The goals of the fifth generation (5G) and beyond (B5G) wireless networks can provide reliable communication among almost all aspects of life through the network with higher date rate, lower latency and ubiquitous connectivity . The security has been identified as an vital factor for wireless communication systems, which has triggered enormous interests from both academia and industry [1, 2]. However, due to the broadcast characteristics of wireless communication, it is difficult to ensure secure communication for the wireless networks without being eavesdropped by un-authored receivers. The conventional methods to ensure the security of wireless communication are to use encryption algorithms, which impose extra computational overhead and system complexity . In addition, with the rapid development of chip and computer technologies, conventional encryption technologies can not provide perfect security.
As an alternative way to ensure security, physical layer security (PLS) has sparked a great deal of research interests . The basic principle of PLS is to exploit the inherent randomness of fading channels to resist the information to be extracted by eavesdroppers . Recently, there are a great of research works investigated the PLS under various fading channels, e.g. see [5, 4, 6, 7] and the references therein. In , a secure transmit-beamforming of the multiple-input multiple-output (MIMO) systems over Rayleigh fading channels was designed, in which the maximal ratio combing (MRC) receivers were adopted to maximize the signal-to-noise ratio (SNR) at main receiver. The authors of  investigated the PLS of artificial noise aided MIMO systems over Rician fading channels. Meanwhile, the secure performance of the classic wiretap model was discussed over the generalized Gamma fading channels and the analytical expressions of the strictly positive secrecy capacity (SPSC) probability and the lower bound for the secrecy outage probability (SOP) were derived . On the other hand, the authors investigated the secrecy performance over
shadowed fading channels of classic Wyner’s wiretap model and the approximate expressions on the lower bound for the SOP in the high SNR region and SPSC probability with the aid of a moment matching method have been obtained.
In actual situations, it is difficult to have direct links between the sources and the destinations due to shadow fading and/or obstacle, so it is indispensable to use the relay to complete the communication [8, 9]. In light of this fact, relaying assisted transmission has been identified as one of the key technologies in the current and future wireless cooperative networks [10, 11]. The signals can be decoded and transferred from the source to the destination by using low cost and low power consumption relay nodes. In general, there are two basic relay protocols: i) amplify-and-forward (AF) [14, 15, 16], and ii) decode-and-forward (DF) [17, 18]. In , the authors studied the ergodic capacity (EC) performance of fixed-gain AF dual-hop (DH) networks and derived two analytical expressions on the EC bound. Extending to multi-hop networks, the authors of 
derived the EC, outage probability (OP) and average symbol error probability by the generalized transformed characteristic function approach. In, the authors investigated the performance of a multi-hop AF communication network over Nakagami-0.5 channels and the closed-form analytical approximate expressions for the OP, ASEP and EC were obtained. To maximize confidentiality, the authors in  investigated the secure performance of multiple DF relay systems.
When deploying multiple relays in the systems, it will incur extra inter-relay (IR) interference and energy consumption. To avoid this problem, relay selection (RS) has been proposed . Among the various RS schemes, optimal relay selection (ORS), suboptimal relay selection (SRS) and MRC are the most prevalent ones [23, 24, 25]. The pioneering work of the ORS scheme has been proposed by Bletsas according to selecting the relay with the largest instantaneous end-to-end SNR . Based on the ORS, the authors in  investigated the symbol error rate (SER) of AF relay systems. To reduce the requirement of channel knowledge, the authors proposed a SRS scheme that the optimal relay is selected according to the link either source-relay or relay-destination . Cognitive radio inspired cooperative relay systems was introduced, and the secure outage performance was studied over independent and non-identically distributed Nakagami- fading channels. In , the authors compared the secrecy outage performance of cognitive radio networks for ORS, SRS with MRC schemes over Nakagami- fading channels.
Although the performance of wireless cooperative networks can be improved by appropriate relay protocols and RS scheme, the operation of wireless communication system is constrained by power shortages of their wireless devices. This happens that in some cases the nodes are deployed in the remote or power limited areas . In light of this context, some energy harvesting (EH) techniques have been proposed to prolong the life of the batteries of such wireless transmission devices [28, 29]. Among the various EH techniques, radio frequency (RF) enabled simultaneous wireless information and power transfer (SWIPT) attracts a lot of attentions because it can overcome the limitations of some other renewable energy resources such as solar energy, wind energy and magnetic induction that can only be used in some specific circumstances . In addition, RF signals are ubiquitous in electromagnetic waves, and EH in RF is green, safe, controllable and reliable . There are usually two common protocols for SWIPT systems: i) time-switching (TS) and ii) power-splitting (PS) [35, 37, 31]. For TS, the authors in 
investigated the outage performance of SWIPT-assisted non-orthogonal multiple access (NOMA) relay systems over Weibull fading channels. Considering PS protocol, the secure performance of two-way relaying systems was researched through a joint-optimization solution over geometric programming and binary particle swarm optimization. Additionally, a large-scale RF-EH technique with PS protocol was adopted, and the OP performance and average harvested energy were analyzed .
The aforementioned studies are based on the assumption of ideal hardware components and perfect channel state information (CSI), which is unrealistic in practical communication systems. In practice, due to component mismatch and manufacturing non-idealities, these monolithic architectures inevitably have defects associated with the RF front-ends, thereby limiting the overall system performance . A typical example of these impairments is the in-phase and quadrature-phase imbalance (IQI), which refers to the mismatches of amplitude and phase between I and Q branches of the transceiver. This will result in incomplete image suppression and ultimately lead to degradation of the performance for the total communication system . Ideally, the I and Q branches of the mixer have an amplitude of 0 and a phase shift of , providing an infinitely attenuated image band; however, in practice, the transceiver is susceptible to some analog front-end damage, and these damages introduce errors in the phase shift resulting in amplitude mismatch between the I and Q branches, thereby damaging the down-converted signal constellation, thereby increasing the corresponding error . Motivated by the above practical concern, several research contributions have studied the systems secure performance in the presence of IQI [41, 9032127, XingwangLI]. Under the assumption of uncorrelation between channel of each subcarrier and its image, Ozdemir et al. in  derived an exact expression for the SINR of OFDM systems with IQI at transceivers. The authors analyzed the impact of joint IQI on the security and reliability of cooperative NOMA for IoT Networks . Considering backscatter communication, Li et al. in [XingwangLI] derived analytical expressions for OP and the intercept probability (IP) of ambient backscatter NOMA systems under IQI. On the other hand, imperfect CSI (ICSI) may be existed due to the presence of channel estimation errors (CEEs) and feedback delay. Therefore, it is of great practical significance to study the impact of ICSI and IQI on the security performance of cooperative networks.
I-a Motivation and Contribution
Motivated by the above discussion, we study the reliability and security of cooperative multi-relay networks in the presence of nonlinear energy harvesters, ICSI and IQI. Under these imperfect conditions, three RS schemes, random relay selection (RRS), SRS, ORS are considered. Specifically, we derive the analytical expressions for the OP and IP. For the security, the direct transmission and cooperative transmission through relay are considered. In this study, we assume that the source and relay nodes of the networks are configured with nonlinear energy harvesters to harvest energy from the nearby power beacon under different saturation thresholds. This is reasonable in some applications, such as internet-of-things (IoT), mesh networks and Ad Hoc networks, etc. The main contributions of this paper are summarized as follows:
Considering IQI and CEEs, we propose three representative RS schemes, namely RRS, SRS and ORS. RRS is considered as a benchmark for the purpose of comparison. In SRS, the optimal relay is selected according to the channel conditions either the or the . In ORS, the optimal relay is selected according to the link qualities both the and the . The major difference between our work and  is that to study the effects of IQI caused by the mismatches of amplitude and phase between I and Q branch.
Different from the most existing research works, we use a more realistic nonlinear EH model due to the nonlinearity of the electronic devices [49, 7264986Schober2015CL]. We have the assumption that nonlinear energy harvesters are equipped at source and relays, which can harvest energy from the nearby power beacon.
For the reliability, we derived the exact analytical expressions for the OP of the proposed system for the three RS schemes. For the security, we consider two typical scenarios that direct transmission and cooperative transmission, the exact closed-form analytical expressions of the IP for the two scenarios are derived.111In some cases, the eavesdropper can simultaneously receive signals from both source and relays. Our work can be easily extended to these cases by combining the received signals from source and relays using the selection combining or MRC.
To obtain more insights, we derived the asymptotic analytical expressions and diversity orders for the OP of the three RS schemes under non-ideal conditions. It reveals that there are error floors for the OP due to the non-zero CEEs, and the OP performance is limited by the IQI parameters.
I-B Organization and Notations
The rest of the paper is organized as follows. In section II, we present a brief introduction of the considered system model. In section III, the reliability of the considered system for the three RS schemes is studied in terms of OP, while the security is analyzed through deriving expressions of the IP. In section IV, some numerical results are provided to verify the correctness of our analysis. Finally, we present a conclusion of this paper in Section V.
We use to define absolute value. The notations and denote the expectation and definition operations, respectively.
defines a complex Gaussian distribution with a mean of
and a variance of. represents the probability and denotes the and , respectively. Finally, is the logarithm.
Ii System Model
We consider a DF multi-relay system as shown in Fig. 1, which deploys one power beacon , one source , relays , one destination , one eavesdropper and all nodes equipped with a single antenna. All nodes are operate in half-duplex (HD) mode. In order to improve the secure performance of the considered system, the SRS and ORS schemes are used to select the optimal relay among the relays, while RRS scheme is presented as a benchmark. The source and all relay nodes are energy-constrained and can harvest energy from nearby according to the TS protocol. It is considered that there is no direct link of due to the blockage or heavy shadowing.
In fact, it is difficult to obtain perfect CSI in the communication process because of the CEEs, and the most common method is to estimate the channel using the training sequence. In this study, the linear minimum mean square error (LMMSE) is adopted here. Thus, the channel can be modeled as
where is estimated channel of the real channel , is the CEE, where is the variance of estimation error and which is considered in two representative channel estimation models: 1) It is a non-negative fixed constant; 2) It is a function of transmit average SNR and can be modeled as , where is the channel estimation quality parameter that indicates the power consumption of the training pilot to obtain CSI; and are the variance of channel gain and transmit average SNR, respectively . We assume that all communication links are subject to Rayleigh fading and path loss .
Typically, IQI is modeled as the phase and/or amplitude imbalance between transceiver I and Q signal paths. As depicts in [46, 47], the asymmetrical IQI model can be considered, where I branch and Q branch are assumed to be ideal and errors, respectively. Here, both transmitter (TX) and receiver (RX) are subject to IQI, in which case the transmitted baseband signals can be expressed as
where is the baseband signal that is transmitted under the conditions of non-ideal I/Q matching with , is the transmit signal of the TX, and are the transmit power at and , respectively; The IQI coefficients are given by , , , , where and denote the amplitude and phase mismatch at TX and RX, respectively [Xingwang2019Electronics]. For ideal conditions, the parameters are set to and .
The entire data transmission is completed in three phases: 1) and relays harvest energy from nearby power beacon ; 2) transmits its own signals to and ; 3) decodes and forwards the signals to and .
1) The first phase: In this phase, and are equipped with nonliear harvesters can reap energy from . The harvested energy at is
where is the energy converse coefficient of harvester at ; is the transmitted power at ; is the channel coefficient between and ; is the time allocation factor for EH, and is the block transmission duration. The harvested energy is used for information transmission in the second phase. The transmit power can be expressed as follows in the case of the nonlinear energy harvester 
where is the saturation threshold of the harvester at .
Similarly, the energy harvested at can be expressed as
where is the energy conversion coefficient of harvester at , and is the channel coefficient from to . The harvested energy at relays is used for the information transmission in the third phase. In the presence of the nonlinear energy harvester, the transmit power at relay is given as follows
where is the saturated threshold of the harvester at .
2) The second phase: In this phase, respectively sends the signals and to and with . Considering IQI and CEEs, the received signals at and can be all written as (7) at the top of next page,
where and are the estimated channel coefficients from transmitter to receiver; and are the complex additive white Gaussian noise (AWGN).
3) The third phase: In the third phase, respectively sends the signals , to and with = . Similarly, the received signals at and can be expressed as (7) at the top of next page.222Note that , and and are the power from and , respectively.
Hence, the received signal-to-interference-plus-noise ratio (SINRs) at , and can be expressed in a unified form as
where , , and .
According to the Shannon’s theorem, the channel capacity can be expressed as follows
where the factor means the data transmission is accomplished in equal two phases.
With DF protocol, the effective end-to-end capacity from to can be expressed as
Iii Performance Analysis
This section analyzes the reliability and security of considered system in the presence of nonlinear energy harvester, IQI and ICSI. The closed-form expressions for the OP under the RRS, SRS, ORS schemes and IP under direct transmission and relay transmission strategies are derived.333The reliability and security are another metrics to characterize the PLS of wireless communication systems without using any secrecy coding, which are formulated by OP and IP . Moreover, the asymptotic behaviors for the OP are analyzed, as well as the diversity orders.
Iii-a Outage Probability Analysis
In this subsection, the expressions for the OP are presented according the three RS strategies considered IQI, ICSI and nonlinear energy harvesters. The OP is defined as the probability that effective channel capacity is below the threshold , which can be expressed as
Iii-A1 Random relay selection
For RRS strategy, the link between and arbitrary one of the relay is selected, and the effective rate can be obtained as
Based on the above discussion, we can obtain the analytical expression for the OP of the RRS strategy in the following theorem.
The analytical expression for the OP of RRS strategy is provided in (13) as shown at the top of next page.
where , , , , , , , , , , , , , , , , , , and .
For , otherwise the OP expressions are equal to 1.
See Appendix A. ∎
To get deeper insights, the asymptotic behavior of non-ideal conditions is investigated at high SNRs in the following corollary.
The asymptotic expression of OP for RRS strategy under non-ideal conditions is given by
where and .
Based on (9), the asymptotic channel capacities of and can be written as
By the definition of OP, the following expression can be obtained as
Utilizing the similar methodology of Appendix A, (14) can be derived. ∎
Furthermore, the diversity order is investigated, which can be defined as :
where is the average SNR and is the asymptotic OP.
The diversity order of OP for RRS scheme in the presence of non-ideal conditions () can be obtained as follows:
Follows trivially by using (18) and the definition of derivative. ∎
From Theorem 1, Corollary 1 and Corollary 2, the following observations can be obtained as: 1) When increases gradually, it can be seen that (13) and (14) are independent of , so the RRS scheme will not change with the increase or decrease of the number of antennas; 2) At high average SNR, is a fixed constant, which results in 0 diversity order. This means that the diversity order can not be improved by increasing the number of relays.
Iii-A2 Suboptimal relay selection
For SRS strategy, the optimal relay is selected according to maximizing the capacity of the link , which can be expressed as
Based on (11) and (20), we have the following Theorem 2.
The analytical expression of OP is provided for SRS strategy in (22) as shown at the top of next page.
where , , , , , , , , , , , , , and .
See Appendix B. ∎
Similarly, the asymptotic behavior of non-ideal conditions is studied of OP for SRS strategy in the high SNR regime.
The asymptotic expression for the OP of SRS strategy under non-ideal conditions is given by
where and .
Then, the diversity order of OP for SRS strategy under non-ideal conditions () is presented in the following corollary.
The diversity order of OP for SRS scheme in the presence of non-ideal conditions () is given by:
From Theorem 2, Corollary 3 and Corollary 4, we can obtain the following information as: 1) when the number of relay increases, it can be concluded from formulas (22) and (23) that the system’s outage performance becomes better under the SRS strategy; 2) From expression (22), it can be obtained that when the is fixed and the transmit power at is in a high state, the OP will cause an error floor; 3) From (24), we can observe that the diversity order of the considered system is zero due to the fixed constant for the OP in the high SNR regime.
Iii-A3 Optimal relay selection
For ORS strategy, the optimal relay is selected according to maximize the capacity of the links both and
According to (11) and (26), Theorem 3 can be obtained as following.
The analytical expression of the OP is provided for the ORS strategy in (27) as shown at the top of next page.
See Appendix C. ∎
Next, the asymptotic behavior for the OP of ORS strategy in the presence of non-ideal conditions is studied.
The asymptotic expression of OP for the ORS strategy under non-ideal conditions is given by
The diversity order of OP for ORS strategy under non-ideal conditions () is following: