1. Introduction
Wireless sensor networks (WSNs) are subject to the constraint of energy storage at each mobile node [
1,
2]. Saving on energy consumption or extending the battery life for sensor nodes has become an important research issue in wireless sensor networks. Recently, energy harvesting (EH) has attracted enormous attention from researchers as a promising cost-effective technique to maximize energy efficiency of a wireless network [
3,
4,
5], especially in wireless sensor networks [
6,
7,
8,
9]. Various energy harvesting sources have been studied, such as natural sources (solar [
10], wind [
11], thermal [
12], etc.), strongly coupled magnetic resonances [
13], etc. Among them, radio frequency (RF) energy from the ambient transmitters is the most popular source for energy harvesting since it can be received more effectively from RF signals [
4]. Because RF signals can convey both energy and information simultaneously, an RF-based energy harvesting technique, called simultaneous wireless information and power transfer (SWIPT), is becoming a more and more promising research topic for WSNs [
14,
15]. The idea of SWIPT was first introduced in 2008 in the seminal paper of Varshney [
16], in which authors proposed an ideal receiver design that is able to simultaneously observe the information and extract power from the same received signal. Zhou et al. [
17] connected Varshney’s idea to practice by proposing two realizable receiver architectures design: time switching (TS) and power splitting (PS). The performance analysis of these two EH protocols was conducted rigorously by Nasir et al. [
18]. The authors derived analytical expressions for the outage probability and the ergodic capacity of one-way amplify-and-forward (AF) relay networks over a Rayleigh fading channel.
The applications of SWIPT to wireless sensor networks have been investigated in recent works. In [
7], the authors proposed a distributed iteration algorithm to solve an energy-efficient cooperative transmission problem for SWIPT in clustered WSNs. In [
9], the authors employed wireless energy harvesting techniques and realistic energy converters in dense and often randomly deployed WSNs, and quantified the potential energy gains that can be achieved in the network. Peng et al. [
19] considered a wireless energy harvesting two-way relay network (TWRN) using power splitting protocol, where the effects of practical hardware impairments were taken into consideration. Mouapi and Hakem [
20] proposed a new approach to defining the specifications of a stand-alone wireless node based on an RF EH system and implemented a hardware circuit to illustrate their energy optimization method. The performance of wireless powered sensor networks for Internet of Things (IoT) application was studied in [
4], where the authors proposed the optimal power allocation to maximize the system throughput and also derived the closed-form of that solution. However, in those papers, the channel gains are assumed either to be constant or to be a Rayleigh distributed random variable.
Wireless communication systems in the real radio environments are not only affected by the short-term fading (multipath), but are also subject to the shadowing effects. The vital issue in studying the performance of energy-harvesting-based wireless networks in these conditions is the outage probability analysis. An important statistical characteristic that can describe the behavior of the wireless channels is the probability density function (PDF) of the signal-to-noise-ratio (SNR) at the receiver output. To derive this PDF for different radio propagation environments is sometimes a difficult mathematical task, especially for complicated channel models such as Nakagami-m or Rician channels. In fact, not only the papers we mentioned about SWIPT for WSN, but most of the other results on outage performance up to now [
21,
22] also focus on Rayleigh fading channels, where we can exploit the fact that the square of channel gain magnitude is exponentially distributed.
As a general comment, it is noted that very few publications about energy harvesting for Rician fading channel exist in the open technical literature. In fact, together with Rayleigh channel, Rician channel should be also considered as one of the small-scale fading models for WSNs, especially for the cases of relatively short range power transfer distance and with existence of a strong line-of-sight (LOS) path [
23]. Recently, Zhao et al. [
24] have derived the capacity expressions for wireless powered communication systems over a Rician fading channel. However, there is no relay in this study. The source directly harvests the energy from the power beacon. In [
25], the authors provided the throughput analysis of relay networks with two energy harvesting protocols (continuous and discrete) over a Rician fading channel, but were only limited in the case of perfect hardware. Furthermore, the paper only provided the integral form of the throughput for the continuous case, so it is not a computationally friendly result. Mishra et al. [
26] did provide impressive results on joint optimization of power allocation, power splitting for EH, and relay placement for SWIPT over a Rician channel. Nevertheless, they only use an approximation expression of outage probability to formulate the optimization problem, and did not consider the hardware impairment at the nodes. In practice, the transceiver hardware is imperfect due to phase noise, I/Q (In-phase/Quadrature) imbalance and amplifier nonlinearities [
27]. The modeling of hardware impairment in system performance analysis has been presented in many works, for instance, in [
28], where the authors analyzed the performance of dual-hop relaying systems in hardware impairment condition, in terms of the capacity, throughput and symbol error rate (SER). We also proposed and evaluated an energy harvesting-based spectrum access model in cognitive radio network with hardware impairment [
29]. Again, these works about hardware impairment considered the Rayleigh channels only.
In our current work, we tackle the problem of investigating the performance analysis of SWIPT for WSN over a Rician fading channel, which takes into account the hardware impairment at source and relay nodes. Specifically, we consider an AF two-way half-duplex energy harvesting relay network model suffering from hardware impairments at all nodes over the Rician fading channels. The exact analytical expressions of the achievable throughput, outage probability, and the exact-form expression for the PDF of SNR at each destination node of a half-duplex AF bidirectional wireless sensor networks over a Rician fading channel are derived rigorously. The main contributions of this paper can be described in more details as follows:
The exact form expression of outage probability and achievable throughput at each destination node with imperfect hardware and in Rician fading environment are derived mathematically.
We derive the exact-form cumulative distribution function (CDF) of the SNR at each destination node, and use this result to derive the integral exact-form of the SER at destination nodes.
We also conduct the asymptotic analysis and provide the approximation of all performance factors mentioned above at high regime.
The analytical results are all confirmed by Monte Carlo simulations. Using the simulation results, the effect of various system parameters on the system performance is carefully studied.
The rest of this paper is organized as follows.
Section 2 describes the system model and the EH protocol that is used in this paper.
Section 3 provides the detailed performance analysis of the system, including exact analysis and asymptotic analysis. The numerical results to validate the analysis are presented in
Section 4. Finally, conclusions are drawn in
Section 5.
4. Numerical Results and Discussion
For the purpose of validation, the correctness of the derived outage probability and SER expressions as well as investigation of the effect of various parameters on the system performance, a set of Monte Carlo simulations are conducted and presented in this section. For each simulation, we first provide the graphs of the outage probability and throughput obtained by the analytical formulas. Secondly, we plot the same outage probability and throughput curves that result from Monte Carlo simulation. To do this, we generate random samples of each channel gain, which are Rician distributed. Using these random samples, the SNR at destination node is calculated and compared with the threshold value γ. The outage probability occurs if this SNR falls below the threshold. By taking the number of cases that divided by the number of samples, we can estimate the outage probability and then the throughput of system. The analytical curve and the simulation one should match together to verify the correctness of our analysis.
The hardware impairment parameters are chosen as
. The ideal hardware impairment situation (
) is also considered as a benchmark performance for simulation. The channel gains are considered as Rician fading with
and with the Rician K-factors equal to 3 for both channels. The transmit power are set to the same value
for both two sources, so that the ratio
varies in the range from 0 to 50 dB. The energy harvesting efficiency is set to be 0.7. The source transmission rate is chosen as 1.5 bps/Hz. From the Shannon’s theorem on capacity of the channel, we can calculate the
threshold as
. All simulation parameters are listed in
Table 1.
4.1. Effects of Various Parameters on the System Performance
Figure 3 and
Figure 4 show the effect of
on the outage probability and throughput of the proposed system, respectively. For this simulation, the utilized parameter settings are:
or
and
. We choose
to consider the case that the duration of energy harvesting and the duration of transmission are balanced. The case
(no hardware impairment) is also introduced for comparison. The first observation is that the outage probability and throughput obtained from mathematical analysis match with the corresponding Monte Carlo simulations. Regarding the effect of
κ, the outage probability decreases and the throughput increases as
κ varies from 0 to 0.2. When
increases, the outage probability and throughput approach the corresponding asymptotic values obtained from analysis. Furthermore, the lower the value of
κ, the faster the outage probability and throughput converge to their asymptotic values.
The effect of hardware impairment level on the outage probability and the achievable throughput at each node is presented more thoroughly in
Figure 5 and
Figure 6. Here,
is set at 20 dB and the transmission rate is fixed at 1.5 bps. Three values of
α are chosen: 0.2, 0.5, and 0.8, corresponding to three cases: the energy harvesting duration is dominant, there is a balance between energy harvesting and information transmission, and the information transmission duration is dominant.
Again, it is observed that the exact-form expressions of outage probability and throughput obtained by the analysis coincide with the ones that are obtained by Monte Carlo simulations. From the numerical results, it is evident that the achievable throughput decreases and the outage probability increases significantly at each destination node when the impairment level
κ increases. In addition, the outage probability tends to reduce at higher time-switching factor. This can be explained because the larger value of
α means more power is used for data transmission. However, this doesn’t mean that the throughput is better for larger
α. In
Figure 6, the throughput performance is improved when
α increases from 0.2 to 0.5, but then degraded when
α increases from 0.5 to 0.8.
Figure 7 and
Figure 8 illustrate more clearly the effect of time-switching factor on the outage and throughput performance. In this simulation, the parameters are chosen as
and
dB (this value is chosen because it is in the middle range of
). The transmission rate varies among three values: 0.5 bps/Hz, 1 bps/Hz and 1.5 bps/Hz, while the time-switching factor varies in the range
. The results confirm what we mentioned just above. There should be a unique value of
α that maximizes the throughput. This is because, when we increase
α initially, there is more power used for transmission, so the outage probability is reduced and the throughput increases correspondingly. However, when
α keeps increasing, the duration of transmission is also reduced, hence, less data is transmitted during a given time interval. As a result, the throughput performance becomes worse.
4.2. Effect of Various Parameters on SER
The purpose of the following simulations is to confirm the correctness of the SER formulas provided in the analysis. First,
Figure 9 presents the effect of the hardware impairment level on the SER performance. In this simulation, the time-switching factor is chosen as
with the same reason as in
Section 4.1, the transmission rate is fixed at 1 bps/Hz, and the ratio
varies in the range from 0 dB to 40 dB. From the results, it is showed that SER decreases to the asymptotic value when the ratio
increases. The Monte Carlo simulation curves overlap with the corresponding analysis curves. This confirms the validity of our analysis. When the hardware impairment level goes higher, the SER also has a larger value, as expected. Furthermore, the SER performance of the QPSK scheme is better than the one of the BPSK scheme in the same simulation condition. This can be explained because the QPSK modulation scheme can transmit two bits in one symbol while the BPSK scheme can only transmit one bit per symbol. Hence, if we have the same constraint on the transmission rate for both methods, the required SNR for maintaining good communication would be smaller for the QPSK method. As a result, the outage probability and the SER for QPSK modulation would be smaller than the ones of BPSK.
In a similar way, the influence of the time-switching factor on the symbol-error-rate at the destination node is illustrated in
Figure 10. The simulation parameters are
bps/Hz, and
varies from 0 dB to 40 dB. Again, the simulation curves match perfectly with the corresponding analysis curves. The SER tends to approach its asymptotic value, and QPSK modulation still provides the better SER performance than BPSK for both values of
α. Note that, for this simulation, the asymptotic values do not depend on the time-switching factor
α. However,
α surely has an effect on the immediate value of SER. In fact, the SER performance should be better with the value of
α that is in the middle of its range. For example, in
Figure 10, the SER value for
is less than the one with
. The explanation is the same as the case of
Figure 8.
4.3. Optimal Time-Switching Factor
As mentioned in
Section 3, the optimal time-switching factor to maximize the achievable throughput of the considered system can be found numerically by using an iterative algorithm such as the golden section search method [
36].
Figure 11 plots the optimal value
for various values of the ratio
at different hardware impairment levels.
It can be observed that the optimal time-switching factor decreases as the ratio increases. This is because, for large , the outage probability tends to reduce, so it is not necessary to use a large amount of energy to transmit data. Reversely, we need to spend more time resources to increase the throughput of the system.
On the other hand, we can learn from this simulation that the optimal α does not change much for different hardware impairment levels. Especially, for small κ, the value of is nearly the same as the one for a perfect hardware case.