Ratio of products of mixture gamma variates with applications to wireless communications systems

The fading scenario of many realistic wireless communication transmission systems, such as, multi-hop communications and spectrum sharing in cognitive radio networks (CRNs), can be modelled by the product and the ratio of the product of the random variables (RVs) of the channel distribution. However, there is no work has been investigated in the literature to provide uniﬁed statistics of the product and the ratio of the products that can be used for a wide range of non-composite and composite fading conditions. Accordingly, in this paper, the statistical properties, namely, probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) of the product and the ratio of the product of independent and non-identically distributed (i.n.d.) mixture Gamma (MG) RVs are derived. A MG distribution has been widely employed to approximate with high accuracy most of the conventional fading models, for example, Rayleigh, Nakagami-m , Nakagami-q (Hoyt), and Nakagami-n (Rician) as well as the generalised composite fading channels, such as, generalised-K ( K G ) , 𝛼 − 𝜇 /gamma, 𝜅 − 𝜇 /gamma, and 𝜂

presence of keyholes [3], and vehicle-to-vehicle (V2V) communications [5].In addition, the performance of wireless communications systems in the presence of co-channel interference (CCI) [6], physical layer security (PLS) [7], spectrum sharing in cognitive radio networks [9], [10] can be investigated by the ratio of products of the variates.
Motivated by the above, many efforts have been devoted in the open technical literature to derive the distribution of the IET Commun.2021; 15:1963-1981.wileyonlinelibrary.com/iet-com1963 product of RVs using different models.For instance, in [11], the statistical properties, namely, the probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF), of the product of N independent and non-identically distributed (i.n.d.) Nakagami-m variates were derived.In [12], the PDF of the product of independent Rayleigh variates was approximated to be expressed in terms of the elementary functions, such as power and exponential.Simple approximate expressions for the PDF and CDF of the cascaded independent Rayleigh, generalised Gamma (GG), Nakagami-m, Gaussian, and Weibull fading channels were given in [13].The statistical characterisations of the product of i.n.d.squared generalised-K (K G ) variates were presented in [3] with applications to the outage probability (OP) and average bit error probability (ABEP) over cascaded fading channels.
On the other hand, several distributions of the ratio of products of RVs have been analysed by the previous works.For example, the performances of the maximal ratio combining (MRC) diversity reception and multi-carrier code-division multiple access (MC-CDMA) system in the presence of an interference were studied in [14] by using the statistics of independent and identically distributed (i.i.d.) exponential variates.In [15], the ratio of products of both Gamma and Weibull RVs were analysed.The cascaded Weibull distribution was reported in [16].The distribution of some mathematical operations, such as, products and ratios, of arbitrary H -function variates that can be used as a unified representation for a large number of fading channels, was derived in [17].
Recently, the product of the RVs of Fisher-Snedecor  [7], [9], generalised  − ,  − , and  − , [18][19][20][21][22], fluctuating two-ray (FTR) [23], and Beaulieu-Xie [24] distributions has been given a special attention by many research articles.This is because these distributions give better fitting to the practical data and provide various composite multipath/shadowing fading conditions.Moreover, they approximately include all the well-known classic fading channels, such as, Rayleigh, Nakagami-m, Rician, exponential distributions as special cases.Additionally, the  −  fading channel is used to model the non-linear environment of communication systems whereas the  −  and the  −  fading channels are employed to represent the line-of-sight (LoS) and the non-LoS (NLoS) scenarios, respectively.Besides, the FTR distribution has been widely utilised to model the fading channel of the millimeterwave (mmWave) communication systems.Hence, the statistics of i.n.d.cascaded Fisher-Snedecor  fading channel which is composite of Nakagami-m/inverse Nakagami-m distributions, were studied in [7] and [9] in terms of the Meijer's G -function and applied in the analysis of the OP and ABEP of amplifyand-forward (AF)-relaying communication systems.The PDF, CDF, and MGF of the product of two i.n.d. −  and  −  variates and their applications to double and composite fading channels were reported in [18] and [19], respectively.Two different expressions of the statistics of non-identical cascaded  −  fading channels were given in [20] and [21] in terms of the Fox's H -function (FHF) and utilised in the analysis of the OP and ABEP, and the average secrecy capacity (ASC), lower bound of secure outage probability (SOP L ) and probability of non-zero secrecy capacity (PNSC) of the PLS, respectively.The product of two envelopes of  − ,  − , and  −  distributions were investigated in [22] via providing infinite series expressions and closed-form results in terms of the multivariate FHF.The distribution of the cascaded FTR variates with non-identical shaping parameters was investigated in [23].The fundamental statistics of two i.n.d.Beaulieu-Xie RVs were provided in [24] in terms of double infinite series as well as Meijer's G -function.
On the other side, the statistics of the ratio of the product of FTR RVs with non-identical shaping indices was derived in [6] and utilised in the analysis of the OP of decode-and-forward (DF) relaying system.The statistics of the distribution of the ratio of the product of arbitrarily distributed N Fisher-Snedecor  variates were presented in [7] and [9] with applications to PLS and CRNs, respectively.The ratio of two  −  RVs was given in [10] and used in the analysis of the outage and delay-limited capacities of the spectrum sharing in CRNs.The results of [10] was then extended by [25] to include multivariate of  −  distribution.Similar to the mathematical results of [22], the statistics of the ratio of two envelopes taken from  − ,  − , and  −  RVs were given in [26] and employed in the study of the PNSC of the PLS over different non-identical fading conditions of V2V communications.In [27], the product and the ratio of products of non-identically distributed GG variates were expressed in terms of the FHF.
Based on the previous studies that are summarised in Table 1, the statistical properties of the distributions of the product and the ratio of the products of an i.n.d.mixture Gamma (MG) variates that have not been yet investigated in the literature, are derived in this article.A MG distribution has been widely employed to approximate with high accuracy most of the conventional fading models, for example, Rayleigh, Nakagami-m, Nakagami-q (Hoyt), and Nakagami-n (Rician) as well as the generalised composite fading channels, such as, generalised-K (K G ),  − /gamma,  − /gamma, and  − /gamma.To this end, unified simple closed-form mathematically tractable expressions of the fundamental statistics of the product and the ratio of the products that can be used for a wide range of noncomposite and composite fading conditions of the realistic wireless communication transmission systems are obtained.Unlike the Gamma [14],  [15], GG [27], and exponential-GG (EGG) [28] distributions, a MG distribution can give low mean-square error (MSE) and Kullback-Leibler (KL) divergence between the exact and the approximated models (please refer to [ [30], Figures 1 and 2]).Additionally, the statistical characterisations of a MG distribution are simpler than that of a mixture of Gaussian (MoG) approach.Consequently, a MG distribution has been widely utilised as a highly accurate approximation for large numbers of fading models [30][31][32][33][34][35][36][37].
The main contributions of this paper are summarised as follows  • The equivalent parameters of a MG distribution for Beaulie-Xie [38] and  −  −  −  shadowed fading channels [39] that are not available in the open literature, are derived.It is worth mentioning that the statistics of these fading channels can not be obtained in exact closed-from due to including the PDF of a single RV on the modified Bessel function of the first kind.Moreover, the using of an infinite series of the modified Bessel function would lead to basic statistics in terms of the multiple infinite series that are converged slowly and unsteadily.These observations were presented in [24] where the PDF, CDF, and MGF of the products of two Beaulie-Xie RVs were included double infinite series.• Capitalising on the above statistics, the OP, ABEP, average symbol error probability (ASEP) for different modulation schemes, and the effective rate (ER) of communications systems and the average area under the receiver operating characteristics (AUC) of energy detection (ED) over nonidentical cascaded fading channels are given in closed-form Organization: Section 2 provides some related works to a MG distribution based fading channel model of wireless communications systems.In Section 3.1, the statistical properties of the product of MG variates are derived, whereas in Section 3.2, the statistics of the ratio of products of MG RVs are given.The equivalent parameters of a MG distribution for Beaulie-Xie and  −  −  −  shadowed fading are provided in Section 4. Based on the results of Section 3.1 and Section 3.2, some performance measures are derived in Section 5 and Section 6, respectively.The numerical results are presented in Section 7. Finally, some conclusions are highlighted in Section 8.

RELATED WORKS TO MG DISTRIBUTION
Motivated by the advantages of a MG model that have been mentioned previously, this distribution has been used by several efforts to represent the fading channels of the wireless communication systems.However, there is no work has been investigated in the literature to provide unified statistics of the product and the ratio of the products that can be used for a wide range of non-composite and composite fading conditions.Accordingly, in this paper, the statistical properties of the product and the ratio of the product of i.n.d.MG RVs are derived.
Table 2 summarises some of the most relevant works related to using a MG distribution in modelling a fading channel, applications in the realistic wireless communication systems, evaluation tools, and performance metrics.For instance, in [30], the OP, ABEP, average channel capacity (ACC), and performance of ED over Nakagami-m,  − , and  −  and other fading channels were analysed.The average detection probability (ADP) and average AUC curve of ED over  − /Gamma,  − /Gamma, and  − /Gamma fading channels were investigated in [31].Furthermore, unified frameworks of the channel capacity under different transmission protocols and the ER of wireless communications systems over  −  − /Gamma fading conditions were presented in [32] and [33], respectively.The performance of the PLS over MG distribution based channel model of both the main and wire-tap channels were analysed in [34] and [35].The sum and the maximum of i.i.d.K G and i.n.d. − /gamma fading channels were derived in [36] and [37], respectively, using a MG distribution with applications to the selection combining (SC) and the MRC diversity receptions.

PRODUCT AND RATIO OF PRODUCTS OF MG VARIATES
The PDF of the instantaneous signal-to-noise ratio (SNR) at i th path,  i , using a MG distribution is expressed as [ [30], equation (1)] where L i stands for the number of terms of l th Gamma component with parameters  l i ,  l i , and  l i .According to [30], the minimum number of terms, L i , that can provide a good matching between the approximate and exact distributions can be computed by using the MSE between the PDF of these distributions.
With the help of [ [41], equation (1.112)], ( 9) can be written as in (2) which completes the proof of the PDF.
The CDF of Y can be deduced after substituting (2) into The MGF of Y can be obtained via inserting (2) in  Y (s) = ∫ ∞ 0 e −sy f Y (y)dy and using [ [41], equation (2.29)] which finishes the proof.□

Ratio of products of i.n.d. MG variates
MG-distributed RVs.Accordingly, the PDF, CDF, and MGF of X are, respectively, deduced as Proof.To compute the Mellin transform for both Y and Z , the methodology that is given in Theorem 1 is used.Thereafter, plugging the result in Substituting ( 13) into [ [43], equation (1.21)], the PDF of X can be expressed as With the aid of the definition of Meijer's G -function [ [41], equation (1.112)], ( 14) is obtained as shown in (10) and the proof is accomplished.
Following the similar steps of deriving ( 3) and ( 4) that are provided in Theorem 1, the CDF and the MGF of X can be deduced as given in (11) and (12), respectively, which completes the proof.□

Beaulieu-Xie fading channels
The Beaulieu-Xie fading model is proposed as a simple representation for multiple specular and diffuse scatter components via introducing a special scale for the non-central chidistribution [38].Moreover, this model unifies the non-central chi,  − , and generalized Rician distributions where the relationship between the latter and Beaulieu-Xie distributions is the same as that between the Nakagami-m and Rayleigh models.Additionally, the Beaulieu-Xie distribution is related to the Nakagami-m model in a similar relationship that is between the Rician and Rayleigh fading models [38].
The PDF of  i , f  i (), over Beaulieu-Xie fading channel can be derived after employing [ [38], equation ( 4)] and performing simple change of variables.Thus, this yields where γi is the average SNR, m i is the fading parameter,  i controls the location and the height of the PDF, and is the modified Bessel function of the first kind and m i − 1 order.When m i = 1, the Beaulieu-Xie fading model reduces to Rician distribution with factor K i =  i ∕Ω i where Ω i controls the spread of the PDF whereas the Rayleigh fading is obtained after plugging m i = 1 and K i = 0.With the aid of the identity [ [42], equation (8.445)], the modified Bessel function of (15) can be written as Plugging ( 16) in ( 15), we have It can be noted that the infinite series of ( 17) can be approximated to the number of terms, L i , that satisfies the required accuracy of the MSE.Accordingly, by matching the PDF of (17) with that of (15), the equivalent parameters of a MG for the Beaulieu-Xie fading channel are expressed as

𝜶 − 𝝀 − 𝜼 − 𝝁 Shadowed fading channels
The  −  −  −  shadowed fading channel is a composite model of  −  −  −  and Nakagami-m distributions.The  −  −  −  was proposed by [39] as a generalised distribution that unifies the  − ,  − , and  −  fading models.Hence, this fading model can be employed to represent the non-linear medium and the non-line-of-sight (NLoS) environment of the wireless communications.Furthermore, the  −  −  −  shadowed can be used to model a wide range of composite multipath/shadowed fading conditions, such as,  − /Nakagami-m,  − /Nakagami-m, and  − /Nakagami-m as explained in Table 3.However, the PDF of the  −  −  −  distribution is included a modified Bessel function which would lead to express the results in terms of non-analytical mathematically complicated functions or include an infinite series.Hence, a MG distribution is used in this effort to approximate the PDF of the composite  −  −  − /Nakagami-m fading channel.
The PDF of  i , f  i (), over  −  −  −  fading channel is given as [ [39], equation (10)] where with ) , The fading parameters are defined as follows. i stands for the non-linearity parameter,  i denotes the correlation coefficient between the quadrature components and in-phase scattered waves,  i indicates the ratio between the power of the quadrature and in-phase scattered components, and  i represents the real extension of the multipath clusters.
The PDF of Nakagami-m distribution is expressed as where m i refers to the shadowing severity index.
According to [ [19], equation ( 4)], the PDF of the product of two RVs can be evaluated by Substituting ( 19) and ( 20) into ( 21), this obtains ) dr. ( Using the substitution z =  i r  i ∕2 into (22), this yields where With the help of a Gaussian-Laguerre quadrature approximation, the integration in (23) , where w l i and z l i are, respectively, the weight factors and abscissas defined in [44].Consequently, ( 23) can be rewritten using (1) with the following parameters

Outage probability
The OP is an important performance metric of the wireless communications systems operating over fading channels and its defined as the probability that the output SNR drops below a predefined threshold value  th [45].Consequently, the OP, P o , of the AF-relay system that is demonstrated in Figure 1, can be calculated by [ [45], equation (1.4)] where F Y (.) is the CDF of the product of i.n.d.MG variates that is given in (3).

Average bit/symbol error probability
In this section, the ABEP and the ASEP for several modulation schemes over cascaded fading channels that are represented by a MG distribution, are derived in exact unified closedfrom expressions.

Proposition 1. The ABEP of coherent BFSK, BPSK, and BFSK with minimum correlation, PC b , over cascaded fading conditions is obtained as
Proof.Plugging (4) in (27) and using the change of the variable x = sin 2  with some mathematical manipulations, we have With the aid of the definition of the Meijer's G -function [ [41], equation (1.112)] and the Fubini's theorem that is applied to interchange the order of the linear and closed integrations, (29) becomes It can be noted that the inner integral of ( 30) can be computed as  28) is yielded and this completes the proof.□

] is the bivariate Meijer's G -function defined in [
Proof.After performing simple mathematical manipulations, (32) can be rewritten as (34) One can see that  1 can be evaluated by following the same methodology of the Proposition 1 as provided in the first term of (33).
For  2 , we substitute ( 4) in (34) and assume x = sin 2 , to obtain Using the definition of the Meijer's G -function [ [41], equation (1.112)] and the Fubini's theorem, the following integral is deduced The inner integral of ( 36) can be expressed as where  (37) and substituting the result into (36) to yield With the aid of [ [46], equation ( 10)],  2 can be written in exact closed-form as shown in the second term of (33) which completes the proof.□
It can be noted that  1 can be calculated by using the same steps of deriving PC b in (28), namely, Proposition 1 as shown in the first term of (42) given at the top of this page.In addition,  2 can be derived by following the same procedure that is employed to derive  2 of (34) after inserting M = 4. Consequently, PMQAM s is obtained in exact closedfrom expression as shown in (42) given at the top of this page.

Effective rate of wireless communications systems
The ER has been proposed to measure the performance of the wireless communications systems under the quality of service (QoS) constraints, such as system delays, that have been neglected by Shannon theorem [48].Hence, this performance metric has been widely analysed over differ-ent fading channels in the open technical literature (see [33] and [49] and references therein).However, there is no work has been achieved to study the ER over cascaded fading channels which is one of our contributions in this paper.
The ER, R, can be calculated by [ [33], equation ( 1)] where A ≜ TB∕ ln 2 with , T , and B are, respectively, the delay exponent, block duration, and bandwidth of the system and {.} stands for the expectation.

AUC of energy detection-based spectrum sensing
The ED technique has been widely employed to perform the spectrum sensing in both cognitive radio (CR) and ultra-wide band (UWB) systems.This refers to its low implementation intricacy where the unlicensed user, namely, secondary user (SU), does not need a prior information about the licensed user (PU) [30], [31], [49,50] as demonstrated in Figure 2. To study the behaviour of an ED, the receiver operating characteristics (ROC) curve which plots the ADP versus the false alarm probability is utilised in many efforts.But, in some cases, this performance metric does not provide a clear result on the superiority of one system on the other.This is due to the intersection between the two compared curves in specific values of ADP and false alarm probability.Based on this observation, the authors in [50] proposed the average AUC curve as another performance metric of an ED via measuring the total area under the ROC.
The average AUC, Ā, can be evaluated by [ [49], equation where A(y) is the AUC in additive white Gaussian noise (AWGN) environment which is expressed as [ [49], equation ] ) denotes the binomial coefficient.The average AUC over cascaded fading conditions can be obtained after plugging (2) and ( 48) in (47) and invoking the fact that ∫ ∞ 0 f Y (y)dy ≜ 1 and [ [41], equation (2.29)].Accordingly, we have To the best of the authors' knowledge, ( 49) is new and there is no work has been devoted to study the average AUC over cascaded fading channels.

Outage probability of multi-hop wireless communications system with CCI
In this section, the OP of the multi-hop wireless communications system with DF relaying and interferer at each hop is analysed.In this system that is shown in Figure 3, the number of hops is K and the signal-to-interference ratio at the input of r- where  1,1 r ,  1,2 r , and  2,1 r are the RVs that represent the impact of the fading, the shadowing, and the co-channel interference, respectively [6].
According to [ [6], equation (23)], the OP of K -hop wireless communication systems, P (K ) o , can be computed by where  th is the threshold value and F X r (.) is the CDF that is given in (11) with N = 2 and M = 1.

Physical layer security
The performance of the PLS over different fading channel models has been given a special consideration by many works in the FIGURE 4 A wireless network consisting of a transmitter S (Alice) and a legitimate receiver D (Bob) in the presence of an eavesdropping attack E (Eve) [51] literature.This is because it is able to fortify the security of the wireless link [22].
In this work, we have assumed the transmitter (S) communicates with the legitimate receiver (D) via the main channel in the presence of an eavesdropper (E), as illustrated in Figure 4. Therefore, the channel state information (CSI) of D can be known by S whereas a perfect knowledge of the CSI of E cannot be assumed at S and hence information-theoretic security cannot be guaranteed.Moreover, both the main and wiretap channels are supposed to be independent and undergo quasi-static fading channels that are modelled by using a MG distribution.Also, we have assumed all S, D and E are equipped with a single received antenna [51].

6.2.1
Lower bound of secure outage probability One of the main performance measurements of the PLS is the secure outage probability (SOP) [22], [34], [35], [40].However, its difficult to derive the SOP in a unified exact closed-form computationally tractable expression [40], [51].Therefore, the lower bound of the SOP (SOP L ) is utilised in many studies in the analysis of the PLS.
The SOP L can be evaluated by [ [7], equation (39)] where is defined in Theorem 2,  D and  E are the received SNR at D and E, respectively, F X (.) is given in (11), and  = 2 R th with R th denotes the target secrecy rate threshold.
In contrast to the SOP L of [ [35], equation (6)] that is given for a single path and [ [38], equation (27)] that is included an infinite series, (53) is derived in a simple unified closed-from expression and it can be applied for a wide range of the cascaded fading conditions.

Probability of non-zero secrecy cpacity
Another metric that is used to analyse the performance of the PLS is the PNSC which means the secrecy capacity is always positive.The PNSC can be readily obtained from (53) as follows PNSC = 1 − F X (1). (54)

Spectrum sharing in cognitive radio networks
As its explained in Section 5.4, in the CRNs, the SU can employ the spectrum of the PU when the interference between the transmitted signal by the SU transmitter (SU-Tx) and the PU receiver (PU-Rx).In this figure, the power gain of the instantaneous channel from the SU-Tx to the PU receiver (PU-Rx) and from the SU-Tx to the SU receiver (SU-Rx) are represented by Z and Y, respectively.
In this work, Z and Y are assumed to be independent with channel coefficients following a MG distribution and subjected to AWGN with zero mean and variance N 0 which stands for the single-sided AWGN.

Outage capacity
The outage capacity (OC), C o , is used to quantify the spectral efficiency over fading channels.It can be defined as the probability of the instantaneous capacity is less than a certain value of the transmission rate R 0 .Accordingly, for the spectrum sharing network that is shown in Figure 2, Q pk } where Pr .stands for the probability and Q pk is the peak received power limits at the PU-Rx side [9].Mathematically, C o can be computed by [ [9], equation (59)] where F X (.) is provided in (11) with N = 1 and M = 1, that is, 2,1 .

Delay-limited capacity
The delay-limited capacity, C d , can be defined as the maximum value of R 0 that is achievable over each of the fading blocks and where Q avg is the average interference power at the PU-Rx and [.] is the expectation operator.Substituting (10) into (56) with N = 1 and M = 1 and recalling [ [41], equation (2.9)], C d is yielded as

ANALYTICAL AND SIMULATION RESULTS
In this section, the numerical results of our derived expressions are compared with their Monte Carlo simulations that are obtained via running 10 6 iterations.In all figures, the minimum number of terms for i th path, L i , that satisfies MSE≤ 10 −5 is chosen to be 15.Additionally, all paths are assumed to be independent but not necessarily identically distributed.For further validations, in some figures, the results for the Rayleigh fading which is a special case of both the  −  −  −  shadowed and the Fisher-Snedecor  fading [9] via inserting m s → ∞ and m = 1 are compared.8 and 9 illustrate the ER with A = 2.5 and the average complementary AUC which is 1 −  for u = 3 versus the average SNR, respectively, and for the same fading parameters of Figure 1.As anticipated, the performance of the system decreases when the number of paths increases.This refers to the increase in the impact of the channel parameters on the transmitted signal which becomes high when N is large.For instance, in Figure 5, at fixed γ = 15 dB, the values of the OP of cascaded Beaulieu-Xie and  −  −  −  shadowed fading channels are reduced by approximately 44.89% and 43.84%, respectively, when N changes from 2 to 1.In the same context, at N = 3, the OP of Beaulieu-Xie fading condition is nearly 0.076 whereas for  −  −  −  shadowed fading channels is roughly 0.029.One can see that the results of Figure 5 have been affirmed by Figures 6 and 7 via carrying out a comparison between the ABEP of BPSK with the ASEP of 4PSK and 8QAM modulation techniques.Moreover, Figure 8 1 The results for format 2 can be easily deduced from format 1 via applying the bilinear transform where  format 2 = Additionally, in Figure 9, the average complementary AUC becomes low when the number of paths deceases.This is due to the increasing in both the missed-ADP and false alarm probability.respectively.Similar to Figure 5, the OP drops when the number of hops, K , reduces.This observation is matched with the results that were presented in [6]. in the Beaulieu-Xie fading channels,  = 0.5 for both D and E whereas for  −  −  −  shadowed fading, the parameters for both D and E are equal where  = 2.5,  = 0.1,  = 0.5, and  = 2.5.From both figures, a substantial degradation in the secrecy performance of the system can be noticed when the value of γE rises.This is because the large improvement in the wiretap channel, namely, S-E link in comparison with the main channel.For example, in Figure 11 and at constant Ψ = 10 dB, the values of the SOP L for γE = 3 dB are higher than that of γE = 0 dB by nearly 65.6% and 82.1% for Beaulieu-Xie and  −  −  −  shadowed fading channels, respectively.This situation is consistent with the results that  were obtained in [34], [35] and [40] but for the K G fading conditions.Figure 13 depicts the OC of the CRNs versus the average SNR for Q pk = 5, 10, and 15 dB, R 0 = 1 bits/Hz, N 0 = 1 W/Hz and the same fading parameters of Figure 11.The figure explains that the OC decreases when Q pk increases.This is because the increase in Q pk gives more flexibil-ity to the SU-Tx in transmitting the signals with higher power.
Figure 14 portrays the delay-limited capacity, C d , versus the average SNR for Q avg = 5, 10, and 15 dB with N 0 = 1 W/Hz.It can be observed that as Q avg increases, C d improves.
In all presented figures, one can observe that the numerical results are perfectly coincided with their simulation counterparts which confirms the correctness of our derived expressions.Moreover, the results of our work for the Rayleigh fading that are plotted in Figures 7, 8 and 14 are in excellent matching with the analysis of [9] which proves the validation of our provided statistical properties.

CONCLUSIONS
In this paper, the statistical characterisations s of the product of i.n.d.MG variates were derived first.Thereafter, the study was extended to include the PDF, CDF, and MGF of the ratio of products of arbitrary distributed MG RVs.Based on these statistics, several performance metrics of wireless communication systems were analysed over cascaded Beaulieu-Xie and  −  −  −  shadowed fading channels.In particular, the OP, ABEP, ASEP, ER, and average AUC of energy detection were provided in simple unified closed-form computationally tractable expressions.Furthermore, the OP of DF relaying of multi-hop wireless communication systems with CCI and the performance metrics of the PLS, namely, the SOP L and PNSC were given.The numerical results for different number of paths and hops as well as the average SNR of the eavesdropper were presented.The outage and delay-limited capacities of the spectrum sharing in CRNs were also studied.These results showed how the performance of the systems diminishes when the aforementioned parameters are increased.Moreover, a comparison between the results of some performance metrics over Rayleigh fading channel which is a special case of  −  −  −  shadowed fading and previous work, namely, [9] was carried out.The derived expressions of this work can be employed for all the fading channels of Table 2 that can be represented by an MG distribution.Additionally, the generalisation of Beaulieu-Xie and  −  −  −  shadowed fading conditions can be exploited to study the behaviour of the wireless communication systems over a variety of fading scenarios that are special cases of these channels, for example,  −  −  −  by plugging m → ∞.

FIGURE 3
FIGURE 3Multi-hop communication system by DF-relay nodes with CCI[6]

Figure 5
Figure5shows the OP of cascaded Beaulieu-Xie and  −  −  −  shadowed (format 1) fading channels with  th = 0 dB for three paths, namely, N = 1, 2, and 3, with (m 1 , m 2 , m 3 ) = (1, 2, 3) for both fading channels1 .In this figure,  1 =  2 =  3 = 0.5 for Beaulieu-Xie fading model, whereas  1 =  2 =  3 = 2.5,  1 =  2 =  3 = 0.1,  1 =  2 =  3 = 0.5, and  1 =  2 =  3 = 2.5 for  −  −  −  shadowed fading channel.For the same simulation parameters of Figure5, Figures6 and 7compare between the ABEP of BPSK with the ASEP of 4PSK and 8QAM modulation schemes over Beaulieu-Xie and  −  −  −  shadowed cascaded fading channels, respectively.Figures8 and 9illustrate the ER with A = 2.5 and the average complementary AUC which is 1 −  for u = 3 versus the average SNR, respectively, and for the same fading parameters of Figure1.As anticipated, the performance of the system decreases when the number of paths increases.This refers to the increase in the impact of the channel parameters on the transmitted signal which becomes high when N is large.For instance, in Figure5, at fixed γ = 15 dB, the values of the OP of cascaded Beaulieu-Xie and  −  −  −  shadowed fading channels are reduced by approximately 44.89% and 43.84%, respectively, when N changes from 2 to 1.In the same context, at N = 3, the OP of Beaulieu-Xie fading condition is nearly 0.076 whereas for  −  −  −  shadowed fading channels is roughly 0.029.One can see that the results of Figure5have been affirmed by Figures6 and 7via carrying out a comparison between the ABEP of BPSK with the ASEP of 4PSK and 8QAM modulation techniques.Moreover, Figure8

FIGURE 6 FIGURE 7
FIGURE 6 ABEP of BPSK and ASEP of 4PSK and 8QAM modulation schemes over cascaded Beaulieu-Xie fading channels

Figure 10
Figure 10 demonstrates the OP of DF relaying based multihop wireless communications systems with multiple interferers for  th = 0 dB.The analysis in this figure is provided for three types of i.n.d.hops which are K = 1 that represents a direct link, K = 2, and K = 3 via using the same fading parameters m for both channels, and , , and  of  −  −  −

FIGURE 10 Ψ
FIGURE 10 OP of DF relaying communication systems over Beaulieu-Xie and  −  −  −  shadowed fading channels with  th = 0 dB

TABLE 1
Survey of works related to product and ratio of products of RVs T w o  − ,  − , and  −  -√ Exact OP and ABEP [27] G G √ √ Exact -FIGURE 1 Cascaded fading channels constructed by AF-relay nodes [9] these unified statistics have not been yet achieved in the open technical literature.• Based on the above results, novel statistical properties of the ratio of the product of i.n.d.MG variates are provided.FIGURE 2 System model of spectrum sharing in CRNs [9]

TABLE 2
Survey of works related to mixture gamma distribution