaneously in the wireless systems [1]. The trade-off between information and energy transmission for point-to-point communication systems was introduced in [2]. The wireless power transfer system refers to the transmission of electrical energy from a power source by means of electromagnetic fields, to an electrical component or a portion of a circuit that consumes electrical power.

The power-splitting techniques for information transmission and energy transmission at each transmitter were developed for simultaneous wireless information and power transfer (SWIPT) systems in [3,4,5,6]. A transmit signal at each transmitter consists of an information-carrying signal and energy-carrying signal for SWIPT systems. At each transmitter, a portion of the transmit power is used for information transmission and the remaining power is used for energy transmission. There is a fundamental trade-off between the information rate and the energy transmission rate according to the power-splitting ratio at each transmitter. For example, as the transmit power for information transmission increases, while the information rate increases and the energy transmission rate decreases.

Spatial-domain SWIPT systems in a point-to-point MIMO channel were investigated [7,8]. Based on the singular value decomposition (SVD) of the MIMO channel, the communication link is transformed to parallel channels that can convey either information or energy using binary variables. The joint eigenchannel assignment and power allocation problem for minimizing the total transmit power subject to rate and energy constraints were studied in [7]. In [8], the authors proposed the joint optimal assignment and power allocation of the available eigenchannels for maximizing the harvested energy while satisfying a minimum information rate requirement.

Recently, power-splitting techniques were extended to multi-user systems for SWIPT systems. The capacity region of a Gaussian multiple access channel (G-MAC) was introduced in [9,10]. For the SWIPT systems, the information-energy capacity region of the two-user MAC without feedback was studied in [3]. The authors in [3] proposed the power-splitting scheme for maximizing the information sum-rate with a minimum energy transmission rate constraint in two-user G-MAC without feedback. The information-energy capacity region for the SWIPT systems in the two-user MAC with feedback was discussed in [4,5]. The work in [5] proposed the power-splitting scheme for maximizing information individual rates and sum-rate given a minimum energy transmission rate constraint. In [6], the authors studied the fundamental limits of decentralized SWIPT systems in the two-user MAC for a case in which a minimum energy rate is required for successful decoding.

This paper studies the energy transmission rate maximization problem subject to an information sum-rate requirement in the two-user MAC. To the best of our knowledge, the optimal power-splitting scheme at each transmitter that maximizes the energy transmission rate for guaranteeing the target information sum-rate has not been proposed in the literature. It is shown that the optimization problem for maximizing the energy transmission rate with sum-rate constraint is strictly a convex problem. To make the optimization problem easier to solve, we reduced the number of constraints of the problem while maintaining the optimality. In addition, we present an efficient method for finding the optimal power-splitting ratio in closed-form.

The remainder of this paper is organized as follows. Section 2 describes the system model. Section 3 provides the proposed optimal power-splitting scheme that maximize the energy transmission rate for a given target information sum-rate. Section 4 shows the numerical results for the proposed optimal power-splitting scheme. Finally, Section 5 concludes the paper.

Notations: Notations $\left|a\right|$ denotes the absolute value of a for any scalar. The transpose of a vector a is denoted by ${\mathbf{a}}^T$. The operator $\mathbb{E}[·]$ indicates the expectation. Also, notations max$(·)$ and min$(·)$ denote the largest value of the arguments and the smallest value of the arguments, respectively.

2. System Model

As shown in Figure 1, consider a two-user memoryless G-MAC in [3]. The receiver consists of the information receiver and energy harvester in [3]. Transmitters 1 and 2 send messages $M_1$ and $M_2$ to the information receiver. The messages $M_1$ and $M_2$ are independent of the noise terms $Z_{1,1},\cdots ,Z_{1,n}$, $Z_{2,1},\cdots ,Z_{2,n}$ and uniformly distributed over the sets ${\mathcal{M}}_1\triangleq \{1,\cdots ,\lfloor 2^{n{R}_1}\rfloor \}$ and ${\mathcal{M}}_2\triangleq \{1,\cdots ,\lfloor 2^{n{R}_2}\rfloor \}$, where $R_1$ and $R_2$ denote the information transmission rates and $n\in \mathbb{N}$ the blocklength. The tth symbol at the transmitter i is given by:

$$\begin{array}{l}{X}_{i,t}=\sqrt{{\beta }_i{p}_i}{U}_{i,t}+\sqrt{(1-{\beta }_i)p_i}{W}_t\mathrm{for}i\in \{1,2\},t\in \{1,2,\cdots ,n\},\hfill \end{array}$$

(1)

where $X_{i,t}$ consists of an information-carrying and energy-carrying signals. Additionally, $U_{1,t}$ and $U_{2,t}$ are independent zero-mean unit-variance Gaussian information-carrying signals, and $W_t$ is a zero-mean unit-variance Gaussian energy-carrying signal known non-causally to all terminals. The symbols $X_{i,t}$ for $i\in \{1,2\}$ satisfy an average input power constraint, $\mathbb{E}\left[X_{i,t}^2\right]=p_i$. In addition, ${\beta }_i\in [0,1]$ is a information carrying-to-transmit power ratio at the transmitter i. Therefore, ${\beta }_i{p}_i$ and $(1-{\beta }_i)p_i$ are the amounts of transmit powers for the information-carrying and energy-carrying, respectively.

The received signals $Y_{1,t}$ and $Y_{2,t}$ at the information receiver and energy harvester, respectively, are written as:

$$\begin{array}{l}{Y}_{1,t}=h_{11}{X}_{1,t}+h_{12}{X}_{2,t}+Z_{1,t},\hfill \\ Y_{2,t}=h_{21}{X}_{1,t}+h_{22}{X}_{2,t}+Z_{2,t},\hfill \end{array}$$

(2)

where $h_{1i}$ and $h_{2i}$ are the corresponding constant non-negative real channel coefficients from transmitter i to the information receiver and energy harvester, respectively. The channel coefficients must satisfy the following ${\mathcal{L}}_2$-norm condition: $j\in \{1,2\},\left|\right|{\mathbf{h}}_j{\left|\right|}^2\le 1$, with ${\mathbf{h}}_j\triangleq {(h_{j1},h_{j2})}^T$ to ensure the principle of conservation of energy. Furthermore, $Z_{j,t}$ is the noise which has a zero-mean and ${\sigma }_j^2$ variance for $j\in \{1,2\}$. The noise $Z_{1,t}$ and $Z_{2,t}$ are realizations of two identically distributed real Gaussian random variables with no particular assumption on the joint distribution of $Z_{1,t}$ and $Z_{2,t}$. The information receiver subtracts first the common randomness and then performs successive decoding to recover the messages $M_1$ and $M_2$. The signal-to-noise ratios (SNRs) are defined as:

$${\gamma }_{ji}=\frac{p_i{\left|h_{ji}\right|}^2}{{\sigma }_j^2}\mathrm{for}(j,i)\in {\{1,2\}}^2.$$

(3)

The information-energy capacity region of the G-MAC was studied in [3] which is the set of all information-energy rate triplets $(R_1,R_2,B)$ that satisfy:

$$\begin{array}{ll}\hfill & 0\le R_1\le \frac{1}{2}{\log }_2(1+{\beta }_1{\gamma }_{11}),\hfill \end{array}$$

(4a)

$$\begin{array}{ll}\hfill & 0\le R_2\le \frac{1}{2}{\log }_2(1+{\beta }_2{\gamma }_{12}),\hfill \end{array}$$

(4b)

$$\begin{array}{ll}\hfill & 0\le R_1+R_2\le \frac{1}{2}{\log }_2(1+{\beta }_1{\gamma }_{11}+{\beta }_2{\gamma }_{12}),\hfill \end{array}$$

(4c)

$$\begin{array}{ll}\hfill & 0\le B\le 1+{\gamma }_{21}+{\gamma }_{22}+2\sqrt{(1-{\beta }_1){\gamma }_{21}(1-{\beta }_2){\gamma }_{22}}\hfill \end{array}$$

(4d)

with $({\beta }_1,{\beta }_2)\in {[0,1]}^2$, where $R_i$ is information rate for the transmitter i and B is the energy transmission rate.

3. Proposed Optimal Power-Splitting Scheme

This section introduces a proposed optimal power-splitting scheme at each transmitter for maximizing the energy transmission rate for guaranteeing the target information sum-rate $R_t$. The optimization problem can be written as:

$$\begin{array}{ll}\left(\mathbf{P}\mathbf{1}\right)\hfill & \underset{{\beta }_1,{\beta }_2}{\mathrm{maximize}}B({\beta }_1,{\beta }_2)\hfill \end{array}$$

(5a)

$$\begin{array}{ll}\hfill & \mathrm{subject}\mathrm{to}0\le {\beta }_1\le 1,\hfill \end{array}$$

(5b)

$$\begin{array}{ll}\hfill & 0\le {\beta }_2\le 1,\hfill \end{array}$$

(5c)

$$\begin{array}{ll}\hfill & R({\beta }_1,{\beta }_2)\ge R_t,\hfill \end{array}$$

(5d)

where $B({\beta }_1,{\beta }_2)=1+{\gamma }_{21}+{\gamma }_{22}+2\sqrt{(1-{\beta }_1){\gamma }_{21}(1-{\beta }_2){\gamma }_{22}}$ and $R({\beta }_1,{\beta }_2)=\frac{1}{2}{\log }_2(1+{\beta }_1{\gamma }_{11}+{\beta }_2{\gamma }_{12})$.

Lemma 1.

Problem $\left(\mathbf{P}\mathbf{1}\right)$ has a unique solution.

Proof of Lemma 1. 

Since problem $\left(\mathbf{P}\mathbf{1}\right)$ is strictly convex, problem $\left(\mathbf{P}\mathbf{1}\right)$ has a unique solution. It can be easily proofed by a standard convex optimization technique [11]. □

Lemma 2.

Problem $\left(\mathbf{P}\mathbf{1}\right)$ is equivalent to problem $\left(\mathbf{P}\mathbf{2}\right)$.

Proof of Lemma 2. 

The functions $B({\beta }_1,{\beta }_2)$ and $R({\beta }_1,{\beta }_2)$ are monotonically decreasing and increasing functions, respectively, with respect to both ${\beta }_1$ and ${\beta }_2$. Therefore, it can be shown that the maximum $B({\beta }_1,{\beta }_2)$ is obtained given that the information sum-rate constraint in Equation (5d) achieves with equality. In addition, since $B({\beta }_1,{\beta }_2)$ is a increasing function in both ${\beta }_1$ and ${\beta }_2$, then ${\beta }_1$ and ${\beta }_2$ for maximizing $B({\beta }_1,{\beta }_2)$ is equivalent to ${\beta }_1$ and ${\beta }_2$ for maximizing $(1-{\beta }_1)(1-{\beta }_2)$. □

By Lemma 1, problem $\left(\mathbf{P}\mathbf{1}\right)$ can be equivalent written as:

$$\begin{array}{ll}\left(\mathbf{P}\mathbf{2}\right)\hfill & \underset{{\beta }_1,{\beta }_2}{\mathrm{maximize}}f({\beta }_1,{\beta }_2)\hfill \end{array}$$

(6a)

$$\begin{array}{ll}\hfill & \mathrm{subject}\mathrm{to}0\le {\beta }_1\le 1,\hfill \end{array}$$

(6b)

$$\begin{array}{ll}\hfill & 0\le {\beta }_2\le 1,\hfill \end{array}$$

(6c)

$$\begin{array}{ll}\hfill & R({\beta }_1,{\beta }_2)=R_t,\hfill \end{array}$$

(6d)

where $f({\beta }_1,{\beta }_2)=(1-{\beta }_1)(1-{\beta }_2)$. According to Equation (6d), ${\beta }_2$ can be determined as:

$$\begin{array}{l}{\beta }_2=\frac{-{\beta }_1{\gamma }_{11}+2^{2R_t}-1}{{\gamma }_{12}}.\hfill \end{array}$$

(7)

By substituting Equation (7) into Equation (6a), $f({\beta }_1,{\beta }_2)$ can be written as:

$$\begin{array}{l}g\left({\beta }_1\right)=-\frac{{\gamma }_{11}}{{\gamma }_{12}}{\beta }_1^2+\frac{{\gamma }_{11}-{\gamma }_{12}+2^{2R_t}-1}{{\gamma }_{12}}{\beta }_1+\frac{{\gamma }_{12}-2^{2R_t}+1}{{\gamma }_{12}},\hfill \end{array}$$

(8)

In addition, by substituting Equation (7) into Equation (6c), the constraint can be written as:

$$\begin{array}{l}\frac{2^{2R_t}-1-{\gamma }_{12}}{{\gamma }_{11}}\le {\beta }_1\le \frac{2^{2R_t}-1}{{\gamma }_{11}}.\hfill \end{array}$$

(9)

The intersection of two constraints for Equations (6b) and (9) is $c_1\le {\beta }_1\le c_2$, where $c_1=\max \left(\frac{2^{2R_t}-1-{\gamma }_{12}}{{\gamma }_{11}},0\right)$ and $c_2=\min \left(\frac{2^{2R_t}-1}{{\gamma }_{11}},1\right)$. Therefore, problem $\left(\mathbf{P}\mathbf{2}\right)$ can be further simplified as:

$$\begin{array}{ll}\left(\mathbf{P}\mathbf{3}\right)\hfill & \underset{{\beta }_1}{\mathrm{maximize}}g\left({\beta }_1\right)\hfill \end{array}$$

(10a)

$$\begin{array}{ll}\hfill & \mathrm{subject}\mathrm{to}{c}_1\le {\beta }_1\le c_2\hfill \end{array}$$

(10b)

In problem $\left(\mathbf{P}\mathbf{3}\right)$, $g\left({\beta }_1\right)$ is a concave function because $\frac{{\partial }^2}{\partial {\beta }_1^2}g\left({\beta }_1\right)=-2\frac{{\gamma }_{11}}{{\gamma }_{12}}<0$. We can obtain b for maximizing $g\left({\beta }_1\right)$ by solving $\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)=0$ which is given by:

$$\begin{array}{l}b=\frac{{\gamma }_{11}-{\gamma }_{12}+2^{2R_t}-1}{2{\gamma }_{11}}.\hfill \end{array}$$

(11)

Furthermore, the two values of ${\beta }_1$ for which $g\left({\beta }_1\right)=0$ are $b_1=\frac{2^{2R_t}-1-{\gamma }_{12}}{{\gamma }_{11}}$ and $b_2=1$. It is observed that $b_1\le c_1$ and $c_2\le b_2$. Therefore, three cases are considered depending on the relation between $c_1$, $c_2$, and b. Figure 2 demonstrates possible three cases with different values of $c_1$, $c_2$, and b. We can exactly choose ${\beta }_1$ for maximizing $g\left({\beta }_1\right)$ for each case.

3.1. Case $c_1\le c_2\le b$

As shown in Figure 2a, $\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)$ can be written as:

$$\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)>0\mathrm{for}{c}_1\le {\beta }_1\le c_2.$$

(12)

According to Equation (12), $g\left({\beta }_1\right)$ is a strictly increasing function with respect to ${\beta }_1$ on $[c_1,c_2]$. Therefore, ${\beta }_1$ for maximizing $g\left({\beta }_1\right)$ is $c_2$.

3.2. Case $c_1\le b\le c_2$

As shown in Figure 2b, $\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)$ can be written as:

$$\left\{\begin{array}{cc}\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)>0\hfill & \mathrm{for}{c}_1\le {\beta }_1<b,\hfill \\ \frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)=0\hfill & \mathrm{for}{\beta }_1=b,\hfill \\ \frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)<0\hfill & \mathrm{for}b<{\beta }_1\le c_2.\hfill \end{array}\right.$$

(13)

According to Equation (13), $g\left({\beta }_1\right)$ is a strictly increasing function with respect to ${\beta }_1$ on $[c_1,b)$ and $g\left({\beta }_1\right)$ is strictly decreasing function with respect to ${\beta }_1$ on $(b,c_2]$. Therefore, ${\beta }_1$ for maximizing $g\left({\beta }_1\right)$ is b.

3.3. Case $b\le c_1\le c_2$

As shown in Figure 2c, $\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)$ can be written as:

$$\frac{\partial }{\partial {\beta }_1}g\left({\beta }_1\right)<0\mathrm{for}{c}_1\le {\beta }_1\le c_2.$$

(14)

According to Equation (14), $g\left({\beta }_1\right)$ is a strictly decreasing function with respect to ${\beta }_1$ on $[c_1,c_2]$. Therefore, ${\beta }_1$ for maximizing $g\left({\beta }_1\right)$ is $c_1$.

From three cases, the optimal ${\beta }_1^{*}$ for maximizing the energy transmission rate is given by:

$${\beta }_1^{*}=\left\{\begin{array}{cc}{c}_2\hfill & \mathrm{for}{c}_1\le c_2\le b,\hfill \\ b\hfill & \mathrm{for}{c}_1\le b\le c_2,\hfill \\ c_1\hfill & \mathrm{for}b\le c_1\le c_2,\hfill \end{array}\right.$$

(15)

where $c_1$, $c_2$, and b are rewritten as:

$$\begin{array}{ll}{c}_1\hfill & =\max \left(\frac{2^{2R_t}-1-{\gamma }_{12}}{{\gamma }_{11}},0\right),\hfill \\ c_2\hfill & =\min \left(\frac{2^{2R_t}-1}{{\gamma }_{11}},1\right),\hfill \\ b\hfill & =\frac{{\gamma }_{11}-{\gamma }_{12}+2^{2R_t}-1}{2{\gamma }_{11}}.\hfill \end{array}$$

(16)

4. Numerical Results

This section numerically investigates the performances of the proposed optimal power-splitting scheme. We compare the proposed optimal power-splitting scheme with the fixed power-splitting scheme. Figure 3 shows the energy transmission rate for the proposed optimal power-splitting scheme and fixed $({\beta }_1,{\beta }_2)$ power-splitting scheme versus ${\gamma }_{12}$ when ${\gamma }_{11}={\gamma }_{21}={\gamma }_{22}=10$ dB and $R_t=1.8$ bps/Hz. It is observed that the energy transmission rate of the proposed optimal power-splitting scheme was greater than that of the fixed $({\beta }_1,{\beta }_2)$ power-splitting scheme. The energy transmission rate of the fixed $({\beta }_1,{\beta }_2)$ power-splitting scheme was constant because ${\beta }_1$ and ${\beta }_2$ were fixed regardless of ${\gamma }_{12}$. As ${\beta }_2$ increased, the energy transmission rate of the fixed $({\beta }_1,{\beta }_2)$ power-splitting scheme decreased.

Figure 4 shows the energy transmission rate for the proposed optimal power-splitting scheme and fixed ${\beta }_1$ power-splitting scheme versus ${\gamma }_{12}$ with different target information sum-rates when ${\gamma }_{11}={\gamma }_{21}={\gamma }_{22}=10$ dB. In the fixed ${\beta }_1$ power-splitting scheme, ${\beta }_1$ was determined by the value of $0\le {\beta }_1\le \frac{2^{2R_t}-1}{{\gamma }_{11}}$ to equal the target information sum-rate and ${\beta }_2$ was determined by ${\beta }_2=\frac{-{\beta }_1{\gamma }_{11}+2^{2R_t}-1}{{\gamma }_{12}}$. It can be seen that the energy transmission rate of the proposed optimal power-splitting scheme was greater than that of the fixed ${\beta }_2$ power-splitting scheme. As ${\gamma }_{12}$ increased and the target information sum-rate decreased, the energy transmission rate of the optimal power-splitting scheme increased.

5. Conclusions

We formulated the optimization problem for maximizing the energy transmission rate with the information sum-rate constraint and presented a method for reducing the problem’s number of constraints while maintaining optimality. This paper presented the optimal power-splitting scheme, which maximized the energy transmission rate for guaranteeing the information sum-rate constraint for SWIPT systems in two-user G-MAC. The proposed optimal power-splitting scheme provided a closed-form solution that required low-complexity computations.