Charge-then-Forward: Wireless Powered Communication for Multiuser Relay Networks

06/26/2018 ∙ by Mengyu Liu, et al. ∙ South China University of Technology International Student Union 0

This paper studies a relay-assisted wireless powered communication network (R-WPCN) consisting of multiple source-destination pairs and a hybrid relay node (HRN). We consider a "charge-then-forward" protocol at the HRN, in which the HRN with constant energy supply first acts as an energy transmitter to charge the sources, and then forwards the information from the sources to their destinations through time division multiple access (TDMA) or frequency division multiple access (FDMA). Processing costs at the wireless-powered sources are taken into account. Our goal is to maximize the sum-rate of all transmission pairs by jointly optimizing the time, frequency and power resources. The formulated optimization problems for both TDMA and FDMA are non-convex. For the TDMA scheme, by appropriate transformation, the problem is reformulated as a convex problem and be optimally solved. For the FDMA case, we find the asymptotically optimal solution in the dual domain. Furthermore, suboptimal algorithms are proposed for both schemes to tradeoff the complexity and performance. Finally, the simulation results validate the effectiveness of the proposed schemes.

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I Introduction

The rapid growth of high-speed data and multimedia services increases energy consumption for better quality-of-services. However, conventional battery-powered communications have to replace or recharge batteries manually to extend their lifetime, which is inconvenient, unsafe, and costly. Recently, radio-frequency (RF) signal enabled wireless power transfer (WPT) has drawn great attention as it essentially provides more cost-effective and green energy supplies for wireless devices, where RF signals are used as the carriers to convey wireless energy to low-power wireless devices.

There are two main directions of WPT among the current related researches. One line of WPT focuses on so-called simultaneous wireless information and power transfer (SWIPT), where the same RF signal carries both energy and information at the same time [1, 2]. Due to the practical limitation of receivers that the received signals cannot be used to perform energy harvesting and information decoding simultaneously, two practical receiver architectures, namely time switching (TS) and power splitting (PS), were proposed in [1]. For TS, the received signal is either used for energy harvesting or information decoding, whereas for PS, the received signal is split into two separate streams with one stream for energy harvesting and the other for information decoding at the same time. SWIPT has been investigated extensively in different systems, e.g., the fading channels [3], relay channels [4, 5, 6, 7] and orthogonal frequency division multiple access (OFDMA) channels [8, 9, 10, 11, 12].

The newly emerging wireless powered communication network (WPCN) is another line of WPT where ambient RF signals are used to power wireless devices [13]. There are two basic applications about WPCN. One is that energy transmitters and information access points (APs) are located separately where energy transmitters transmit energy to wireless devices and then wireless devices transmit their information to APs using their harvested energy from energy transmitters [14]. Another application is that a hybrid AP (HAP) performs the roles of energy transmitter and AP integrally. For instance, a “harvest-then-transmit” protocol was proposed in [15], where HAP first broadcasts wireless energy to all wireless devices in the downlink and then wireless devices utilize the harvested energy to transmit their independent information to HAP in the uplink based on TDMA. Different from the HAP in SWIPT that coordinates wireless energy and information, the HAP in WPCN only broadcasts wireless energy.

An important application for WPCN lies in relay-assisted WPCN (R-WPCN), where relays are used to assist information transmission in R-WPCN. There are two categories among the current related works about R-WPCN: one is source powering relay [16, 17, 18] and the second is relay powering source [19, 20]. For the first category, i.e., source powering relay, the authors in [16] proposed a “harvest-then-cooperate” protocol, where both source and relay can harvest energy from the RF signals from a base-station. A two-user R-WPCN was studied in [17] where a nearer user to HAP harvests energy sent by HAP and relays information of the farther user in half-duplex. In [18], the full-duplex relay not only is powered by the source but also harvests energy from itself by energy recycling. As for the category of relay powering source, the throughput maximization problem was investigated in [19], where the source can harvest energy from the access point and/or relay before information transmission. The authors in [20] studied the channel capacity subject to an additional energy transmission cost at the energy harvesting sources. Note that both [19] and [20] considered a single source-destination pair.

Fig. 1: System model of the considered multiuser R-WPCN.

In this paper, we consider a new R-WPCN consisting of multiple source-destination pairs assisted by a single hybrid relay node (HRN), as shown in Fig. 1. We assume that the HRN in this paper has constant energy supply, while the sources nodes have no embedded power supply so that they have to be powered by the HRN before information transmission, i.e., “harvest-then-transmit” protocol is applied at the sources. The HRN thus acts double roles, one for an energy transmitter and the other for an information helper. That is, the HRN first charges the sources and then forwards their information, i.e., “charge-then-forward” protocol is considered at the HRN.

As the considered HRN has double roles, i.e., energy transmitter and information helper, energy charging and information forwarding of the HRN are mutually influenced and restricted since the HRN’s total energy is fixed. That is, encouraging the energy charging will increase the transmit power of sources at the first hop but decrease the information forwarding at the second hop. How to find the optimal tradeoff that maximizes the system sum-rate is non-trivial. In addition, based on TDMA and FDMA for multiuser information transmission, the network resources, like time, power, and frequency, are highly coupled and the formulated optimization problems are non-convex and difficult to solve. Moreover, we consider the processing cost at the wireless-powered sources, which further complicates the problems.

The main contributions of this paper are summarized as follows:

  • We consider a new multiuser R-WPCN based on a “charge-then-forward” relaying protocol, where the HRN first powers the energy-free sources and then forwards the information from the sources to their destinations by TDMA and FDMA. Processing cost is considered at the wireless-powered sources.

  • Depending on whether TDMA or FDMA is adopted, we formulate two optimization problems respectively for sum-rate maximization, which are both non-convex. We propose efficient algorithms to find the optimal solutions. In addition, suboptimal algorithms are proposed for both schemes to tradeoff the complexity and performance.

  • We provide some useful insights into the R-WPCN. For example, the time of WPT should be as small as possible so that the time for wireless information transmission (WIT) can be maximized for sum-rate maximization. In addition, due to the doubly distance-dependent signal attenuation for both WPT and the first hop of WIT, it is shown that the sum-rate decreases when the HRN moves from the sources to the destinations.

The remainder of this paper is organized as follows. In Section II, we introduce the system model of multiuser R-WPCN and problem formulations based on TDMA and FDMA, respectively. Section III presents the optimal and suboptimal resource allocation algorithms for the TDMA based problem. In the next, the asymptotically optimal and suboptimal algorithms for the FDMA based problem are presented in Section IV. In Section V, we evaluate the performance of proposed algorithms by simulations. Finally, Section VI concludes the paper.

Ii System Model and Problem Formulation

As shown in Fig. 1, we consider a general two-hop R-WPCN with multiple source-destination pairs as well as a HRN which not only transfers energy to the sources but also forwards information from the sources to the destinations. All nodes are equipped with a single antenna. We assume that the HRN is half-duplex due to the practical consideration, and there is no direct link between each source-destination pair due to the shielding effect caused by obstacles. As a result, each pair needs the assistance of the HRN to forward information. In this paper, we consider that the HRN has a constant energy supply while the source nodes have no embedded energy and thus have to harvest energy for information transmission. In addition, we assume that each source has the energy harvesting function to store the energy. In particular, we consider the “charge-then-forward” relaying protocol to coordinate power and information transfer, in which the HRN first acts as a wireless power beacon to charge the sources then as a helper for forwarding their information. Specifically, the whole transmission is divided into two continuous phases. The first phase is used for WPT conducted by the HRN. The second phase is WIT, i.e., the sources use the harvested energy to transmit their independent information to their destinations via the assistance of the HRN in the second phase based on TDMA or FDMA. The sources do not store the harvested energy for future, i.e., all the energy harvested during WPT phase is used for WIT.

The global channel state information (CSI) of the network is assumed to be known at the HRN where the central processing task is embedded. RF power transfer crucially depends on the available CSI of the nodes, which needs additional resources to acquire and the straightforward way is channel estimation via pilot signals, similar to conventional wireless communication systems. In our R-WPCN, whether the sources transmit pilots and the HRN estimates CSI, or the HRN transmits pilots and the sources estimate CSI, the sources are required to have initial energy to transmit/decode the pilot signals at the beginning of the training phase (before WPT in transmission phase). Thus it is reasonable to assume that the wireless-powered sources reserve some circuit power at the beginning for channel estimation, since the energy used to channel estimation is much smaller than that of information transmission in practice. CSI acquisition in WPT systems is very important but seems to be beyond of the scope of this paper. In this paper, we consider a block fading wireless environment so that the channel impulse response can be treated as time invariant in the block duration. As a result, the channel gains within the block duration remain unchanged (but can vary in different block durations). For convenience, we assume that the transmission time of each block is normalized to be unit.

Ii-a TDMA Case

(a) TDMA
(b) FDMA
Fig. 2: The “charge-then-forward” relaying protocol based on (a) TDMA and (b) FDMA information transmission.

We first consider the case of TDMA-based information transmission as shown in Fig. 2(a). The total transmission time is divided into time slots where the first time slot, say slot 0 with time duration , is allocated for WPT and all source nodes harvest energy from the HRN, while the rest slots are assigned to WIT of the pairs where each pair is allocated time duration. Moreover, for each pair’s information transmission, is further divided into two equal sub-slots with for the first hop and the rest for the second hop. By normalizing the whole time to be unit, we have

(1)

In addition, the power of the HRN used at slot is denoted as . Besides, we consider that there is a peak power constraint on , i.e., Denote the maximum transmit power of the HRN as , then the energy constraint at the HRN is given by

(2)

where is the transmission time of the HRN in the second hop for each pair . Note that the terms of power and energy are interchangeably used here since the duration of each block is normalized to be unit.

We consider energy accumulation for TDMA case, i.e., source harvests and accumulates energy from the previous slots, i.e., slot 0 to slot . The channel power gain from HRN to source for WPT and the channel power gain from source to source are denoted by and , respectively. Denote source ’s transmit power at slot as . Then the harvested energy of source can be expressed as

(3)

which comprises three parts: the first term is the energy harvested in WPT phase, and the last two terms correspond to the energy harvested from the HRN and sources in the previous WIT phase. Here is the energy conversion efficiency at the sources. As a result, the energy causality constraint at source is given by

(4)

where is the time of the first hop for the information transmission of pair and is the non-zero energy processing cost at source

. Moreover, the additional White Gaussian noise (AWGN) at each node is modeled as circularly symmetric complex Gaussian (CSCG) random variable with zero mean and variance

. Denote the channel power gains for the first and second hops of WIT for pair as and , respectively. Using decode-and-forward (DF) relaying strategy, the achievable rate for each pair is given by

(5)

Our objective is to maximize the sum-rate of all pairs by jointly optimizing the time allocation, the transmit power of sources and HRN. Let , , and , the problem can be mathematically formulated as

(6a)
(6b)

Problem (P1) is non-convex since the rate expression (5) is not jointly concave in the variables. We will optimally solve this problem in Section III.

Ii-B FDMA Case

We also consider the case of FDMA-based information transmission for the multiple pairs as shown in Fig. 2(b), where the total time is divided into two time slots, i.e., slot 0 and slot 1 utilized for WPT (using energy signals) and WIT, respectively. The time duration of slot 0 and slot 1 are denoted by and with

(7)

We assume that the HRN broadcasts energy signals over the entire bandwidth in the phase of WPT, while information signals are conveyed by using FDMA over subcarriers (SCs) in the next phase of WIT. For information transmission, we define a binary SC allocation variable with representing that SC is allocated to pair for WIT and otherwise. Each SC is allocated to at most one pair at slot 1 for WIT to avoid interference. The constraint can be expressed as

(8)

The channel power gains for WPT of source , the first and second hops of pair over SC for WIT are denoted as , and , respectively. The transmit power of HRN for WPT at slot 0 is denoted as , and the power of the HRN for forwarding pair ’s information on SC at slot 1 is . Note that WPT is conducted over the entire bandwidth, thus there is no index for and . The total transmit energy constraint of the HRN is thus given by

(9)

where represents the transmission time of the HRN in the second hop.

Moreover, we define source ’s transmit power on SC at slot 1 as . Different from TDMA case, since all sources transmit their information at the same time in slot 1, the harvested energy of sources are only from HRN during WPT phase. Therefore, the energy constraint at source is given by

(10)

where the represents the time of the first hop during information transmission.

The achievable rates of the first and second hops for pair over SC can be respectively written as:

(11)
(12)

The achievable rate of pair by using DF relaying strategy is the minimum of the rates achieved in the two hops, which can be expressed as

(13)

Our goal is maximizing the sum-rate of all transmission pairs by varying the transmit power of the sources and HRN, SC assignment and time allocation. Let , , , and , the optimization problem can be mathematically formulated as

(14a)
(14b)
(14c)

Problem (P2) is also non-convex since both binary and continuous variables are involved, which is a mixed-integer programming problem. The asymptotically optimal solution for Problem (P2) will be obtained in Section IV.

Iii Resource Allocation in TDMA Case

In this section, we study the TDMA case by solving Problem (P1). Problem (P1) is not convex and cannot be solved in its original form. Therefore, to make the problem tractable, we introduce a set of new variables and . Clearly, and can be viewed as the actual transmit energy of the sources and HRN, respectively. Problem (P1) is equivalent to the following problem:

(15a)
(15b)
(15c)
(15d)
(15e)
(15f)
(15g)

where

(16)
(17)

Since constraint (15d) is convex and the other constraints of Problem (P1) are affine, Problem (P1) is convex in its current form. In the literature [21, 22, 23, 24], the first-order method can be used to solve these non-convex problems by approximating the non-convex objective functions and constraints into convex ones. However, in this paper, by appropriate variable transformation, Problem (P1) is reformulated to be convex, which can thus be optimally solved by applying the Lagrange duality method, as will be shown next.

We first introduce non-negative Lagrangian multipliers and associated with the rate constraint (15d), associated with the energy causality constraints (15e) and (15f). In addition, non-negative Lagrangian multipliers and are associated with the total time constraint (15b) and total energy constraint at HRN (15c). Then, the Lagrangian of Problem (P1) is given by

(18)

Denote as the set of satisfying the primary constraints, then the dual function of Problem (P1) is given by

(19)

To compute the dual function , we need to find the optimal to maximize the Lagrangian under the given dual variables . In the following we present the derivations in detail.

Iii-a Optimizing for Given

Iii-A1 Maximizing Lagrangian over

The part of the dual function with respect to the rate variable is given by

(20)

To make sure that the dual function is bounded, we have . In such case, [25] and we obtain that . Note that such that is non-negative. By substituting these results above into (18), the Lagrangian can be rewritten as:

(21)

Iii-A2 Maximizing Lagrangian over and

Observing the Lagrangian in (21), we find that the dual function in (19) can be decomposed into independent functions:

(22)

where

(23)

with

(24)

For given dual point , maximizing (21) over is equivalent to solving (23) for From (III-A2), the partial derivatives of with respect to and can be given by (25) and (26) on the top of the next page.

(25)
(26)

Given , the optimal energy variables and that maximize can be obtained by setting and and are given by (27) and (28) on the top of the next page.

(27)
(28)

With given and , we can easily prove that is a decreasing function of . As a result, the optimal with given and can be found by a simple bisection search over

To summarize, for , Problem (23) can be solved by iteratively optimizing between and with one of them fixed at one time, which is known as block-coordinate descent (BCD) method.

Iii-A3 Maximizing Lagrangian over and

Next, we study the solution of Problem (23) for

, which is a linear programming problem (LP). From (

III-A2), to maximize we have

(29)
(30)

Iii-B Optimizing Dual Variables

As a dual function is always convex [26], we adopt the ellipsoid method to simultaneously iterate the dual variables to the optimal ones by using the defined subgradients as follows:

(31)

Iii-C Discussion on Optimality and Complexity

The optimal , and for are obtained at optimal , then the optimal is given by . With , and , Problem (P1) becomes a LP with variable . The optimal value of is obtained by solving this LP.

To summarize, the algorithm to solve Problem (P1) is given in Algorithm 1. The time complexity of steps 3-7 is of order . The complexity of step 9 is . Therefore, the complexity of steps 3-9 is given by . Note that step 10 iterates to converge, where is the number of dual variables and in our case. Thus the complexity of steps 1-10 is . The time complexity of the LP is . Therefore, the complexity of Algorithm 1 is .

1:  Initialize .
2:  repeat
3:     Initialize .
4:     repeat
5:        Compute and by (27) and (28), respectively.
6:        Obtain with given and by bisection search.
7:     until improvement of converges to a prescribed accuracy.
8:     Compute and by (29) and (30), respectively.
9:     Update according to the ellipsoid method via (31).
10:  until  converge to a prescribed accuracy.
11:  Set , , for , and .
12:  Obtain by solving Problem (P1) with , and.
Algorithm 1 Optimal Algorithm for Problem (P1)
Proposition iii.1

For the TDMA case with source-destination pairs and , the maximum sum-rate by solving Problem (P1) is achieved by .

Proof: Clearly, we have and ; otherwise, no energy will be harvested at the sources. Since the objective function of Problem (P1) is an increasing function of for from constraint (15d), when it comes to the extreme case with , for any given and satisfying constraints (15c), (15e) and (15f), the optimal solution must be achieved by according to constraint (15b). In this case, and are required to guarantee positive harvested energy at the sources. The proof is thus completed.

Proposition iii.2

For the TDMA case with source-destination pairs and finite , the maximum sum-rate for Problem (P1) is achieved by .

Proof: Please refer to Appendix A.

By Proposition III.1 and Proposition III.2, it can be inferred that Problem (P1) is actually a problem of energy and time allocation at the HRN, i.e., allocating energy and time for WPT and each WIT. Therefore, for any given energy allocated for WPT (i.e., ), the HRN should charge the sources at its maximum available power (i.e., ), so that the time used for WPT can be as small as possible and more time can be allocated to WIT due to the sum-rate maximization goal. In particular, when , the portion of transmission time for WPT should asymptotically go to zero, which means that the sources can harvest sufficient energy in a sufficiently small time and almost whole time is allocated to WIT.

Iii-D Suboptimal Algorithm

The complexity of the optimal algorithm becomes high as the number of pairs increases, mainly due to the dual updates. By simplifying the system model and eliminating the dual updates, in this section, we present an efficient suboptimal algorithm which significantly reduces the complexity.

At first, in WIT phase, the received power at each source in other periods is from the relay and other sources, which are both small. Specifically, the received energy from other sources is negligible due to the double energy decay, i.e., the energy decay of relay-to-source and then source-to-source. As DF relaying protocol is adopted, the transmission power of the relay could relatively match the source’s transmit power, and thus the relay’s transmit power for forwarding is also small. As a result, in this section, we consider that the harvested energy at the sources is only from the WPT phase. With give , the transmit power of source can be given by

(32)

Second, due to Proposition III.2, we let . Moreover, we assume that the equal power allocation (EPA) at the HRN in the WIT phase, the transmit power at the HRN for pair is thus given by

(33)

Third, due to the energy decay in the WPT phase, the transmit power of sources may be small, thus the performance of this considered dual-hop relaying system may depend on the rate of first hop under most cases. As a result, in this section, we only focus on maximizing the sum rate of the first hop. Therefore, we have the following problem:

(34)

where .

Proposition iii.3

The optimal solution of Problem (34) with given is given by

(35)
(36)

Proof: Please refer to Appendix B.

With given , we can obtain a set of by (33), (35) and (36). Then, the optimal maximizing the sum-rate can be found by the one-dimensional search.

1:  Divide in with fixed step .
2:  for  each  do
3:     Compute the time allocation for WIT according to (36).
4:     Compute the power allocation for WIT and according to (33) and (35), respectively.
5:     Compute the sum-rate according to (5) with given .
6:  end for
7:  Choose the optimal that has the maximum sum-rate.
Algorithm 2 Suboptimal Algorithm for Problem (P1)

To summarize, the above suboptimal algorithm is given in Algorithm 2. The complexity of steps 3-5 is . The complexity for searching is . Therefore, the whole complexity of Algorithm 2 is , which is linear in and much lower than that of the optimal algorithm in above subsection.

Iv Resource Allocation in FDMA Case

Problem (P2) is a mixed integer programming and thus is NP-hard and non-convex. However, it has been shown that the duality gap of the resource allocation problems in FDMA systems becomes zero when the number of SCs goes to large [27, 28]. This means that the optimal solution obtained in dual domain is equivalent to the optimal solution of the original non-convex problem due to the zero duality gap. Thus we solve Problem (P2) in dual domain.

At first, we introduce non-negative Lagrangian multipliers and corresponding to the two rates of the first and second hops in (14b), and associated with the energy causality constraint (10). Moreover, , are introduced to associate with the total time constraint (7) and total energy constraint (9), respectively. Then the dual function of Problem (P2) can be defined as

(37)

where is the set of all primal variables satisfying the constraints, and the Lagrangian of Problem (P2) is

(38)

Computing the dual function requires to determine the optimal for given dual variables . In the following we present the derivations in detail.

Iv-a Optimizing for Given

Iv-A1 Maximizing Lagrangian over

Similar to TDMA case, the part of dual function with respect to is given by

(39)

To make sure that the dual function is bounded, we have . In such case, and we obtain that Note that to make sure that is non-negative. By substituting the result above into (IV), we have

(40)

Iv-A2 Maximizing Lagrangian over , , and

Observing the Lagrangian in (IV-A1), we can rewrite (IV-A1) as follows: