Overlay CR-NOMA Assisted Intelligent Transportation System Networks with Imperfect SIC and CEEs

I.   Introduction

As the key enabling technology of the future intelligent transportation systems (ITS), Internet-of-vehicles (IoV) will greatly promote the development of society towards intelligence and informatization [1], [2]. Generally, IoV communication includes vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I) and the vehicle-to-person (V2P) [3]. With the rapid development of the sixth generation (6G) mobile communication, the IoV of ITS networks driven by 6G can achieve higher data transmission rate, lower latency and higher quality-of-service (QoS) [4], [5]. However, it also faces great challenges, such as spectrum shortage and large-scale connections [6]. In order to effectively solve the above problems, some promising technologies have been proposed, such as non-orthogonal multiple access (NOMA) [7] and cognitive radio [8].

NOMA is considered as a key technology in 6G since it can improve the spectral efficiency and reduce the latency by serving massive devices in the same resource block (time/frequency/code) using power multiplexing [9]. Moreover, at the transmitting side, the transmitter allocates different power to different users according to their channel conditions and carries out superposition coding. At the receiving side, the signal is detected through the successive interference cancelation (SIC) technology [10]. And specifically, NOMA can ensure fairness among the served users by allocating different users with different powers [11].

There are numerous excellent research focusing on the investigations of NOMA in wireless networks [12]-[18]. The authors in [12] proposed a general framework to evaluate a downlink NOMA system performance. The authors in [13] studied the reliable performance of a downlink NOMA system under the conditions of second-order statistics of channel state information (CSI). In [14], Lu et al. designed a multi-carrier NOMA system for video transmission to meet the growing demands for video services such as massive traffic and low latency. The authors in [15] investigated the performance of NOMA-enabled unmanned aerial vehicle (UAV) relay networks by deducing the outage probability (OP) and ergodic capacity for both amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols at UAV. The authors in [16] studied the effective capacity of a reconfigurable intelligent surfaces aided NOMA system. Xu et al. [17] proposed a cognitive orthogonal frequency-division multiplexing-NOMA network to increase the system capacity. To support ultra-reliability, high throughput and multiple concurrent connections, reference [18] investigated the variation of the diversity order of OP relative to the transmit power in a hybrid automatic repeat request assisted NOMA system.

Cognitive radio (CR) is another promising technology to improve the frequency spectrum utilization [8], [19]. In the CR networks, the secondary network is allowed to selectively access the authorized frequency spectrum of the primary network to solve the problem of insufficient spectrum [20]. According to the spectrum access paradigms, interweave, underlay, overlay are the three most popular CR models [21]. In interweave model, the secondary user (SU) is allowed to access the authorized frequency bands only when primary user (PU) does not occupy it. Underlay cognitive radio allows the primary and secondary users to transmit messages simultaneously in the same frequency band, but the interference to the primary user needs to be less than a predefined value [22]. In overlay mode, the secondary network uses part of the energy to help the primary network transmit and obtain the right to access the authorized spectrum, and it improves the performance of the primary network while realizing the simultaneous transmission of the primary and secondary networks [23].

Scanning the technical literature of recent years, the performance of CR network was discussed in many literature since CR technology can mitigate the shortage of spectrum resources to a certain extent [24]-[28]. In [24], the authors derived the analytical expression of OP and ergodic rate (ER) so as to compare the throughput performance of CR networks based on interweave and underlay. In more detail, to maximize the throughput of SU, Wang et al.. [25] proposed a channel-and-sensing-aware channel access strategy for an interweave cognitive network. In [26], the authors discussed the influence of improper Gaussian signaling on the underlay CR network performance. The authors in [27] designed a novel scheme of spatial modulation in overlay CR network and analyzed the system performance by calculating average symbol error rate. The authors in [28] designed a scheme about optimal transmitting power for underlay CR network to minimize the average symbol error probability.

To address the challenges of the exponentially growing demand for mobile traffic [29], many researchers introduced the CR technology into the NOMA networks, see references [30]-[34]. Liu et al. [30] deduced the analytical expression of OP by means of stochastic geometry in order to characterize the reliable performance of the proposed CR-NOMA network. Wei et al. [31] deduced the analytical expressions of the secrecy sum rate to study the secrecy performance of a NOMA-enabled underlay CR network. The authors in [32] analyzed the performance of a NOMA assisted underlay CR network by deriving the analytical expression of pairwise error probability of SU. The authors in [33] proposed a novel spectrum sharing framework for multiuser CR-NOMA network to effectively improve the spectrum efficiency. In [34], the authors optimized the power allocation of the proposed spectrum leasing scheme for CR network so that the QoS of the primary network has been met while maximizing the performance of the secondary network.

Based on the above discussions, the existing studies are mostly conducted under ideal conditions. However, the actual communication process is generally carried out under the non-ideal conditions, such as imperfect successive interference cancellation (ipSIC) and non-ideal CSI. ipSIC is produced by channel state information unattainable, receiver performance limitation or synchronization errors, residual impairments and error propagation during transmission, etc. [35]-[40], and the channel estimation errors (CEEs) result in non-ideal CSI. Chen et al. [36] designed a novel algorithm to reduce the influence of ipSIC on the considered system performance. In [37], the authors investigated the reliable and ergodic performance of a multi-input multi-output (MIMO) interference networks under the condition of non-ideal CSI. Yang et al. [38] studied the effect of non-ideal CSI on secrecy performance of multi-user massive MIMO networks. The authors in [39] considered nonlinear multi-objective optimization problem under ipSIC in order to maximize the sum capacity and minimize the total transmit power under the constraint of QoS. Further, the authors in [40] analyzed the influence of ipSIC, non-ideal CSI and imperfect timing synchronization on a space time block code-based NOMA network simultaneously.

Although many researchers have conducted studies on NOMA, CR, imperfect SIC and non-ideal CSI, the joint impacts of the above factors on system performance are rarely studied. Sun et al. in [41] modeled a NOMA vehicular communication network, and deduced the analytical expression for the OP of the considered network. However, the non-ideal CSI, ipSIC and CR network were not involved. On the premise of ensuring quality-of-service, Xiao et al in [42] optimized transmission time and power control for the proposed radio-frequency-powered CR network to achieve maximum energy efficiency. Unfortunately, NOMA was not taken into consideration. The authors in [43] studied a CR-NOMA network and proposed a new dynamic power transmission scheme to guarantee the QoS of the considered system. The fly in the ointment was that non-ideal CSI has not been taken into account. Li et al. assessed the reliability and security of cooperative NOMA and non-cooperative NOMA systems by calculating secrecy outage probability and connection outage probability in [44]. However, ipSIC and non-ideal CSI were not taken into account. Luo et al. analyzed the performance of a cognitive NOMA system, and deduced the analytical expression of OP and throughput under high SNR regions in [45], but the non-ideal CSI was not considered. The authors in [46] analyzed and optimized the OP and throughput of the proposed underlay CR-NOMA in the presence of ipSIC. However, non-ideal CSI was not included. In [47], the authors proposed a CR assisted-NOMA network based on ITS and evaluated the reliability of the considered network by analyzing OP and throughput. Nonetheless, the ergodic performance of the network was not considered. The authors in [48] introduced wireless power transmission into CR-NOMA network to solve the challenges of energy consumption and large-scale connection in ITS. The fly in the ointment was that non-ideal CSI has not been mentioned.

In order to fill these bridges, considering CEEs and ipSIC, we introduce a NOMA-based overlay CR (OCR) ITS system. In the considered system, the SR utilizes the spectrum resources of the primary network to transmit its own information. In return, the SU acts as a relay to transmit the information of the primary network to the primary receivers. The main contributions of this paper are as following.

• We propose a novel NOMA-based OCR system for the ITS, which consists of a primary vehicle network and a secondary vehicle network. In the primary network, the primary transmitter sends superimposed information to both the primary and secondary receiving vehicles. In the secondary network, the secondary receiving vehicle receives their own information and acts as the relay of the primary network to transmit information to the primary receiving vehicles. Moreover, in order to make the work more realistic, we consider the CEEs and ipSIC. Specifically, two channel estimation models are considered: i) The CEEs is a fixed constant; ii) The CEEs is a function related to the SNR.

• We evaluate the reliability performance of the system under consideration by deducing the analytical expressions of the OP. In order to obtain more precise insights, the asymptotic expressions of the OP at high SNR regions are analyzed. The derivations show that increasing transmit SNR enhances the reliability of the consideration system and the OP exists an error floor because of the constant estimation error.

• The ergodic sum rate (ESR) of primary and secondary network are derived through a series of calculations to evaluate the ergodicity of the ITS system. In addition, the asymptotic expressions of ESR at high SNR regions are also performed. These results indicate that increasing SNR enhances the system ergodicity and the value of ESR tends to a constant at high SNR regime.

Here are some notations about this paper. $E\left( \cdot \right)$ stands for expectation operation. ${\cal {C}}{\cal {N}}(\mu ,{\sigma ^2})$ represents the complex Gaussian random variable with mean $\mu$ and variance ${\sigma ^2}$. In addition, ${F_X}( \cdot )$ and ${f_X}( \cdot )$ denote the cumulative distribution function (CDF) and the probability dense function (PDF) respectively. $E_i(x)$ is the exponential distribution function, and can be expressed as $E_i(x) = \int_0^\infty {\frac{{{e^\rho }}}{\rho }} {{\rm{d}}}\rho$.

II.   System Model

As illustrated in Fig.1, we consider a NOMA-based OCR system of ITS, which consists of a primary transmitter (PT), a marginal vehicle (PR1), a stronger vehicle (PR2) in primary network, a secondary transmitter (ST), and a secondary receiving vehicle (SR) in secondary network. It is assumed that: i) All nodes are equipped with one antenna; ii) All channels follow the independent Rayleigh fading.

Figure  1.  System model.

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It is difficult to obtain perfect CSI due to some practical factors, so the channel estimation method is introduced. By using linear minimum mean square error (LMMSE), the channel coefficient can be written as follows [49]:

where ${\hat h_i}$ represents the estimated channel coefficient, ${e_i} \sim {\cal {C}}{\cal {N}}(0,\sigma _{{e_i}}^2)$ represents the estimated error. ${\hat h_i}$ and ${e_i}$ is independent and orthogonal because of the orthogonality of the LMMSE algorithm. The estimated channels are denoted as ${\hat h_{p{p_1}}}\sim {\cal {C}}{\cal {N}}(0,{\lambda _{p{p_1}}})$, ${\hat h_{p{p_2}}}\sim {\cal {C}}{\cal {N}}(0,{\lambda _{p{p_2}}})$, ${\hat h_{p{s_1}}}\sim {\cal {C}}{\cal {N}}(0,{\lambda _{p{s_1}}})$, ${\hat h_{s{s_1}}}\sim {\cal {C}}{\cal {N}}(0,{\lambda _{s{s_1}}})$, ${\hat h_{{s_1}{p_1}}}\sim {\cal {C}}{\cal {N}}(0, {\lambda _{{s_1}{p_1}}})$, ${\hat h_{{s_1}{p_2}}}\sim {\cal {C}}{\cal {N}}(0,{\lambda _{{s_1}{p_2}}})$.

As in [50], we consider two channel estimation models:

1) $\sigma _{{e_i}}^2$ is a fixed constant, which is independent of the average SNR.

2) $\sigma _{{e_i}}^2$ is a function associated with the average SNR and the expression is $\sigma _{{e_i}}^2 = {\Omega _i}/(1 + \delta \gamma {\Omega _i})$, where ${\Omega _i}$ is the variance of ${h_i}$, $\gamma$ is the transmit SNR, while $\delta \geq 0$ represents the quality of channel estimation.

The whole communication process is divided into two slots.

1   The first slot

1   Received signals at PR1

PT transmits the superimposed signal to PR1, PR2 and SR through power domain multiplexing, and thus, the signal received at PR1 can be written as

where ${x_{p,1}}$ and ${x_{p,2}}$ are the signals for PR1 and PR2 with $E( {{{\left| {{x_{p,1}}} \right|}^2}} ) = 1$ and $E( {{{\left| {{x_{p,2}}} \right|}^2}} ) = 1$. ${{P_p}}$ is the transmit power of PT, ${{\alpha _1}}$ and ${{\alpha _2}}$ are the power allocation coefficients of the transmitting information ${x_{p,1}}$ and ${x_{p,2}}$ with satisfying ${\alpha _1} + {\alpha _2} = 1$, respectively. ${n_{p{p_1}}}$ is the complex Gaussian noise and it follows ${n_{p{p_1}}}{\sim} {\cal {C}}{\cal {N}}(0,{ N_n })$.

PR1 decodes ${x_{p,1}}$ and regards ${x_{p,2}}$ as a noise. The received signal-interference-plus-noise ratio (SINR) can be written as

where ${\rho _p} = \frac{{{P_p}}}{{{ N_n }}}$ is the transmit SINR at PT.

2   Received signals at PR2

The signal received in PR2 can be expressed as

where ${n_{p{p_2}}}\sim {\cal {CN}}(0,{ N_n })$.

PR2 first decodes ${x_{p,1}}$ then decodes ${x_{p,2}}$ by using SIC on the basis of NOMA protocol. When decoding ${x_{p,1}}$, the SINR at PR2 can be represented as

In practice, the receiver always have some types of errors in transmitting and detecting process, such as synchronization error, residual impairments or intrinsic constraints, therefore we assume ipSIC happens at PR2. The SINR of decoding ${x_{p,2}}$ at PR2 can be denoted by

where ${{\zeta _1}}$ reprsents the coefficient of imperfect SIC and $0 \leq {\zeta _1} \leq 1$. ${\zeta _1} =0$ and ${\zeta _1}=1$ indicate perfect SIC and no SIC, respectively.

3   Received signals at SR

The signal received at SR can be denoted as

where ${n_{p{p_3}}}\sim {\cal {C}}{\cal {N}}(0,{ N_n })$.

SR needs to decode the message ${{x_{p,1}}}$ of PR1 before decoding the message ${{x_{p,2}}}$ of PR2. The SINR of decoding ${{x_{p,1}}}$ and ${{x_{p,2}}}$ respectively at SR can be represented as

where ${{\zeta _2}}$ reprsents the coefficient of imperfect SIC and $0 \leq {\zeta _2} \leq 1$.

2   The second slot

During the second slot, SR receives the message ${x_s}$ from ST, while acting as a relay to decode and forward the messages ${x_{p,1}}$ and ${x_{p,2}}$ to PR1 and PR2 ¹.

The two phases of receiving and decoding-and-forward the message are performed in the same resource block.

1   Received signals at SR

The signal received at SR can be written as

where ${x_s}$ denotes the message for SR with $E( {{{\left| {{x_s}} \right|}^2}} ) = 1$, ${{P_s}}$ represents the transmit power of ST, and ${n_{s{s_1}}}\sim {\cal {C}}{\cal {N}}(0,{ N_n })$.

When decoding ${x_s}$, the SINR at SR can be expressed as

where ${\rho _s} = \frac{{{P_s}}}{{{ N_n }}}$ represents the transmit SINR of ST.

2   Received signals at PR1

The received signals at PR1 can be denoted by

where ${n_{_{{s_1}{p_1}}}}\sim {\cal {C}}{\cal {N}}(0,{ N_n })$, ${{P_{{s_1}}}}$ is the overall transmit power of SR, ${{\alpha _3}}$ and ${{\alpha _4}}$ are the power allocation coefficients of ${x_{p,1}}$ and ${x_{p,2}}$, respectively, and satisfying ${\alpha _3} + {\alpha _4} = 1$.

PR1 only decodes ${x_{p,1}}$ by considering other signals as interference. The SINR at PR1 can be computed as

where ${\rho _{{s_1}}} = \frac{{{P_{{s_1}}}}}{{{ N_n }}}$ represents the transmit SINR of SR.

3   Received signals at PR2

The received signals at PR2 can be represented as

where ${n_{{s_1}{p_2}}} \sim {\cal {C}}{\cal {N}}(0,{ N_n })$.

PR2 needs to decode the message ${x_{p,1}}$ of marginal user before decoding its own message ${x_{p,2}}$. Therefore, the SINR of decoding ${x_{p,1}}$ and ${x_{p,2}}$ at PR2 can be respectively computed as

where ${\zeta _3}$ represents the coefficient of ipSIC, and it satisfies $0 \leq {\zeta _3} \leq 1$.

III.   Performance Analysis

The OP and ESR are the two important evaluation metrics of wireless communication systems. In this section, we study the reliability and ergodicity by calculating OP and ESR, and analyze the asymptotic expressions of OP and ESR at high SNR region.

1   Outage probability analysis

1   Outage probability for PR1

There are three conditions that the outage event occurs at PR1 according to the NOMA protocol: i) PR1 fails to decodes ${x_{p,1}}$ in direct link; ii) SR fails to decode the message ${x_{p,1}}$ and ${x_{p,2}}$ from PT; iii) SR decodes the ${x_{p,1}}$ and ${x_{p,2}}$ successfully, but PR1 cannot decode the message ${x_{p,1}}$ successfully. Therefore, the OP at PR1 can be computed as

where ${\gamma _{{\rm{th}}1}}$ and ${\gamma _{{\rm{th}}2}}$ is the target SINR of ${x_{p,1}}$ and ${x_{p,2}}$, respectively.

Theorem 1　The analytical expression of OP at PR1 is presented as

if $0 < {\gamma _{{\rm{th1}}}} < \min \left( {\frac{{{\alpha _3}}}{{{\alpha _4}}},\frac{{{\alpha _1}}}{{{\alpha _2}}}} \right)$ and $0 < {\gamma _{{\rm{th}}2}} < \frac{{{\alpha _2}}}{{{\zeta _2}{\alpha _1}}}$, otherwise $P_{{\rm{ou}}t}^{PR1}$ is equal to 1, where ${M_1} = \frac{{{\gamma _{{\rm{th}}1}}(\sigma _{{e_{p{p_1}}}}^2{\rho _p} + 1)}}{{{\rho _p}({\alpha _1} - {\alpha _2}{\gamma _{{\rm{th}}1}})}}$, ${M_2} = \frac{{{\gamma _{{\rm{th}}1}}(\sigma _{{e_{p{s_1}}}}^2{\rho _p} + 1)}}{{{\rho _p}({\alpha _1} - {\alpha _2}{\gamma _{{\rm{th}}1}})}}$, ${M_3} = \frac{{{\gamma _{{\rm{th}}2}}(\sigma _{{e_{p{s_1}}}}^2{\rho _p} + 1)}}{{{\rho _p}({\alpha_2} - {\zeta _2}{\alpha _1}{\gamma _{{\rm{th}}2}})}}$, ${M_4} = \max ({M_2}, {M_3})$, and ${M_5} = \frac{{{\gamma _{{\rm{th}}1}}(\sigma _{{e_{{s_1}{p_1}}}}^2{\rho _{{s_1}}} + 1)}}{{{\rho _{{s_1}}}({\alpha _3} - {\alpha _4}{\gamma _{{\rm{th}}1}})}}$.

Proof　See Appendix A.

Corollary 1　At high SNRs (${\rho _p} \to \infty$, ${\rho _{{s_1}}} \to \infty$), the OP approximate expression of PR1 can be expressed as

where ${M_6} = \frac{{{\gamma _{{\rm{th}}1}}\sigma _{{e_{p{p_1}}}}^2}}{{{\alpha _1} - {\alpha _2}{\gamma _{{\rm{th}}1}}}}$, ${M_7} = \frac{{{\gamma _{{\rm{th}}1}}\sigma _{{e_{p{s_1}}}}^2}}{{{\alpha _1} - {\alpha _2}{\gamma _{{\rm{th}}1}}}}$, ${M_8} = \frac{{{\gamma _{{\rm{th}}2}}\sigma _{{e_{p{s_1}}}}^2}}{{{\alpha _2} - {\zeta _2}{\alpha _1}{\gamma _{{\rm{th}}2}}}}$, ${M_9} = \max ({M_7},{M_8})$, and ${M_{10}} = \frac{{{\gamma _{{\rm{th}}1}}\sigma _{{e_{{s_1}{p_1}}}}^2}}{{{\alpha _3} - {\alpha _4}{\gamma _{{\rm{th}}1}}}}$.

2   Outage probability for PR2

There are three conditions when PR2 encounters outage event: i) PR2 cannot decodes ${{x_{p,1}}}$ and ${{x_{p,2}}}$ from the direct link successfully; ii) SR cannot decodes ${{x_{p,1}}}$ and ${{x_{p,2}}}$ from PT successfully; iii) SR decodes ${{x_{p,1}}}$ and ${{x_{p,2}}}$ from PT successfully, but PR2 fails to decode ${{x_{p,1}}}$ and ${{x_{p,2}}}$ from SR. Thus we can calculate the OP at PR2 as

Theorem 2　The analytical expression of OP at PR2 can be written as

if $0 {<} {\gamma _{{\rm{th}}1}} {<} \min \left( {\frac{{{\alpha _1}}}{{{\alpha _2}}},\frac{{{\alpha _3}}}{{{\alpha _4}}}} \right)$ and $0 {<} {\gamma _{{\rm{th}}2}} {<} \min \left( \frac{{{\alpha _2}}}{{{\zeta _1}{\alpha _1}}}, \frac{{{\alpha _4}}}{{{\zeta _2}{\alpha _3}}}, \frac{{{\alpha _4}}}{{{\zeta _3}{\alpha _3}}} \right)$, otherwise $P_{{\rm{out}}}^{PR2}$ is equal to 1, where ${M_{11}} = \frac{{{\gamma _{{\rm{th}}1}}(\sigma _{{e_{p{p_2}}}}^2{\rho _p} + 1)}}{{{\rho _p}({\alpha _1} - {\alpha _2}{\gamma _{{\rm{th}}1}})}}$, ${M_{12}} = \frac{{{\gamma _{{\rm{th}}2}}(\sigma _{{e_{p{p_2}}}}^2{\rho _p} + 1)}}{{{\rho _p}({\alpha _2} - {\zeta _1}{\alpha _1}{\gamma _{{\rm{th}}2}})}}$, ${M_{13}} = \max ({M_{11}}, {M_{12}})$, ${M_{14}} = \frac{{{\gamma _{{\rm{th}}1}}(\sigma _{{e_{{s_1}{p_2}}}}^2{\rho _{{s_1}}} + 1)}}{{{\rho _{{s_1}}}({\alpha _3} - {\alpha _4}{\gamma _{{\rm{th}}1}})}}$, ${M_{15}} = \frac{{{\gamma _{{\rm{th}}2}}(\sigma _{{e_{{s_1}{p_2}}}}^2{\rho _{{s_1}}} + 1)}}{{{\rho _{{s_1}}}({\alpha _4} - {\zeta _3}{\alpha _3}{\gamma _{{\rm{th}}2}})}}$, and ${M_{16}} = \max ({M_{14}},{M_{15}})$.

Proof　See Appendix B.

Corollary 2　At high SNRs (${\rho _p} \to \infty$, ${\rho _{{s_1}}} \to \infty$), the approximate expression of PR2 having outage event is

In (22), ${M_{17}} = \frac{{{\gamma _{{\rm{th}}1}}\sigma _{{e_{p{p_2}}}}^2}}{{{\alpha _1} - {\alpha _2}{\gamma _{{\rm{th}}1}}}}$, ${M_{18}} = \frac{{{\gamma _{{\rm{th}}2}}\sigma _{{e_{p{p_2}}}}^2}}{{{\alpha _2} - {\zeta _1}{\alpha _1}{\gamma _{{\rm{th}}2}}}}$, ${M_{19}} = \max ({M_{17}}, {M_{18}})$, ${M_{20}} = \frac{{{\gamma _{{\rm{th}}1}}\sigma _{{e_{{s_1}{p_2}}}}^2}}{{{\alpha _3} - {\alpha _4}{\gamma _{{\rm{th}}1}}}}$, ${M_{21}} = \frac{{{\gamma _{{\rm{th}}2}}\sigma _{{e_{{s_1}{p_2}}}}^2}}{{{\alpha _4} - {\zeta _3}{\alpha _3}{\gamma _{{\rm{th}}2}}}}$, and ${M_{22}} = \max ({M_{20}},{M_{21}})$.

3   Outage probability for SR

The outage event will happen when SR fails to decode the message ${x_s}$, that is

where ${\gamma _{{\rm{th}}s}}$ represents the target SINR of decoding the message ${x_s}$.

Theorem 3　The analytical expression of OP at SR can be written as

where ${M_{23}} = \frac{{{\gamma _{{\rm{th}}s}}(\sigma _{{e_{s{s_1}}}}^2{\rho _s} + 1)}}{{{\rho _s}}}$.

Proof　See Appendix C.

Corollary 3　At high SNRs (${\rho _s} \to \infty$), the approximate expression of SR having outage event is

To gain further insight, we discuss the diversity order of PR1, PR2 and SR. According to [7], the diversity order is defined as

Corollary 4　The diversity orders of PR1, PR2 and SR can be represent as

Remark 1　From Corollary 1, Corollary 2 and Corollary 3, we can find that when the transmitted SNR approaches infinity, the asymptotic OPs of PR1, PR2 and SR become a fixed constant, indicating that the OP has an error floor, yielding the diversity order to be 0.

2   Ergodic sum rate analysis

1   Ergodic sum rate for primary network

The ESR of primary network can be expressed as

The ergodic rates (ERs) for PR1 and PR2 are inferred as follows.

2   Ergodic rate for PR1

The ER of PR1 can be written as

where $W = \min ({u_1},{u_2},{u_3})$, ${u_1} = \gamma _{{x_{p,1}}}^{p{p_1}}$, ${u_2} = \gamma _{{x_{p,1}}}^{p{s_1}}$, and ${u_3} = \gamma _{{x_{p,1}}}^{{s_1}{p_1}}$.

Theorem 4　The analytical expression of ER at PR1 can be written as

where ${\phi _i} = \cos \left( {\frac{{(2i - 1)\pi }}{{2k}}} \right)$, ${M_{25}}{\rm{ = }}\frac{{w(\sigma _{{e_{p{p_1}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{p_1}}}{\rho _p}({\alpha _1} - w{\alpha _2})}}$, ${M_{26}}{\rm{ = }}\frac{{w(\sigma _{{e_{p{s_1}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{s_1}}}{\rho _p}({\alpha _1} - w{\alpha _2})}}$, ${M_{27}}{\rm{ = }}\frac{{w(\sigma _{{e_{{s_1}{p_1}}}}^2{\rho _{{s_1}}} + 1)}}{{{\lambda _{{s_1}{p_1}}}{\rho _{{s_1}}}({\alpha _3} - w{\alpha _4})}}$, ${M_{28}} = \frac{{N({\phi _i} + 1)(\sigma _{{e_{p{p_1}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{p_1}}}{\rho _p}(2{\alpha _1} - {\alpha _2}N({\phi _i} + 1))}}$, ${M_{29}} = \frac{{N({\phi _i} + 1)(\sigma _{{e_{p{s_1}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{s_1}}}{\rho _p}(2{\alpha _{_1}} - {\alpha _2}N({\phi _i} + 1))}}$ and ${M_{30}} = \frac{{N({\phi _i} + 1)(\sigma _{{e_{{s_1}{p_1}}}}^2{\rho _{{s_1}}} + 1)}}{{{\lambda _{{s_1}{p_1}}}{\rho _{{s_1}}}(2{\alpha _3} - {\alpha _4}N({\phi _i} + 1))}}$.

Proof　See Appendix D.

Corollary 5　At high SNRs (${\rho _p} \to \infty$, ${\rho _{{s_1}}} \to \infty$), we can calculate the ER asymptotic expression of PR1 as

where ${M_{31}} = \frac{{N({\phi _i} + 1)\sigma _{{e_{p{p_1}}}}^2}}{{{\lambda _{p{p_1}}}(2{\alpha _1} - {\alpha _2}N({\phi _i} + 1))}}$, ${M_{32}} = \frac{{N({\phi _i} + 1)\sigma _{{e_{p{s_1}}}}^2}}{{{\lambda _{p{s_1}}}(2{\alpha _{_1}} - {\alpha _2}N({\phi _i} + 1))}}$ and ${M_{33}} = \frac{{N({\phi _i} + 1)\sigma _{{e_{{s_1}{p_1}}}}^2}}{{{\lambda _{{s_1}{p_1}}}(2{\alpha _3} - {\alpha _4}N({\phi _i} + 1))}}$.

3   Ergodic rate for PR2

The ER of PR2 can be represented as

where $Z = \min ({v_1},{v_2},{v_3})$, ${v_1} = \gamma _{{x_{p,2}}}^{p{p_2}}$, ${v_2} = \gamma _{{x_{p,2}}}^{p{s_1}}$, and ${v_3} = \gamma _{{x_{p,2}}}^{{s_1}{p_2}}$.

Theorem 5　The analytical expression of ER at PR2 is given by

where ${M_{34}} = \frac{{z(\sigma _{{e_{p{p_2}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{p_2}}}{\rho _p}({\alpha _2} - z{\zeta _1}{\alpha _1})}}$, ${M_{35}} = \frac{{z(\sigma _{{e_{p{s_1}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{s_1}}}{\rho _p}({\alpha _2} - z{\zeta _2}{\alpha _1})}}$, ${M_{36}} = \frac{{z(\sigma _{{e_{{s_1}{p_2}}}}^2{\rho _{{s_1}}} + 1)}}{{{\lambda _{{s_1}{p_2}}}{\rho _{{s_1}}}({\alpha _4} - z{\zeta _3}{\alpha _3})}}$, ${M_{37}} = \frac{{{N_2}({\phi _i} + 1)(\sigma _{{e_{p{p_2}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{p_2}}}{\rho _p}(2{\alpha _2} - {\alpha _1}{N_2}({\phi _i} + 1){\zeta _1})}}$, ${M_{38}} = \frac{{{N_2}({\phi _i} + 1)(\sigma _{{e_{p{s_1}}}}^2{\rho _p} + 1)}}{{{\lambda _{p{s_1}}}{\rho _p}(2{\alpha _2} - {\alpha _1}{N_2}({\phi _i} + 1){\zeta _2})}}$ and ${M_{39}}$ is expressed as ${M_{39}} = \frac{{{N_2}({\phi _i} + 1)(\sigma _{{e_{{s_1}{p_2}}}}^2{\rho _{{s_1}}} + 1)}}{{{\lambda _{{s_1}{p_2}}}{\rho _{{s_1}}}(2{\alpha _4} - {\alpha _3}{N_2}({\phi _i} + 1){\zeta _3})}}$.

Proof　See Appendix E.

Corollary 6　At high SNRs (${\rho _p} \to \infty$, ${\rho _{{s_1}}} \to \infty$), we can calculate the ER asymptotic expression of PR2 as

where ${M_{40}}$ and ${M_{41}}$ are ${M_{40}} = \frac{{{N_2}({\phi _i} + 1)\sigma _{{e_{p{p_2}}}}^2}}{{{\lambda _{p{p_2}}}(2{\alpha _2} - {\alpha _1}{N_2}({\phi _i} + 1){\zeta _1})}}$ and ${M_{41}} = \frac{{{N_2}({\phi _i} + 1)\sigma _{{e_{p{s_1}}}}^2}}{{{\lambda _{p{s_1}}}(2{\alpha _2} - {\alpha _1}{N_2}({\phi _i} + 1){\zeta _2})}}$, respectively, ${M_{42}}$ is ${M_{42}} = \frac{{{N_2}({\phi _i} + 1)\sigma _{{e_{{s_1}{p_2}}}}^2}}{{{\lambda _{{s_1}{p_2}}}(2{\alpha _4} - {\alpha _3}{N_2}({\phi _i} + 1){\zeta _3})}}$.

Theorem 6　The ESR of primary network can be expressed as

Corollary 7　At high SNRs (${\rho _s} \to \infty$), we can simplify the ESR asymptotic expression of primary network as

4   Ergodic sum rate for secondary network

The ESR of secondary network can be expressed as

Theorem 7　The ESR of secondary network is given by

Proof　See Appendix F.

Corollary 8　At high SNRs (${\rho _s} \to \infty$), we can simplify the ESR asymptotic expression of secondary network as

Remark 2　From Corollary 7 and Corollary 8, we found that when the transmitted SNR approaches infinity, the asymptotic ESRs of primary network and secondary network become a fixed constant, implying that there exists a ceiling for ESR due to CEEs.

IV.   Numerical Results and Discussion

In this section, we offer some numerical results to verify the correctness of the analysis in Section III, and these results are obtained based on the ${10^6}$ Mente Carlo simulations. Unless otherwise specified, the parameter values are set as in Table 1.

Table  1.  Parameters for numerical results

Channel estimation error	$\sigma _{ {e_{p{p_1} } } }^2 = \sigma _{ {e_{p{p_2} } } }^2 = \sigma _{ {e_{p{s_1} } } }^2 = \sigma _{ {e_{s{s_1} } } }^2 = \sigma _{ {e_{ {s_1}{p_1} } } }^2 = \sigma _{ {e_{ {s_1}{p_2} } } }^2 = 0.01$
Estimated channel coefficient	$\{ {\lambda _{p{p_1}}},{\lambda _{p{p_2}}},{\lambda _{p{s_1}}},{\lambda _{s{s_1}}},{\lambda _{{s_1}{p_1}}},{\lambda _{{s_1}{p_2}}}\} {\rm{ = \{ 0}}{\rm{.1,0.5,2,0.1,4,3\} }}$
Power allocation coefficient	${\alpha _1} = {\alpha _3} = 0.9 ， {\alpha _2} = {\alpha _4} = 0.1$
Noise power	$N_n = 1$
Imperfect SIC	$\{ {\zeta _1},{\zeta _2},{\zeta _3}\} = 0.001$
Targeted date rate	$\{ {\gamma _{{\rm{th}}1 } },{\gamma _{{\rm{th}}2 } },{\gamma _{{\rm{th}}s } }\} = \{ 1,1,2\}$

 | Show Table

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Fig.2 shows the OP curves of the users under the ideal and non-ideal conditions. For the purpose of comparison, the parameter settings under the ideal conditions are provided, and the variances of CEEs are $\sigma _{{e_{p{p_1}}}}^2 = \sigma _{{e_{p{p_2}}}}^2 = \sigma _{{e_{p{s_1}}}}^2 = \sigma _{{e_{s{s_1}}}}^2 = \sigma _{{e_{{s_1}{p_1}}}}^2 = \sigma _{{e_{{s_1}{p_2}}}}^2 = 0$. With transmit SNR increasing, the OPs of PR1, PR2, SR decrease gradually and finally converge to fixed values, which leads to error floors. This means that when there are CEEs, the transmit power is not always beneficial to the reliable performance. Under ideal conditions, the OPs decrease with the SNR. In addition, the OP value of each user under the ideal condition is smaller than that under the non-ideal condition. This indicates that the existence of ipSIC and CEEs reduce the reliability of the system.

Figure  2.  The OP of the users in ideal and non-ideal conditions.

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Fig.3 provides the curves of OP versus SNR under two channel estimation models: i) the channel estimation error is a fixed constant; ii) the channel estimation error is a function of the SNR. As can be seen from Fig.3, the theoretical analysis curves are in consistence with the simulation results, validating the accuracy of our analysis. For the first model, the OP is a fixed constant in the high SNR regimes, while for the second mode, the OP decrease as the SNR grows large. This is because that when the SNR grows large, the estimated channel gradually approaches the real channel, yielding improves reliable performance.

Figure  3.  The OP of the users in different channel estimation error models.

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Fig.4 plots the OPs of users versus power allocation parameter ${{\alpha _2}}$ for different target rates. It is clear that the OP first decreases and then increases with ${{\alpha _2}}$, we can draw the conclusion from equations (18) and (21). Fig.4 also indicates that when the target rates ${\gamma _{th1}}$ and ${\gamma _{th2}}$ are smaller, the OP of the considered system is smaller and the reliability of the system is higher. In addition, the optimal value of OP is insensitive to the target rates ${\gamma _{th1}}$ and ${\gamma _{th2}}$.

Figure  4.  OP versus ${{\alpha _2}}$ for different ${\gamma _{{th}1}}$, ${\gamma _{{th}2}}$.

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Fig.5 shows the ESR versus SNR. We set $N = 8$, ${N_2} = 100$, and $k = {k_2} = 100$. From Fig.5, we can observe that the ESR increases gradually with the SNR and then tends to a fixed constant under the non-ideal condition. In contrast, under the ideal condition ($\sigma _{{e_{p{p_1}}}}^2 = \sigma _{{e_{p{p_2}}}}^2 = \sigma _{{e_{p{s_1}}}}^2 = \sigma _{{e_{s{s_1}}}}^2 = \sigma _{{e_{{s_1}{p_1}}}}^2 = \sigma _{{e_{{s_1}{p_2}}}}^2 = 0$ and ${\zeta _1} = {\zeta _2} = {\zeta _3} = 0$), the ESR grows linearly with the SNR. At high SNR regions, the analytical and asymptotic values of the ESR are basically coincident.

Figure  5.  The ESR under ideal and non-ideal condition.

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Fig.6 represents the relationship between ESR and transmitting SNR under the above mentioned two CEEs models. In this simulation, we set $N = 8$, ${N_2} = 100$, $k = {k_2} = 100$ and $\delta = \{ 0.1,1\}$. we can see from Fig.6 that the ESR has a ceiling at high SNR regions when estimation error is a fixed constant, and the ESR decreases with $\delta$ when the estimation error is a unary function. Changes in $\delta$ have a negligible effect on the system ESR value in the low SNR region.

Figure  6.  The ESR in different channel estimation error models.

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The relationship between ER and ${{\alpha _2}}$ under different SNRs is depicted in Fig.7. In simulation, we set $N = 8$, ${N_2} = 100$, $k = {k_2} = 100$ and ${\rm{SNR}}=\left\{ {10,15} \right\}$ dB. The ER of PR1 decreases as the power allocation parameter ${{\alpha _2}}$ increases, while the ER of PR2 increases with ${{\alpha _2}}$, which can be inferred from equations (30) and (33). Generally speaking, the larger ${{\alpha _2}}$ is, the easier to decode ${x_{p,2}}$ and the more difficult to decode ${x_{p,1}}$. That is, the combination of ${x_{p,1}}$ and ${x_{p,2}}$ could explain the changes in the curves.

Figure  7.  Ergodic Rate versus ${{\alpha _2}}$ for different SNR.

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V.   Conclusions

To improve the spectral efficiency, we proposed a NOMA-based OCR system for ITS. The reliability and ergodicity of the proposed system were analyzed by considering ipSIC and two channel estimation models. We derived the analytical expressions of the OP and ESR, as well as the asymptotic expressions of OP and ESR at the high SNR region. A series of simulations were carried out to verify the accuracy of the analysis. In addition, the effect of a series of related parameters on the system performance were investigated through simulation.

Appendix A. Proof of Theorem 1

The proof starts by substituting equations (3), (8), (9), (13) into (17), the OP for PR1 can be written ${\rm{as}}$

where ${I_1}$ and ${I_2}$ are calculated as follows:

and

Appendix B. Proof of Theorem 2

The proof starts by substituting equations (5), (6), (8), (9), (15), (16) into (20), the OP for PR2 can be written as

where ${I_3}$ and ${I_4}$ are calculated as follows:

and

Appendix C. Proof of Theorem 3

The proof starts by substituting equation (11) into (23), the OP for SR can be written as

Appendix D. Proof of Theorem 4

Based on the definition of expectation, equation (28) can be further calculated as

It is assumed that ${f_W}(w)$, ${{F_W}(w)}$ are the PDF and CDF of random variable W respectively. ${F_W}(w)$ can be computed as

We can figure out the ER of PR1 by plugging (D-2) into (D-1),

It is difficult to acquire an accurate closed-formed solution. Alternatively, we obtain the approximation with the aid of Gaussian-Chebyshev quadrature as [51]

Appendix E. Proof of Theorem 5

It is assumed that ${F_Z}(z)$ is the CDF of random variable $Z$. We can obtain the expression by calculating as follows:

The ER of PR2 can be written as

Since exact closed-formed expressions are not available, considering that when $N_2$ is a large number, we have the following equation according to Gaussian-Chebyshev quadrature.

Appendix F. Proof of Theorem 7

Letting $\theta = r_{{x_s}}^{s{s_1}}$ to facilitate the calculation, we can obtain the PDF of $\theta$ as

By inserting (F-1) into (37), equation (38) can be acquired.