Performance analysis of inverse successive interference cancellation in NOMA-based communications

Abstract

Non-orthogonal multiple access (NOMA) is a promising solution to enhance the spectral efficiency of sixth generation (6G) networks. NOMA enables multiple users to be concurrently served, relying on superposition coding at the transmitter and successive interference cancellation (SIC) at the receivers. However, in practice, hardware impairments and channel estimation errors will lead to incorrect signal ordering. This study analyses inverse successive interference cancellation (ISIC) in downlink power-domain NOMA networks. In the worst case, ISIC will inversely order the N received signals, compared to conventional SIC. First, ISIC will decode the signal with the minimum allocated power, while treating the other $N-1$ signals as interference. This process is repeated $N-1$ times and in the last step, the signal with the maximum allocated power will be decoded without interference. Various key performance metrics for ISIC are analytically derived, including the outage probability, throughput, error probability, ergodic rate, achievable sum-rate, energy and spectral efficiency. In order to provide further insights on ISIC, an asymptotic analysis is also provided. Comparisons with SIC reveal that for the signal with the second-lowest power allocation level, ISIC achieves better performance, while the performance of the information signal with the minimum allocated power is degraded. Overall, our findings shed light on the performance of NOMA in practical settings where SIC can not be correctly performed and focuses on the worst-case where the decoding order at the receiver is completely inversed.

1 Introduction

The remarkable growth of wireless data traffic and highly-demanding Internet-of-Things (IoT) use cases necessitate the support of heterogeneous networks, i.e. with coexisting humans and machines with higher capacity and energy-efficient resource allocation [1,2,3]. In addition, the development of sixth generation (6G) networks will be based on novel communication techniques, departing from conventional approaches. Among these techniques, non-orthogonal multiple access (NOMA) enables increased resource allocation flexibility over orthogonal multiple access (OMA), allowing radio-resources to be shared by multiple users by exploiting their channel power level and rate asymmetries [4, 5]. Also, focus has been given to energy-efficient resource allocation towards optimised and integrated-access-and-backhaul solutions in both uplink and downlink NOMA-based heterogeneous networks [6]. The surveys in [7,8,9] presented several challenges for successfully implementing NOMA, including power allocation optimization, energy efficiency, hardware and channel estimation imperfections, and integration with full-duplex transceivers. In NOMA, the transmitter sends super-imposed information signals allocating more power to users with weak channels and less power to users with strong channels. [10,11,12,13]. In contrast to OMA where each channel is allocated to a single user, power domain (PD)-NOMA concurrently serves the users and allocates smaller power fractions to the users with strong channel conditions and larger fractions to users with weaker channels. When decoding the received signal, the receiver adopts successive interference cancellation (SIC) and first decodes the signal of the strongest user, considering the other signals as interference. Then, the decoded signal is subtracted and this process is repeated, until the last signal, i.e. of the weakest user, is decoded without interference [7, 14].

1.1 Background

In the literature, several improvements to conventional SIC have been proposed, adopting advanced signal decoding techniques to provide better performance for downlink NOMA networks. The authors in [15] presented parallel interference cancellation (PIC) for user decoding and simulations showed that it outperformed conventional SIC. The study in [16] proposed advanced SIC, mapping the received signals into sub-groups for achieving the highest rates in each one. The analytical and simulation results showed that advanced SIC provided better outage performance over conventional SIC. Furthermore, the authors in [17] proposed a universal receiver, relying on a general turbo-like iterative receiver structure, a universal expectation propagation algorithm detector and hybrid PIC. It was observed that the proposed receiver offered improved block error rate (BLER) with low complexity and fast convergence, always outperforming conventional codeword level MMSE-PIC receiver. In order to minimize the error propagation in the interference cancellation process, the paper in [18] presented multiple decision-aided SIC techniques. Contrary to the selection of one codeword for each user, the proposed techniques introduced multiple candidate codewords to overcome error propagation and outperformed conventional SIC with reduced computational complexity. The study in [19] proposed advanced SIC for outage balancing of the consequently decoded users in NOMA systems. Here, the BS treated the unsuccessfully decoded signals as interference for the consequent users and even when an outage occurs for previously decoded users, it still tried to decode the remaining signals. The paper in [20] provided performance comparisons of three potential interference cancellation schemes, i.e. independent single-user decoding, soft-in soft-out PIC, and hybrid interference cancellation, for NOMA systems. The performance of these schemes was evaluated, considering the error propagation, decoding delay, and complexity, highlighting that hybrid interference cancellation provides a trade-off among complexity and performance with increased flexibility. Another work provided performance comparisons of different NOMA receivers, i.e. SIC and joint decoding techniques for single-block transmissions over a two-user Gaussian multiple-access channel with partial CSI at the transmitter [21]. Results showed that joint decoding provided a sum-rate gain of up to 10% over SIC. A joint modulation and detection-based transceiver structure was presented in [22], and compared with codeword-level SIC, symbol-level SIC, and ideal SIC. Performance comparisons showed that codeword-level SIC and the proposed transceiver design offered improved performance even for small power ratio difference among the cell center user and the cell edge user. In addition, the authors in [23] proposed an improved signal detection algorithm for detecting the stronger user and employed NOMA in a smaller power allocation coefficients range among users, ensuring better detection performance against conventional SIC.

Other works studied uplink NOMA topologies and developed advanced SIC schemes. A SIC enhancement was proposed in [13], namely triangular SIC technique for asynchronous NOMA systems. Here, SIC uses a triangular pattern for interference cancellation of the interfering users to the desired user. BER and capacity performance in an uplink NOMA system and comparisons with an OMA network, relying on orthogonal frequency division multiple access (OFDMA) are presented. Results showed that triangular SIC outperforms OFDMA while the number of iterations differs, depending on the modulation level and the detection method. In addition, the authors in [24] proposed dynamic decoding ordering for SIC, relying on instantaneous channel state information (CSI), for uplink NOMA systems. The dynamic SIC receiver showed significant improvement over SIC with fixed user ordering, based on statistical CSI, with increased estimation requirements. The dynamic decoding ordering was also evaluated in an opportunistic multi-relay uplink NOMA network in order to support users with different rate requirements [25, 26], offering significant sum-rate gains over OMA. Another uplink NOMA network was studied in [27], proposing SIC with model-driven deep learning in scenarios where noise impacts the channel estimation process and incomplete detection and cancellation in SIC process. Performance comparisons revealed that the proposed learning-aided SIC improved the BER performance without incurring high training and implementation complexity.

Focusing on practical issues, a number works focused on analyzing the performance of NOMA under hardware impairments and channel estimation errors. In this context, the work in [28] extended [24] by considering channel estimation errors in the performance analysis, causing additional interference on the desired signal and incorrect decoding order. Simulations showed that the dynamic-ordered SIC receiver outperforms conventional SIC when highly accurate CSI estimation was performed. Next, the authors in [28] investigated a two-user uplink NOMA network, based on dynamic SIC with imperfect CSI due to channel estimation errors. Also, the study in [29] analysed the impact of imperfect CSI due to estimation errors and antenna pointing error on the ergodic capacity performance of NOMA-based satellite networks with randomly deployed mobile terminals. Next, the work in [30] compared the queuing delay of NOMA and OMA in a two-user scenario with imperfect CSI, concluding that in such cases, NOMA can lead to increased delays. Power allocation was the topic of [31] in NOMA networks with channel estimation errors. NOMA-based algorithms were developed, leveraging the varying QoS requirements of the users and offer improved rates over other OMA and NOMA solutions. The impact of outdated CSI was examined in [32] and a hybrid NOMA/OMA communication strategy was devised, relying on distributed implementation to overcome centralised coordination. In this way, the impact of outdated CSI was addressed and higher sum-throughput was ensured. The study in [33] examined the performance of uplink NOMA-based underlay scheme and employs CSI-based NOMA/OMA mode switching under imperfect SIC, surpassing standalone OMA and NOMA.

1.2 Contributions

Taking into consideration that the successful integration of NOMA in wireless networks demands that practical issues will be carefully addressed, in this paper, the performance of SIC when the signal decoding order is inversed due to hardware or channel estimation imperfections is analysed. Without loss of generality, we focus on a single-source single-destination network where the source multiplexes N signals in each transmission. In the worst case, SIC will inversely order the N information signals. More specifically, ISIC orders the power allocation coefficients in the inverse order, initially decoding the information with the smallest power coefficient and considers the other signals as interference. Next, the decoded information is subtracted and the receiver moves on to decode the second-lowest power allocated information, treating the remaining signals as interference. ISIC will follow the inverse order of decoding, until the last signal is decoded interference-free. This work extends [34] with additional theoretical results, asymptotic analysis and power allocation optimization while presenting extensive simulation results to shed light to ISIC performance. In this context, the main contributions of this study are summarised as follows:

• Towards examining NOMA’s performance under non-ideal settings, this paper presents the case of ISIC where inverse user ordering takes place. In this context, ISIC and SIC are presented, assuming perfect interference cancellation, i.e. no residual interference and perfect CSI when performing power allocation. This benchmark comparisons allows us to thoroughly analyze the performance of ISIC and highlight its advantages and disadvantages over SIC.

• The outage probability, throughput, error probability, ergodic rate, and achievable sum-rate metrics are analysed for both SIC and ISIC.

• ISIC is also examined under asymptotically high signal-to-noise (SNR) ratio, deriving the asymptotic expressions for the considered performance metrics.

• Furthermore, Lagrangian-based optimization is presented, aiming to obtain the optimal power allocation coefficients for ISIC.

• The analytical and asymptotic results are verified through Monte Carlo simulations, evaluating the impact of different network parameters.

1.3 Outline and notation

This work is structured as follows. Section 2 provides the system and channel models details. The theoretical analysis of ISIC, considering the outage probability, throughput, error probability, ergodic rate, and achievable sum-rate metrics is given in Sect. 3. Power coefficient optimization is investigated in Sect. 4 while Sect. 5 includes the numerical results. Conclusions and future directions are discussed in Sect. 6.

1.3.1 Notations

Superscript ”i” represents the inverse. $\mathrm log$ terms are base 2 unless stated otherwise. $D_{n}(z)$ is Parabolic cylinder function [35]. $\Gamma (.)$ is the complete Gamma function [35, Eq. (8.310.1)]; $G^{m,n}_{p,q}[.]$ is the Meijer’s G function [36, Eq. (21)]; $G^{m,n:m_{1},n_{1}:m_{2},n_{2}}_{p,q:p_{1},q_{1}:p_{2},q_{2}}[.]$ is the extended generalised bi-variate Meijer’s G function [37, Eq. (13)]; The operators ${\mathbb {E}}[.]$ and ${\text{Pr}}(.)$ represent the expectation and probability, respectively. Finally, $f_g(.)$ and $F_g(.)$ denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable (RV) g, respectively.

Table 1 List of acronyms

Table 2 List of notation

2 System model

To study ISIC’s performance, a downlink PD-NOMA network is considered. According to the downlink model, a BS transmits a waveform, which contains multiple information signals in a super-imposed form, towards a single mobile terminal (MT), following the PD-NOMA principle. In this topology, each network node is equipped with a single omnidirectional antenna. The fading channel coefficient, representing the channel impulse response between the BS and the MT in each time-slot h is a complex Gaussian RV, characterized by $\sigma ^{2}$ variance and zero mean, $\left( h \sim \mathcal{C}\mathcal{N}\left( 0, \sigma ^{2}\right) \right)$. The amplitude of h is distributed according to Rayleigh distribution, i.e. a non line-of-sight setting is assumed. The transmitted super-imposed signal is described by $\sum _{k=1}^{N}\sqrt{\beta _{k}P_{s}}x_{k}$ where $\beta _{k}$ denotes the power allocation coefficient, $P _{s}$ represents the transmit power, and $x _{k}$, ${\mathbb {E}}\left[ |x _{k}|^{2}\right] =1$, is the transmitted information. The summation of the power allocation coefficients and their orders are given as $\sum _{k=1}^{N}\beta _{k}=1$ and $0<\beta _{k}<\beta _{t}<\beta _{N}<1$, respectively. At the end of an arbitrary time-slot, the MT receives the following signal:

$$\begin{aligned} Z_{r}=\sum _{k=1}^{N}\sqrt{\beta _{k}P_{s}}hx_{k}+w_{m}, \end{aligned}$$

(1)

where $w_{m}$ represents the additive white Gaussian noise (AWGN) with $\sigma ^{2}$ variance, at the MT. Considering that correct user ordering is not possible due to hardware or channel estimation imperfections, the MT will employ SIC with inverse signal ordering for its decoding process. Different from SIC, ISIC firstly decodes the information signal, which has the smallest power allocation coefficient, treating the other signals as interference. Towards this end, based on the power allocation coefficients order $0<\beta _{k}<\beta _{t}<\beta _{N}<1$, the achievable rates are obtained as:

$$\begin{aligned} R_{x_{k}}^{i}&=\mathrm log\left( 1+\frac{\beta _{k}\gamma _{x}}{\sum _{t=k+1}^{N}\beta _{t}\gamma _{x}+1}\right) , \end{aligned}$$

(2)

$$\begin{aligned} R_{x_{N}}^{i}&=\mathrm log\left( 1+\beta _{N}\gamma _{x}\right) , \end{aligned}$$

(3)

where $\gamma _{x}=\frac{P_{s}|h|^{2}}{\sigma ^{2}}$. The operation of SIC in an arbitrary time-slot is summarised in Algorithm 1.

In a similar way, in the case of SIC, invoking (1) and also, considering the order of power allocation coefficients, the following achievable rates can be obtained:

$$\begin{aligned} R_{x_{k}}&=\mathrm log\left( 1+\beta _{k}\gamma _{x}\right) , \end{aligned}$$

(4)

$$\begin{aligned} R_{x_{N}}&=\mathrm log\left( 1+\frac{\beta _{N}\gamma _{x}}{\sum _{t=k+1}^{N}\beta _{t}\gamma _{x}+1}\right) . \end{aligned}$$

(5)

Remark 1

In this study, a thorough performance comparison of ISIC/SIC techniques in the context of NOMA are investigated under perfect conditions, in terms of SIC decoding, i.e. no residual interference and without considering outdated CSI issues while performing power allocation to the different signals [38, 39]. In this way, the performance of each interference cancellation technique can be more easily analyzed, shedding light on their advantages and disadvantages for varying SNR conditions. As a next step, the system model should incorporate issues related to residual interference after subtracting each signal during ISIC/SIC and imperfections during the power allocation process due to outdated CSI.

Algorithm 1

Inverse Successive Interference Cancellation

3 Performance analysis

Analytical and asymptotic derivations of various types of performance metrics are presented in this section. In this regard, the outage probability, throughput, error probability, ergodic rate, achievable sum-rate, energy efficiency, and spectral efficiency performance metrics are analysed for both ISIC and SIC.

3.1 Outage probability

The analytical derivation of the NOMA transmission’s outage performance can be defined as the CDF of the received SNR/signal-to-interference noise ratio (SINR) evaluated at the target threshold rate, $R_{th} = \log (1+\gamma _{th})$ where $\gamma _{th}$ is the target SNR/SINR value. By means of this definition, the CDFs of (2)–(5) can be obtained as:

Proposition 1

The $F_{x_{k}}^{i}\left( \gamma _{\text{th}}\right)$ and $F_{x_{N}}^{i}\left( \gamma _{\text{th}}\right)$ and also $F_{x_{k}}\left( \gamma _{\text{th}}\right)$ and $F_{x_{N}}\left( \gamma _{\text{th}}\right)$ can be calculated as in the following Proposition.

$$\begin{aligned} F_{x _{k}}^{i}\left( \gamma _\text{th}\right)&=1-e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{k} -\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) }, \end{aligned}$$

(6)

$$\begin{aligned} F_{x _{N}}^{i}\left( \gamma _\text{th}\right)&=1-e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\beta _{N}}\right) }, \end{aligned}$$

(7)

$$\begin{aligned} F_{x _{k}}\left( \gamma _\text{th}\right)&=1-e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\beta _{k}}\right) }, \end{aligned}$$

(8)

$$\begin{aligned} F_{x _{N}}\left( \gamma _\text{th}\right)&=1-e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{N} -\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) }, \end{aligned}$$

(9)

where $\Omega _{h}$ is mean of $|h|^2$ and $\gamma _{th}=2^{R}-1$.

Proof

See Appendix 1. $\square$

3.2 Throughput analysis

The throughput performance of ISIC and SIC can be calculated by means of [40, Eq. (15(a))] as in (10) and (11), respectively:

$$\begin{aligned} \tau _{x_{k}}^{i}&=\gamma _{th}\left( 1-F_{x_{k}}^{i}\right) , \forall _{k}=1,...,N, \end{aligned}$$

(10)

$$\begin{aligned} \tau _{x_{k}}&=\gamma _{th}\left( 1-F_{x_{k}}\right) . \end{aligned}$$

(11)

By plugging the CDFs from (6), (7), (8), and (9), into (10) and (11), the following expressions can be obtained:

$$\begin{aligned} \tau _{x_{k}}^{i}&=\gamma _{th}\left( e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h} \left( \beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) }\right) , \end{aligned}$$

(12)

$$\begin{aligned} \tau _{x_{N}}^{i}&=\gamma _{th}\left( e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\beta _{N}}\right) }\right) , \end{aligned}$$

(13)

$$\begin{aligned} \tau _{x_{k}}&=\gamma _{th}\left( e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\beta _{k}}\right) }\right) , \end{aligned}$$

(14)

$$\begin{aligned} \tau _{x_{N}}&=\gamma _{th}\left( e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{N} -\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) }\right) . \end{aligned}$$

(15)

Note that utilizing (10) and (11), the achievable sum throughput can be formulated for ISIC and SIC, as $\sum _{k=1}^{N}\tau _{x_{k}}^{i}$ and $\sum _{k=1}^{N}\tau _{x_{k}}$, respectively.

3.3 Error probability

The ISIC’s error probability performance is investigated in this subsection. Moreover, in order to conduct a performance comparison, this subsection also provides error probability results for SIC. To this end, with the help of [41, Eq. (27)], the error probability formulation can be written, as in following expression, also noting that $a=b=1$ and $a=b=2$ correspond to binary phase-shift keying (BPSK) and quadrature phase-shift keying (QPSK), correspondingly:

$$\begin{aligned} \bar{P_{e}} = \frac{a}{2}\sqrt{\frac{b}{\pi }} \int _0^\infty \frac{{\text{exp}}\left( -bx\right) }{\sqrt{x}}F_{X}(x){dx}. \end{aligned}$$

(16)

Utilizing (16), the analytical expressions of the error probability for ISIC and SIC are given in Proposition 3.

Proposition 2

$\bar{P_{e}}_{x_{k}}^{i}$, $\bar{P_{e}}_{x_{N}}^{i}$, $\bar{P_{e}}_{x_{k}}$ and $\bar{P_{e}}_{x_{N}}$ can be calculated as:

$$\begin{aligned} \bar{P_{e}}_{x_{k}}^{i, \infty }&=\frac{1}{2\sqrt{\pi }}\Bigg [\sqrt{\pi }\nonumber \\&-\left( 2\left( \frac{1}{P_{s}\Omega _{h}}+1\right) \right) ^{-\frac{1}{4}}\sqrt{\pi }\mathrm exp \left( \frac{\left( \frac{1}{P_{s}\Omega _{h}}\sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{k}}\right) ^{2}}{8\left( \frac{1}{P_{s}\Omega _{h}}+1\right) }\right) \nonumber \\&\times D_{-\frac{1}{2}}\left( \frac{\frac{1}{P_{s}\Omega _{h}}\sum _{t=k+1}^{N} \frac{\beta _{t}}{\beta _{k}}}{\sqrt{2\left( \frac{1}{P_{s}\Omega _{h}}+1\right) }}\right) \Bigg ], \end{aligned}$$

(17)

$$\begin{aligned} \bar{P_{e}}_{x_{N}}^{i}&=\frac{1}{2\sqrt{\pi }}\Bigg [\sqrt{\pi }-\sqrt{\pi } \left( \frac{1}{P_{s}\Omega _{h}\beta _{N}}+1\right) ^{-\frac{1}{2}}\Bigg ], \end{aligned}$$

(18)

$$\begin{aligned} \bar{P_{e}}_{x_{k}}&=\frac{1}{2\sqrt{\pi }}\Bigg [\sqrt{\pi }-\sqrt{\pi } \left( \frac{1}{P_{s}\Omega _{h}\beta _{k}}+1\right) ^{-\frac{1}{2}}\Bigg ], \end{aligned}$$

(19)

$$\begin{aligned} \bar{P_{e}}_{x_{N}}^{\infty }&=\frac{1}{2\sqrt{\pi }}\Bigg [\sqrt{\pi }\nonumber \\&-2\left( \frac{1}{P_{s}\Omega _{h}}+1\right) ^{-\frac{1}{4}}\sqrt{\pi }\mathrm exp \left( \frac{\left( \frac{1}{P_{s}\Omega _{h}}\sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{N}}\right) ^{2}}{8\left( \frac{1}{P_{s}\Omega _{h}}+1\right) }\right) \nonumber \\&\times D_{-\frac{1}{2}}\left( \frac{\frac{1}{P_{s}\Omega _{h}} \sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{N}}}{\sqrt{2\left( \frac{1}{P_{s}\Omega _{h}}+1\right) }}\right) \Bigg ]. \end{aligned}$$

(20)

Proof

See Appendix 2. $\square$

3.4 Ergodic rate

Here, the ergodic rate (ER) performance of the ISIC and SIC cases is presented. In this regard, [41, Eq. (38)] is used for the analytical derivations:

$$\begin{aligned} ER\left( \gamma \right) =\frac{1}{{\text{ln}}2}\int _0^\infty \frac{1-F_{X}\left( \gamma \right) }{1+\gamma }d_{\gamma }. \end{aligned}$$

(21)

Utilizing (21), the analytical representations of the ergodic rate are given as in the next proposition.

Proposition 3

$ER_{x_{k}}^{i}$, $ER_{x_{N}}^{i}$, $ER_{x_{k}}$, and $ER_{x_{N}}$ can be calculated as:

$$\begin{aligned} ER_{x_{k}}^{i, \infty }&=\frac{\pi }{{\text{ln}}2}\left( \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}{P_{s}\Omega _{h}\beta _{k}}\right) ^{-1}\nonumber \\&\times G_{1,1:2,2:1,1}^{1,0:1,1:1,1}\Bigg ({\begin{array}{c} {1}\\ {3} \end{array}}\ \Bigg \vert \ {\begin{array}{c} {0, 0.5}\\ {0. 0.5} \end{array}}\ \Bigg \vert \ {\begin{array}{c} {0}\\ {0} \end{array}}\ \Bigg \vert \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}, \frac{P_{s}\Omega _{h}\beta _{k}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1} \Bigg ), \end{aligned}$$

(22)

$$\begin{aligned} ER_{x_{N}}^{i}&=\frac{1}{{\text{ln}}2}\left[ P _{s}\Omega _{h}\beta _{N}G_{2,1}^{1,2}\left( P_{s}\Omega _{h}\beta _{N}\Bigg \vert \ {\begin{array}{c} {0,0}\\ {0} \end{array}}\ \right) \right] , \end{aligned}$$

(23)

$$\begin{aligned} ER_{x_{k}}&=\frac{1}{{\text{ln}}2}\left[ P _{s}\Omega _{h}\beta _{k}G_{2,1}^{1,2}\left( P_{s}\Omega _{h}\beta _{k}\Bigg \vert \ {\begin{array}{c} {0,0}\\ {0} \end{array}} \ \right) \right] , \end{aligned}$$

(24)

$$\begin{aligned} ER_{x_{N}}^{\infty }&=\frac{\pi }{{\text{ln}}2}\left( \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}{P_{s}\Omega _{h}\beta _{N}}\right) ^{-1}\nonumber \\&\times G_{1,1:2,2:1,1}^{1,0:1,1:1,1}\Bigg ({\begin{array}{c} {1}\\ {3} \end{array}}\ \Bigg \vert \ {\begin{array}{c} {0, 0.5}\\ {0. 0.5} \end{array}} \ \Bigg \vert \ {\begin{array}{c} {0}\\ {0} \end{array}} \ \Bigg \vert \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}, \frac{P_{s}\Omega _{h}\beta _{N}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1} \Bigg ). \end{aligned}$$

(25)

Proof

See Appendix 3. $\square$

3.5 Achievable sum-rate

This subsection investigates the achievable sum-rate (ASR) performance of the ISIC and SIC cases. In this regard, utilizing [41, Eq. (32)], the CDF-based ASR can be formulated as:

$$\begin{aligned} ASR^{i}&=\sum _{k=1}^{N}{\mathbb {E}}\left[ {\text{log}}(1+\gamma _{x_{k}}^{i})\right] , \end{aligned}$$

(26)

$$\begin{aligned} ASR&=\sum _{k=1}^{N}{\mathbb {E}}\left[ {\text{log}}(1+\gamma _{x_{k}})\right] . \end{aligned}$$

(27)

Utilizing the results of Proposition 3, ${ASR}^{i}$ and ASR are obtained as:

$$\begin{aligned} ASR^{i}&=\frac{\pi }{{\text{ln}}2}\left( \frac{P_{s}\Omega _{h} \sum _{t=k+1}^{N}\beta _{t}+1}{P_{s}\Omega _{h}\beta _{k}}\right) ^{-1} \nonumber \\&\times G_{1,1:2,2:1,1}^{1,0:1,1:1,1}\Bigg ({\begin{array}{c} {1}\\ {3} \end{array}}\ \Bigg \vert \ {\begin{array}{c} {0, 0.5}\\ {0. 0.5} \end{array}}\ \Bigg \vert \ {\begin{array}{c} {0}\\ {0} \end{array}} \ \Bigg \vert \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}, \frac{P_{s}\Omega _{h}\beta _{k}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1} \Bigg ) \nonumber \\&+\frac{1}{{\text{ln}}2}\left[ P _{s}\Omega _{h}\beta _{N}G_{2,1}^{1,2}\left( P_{s}\Omega _{h}\beta _{N}\Bigg \vert \ {\begin{array}{c} {0,0}\\ {0} \end{array}} \ \right) \right] , \end{aligned}$$

(28)

$$\begin{aligned} ASR&=\frac{1}{{\text{ln}}2}\left[ P _{s}\Omega _{h}\beta _{k}G_{2,1}^{1,2}\left( P_{s}\Omega _{h}\beta _{k}\Bigg \vert \ {\begin{array}{c} {0,0}\\ {0} \end{array}}\ \right) \right] \nonumber \\&+\frac{\pi }{{\text{ln}}2}\left( \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}{P_{s}\Omega _{h}\beta _{N}}\right) ^{-1} \nonumber \\&\times G_{1,1:2,2:1,1}^{1,0:1,1:1,1}\Bigg ({\begin{array}{c} {1}\\ {3} \end{array}}\ \Bigg \vert \ {\begin{array}{c} {0, 0.5}\\ {0. 0.5} \end{array}} \ \Bigg \vert \ {\begin{array}{c} {0}\\ {0} \end{array}} \ \Bigg \vert \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}, \frac{P_{s}\Omega _{h}\beta _{N}}{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1} \Bigg ). \end{aligned}$$

(29)

3.6 Energy efficiency

Here, an energy efficiency (EE) analysis for ISIC and SIC is given. EE is defined as the throughput performance over the total power consumption [42, Eq. (3)]. Thus, utilizing (10) and (11), the EE for ISIC can be formulated as:

$$\begin{aligned} {EE^{i}}=\frac{\sum _{k=1}^{N} \tau ^{{\text{i}}}_{x_{k}}}{P_s}, \end{aligned}$$

(30)

and for SIC:

$$\begin{aligned} {EE}=\frac{\sum _{k=1}^{N} \tau _{x_{k}}}{P_s}. \end{aligned}$$

(31)

3.7 Spectral efficiency

The spectral efficiency (SE) of the transmission is measured in bits per second per Hz [43, 44]. In this regard, utilizing (10) and (11), the SE of ISIC can be formulated as:

$$\begin{aligned} SE^{i}=\frac{\sum _{k=1}^{N} \tau ^{{\text{i}}}_{x_{k}}}{B}, \end{aligned}$$

(32)

and for SIC as:

$$\begin{aligned} SE=\frac{\sum _{k=1}^{N} \tau _{x_{k}}}{B}, \end{aligned}$$

(33)

where B represents the occupied bandwidth.

3.8 Asymptotic analysis

This subsection focuses on the transmission’s performance in the asymptotic SNR regime. The ${\text{exp}}(x)$ term can be approximated to: $1+x$, $x\rightarrow 0$ [35] by using the Taylor series expansion. Utilizing the approximation in (6) and (7) and also (8) and (9), the asymptotic expressions for the ISIC and SIC cases can be obtained as:

$$\begin{aligned} F_{x _{k}}^{i,\infty }\left( \gamma _{{\text{th}}}\right)&=\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) , \end{aligned}$$

(34)

$$\begin{aligned} F_{x _{N}}^{i,\infty }\left( \gamma _{{\text{th}}}\right)&=\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\beta _{N}}\right) , \end{aligned}$$

(35)

$$\begin{aligned} F_{x _{k}}^{\infty }\left( \gamma _{{\text{th}}}\right)&=\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\beta _{k}}\right) , \end{aligned}$$

(36)

$$\begin{aligned} F_{x _{N}}^{\infty }\left( \gamma _{{\text{th}}}\right)&=\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{N}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) . \end{aligned}$$

(37)

Regarding the asymptotic derivations of $ER_{x_{N}}^{i}$ and $ER_{x_{k}}$, utilizing [45, Eq. (07.34.03.0396.01)] and ${\text{exp}}(x)\approx 1+x$, $x\rightarrow 0$ [35], $ER_{x_{N}}^{i}$ and $ER_{x_{k}}$ can be obtained as: $\frac{1}{{\text{ln}}2}\left[ \left( 1+\frac{1}{P_{s}\Omega _{h}\beta _{N}}\right) \Gamma \left( 0,\frac{1}{P_{s}\Omega _{h}\beta _{N}}\right) \right]$ and $\frac{1}{{\text{ln}}2}\left[ \left( 1+\frac{1}{P_{s}\Omega _{h}\beta _{k}}\right) \Gamma \left( 0,\frac{1}{P_{s}\Omega _{h}\beta _{k}}\right) \right]$, respectively. Note that the obtained results are also valid for $ASR^{i}$ and ASR, which are (28) and (29).

4 Optimization

This section deals with ISIC’s power allocation coefficients optimization, in order to minimize the performance loss when compared to SIC.

4.1 Optimal power allocation coefficients

The power allocation optimization problem and its constraints are presented as follows:

$$\begin{aligned}&\underset{\gamma _{th}}{\text {minimize}}{} & {} \mathrm {F_{x_{k}}^{i,\infty }(\gamma _{\text{th}})}~{\text{and}}~ \mathrm {F_{x_{N}}^{i, \infty }(\gamma _{{\text{th}}})} \\&\text {subject to}{} & {} P_{s}=P_{T}, 0<P_{s},\\{} & {} &\sum _{k=1}^{N}\beta _{k}=1~ {\text{and}}~\beta _{k}<\beta _{N}. \end{aligned}$$

(38)

where $P_{s}=\left( \beta _{k}+\sum _{t=k+1}^{N}\beta _{t}\right) P_{T}$ and $\left( \beta _{k}+\sum _{t=k+1}^{N}\beta _{t}\right) \epsilon \left( 0,1\right)$. Utilizing these definitions in the derived asymptotic expression of (6), and also differentiating with respect to $\beta _{k}$ and $\sum _{t=k+1}^{N}\beta _{t}$, the following derivations can be obtained:

$$\begin{aligned} \frac{\partial {\mathcal {L}}\left( F_{{\text{x}}_{k}}^{i,\infty }\left( \gamma _{{\text{th}}}\right) \right) }{\partial \beta _{k}}&=\nonumber \\&-\chi \Bigg (\left( \beta _{k}+\sum _{t=k+1}^{N}\beta _{t}\right) P_{T} \left( \beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) \Bigg )^{-2} \nonumber \\&\times \left( 2P_{T}\beta _{k}+\sum _{t=k+1}^{N}\beta _{t}P_{T}-\gamma _{th}\left( \sum _{t=k+1}^{N}\beta _{t}P_{T}\right) \right) , \end{aligned}$$

(39)

$$\begin{aligned} \frac{\partial {\mathcal {L}}\left( F_{{\text{x}}_{k}}^{i,\infty }\left( \gamma _{{\text{th}}}\right) \right) }{\partial \sum _{t=k+1}^{N}\beta _{t}}&=\nonumber \\&-\chi \Bigg (\left( \beta _{k}+\sum _{t=k+1}^{N}\beta _{t}\right) P_{T} \left( \beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) \Bigg )^{-2}\nonumber \\&\times \left( P_{T}\beta _{k}-\gamma _{th}\left( \beta _{k}P_{T}-2P_{T}\sum _{t=k+1}^{N}\beta _{t}\right) \right) , \end{aligned}$$

(40)

where $\chi =\left( \frac{\gamma _{th}}{\Omega _{h}}\right)$. Utilizing the $\left( 2\beta _{k}P_{T}+\sum _{t=k+1}^{N}\beta _{t}P_{T}\right)$ term in (39) and setting to zero and also invoking the constraints $\beta _{k}<\sum _{t=k+1}^{N}\beta _{t}$ and $\beta _{k}+\sum _{t=k+1}^{N}\beta _{t}=1$, the $\beta _{k}$ and $\sum _{t=k+1}^{N}\beta _{t}$ can be obtained as 1/3 and 2/3, respectively. Note that utilizing $\left( \beta _{k}P_{T}-2P_{T}\sum _{t=k+1}^{N}\beta _{t}\right)$, the $\beta _{k}$ and $\sum _{t=k+1}^{N}\beta _{t}$ terms can be obtained as 2/3 and 1/3, respectively. However, these results are not consistent with the constraints. Therefore, the firstly derived results, i.e. $\beta _{k}=1/3$ and $\sum _{t=k+1}^{N}\beta _{t}=2/3$ are considered for the optimised ISIC performance analysis. Regarding the SIC-based optimal power allocation coefficients, by defining $P_{s}=\left( \beta _{N}+\sum _{t=k+1}^{N}\beta _{t}\right) P_{T}$ and substituting into (40) and also differentiating with respect to $\sum _{t=k+1}^{N}\beta _{t}$, $\sum _{t=k+1}^{N}\beta _{t}=1/3$ and $\beta _{N}=2/3$, which are consistent with the constraints, can be obtained.

5 Numerical results

Here, Monte Carlo-based computer simulations are conducted for verifying the analytical and asymptotic derivations. To evaluate the NOMA transmission’s outage performance, 0.5 and 1 bps/Hz target rates are considered. Moreover, the optimal power allocation coefficients, $\beta _{k}=1/3$ and $\sum _{t=k+1}^{N}\beta _{t}=2/3$ are considered for the ISIC analysis. Regarding the SIC-based optimal power allocation coefficients, $\sum _{t=k+1}^{N}\beta _{t}=1/3$ and $\beta _{N}=2/3$ are utilised. The number of information signals N is set to 2. The total transmit power $P _{T}$ is set to $P _{s}$.

5.1 Outage probability

SIC and ISIC outage performance comparisons for $R=0.5$ bps/Hz are presented in Fig. 1. Figure 1 reveals that ISIC provides slightly better outage performance for $x_{2}$, compared to SIC. This observation occurs because in ISIC, the $x_{2}$ term in (3) does not experience interference while $x_{2}$ in (5) is interfered by $x_{1}$. However, ISIC leads to worse outage performance for $x_{1}$, compared to SIC, as the interference term in (2), which is $\beta _{2}\gamma _{x}$, degrades the decoding process of $x_{1}$. In order to reach $10^{-3}$ outage, the SNRs for $x_{1}$ and $x_{2}$ are 30 and 28 dB for SIC and 36 and 27 dB for ISIC. Moreover, the theoretical expressions of (6)–(9) and (34)–(37), are in agreement with the simulation results.

Fig. 1

SIC and ISIC outage performance comparison for $R=0.5$ bps/Hz

Then, SIC and ISIC outage performance for $R=1$ bps/Hz is presented in Fig. 2. In reference to Fig. 2, the $x_{2}$ in the ISIC cases has better outage performance, compared to the SIC case, as it was also observed in Fig. 1. However, in ISIC’s case, $x_{1}$, (2), cannot be decoded, since its achievable rate as provided by (2) falls below the more stringent target rate $R=1$ bps/Hz. It can be observed that in order to reach an outage value of $10^{-3}$, the required SNR for $x_{2}$ in the case of ISIC is around 32 dB while for $x_{1}$ and $x_{2}$ in the case of SIC, SNR values of 35 dB must be reached.

Fig. 2

SIC and ISIC outage performance comparison for $R=1$ bps/Hz

5.2 Throughput

SIC and ISIC throughput performance for $R=0.50$ bps/Hz is presented in Fig. 3. Figure 3 reveals that when ISIC is adopted, $x_{2}$ exhibits slightly higher throughput, compared to SIC. However, in SIC, $x_{1}$ achieves significantly higher throughput over ISIC. More specifically, in order to achieve 0.3 bps/Hz, the required SNR for $x_{1}$ and $x_{2}$ are 6 and 4 dB, and 14 and 3 dB for SIC and ISIC, respectively. It is also observed that the obtained results are consistent with (12)–(15).

Fig. 3

SIC and ISIC throughput performance comparison for $R=0.50$ bps/Hz

When a more strict rate requirement of $R=1$ bps/Hz is imposed, the SIC and ISIC throughput performance when is presented in Fig. 4. As shown in Fig. 4, in ISIC, $x_{2}$ achieves higher throughput than SIC. It is also shown that in ISIC, $x_{1}$ cannot be decoded. In order to achieve 0.3 bps/Hz, the required SNR for $x_{2}$ in the case of ISIC is 1 dB while for SIC, an SNR value equal to 4 dB is needed for both signals. These observations are consistent with (12)–(15).

Fig. 4

SIC and ISIC throughput performance comparison for $R=1$ bps/Hz

5.3 Ergodic rate

The ergodic rate performance of SIC and ISIC is depicted in Fig. 5, revealing that in ISIC, $x_{2}$ achieves slightly higher ergodic rate over $x_{1}$ of the SIC case. For example, in order for iSIC’s $x_{2}$ to reach an ergodic rate of 8 bps/Hz, an SNR value of 27 dB and is needed while SIC’s $x_{1}$ requires 32 dB. These differences occur because of the decoding ordering and the interference experienced while decoding the respective signals, as expressed in (3) and (4). Conversely, SIC’s $x_{2}$ achieves higher ergodic rate, compared to the ISIC’s $x_{1}$, as a result of the decoding ordering and the corresponding interference terms that are included in (2) and (5). Note that SIC’s $x_{2}$ and ISIC’s $x_{1}$ saturate at high SNR regimes. This saturation occurs, since the denominator terms in (2) and (5) dominate the decoding process and lead to an ergodic rate ceiling. Again, these results are in agreement with (22), (23), (24), and (25).

Fig. 5

SIC and ISIC ergodic rate performance comparison

5.4 Sum-rate

The achievable sum-rate performance of SIC and ISIC is presented in Fig. 6. According to Fig. 6, SIC and ISIC offer similar achievable sum-rate performance. To achieve 6 bps/Hz achievable sum-rate, the required SNR is 20 dB for both SIC and ISIC. This almost identical performance occurs because of the summation of (2)–(3) and (4)–(5). Note that in low SNR, approximations, which are used in Proposition 3, result in coding gain losses on the asymptotic analysis. (28) and (29) are in close agreement with the performance results.

Fig. 6

SIC and ISIC achievable sum-rate performance comparison

5.5 Error probability

Error probability results for SIC and ISIC are presented in Fig. 7, highlighting that ISIC’s $x_{2}$ achieves lower error probability than SIC’s $x_{1}$. As an example, to reach a $10^{-3}$ error probability, the required SNR values must be equal to 25 and 29 dB for ISIC’s $x_{2}$ and SIC’s $x_{1}$, respectively. This performance difference occurs due to the different decoding ordering and interference terms, as presented in (3) and (4). For similar reasons, SIC’s $x_{2}$ experiences lower error probability, compared to ISIC’s $x_{1}$, as expressed in (2) and (5). Note that both SIC’s $x_{2}$ and ISIC’s $x_{1}$ saturate at high SNR, since the denominator terms in (2) and (5) dominate the decoding process and cause saturation. The obtained results are in agreement with (17), (18), (19), and (20).

Fig. 7

SIC and ISIC error probability performance comparison

5.6 Energy efficiency

Next, energy efficiency results, in terms of bits/J/Hz for SIC and ISIC are presented in Fig. 8, showing that ISIC’s $x_{2}$ achieves slightly better energy efficiency performance over SIC’s $x_{2}$ for $P_{s}={0,1,2}$ dBW. However, SIC’s $x_{1}$ achieves improved energy efficiency, compared to ISIC’s $x_{1}$ for $P_{s}={0,1,2}$ dBW. Figure 8 also reveals that better energy efficiency results are obtained at $P_{s}={0}$ dBW, compared to $P_{s}={1,2}$ dBW for SIC- and ISIC-based $x_{1}$ and $x_{2}$. As an example, for 25 dB, the case of $P_{s}={0}$ dBW achieves 0.4 bits/J/Hz, the $P_{s}={1}$ dBW case achieves 0.34 bits/J/Hz, and the $P_{s}={2}$ dBW case achieves 0.25 bits/J/Hz.

Fig. 8

SIC and ISIC energy efficiency performance comparison for R = 0.5 bps/Hz

Likewise, the energy efficiency performance of SIC and ISIC for $R=1$ bps/Hz is presented in Fig. 9. According to Fig. 9, ISIC’s $x_{2}$ achieves better energy efficiency, compared to SIC’s $x_{2}$ for $P_{s}={0,1,2}$ dBW. Still, while SIC’s $x_1$ reaches an energy efficiency of 1, 0.8, and 0.6 bits/J/Hz at 25 dB, ISIC’s $x_1$ cannot be decoded for all $P_s$ cases.

Fig. 9

SIC and ISIC energy efficiency performance comparison for R = 1 bps/Hz

6 Conclusions and future directions

6.1 Conclusions

Non-orthogonal multiple access (NOMA) is a viable technique to improve the performance of wireless networks and improve their spectral efficiency. However, in practice, correct ordering of the information signals at the receiver might not be possible, due to hardware impairments or estimation errors. In such cases, the receiver will erroneously order the signals and in the worst case, inverse ordering might occur prior to successive interference cancellation (SIC). In this work, we aim to shed light on the performance of inverse SIC (ISIC) when a base station (BS) transmits a waveform, which contains multiple super-imposed information signals to a single receiver. The outage probability, throughput, error probability, ergodic rate, and achievable sum-rate metrics have been utilised to measure ISIC’s performance. Analytical derivations are also provided and verified by means of Monte Carlo-based simulations. The asymptotic analysis is also given to extend the derived analytical results. Moreover, under ISIC, Lagrangian-based optimization techniques are utilised to obtain the optimal power allocation coefficients. According to the obtained performance results, for the signal with the second-lowest power allocation, ISIC achieves better performance while it severely degrades the outage and throughput performance of the signal with the minimum power allocation, compared to conventional SIC. Results also revealed that when higher rate requirements are imposed, ISIC leads to severe degradation to the least-power allocated information signal’s outage and throughput performance while the performance for the second-lowest power allocated information signal is improved. In conclusion, the results of this study show the impact of imperfections on SIC’s performance in NOMA networks, in the worst case when signal ordering is inversed.

6.2 Future directions

This work can be extended in various interesting directions:

• First, the performance of ISIC can be examined in more complex topologies, e.g. relay- and re-configurable-intelligent surface-assisted topologies, such as those in [46,47,48] where the impact of imperfections can be more pronounced due to error propagation.

• Furthermore, the use of machine learning-aided solutions to tackle practical issues related to channel estimation can be studied, e.g. relying on reinforcement learning where feedback information is exploited to perform power allocation [49].

• Another important aspect that must be studied includes more advanced interference scenarios, such as topologies with inter-cell interference and malicious jammers, aiming to reduce communication quality [50,51,52,53]

• Finally, Internet-of-Things deployments form the basis of various application domains, e.g. Industry 4.0, smart cities, and precision agriculture. Here, low-complexity NOMA-based strategies should be devised to accommodate resource-constrained devices, in terms of energy and hardware capabilities. In this area, integrating energy harvesting in the context of NOMA represents an attractive direction [54, 55].

Appendices

Appendices

Proof of proposition 3.1

Revisiting (4) and considering the logarithm properties, below analytical derivations are obtained.

$$\begin{aligned} F_{x_{k}}^{i}\left( \gamma _{th}\right)&={\text{P}}_{r}\left( \frac{\gamma _{x}\beta _{k}}{\sum _{t=k+1}^{N}\beta _{t}\gamma _{x}+1}\le \underbrace{2^{R}-1}_{\gamma _{th}}\right) \nonumber \\&={\text{P}}_{r}\left( \gamma _{x}\le \left( \frac{\gamma _{th}}{\beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}}\right) \right) \nonumber \\&=1-{\text{P}}_{r}\left( \gamma _{x}\ge \left( \frac{\gamma _{th}}{\beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}}\right) \right) \nonumber \\&=1-{\text{P}}_{r}\left( 1-F_{\gamma _{x}}\left( \frac{\gamma _{th}}{\beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}}\right) \right) \nonumber \\&=1-e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{k}-\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }\right) }. \end{aligned}$$

(41)

Note that considering the similar methodologies as in (41), the other CDFs, which are (3), (4), and (5), can also be obtained.

Proof of proposition 3.3

Substituting (6) into (16), the following result can be obtained, noting that BPSK modulation is considered in the derivations:

$$\begin{aligned} \bar{P_{e}}_{x_{k}}^{i}&= \frac{1}{2\sqrt{\pi }}\left( \int _0^\infty x^{-\frac{1}{2}}{\text{exp}}\left( -x\right) dx \right. \nonumber \\&\left. -\int _0^\infty x^{-\frac{1}{2}}e^{-\gamma _{th}\left( \frac{1}{P_{s}\Omega _{h}\left( \beta _{k} -\gamma _{th}\sum _{t=k+1}^{N}\beta _{t}\right) }+1\right) }dx\right) . \end{aligned}$$

(42)

Utilizing [35, Eq. ($3.326.2^{10}$)], the first integral in (42) can be solved as $\Gamma \left( \frac{1}{2}\right) =\sqrt{\pi }$. By performing some mathematical modifications and utilizing $\frac{1}{1-x}\approx 1+x$, $x\rightarrow 0$ [35], the second integral in (42) becomes:

$$\begin{aligned} \int _0^\infty x^{-\frac{1}{2}}e^{-x\left( 1+\frac{1}{P_{s}\Omega _{h}}\right) -x^{2} \left( \frac{1}{P_{s}\Omega _{h}}\sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{k}}\right) }dx. \end{aligned}$$

(43)

Modifying [35, Eq. (3.462.1)], as $\int _{0}^{\infty } \gamma _{th}^{v-1}e^{-\gamma \gamma _{th}^{2}-\beta \gamma _{th}}d\gamma _{th}$ =$\left( 2\beta \right) ^{-\frac{v}{2}}\Gamma (v){\text{exp}}\left( \frac{\gamma ^{2}}{8\beta }\right) D_{-v}\left( \frac{\gamma }{\sqrt{2\beta }}\right)$, where $v=\frac{1}{2}$, $\beta =\left( \frac{1}{P_{s}\Omega _{h}}+1\right)$, and $\gamma =\left( \frac{1}{P_{s}\Omega _{h}}\sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{k}}\right)$, the final expression can be obtained, as in (17). In a similar way, substituting (7) into (16) and solving the integrals by means of [35, Eq. ($3.326.2^{10}$)], the final result can be obtained as in (18). Likewise, the EP expressions for SIC can be obtained, as in (19) and (20). Note that since the obtained result in (20) does not perfectly match with the Monte Carlo-based simulations, the SIC-based modified integral expression, which is $\frac{1}{2\sqrt{\pi }}\left( \int _0^\infty x^{-\frac{1}{2}}{\text{exp}}\left( -x\right) dx-\int _0^\infty x^{-\frac{1}{2}}e^{-x\left( 1+\frac{1}{P_{s}\Omega _{h}}\right) -x^{2}\left( \frac{1}{P_{s}\Omega _{h}}\sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{N}}\right) }dx\right)$, is presented numerically in Fig. 7. Note that [56] is utilised for the parabolic cylinder function’s implementation in the computer simulations.

Proof of proposition 3.4

Substituting (6) into (21) and utilizing ${\text{exp}}(-x)\approx 1-x$ [35] and also performing basic mathematical modifications, thefollowing expression can be obtained.

$$\begin{aligned} ER_{x_{k}}^{i,\infty }\left( \gamma \right)&=\frac{1}{{\text{ln}}2}\int _0^\infty \left( 1\right. \nonumber \\&\left. -\gamma \left( \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}{P_{s} \Omega _{h}\beta _{k}}\right) \right) \left( 1-\gamma \sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{k}}\right) ^{-1}\nonumber \\&\times \left( 1+\gamma \right) ^{-1}d_{\gamma }. \end{aligned}$$

(44)

Utilizing [36, Eq. (9, 10)], (44) can be written as

$$\begin{aligned} ER_{x_{k}}^{i,\infty }\left( \gamma \right)&=\frac{\pi }{{\text{ln}}2}\int _0^\infty G_{1,1}^{1,0}\Bigg (\gamma \left( \frac{P_{s}\Omega _{h}\sum _{t=k+1}^{N}\beta _{t}+1}{P_{s}\Omega _{h}\beta _{k}}\right) \ \Bigg \vert {\begin{array}{c} {2}\\ {0} \end{array}} \Bigg ) \nonumber \\&\times G_{2,2}^{1,1}\Bigg (\gamma \left( \sum _{t=k+1}^{N}\frac{\beta _{t}}{\beta _{k}}\right) \ \Bigg \vert {\begin{array}{c} {0, 0.5}\\ {0, 0.5} \end{array}} \Bigg )G_{1,1}^{1,1}\Bigg (\gamma \Bigg \vert {\begin{array}{c} {0}\\ {0} \end{array}} \Bigg )d_{\gamma }. \end{aligned}$$

(45)

Utilizing [37, Eq. (13)] and setting $\alpha$ term to 1, (45) can be solved, as in (22). Likewise, substituting (7) into (21) and utilizing [36, Eq. (10, 11)] and also utilizing [36, Eq. (21)], the final result can be obtained, as in (23). The $\alpha$ is equal to 1 in [36, Eq. (21)]. Following a similar procedure, as in (22) and (23), $ER_{x_{k}}^{i,\infty }$ and $ER_{x_{N}}$ can be obtained, as in (24) and (25), respectively.