I Introduction
Ia Motivation
The limited lifetime of batterypowered wireless devices (WD)s in a conventional wireless communication network has always been a fundamental bottleneck for which the only solution has been to manually replace or recharge the batteries after depletion. Yet, there exists applications where doing so is laborious or even impractical^{1}^{1}1Consider, for example, a large number of WDs deployed in a large area, or a sensor implanted in the human body.. Wirelesspowered communication (WPC) has recently emerged as a promising solution to prolong the lifetime of such energyconstrained WDs [1, 2]. WPC can be used in a variety of applications, such as IOT networks, RFID systems, and wireless sensor networks (WSNs) [3, 4]. It uses RFenabled wireless energy transfer (WET) [1] as a means to wirelessly supply the energy of WDs, thus enabling them to function seamlessly without the need of battery replacement/ recharging. WET can be defined as the use of electromagnetic (EM) waves to transfer energy from an energy transmitter (ET) to an energy receiver (ER) over the air [2]. As with many new technologies, WET raises its own issues however. First, since the power signal from the transmitter is severely attenuated over distance, the problem of transferring sufficient energy over even moderately large distances is not trivial. Second, in many cases there are multiple ERs which could all be mobile. Therefore, the scheme needs to be adaptable to multiple receivers and at the same time robust to mobile scenarios [1]. When multiple antennas are available at the ET, the solution is to carefully weight the transmitted signals from different antennas in such a way that they superimpose constructively at the ERs and destructively everywhere else [3]. This technique, referred to as energy beamforming (EB), results in the concentration of the transmitted wave into narrow beams, and thus enables the ET to deliver ample energy to the ERs [2, 5, 6]. WET has been studied in a number of works. In [7]
, the probability of outage, and in
[8, 9, 10],also 15201087 channel acquisition and training methods for WET systems have been studied. In [6] fairnessaware EB methods were investigated and in [2] a pareto optimal energy beamformer (EB) was derived. In what follows, we concentrate on a crucial application of WET, namely, WPC.IB Wireless Powered Communication
WPC is the result of using WET in a wireless communication network whose purpose is wireless information transmission (WIT). There are currently two different lines of research in WPC: simultaneous wireless information and power transfer (SWIPT) [5] and 04595260 and wireless powered communication network (WPCN) [2, 11]also 06954434. In SWIPT, both energy and information are carried via the same RF signal, while in WPCN the AP transfers energy to the WDs in downlink (DL), and the WDs perform uplink (UL) WIT using the harvested energy. This means that, conceptually in SWIPT, data is transmitted to the ER, whereas it is transmitted by the ER in case of WPCN. Although in both cases of WPCN and SWIPT, the harvested energy decays rapidly with respect to the distance between the wireless device (WD) and the AP, power constraints for WPCN are more stringent than those of SWIPT. This is so because, while in SWIPT the harvested energy is only needed to keep the WDs alive for energy harvesting (EH) and information decoding (ID), in WPCN the transmitted power in the UL WIT is supplied by the DL WET. This is fundamentally more difficult as the WD is now required to actively transmit, rather than silently “listen”. WPCNs have been recently studied in the literature. References [1] and [3], for instance, provide overviews of possible WPCN configurations and the techniques employed in such networks. In what follows we briefly discuss the challenges encountered in designing a WPCN with regard to our design paradigm and introduce some of the relevant papers that discuss such issues.
IB1 Duplexing
The most fundamental challenge in WPCN (and SWIPT) design is the socalled energyinformation tradeoff. This tradeoff exists because energy and information both use the same communication resources, such as time and bandwidth (BW). Since in WPCNs information and energy are transferred in different directions, this tradeoff is achieved via duplexing techniques in such networks.

When timedivision duplexing (TDD) is used [2, 11, 12, 15, 13, 14, 16, 17, 18, 19, 20]also 06954434, the challenge is to determine the optimal time lengths during which the UL WIT and the DL WET occur while the advantage is that channel reciprocity may be taken advantage of for channel state informtion (CSI) acquisition at the transmitter when multiple antennas are available there [21]. Yet, due to the orthogonal time allocations of the DL and UL channels, WET can not occur continuously. This is a restriction for energy transmitters having a peak transmit power, limiting the total amount of delivered energy.

Although frequency division duplexing (FDD) has not been thoroughly studied in WPCNs, it has been more successfully implemented in wireless communication networks in general [21]. When used in WPCNs [23, 22, 20], the total available BW should be optimally allocated to the DL and UL channels. However, since channel receiprocity does not apply to such a system, the CSI must be fed back to the access point [24, 21, 22, 23]. Hence, the optimum design should specify how much CSI feedback is needed to achieve the best performance. In this scheme, however, the energy may be continuously transmitted, posing less restriction to the peak transmit power.
IB2 Multiple Access
While singledevice scenarios have been considered in [22, 23, 25, 16, 19], WPCNs may also be used for multiple WDs [2, 14, 13, 17, 18]also 06954434, 07386616. When multiple devices are considered, multiple access techniques should be utilized for the UL channels. In [12, 14, 18] TDMA is used for which the challenge is to optimally partition the frame length into orthogonal time slots serving different WDs. When the HAP is equipped with multiple antennas, spacedivisionmultipleaccess (SDMA) may be utilized [2, 11, 15]06954434 which enjoys a higher spectrum efficiency than TDMA. Finally, in [13] nonorthogonal multiple access (NOMA) technique has been employed. Note that due to the intrinsic broadcast property of wireless transmission, the transmitted energy in the DL can be harvested by all WDs and multiple access techniques are pointless for WIT. Nevertheless, when the HAP is equipped with multiple antennas, EB may be utilized to generate multiple beams aimed at multiple WDs. This may be viewed as a multiple access technique for energy where the purpose is to send desired amounts of energy to each WD.
IB3 Fairness
WPCNs can have separate APs for the purpose of information reception and energy transmission for which case the APs are referred to as data access points (DAPs) and energy access points (EAPs) respectively [25]. On the other hand, the DAP and EAP may be collocated in which case it is called a hybrid access point (HAP). References [11, 2, 12, 13, 16, 17]also 07386616, 06954434 consider a system with a HAP, while in [25, 14, 22] both scenarios are considered. When HAPs are emplyed, the WDs located far from the HAP, harvest less energy in each block compared to the closer WDs. Furthermore, if they are to enjoye the same SINR and hence throughput, WDs located far from the HAP should consume more power when transmitting data in the UL compared to closer WDs [3]. This creates a severely unfair situation referred to as the doubly nearfar effect [11] which substantially starves far WDs of rate if rate fairness is not considered. To address the this issue, in [11, 2] maximizing the minimum rate, among the WDs is considered.
IC This Paper
In this paper, a FDD multipleuser multipleinputsingleoutput (MUMISO) WPCN consisting of one HAP with multiple antennas and a set of distributed singleantenna WDs, as illustrated in Fig. [fig.schematic], is studied.
We try to address all of the aforementioned challenges for a FDD MUMISO WPCN; specifically

Given a certain amount of available BW, the maximum allowed power spectrum density, and a fixed frame length, the optimal UL and DL BWs as well as the optimal amount of CSI feedback time lengths are calculated.

It is shown that, under finiterate CSI feedback, the beamformer for this problem can be pareto optimal when each WD sends at least one feedback bit.

To ensure rate fairness among the WDs, the optimization is performed such that the minimum throughput among the WDs, referred to as the minimum WIT data rate, is maximized. In order to achieve this, the HAP allocates more wireless power to the WDs located farther away from the HAP relative to the others.

We will define a metric called the fairness radius which describes a circular boundary around the HAP. We will show that the WDs whose distances to the HAP are greater or equal to this radius achieve equal data rates and those whose distances are less than this radius attain higher data rates than the rest of the WDs.

It is shown that as the number of HAP antennas goes to infinity, the optimal CSI feedback phase time length ratio, the optimal DL BW ratio, and the fairness radius tend to zero.
Uppercase and Lowercase: We will use bold lowercase and uppercase letters for column vectors and matrices respectively, nonbold lower or uppercase Latin or Greek letters for scalars, and caligraphic letters for sets. Matrices and Vectors: For any arbitrary m by n matrix (mvector)
, and represent its transpose and conjugate transpose respectively. Defining sets , and , submatrix (subvector ) is a matrix (vector) whose elements are () in the original column and row order (original order). Standard Vectors: , , and will be used to represent allone, allzero and the  canonical basis vector of (a vector of all zeros, except for the  place, where it is one), whilewill represent as an identity matrix of order
. Hadamard Product: For two vectors and of the same dimension, represents the elementwise or the Hadamard product. Vector Operations: In a small abuse of notation, we interpret vector exponentiation in an elementwise fashion; i.e. for and , vectors and are vectors of the same dimension as for which . Positive Operator: Furthermore, for vector , is a vector of the same dimension for which and for vector , is a diagonal matrix of order for which . Ineuality overload: We overload inequality symbols to apply to real vectors of the same dimension in a componentwise fashion. That is, for , () means () . Expectation: Finally, stands for the statistical expectation operator and is used to represent the pnorm of vector . In section II, we present the system model which consists of the frame structures and data rates, transfer and consumption of power to and in the WDs, as well as the representation of the UL and DL channels in the system. In addition, a table of notations will be provided for future reference. We next solve what we call the forward problem; that is, we derive an explicit expression for the UL WIT data rates of all WDs in terms of all parameters of the problem. In order to do so, we need to study the UL and DL channels separately, and then equate the harvested power in the DL to the consumed power in the UL, tying together the two channels. Based on the results obtained, in section IV of the paper, these parameters are optimized such that the minimum UL WIT data rate among all WDs is maximized. We then analyze the performance of the system in asymptotic regime when the number of antennas goes to infinity. The next section provides simulation results to examine the truthfulness of our analytical findings as well as to offer the reader some interesting intuition. The paper is finally concluded in section VII.Ii System Model
Consider a singlecell FDD WPCN consisting of a HAP equipped with antennas and singleantenna WDs denoted by . We assume that the WDs are sorted in an increasing order of their distances to the HAP. We begin describing the model of this system with its frame structure which explains how different phases of information or power transmission are scheduled. Then, we describe power transmission: the HAP sends a certain amount of power to each WD and the WDs harvest a portion of that power along with some additional interference power and utilize it to send data and feedback to the HAP.. Finally, the UL and DL communication channels are characterized in the last subsection.
Iia Frame Structure and Data Rates
The HAP spends the whole frame length, designated as , transmitting wireless energy to the WDs. Using the harvested energy and through SDMA, all WDs, on the other hand, send data to the HAP in time , and send CSI feedback in time , where is the CSI feedback phase time length ratio and should satisfy . We represent the  WD average UL data rate associated with the CSI feedback phase and the WIT phase by and respectively, where is the total UL data rate for the  WD. In the following sections, we will be switching back and forth between scalar notations, , , and ; and , , and frequently. It is further assumed that the HAP and all WDs are perfectly synchronized. The UL and DL frame structures are shown in Fig. [fig.frame].
IiB Power and Energy
We assume the HAP has access to a permanent wired power supply but the WDs only receive wireless energy from the HAP through WET. In particular, the HAP continuously transmits modulated power signals [1] to the WDs with a maximum power spectral density of in a BW of , where is the DL BW ratio and is the total BW. We assume the power is transmitted with maximum power spectral density and so the total DL transmit power is . However, the total DL transmit power should not exceed maximum HAP power budget . Moreover, the distribution of the harvested power among the WDs is not uniform. In fact, as will be shown later, the HAP should send more power to WDs located farther from the HAP than those closer to it. This is achieved through energy allocation weight vector which, because of the positivity of energy and the total DL power constraint, should satisfy , and respectively. On the other hand, the expected harvested energy by wireless devices 1 through is represented by expected harvested energy vector . In our model, data transmission by WDs consumes all of the harvested power which is safe to assume for most lowpower circuits. In addition, because of the zeroforcing (ZF) receivers we will later employ at the HAP, the UL data transmission of the WDs cause no interference to each other and hence there is no reason not to transmit with maximum power [24]06954434. As a result, the WDs transmit with constant UL power vector which means that the UL WIT and CSI feedback phases use equal powers. As a result, the energy consumed in these phases becomes proportional to their time lengths, i.e. , and respectively. Note that we assume there is no power loss in the system, i.e. the AP and the WDs transmit and receive with absolute efficiency respectively. In addition, our analysis only holds at the steady state where in each frame the amount of energy the WDs harvest is equal to the amount of energy they transmit.
IiC Uplink and Downlink Channels
We assume no dominant lineofsight propagation path between the HAP and the WDs exist and therefore we adapt the Rayleigh fading model for both the DL and UL channels. In what follows we describe how the UL and DL channels are described using this model. Assuming are component UL channel vectors, we define the UL channel matrix by compiling into a matrix where is modeled as
in which is the UL Rayleigh fading coefficient matrix satisfying , where
stands for circularly symmetric complex gaussian (CSCG) random variable with mean
and variance
, stands for “distributed as”, and is the component largescale fading coefficient vector each element of which, , is assumed to be a known constant both at the HAP and modeling channel path loss between the HAP and [2, 27]. We assume the following longterm fading model holds(1) 
where is a constant representing attenuation at the reference distance , is the pathloss exponent and is the component distance vector of the WDs to the HAP. Similarly, let us assume are component DL channel vectors and define the DL channel matrix by compiling into a matrix and model as where is the DL Rayleigh fading coefficient matrix satisfying [2]. The DL channel vector for
is estimated and then sent back to the HAP via CSI feedback. We assume that the DL channel vector is estimated sufficiently well at each WD such that channel estimation error incurs negligible performance degradation. The HAP receives the quantized, and normalized fedback DL channel vector
which we call fedback DL channel vector for brevity. For the sake of consistency of notation, these vectors are represented as columns of quantized, and normalized fedback DL channel matrix which we call fedback DL channel matrix for brevity. Lastly, we assume that the BW allocated to DL and UL channels are and respectively and that both channels are quasistatic flatfading, where , and consequentlyare constant during each block, but can change from one block to another in accordance with the fading probability density function (PDF). The latter assumption, called
block fading, is reasonable to make in systems with stationary nodes or systems where nodes move at walking speed. Parameter Description Dimension M Number of HAP antennas K Number of WDs CSI feedback phase time length ratio CSI feedback phase time length ratio Total UL data rate UL WIT data rate UL Feedback data rate maximum power spectral density Total Bandwidth Maximum HAP power budget energy allocation weight vector UL power vector expected harvested energy vector UL channel vector DL channel vector UL channel matrix DL channel matrix UL Rayleigh fading coefficient matrix DL Rayleigh fading coefficient matrix largescale fading coefficient vector fedback DL channel vector fedback DL channel matrix constant attenuation at the reference distance pathloss exponentIii The Forward Problem
In this section, we analyze the UL and DL channels to solve the forward problem of calculating the WIT data rate for every WD in term of the optimization varaiables. In other words, we will derive an explicit formula for . Note that vector is a UL parameter. On the other hand, while and are related to both DL and UL, is related to DL only. As a result, to arrive at the desired formula, we will first examine the UL and DL channels separately, and then we will combine the results to arrive at the final explicit formula.
Iiia Uplink Transmission
As mentioned earlier, the UL transmission consists of two phases, namely a CSI feedback phase of length and a WIT phase of length which we will analyze in this subsection. In particular, we derive a formula for the total UL data rate of the  WD and express the DL channel vector error incurred through feedback in terms of CSI feedback phase time length ratio .
IiiA1 Wireless Information Transmission
Let be the component received complex baseband signal vector at the HAP in the WIT or CSI feedback phase
(2) 
where is the component UL information carrying signal vector, and is the component UL noise vector, where is the UL noise variance. A linear detector is used at the HAP to detect the signal transmitted by WDs. Here, we use the ZF detector
(3) 
which yields component detected signal vector given by
(4) 
Letting denote the  column of , the  WD’s detected signal, shown by can be expressed as
(5) 
using which we can calculate , the signaltointerferenceplusnoiseratio (SINR) for
(6) 
and the achievable total UL data rate for can therefore be written as , where is the UL BW. A lower bound for the total UL data rate for is
(7) 
where
(8) 
Following a procedure similar to that in [2, lemma 4], we can simplify this expression to
(9) 
It should be emphasized that the analysis in this subsection applies to both phases in the UL transmission. In the rest of the paper, we will assume and use and in lieu of diacritical characters and . Therefore, using the definition for and ([eqShannon])
(10) 
IiiA2 CSI Feedback
In the CSI feedback phase, the directions of the estimated channel vectors are fed back to the HAP via CSI feedback. This is done using a socalled codebook, known both to the HAP and the WDs. Consequently, the WDs only need to send the index of the closest code (vector) to the HAP, hence feedingback the channel information. Note that, generally, the optimal vector quantizer for this problem is not known. One approach for creating the codebook is to choose all of the quantization vectors independently from the isotropic distribution on a unit sphere of Mdimensions [28, 24], also Jindal11a technique referred to as Random Vector Quantization (RVQ). RVQ is easy to analyze and its performance is very close to optimal quantization [24]. We assume that the number of feedback bits for is given by and define the DL channel vector feedback quantization error for as
(11) 
In [24] also LoveRef it was shown that
(12) 
where is the beta function defined by where is the gamma function. This upper error bound will be used in the DL section to derive the harvested energy formula.
IiiB Downlink Transmission
The DL transmission consists of a DL WET phase only where the DL transmission power is transferred via component beamforming vector , where we assume . This vector is used to adjust the energy transmit direction adaptively according to the instantaneous CSI of each frame [12]. WDs 1 through receive the complex baseband signals where vector is expressed as
(13) 
in which is used to represent the WD noise vector whose elements we assume are negligible. As a result, the expected harvested energy vector is
(14) 
In words, the expected harvested energy by is proportional to the square of the vector projection of the DL channel vector onto the beamforming vector [6]. We now need to find the pareto optimal energy beamformer for this problem which maximizes the harvested energy for a given energy allocation weight vector. This problem is formally defined as the following vector optimization problem
(15a)  
subject to  (15b) 
This is an optimization problem with a vectorvalued objective function for which the set of achievable objective values does not have a maximum element, but rather a set of maximal elements, hence the name pareto optimal [26]. In lemma [lem:pareto.optimal.beamformer] we obtain such a pareto optimal beamformer and derive a simple sufficient condition of its pareto optimality. The pareto optimal beamformer can be written as a linear combination of the normalized fedback channel estimates
(16) 
provided that each WD sends at least one feedback bit. Proof: See Appendix [app.pareto.optimal.beamformer]. Equation ([eqParetoOptimalBeamformer]) means that the allocated energy should be sent along the quantized, and normalized fedback DL channel vectors of WDs. Therefore, in the absence of any feedback error, the pareto optimal beamformer becomes a linear combination of the set of DL channel vectors. Note that when the HAP intends to send energy to a single WD, for example wireless device 1, we have , and the beamforming vector reduces to , which maximizes the harvested energy for a single WD and is known as maximum ratio transmission (MRT) [12]. Therefore ([eqParetoOptimalBeamformer]) is a direct extension of MRT. We expect that the more power the HAP allocates to a DL channel vector, the more power the WD corresponding to that specific DL channel vector receives. This is in fact true and is verified in lemma [lem:EnergyVector] where we derive the amount of expected energy harvested by WDs.In lemma [lem:EnergyVector] we derive the amount of expected energy harvested by WDs. The expected harvested energy vector is given by
(17) 
where is the mixing power matrix for which and whose offdiagonal elements are all one. Proof: Following a similar procedure as that in [2, lemma 2] and using [29]
(18) 
we can derive
(19) 
which can be compactly written as ([eqEnergyVectorTwo]). Two observations are in order. First, from the structure of the mixing power matrix , not to be confused with the number of HAP antennas , we realize that, assuming DL channel vector feedback quantization error for a particular WD is negligible, the energy harvested from the beam aimed at this WD is multiplied by the number of HAP antennas , while the power harvested from beams aimed at other WDs is multiplied by one. We call the first and the second term the beamed energy and the interference energy respectively. It is because of this multiplication factor that the HAP can control the distribution of power among the WDs. Second, whether the multiplication factor is effective depends upon the amount of DL channel vector feedback quantization error as a large error makes the diagonal elements diminish. In the extreme case, when , the beamforming vector becomes essentially random with respect to the channel vector and hence the diagonal element for becomes 1. This is verified mathematically if we note that from ([eqFeedbackErrorTwo]), the expected value of the DL channel vector feedback quantization error squared is equal to at . When the feedback is eliminated for all the WDs, the harvested energy becomes , at which point the HAP totally fails to control the distribution of energy among the WDs.
IiiC Solution to the Forward Problem
So far, we have analyzed the DL and UL channels separately. Yet, these channels are coupled through energy. In this subsection we combine the DL and UL formulas to obtain an explicit expression of the total UL data rate for every WD in terms of our decision variables , and . Using ([eqSINRThree]) and ([eqEnergyVectorTwo]), the SINR vector can be written as
(20) 
Let us decompose to arrive at a more intuitive formula
(21) 
where s diagonal and offdiagonal elements are and one respectively and . Then can be written as where
(22) 
(23) 
We can express as
(24) 
where
(25) 
Combining equations ([eqShannon]), ([eqFeedbackErrorTwo]), and ([eqLambdaLossTwo]) we get
(26) 
This equation means that the overall effect of CSI feedback is to reduce the SINR by which is a decreasing function of . This equation, however, is implicit, because the WIT data rate loss for a WD is affected by the number of CSI feedback bits that is being transmitted which, itself depends upon the total UL data rate for that particular WD. In lemma [lem:UL.data.rate], we derive an explicit formula for the UL data rate loss of every WD, from which the WIT data rate may be easily calculated. The DL channel vector feedback quantization error for is equal to
(27) 
Proof: See Appendix [app.UL.data.rate]. Using ([eqDataRateWIT]), ([eqLambdaLossTwo]), and ([eqFeedbackErrorThree]) we can get an explicit formula relating the WIT data rate of every WD in terms of our decision variables
(28) 
We can now proceed to optimization.
Iv Throughput Optimization
To optimize the throughput and improve the WIT data rates while attaining fairness, we propose to maximize the minimum WIT data rate among all WDs 07386616; that is, to solve the following optimization problem
(29a)  
subject to  (29b)  
(29c)  
(29d)  
(29e)  
(29f) 
As can be seen, the optimization variables are CSI feedback phase time length ratio , DL BW ratio , and energy allocation weight vector which are interrelated as follows: The amount of energy harvested by every WD in the WET phase is basically controlled by energy allocation weight vector and DL BW ratio but is also affected by CSI accuracy. On the other hand, CSI accuracy is dependent upon the DL channel vector feedback quantization error which is, in turn, determined by feedback data rate . The feedback and WIT data rates for every WD depend on the length of the feedback and WIT phases, i.e. and respectively, the UL channel BW , and the corresponding SINR at the HAP. Finally, the SINR at the HAP is related to the harvested energy by the WD in question. Solving ([prOptimizationOne]) efficiently depends upon the fact that whether or not the problem is convex, which, in turn, requires ([eqXiPositivityCostFunc]) to be a concave function. Proving the concavity of ([eqDataRateWITThree]), however, is difficult. Instead, in order to prove the existence and uniqueness of the solution of ([prOptimizationOne]), we use ([eqFundamentalOne]) and assume it is solved recursively and therefore the data rate at the previous recursion is constant; making ([eqFundamentalOne]) explicit at each iteration. Then, we prove the convexity of the WIT data rate equation at each recursion and show the convergence of recursions through simulations. The proof for concavity of ([eqFundamentalOne]) is given in the following lemma. When solved recursively, ([eqFundamentalOne]) is concave with recpect to , , and . Proof: Using ([eqDataRateWIT]) and ([eqLambdaOne])
(30) 
Note that
(31) 
therefore, assuming is constant, is a nonnegative concave function of . This means that is a product of nonnegative affine with concave functions. As a result, it is a log concave function. In addition, it can be shown that and therefore its product with affine functions; that is is logconcave w.r.t. , , and too. In what follows we proceed to find the optimal values of our decision variables one by one.
Iva Energy Allocation Weight Vector
As distances of the WDs from the HAP can be widely different, DL signal attenuation and hence the harvested power varies greatly among different WDs. Moreover, due to the UL signal attenuation, farther WDs from the HAP have to transmit with greater power so that the received signal at the HAP has the same SINR, a problem referred to as the “double nearfar” [2], or “doubly nearfar” effect [3, 11]. Thus, the energy allocation weight vector needs to be chosen in such a way as to cancel out this problem. Suppose that we can partition the WD index set into two sets and where and . In theorem [thm:OptimalXiup], the optimal energy allocation weight vector for this problem is derived. Before that, however, lemma [lem:ZeroXiup] characterizes . The only case where the WIT datarate for a WD is different from the others is when its energy allocation weight coefficient is zero. Proof: Forcing the WIT datarates for all WDs to be equal demands that the SINR at the HAP be the same for every WD
(32) 
where is the common SINR value. Note, however, that this equation neglects the nonnegativity of . The fact that some elements of should be negative to ensure fairness means that fairness cannot be achieved for such WDs. The best choice, then, is to set the energy allocation coefficient of such WDs to zero. Based on this lemma, we can define sets and as follows
(33a)  
(33b) 
where superscripts and stand for fair and unfair respectively. Assuming sets and are known, we now proceed to calculate the optimal energy allocation weight vector The optimal energy allocation weight vector is given by
(34a)  
(34b) 
Proof: In lemma [lem:ZeroXiup], we explained why the energy allocation vector of is set to zero. On the other hand, for every , the WIT data rate and as a result, the SINR should be the same at the HAP
(35) 
where is the common SINR whose value is determined upon normalization of vector. The solution to this equation is ([eqOptimalXiupfair]). Note that the condition for paretooptimality of the beamformer (that is, each WD feeding back at least one bit) ensures invertibility of and its submatrices. The procedure by which and are determined is as follows. We begin by setting , and . After computing the energy allocation weight vector, some of its elements may be negative. If this is the case, then their indices should be added to and excluded from and the vector should be recomputed. This process is repeated until the resulting energy allocation weight vector is nonnegative. According to ([eqLongTermFading]), as the distance of to the HAP decreases, the largescale fading coefficient is increased. In a network having more than one WD, this, according to ([eqOptimalXiupfair]), leads to a decrease in the corresponding energy allocation weight coefficient . As moves even closer to the HAP, reaches zero, from which point onward and moving the device closer to the HAP has no effect on . On the other hand, doing so will further increase , which, according to ([eqDataRateWITTwo]) increases . Qualitatively, bringing closer to the HAP without changing will result to receive more power than needed which gives rise to a higher maximum WIT data rate than other WDs. According to these definitions, ([eqLambdaMax]), ([eqLambdaLossOne]), ([eqFundamentalOne]), and assuming (This condition makes sure the diagonal elements of matrix are positive.)
(36a)  
(36b) 
where is the common data rate in the fair region. This can be justified because, in addition to the beamed energy controlled by , every WD receives some interference energy as well. At very low distances, the interference energy a WD receives alone might be sufficient or even more than sufficient to power its UL transmission so as to achieve the desired UL WIT data rate. Hence the corresponding energy allocation weight coefficient becomes zero at such distances. Geometrically, as shown in Fig. [figFairnessFig], the area around the HAP can be divided into two regions: the unfair region defined by and the fair region defined by where is the distance to the HAP and is the fairness radius defined by the following theorem. The fairness radius is given by
(37) 
Proof: The fairness radius is defined to be the distance at which the interference energy is exactly equal to the energy needed for the WDs placed at this distance to achieve the intended UL WIT data rate. In order to calculate this distance, we first simplify using the ShermanMorrison formula
where . Next, we set the  element of in ([eqOptimalXiupfair]) to zero
Simplification gives
Substituting the path loss model for , ([eqFairnessRadiusOne]) results. Note that, for a given system, the fairness radius is not fixed and depends on the distances of the WDs to the HAP. Assuming the availability of full CSI at the HAP, and therefore
Since , this equation may further be approximated by
(38) 
This formula simply means that the fairness radius roughly only depends on the maximum distance of the WDs to the HAP. The closer the farthest WD is to the HAP, the lower this radius becomes and vice versa.
IvB Downlink Bandwidth Ratio
Now that the optimum for a particular choice of and has been found, we can use this information to simplify the optimization problem ([prOptimizationOne]) Having computed , the optimal and can be calculated from the following simplified optimization problem
(39a)  
subject to  (39b) 
Proof: From ([prOptimizationOne]), and are the maximizers of . Yet, assuming and considering ([eqRMaxKSet]),
On the other hand, it is clear that the fair region always includes the farthest WD from the HAP, i.e. which completes the proof. Increasing the DL BW by increasing DL BW ratio increases the harvested power by the WDs which leads to a higher SNR at the HAP and ultimately a higher WIT data rate. Nevertheless, increasing the DL BW at the same time decreases the UL BW which directly decreases the WIT data rate. Hence, an optimal value for the DL BW ratio exists which we will subsequently find. In the following theorem, to show the dependence on explicitly, instead of , , and , we will use , , and respectively. The optimal DL BW ratio is given by
(40) 
where is the principal branch of the Lambert W function. Proof: Taking the derivative of ([eqShannon]) for with respect to , we get
Setting yields the solution to which is
where the principal branch of the LambertW function is used because its argument is always positive. Now, imposing ([eqBetaConstraintEq]) upon this equation gives ([eqOptimalBeta]). It can be shown that for we have . On the other hand, we earlier assumed . Therefore, ([UeqBetaLimitEq]) is satisfied as well. In a powerconstrained system, the first argument of the min operator and in a BWconstrained system the second argument takes hold.
IvC CSI Feedback Phase Time Length Ratio
Increasing improves CSI knowledge at the HAP which increases the WIT data rate. At the same time, however, it decreases the WIT phase time length which lowers the WIT data rate. So, an optimal value for exists which we will subsequently find. Using ([eqOptimizationTwo]), we derive an iterative solution for the optimal CSI feedback phase time length ratio in theorem [thm:optimal.alpha]. Nevertheless, before that, we shall prove the optimization problem ([eqOptimizationTwo]) is convex w.r.t ^{2}^{2}2Note that in lemma [lem:convexity1] we showed that the problem is quasiconvex. But quasiconvexity is not enough for the convergence of an iterative solution to a global optimum.. Optimization problem ([eqOptimizationTwo]) is convex w.r.t . Proof: Please refer to Appendix [app.convexity.2]. Now we can proceed to find the optimal . The optimal , denoted by is given by
(41) 
where
(42a)  
(42b)  
(42c) 
and is the iteration number. Note that, in practice, a few iterations are sufficient. Proof: Rewriting the  element of ([eqFundamentalOne]) in terms of gives
Taking derivative with respect to results in
But
Substituting
and setting results in
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