Performance Analysis of Beacon-Assisted Wireless Powered Communications With Spatially Random Sensors

In this work, we present the performance evaluation of a power beacon-assisted wireless-powered communication system in the presence of multiple co-channel interferers. We consider and compare three distinct interference modeling scenarios, namely the totally induced interference is approximated by the Gamma distribution, interferers are distributed according to a Poisson point process and interferers are distributed according to a binomial point process. Analytical expressions for the outage probability and the ergodic capacity of the considered system under each interference scenario are derived. Furthermore, optimization problems are formulated to determine the optimum switching time maximizing the achieved ergodic capacity. Numerical results accompanied with Monte Carlo simulations are presented to corroborate the proposed theoretical analysis.


I. INTRODUCTION
Radio frequency (RF) energy harvesting (EH) using wireless power transfer (WPT) is a promising solution for providing stable energy supply and improving the lifetime of energy-constrained wireless devices (WDs) [1], [2]. In a wireless powered communication (WPC) system, energy nodes (ENs) transmit wireless energy to WDs in the downlink, and the WDs use the harvested energy to transmit their own data to information access points (APs) in the uplink [3]. There are three main forms of WPT, namely, power beacons (PBs) or macro base stations (BSs) [4], hybrid access points (HAPs) [5], and simultaneous wireless information and power transfer (SWIPT) [6].
The performance of WPC networks has been analyzed in several research works using performance metrics such as the outage probability (OP), the ergodic capacity (EC), The associate editor coordinating the review of this manuscript and approving it for publication was Jiankang Zhang . the average bit error rate (BER) and the optimum switching time [2]. Most works consider only additive white Gaussian noise (AWGN) at the receiver [4], [7] whereas some analyses include the presence of co-channel interference (CCI) [5], [8], [9]. In the latter case, there are works that consider a single interferer [2] and others that consider the effect of multiple interferers [10]. When WPT is performed in the same frequency as information delivery, CCI occurs from both other WDs transmissions as well as other ENs [5]. Moreover, in some works, CCI is used as a source of EH [7], [8]. However, the most reliable source for WPT are dedicated RF sources such as macro BSs or PBs [4]. The PBs are dedicated ENs that can deliver energy to users in both the isotropic as well as directed energy transfer modes [3]. In some works the power transfer efficiency was improved by considering EH from both dedicated as well as ambient RF sources [10]. Moreover, PB may be empowered to perform communication tasks such as channel estimation, digital beamforming, and spectrum sensing among others.
For example, in order to enhance the power transfer efficiency, PBs with multiple antennas that apply beamforming schemes have been considered into the power transfer process [2], [9]. However, beamforming requires channel state information (CSI) available at the PB, which is difficult to accomplish in practical systems since the channel is a multi-dimensional time-varying random matrix and there are energy restrictions of the devices [11]. Therefore various analyses with multiple antennas take into account the effects of imperfect CSI at the PB [9], [12], or use statistical CSI beamforming for massive antenna array at the BS and terminals that appear to be spatially clustered [11].
In all cases, CCI is considered as one of the biggest obstacles in wireless networks and its impact on the receiver's performance needs to be considered [13]. In order to provide a realistic performance evaluation, stochastic geometry has emerged as a useful tool for performance analysis purposes [14]. Under a stochastic geometry approach, it is assumed that node positions are distributed according to a point process [15]. Representative examples include the Poisson point process (PPP) [14], used to model unplanned (random/independent) deployment of nodes over an infinite plane, and the binomial point process (BPP), used to model node deployment over a finite area [16], [17].
In this paper, we analyze the performance of single antennas beacon-assisted WPC system with spatially random WDs distributed according to a PPP or a BPP and operating in the presence of Nakagami-m fading. Throughout this analysis, we consider the time-slotted ''harvest-then-transmit'' model [4], [9]. The performance of the considered system is carried out in terms of the OP and the EC. The paper's contributions can be summarized as follows: • We derive integral representations of the exact OP and EC of WPC systems using dedicated PBs and the time splitting protocol under three CCI models, namely, CCI is approximated by a Gamma distributed random variable [18], in a scenario where the locations of the interfering nodes are modeled by a homogeneous PPP, and a scenario in which the random placement of the interfering nodes are modeled by the BPP distribution.
• For each interference scenario, we derive novel closed-form expressions for the n-th moment of the received signal-to-interference and noise ratio (SINR) at the receiver. Based on these results approximate closed-form expressions for the EC are presented.
• For each interference scenario, using the results for the n-th moment of the received SINR, we obtain analytical expressions for the optimum energy harvesting fraction of time that maximizes the attained EC. The remainder of the paper is organized as follows.
In Section II, we present the WPC system model and derive the received SINR. In section III we derive exact expressions for the OP and EC in terms of the moment generating functions (MGFs) of the received signal and the aggregate interference, for three widely used interference models.
Moreover, an accurate approximation of the EC in terms of the moments of the received SINR is derived and applied to each CCI model. Various performance evaluation results accompanied with Monte Carlo simulations are presented to substantiate the proposed analysis for each scenario. Finally, Section IV concludes this paper.

II. SYSTEM MODEL
We consider a wireless powered point-to-point communication system in which a source S communicates with a destination D, as shown in Fig. 1. Node S is an energy-constrained node which relies on external charging through power transfer, e.g. from dedicated PBs. The considered system employs a single dedicated PB which may also be from a selection of a finite group of multiple PBs.
The PB is located at a distance d P from the source S and the channel coefficient |h P | between PB and S is assumed to be a Nakagami-m distributed random variable (RV) with parameters m P and P . The destination node, D, is located at a relatively short distance d D from the source. Assuming a fixed location of D the channel coefficient |h D | between S and D follows the Nakagami-m distribution with parameters m D and D . In the considered system, a number of interferers around D are also present, distributed as a point process I with density of λ I . Again, the channel coefficient |h I k | between the k-th interferer and D is characterized by Nakagami-m fading with fading parameter m I . All nodes are equipped with a single antenna.
A two-stage (harvest-then-transmit) communication protocol is employed in time of T symbols. The considered protocol consists of two phases. During the first phase (harvest phase) of duration τ T , where 0 < τ < 1, S harvests energy from the PB. In the second phase (transmit phase) of duration (1 − τ )T , S transmits information to D using the harvested energy. The SINR at the receiver can be expressed as [9, eq. (5)] where 0 < η < 1 is the energy harvesting efficiency, v is the path loss exponent of all links, P P is the transmitting power of the PB, N 0 is the AWGN power spectral density and the total interference power-tonoise ratio (INR), with P I being the transmit power of each interferer.

III. MAIN RESULTS
Hereafter, we derive analytical results for the moments of the SINR at the receiver, the OP and the EC under various interference modeling scenarios. Let us first define X ≜ (τ )|h P | 2 |h D | 2 . By following an MGF-based approach, exact analytical expressions for the OP and the EC can be derived provided that the MGFs of X and I are readily available. Specifically, the OP is defined as the probability that the instantaneous SINR falls below a pre-defined outage threshold, namely By employing the inversion theorem [21], P out can be deduced as Moreover, exact EC, given by Note that the above integrals can be efficiently evaluated numerically using standard routines available in mathematical software packages, e.g. Mathematica. Let alsoμ(n) = E⟨γ n D ⟩ be the n th moment of γ D . Using a Taylor series expansion of log(1 + γ D ) aroundμ(1) and taking its expectation, an accurate approximation for the EC can be deduced as This approximation turns out to be tight for a wide range of system parameters. Moreover, (6) can be efficiently used to evaluate the optimal τ maximizing EC. To this end, the first derivative with respect to τ is set to zero and the optimal value of τ , τ * can be easily obtained numerically. The following scenarios are investigated next.

1) SCENARIO 1-DEDICATED PB AND GAMMA DISTRIBUTED INTERFERENCE
In this case, it is assumed that node S harvests power from a dedicated PB and the aggregate interference can be approximated as a Gamma distributed random variable with parameter m I [13]. Thus, the PDF of I can be expressed as where β I = I /m I and I = E⟨I ⟩. The MGF of I is given as Moreover, the RVs |h P | 2 and |h D | 2 also follow the Gamma distribution with PDFs given as and respectively, where m i > 0 are parameters corresponding to the fading severity and Therefore, the random variable Y = |h P | 2 |h D | 2 follows a double Gamma distribution with MGF given by [23] Using [24], the CDF of Y can be expressed in closed-form as In the presence of CCI, the exact OP can then be obtained in a straightforward manner using the MGF-based approach and the inversion theorem in (4). In addition, we derive an alternative exact PDF-based expression for the OP.
Proposition 1: Under Scenario 1, the OP of the considered system is given as (13) where and is the transmit INR.
Proof: Observe that P out (γ th ) can be expressed as where F and H are given by (14) and (15), respectively. This probability can be further expressed as Using (12) and (7) and after performing some straightforward manipulations, (17) yields (13), thus completing the proof. Note that the OP can be numerically evaluated using (13) in an efficient manner, due to the exponentially decaying factor e −x/β I . Thus, (13) can be evaluated using standard built-in routines, available in popular software packages such as Mathematica, or by employing Gaussian quadrature techniques.
In the following, accurate closed-form approximations to the OP will also be obtained. To this end, an analytical expression for the n th moment of γ D will be deduced first. The following result holds.
Proposition 2: The n th moment of γ D can be expressed in closed-form as Proof: Using (1), the n th moment of γ D can be obtained asμ Because of the independence of |h P | and |h D |, the expectation E |h P | 2n |h D | 2n can be deduced as [23] E |h P | 2n Moreover, it holds that By employing [19, eq. (9.211/4)], (21) can be readily evaluated yielding (18), thus completing the proof. Using (21), the CDF of γ D can be approximated by the CDF of an α-µ distribution as [25] In order to render (22) an accurate approximation, momentbased estimators for α, µ and x are used, obtained as [25] This approximation is highly accurate for a wide range of system parameters. Fig. 2 depicts the EC and OP versus transmitting SNR P P /N 0 dB for various values of v, assuming η = 0.5, τ = 0.5, d P = d D = 10 m, m P = 2, m D = 1.5, m I = 1.2, P = D = 1, γ th = 0 dB, and P I /N 0 = 5 dB. As it can be observed, simulations perfectly agree with the exact solution. Moreover the α-µ approximation for the OP is very accurate for all considered system parameters. Also, the moment-based approximation for the EC provides a tight bound in the entire SNR region for all considered cases. Fig. 3 depicts exact and approximate EC for v = 2.5, the same system parameters, P P /N 0 = 45 dB and various values of τ . As it is evident there exists an optimal value, τ * , maximizing the EC. This value has been computed using the moment based approximation. It is evident that τ * closely approximates the exact optimal EC.

2) SCENARIO 2-DEDICATED PB AND PPP INTERFERENCE AT D
In this case, the statistical characteristics of the aggregate interference depend on system parameters such as propagation effects, location of interferers, mobility patterns, and user activity. Assuming that the interfering signals sum incoherently, the aggregate interference can be modeled as a shot-noise process. Under certain conditions, this process is modeled as an alpha-stable process [26], [27], [28] with MGF given by [29] M I (s) = exp −qs 2/v (26) VOLUME 11, 2023  where q = π λ I (1 − 2/v) (m I + 2/v)(P I /m I ) 2/v / (m I ) (27) and m I is the fading parameter of each interferer. Next, a closed-form expression for the n th moment of the received SINR will be deduced. Proposition 3: The n th moment of γ D can be expressed in closed-form as Proof: In order to obtain an analytical expression for µ(n) the expectation E (1 + I ) −n should be evaluated. Using the expression The integral in (31) The resulting expectation is recognized as the Laplace transform of an H-function, which, by employing [20, eq. (2.19)], can be evaluated in closed-form as By employing [20, eq. (1.140)], the H-function in (33) can be expressed in terms of the Wright hypergeometric function, yielding (28), thus completing the proof. Note that, in general, the H-function can be evaluated numerically in an efficient manner using the Matlab script available in [30]. Nevertheless, it turned out that the Wright function, can be evaluated efficiently numerically by truncating the corresponding infinite series, requiring rather few terms for achieving sufficient numerical accuracy. For example, 60 terms are in general sufficient to achieve accuracy up to the fourth significant decimal digit.
Therefore, exact expressions for the OP and EC under the PPP interference scenario can be obtained from (4) and (5) using (11) and (26). Moreover, a moment based approximation for the EC can be obtained from (6) using the result in (28).

3) SCENARIO 3-DEDICATED PB AND BPP INTERFERENCE AT D
In this case we assume that the considered system is deployed in a service are of radius R that contains K independent and identically distributed transmitting nodes from a BPP denoted by K. Node D is located at the center of a circular subregion K B of radius B and has a communication range B so that it can receive transmissions from all transmitting nodes located within the sub-region K B . The radius B is chosen to be sufficiently large such that the power of interfering signals outside this circular area decays to a negligibly low level at the receiver. We also assume that there exists an exclusion region of radius A around D, from which interfering transmissions are prohibited [16], [17].
Since the interfering nodes are uniformly distributed, the distances r k of the kth interferer to D are independent and identically distributed random variables with PDF given as [16] There is a total of K interfering transmitters in K. Of these, the number that falls in the sub-region K B of radius B is denoted by K B (B) and the probability that K B (B) = k follows a binomial distribution [16] with probability mass function given by where p = (B 2 − A 2 )/R 2 denotes the ratio of the area of the circle centered at D whose radius is B, to the total service area of radius R.
The total receive INR, I , can be expressed as where ξ 2 k is the fading INR of the kth interferer at D. Assuming Nakagami-m fading, ξ 2 k follows the Gamma distribution with PDF where I = P I /N 0 is the transmit INR which is the same for all interfering nodes. In the following, a closed-form expression for the MGF of the aggregate interference will be deduced. 2 Proposition 4: The MGF of the aggregate INR can be expressed in closed-form as where Proof: The MGF of the aggregate INR can be obtained as Note that the MGF expressions available in [16] and [17] hold for Rayleigh interferers only.
The PDF of r −v can be obtained from the PDF of r by performing a change of variables, as Thus, M I (s) can be written as which, using the definition of the MGF can be further written as Assuming Nakagami-m fading, the MGF of ξ 2 i is given as Consequently, by employing the binomial theorem, the MGF of the aggregate interference can be expressed as (38) with By employing [19, eq. (3.194.1)] and [19, eq. (9.131.1)], q(s) can be expressed as (39) thus completing the proof. Using (4) and (5) along with (11) and (38), the exact OP and EC can be easily evaluated. In order to obtain an analytical expression for the optimal τ maximizing EC, an approximate closed-form expression for the moments of the SINR is derived next using a moment-based approach. By following a similar line of arguments as in the proof of Proposition 3, the evaluation of the n th moment of γ D requires the evaluation of the expectation E (1 + I ) −n . Nevertheless, an analytical expression for this expectation is very difficult -if not impossible to be obtained in closed-form. To this end, we proposed to use the approximation The required n th moment of I can be obtained as As it will become evident, this approximation turns out to be quite tight for a wide range of system parameters. Also, using (6) an accurate approximation to the EC and the optimal τ can be easily obtained. In order to evaluate the nth derivative of M I (s), the nth derivative of q(s) should be computed. The following result holds.

4) SCENARIO 4-DISTRIBUTED PBs AND GAMMA DISTRIBUTED INTERFERENCE
Here, we consider the scenario presented in [31] and [32] in which only one PB among a cluster of PBs is selected to activate each time for the reduction of computational complexity and energy cost. Therefore, the best PB is selected among N PBs according to the links from PB i , i = 1, . . . , N to S in order to achieve maximum instantaneous harvested energy. Thus, the link that is used for EH is the strongest one [33], i.e. where |h P i |, i = 1, . . . , N are independent and identically distributed random variables with parameters m P and P . Next, by assuming that the effect of noise is negligible (N 0 ≈ 0), analytical expressions for the OP and EC in interference-limited case will be obtained. where λ D = m D /( D m I ).
Proof: Observe that Due to the independence of |h P i | 2 it holds that Moreover, the ratio Y = |h D | 2 |h I | 2 follows a Fisher-Snedecor F-distribution with PDF given by Proof: Since an exact expression for the EC is very difficult to be evaluated, we propose to approximate the instantaneous signal-to-interference ratio (SIR) as Also, it holds that and Thus, C can be approximated as

IV. CONCLUSION
In this paper, a performance evaluation of beacon-assisted wireless powered communications has been presented under different interference scenarios. To this end, rapidly convergent integral expressions as well as closed-form approximations have been presented for the OP and EC performances. Extensive numerically evaluated and computer simulation results have been presented and compared, and a good match has been observed.