On the Performance of Uplink Power-Domain NOMA with Imperfect CSI and SIC in 6G Networks

I. INTRODUCTION

SIXTH generation (6G) services will require more bandwidth, faster speeds, lower latency, and higher reliability. To compensate the aforementioned demands, there is a need to employ different types of multiple access techniques. In this regard, the non-orthogonal multiple access (NOMA) technique has become an efficient solution for the ever-increasing demands of mobile networks. This is because NOMA is superior to existing orthogonal multiple access (OMA) techniques, in terms of utilizing the spectrum efficiently and enabling massive connectivity [1], [2]. In this context, NOMA has emerged as a key technology for 6G networks where coexisting user, machines and autonomous devices will coexist. Contrary to traditional OMA schemes, NOMA allows multiple users to share the same spectral resources by exploitating channel and rate asymmetries. In the uplink, the use of NOMA faces completely different challenges compared to the downlink case. More specifically, in the uplink the interference among users transmitting simultaneously must be efficiently managed, while in the downlink, the base station controls the transmissions to multiple users. Recently, NOMA has been also proposed in important 6G settings, such as those related to reflecting intelligent surfaces (RISs) and integrated satellite-aerial-terrestrial networks, highlighting its importance as a spectral-efficient communication technique with improved fairness [3].

In recent years, various studies have investigated the performance of uplink NOMA systems. However, there is a need to consider more realistic scenarios, where imperfect channel state information (CSI) and successive interference cancellation (SIC) may degrade the system performance. In this context, the study in [4] investigates the performance of two users, requiring different quality of service (QoS), with dynamic SIC in uplink NOMA in the presence of imperfect CSI that is caused by channel estimation error (CEE). In addition, the authors in [5] investigate the ergodic rate (ER) performance of NOMA-based uplink satellite networks with randomly deployed mobile terminals (MTs), considering the location information, imperfect CSI due to CEE and antenna pointing error. Next, the work in [6] seeks an answer to whether or not uplink NOMA can reduce the queuing delay, compared to OMA. The authors consider a two-user scenario and take into consideration the imperfect CSI, stemming from CEE in the performance analysis, concluding that NOMA with SIC decoding might not be suitable for low-latency applications. The optimization of power allocation is the focus of [7] in uplink/downlink NOMA networks in the presence of CEE. The proposed algorithms exploit the heterogeneity in QoS requirements of the users and ensure improved rates over other OMA and NOMA approaches. Then, a new receiver design is proposed in [8], based on forward error correction codes, for a two-user uplink NOMA network in the presence of CEE. Simulation results demonstrate that the proposed receiver can efficiently improve SIC performance and increase the robustness against CSI errors over legacy designs. The study in [9] examines the performance of uplink NOMA-based underlay scheme and employs CSI-based NOMA/OMA mode switching under imperfect SIC, surpassing standalone OMA and NOMA. When feedback constraints degrade network performance, [10] proposes interference-aware resource allocation in multi-layer uplink NOMA network, in the presence of feedback constraints. In this setting, the optimal power allocation policy is derived through the maximization of aggregate data rate of the mobile users of the desired tier, under a constraint, considering the total amount of interference at the first tier, as well as CSI feedback constraints.

Other works propose modified NOMA schemes, exploiting spatial modulation (SM) and media-based modulation (MBM). The work in [11] examines the bit-error rate (BER) performance of an MBM-aided uplink NOMA with two users in the presence of hardware impairments and CEE and shows improved performance over spatial modulation-aided and conventional NOMA schemes. Another work in [12] studies the BER performance of spatial media-based modulation-aided uplink multiple-input multiple-output (MIMO) NOMA with imperfect CSI that is caused by CEE, achieving worse BER performance over MBM-NOMA, SM-NOMA and MIMOaided NOMA. The paper in [13] investigates the performance of SM-aided uplink NOMA without SIC at the BS under both perfect, and imperfect CSI cases, showing that SIC-free SM-NOMA improves network performance over conventional NOMA, in terms of spectral and energy efficiency, and BER.

Finally, some works focus on relay-aided uplink NOMA networks with practical constraints. The work in [14] considers an uplink two-user half-duplex (HD) relay-assisted NOMA network. The authors utilize frequency-selective multipath fading channels and investigate the ER performance of two NOMA users. Instead of perfect CSI, the authors also consider that NOMA users have statistical CSI while imperfect SIC is performed at BS. Next, the paper in [15] analyzes the outage performance of full-duplex (FD) decode-and-forward (DF) relay-aided uplink NOMA network with imperfect SIC and CEE while [16] studies the ER performance of a cooperative uplink NOMA network, where three users communicate with the BS in either direct or relay-based mode with the help of an HD DF relay, relying on statistical CSI. Finally, the study in [17] relies on opportunistic relaying and buffering to enhance the reliability of uplink NOMA in settings where users and IoT devices coexist and CSI at the transmitters is not available.

It is evident that performance evaluation for NOMA-based communication under practical limitations has been examined in various settings with some recent works focusing on integrated satellite-terrestrial networks [18], [19]. However, although the imperfect CSI and SIC issues are the same in both these works and ours, herein, we tackle a completely different problem, as we do not consider an integrated satelliteterrestrial network and more importantly, we focus on the uplink transmission. Thus, in this work, a realistic uplink NOMA network model is adopted, where MTs in the cellular coverage area are randomly deployed [20]. Assuming practical limitations, in terms of CSI and SIC, we provide the following contributions:

· Departing from the current state-of-the-art in uplink NOMA studies, this paper takes into consideration the imperfect CSI that is caused by CEE and feedback delay (F-D). In addition, the imperfect SIC effects on the system performance are examined.

· The adopted system model is investigated in terms of outage probability (OP), error probability (EP), ER, throughput, energy efficiency (EE), and spectral efficiency (SE). For these metrics, exact and asymptotic closed-form expressions are derived, and corroborated with Monte- Carlo-based simulations.

· Furthermore, to provide further insights on the asymptotic results, a diversity-order analysis is presented.

· By leveraging the closed-form expressions and to improve the outage performance, Lagrangian multipliers are utilized for optimizing the power allocation coefficients. The theoretical and simulation findings of this study facilitate the adoption of optimized parameters to improve the performance of NOMA-based uplink networks, overcoming the detrimental impact of practical constraints in the sense of CEE and F-D.

Overall, the system design insights obtained from our theoretical analysis and its numerical validation demonstrates the need for power allocation optimization towards mitigating the degrading effects of imperfect CSI and SIC in uplink NOMA networks. By employing the Lagrangian Multiplier technique, we show that key performance metrics such as outage probability, error probability, ergodic rate, and throughput can be improved. In this respect, optimization is vital to maintain the system performance under practical constraints, ensuring reliable communication in 6G networks.

In what follows, the uplink NOMA system model is introduced and details on the channel model are presented in Section II. In Section III, theoretical analysis is conducted to measure the system’s performance. In Section IV, transmit power and power allocation coefficient optimization is presented. Then, Section V verifies the theoretical findings, using Monte-Carlo-based simulations. Finally, Section VI concludes the paper and provides future directions.

Notations: [TeX:] $$G_{p, q}^{m, n}[.]$$ is the Meijer’s G-function [21, Eq. (21)] and [TeX:] $$G_{p, q: p_1, q_1: p_2, q_2}^{m, n: m_1, n_1: m_2, n_2}[.]$$ denotes the extended generalized bivariate Meijer’s G-function [22, Eq. (13)]. Furthermore, [TeX:] $$\Gamma(.)$$ denotes the complete Gamma function [23, Eq. (8.310.1)]. All log are based 2, unless stated otherwise. [TeX:] $$\mathrm{F}_h(.)$$ represents the cumulative distribution function (CDF) of a random variable (RV) h while [TeX:] $$\mathrm{f}_h(.)$$ denotes the h RV’s probability density function (PDF).

II. SYSTEM MODEL &CHANNEL STATISTICS

Fig. 1 depicts an uplink NOMA network topology, characterized by imperfect CSI and SIC. In this topology, a single antenna equipped multiple MTs, being randomly deployed, aim to transmit towards a single antenna equipped BS. The wireless channel coefficient among jth MT → BS is denoted by [TeX:] $$h_j, \forall_j=1, \cdots, N.$$ Here, [TeX:] $$h_j$$ follows a circularly symmetric complex Gaussian distribution, having zero mean and variances [TeX:] $$\left.\Omega_{h_j}$$. (i.e. [TeX:] $$h_j \sim \mathcal{C N}\left(0, \Omega_{h_j}\right) \text {. }$$) Amplitude of [TeX:] $$\left|h_j\right|$$ follows Rayleigh distribution. Likewise, kth, [TeX:] $$\forall_k=j+1, \cdots, N-1,$$ MT → BS and Nth MT → BS are [TeX:] $$h_k \sim \mathcal{C N}\left(0, \Omega_{h_k}\right)$$ and [TeX:] $$h_N \sim \mathcal{C N}\left(0, \Omega_{h_N}\right),$$ respectively. Also, in this uplink network, the MT with the best wireless channel conditions is the one, being located nearest to the BS. In this case, the MT channel gains are sorted as [TeX:] $$\left|h_1\right|^2 \geq, \cdots, \geq\left|h_N\right|^2$$ and the order of the power allocation coefficients are defined as [TeX:] $$1\gt \beta_j \geq \beta_k \geq, \cdots, \geq \beta_N\gt 0, \sum_{j=1}^N \beta_j=1.$$ The distance between the jth MT and the BS is denoted as [TeX:] $$d_j$$ and the corresponding distances are sorted as [TeX:] $$d_1 \leq d_2 \leq, \cdots, \leq d_N.$$ Therefore, the MTs’ channel variances can be assumed to be as [TeX:] $$\Omega_{h_j}=d_j^{-\nu}, \Omega_{h_k}=d_k^{-\nu},$$ and [TeX:] $$\Omega_{h_N}=d_N^{-\nu},$$ where ν is the path-loss exponent and takes values between 2 and 6 [24]. [TeX:] $$h_l, h_l \sim \mathcal{C N}\left(0, \lambda_l \Omega_{h_l}\right)$$ represents the imperfect SIC term and [TeX:] $$\lambda_l,\left(0 \leq \lambda_l \leq 1\right),$$ represents the level of the residual interference that is caused by the imperfect SIC at BS [25], [26]. In particular, when [TeX:] $$\lambda_l$$ is is equal to zero and one, perfect and imperfect SIC conditions can be achieved, respectively.

A central-unit (CU) controls the information exchange process and before the communications starts, the CU transmits a training sequence to the destination terminals via the BS and the destination terminals transmit back the estimated CSI to the CU via the BS. In this work, a widely used imperfect CSI modelling, capturing the impact of imperfect CSI on system performance is adopted [4], [27], [28]. In greater detail, these works have modeled imperfect CSI by considering different types of impairments in the channel estimation process, i.e. CEE and F-D.

The training sequence-based CEE model for the jth MT → BS can be formulated as: [TeX:] $$h_j=\delta_e \hat{h}_j+\varepsilon_{h_j}.$$ Here, [TeX:] $$h_j$$ is the true channel and [TeX:] $$\hat{h}_j, \hat{h}_j \sim \mathcal{C} \mathcal{N}\left(0, \Omega_{\hat{h}_j}\right),$$ is estimated channel of [TeX:] $$h_j$$ and [TeX:] $$\varepsilon_{h_j} \sim \mathcal{C} \mathcal{N}\left(0, \Omega_{\varepsilon_{h_j}}\right)$$ is the estimation error. [TeX:] $$\delta_e$$ is the training sequences power. In a similar way, the F-D model for the jth MT → BS can be formulated as: [TeX:] $$h_j=\delta_f \hat{h}_j+\sqrt{1-\delta_f^2} \epsilon_{h_j},$$ where [TeX:] $$\delta_f$$ is the correlation coefficient of the true and estimated channel gains. The [TeX:] $$\delta_f$$ term takes value between 0 and 1 [29], [30]. So as to make analysis more compact, the CEE and F-D cases are presented as [TeX:] $$\left(\delta_s \hat{h}_j+\kappa_s \epsilon_{h_j}\right),$$ where [TeX:] $$s \in\{\mathrm{CEE}, \mathrm{F}-\mathrm{D}\},$$ [TeX:] $$\delta_{\mathrm{CEE}}=\delta_e,$$ [TeX:] $$\delta_{\mathrm{F}-\mathrm{D}}=\delta_f, \kappa_{\mathrm{CEE}}=1, \kappa_{\mathrm{F}-\mathrm{D}}=\sqrt{1-\delta_f^2}.$$

The signal that the BS receives is written as

where [TeX:] $$\mathrm{P}_{\mathrm{s}}$$ represents the MTs’ transmit power and [TeX:] $$x_j,$$ [TeX:] $$\mathbb{E}\left[\left|x_j\right|^2\right]=1,$$ is the information signal of the jth MT. [TeX:] $$w_r$$ represents the additive white Gaussian noise (AWGN), experienced at the BS, characterized by [TeX:] $$\sigma^2$$ variance. The achievable rates for the jth and Nth users can be obtained as

where [TeX:] $$\Delta=\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{j}}}}^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \delta_{\mathrm{s}}^2\left|\hat{\mathrm{h}}_{\mathrm{k}}\right|^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma^2_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}+\sum_{\ell=1}^{\mathrm{j}-1} \mathrm{P}_{\mathrm{s}} \beta_{\ell}\left|\mathrm{h}_{\ell}\right|^2+\sigma^2.$$ Note that during the SIC process of each user, there will be some level of residual interference due to imperfect SIC. In this regard, [TeX:] $$\sum_{\ell=1}^{q-1} \sqrt{\mathrm{P}_{\mathrm{s}} \beta_{\ell}} h_{\ell}.$$ [TeX:] $$q \in\{j, N\}, h_{\ell} \sim \mathcal{C N}\left(0, \lambda_{\ell} \Omega_{h_{\ell}}\right)$$ represents the imperfect SIC term. Also note that since the BS starts decoding the users based on the allocated power coefficients, the first user, j = 1, will not experience any residual interference that is caused by imperfect SIC.

III. THEORETICAL ANALYSIS

This section includes the outage probability, error probability, ergodic rate, throughput, energy and spectral efficiency analytical derivations. In addition, an asymptotic analysis is conducted and the diversity order is extracted.

A. Outage Probability

The OP expresses the probability that the network’s achievable-rate is not sufficient to support the desired target rate [TeX:] $$R_{j, N}.$$ More specifically, in this case, the CDF of the received SNR/signal-to-interference-plus-noise ratio (SINR) measured at the desired threshold rate, [TeX:] $$\gamma_{t h, j, N}$$ leads to the OP expressions.

Proposition 1. The OP expressions for the jth and Nth users can be expressed as

Proof. See Appendix

B. Error Probability

This subsection studies the EP performance of uplink NOMA, being necessary to assess the performance of the network when different modulation orders are adopted. With the help of [31], CDF-based EP performance metric can be formulated as

where a = b = 1 represents the BPSK modulation while a = b = 2 corresponds to the QPSK modulation. It should be noted that BPSK modulation is considered in the performance analysis. The EP theoretical expressions are included in Proposition 2, below.

Proposition 2. The EP expressions for the jth and Nth users can be calculated as in (7) and (8), given at the top of the following page, where [TeX:] $$\phi=\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{j}}}}^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}^2+\beta_{\mathrm{j}} \delta_{\mathrm{s}}^2 \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}+1.$$

C. Ergodic Rate

This subsection presents analytical derivations for ER for the uplink NOMA-based topology. The ER is another key metric to evaluate the performance of the uplink NOMA network, and it is defined as the maximum achievable rate averaged over all the fading blocks. By means of [31], the CDF-based ER can be formulated as

Proposition 3. The ER expressions for jth and Nth users are calculated by using (10) and (11), given at the top of the following page.

Proof. See Appendix

D. Throughput Analysis

The throughput performance of the uplink NOMA network in the presence of imperfect CSI and SIC is investigated here. Considering [32, Eq. (15(a))], the CDF-based throughput performance is formulated as

Substituting (4) and (5) into (12), the throughput expressions for jth and Nth users can be obtained as follows

E. Energy Efficiency Analysis

A key performance indicator for the sustainable operation of the uplink NOMA network is EE. More specifically, here, EE is defined as the ratio of the system throughput performance over the total transmit power that is employed by the transmitter [33]. By means of [33, Eq. (3)], the EE can be formulated as

Substituting the related CDF expressions into (15), the final expressions can be obtained.

F. Spectral Efficiency Analysis

In the considered setup, NOMA is adopted to increase the SE of uplink communication by simultaneously serving multiple users over the same wireless channel. Thus, below, we provide the SE of the uplink NOMA network. By means of [34], [35], the SE can be formulated as

where B is the bandwidth.

G. Asymptotic Analysis

This section presents asymptotic derivations of the previously obtained performance metrics’ analytical derivations.

1) Outage probability: Employing [TeX:] $$(1+\mathrm{x})^{-\mathrm{M}} \approx(1-\mathrm{Mx})$$ and [TeX:] $$\exp (\mathrm{x}) \approx(1+\mathrm{x}), \mathrm{x} \rightarrow 0$$ [23] and utilizing in (4) and (5), following asymptotic expressions can be obtained.

2) Error probability:

Proposition 4. The asymptotic EP expressions for jth and Nth users are derived through (19) and (20), presented at the top of the next page.

Proof. See Appendix

3) Ergodic rate:

Proposition 5. The asymptotic ER expressions for jth and Nth users are calculated as in (21) and (22) at the top of the next page.

Proof. See Appendix

4) Throughput analysis: Considering [TeX:] $$(1+\mathrm{x})^{-\mathrm{M}} \approx(1-\mathrm{Mx})$$ and [TeX:] $$\exp (\mathrm{x}) \approx(1+\mathrm{x}), \mathrm{x} \rightarrow 0$$ [23] and also utilizing in (13) and (14), the asymptotic throughput expressions for jth and Nth users can be obtained. The final result is omitted due to space limitation but utilized in the numerical results section.

5) Energy efficiency analysis: Regarding the asymptotic expression of energy efficiency, utilizing the derived asymptotic throughput expressions in (15), the asymptotic energy efficiency expressions for jth and Nth users can be obtained. The final result is also omitted due to space limitation but utilized in the numerical results section.

H. Diversity order analysis

The diversity order and the coding gain relation can be expressed as [36]

where [TeX:] $$G_d$$ term represents the diversity order and [TeX:] $$G_c$$ represents the coding gain. Regarding the diversity order analysis of [TeX:] $$\bar{\mathrm{P}}_{\text {out }}^{\mathrm{j}^{\mathrm{th}}, \infty}\left(\gamma_{\mathrm{th}, \mathrm{j}}\right),$$ under CEE, when [TeX:] $$\mathrm{P}_\mathrm{s}$$ and [TeX:] $$\delta_e$$ go to [TeX:] $$\infty$$ in (17), [TeX:] $$\sigma_{\varepsilon_{{h}_{j,k}}}^2$$ term, which proportionally decreases with SNR, degrades the numerator of [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{j}}}}^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}^2+1}{\beta_j \delta_s^2 \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}}$$ and the [TeX:] $$\gamma_{t h_j}$$ become dominant. Therefore, the power of [TeX:] $$\gamma_{t h_j}$$, which is equal to 1 yields the diversity order. In F-D scenario, since [TeX:] $$\sigma_{\varepsilon_{h_{j, k}}}^2$$ term is not proportional with SNR cannot degrade the [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma^2_{\varepsilon_{\mathrm{h}_{\mathrm{j}}}}+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}^2+1}{\beta_j \delta_s^2 \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}}.$$ Thus, [TeX:] $$\gamma_{t h_j}$$, becomes negligible and the power of [TeX:] $$\gamma_{t h_j}$$, which is equal to zero, yields the diversity order.

Regarding the diversity order analysis of [TeX:] $$\bar{\mathrm{P}}_{\text {out }}^{\mathrm{N}^{\mathrm{th}}, \infty}\left(\gamma_{\mathrm{th}_{\mathrm{N}}}\right)$$, with CEE, when [TeX:] $$\mathrm{P}_{\mathrm{s}} \text { and } \delta_e$$ go to [TeX:] $$\infty$$ in (18), [TeX:] $$\sigma_{\varepsilon_{h_N}}^2$$ term, which proportionally decreases with SNR, degrades the numerator of [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{N}} \kappa_{\mathrm{s}}^2 \sigma^2_{\varepsilon_{\mathrm{h}_{\mathrm{N}}}}+1}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{N}}}$$ and [TeX:] $$\gamma_{\operatorname{th}_{\mathrm{N}}}$$ become dominant and the power of [TeX:] $$\gamma_{\operatorname{th}_{\mathrm{N}}}$$, which is equal to 1, yields the diversity order. In F-D scenario, since [TeX:] $$\sigma_{\varepsilon_{h_N}}^2$$ is not proportional to SNR, the [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{N}} \kappa_{\mathrm{s}}^2 \sigma^2_{\varepsilon_{\mathrm{h}_{\mathrm{N}}}}+1}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{N}}}$$ cannot be degraded and become dominant. Therefore, the [TeX:] $$\gamma_{t h_N}$$ becomes negligible and the power of [TeX:] $$\gamma_{t h_N}$$, which is equal to zero yields the diversity order.

For imperfect SIC scenario, when [TeX:] $$\mathrm{P}_{\mathrm{s}} \rightarrow \infty, \frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}} \sum_{\ell=1}^{\mathrm{N}-1} \beta_{\ell}}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{N}}} \beta_{\mathrm{N}} \delta_{\mathrm{s}}^2}$$ term cannot approximate to zero and [TeX:] $$\gamma_{t h_N}$$ become negligible. Therefore, the power of [TeX:] $$\gamma_{t h_N}$$, which is equal to zero, yields the diversity order.

IV. POWER ALLOCATION OPTIMIZATION

Here, the steps to acquire the optimal transmit power and power allocation coefficients are given. The Lagrangian multiplier technique is utilized to minimize the outage performance subject to available resources. So as to minimize the system outage performance, the objective function and its constraints can be written as

where [TeX:] $$\mathrm{P}_{\mathrm{s}}=\left(\beta_{\mathrm{j}}+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \beta_{\mathrm{k}}\right) \times \mathrm{P}_{\mathrm{T}} / \mathrm{N}$$ and [TeX:] $$\left(\beta_j+\sum_{k=j+1}^{N-1} \beta_k\right) \in(0,1).$$ Substituting these expressions into (17), the following expression can be obtained

where [TeX:] $$A=\kappa_s^2 \sigma_{\varepsilon_{h_j}}^2, B=\kappa_s^2 \sigma_{\varepsilon_{h_k}}^2, \text { and } C=\Omega_{h_j} \delta_s^2.$$ Differentiating (25) with respect to [TeX:] $$\beta_j$$ and setting the obtained result to zero, the following expressions can be obtained.

Utilizing [TeX:] $$2 \beta_j \frac{\mathrm{P}_{\mathrm{T}}}{N} C+\sum_{k=j+1}^{N-1} \beta_k \frac{\mathrm{P}_{\mathrm{T}}}{N} C$$ and setting to zero [TeX:] $$\beta_j$$ and [TeX:] $$\sum_{k=j+1}^{N-1} \beta_k$$ can be obtained as 1/3 and 2/3, respectively. However, the obtained results do not adhere to the constraints. In this regard, following similar procedures and differentiating with respect to [TeX:] $$\sum_{k=j+1}^{N-1} \beta_k$$ and setting the obtained result to zero, [TeX:] $$\beta_j$$ and [TeX:] $$\sum_{k=j+1}^{N-1} \beta_k$$ can be obtained as 2/3 and 1/3, respectively. Thus, these results adhere to the constraints and can be easily used for N = 2 scenario. However, the optimized values of power allocation coefficients for N ≥ 3 are still indefinite in this system model. To clarify it, by expanding [TeX:] $$\sum_{k=j+1}^{N-1} \beta_k$$ as [TeX:] $$\beta_k+\sum_{m=k+1}^{N-2} \beta_m,$$ where [TeX:] $$\beta_j \geq \beta_k \geq \sum_{m=k+1}^{N-2} \beta_m,$$ and redefining the [TeX:] $$\mathrm{P}_\mathrm{s}$$ constraint as [TeX:] $$\mathrm{P}_{\mathrm{s}}=\left(\beta_{\mathrm{j}}+\beta_{\mathrm{k}}+\sum_{\mathrm{m}=\mathrm{k}+1}^{\mathrm{N}-2} \beta_{\mathrm{m}}\right) \mathrm{P}_{\mathrm{T}} / \mathrm{N}$$ and substituting into (17), and finally differentiating the obtained result with respect to [TeX:] $$\beta_j, \beta_k \text {, and } \sum_{m=k+1}^{N-2} \beta_m \text {, }$$ the optimized power allocation coefficients can be obtained as 1/3 for each user.

V. NUMERICAL RESULTS

In this section, the numerical results of the uplink PDNOMA network performance, characterized by imperfect CSI and SIC are presented. The previously obtained analytical and asymptotic results are verified through Monte-Carlo simulations using Matlab and generating [TeX:] $$10^6$$ channel realizations. Since the MTs are randomly deployed in the cellular coverage area, the Euclidean distance is utilized. In this regard, the jth MT → BS and Nth MT → BS distances are set to 10 and denoted as [TeX:] $$d_j \text { and } d_N \text {, }$$ respectively. The path-loss exponents for the first and the last users are considered as 2,3. It is also considered that the system model has 2 users, N = 2, and the near and far users’ optimized and nonoptimized power allocation coefficients are considered as 2/3, 1/3 and 9/10, 1/9, respectively as presented in Section IV. The target threshold rate, [TeX:] $$R_{t h, j, N} \text {, }$$ is set to 0.2 bps/Hz. and 0.001 bps/Hz. for near and far users. Regarding the imperfect CSI that is caused by the CEE, the training sequence power, [TeX:] $$\delta_e \text {, }$$ is considered as inversely proportional to SNR. It is expected that for under low SNR values, the training sequences power remains low and high CEE variance is observed. Likewise, for high SNR values, high training sequences power is obtained while CEE’s variance remains low. Then, for the F-D case, the correlation coefficient, [TeX:] $$\delta_f$$ is set to 0.9997. The residual interference level, [TeX:] $$\lambda_l,$$ is set to 0.5 in imperfect SIC at BS.

A. Outage Probability

Fig. 2 presents the [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2 \text {'s }$$ OP performance evaluation in uplink PD-NOMA network along with CEE for N = 2. Fig. 2 presents two types of information, which are perfect CSI and imperfect CSI along with optimized and nonoptimized outage performance evaluations of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ According to Fig. 2 for low SNR, the CEE causes system coding gain losses in optimized and non-optimized scenarios. However, the CEE’s detrimental effect becomes negligible for high SNR values. These observations occur because of the training sequence power is low and CEE’s variance is high for low SNR. Then, for high SNR, the training sequence power is high and the CEE’s variance become at negligible levels. However, it is also observed from Fig. 2 that [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2 \text {'s }$$ outage performance curves saturate in the high SNR regimes. Observed saturation occurs for different reasons for both users. [TeX:] $$\mathrm{MT}_1 \text {'s }$$ saturation originates from the order of decoding process while the saturation in [TeX:] $$\mathrm{MT}_2$$ is due to the imperfect SIC (ISIC) effect. In greater detail, the nonoptimized scenario achieves slightly better outage performance compared to its counterpart optimized scenario for [TeX:] $$\mathrm{MT}_1 \text {'s }$$ outage performance. However, optimized [TeX:] $$\mathrm{MT}_2$$ achieves a better outage performance than non-optimized [TeX:] $$\mathrm{MT}_2$$. These observations occur because of the power allocation coefficient differences. The obtained results verify the analytical expressions, (4) and (5), and asymptotic derivations, (17) and (18).

Fig. 3 presents the [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2 \text {'s }$$ outage performance along with F-D for N = 2. Fig. 3 presents two types of information, corresponding to perfect CSI and imperfect CSI, caused by F-D along with optimized and non-optimized outage performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2.$$ In reference to Fig. 3, the F-D does not have any disruptive effect on the system performance for low SNR but causes coding gain losses for high SNR, compared to the perfect CSI case. This observation occurs because of the correlation coefficient differences between the two cases. Moreover, order of decoding process and ISIC cause additional system coding gain losses on the [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2 \text {'s }$$ outage performance, respectively. Results also show that because of the power allocation coefficient differences, the non-optimized scenario achieves slightly better outage performance than the optimized scenario for [TeX:] $$\mathrm{MT}_1$$ while optimized [TeX:] $$\mathrm{MT}_2$$ outperforms the non-optimized counterpart. The obtained results can also be tractable with analytical derivations, (4) and (5), and asymptotic derivations, (17) and (18).

Fig. 4 presents the outage probability performance comparison of [TeX:] $$\mathrm{MT}_1, \mathrm{MT}_2 \text {, and } \mathrm{MT}_3$$ along with perfect CSI/CEE and optimized, as well as non-optimized scenarios. Since the MTs have a different QoS requirements, following target rates, [TeX:] $$R_{t h, j}=0.2 \mathrm{bps} / \mathrm{Hz}, R_{t h, k}=0.001 \mathrm{bps} / \mathrm{Hz}, R_{t h, N}=0.00001 \mathrm{bps} / \mathrm{Hz},$$ are considered for the near, middle, and far users, respectively. The Euclidean distance between jth MT → BS, kth MT → BS, and Nth MT → BS are set to 10 and denoted as [TeX:] $$d_j, d_k, \text { and } d_N,$$ respectively. The path-loss exponents for the near, middle, and the far users are considered as 2, 3, and 4, respectively. According to Section IV, the optimized power allocation coefficients for each terminal is set to 1/3. Regarding the non-optimized power allocation coefficients, 5/10, 3/10, and 2/10, are considered for the near, middle, and the far users, respectively. Because of the number of mobile terminals, locations of the MTs, and the different power allocation coefficients, Fig. 4, N = 3, shows performance differences compared to Fig. 2, which is N = 2. Similarly in Fig. 2, the obtained results in Fig. 4 show that CEE causes system coding gain losses compared to perfect CSI counterpart in low SNR regimes. The detrimental effect of the CEE becomes at negligible levels in high SNR regimes. Because of the order of SIC process and detrimental effect of the imperfect SIC, outage performance curves tend to saturate in high SNR regimes. Lastly, optimized and nonoptimized power allocation coefficients have different effects on each users’ outage performances. For instance, the nonoptimized [TeX:] $$\mathrm{MT}_1$$ achieves a slightly better performance than optimized [TeX:] $$\mathrm{MT}_1$$. This is because in non-optimized case, [TeX:] $$\mathrm{MT}_1$$ is allocated with large power allocation coefficient while the optimized case is allocated with a lower power allocation coefficients. Unlike [TeX:] $$\mathrm{MT}_1$$, in the non-optimized case, [TeX:] $$\mathrm{MT}_2$$ and [TeX:] $$\mathrm{MT}_3$$ were allocated lower power allocation coefficients, 3/10, 2/10, respectively. In contrast, the optimized case allocated coefficients of 1/3. Consequently, the optimized case for [TeX:] $$\mathrm{MT}_2 \text{ and } \mathrm{MT}_3$$ achieved a slightly better outage performance.

Next, Fig. 5 plots the outage probability performance comparison of [TeX:] $$\mathrm{MT}_1, \mathrm{MT}_2, \text { and } \mathrm{MT}_3$$ along with perfect CSI/F-D and optimized, as well as non-optimized scenarios. A similar system configuration is considered as in Fig. 4 for fair comparison. According to obtained results in Fig. 5, F-D does not have any detrimental effect on the users’ outage performance in low SNR regimes but causes system coding gain losses in addition to imperfect SIC in high SNR regimes. Fig. 5 also shows that because of the order of decoding process and also different QoS requirements, [TeX:] $$\mathrm{MT}_3$$ achieves a better outage performance compared to its counterparts [TeX:] $$\mathrm{MT}_1, \text { and } \mathrm{MT}_2.$$ Lastly, optimized and non-optimized power allocation coefficients have different effects on each users’ outage performances.

Fig. 6 plots outage probability performance comparison of orthogonal multiple access (OMA) and NOMA techniques in the presence of CEE. Fig. 6 reveals that CEE causes system coding gain losses on OMA and NOMA techniques in low SNR regimes. The detrimental effect of the CEE becomes at negligible levels on OMA and NOMA techniques in high SNR regimes. Because of the ISIC, the NOMA-based performance curves tend to saturate in high SNR regimes, whereas the OMA-based curves do not exhibit such saturation in high SNR regimes. Fig. 6 also reveal that although utilizing same target rates, the NOMA-based performance curves achieves slightly better performance compared to OMA counterpart because of the pre-log penalty factor of OMA technique [37]. The optimized and non-optimized OMA curves achieves a similar performance behaviours as in the NOMA counterpart.

Fig. 7 plots outage probability performance comparison of OMA and NOMA techniques in the presence of F-D. According to obtained results, F-D does not have any detrimental effect on the OMA and NOMA systems in low SNR regimes. However, F-D slightly causes system coding gain losses on OMA and NOMA in high SNR regimes. Differently from OMA, the NOMA-based performance curves, perfect and imperfect CSI, tend to saturates in high SNR regimes. This observation occurs because of the imperfect SIC process. The F-D causes additional system coding gain losses on the perfect CSI in OMA and NOMA systems. As it is explained in fig 6, the NOMA-based performance curves achieves slightly better performance compared to OMA counterpart because of the pre-log penalty factor of OMA technique.

B. Error Probability

Fig. 8 presents the EP performance of [TeX:] $$\mathrm{MT}_1, \mathrm{MT}_2$$ along with perfect CSI/CEE and optimized, as well as non-optimized scenarios. Results show that CEE has a disruptive effect on the system EP performance for low SNR and causes system coding gain losses but does not have any disruptive effect under high SNR values. Results also show that the order of decoding process and ISIC cause system coding gain losses and saturation on the EP performance of [TeX:] $$\mathrm{MT}_1, \mathrm{MT}_2$$, respectively. Results also reveal that because of the power allocation coefficient differences, the non-optimized scenario achieves slightly better EP performance compared to its counterpart optimized scenario for [TeX:] $$\mathrm{MT}_1$$ while optimized [TeX:] $$\mathrm{MT}_2$$ achieves better EP performance than non-optimized [TeX:] $$\mathrm{MT}_2.$$ The obtained results match with (7), (8) and (19), (20) providing the analytical and asymptotic derivations, respectively.

Next, Fig. 9 presents results for EP performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ along with perfect/imperfect CSI that is caused by FD and optimized/non-optimized scenarios. Results show that F-D does not have any disruptive effect on the system EP performance under low SNR values. However, for high SNR, the F-D causes system coding gain losses on the optimized and non-optimized EP performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$. Results also show that the order of decoding process and ISIC cause system coding gain losses and saturation on the EP performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$, respectively. Finally, results also reveal that because of the power allocation differences, the nonoptimized [TeX:] $$\mathrm{MT}_1$$ achieves better EP performance compared to optimized [TeX:] $$\mathrm{MT}_1$$. Conversely, the optimized [TeX:] $$\mathrm{MT}_2$$ achieves slightly better EP performance compared to non-optimized [TeX:] $$\mathrm{MT}_2$$. The observed results agree with (7), (8) and (19), (20) of the analytical and asymptotic expressions, respectively.

C. Ergodic Rate

Fig. 10 presents the ER performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ along with perfect CSI/CEE and optimized, as well as nonoptimized cases. Results in Fig. 6 reveal that the CEE causes system coding gain losses for low SNR values on the perfectly acquired CSI-based ER performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$. However, because of the training sequence power and estimation error differences, the CEE’s detrimental effects become negligible in the high SNR regime. In addition, results also show that in this case, the order of decoding process and ISIC cause system coding gain losses and saturation on the ER performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$, respectively. Results also show that the non-optimized [TeX:] $$\mathrm{MT}_1$$ achieves a better ER performance compared to optimized [TeX:] $$\mathrm{MT}_1$$ while optimized [TeX:] $$\mathrm{MT}_2$$ provides better ER performance compared to non-optimized counterpart. The observed results agree with (10), (11) and (21), (22) of the analytical and asymptotic expressions, respectively.

Fig. 11 presents the ER performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ along with perfect CSI/F-D and optimized/non-optimized deployments. Results reveal that F-D does not have any detrimental effect on the optimized and non-optimized [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ in the low SNR regime. However, the F-D causes system coding gain losses on the perfectly acquired CSI for high SNR values. Moreover, results also show that in this case, the order of decoding process and ISIC cause system coding gain losses and saturation on the ER performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$, respectively. Results also reveal that the because of the power allocation coefficient differences, the non-optimized [TeX:] $$\mathrm{MT}_1$$ achieves a better ER performance compared to the optimized [TeX:] $$\mathrm{MT}_1$$ for medium and high SNR values. Conversely, the optimized [TeX:] $$\mathrm{MT}_2$$ provides better ER performance compared to non-optimized counterpart. These observations follow (10), (11) and (21), (22) of the analytical and asymptotic expressions, respectively.

D. Throughput

Figs. 12 and 13 show the throughput performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ for both perfect and imperfect CSI due to CEE and optimized, as well as non-optimized cases. Results reveal that CEE causes coding gain losses for low SNR for the optimized and non-optimized scenarios. However, the CEE’s effect becomes negligible on the optimized and non-optimized [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ for high SNR. Moreover, it is also observed that the order of decoding process and ISIC cause coding gain losses and saturation on the throughput performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$, respectively. Furthermore, results show that due to power allocation differences, the non-optimized [TeX:] $$\mathrm{MT}_1$$ achieves slightly better throughput than the optimized [TeX:] $$\mathrm{MT}_1$$ while optimized [TeX:] $$\mathrm{MT}_2$$ outperforms the non-optimized counterpart. These observations are consistent with (13), (14) and derived asymptotic expressions.

Figs. 14 and 15 present the throughput performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ along with perfect/imperfect CSI that is caused by F-D and both optimized and non-optimized communication parameters. Results in Figs. 14 and 15 reveal that F-D does not have any disruptive effect on the throughput performance of optimized and non-optimized [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$ for low SNR values. However, in the high SNR regimes, the F-D causes system coding gain losses on the throughput performance of optimized and non-optimized [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$. Further, it is also observed from Figs. 14 and 15 that in this case, the order of decoding process and ISIC cause system coding gain losses and saturation on the throughput performance of [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2$$, respectively. Results also reveal that because of the power allocation coefficient differences, the non-optimized [TeX:] $$\mathrm{MT}_1$$ achieves slightly better throughput performance than its counterpart optimized [TeX:] $$\mathrm{MT}_1$$. However, optimized [TeX:] $$\mathrm{MT}_2$$ achieves slightly better throughput performance than non-optimized [TeX:] $$\mathrm{MT}_2$$. These observations are consistent with (13), (14) and derived asymptotic expressions.

E. Energy Efficiency

Figs. 16 and 17 present the EE performance of [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$ along with perfect/imperfect CSI that is caused by CEE and optimized, as well as non-optimized network deployment. Results reveal that the CEE causes system coding gain losses in the low SNR regime on the EE performance of both optimized and non-optimized [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$. However, the CEE’s disruptive effects become negligible for high SNR. Results also show that in this case, the order of decoding process and ISIC cause system coding gain losses and saturation on the EE performance of [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$, respectively. Moreover, because of the power allocation coefficient differences, the non-optimized [TeX:] $$\mathrm{MT}_1$$ achieves slightly better EE performance compared to its counterpart optimized [TeX:] $$\mathrm{MT}_1$$ while optimized [TeX:] $$\mathrm{MT}_2$$ outperforms the non-optimized [TeX:] $$\mathrm{MT}_2$$. Results also reveal that [TeX:] $$\mathrm{P}_{\mathrm{s}}=0 \mathrm{dBW}$$ achieves slightly better EE performance for [TeX:] $$\mathrm{MT}_1 \text { and } \mathrm{MT}_2 \text { than } \mathrm{P}_{\mathrm{s}}=1 \mathrm{dBW}.$$

Figs. 18 and 19 depict the EE performance of [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$ along with perfect/imperfect CSI that is caused by FD and optimized, as well as non-optimized uplink NOMA communication. Results in Figs. 18 and 19 reveal that F-D does not have any detrimental effect on the EE performance of both optimized and non-optimized [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$ under low SNR. However, in the high SNR regimes, F-D slightly causes coding gain losses on the optimized/non-optimized [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$. Moreover, for high SNR, the order of decoding process and ISIC cause system coding gain losses and saturation on the EE performance of [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$, respectively. Results also reveal that [TeX:] $$\mathrm{P}_{\mathrm{s}}=0 \mathrm{dBW}$$ achieves slightly better EE performance than [TeX:] $$\mathrm{P}_{\mathrm{s}}=1 \mathrm{dBW}$$ for optimized, as well as non-optimized [TeX:] $$\mathrm{MT}_1 \text{ and } \mathrm{MT}_2$$ for both low and high SNR values.

VI. CONCLUSIONS &FUTURE DIRECTIONS

This work provided a detailed analysis of uplink NOMA performance under practical impairments related to the channel estimation and interference cancellation processes. In this context, several important metrics, such as outage and error probability, ergodic rate, throughput, energy and spectral efficiency were analytically and asymptotically derived and validated through Monte-Carlo simulations. Our findings show that these impairments can have a significant impact on the communication performance, especially in low and high SNR regimes. In order to improve the performance, we presented the optimization of power allocation using the Lagrangian multipliers technique, obtaining improved outage probability, error probability, ergodic rate, and throughput. Compared to traditional OMA and other NOMA schemes, our optimized NOMA-based approach provides enhanced performance, making it a viable solution for uplink communication. In conclusion, our findings aimed at highlighting the potential of NOMA in real-world deployments, as the impact of practical issues was thoroughly analyzed and mitigated through proper power allocation.

Regarding possible future directions for our study, machine learning techniques can be utilized to estimate the channel parameters and minimize system overheads and feedback delays. Also, adopting and evaluating the integration of aerial communications on the performance of uplink and downlink NOMA deployments where unmanned aerial vehicles provide additional degrees of freedom in network deployment represents another interesting research direction [38].

VII. ACKNOWLEDGEMENT

This work was supported by HORSE project, funded by the Smart Networks and Services Joint Undertaking (SNS JU) under the European Union’s Horizon Europe research and innovation programme under Grant Agreement Number 101096342 (www.horse-6g.eu).

APPENDIX

PROOF OF PROPOSITION 1

Revisiting (2) and utilizing logarithm and probability properties, the following expressions can be obtained.

Substituting related CDF expression into (27), following expressions can be obtained.

where [TeX:] $$\Delta=\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{{\mathrm{h}}_{\mathrm{j}}}}^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \delta_{\mathrm{s}}^2\left|\hat{\mathrm{h}}_{\mathrm{k}}\right|^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}^2+\sum_{\ell=1}^{\mathrm{j}-1} \mathrm{P}_{\mathrm{s}} \beta_{\ell}\left|\mathrm{h}_{\ell}\right|^2,$$ [TeX:] $$\gamma_{\hat{x}_j}=\mathrm{P}_{\mathrm{s}}\left|\hat{\mathrm{h}}_{\mathrm{j}}\right|^2 / \sigma^2, \gamma_{\hat{x}_k}=\mathrm{P}_{\mathrm{s}}\left|\hat{\mathrm{h}}_{\mathrm{k}}\right|^2 / \sigma^2, \gamma_{x_{\ell}}=\mathrm{P}_{\mathrm{s}}\left|\mathrm{h}_{\ell}\right|^2 / \sigma^2,$$ [TeX:] $$\gamma_{\hat{x}_K}=\sum_{k=j+1}^{N-1} \gamma_{\hat{x}_k}, \gamma_{x_L}=\sum_{\ell=1}^{j-1} \gamma_{x_{\ell}},$$ and [TeX:] $$f \gamma_{\hat{x}_j}(\gamma)=\frac{1}{P_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}} e^{-\frac{\gamma}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}}}.$$ Note that summation of M independent identically distributed (i.i.d) Rayleigh distribution becomes Gamma distribution. In this regard, [TeX:] $$f \gamma_{\hat{x}_K}(\gamma)=\frac{\gamma^{N-2}}{\left(\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{k}}}\right)^{N-1}(N-2)!} e^{-\frac{\gamma}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{k}}}}}$$ and [TeX:] $$f \gamma_{x_L}(\gamma)=\frac{\gamma^{j-2}}{\left(\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}}\right)^{j-1}(j-2)!} e^{-\frac{\gamma}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}}}}$$ [39]. Substituting related PDF expressions into (28) and solving the integral expression by means of [23, Eq. (3.351.3)] and final result is presented in (4). Likewise, following a similar procedures, [TeX:] $$\bar{\mathrm{P}}_{\text {out }}^{N^{\text {th }}}\left(\gamma_{\text {th}_\mathrm{N}}\right)$$ is obtained as in (5).

PROOF OF PROPOSITION 2

Substituting (4) into (6) and using distributive properties, following expression can be obtained.

The first integral expression can be solved with the help of [23, Eq. (3.326.2¹⁰)] as [TeX:] $$\Gamma(1 / 2)=\sqrt{\pi}.$$ Regarding the second integral expression, by means of [21, Eq. (10 and 11)], (29) can be written as

where [TeX:] $$\Psi=\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{j}}}}^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma^2_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}+1.$$ By means of [22, Eq. (13)], (30) is solved as in (7). Note that α term is set to 1/2 in (7).

Likewise, following similar procedures and utilizing [21, Eq. (10, 11, and 21)], the Nth user’s EP can be obtained as in (8). Note that α term is set to 1/2 in (8).

PROOF OF PROPOSITION 3

By means of [40, Eq. (4)], (9) can be re-written as [TeX:] $$\mathrm{ER}^{\mathrm{j}^{\mathrm{th}}}=\frac{1}{\ln 2} \int_0^{\infty} \frac{1-\mathrm{F} \gamma_{\mathrm{x}_{\mathrm{j}}}\left(\gamma_{\text {th }}\right)}{1+\gamma_{\text {th }}} \mathrm{d} \gamma_{\text {th }}.$$ Substituting (4) into the CDF-based ER formula, following expression can be obtained.

By means of partial fraction decomposition technique, (31) can be written as

where [TeX:] $$A_i=\lim _{y \rightarrow-\frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}} \sum_{\ell=1}^{j-1} \beta_{\ell}}} \frac{\partial^{j-1-i}}{(j-1-i)!\partial y^{j-1-i}}\times\left(1+y \frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}} \sum_{\ell=1}^{j-1} \beta_{\ell}}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}}\right)^{j-1} \Phi,$$ [TeX:] $$D=\lim _{y \rightarrow-1} \frac{\partial}{\partial y}(y+1) \Phi,$$ and [TeX:] $$\Phi=\left[\frac{1}{\left(1+\frac{\gamma_{t h_j} \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}} \sum_{\ell=1}^{j-1} \beta_{\ell}}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}}\right)^{j-1}\left(\gamma_{t h_j}+1\right)}\right].$$

By using the distributive properties and utilizing [21, Eq. (10 and 11)] following expression can be obtained.

By means of [22, Eq. (13)], the final expression can be obtained as in (10). Note that α term in [22, Eq. (13)] is set \to 1. Likewise, following similar procedures, the Nth user’s ER expression can be obtained as in (11). Also note that the α term is set to 1 in [22, Eq. (13)] for the analytical derivations of Nth user’s ER.

PROOF OF PROPOSITION 4

Revisiting (30), when [TeX:] $$P_s \rightarrow \infty,$$ [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{k}}} \sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \beta_{\mathrm{k}} \delta_{\mathrm{s}}^2}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}},$$ [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}} \sum_{\ell=1}^{\mathrm{j}-1} \beta_{\ell}}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}}$$ terms in the first and second Meijer’s G functions in (30) approximate to zero. Therefore, by means of [41, Eq. (07.34.06.0040.01)], first and second Meijer’s G functions in (30) can be obtained as 1. Further, replacing the newly obtained results into (30) and utilizing [21, Eq. (24)] in (30), [TeX:] $$\left(\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{j}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{j}}}}^2+\sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \mathrm{P}_{\mathrm{s}} \beta_{\mathrm{k}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{k}}}}^2+\beta_{\mathrm{j}} \delta_{\mathrm{s}}^2 \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}+1}{\beta_j \delta_{\mathrm{s}}^2 \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}}}\right)^{-1 / 2} \Gamma(1 / 2)$$ can be obtained. Note that α−n term in [21, Eq. (24)] is set to 0.5. The final asymptotic EP expression for the jth user is presented in (19).

Regarding the asymptotic expression of the Nth users, when [TeX:] $$P_s \rightarrow \infty,$$ since the CEE variance is inversely proportional to SNR, the CEE variance, which is [TeX:] $$\sigma_{\varepsilon_{h_N}}^2,$$ in the denominator of [TeX:] $$\frac{\sum_{\ell=1}^{N-1} \beta_{\ell} \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}}}{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{N}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{N}}}}^2+1}$$ term in Meijer’s-G function in (8) become negligible for high SNR. In addition, the [TeX:] $$\sum_{\ell=1}^{N-1} \beta_{\ell} \Omega_{h_{\ell}}$$ term in the numerator of [TeX:] $$\frac{\sum_{\ell=1}^{N-1} \beta_{\ell} \mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}}}{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{N}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{N}}}}^2+1}$$ term in Meijer’s-G function, the numerator become dominant and the expression goes to [TeX:] $$\infty$$. In this regard, utilizing [42, Eq. (41)], the final asymptotic expression can be obtained as in (20).

PROOF OF PROPOSITION 5

Regarding the asymptotic expression of jth user’s ER expression, revisiting (33), when [TeX:] $$P_s \rightarrow \infty,$$ [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{k}}} \sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}} \beta_{\mathrm{k}} \delta_{\mathrm{s}}^2}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}},$$ [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\ell}} \sum_{\ell=1}^{\mathrm{j}-1} \beta_{\ell}}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}}$$ terms in the second and third Meijer’s G functions in (33) approximate to zero. Therefore, by means of [41, Eq. (07.34.06.0040.01)], second and third Meijer’s G functions in (33) can be obtained as 1. Replacing the newly obtained results into (33) and solving the integral expression with the help of [21, Eq. (24)], the following expression is obtained:

Note that α-n term is set to 1 in [21, Eq. (24)]. The final expression can be obtained as in the first part of (21). When it is come to the second integral expression of (33), when [TeX:] $$P_s \rightarrow \infty,$$ [TeX:] $$\frac{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{k}}} \sum_{\mathrm{k}=\mathrm{j}+1}^{\mathrm{N}-1} \beta_{\mathrm{k}} \delta_{\mathrm{s}}^2}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{j}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{j}}}$$ term in the second Meijer’s G functions in the second integral expression of (33) approximate to zero. Therefore, by means of [41, Eq. (07.34.06.0040.01)], second Meijer’s G function in the second integral expression of (33) can be obtained as 1. Replacing newly obtained result into the second integral expression of (33) and solving the integral expression with the help of [21, Eq. (21)] the following expression can be obtained.

when [TeX:] $$P_s \rightarrow \infty,$$ analytical expression in the Meijer’s G function in (35) approximate to zero. Therefore, by means of [41, Eq. (07.34.06.0040.01)], Meijer’s G function in (35) can be obtained as 1. Therefore, analytical expression before the Meijer’s G function can be the final result for the asymptotic expression of the second part of the integral expression in (33). Utilizing (34) and newly obtained result, the final asymptotic ER expression for the jth user can be obtained as in (21).

Regarding the asymptotic expression of Nth user’s ER expression, utilizing [41, Eq. (07.34.06.0040.01)] and [21, Eq. (21)] in the derived ER integral expression of Nth user following expression can be obtained. [TeX:] $$\left(\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{N}} \kappa_{\mathrm{s}}^2 \sigma^2_{\varepsilon_{\mathrm{h}_{\mathrm{N}}}}+1}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{N}}}\right)^{-1}G_{2,1}^{1,2}\left(\left.\frac{\mathrm{P}_\mathrm{s} \Omega_\mathrm{h} \delta^2_s \beta_\mathrm{N}} {\mathrm{P}_{\mathrm{s}} \beta_\mathrm{N} \kappa^2_\mathrm{s}\sigma^2_{\varepsilon_{\mathrm{h}_\mathrm{N}}} + 1} \right\rvert\, \begin{array}{c} {0,0} \\ {0, -} \end{array}\right).$$ By means of [41, Eq. (07.34.06.0040.01)], the obtained result further simplified as [TeX:] $$\left(\frac{\mathrm{P}_{\mathrm{s}} \beta_{\mathrm{N}} \kappa_{\mathrm{s}}^2 \sigma_{\varepsilon_{\mathrm{h}_{\mathrm{N}}}}^2+1}{\mathrm{P}_{\mathrm{s}} \Omega_{\mathrm{h}_{\mathrm{N}}} \delta_{\mathrm{s}}^2 \beta_{\mathrm{N}}}\right)^{-1} .$$ The final expression can be obtained as in (22).