I Introduction
To mitigate the impacts of the multipath fading and shadowing on the performance of wireless communications systems, diversity reception techniques have been used in the open technical literature. Selection combining (SC) approach has been considered as an efficient diversity scheme to improve the signaltonoiseratio (SNR) at the receiver side. This is because it’s a noncoherent combining technique where the branch with a high SNR is selected among many branches [1]. The statistical properties, namely, the probability density function (PDF), the cumulative distribution function (CDF), and the moment generating function (MGF), of the maximum of random variables (RVs) of the fading channels are widely employed to study the SC diversity [2][5]. In this context, the SC receivers over independent and nonidentically distributed (i.n.i.d.) generalized
fading channels was investigated in [2]. The authors in [3] studied the average bit error probability (ABEP) of SC technique with i.n.i.d. branches over shadowed fading channels. In [4], the PDF, the CDF, and the MGF of the maximum of /gamma RVs were derived and used in the analysis of average channel capacity (ACC) of wireless communications systems. Based on the results of [4], the behaviour of energy detection (ED) that is one of the most utilised spectrum sensing methods was analysed in [5] by providing unified expressions for the average detection probability (ADP) and the average area under the receiver operating characteristics (ROC) curve (AUC).More recently, the FisherSnedecor fading channel has been proposed as a composite of Nakagami/inverse Nakagami distributions to model devicetodevice (D2D) fading channels at 5.8 GHz in both indoor and outdoor environments [6]. In contrast to the generalised fading channel, the statistics of the FisherSnedecor fading channel are expressed in simple analytic functions. Furthermore, it includes Nakagami, Rayleigh, and onesided Gaussian as special cases. In addition, the FisherSnedecor fading channel can be utilised for both lineofsight (LoS) and nonLoS (NLoS) communications scenarios with better fitting to the empirical measurements than the generalised () fading model. The authors in [7] derived the basic statistics of the sum of i.n.i.d. FisherSnedecor RVs with applications to maximal ratio combining (MRC) receivers. The ADP and the average AUC of ED with square law selection (SLS) branches over arbitrarily distributed FisherSnedecor fading channels were given in [8]. The product of multiple FisherSnedecor RVs, namely, cascaded fading model, was addressed in [9].
To the best authors’ knowledge, the statistical characteristics of the maximum of i.n.i.d. FisherSnedecor variates have not been yet reported in the open literature. Motivated by this and based on the above observations, this paper derives exact analytic closedform mathematically tractable of the PDF and the MGF of the maximum of i.n.i.d. FisherSnedecor RVs. To this end, the performance of SC scheme is analysed by deriving the ABEP, the ACC, the ADP and the average AUC of ED in terms of the multivariate Fox’s function.
Ii The PDF and MGF of the Maximum I.N.I.D. FisherSnedecor Variates
The CDF of the received instantaneous SNR, , at th branch of a SC receiver over FisherSnedecor fading channel is expressed as [6, eq. (11)]
(1) 
where , for , , , , and stand for the multipath index, the shadowing parameter, the number of diversity branches, and the average SNR, respectively, is the beta function [10, eq. (8.380.1)] and is the Gauss hypergeometric function [10, eq. (9.14.1)].
Recalling the identity [11, eq. (1.132)] and performing some mathematical simplifications with the aid of [10, eq. (8.384.1)] and [10, eq. (8.331.1)], (1) can be equivalently rewritten as
(2) 
where is the gamma function and is the univariate Fox’s function defined in [11, eq. (1.2)].
Proposition 1
Let all RVs, , follow i.n.i.d. FisherSnedecor distribution. Thus, the PDF of is given as
(3) 
where and is the multivariate Fox’s function [11, eq. (A.1)]. An efficient MATLAB code that is readily implemented by [12] to compute the multivariate Fox’s function is used in this work. This because this function is not yet available as a builtin in the popular software packages such as MATLAB and MATHEMATICA.
Proof:
The CDF of the maximum i.n.i.d. variates can be computed by [1]
(4) 
Substituting (2) into (4), yielding
(5) 
After using the definition of the single variable Fox’s function [11, eq. (1.2)], (5) can be expressed in multiple Barnestype closed contours as
(6) 
where and is the th suitable contours in the plane from to with is a constant value.
Differentiating (6) with respect to to obtain , i.e. and then employing the identity [10, eq. (8.331.1)]. Thus, this yields
(7) 
With the help of [11, eq. (A.1)], (7) can be written in exact closedform expression as in (3), which completes the proof.
Proposition 2
The MGF of , , is given as
(8) 
Proof:
The MGF can be calculated by plugging (6) into where denotes the Laplace transform. Hence, we have
(9) 
The Laplace transform in (8) is recoded in [10, eq. (3.381.4)]; thus, can be derived as
(10) 
Again, with the aid of [11, eq. (A.1)], (8) is deduced and the proof is accomplished.
Iii Performance of SC over NonIdentically Distributed FisherSnedecor Fading Channels
Due to the space limitations, the following unified framework can be utilised
(11) 
where and are the average and the conditional of the performance metric, respectively.
Substituting (7) into (11), we have
(12) 
Iiia Average Bit Error Probability
The ABEP can be evaluated by [1]
(13) 
where is the Gaussian function presented in [1, eq. (4.1)] and represents the modulation parameter. For example, for binary phase shift keying (BPSK), while for binary frequency shift keying (BFSK).
Inserting (7) in (13) and invoking the identity [13, eq. (13)], of (12) is obtained as
(14) 
where follows [11, eq. (2.8)].
Next, plugging (14) in (12), performing some mathematical straightforward simplifications and using [11, eq. (A.1)], is obtained as
(15) 
IiiB Average Channel Capacity
According to Shannon theory, the ACC, , can be computed by
(16) 
where is the bandwidth of the channel.
Inserting (7) in (16), of (12) for becomes
(17) 
where follows after employing [10, eq. (4.293.10)] and making use of the properties [10, eq. (8.334.3)] and [10, eq. (8.331.1)].
Now, substituting of (17) into (12) and doing some algebraic manipulations, is yielded as follows
(18) 
It can be noted that (18) reduces to [15, eq. (18)] for .
IiiC ED with SC over FisherSnedecor fading conditions
IiiC1 Average Detection Probability
The ADP can be evaluated by [9, eq. (9)/eq. (4)]
(19) 
where is the threshold value, stands for the timebandwidth product and is the generalized Marcum function.
(22) 
(26) 
It can be observed that the generalized Marcum function can be expressed as
(20) 
where and arise after employing [1, eq. (4.60)] and then respectively utilising the properties [11, eq. (1.39)] and [14, eq. (03.02.26.0067.01)] for the exponential function and , which represents the modified Bessel function of the first kind and thorder. Using the definition of the univariate Fox’s function [11, eq. (1.2)] and solving the integral of , then follows in terms of the contour integral form where and are the suitable closed contours in the complex plane.
Now, plugging (7) and (20) into (19) and using the fact that , of (12) is deduced as follows
(21) 
Recalling [10, eq. (3.381.4)] for the inner integral of (21), substituting the result into (12) and making employ of [11, eq. (A.1)], then is obtained as shown on the top of this page.
In contrast to [9, eq. (14)] and [16, eq. (14)] that are derived for no diversity scenario in terms of the infinite series, (22) for can be obtained in exact closedfrom computationally tractable expression in terms of a single variable Fox’s function.
IiiC2 Average AUC
The average AUC is a single figure of merit that can be used in the analysis of performance of the ED when the plotting of the ADP versus the probability of false alarm, namely, ROC, doesn’t provide a clear insight into the behaviour of the system.
The average AUC, , can be calculated by [9, eq. (36)]
(23) 
where is the AUC at the instantaneous SNR.
The is given as [9, eq. (35)]
(24) 
where denotes the binomial coefficient.
Substituting (24) and (7) into (23) and invoking , we have of (12) as
(25) 
Utilising [10, eq. (3.381.4)] to evaluate the integral of (25) and plugging the result in (12), we have a closedform expression of as given on the top of this page.
Iv Analytical and Simulation Results
In this section, to validate our derived PDF and MGF of the maximum of i.n.i.d. FisherSnedecor variates, the ABEP, the ACC, the ADP, and the average AUC of SC diversity are analysed. The Monte Carlo simulations that are obtained via generating realizations for each RV are compared with the analytical results. In all figures, the multivariate Fox’s function has been evaluated by the MATLAB code that was implemented by [12]. Additionally, the solid lines corresponds to the simulations results whereas the markers represents the numerical results. Three scenarios of the shadowing impact, which are heavy, moderate, and light shadowing are studied by using , and , respectively.
Figs. 1, 2, and 4 illustrate the ABEP for BPSK, the normalised ACC, and the complementary AUC () with single receiver, dual, and triple SC branches over i.n.i.d. FisherSnedecor fading channels versus the average SNR per branch, , respectively, for different scenarios of the fading parameters. In the same context, Fig. 3 explains the complementary ROC, which plots the average probability of misseddetection () versus the probability of false alarm for and dB^{1}^{1}1Here, represents the upper incomplete gamma function [10, eq. (8.350.2)].. As anticipated, the performance of the communication systems becomes better when the SC diversity is employed and monotonically improves with the increasing in the number of diversity branches. The reason has been widely presented in the literature, which is the received average SNR of SC scheme is higher than the nodiversity and its increases when is used rather than . For comparison purpose, the scenario and that was studied in [7, Fig. 3], has been utilised here. As expected, the MRC diversity provides less ABEP than the SC branches but with high implementation complexity.
In all provided figures, the perfect matching between the numerical results and their Monte Carlo simulation counterparts can be observed, which confirms the validation of our derived expressions.
V Conclusions
In this paper, the PDF and the MGF of the maximum of not necessarily identically distributed FisherSnedecor RVs were derived in terms of the multivariate Fox’s function that has been widely used and implemented in the literature. These statistics were then employed to analyse the performance of SC diversity with nonidentically distributed branches. To be specific, the ABEP, the ACC, the ADP, and the AUC of ED technique were obtained in exact mathematically tractable closedform expressions. Comparisons of our results with previous works that were achieved by using a single receiver and MRC scheme as well as the numerical and simulation results for different scenarios have been carried out via using the same simulation parameters.
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