Extended α-η-μ Fading Distribution: Statistical Properties and Applications

In this paper, the impact of the non-linearity of the propagation medium on the extended <inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> fading distribution is studied. In particular, the extended <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> fading model in which the parameter <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> represents the non-linearity of the propagation environment is presented via providing the exact and asymptotic of the statistical properties, namely, the probability density function (PDF), cumulative distribution function (CDF), and generalized-moment generating function (G-MGF). To this effect, exact closed-form mathematically tractable expressions of the outage probability (OP), average symbol error probability (ASEP), amount of fading (AoF), channel quality estimation index (CQEI) and effective rate (ER) are obtained. The asymptotic behaviour at high average signal-to-noise (SNR) values is also analysed to gain further insights into the influence of the index <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> on the performance metrics of the wireless communication systems. Moreover, the average channel capacity (ACC) under different adaptive transmission techniques, such as, optimum rate and adaptation (ORA), capacity of the channel with inversion and fixed rate (CIFR), and truncated inversion and fixed rate (TIFR) are derived. The validation of the derived expressions is verified via comparing the numerical results with the Monte-Carlo simulations as well as some previous works for different scenarios.


I. INTRODUCTION
The κ-µ, η-µ [1], and α-µ [2] distributions have been widely utilised to model the multipath fading of the wireless communications channels. This is because these fading models unify most of the classical distributions, namely, Rayleigh, Nakagami-m, and Rician, but with better fitting to the realistic measurements.
Recently, several models of the η-µ fading that is used for non-line-of-sight (NLoS) communications scenario have been reported in the technical literature. For instance, the statistical characterization of the composite η-µ/ Gamma fading channel was derived in [3]. The probability density function (PDF) and cumulative distribution function (CDF) The associate editor coordinating the review of this manuscript and approving it for publication was Xiaofan He . of both the signal envelope and instantaneous signal-to-noise ratio (SNR) of α-η-µ distribution were given in [4]. The performance of the wireless digital communications systems over α-η-µ fading model in terms of the outage probability (OP), average symbol error probability (ASEP) for various modulation schemes, and average channel capacity (ACC) was studied in [5] and [6]. The average probability of energy detection based spectrum sensing over α-η-µ fading channels was derived in [7]. Moreover, the effective rate (ER), i.e. capacity, in α-η-µ fading condition was given in [8]. The authors of [9] analysed the physical layer security over α-η-µ fading scenario in terms of the lower bound secure outage probability (SOP). In [10], the ACC of a mobile user over α-η-µ fading channel was investigated via employing the random waypoint (RWP) mobility approach. The composite of α-η-µ/Gamma and α-η-λ-µ/Gamma distributions were VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ investigated in [11] and [12], respectively, with applications to the ACC with optimum rate adaptation (ORA) and constant transmit power, channel inversion with fixed rate (CIFR), and truncated inversion and fixed rate (TIFR) adaptive transmission protocols. The α-η-κ-µ fading model was suggested in [13] as a unified representation for both α-η-µ and κ-µ distributions. Additionally, the main advantage of this model is manifested in providing better fitting to the empirical data in a mmWave propagation medium than Rayleigh, Rice, • Deriving the PDF, CDF, and generalised-MGF (G-GMF) of both the envelope and instantaneous signal-to-noise ratio (SNR) over extended α-η-µ fading distribution in novel exact closed-form mathematically acceptable formats. To the best of the authors' knowledge, there is no effort in the state-of-the-art has been dedicated to study the effect of the non-linearity of the propagation media on the extended η-µ distribution.
• Providing the asymptotic expressions of the derived statistics at high average SNR regime to get more insights into the influence of the fading indices on the system behaviour.
• Using the above statistical properties, the performance of the wireless communication systems over extended α-η-µ fading channel is analysed in terms of the OP, ASEP, AoF, channel quality estimation index (CQEI) and ER. To this effect, both the exact and asymptotic expressions of the performance metrics are novel, simple and computationally tractable.
• Studying the ACC over extended α-η-µ fading channels with different adaptive transmission strategies. In particular, the average capacity of channel with ORA and constant transmit power, CIFR, and TIFR transmission strategies are studied.
Organization: Section II provides some fundamental information about the extended η-µ fading distribution. In Section III, the the principle work of the proposed α-η-µ fading distribution is explained as well as the statistical properties of this model are given. Section IV analyses the performance of wireless communications systems in terms of the OP, ASEP, AoF, and ER. In this section, the ACC under ORA, CIFR, and TIFR transmission techniques are also derived. Section V presents the Monte Carlo simulations and numerical results for different scenarios. Section VI highlights some conclusion remarks about the proposed model. Notations

II. THE EXTENDED η-µ FADING MODEL
The received signal envelope for the extended η-µ fading distribution, R Eηµ , is given by [17] where µ x and µ y are the number of the multipath clusters of the in-phase and quadrature components, respectively, whereas X 2 l and Y 2 l that are mutually independent Gaussian RVs with zero mean, namely, E[X l ] = E[Y l ], represent the in-phase and quadrature components of the cluster l, respectively.
The PDF of R Eηµ , f R Eηµ (r), is expressed as [17, eq. (14)] where µ = (µ x + µ y )/2 stands for the total number of the multipath clusters,r 2 = E[R 2 Eηµ ] is the average power of the fading envelope, the parameter p indicates the clustering imbalance between the in-phase and quadrature components, Similar to the η-µ fading model, the extended η-µ distribution encompasses two formats. In Format I, p represents the ratio between the number of the multipath clusters of the components, namely, p = µ x /µ y with µ x = 2µp/(1 + p) and µ y = 2µ/(1 + p). Moreover, η is the ratio of the powers of the in-phase and quadrature components, i.e., η = On the other side, in Format II, p stands for the normalized difference of the number of the multipath clusters of the in-phase and quadrature components, namely, that means η represents the normalized difference of the powers of the multipath clusters with x = (1 + η)r 2 and It is worth mentioning that Format I and Format II are related to each other by η I = (1 + η II )/(1 − η II ) and p I = (1 + p II )/(1 − p II ). Hence, Format I is used in this work.

III. THE EXTENDED α-η-µ FADING DISTRIBUTION
In this section, the extended α-η-µ fading distribution is proposed via adding the impact of the non-linearity of the propagation medium that is represented by the parameter α. Interestingly, this distribution comprises several models as particular cases, such as, α-µ and extended η-µ distributions. The former model is obtained via inserting p = η or p = 1 and a specific value for η whereas the latter arises by making α = 2. In addition to that, the extended α-µ fading model that has not been yet reported by the previous works, can be also considered as special case of the α-η-µ fading distribution. All the special cases of the α-η-µ fading and the corresponding values of their fading parameters p, α, η, and µ are described in Table 1.
A. PDF OF THE EXTENDED α-η-µ DISTRIBUTION Theorem 1: Let α, η, µ, p,r α , r ∈ R + where r is a random variable (RV) that denotes the instantaneous signal power of the fading envelope, R, of the extended α-η-µ distribution. Accordingly, the PDF of envelope, R, can be expressed as where Proof: The envelope R can be written as Comparing (1) with (5), one can deduce that Now, let's define the received signal power of the fading envelope of the extended η-µ fading as Employing the transformation of the RV of (7) in (2), this yields The instantaneous signal power of the envelope, r, under α-η-µ distribution can be found using To compute E[χ
Lemma 2: To avoid the infinite series expressions of the CDF and G-MGF as well as the performance metrics, the PDF of (12) can be rewritten in terms of the product of two univariate FHFs as where (ϕ, δ, β) = (p, 1, p − η) for η < p whereas (ϕ, δ, β) = (1, p/η, η − p) for η > p. 1 Proof: The exponential function can be written in terms of the FHF via recalling the identity [23, eq. (1.39)] In addition, the confluent hypergeometric function can be expressed in terms of the FHF via invoking the property [23, eq. (1.130)] .
It is worth noting that (17) is not applicable for η > p because the integral of the FHF converges when the arguments are positive. Consequently, the property that is given in (16) can be used after recalling the identity 1 One can notice that both (17) and (18) can be unified in a single format with three different parameters as in (14) and this completes the proof. Fig. 1 illustrates the PDF of the envelope of extended α − η −µ fading with α = 4.5 and µ = 3.5. From this figure, one can see that the increasing in α from 2 (extended η-µ) to 4.5 (extended α-η-µ) leads to increase the PDF as well as shift it rightwards which means better fading scenario. In the same context, when p decreases from 3 to 0.3, the PDF reduces and shifts leftwards. Fig. 1 also demonstrates some special cases of the extended α-η-µ fading distribution that are explained in Table 1.
The CDF of R, F R (r), and the CDF of γ , F γ (γ ), under extended α-η-µ fading model can be written in exact closed-form expression as given in (19) and (20), respectively, shown at the bottom of the next page.
The asymptotic of the CDF of (20) atγ → ∞, F ∞ γ (γ ), is expressed as Proof: Following the same steps in Lemma 2 for (3) and making use of the definition of the FHF [23, eq. (1.2)] that is given in (22) shown at the bottom of the page in which T represents the suitable contour in the t-plane from -i∞ to +i∞ with is a constant value, we have (23) as shown at the bottom of the page. Now, inserting (23) in F R (r) = r 0 f R (r)dr and changing the order of the integrals, the linear integral can be easily evaluated. Thereafter, doing some mathematical manipulations via using the identity (x + 1) = x (x) [ Plugging (23) in f γ (γ ) = r 2 /γ γ f R ( γr 2 /γ )/2, we have (24), as shown at the bottom of the page. Then, substituting (24) into F γ (γ ) = γ 0 f γ (γ )dγ and employing the same procedure of (19), (20) is derived.   Fig. 2 where the increasing in α and/or p would lead to diminish the CDF which refers to better fading condition. Also, a comparison with [2] and [17] for α-µ and extended η-µ, respectively, is also carried out in Fig. 2. As it can be observed, the envelope CDF of (19) is in an excellent matching with the results of the previous works [2] and [17] which verifies the correctness of our derived CDF.  (25) shown at the bottom of the page.

IV. PERFORMANCE EVALUATION OF WIRELESS COMMUNICATIONS SYSTEMS OVER EXTENDED α-η-µ FADING MODEL A. OUTAGE PROBABILITY
The probability of reducing the output SNR to less than a certain threshold value, ψ, is called the outage probability. Consequently, the OP, P o , can be evaluated by [26, eq. (1.4)] where F γ (.) is presented in (20). The asymptotic of the OP, P ∞ o , can be deduced from (21)
Similarly, the asymptotic expression of the ASEP for the non-coherent detection schemes at highγ values,P N ,∞ s , can be derived from (30) and (26) after inserting k = 0 and s = a 2 . Hence, one can observe that G d is also proportional to α and µ where G d = αµ/2 and G c = where A 1 and A 2 that represent the modulation parameters are given in Table 2 performing some mathematical simplifications, this yields From (35), it is clear that G d = αµ/2 and G c =

C. AMOUNT OF FADING AND CHANNEL QUALITY ESTIMATION INDEX
The AoF, AF, can be computed by [26, eq. (1.27)] where E[γ ] and E[γ 2 ] that represent the first and second moments of γ , respectively, can be deduced from (25) via inserting k = 2 and k = 1, respectively, along with s = 0. Thus, after plugging (18) in (27) Alternatively, (37) can be rewritten in terms of the hypergeometric function via recalling the identity [23, eq. (1.132)] Consequently, this yields The CQEI, CQEI, that is defined as an effective metric that gives insights on the AoF of at a specific values of SNR, can be expressed as [29] CQEI = AF The ER of wireless communications systems is measured under imperfect quality of service (QoS) where several restrictions such as system delay and block duration are taken into account [30]. The ER, R is given by [30, eq. (8)] where A θ T B/ln(2) with θ represents the delay exponent whereas T and B indicate the time and bandwidth of the channel, respectively. The ER over extended α − η − µ fading model can be derived via inserting (24) in (41)  The asymptotic behaviour of ER at highγ , R ∞ , can be analysed by where (43) arises after substituting (13) into (41) and making use of [22, eq. (3.194.3)].

E. AVERAGE CHANNEL CAPACITY UNDER DIFFERENT ADAPTIVE TRANSMISSION PROTOCOLS 1) OPTIMUM RATE ADAPTATION WITH CONSTANT TRANSMIT POWER
The normalized ACC under ORA strategy,C ORA , can be determined by [31, eq. (20)] Substituting (20)   , φβ .

3) TRUNCATED INVERSION AND FIXED RATE
The normalized ACC with TIFR,C TIFR , can be calculated by [31, eq. (85)] where γ 0 is a certain threshold value that is selected to achieve maximumC TIFR .
The integral of (47) can be rewritten as where It can be observed that I 1 can be deduced from (37) or (39) with k = −1. In addition, I 2 can be derived via following the same procedure of the CDF in Theorem 2 to yield (49) given at the bottom of the page.

V. NUMERICAL AND SIMULATION RESULTS
In this section, the numerical results for different scenarios are presented to understand the impact of the fading parameters α and p of the extended α-η-µ fading distribution on the performance metrics of the wireless communications systems. In addition, to verify the validation of the derived expressions, the analytical results are compared with the Monte Carlo simulations that are generated via using 10 7 realizations. To achieve further validations for our derived expressions, several comparisons with some special cases of extended α-η-µ fading, such as, η-µ and Nakagami-2.5, that were investigated in the previous works, have been carried out. 3 In all figures, the markers stand for the numerical results, whereas the solid and dashed lines denote their simulation and asymptotic counterparts, respectively. Moreover, in this work, the parameters η and µ are assumed to be constant where η = 3 and µ = 2.5. This is because the influence of these parameters has been widely studied in the literature (please refer to [1], [3], [6], [8], [11], and [17]).
Figs. 3 and 4 demonstrate the OP with ψ = 5 dB and ASEP for DBPSK modulation, respectively, versus average SNR,γ , over extended α − η − µ fading model for arbitrary values of α and p. As expected, both figures explain that the OP and ASEP diminish when α or/and p increase. This is because the increasing in α leads to high power exponent of the sum of multipath components. On the other side, the increasing in p means the number of the multipath clusters of the in-phase components is higher than that of the quadrature components of the same signal. For instance, in Figs. 3 and 4, when α changes from 1.5 to 2.5 at p = 1 (fixed) andγ = 15 dB (fixed), the OP is reduced by approximately 94% and .  the ASEP is decreased by nearly 95%. In the same context, at constantγ = 15 dB and α = 2.5, the OP and ASEP for p = 3 are roughly 27% and 30%, respectively, less than that of p = 1. However, the effect of α on the performance measures is more pronounced of that of p. This confirms the correctness of our derived expressions of the diversity gain for both the OP and ASEP that is proportional to α and µ, namely, G d = αµ/2. Hence, the overall system performance greatly degrades with the dropping in the value of α and vice versa. Fig. 5 depicts the exact and asymptotic results of the ASEP for various modulation schemes versus average SNR,γ , for α = 1.5 and p = 0.3. As anticipated and widely presented in the technical literature, the C-BPSK scheme outperforms the other formats. On the contrary, the 8-QAM technique has  higher ASEP than that of 4-FSK, NC-BFSK, C-BFSK, and DBPSK schemes. Fig. 6 plots the CQEI versus average SNR,γ , over extended α − η − µ fading for different values of α and p.
As it can noted, in this figure, that the CQEI drops when α or/and p increase. This is because the AoF diminishes as α or/and p become high. Fig. 7 shows the exact and asymptotic results of the ER versus average SNR,γ , over extended α − η − µ fading for A = 3.5 and arbitrary numbers of α and p. Similar to Figs. 3 and 4 as well as for the same reasons, the ER improves when α or/and p become large. For example, at p = 1 (fixed) andγ = 15 dB (fixed), the ER for α = 2.5 is roughly increased by 50% when compared to α = 1.5. In the same context, the ER for p = 3 is nearly improved by 2.8% when compared to p = 1 at constant α = 2.5.
It is worth mentioning that the influence of p on the ACC with different transmission strategies has not been studied in [17] and [21]. Hence, Fig. 8 illustrates the normalized ACC with ORA, TIFR, and CIFR transmission techniques over extended α − η − µ fading model for α = 1.5, p = 0.05, and p = 1. From this figure, it is clear that as p increases, the ACC becomes high and for the same reason that has been shown for the results of Figs. 3 and 4. Furthermore, the ORA has the superiority on the TIFR which is better than CIFR. These observations are consistent with the results of [3] and [12] which prove the correctness of the provided expressions in this work.
For further insights about the proposed fading model, Figs. 3, 4, 6, and 7 also include the results for η-µ, Nakagami-2.5, and Rayleigh fading channels that are special cases of the extended α − η − µ fading model as previously explained in Table 1. From these figures, it can be noted that the metrics over the extended α − η − µ fading condition with α = 2.5 have better performance than the Nakagami-2.5 fading channel. This refers to the low value of α of the Nakagami-2.5 fading condition (α = 2). This provides another confirmation about the influence of the non-linearity of the propagation media on the system behaviour. Besides, one can see that the system has worse performance over Rayleigh fading channel. This is because the number of the multipath clusters of the Rayleigh fading channel that arrives at the receiver is 1 (µ = 1) whereas µ = 2.5 for the extended α − η − µ fading model and Nakagami-2.5. Therefore, the performance metrics over Rayleigh fading channel have lower diversity gain in comparison with the other scenarios. Also, it is obvious, in all figures, that the numerical results are in excellent matching with the asymptotic behaviour at high average SNR,γ , as well as the Monte Carlo simulations.

VI. CONCLUSION
This paper was devoted to analyse the impact of the non-linearity propagation environment on the extended η-µ fading distribution. Accordingly, the extended α-η-µ fading was proposed as a new fading model via deriving the exact and asymptotic expressions of the PDF, CDF, and G-MGF of both the signal envelope and instantaneous SNR. Based on these statistics, exact closed-form mathematically tractable expressions of the OP, ASEP for different modulation formats, AoF, CQEI, and ER were provided. Moreover, the asymptotic expressions at high average SNR of the OP, ASEP, and ER were also given. Additionally, the analysis of the ACC with ORA, CIFR, and TIFR transmission protocols was presented. From the presented results, a degradation in the performance metrics can be noticed when α or/and p reduce. However, the parameter α has higher impact than p and this has been proved by the diversity gain which is proportional to α. A comparison between the numerical results and Monte Carlo simulation was performed to verify the validation of the derived expressions. The derived expressions of this work can be applied for number of fading channels that are special cases of the extended α-η-µ fading model. For instance, the performance analysis of the wireless communications systems over Nakagami-m fading channel with better fitting to the practical data than the classical model can be obtained by using the extended α-η-µ fading condition with the fading parameters of Table 1. In addition, the ACC with different adaptive power transmission policies over extended η-µ fading that has not been yet analysed in the literature, can be studied by utilising the expressions of Section IV.E after inserting α = 2.