Performance of NOMA-based mmWave D2D Networks under Practical System Conditions

This work elaborates the analysis on ergodic capacity, coverage probability, and average throughput for multi-user non-orthogonal multiple access (NOMA) based device-to-device communication networks, which operate in the millimeter-wave spectrum range and are constrained by practical system imperfections such as residual transceiver hardware impairments, imperfect channel state information, and non-ideal successive interference cancellation. More importantly, we consider that the proposed network model is limited by independent and non-identically distributed interference noises emerging from neighboring device nodes. Computationally effective and comprehensive closed-form expressions are delivered to evaluate the ergodic capacity with its upper and lower bounds, as well as coverage probability and average throughput expressions. Furthermore, the asymptotic analysis of ergodic capacity and coverage probability at high and low signal-to-noise-ratio regimes are analyzed and the corresponding closed-form expressions are presented. Valuable discussions on the fairness-based power allocation scheme for NOMA users have been provided. Moreover, a thorough Monte Carlo simulation is carried out to validate the corresponding analytical findings. Finally, simulation results have revealed that the system impairments aforementioned herein cause an ergodic capacity saturation phenomenon. Especially, interference plays a significant role as a performance limitation factor for the ergodic capacity and coverage probability.


I. INTRODUCTION
O NE of the main aims of fifth-generation (5G) communication systems is to support connections of up to 50 billion devices by 2025 [1]. To this extent, the deviceto-device (D2D) communication [2]- [4], millimeter wave (mmWave) technology [5]- [8], and non-orthogonal multiple access (NOMA) [9]- [13] (enabling concurrent device connections) can fulfill this goal. The NOMA is particularly useful for efficient utilization of network resources such as time, code, frequency, as well as radio frequency (RF) chains dedicated per user equipment (UE). For instance, the in-field trials conducted by NTT DOCOMO (Japan) and Medi-aTek (Taiwan) demonstrated a 2.3-times spectral efficiency (SE) improvement for smartphone-sized NOMA users in comparison to the single-user multiple-input multiple-output (MIMO) scenario [14].

A. RELATED WORKS
In NOMA-based D2D networks, multiple and nearby located devices' signals are compounded in the power domain to boost their SE. The D2D-empowered NOMA networks also demonstrated the increase in energy efficiency, data offloading, and ability to massively connect devices [15]. The authors in [15] studied the user-clustering and power assignment optimization in the network comprising cellular users and underlay NOMA-enabled D2D users. Next, the work [4] presented a novel resource allocation algorithm by using a many-to-one side matching theory to achieve nearoptimal sum-rate performance for underlay NOMA-based D2D networks. Moreover, the authors in [16] developed an interlay mode to accommodate power-domain multiplexing for both cellular and D2D users. Next, capacity scaling of a NOMA-D2D-based cooperative relay system for the twouser scenario was investigated in [17] over Rayleigh fading channels. To conclude, it is pertinent to mention that all the aforementioned recent works considered ideal system configurations for the NOMA-based D2D networks.
It is well-known that the RF chains considerably contribute to the cost of communication systems due to their high power consumption and cost of analog-to-digital converters (ADCs)/ digital-to-analog converters (DACs) in RF chains [18]. At the same time, low-grade transceiver equipment is often used in modern high-rate systems to reduce their CAPEX costs. This inevitably adds residual transceiver hardware impairments (RTHIs) from the phase noise [19], [20], amplifier non-linearity [21], and in-phase/quadrature-phase imbalance [22], [23] to the received signal. The RF front-end imperfections have a more negative impact on the mmWave spectrum range than in the conventional frequency range [24] due to an inverse relationship between the transmission rate and RTHI level [25].
Although the assumption of an ideal transceiver is acceptable for systems with a low data rate, this assumption is not valid for high-speed communication systems [26]. Comparative analysis of 60 GHz commercial RF front-ends for orthogonal frequency division multiplexing (OFDM) systems was presented in [27] for intrinsic RF hardware imperfections and their effect on the system performance. Moreover, the authors proposed a practical link budget analysis to evaluate Error Vector Magnitude (EVM). According to [28], RTHI limits the capacity at the UE side with multiple antennas. Oppositely, the effects of RTHI and inter-cluster interference are less severe at the base stations that employ massive MIMO antennas. Therefore, it is essential to model the RTHI noise and interference in D2D networks.
Recent works on the RTHI, imperfect channel state information (CSI), and non-ideal successive interference cancellation (SIC) have considered the cooperative NOMA networks in [29]- [36]. For instance, the authors in [29] studied the two-user uplink NOMA-based mmWave networks and proposed a joint power control and beamforming technique. Moreover, the authors in [30] evaluated the ergodic capacity for amplify-and-forward (AF) and decode-and-forward (DF) MIMO relay systems with an arbitrary number of antennas and by considering the effect of RTHI. Similarly, a NOMAbased AF relay network constrained by RTHIs was studied in [31], where the outage probability along with the ergodic capacity performance was investigated. Moreover, the recent works [32] and [33] studied the RTHI and imperfect SIC while incorporating the energy harvesting capabilities in the overlay cognitive NOMA networks over Nakagami-m fading environment.
It was shown that the NOMA-based system outperforms the traditional orthogonal multiple access (OMA) system at the medium-to-high signal-to-noise (SNR) regime. Similarly, an approximate expression for the ergodic capacity was provided in the recent work [34] for both cooperative and noncooperative NOMA networks considering the RTHI, imperfect CSI [37] and non-ideal SIC [38]. The authors in [35] studied the security aspects in NOMA-based AF relay cooperative networks by considering the RTHI noise, where the exact and asymptotic outage probability and intercept probability analysis were provided. The authors in [36] studied the cooperative NOMA-based coverage probability analysis for a mmWave network given an ideal system configuration; three potential relay selection schemes were proposed to justify the advantage of the cooperative NOMA approach over the traditional OMA one.
Next, the ergodic capacity analysis for NOMA-based uplink satellite networks with randomly deployed end-users under imperfect CSI and antenna-pointing errors was studied in [39]; however, this work did not study the other system impairments. Likewise, the work [40] studied the ideal NOMA network with two end-users and presented the outage probability and approximate ergodic capacity formulas.
The authors in [41] studied a different aspect of practical system imperfection in NOMA networks such as arbitrary user mobility profile at various antenna array ranges given the short packet transmission regime. The other authors in [42] introduced the heterogeneous mobility profile to group users that move at a certain speed by using a novel NOMA orthogonal frequency space modulation. Furthermore, the authors in [43] proposed a detection method and channel estimation algorithm for sky-ground uplink NOMA, where aerial users are moving while terrestrial users are static. The proposed method improved both the channel estimation and time-varying successive interference cancellation.
It is noted that the contemporary research in the field of NOMA-based mmWave D2D networks either has studied the ideal transceiver hardware or does not consider independent and non-identically distributed (i.n.i.d.) interference noise induced by neighboring device nodes. However, the authors in [44] discussed the importance of inter-cell and intra-cell interference analysis as a main limiting factor for NOMAbased 5G networks. On the other hand, they did not present an analytical performance analysis. Moreover, the authors in [45] considered a simple yet not accurate interference noise model for underlay cognitive radio NOMA networks, where several interfering nodes are treated as a Gaussian noise by applying a central limit theorem (CLT).
To the best of the authors' knowledge, little attention has been paid to studying the impact of interference in mmWave D2D NOMA networks. This problem, as well as the other realistic system impairments (i.e., RTHI and imperfect CSI/SIC), are worth being addressed in forthcoming massively connected communication systems; therefore, these issues mentioned above are the main investigation aspects of this paper.

B. MAIN CONTRIBUTION
Motivated by the abovementioned discussion, in this work, we aim to study the essential sources of system impairments that degrade the performance of multi-user NOMA-based D2D mmWave networks and develop a unified framework to evaluate their performance. For statistical RTHI modeling, we adopt the power-dependent additive white Gaussian noise (AWGN) model to describe the aggregate effect of transceiver impairments due to its analytical tractability as well as its theoretical and practical validity supported by [26], [46]- [49]. Moreover, we incorporate an analytically tractable model for i.n.i.d. interference noise terms over Nakagmai-m fading channels into our system model. To the best of the authors' knowledge, no prior work derived the closed-form expressions for the ergodic capacity and coverage probability analysis considering M i.n.i.d. interference noises. Since channel conditions and signal power of interfering nodes may vary depending on their locations, it is important to consider i.n.i.d. Gamma variates to represent channel gains of Nakagami-m channel amplitudes. The authors in [50] presented a single Gamma approximation PDF for the summation of M i.n.i.d. Gamma RVs. This work enabled us to present all channel noises (a summation of M i.n.i.d. interference, RTHI, and imperfect CSI/SIC) as a single Gamma RV. This approach substantially simplified the analysis of the proposed system model.
The key contributions of this work are summarized as • Different from [34] and [39], we develop a practical framework for NOMA-based mmWave D2D networks while considering the compound effect of RTHI, imperfect CSI/SIC, and interference from side/back lobe antenna gains of the interfering nodes in the multi-user NOMA D2D network. • In contrast to [34] and [40], where the authors delivered the approximate ergodic capacity analysis of NOMA-based networks due to the mathematical complexity of the exact analytical derivations, this work delivers compact and insightful closed-form expressions for the ergodic capacity and corresponding lower and upper bounds along with the asymptotic analysis at the high and low SNR regimes. • Closed-form expressions for the coverage probability and average throughput have been presented in this work. To obtain deeper insights, the asymptotic behavior of the coverage probability is investigated at the high-SNR regime. • The individual effect of each system impairment (i.e., RTHI, imperfect CSI/SIC, and M i.n.i.d. interference noise) is studied separately and compared with its ideal counterpart. Given a non-ideal system configuration, the closed-form power allocation (PA) scheme is proposed to ensure fairness among NOMA users. Finally, thorough Monte-Carlo simulations validate the correctness of all derived analytical expressions and demonstrate the advantage of the NOMA-based network over the OMA one.

C. NOTATIONS AND PAPER ORGANIZATION
Throughout the paper, the expectation operator is denoted as E{·}, G m,n p,q z b1,··· ,bq a1,··· ,ap represents the Meijer G-function, and X ∼ CN µ, σ 2 stands for the circularly symmetric complex Gaussian random variable (RV) X with mean µ and σ 2 variance. In addition, F X (·) and f X (·) symbolize the cumulative distribution function (CDF) and probability density function (PDF) of RV X, correspondingly. Moreover, Γ(m, β) is an incomplete Gamma function with m fading and β > 0 scale parameters, and ψ(·) is the digamma function [51, (8.360.1)]. Pr(C) is the probability of an event C. Gamma RV X with m fading and β scale parameters is denoted as X ∼ Gamma(m, β). The remainder of the paper is organized as follows. Section II defines the system model of a realistic NOMAbased mmWave D2D network limited by imperfect SIC/CSI, RTHIs, and interference. Next, Section III presents generic formulas to evaluate the ergodic capacities with its lower and VOLUME 4, 2016 upper bounds for the proposed system model. Furthermore, Section IV evaluates the ergodic capacity with its lower and upper bounds and corresponding asymptotic analysis for the NOMA-based mmWave D2D network under study over Nakagami-m fading channels. Section V presents the closedform expression for the coverage probability and its high-SNR approximation. Section V-B showcases the closed-form expression for the average throughput formula for the system model under study. Furthermore, in Section VI, the fairness aspects of the power allocation are discussed and the closedform solution is proposed. The main result discussions are drawn in Section VII, and Section VIII summarizes the key points of the work.

II. SYSTEM MODEL
Consider a downlink NOMA-based D2D mmWave network that uses analog beamforming to establish direct communication links for N NOMA users, denoted by U i , i ∈ A = {1, 2, . . . , N }. All device nodes are equipped with MIMO antennas and non-ideal transceiver hardware. The source node (S) applies the NOMA principle to communicate with N device nodes located at distances, denoted by d i , i ∈ A. It is assumed that each reference NOMA user is surrounded by M interfering user nodes 1 . The side/back lobes of the other device nodes create interference to the end-users. A base station controller unit is employed to arbitrate the coordination and beam alignment between the communicating device nodes to avoid interference. We assume an ideal beam alignment between device nodes for analytical tractability as in [3] and [52]. This way, one can focus more on the other types of system non-idealities such as RTHI, imperfect CSI/SIC, and interference. For more details on the beam misalignment, one can refer to [53], [54].
We model the mmWave channel by using Nakagami-m fading along with analog beamformed user association as in [55] and [56], where a fading parameter m models the line-of-sight (LOS)/non-LOS components of mmWave communication. In addition, Nakagami-m distribution scaled by analog beamforming gain is a mathematically tractable solution to model mmWave channel that simplifies the analysis of complex system models.
Inspired by [52], a sectored antenna array pattern is used to model the analog beamformed antenna array as where θ and θ b are the angles of a boresight direction and antenna beamwidth, accordingly. G s and G m indicate the gains related to the side/back and main lobes, respectively. In addition, a linear minimum mean square error is used to estimate the communication channel between S and U i as h i = h i + i , where h i ,h i , and i denote the channel amplitude, estimated channel amplitude, and estimation error with i ∼ CN 0, σ 2 ei , respectively, where σ 2 ei approximates the error in the channel estimation. The NOMA technology allows S to transmit a superimposed signal x = N n=1 √ α n P x n towards N NOMA nodes, where x n is the signal designated for the nth user, α n is the PA coefficient with a unit power property, given by N n=1 α n = 1, and P is the average signal power. The PA coefficient ensures fairness during the power allocation process for U i . In this paper, similar to [57], we allocate the power based on the quality of service (QoS) requirements imposed on the NOMA users. For example, a higher PA coefficient is assigned to a user with a higher QoS requirement, while a user with a lower QoS is devoted to a smaller PA coefficient. Therefore, we order the PA factors for the NOMA users as α 1 > α 2 > . . . > α n > . . . > α N .
Hence, the received signal at a reference receiver, U i , is given as where τ is the path loss exponent (PLE) of the main D2D link, τ is the PLE of the interference links 2 , d i and d ik are the distances between S to U i and between U i to interfering node k, respectively. Moreover, G k is a gain factor from an interfering node k, which could either land to the reference receiver's side/back lobe with the main lobe or side/back lobe (G m or G s ). In (2), h i is the channel between the D2D pair, while g ik is the channel between the kth interference node and the reference receiver i within a given cluster, h i and g ik are both Nakagami-m distributed channel amplitudes. Additionally, I k is the average signal power from the interfering node k, RTHI noise components are modeled as µ i ∼ CN 0, κ 2 i P andμ k ∼ CN 0,κ 2 k I k , where κ i andκ k are the RTHI levels measured by EVM [58]. The EVM metric is frequently applied to measure a mismatch between intended and actual signals. An ideal RF transceiver would have an EVM value equal to zero and lower values of EVM signify a higher quality of RF transceiver hardware [31]. Moreover, w i ∼ CN 0, σ 2 i is the AWGN term. Furthermore, the signal-to-interference-noise-distortion ratio (SINDR) at U i to decode the signal x j given j ≤ i can be formulated by considering the RTHI, interference, and imperfect CSI/SIC as where ξ r α r for imperfect SIC range, 0 ≤ ξ r ≤ 1, such that ξ r = 0 represents ideal SIC and ξ r = 1 stands for non-ideal SIC. We make the following variable substitutions for simplification purposes of SINDR as where Note that U 1 decodes its own signal x 1 and treats other x n messages as a noise with Φ 1 = N t=2 α t andΦ 1 = 0 parameters. The user U N decodes the x N signal by applying SIC and by evaluatingΦ N = N −1 r=1 ξ r α r .

III. GENERIC ERGODIC CAPACITY WITH BOUNDS
This section analyzes the ergodic capacity of NOMA-based mmWave D2D networks by considering RTHI, imperfect CSI/SIC, and i.n.i.d. interference noises. Inspired by [59], the general expressions for the ergodic capacity with its upper and lower bounds for a NOMA-based mmWave D2D network constrained by system imperfections in arbitrary fading channels are formulated below.

A. ERGODIC CAPACITY
By definition, the ergodic capacity is given by . (5) By using the property of a logarithm function, (5) can be further written as a difference of two terms . (6) We make the following variable substitutions in (6) for simplification purposes: that represents the summation of (M +1) i.n.i.d. Gamma RVs and Gamma RVs with different fading and scale parameters. Hence, we get the following simplified representation for the ergodic capacity as

B. LOWER ERGODIC CAPACITY
By using the concavity property of log 2 (1 + υ exp(x)) for variable x with υ > 0 and Jensen's inequality [51, (12.41)], the ergodic capacity is lower-bounded as . (8) Next, by exploiting the property of a logarithmic function, we may re-write the lower bound ergodic capacity as

C. UPPER ERGODIC CAPACITY
Since a logarithm is a concave function, Jensen's inequality is applied to evaluate the generic upper ergodic capacity as . (10)

IV. ERGODIC CAPACITY WITH BOUNDS IN NAKAGAMI-M FADING CHANNEL
This section demonstrates the approximated closed-form expressions for the ergodic capacity with its upper and lower bounds in for NOMA-based mmWave D2D network with system impairments over Nakagami-m fading channels.
Lemma 1 below closely approximates PDF of a summation of Q i.n.i.d. Gamma variates. Its proof and validation are presented in detail by the authors in [50]. The ergodic capacity and ergodic capacity bounds are evaluated with the aid of Lemma 1.
Lemma 1: Consider X t (t = 1, · · · , Q) are non-negative i.n.i.d. Gamma RVs with m t fading and β t scale parameters, accordingly. The probability density function (PDF) of Z = Q t=1 X t is given as where an approximated scale parameter of a single Gamma function,β, is evaluated by solving a set of equations A shape parameter for the single Gamma function is evaluated from m z = μ βz . We derive an ergodic capacity with the aid of Lemma 1 to represent the summation of (M + 1) i.n.i.d. Gamma variates.
where {β t ,β v } and {m t ,m v } are the scale and fading pa- Gamma representations of the sum of (M +1) i.n.i.d. Gamma distributed noise components, accordingly. Proof: Full proof is relegated to Appendix A.
There are other PDFs available that could be used to represent a summation of M i.n.i.d. Gamma variates. For instance, the work in [60] developed a summation PDF in terms of Fox H-function, which is mathematically very challenging to be integrated into the computation of ergodic capacity. In fact, the author in [61] represented the PDF for summation of M i.n.i.d. Gamma RVs as the semi-infinite summations with a recursive function. However, it is computationally intensive to use this PDF in our system model as well. It is possible to derive the compact ergodic capacity formula by using a single Gamma representation for the summation of M i.n.i.d. Gamma RVs [50].
Proposition 2 below presents a closed-form expression for the lower bound of ergodic capacity. Proposition 2: The ergodic capacity for a NOMA-based mmWave D2D network with RTHI, imperfect CSI/SIC, and interference noise is lower bounded bỹ Proof: See Appendix B. Similarly, Proposition 3 presents an upper ergodic capacity for the proposed system model. Proposition 3: An upper bound of the ergodic capacity for a given NOMA-based mmWave D2D network with system impairments and M interference noise terms is evaluated by using (10) as Proof: The proof is relegated to Appendix C.

A. ASYMPTOTIC ANALYSIS
System imperfection noises, such as RTHI and imperfect CSI/SIC, are proportional to the transmit power. Therefore, it is particularly interesting to analyze the ergodic capacity at the high-SNR regime to elaborate their effect on the capacity ceiling. Proposition 4 below provides the closedform expression for asymptotic ergodic capacity formulas at the high-SNR regime.
Proof: The proof is in Appendix D. Next, by using the SINDR in terms of SN R i given in (38) and by taking the limit of the ergodic capacity when SNR tends to zero, we obtain low-SNR ergodic capacity The ergodic capacity in the high-SNR regime is particularly important to verify whether the system impairments contribute to limit the performance of high-rate systems.

V. COVERAGE PROBABILITY
In the following section, the coverage probability analysis for the proposed system model has been performed. The coverage probability evaluates the probability of the received signal, x j being correctly decoded at the reference receiver, U i , which is calculated by the probability of the SINDR being higher at the predefined rate threshold, v = 2 R th − 1 > 0, where R th is the communication rate threshold.
Proposition 5: For NOMA-based mmWave D2D network constrained by N i.n.i.d. interference, imperfect CSI/SIC, and RTHI, coverage probability is evaluated by applying (17) and Lemma 1 to approximate the summation of N interference noise as a single Gamma RV and shown on the top of the next page, where .
Proof: A full derivation is shown in Appendix E.

A. ASYMPTOTIC ANALYSIS: HIGH-SNR APPROXIMATION
In this subsection, we present the coverage analysis using asymptotic high-SNR behaviour for the NOMA-based mmWave D2D networks under interference noise, RTHI, and imperfect CSI/SIC. By considering the high-SNR regime for (39), the asymptotic coverage probability is evaluated as where X [i] ∼ Gamma(m i , β i ) and generic CDF of Gamma , that may be represented in a series form as Next, by using (19) and (20), the asymptotic coverage probability at high-SNR region is given as

B. AVERAGE THROUGHPUT ANALYSIS
In this subsection, the average throughput evaluation formula is given for the NOMA-based mmWave D2D network under system impairments. Proposition 6: Consider a NOMA-based mmWave D2D network that is constrained by RTHI, imperfect CSI/SIC noise, and i.n.i.d. interference noise. The average throughput formula is given as Proof: By definition, the average throughput is the multiplication of the ergodic capacity by the coverage probability as c [62]. Hence, the average throughput closed-form expression is found by multiplying (12) and (17) that stand for the ergodic capacity and coverage probability, correspondingly.

VI. FAIRNESS-BASED POWER ALLOCATION
In addition to the presented comprehensive performance analysis, in this section, we obtain a closed-form adaptive PA solution based on the fair treatment of NOMA users to provide them with equal ergodic capacity/coverage/throughput metrics to NOMA users. The previously derived analytical findings on the ergodic capacity, coverage probability, and average throughput performance metrics are based on fixed pa factors, where users are allocated power depending on their channel conditions. Hence, users obtain different ergodic capacity and coverage probability results. The more obvious way of finding the optimal PA for NOMA users is to retrieve them from the closed-form expressions of ergodic capacity in (12) and coverage probability in (18). However, due to the complexity of those equations, the derivation of optimal PAs becomes very challenging or even intractable. Therefore, we equalize the SINDR values of NOMA users to the optimal SINDR, denoted byγ. It is pertinent to note that NOMA is recommended for use only in the case of two users since the NOMA-induced processing complexity nonlinearly depends on the number of active user devices [63]. The system complexity aspects become more essential when the SIC-caused error propagation happens [64]. Moreover, we show in Fig. 2 that NOMA networks provide the best capacity performance for N = 2 compared to the cases when N > 2. With that, the design of a closed-form optimal PA scheme becomes infeasible for the considered system model, with N > 2. Hence, considering above-mentioned discussions, we provide a closed-form optimal PA solution for a two-user case with γ 1 and γ 2 . Thus, to find the optimal PA and SINDR, we formulate the problem as follows The corresponding solution for (23) is found by making γ 1 =γ and γ 2 =γ, which can be written as α 2 ρ 2 |h 2 | 2 ρ 2 |h 2 | 2 (ξ 1 α 1 + κ 2 2 ) + Λ [2] + Z [2] =γ.

VII. RESULTS DISCUSSION
This section provides some numerical examples to corroborate our analytical derivations on the ergodic capacity, coverage probability, and average throughput under different levels of system imperfections such as RTHI, imperfect CSI/SIC, and i.n.i.d. interference noise. These findings are fully validated through the averaged results via Monte-Carlo simulation. We assume the following simulation parameters unless it stated otherwise: each reference U i is surrounded by M interference users that are located at different radial distances 3 , i.e., d ik = {8; 15; 22} m, from U i ; all device nodes are equipped with L = 16 antennas that produce a main lobe gain of G m = L = 16 in linear scale (or G m = 12 in dB scale) and a side lobe gain calculated using G s = 0.7746 = −1.1092 dB [3, Table I]; all channel links are LOS and interfering nodes hit the side lobes of the reference U i either by the main or side lobes; an equal ratio of main and side lobes of interferers; finally, the remaining simulation parameters are drawn in Table 1, which mostly follow the ones in [34], [65], [66]. The Figs. 2 − 7 and Figs. 9−15 are plotted as a function of transmit SNR by setting the AWGN noise variance to σ 2 i = −80 dBm as in [67], [68]. To obtain a full vision on the system performance, we study both fixed (in Figs. [2][3][4][5][6][7][8] and fair, that is, adaptive, (in Figs. 9-15) PA schemes. The selection of the fixed or fair PA schemes is a matter of the system preferences.

A. FIXED POWER ALLOCATION
In Fig. 2, we evaluate the sum-rate of normalized ergodic capacity both for the multi-user NOMA and multi-user OMA networks with N = 2, 3, 4 users for ideal system parameters (given by κ i = 0, σ 2 ei = 0 and ξ i = 0) in the presence of M = 24 i.n.i.d. interfering nodes. In this figure, we assume the fixed PA coefficients for the two-user NOMA network are set to α 1 = 0.8 and α 2 = 0.2 [34], whereas, for the three and four-user NOMA networks, the fixed power coefficients are evaluated as α i = 2 N −i P 2 N −1 [69]. Based on this figure, one could witness a minor advantage of the two-user NOMA network at the mid-SNR region as opposed to the N = 3 and N = 4 counterparts. To support this, the authors in 3 This geometry for interfering nodes is simulation-specific; however, the closed-form expressions for performance metrics are valid for any fixed location of i.n.i.d. interfering nodes. The users' distribution is beyond the scope of this paper.  [34] also demonstrated that the NOMA networks have the best performance for the two-user case. Therefore, we will carry out further numerical analysis for the two-user NOMA network. Similarly, the two-user OMA network also performs better than the three-and four-user OMA networks. From  Fig. 2, the overall ergodic capacity for the two-user NOMA network performs higher than the two-user OMA network.
For example, at the transmit SNR of 50 and 60 dB, the sumrate of ergodic capacity for the NOMA network achieves 5.55 and 8.96 bits/s/Hz, whereas the OMA network supports 4 and 7 bits/s/Hz, respectively. In Fig. 3, we make a comparison of the normalized er- godic capacities between NOMA U 1 /U 2 and OMA 4 U 1 /U 2 users as the transmit SNR ranges from 20 to 40 dB in an ideal NOMA network with ideal transceiver hardware, perfect CSI/SIC, as well as under the presence of AWGN and interference, i.e., κ i = 0, σ 2 ei = 0, ξ i = 0, and M = 24. From the NOMA results, one can notice that U 2 outperforms U 1 which can be explained by the SIC implementation at U 2 . At the same time, U 1 experiences saturation above the transmit SNR of 25 dB since the signal of U 2 contributes to the SINDR of U 1 as an additional noise power, and it has the more substantial effect at higher SNR values. Compared to the OMA users, the NOMA ones are characterized by higher ergodic capacity, i.e., NOMA U 2 outperforms OMA U 2 by at least 1 bits/s/Hz while NOMA U 1 outperforms OMA U 1 up to 32.5 dB. However, when the transmit SNR increases further, OMA U 1 starts outperforming NOMA U 1 mainly due to the interference caused by NOMA U 2 . Fig. 3 also presents the plots for the lower and upper bounds for the ergodic capacity for NOMA U 1 /U 2 . The lower and upper bounds of ergodic capacity are tight to ergodic capacity. Therefore, these formulas could be alternatively used to analyze the ergodic capacity accurately.
In Figs. 4-7, we study each system impairment (i.e., RTHI, imperfect CSI/SIC) and interference separately from the rest to quantify their effect on the ergodic capacity of the considered NOMA-based system given a fixed PA method between NOMA users.  rise above 37 dB, the ergodic capacities become sensitive to RTHI. It appears that U 2 is more susceptible to higher κ i values. On the one hand, the ergodic capacity degradation from the perfect hardware case to κ i = 0.1 case is 23% and κ i = 0.2 is 49.48%, accordingly, at SNR = 50 dB for U 2 .
On the other hand, the ergodic capacity diminishes for ideal hardware at κ i = 0.1 to 6.575% and at κ i = 0.2 to 8.44% for U 1 given SNR = 50 dB. Due to the proportionality of the RTHI level to transmit power and the fact that the power level of U 2 is higher than U 1 , the RTHI level has more effect on U 2 . In Fig. 4, we also plot the asymptotic ergodic capacity results when the transmit SNR approaches infinity, i.e., high-SNR approximation. In this case, RTHI creates an ergodic capacity ceiling. For instance, when κ i = 0.1, U 1 is bounded by 2.3 bits/s/Hz and U 2 by 4.4 bits/s/Hz, correspondingly. In Fig. 5, we present the ergodic capacity performance versus transmit SNR for different imperfect CSI variance values, σ 2 ei = {0, 005, 0.01, 0.05}. Both U 1 /U 2 operate at similar ergodic capacity rates when the transmit SNR is below 40 dB. However, it is observed that U 2 is more influenced by erroneous CSI as opposed to U 1 . This matter can also explained by the proportionality of CSI error to the transmit power of U 2 as it was studied above in the discussion of Fig.  4. One can observe that the erroneous CSI starts deteriorating the performance of U 2 for the transmit SNR above 55 dB. Even a small growth of the error variance, i.e., from the ideal case to σ 2 ei = 0.005, bounds the normalized ergodic capacity at 7.3 bits/s/Hz, whereas at the ideal CSI case for SNR = 75 dB the ergodic capacity reaches to 11.7 bits/s/Hz. Likewise, the ergodic capacity is bounded at 6.1 bits/s/Hz for σ 2 ei = 0.01. In addition, considering the performance of U 1 , there is ∼0.8 bits/s/Hz difference between the ideal CSI and erroneous CSI plots when the transmit SNR is above 50 dB.
In Fig. 6, we study the impact of imperfect SIC on the performance of U 2 . Therefore, the SIC factor varies as ξ i = When the transmit SNR is below 40 dB, there is no obvious evidence of the negative effect of imperfect SIC on the ergodic capacity performance. However, when the SNR goes beyond 40 dB, a detrimental effect of imperfect SIC becomes apparent and puts the following ergodic capacity bounds: ξ i = {0.005, 0.01, 0.05} correspond to the capacity ceilings given by {5.67, 4.7, 2.59} bits/s/Hz, respectively. Considering the ideal SIC case with 11.74 bits/s/Hz achievable at 75 dB, one can draw the following conclusions: ξ i = 0.005 plot shows 2 times, ξ i = 0.01 plot demonstrates 2.5 times, and ξ i = 0.05 plot results in 4.55 times performance degradation. This figure clearly shows that the error SIC has a significant impact on the normalized ergodic capacity.
In Fig. 7, we vary the number of interfering nodes as M = {0, 12, 24, 48} and set ξ i = 0, κ i = 0, and σ 2 ei = 0. From this figure, U 2 notably outperforms U 1 . Moreover, the increasing number of interfering nodes significantly degrades the ergodic capacity performance of U 2 at all SNR regions as opposed to the "no interference" case. When SNR is above 18 dB, the curve of M = 0 plot is higher at least to 6 bits/s/Hz in comparison to M = {12, 24, 48} case. Moreover, U 1 is saturated at 2.3 bits/s/Hz even in the "no interference" case due to the interference from U 2 . From this figure, one can note that interference is a crucial capacity limitation factor in the NOMA-based mmWave D2D mmWawe networks.
In Fig. 8, we plot the normalized ergodic capacity versus the PA coefficient, α 1 , for U 1 /U 2 at the transmit SNR values of 30, 40 and 50 dB and ξ i = 0, κ i = 0, σ 2 ei = 0, M = 24. From this figure, one can notice that the higher α 1 values increase the normalized ergodic capacity of U 1 , as opposed to U 2 that experiences the capacity degradation. Intersection points between U 1 /U 2 curves represent a fair value for α 1 . More precisely, at 30 dB, fair PA coefficients are given by α 1 = 0.8 and α 2 = 0.2. Similarly, the transmit SNRs of 40 dB and 50 dB correspond to the allocation coefficients α 1 = 0.84; α 2 = 0.16 and α 1 = 0.91; α 2 = 0.09, respectively. Therefore, it becomes obvious that the PA scheme needs adaptiveness to maintain some fairness among the users.

B. ADAPTIVE POWER ALLOCATION
In Figs. 9-15, we investigate the ergodic capacity under an optimal fairness-based PA scheme (please refer to Section VI) over all the SNR regions. Our goal is to compare the normalized ergodic capacity of NOMA users at the fair (optimal) and fixed (non-optimal) PA coefficients. If the fairness-based PA coefficients, α i , are adjusted based on the received powers of U 1 and U 2 , the fixed PA coefficients are set as α 1 = 0.8 and α 2 = 0.2.
In Fig. 9, we set ξ i = 0, σ 2 ei = 0, M = 24 and vary κ i = {0.05, 0.16, 0.25} for both fair and fixed PA schemes. From this figure, one can witness the lack of fairness in resource allocation for U 1 /U 2 in the fixed PA scheme, which leads to a considerable difference in their performance. For instance, U 2 grows as SNR increases at κ i = 0.05 and U 1 plots start saturating above 55 dB for κ i = {0.05, 0.16, 0.25} near 2 − 2.25 bits/s/Hz. However, when the distortion noise level increases to κ i = 0.16 for U 2 , the ergodic capacity saturates around 3.2 bits/s/Hz. Similarly, we notice that U 2 saturates at 2 bits/s/Hz for κ i = 0.25. When the fair PA scheme is applied, both users obtain the similar capacity performance, i.e., {4.3, 2.7, 2.1} bits/s/Hz, in the high-SNR regime for κ i = {0.05, 0.16, 0.25}, respectively. Both fair and fixed PA schemes obtain very similar normalized ergodic capacity sum-rate performances.
In Fig. 10, the individual normalized ergodic capacity is under study at ξ i = 0, κ i = 0, M = 24, and σ 2 ei = {0.001, 0.05, 0.1} for both fair and fixed PA schemes. As seen from this figure, the fair PA scheme provides even ergodic capacity performance for U 1 /U 2 as opposed to the fixed scheme.   [3], [52], we consider m i = 4 and τ = 2 for the LOS links and m i = 2 and τ = {3, 4} for the NLOS links. As expected, the LOS parameters play a crucial role in the ergodic capacity performance for both fixed and fair PA schemes, especially at low and medium SNR regions. The worst performance was achieved in the NLOS plots with m i = 2 and τ = 4, even a unit difference in PLE made a substantial distinction in the ergodic capacity in comparison to the case with m i = 2 and τ = 3. For instance, for U 2 , the τ = 3 plot reaches 2.5 bits/s/Hz at 60 dB, whereas the τ = 2 plot attains 0.1332 bits/s/Hz. Similarly, the τ = 3 and τ = 2 plots achieve 5.625 bits/s/Hz and 0.976 bits/s/Hz at 70 dB, respectively. Moreover, for U 2 , the LOS plot with m i = 4 VOLUME 4, 2016 In Fig. 12, we investigate the coverage probability versus the transmit SNR for the multi-user NOMA network with N = {2, 3, 4} users that apply the fair PA scheme and twouser OMA network given ξ i = 0, σ 2 ei = 0, κ i = 0, and M = 24. When P [i] c = 0.1, there is a 5 dB performance degradation from the NOMA N = 3 to N = 2 case, and similarly, there is 3.7 dB drop from the NOMA N = 3 to N = 4 scenario. It is imperative to note that a two-user NOMA network has shown the advantage over the three-/four-user NOMA network. From this figure, OMA U 1 shows a higher coverage probability performance than the optimized NOMA U i . However, optimized NOMA U i performs better than the OMA U 2 .
In Fig. 13 For the low rate requirements (up to 0.5 bits/s/Hz), the system obtains the same performance irrespective of the amount and type of considered impairments. However, at higher rate thresholds, one can observe that the coverage metric degrades fast and non-linearly (even for the ideal case) as R th increases. For instance, for the rate thresholds given by R th = {0.6, 0.8, 1} bits/s/Hz, the system with ideal settings achieves the coverage probability of P {0.8, 0.75, 0.56}, accordingly. Moreover, it is noted that the hardware imperfections have the most significant impact on the coverage metric. Imperfect SIC has less influence on the coverage probability. At the same time, it is apparent that the considered system model is more robust to the imperfect CSI compared to the other impairments.
Finally, in Fig. 15, we study the average throughput performance versus the transmit SNR given ξ i = 0, σ 2 ei = 0, M = 24, and κ i = {0, 0.1, 0.2, 0.3} for the two-user NOMA network. From this figure, we notice that the RTHI level greatly influences the average throughput that causes performance saturation at the SNR above 40 dB. Below 40 dB all the plots demonstrate similar average throughput performance. For instance, a plot with κ i = 0.1 saturates at average throughput, 3.4 bits/s/Hz, κ i = 0.2 saturates at 2.4 bits/s/Hz, and κ i = 0.3 saturates at 1.8 bits/s/Hz. In the mid-SNR region (at 55 dB), the ideal case results in 3.65 bits/s/Hz, κ i = 0.1 at 2.96 bits/s/Hz, κ i = 0.2 at 2.96 bits/s/Hz, and κ i = 0.3 at 1.7 bits/s/Hz of average throughput, correspondingly. Hence, there is a twice performance degradation from the ideal case to the κ i = 0.3 case. This figure justifies the significance of the RTHI impact on the high-rate systems.

VIII. CONCLUSION
In this work, we have studied the integrated model of 5G technologies as D2D communication and NOMA operating on mmWave frequencies by considering practical system limitations such as RTHI, imperfect CSI/SIC, and i.n.i.d. interference noises. These system imperfections inevitably limit the ergodic capacity performance of future communication systems; especially, interference has a strong influence on the ergodic capacity. Interference may be combated by increasing the number of antenna elements per device node, introducing additional costs to the network. The simulation results of this work carefully investigated each system im-pairment separately and proved the importance of practical system settings. We obtained mathematically tractable ergodic capacity formulas with their tight upper and lower bounds as well as asymptotic ergodic capacity expressions that provide valuable insights into the effect of each system impairment. Moreover, we have derived comprehensive coverage probability and average throughput formulas that enable further performance analysis of the proposed system model. The user-fairness-based PA scheme presented in this work provides a fair resource allocation for all NOMA users. .

APPENDIX A DERIVATION OF THE ERGODIC CAPACITY
The closed-form expression for ergodic capacity in (12) is evaluated by using (11) and (7), as shown in (31) Now, by using (22) and (23), we obtain the ergodic capacity formula in (12).

APPENDIX B DERIVATION OF THE LOWER ERGODIC CAPACITY
To compute the lower ergodic capacity, it is essential to find the expressions E{ln(a Moreover, E{ln Λ [i] + V [i] } is evaluated similarly to (34). Hence, the final expression for the lower ergodic capacity is displayed in (13).

APPENDIX C DERIVATION OF THE UPPER ERGODIC CAPACITY
Next, we evaluate the second component as By applying (35) and (36) into (10), we obtain the expression for the upper ergodic capacity in (14).

APPENDIX D DERIVATION OF THE ASYMPTOTIC ERGODIC CAPACITY AT HIGH-SNR REGIME
To evaluate the asymptotic ergodic capacity formula at high-SNR, regime the SINDR formula in (3) is revisited. Next, by using the definition of transmit SNR, SNR i = ρi σ 2 i , one could rewrite the SINDR in terms of SNR as where X [i] = |h i | 2 and η represents the interference term as η = M k=1 |g ik | 2 1 +κ 2 k ρ ik . Now, let us divide both the numerical and denominator parts in (37) by SNR i to obtain the simplified representation of the SINDR as First, we find the SINDR at high-SNR regime by taking the limit of the SINDR in (38) when the SNR tends to infinity as By applying the high-SNR approximated SINDR in (39) and by using Jensen's inequality, we derive the asymptotic ergodic capacity as The evaluation of the denominator Ξ in (40) is given as Derivation of the numerator part Ξ is obtained identically to (35) and written as E{α j X [i] } = α j β i m i . Hence, we present the final asymptotic ergodic capacity (i.e., capacity ceiling) in (15).

APPENDIX E DERIVATION OF THE COVERAGE PROBABILITY
The proof of Proposition 4 that presents the coverage probability for the proposed system model is calculated below ; otherwise, it equals to 0. The CDF of Gamma RV is given in (20).
Now, (42) may be evaluated as The integral in (44) is evaluated by using [51, (8.310.1)] and the closed-form expression for the coverage probability is obtained in (18).