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Power allocation for SWIPT in Kuser interference channels using game theory
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 27 (2018)
Abstract
A simultaneous wireless information and power transfer system in interference channels of multiusers is considered. In this system, each transmitter sends one data stream to its targeted receiver, which causes interference to other receivers. Since all transmitterreceiver links want to maximize their own average transmission rate, a power allocation problem under the transmit power constraints and the energyharvesting constraints is developed. To solve this problem, we propose a game theory framework. Then, we convert the game into a variational inequalities problem by establishing the connection between game theory and variational inequalities and solve the variational inequalities problem. Through theoretical analysis, the existence and uniqueness of Nash equilibrium are both guaranteed by the theory of variational inequalities. A distributed iterative alternating optimization waterfilling algorithm is derived, which is proved to converge. Numerical results show that the proposed algorithm reaches fast convergence and achieves a higher sum rate than the unaided scheme.
Introduction
Simultaneous wireless information and power transfer (SWIPT), which transports both information and energy simultaneously by the same radiofrequency (RF) signal, has caused great concern in both academic and industrial fields and offers great convenience to wireless terminals [1–3]. According to [3], time switching (TS) and power splitting (PS) are two practical receiver designs. As a PS receiver plays a significant role in SWIPT, it divides the received signal into two signal flows, one for energy harvesting (EH) and the other for information decoding (ID) [4, 5]. Based on the position relationship between the ID receiver and the EH receiver, which is spatially separated or colocated, there are two types of SWIPT networks [6].
Recently, the study of realizing SWIPT in the interference channels (IFC) has received considerable attention. As an extra energy source in the IFC, the crosslink signals are salutary to EH [7]. However, the crosslink signals are harmful to information transmission, which brings new challenges to the transmission designs. The authors of [8] investigated an adaptive resource allocation scheme called proportionalfair power allocation (PFPA) in multiuser OFDM systems for fair share of resources and efficient operation. In [9], the authors solved a sum rate maximization problem in a twouser IFC where the two receivers can simultaneously decode information and harvest energy. The author of [10] found a necessary condition of the optimal transmission strategy considering three different scenarios according to the receiver mode in a Kuser IFC. In [11], the authors divided an optimal robust secure beamforming and power splitting scheme to minimize the total transmit power while satisfying the constraints on the minimum amounts over IFC. The authors of [12] jointly designed the allocating transfer power and receive PS coefficient for a twolink SWIPT system in IFC. In [13], a new transmission strategy was derived to maximize the energy beamforming and minimize the leakage beamforming in a twouser MIMO IFC. In [14], the authors derived a hybrid algorithm comprised of a linear combination of maximum ratio transmission (MRT) and zeroforcing (ZF) beamforming to minimize required power in a Kuser IFC network. To minimize the total transmission power, [15] proposed a joint beamforming and power splitting algorithm based on secondorder cone programming (SOCP) relaxation in a Kuser IFC network. Based on PS scheme, the work [16] considered a multiuser SWIPT interference system and studied joint transceiver design to minimize the total transmit power. Reference [17] proposed a synchronous power descending (SPD) algorithm to updates each links’ transmit power and PS ratio in a IFC SWIPT system. However, to the best of our knowledge, an iterative waterfilling algorithm based on a game theory to solve the power allocation problem in IFC SWIPT systems with K direct links to maximize its sum rate has not yet been studied.
In this paper, we devise an iterative waterfilling algorithm based on game theory to solve the formulated power allocation problem for the IFC SWIPT systems. The main contributions of this paper are listed in the following:

1.
A K transceiver links SWIPT system in interference channels is developed. In particular, each transmitterreceiver link is modeled as a strategy player who chooses its transmit power to maximize its individual rate.

2.
A game framework is formulated to solve the proposed power allocation problem. Then, we convert the game into a variational inequality (VI) problem using VI theory and analyze the existence and uniqueness of the Nash equilibrium (NE). In addition, a pricing mechanism is introduced to keep a balance between EH and ID.

3.
Different from that of reference [18], an alternating optimization (AO) method is employed because of the coupled variables and nonconvexity, which converts the original problem into two subproblems.

4.
To deal with the formulated subproblem, a distributed iterative waterfilling algorithm (IWFA) is devised to optimize the power allocation problem.
Problem formulation and power allocation method
Problem formulation
Gametheoretic framework formulation
An SWIPT system in the interference channels consisting K sourcedestination links is considered. In this system, each link includes a singleantenna source node S and a singleantenna destination node D equipped with a power splitter, which splits the received signal into two streams, one for EH and the other for ID. Moreover, it is assumed that perfect channel state information (CSI) is available, only one singledata stream is transmitted and all nodes operate in halfduplex mode. As illustrated in Fig. 1, each source node S_{ j } transmits data to its own receiver D_{ j } at the same time. The received signal at D_{ k } can be expressed as
where p_{ k } and p_{ j } are the transmit power of S_{ k } and S_{ j }, s_{ k } and s_{ j } are the transmit symbols of S_{ k } and S_{ j } with \(\mathbb {E}\{s_{k}^{2}\}=1, \mathbb {E}\{s_{j}^{2}\}=1\), h_{ kk } and h_{ jk } are the channel gains from S_{ k } to D_{ k } and S_{ j } to D_{ j }, and \(n_{ak} \thicksim \mathcal {CN}(0, \sigma _{ak}^{2})\) is the additive white Gaussian noise (AWGN) introduced by the receiver antenna at D_{ k }, and K={1,…,K}. The transmit power constraint of each node can be given by
where \( p_{j}^{max}\), j∈K denotes the peak power of each user.
A portion, β_{ j }∈(0,1), of the received signal is allocated for ID. Then, the signal for ID at D_{ k } can be expressed as
where \(n_{all} = \sqrt {\beta _{k}}{n_{ak}} + n_{k}\) is the overall noise at D_{ k } with covariance \(\sigma _{all}^{2}\) and n_{ k } is the AWGN originating from the power splitter with covariance \(\sigma _{k}^{2}\). Meanwhile, the power of the desired signal received by D_{ k } for ID can be given as
Dividing the desired signal power by the interference and noise power, the signaltointerferencenoiseratio (SINR) for the kth link between S_{ k } and D_{ k } can be expressed as
Accordingly, the achievable rate of the kth link is
where \(\mathbf {p}_{(k)}\triangleq (p_{j})_{j\neq k}\) is the set of allocating power of all users except the kth one.
A portion, 1−β_{ j }, of the signal received at D_{ k } is for EH and the power collected at D_{ k } follows the constraints
where e_{ k } is the power threshold at D_{ k }.
To maximize the achievable sum rate, a rate maximization problem with power constraint can be stated as follows
where p=[p_{1},p_{2},...,p_{ K }]^{T} represents the power allocation strategy of all users, and \(\mathcal {P} \triangleq \left \{\mathbf {p}0\leq p_{j} \leq p_{j}^{max},\right.\left.\left (1\,\,\beta _{k}\right)\sum _{j=1}^{K}h_{jk}^{2}{p_{j}}\geq {e_{k}}\right \}\) is the set of power constraints.
Since the nonconvex problem (8) is difficult to solve, we devise a distributed framework based on game theory to solve the power allocation problem. Specifically, we consider the scenario where each user maximizes its own rate selfishly via allocating the transmit power. Then, the game \(\mathcal {G}\) is formulated as
where \(\mathcal {P}_{k} \triangleq \left \{{p_{k}}0\leq p_{k} \leq p_{k}^{max}, \left (1\beta _{k}\right)\left (h_{kk}^{2}{p_{k}}+{\vphantom {\sum _{j=1, j\neq k}^{K}}}\right.\right.\left.\left.\sum _{j=1, j\neq k}^{K}h_{jk}^{2}{p_{j}}\right)\geq {e_{k}}\right \}\) is the feasible set of the kth user.
It can be observed from (9) that the problem is still nonconvex and difficult to solve because the objective function r_{ k } is nonconvex and the constraint \(\mathcal {P}_{k}\) of the kth user is coupled. Therefore, we redefine the objective function as
To let the optimization problem more decentralized and keep a balance between EH and ID, a pricing mechanism is introduced through a punishment in the payoff function. We define the pricing factor as \({\alpha } =\left (\alpha _{j}\right)_{j=1}^{K} \), where α_{ j } represents the penalty of the jth user. From (7), we define a linear function about the power constraint of the kth link,
where A=(1−β_{ k })h_{ kk }^{2} and \(B=\left (1\beta _{k}\right)\sum _{j=1,j\neq k}^{K}h_{jk}^{2}{p_{j}}{e_{k}}\). Then, we define the total penalty at the kth user as
where \(\zeta _{k}\triangleq {\alpha _{k}}\left (1\beta _{k}\right)h_{kk}^{2}\).
Considering the penalty, the payoff function of the kth user can be given as
Consequently, the original game \(\mathcal {G}_{1}\) can be formulated as a game
Our purpose is determining a NE point {p^{∗},α^{∗}} in the feasible set to minimize the objective function v_{ k } in \(\mathcal {G}_{2}\), which satisfies the following condition
VI problem formulation
It can be observed from (14) that the objective function and the constraints of the game \(\mathcal {G}_{2}\) involves coupled variables, then we derived an AO method to solve this problem, which converts the original problem into two subproblems. In this section, we analyze the NE point of the game \(\mathcal {G}_{2}\) using VI theory, optimize the power allocation problem using IWFA algorithm, and update the price vector using variablestep projection scheme.
In this subsection, we focus on finding the optimal power strategy p^{∗} of the game \(\mathcal {G}_{2}\) for the given price α. First, we rewrite the original game \(\mathcal {G}_{2}\) into a VI problem \(VI(\mathcal {P}_{k}, V_{k})\), which is denoted to find a power strategy \(p^{*} \in \mathcal {P}_{k}\) satisfying the following condition
where
is the gradient ▽v_{ k }. Then, we establish the relation between the formulated game \(\mathcal {G}_{2}\) and the VI problem \(VI\left (\mathcal {P}_{k}, V_{k}\right)\).
Proposition 1
The game \(\mathcal {G}_{2}\) is equivalent to the problem \(VI\left (\mathcal {P}_{k}, V_{k}\right)\).
Proof
A proof is given in Appendix Appendix A. □
The game \(\mathcal {G}_{2}\) can be rewritten as \(VI\left (\mathcal {P}_{k}, {V_{k}}\right)\) after the proof is completed. And the existence and uniqueness of the NE can be analyzed by studying the VI problem \(VI\left (\mathcal {P}_{k}, {V_{k}}\right)\).
Power allocation method
Analysis of the NE
The following theorem proposed in [19] is commonly used to verify the existence of the NE.
Theorem 1
A NE exists in a \(VI\left (\mathcal {A}, F\right)\) problem if the set \(\mathcal {A}\) is convex and compact; the function F is continuous in its feasible set.
After investigating the properties of the set \(\mathcal {A}\) and the function F of the VI problem, we have the following proposition regarding the existence of the NE.
Proposition 2
The game \(\mathcal {G}_{2}\) possesses at least one NE.
Proof
A proof is given in Appendix Appendix B. □
Based on ([20], Eq. 13), the \(VI\left (\mathcal {A}, F\right)\) admits a unique solution if F is strongly monotone on \(\mathcal {A}\). The definition of strongly monotone is provided in the following.
Definition 1
Given a mapping \(F: \mathcal {A} \subseteq \mathbb {R}^{n} \rightarrow \mathbb {R}^{n}\), if the set \(\mathcal {A}\) is convex and there exists a constant c>0 satisfying the following condition, F is strongly monotone.
We analyze the uniqueness of the NE by proving a sufficient condition for the strong monotonicity of V_{ k }. To prove the strong monotonicity of V_{ k }, the second derivative of v_{ k }(p) can be given as
we have the following proposition regarding the uniqueness of the NE.
Proposition 3
V_{ k } is strongly monotone in its feasible set. Furthermore, the game \(\mathcal {G}_{2}\) has the unique NE.
Proof
A proof is given in Appendix Appendix C. □
The above analysis about the existence and uniqueness of the NE shows that the NE of game \( \mathcal {G}_{2}\) always exists and admits its uniqueness for the given price factor. To achieve the unique NE, we use a waterfilling method based on the best response. For any fixed p_{−k} and α, the NE point of the game \(\mathcal {G}_{2}\) is the fixedpoint of the waterfilling mapping, which can be expressed as
where the majorization notion in p^{∗}(α) represents p(α) after optimization, \([\!x]_{a}^{b}\!\triangleq min(max(x, a)b)\) with 0≤a≤b and μ_{ k }≥0 is chosen to satisfy the power constraint (2).
Optimal α with variablestep projection scheme
In this section, we discuss the optimization of α with p fixed. For the optimal \(p_{k}^{*}({\alpha })\), we rewrite Eq. (11) as
To optimize α, we introduce a nonlinear complement problem (NCP), which is to find the price vector such that
The NCP is an equivalent form to VI problem. We use the wellknown variablestep projection scheme to solve this NCP, which is described in the following algorithm.
Distributed iterative algorithm
Denote the allocating power and the pricing factor of the kth user at the nth iteration as \(p_{k}^{(n)}(\mathbf {\alpha })\) and α^{(n)}, respectively. To achieve the unique NE, a distributed iterative algorithm with AO is summarized as the Proposed Iterative AO Algorithm in Table 1. It is worth noting that such the AO scheme guarantees the local optimum.
It can be seen that the price α is computed by the variablestep projection scheme and ε_{ n } is the nth iteration step size. As mentioned in [21], we could choose a sufficiently small value to assign to the step size ε_{ n }. The notion [·]^{+} represents that the value is meaningful when it is larger than zero, and let the value be zero when it is less than zero. And the proofs of convergence property about α and p_{ k }(α) are similar to the Theorems 6 and 10 in [18].
Numerical results and discussions
It is assumed that the channels between all links are mutually independent Rayleigh fading and the freespace propagation pathloss coefficient is 2. And random channels with 100 slots are generated. The variances of the noises are \(\sigma _{all}^{2} = \sigma ^{2}\), \(p_{k}^{max} = P_{max}\), and β_{ k }=0.5.
We study the convergence property of IWFA in a fourlink network. Figure 2 shows the allocating power of four users versus iterations considering two different initial points: (1.1,1.5,1.2,1.3) and (1.4,1.3,0.8,1.0) under the same conditions P_{ max }=25 and σ^{2}=1. It can be observed that all users’ allocating power converge to the same points (1.28,1.39,1.00,1.41). After several similar attempts, we have that the IWFA quickly converges to the unique NE from different initial points.
Figure 3 provides the results of sum rate versus the interlink distance dL. Different conditions of P_{ max }∈{20dB,30dB} and σ^{2}∈{0.5,1.0} are simulated, respectively. There are two schemes for comparison. Scheme 1 is the unaided scheme without iteration, and scheme 2 is IWFA scheme. We consider a linear topology where every transmitterreceiver link is parallel to each other. It can be seen that the sum rate increases as the dL increasing for two schemes. Then, the IWFA method outperforms to the unaided method under all conditions. In addition, the system performance is better when the AWGN is smaller and the maximum transmit power is bigger.
Figure 4 shows results of the average transmit power versus the harvest energy threshold e_{ k }. Different conditions of K∈{3,4} and the different initial pricing α∈{0.5,0.6,0.7} are simulated, respectively. It can be observed that the average transmit power decreases as the harvest energy threshold e_{ k } increases for our proposed scheme, and the system requires less average transmit power with the increase of the number of links. We can also see that the average transmit power decreases as α increasing. In addition, the average transmit power of our proposed IWFA scheme always smaller than the unaided scheme when the number of links is the same.
Figure 5 provide results of the average bit error rate (BER) versus SINR. As we can see, our propose IWFA scheme is always better than the unaided scheme in BER performance. The reason is that the unaided is the simplest in computing complication with no iteration involved.
Conclusions
In this paper, a power allocation problem was solved for a SWIPT system in Kuser interference channels using the framework of game theory. We rewrote the formulated game as a variational inequality problem to analyze the NE of the game. Furthermore, we provided a distributed iterative algorithm with AO scheme to solve the formulated problem and update the price factor. Numerical results demonstrated that the proposed IWFA scheme can attains more sum rate and requires less transmit power than the unaided scheme under the same conditions of P_{ max } and σ^{2}.
Appendix A
Proof of Proposition 1
The relationship between game and VI is usually verified using the following theorem proposed in the reference [19].
Theorem 2
A given game \(\mathcal {G} = \langle {\mathcal {N}, \left \{\mathcal {A}_{n}\right \}, \{f_{n}(x)\}}\rangle \) is equivalent to \(VI(\mathcal {A}, F)\) if the following two conditions hold: (i) The strategy set \(\mathcal {A}_{n}\) is closed and convex; (ii) the payoff function is convex and continuously differential for \(x \in \mathcal {A}_{n}\).
Now, let us prove the relationship between the game \(\mathcal {G}_{2}\) and the VI problem \(VI\left (\mathcal {P}_{k}, V_{k}\right)\) using Theorem 2. First, note that the strategy set \(\mathcal {P}_{k}\) in (14b) is convex. Then, the payoff function v_{ k }(p_{ k },p_{−k}) in (14a) is continuously differentiable in its feasible set. Finally, the payoff function v_{ k }(p_{ k },p_{−k}) is convex in its feasible set because the logarithmic function is always concave and ζ_{ k }p_{ k } is a linear function. Both conditions in Theorem 2 are satisfied. Thus, the problem \(VI\left (\mathcal {P}_{k}, {V_{k}}\right)\) is equivalent to the formulated game \(\mathcal {G}_{2}\).
Appendix B
Proof of Proposition 2
For problem \(VI\left (\mathcal {P}_{k}, s{V_{k}}\right)\), the strategy set \(\mathcal {P}_{k}\) is convex and compact since the items \(0\leq p_{k} \leq p_{k}^{max}\) and \(\left (1\beta _{k}\right)\left (h_{kk}^{2}{p_{k}}+\sum _{j=1, j\neq k}^{K}h_{jk}^{2}{p_{j}}\right)\geq {e_{k}}\) of \(\mathcal {P}_{k}\) are linear functions.
It is can be observed from Eq. (17) that \(\phantom {\dot {i}\!}{V_{k}}=\bigtriangledown _{p_{k}}v_{k}(p_{k},\mathbf {p}_{k}; \alpha _{k})\), we can prove the continuity of V_{ k } by computing its first derivative. And we can see that its first derivative is exists; furthermore, the utility function V_{ k } is continuous in \(\mathcal {P}_{k}\).
Thus, we know that the VI problem admits at least one solution according to Theorem 1 and the NE existence is proved.
Appendix C
Proof of Proposition 3
From (17), we have that \(\bigtriangledown _{p_{k}}v_{k}(p_{k},{\mathbf {p}}_{k}) = V_{k}\). Consider two different power \(p_{k}^{(1)}\) and \(p_{k}^{(2)}\) in the strategy set. According to the meanvalue theorem, we have
Define \(d_{k} = p_{k}^{(1)}p_{k}^{(2)}\) and \(d_{j} = p_{j}^{(1)}p_{j}^{(2)}\), we have
since \({d_{k}}\sum _{j \neq k}^{K} \bigtriangledown _{p_{k}, p_{j}}^{2}{v_{k}(z_{k})}{d_{j}} > 0\). Then, we have
Then, the strongly monotonicity of V_{ k } can be proved as
where \(c_{sm} =\sum _{k=1}^{K}\bigtriangledown _{p_{k}}^{2}{v_{k}(z_{k})} > 0\) is the strongly monotone constant. Therefore, the Eq. (18) always hold, which completes the proof.
Abbreviations
 AO:

Alternating optimization
 AWGN:

Additive white Gaussian noise
 CSI:

Channel state information
 EH:

Energy harvesting
 ID:

Information decoding
 IFC:

Interference channels
 IWFA:

Iterative waterfilling algorithm
 MRT:

Maximum ratio transmission
 NCP:

Nonlinear complement problem
 NE:

Nash equilibrium
 PFPA:

Proportionalfair power allocation
 PS:

Power splitting
 RF:

Radiofrequency
 SINR:

Signaltointerferencenoiseratio
 SOCP:

Secondorder cone programming
 SPD:

Synchronous power descending
 SWIPT:

Simultaneous wireless information and power transfer
 TS:

Time switching
 VI:

Variational inequality
 ZF:

Zeroforcing
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Acknowledgements
The authors thank the National Natural Science Foundation of China (Grants Nos. 61629101, 61471067) and National Science and Technology major projects (Grant No. 2015ZX03002008).
Funding
This work was supported by the National Natural Science Foundation of China (Grants Nos. 61629101, 61471067) and the National Science and Technology major projects (Grant No. 2015ZX03002008).
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First, we formulated a power allocation problem, designed a scheme to optimize the power allocation strategy, performed numerical simulations, and prepared the initial draft as well as the revision. Then, we modified the solution pattern, verified the mathematical derivations, checked and analyzed the simulation results, and improved the writing. All authors read and approved the final manuscript.
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Wen, Z., Liu, Y., Liu, X. et al. Power allocation for SWIPT in Kuser interference channels using game theory. EURASIP J. Adv. Signal Process. 2018, 27 (2018). https://doi.org/10.1186/s1363401805477
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Keywords
 Distributed algorithm
 Game theory
 Interference channels
 Power allocation
 Simultaneous wireless information and power transfer
 Variational inequality theory