# Selection Combining Scheme over Non-identically Distributed Fisher-Snedecor F Fading Channels

In this paper, the performance of the selection combining (SC) scheme over Fisher-Snedecor F fading channels with independent and non-identically distributed (i.n.i.d.) branches is analysed. The probability density function (PDF) and the moment generating function (MGF) of the maximum i.n.i.d. Fisher-Snedecor F variates are derived first in terms of the multivariate Fox's H-function that has been efficiently implemented in the technical literature by various software codes. Based on this, the average bit error probability (ABEP) and the average channel capacity (ACC) of SC diversity with i.n.i.d. receivers are investigated. Moreover, we analyse the performance of the energy detection that is widely employed to perform the spectrum sensing in cognitive radio networks via deriving the average detection probability (ADP) and the average area under the receiver operating characteristics curve (AUC). To validate our analysis, the numerical results are affirmed by the Monte Carlo simulations.

## Authors

• 9 publications
• 10 publications
• ### Performance of Cognitive Radio Systems over κ-μ Shadowed with Integer μ and Fisher-Snedecor F Fading Channels

In this paper, we analyze the performance of cognitive radio (CR) system...
07/25/2018 ∙ by Hussien Al-Hmood, et al. ∙ 0

• ### EVM analysis of an Interference Limited SIMO-SC System With Independent and Correlated Channels

In this paper, we derive the error vector magnitude (EVM) in a selection...
11/26/2018 ∙ by Sudharsan Parthasarathy, et al. ∙ 0

• ### Ratio of Products of Mixture Gamma Variates with Applications to Wireless Communications

In this paper, the statistical properties of the product of independent ...
07/17/2020 ∙ by Hussien Al-Hmood, et al. ∙ 0

• ### Entropy and Energy Detection-based Spectrum Sensing over F Composite Fading Channels

In this paper, we investigate the performance of energy detection-based ...
07/16/2018 ∙ by Seong Ki Yoo, et al. ∙ 0

• ### An Accurate Sample Rejection Estimator for the Estimation of Outage Probability of EGC Receivers

In this work, we evaluate the outage probability (OP) for L-branch equal...
03/13/2019 ∙ by Nadhir Ben Rached, et al. ∙ 0

• ### Asymptotic maximal order statistic for SIR in κ-μ shadowed fading

Using tools from extreme value theory (EVT), it is proved that the limit...
06/14/2018 ∙ by Athira Subhash, et al. ∙ 0

• ### Ergodic Capacity of Composite Fading Channels in Cognitive Radios with the Product of κ-μ and α-μ Variates

In this study, the product of two independent and non-identically distri...
12/12/2017 ∙ by He Huang, et al. ∙ 0

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## 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 signal-to-noise-ratio (SNR) at the receiver side. This is because it’s a non-coherent 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 non-identically 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 Fisher-Snedecor fading channel has been proposed as a composite of Nakagami-/inverse Nakagami- distributions to model device-to-device (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 Fisher-Snedecor fading channel are expressed in simple analytic functions. Furthermore, it includes Nakagami-, Rayleigh, and one-sided Gaussian as special cases. In addition, the Fisher-Snedecor fading channel can be utilised for both line-of-sight (LoS) and non-LoS (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. Fisher-Snedecor 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 Fisher-Snedecor fading channels were given in [8]. The product of multiple Fisher-Snedecor 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. Fisher-Snedecor variates have not been yet reported in the open literature. Motivated by this and based on the above observations, this paper derives exact analytic closed-form mathematically tractable of the PDF and the MGF of the maximum of i.n.i.d. Fisher-Snedecor 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. Fisher-Snedecor F Variates

The CDF of the received instantaneous SNR, , at th branch of a SC receiver over Fisher-Snedecor fading channel is expressed as [6, eq. (11)]

 Fγi(γ) =ΞmiiγmimiB(mi,msi)2F1(mi+msi,mi;1+mi;−Ξiγ) (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

 Fγi(γ)= ΞmiiγmiΓ(mi)Γ(msi) ×H1,22,2[Ξiγ∣∣∣(1−mi−msi,1),(1−mi,1)(0,1),(−mi,1)] (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. Fisher-Snedecor distribution. Thus, the PDF of is given as

 fγ(γ)=(L∏i=1ΞmiiΓ(mi)Γ(msi)) ×γΩ−1H0,1:[1,2]i=1:L1,1:[2,2]i=1:L[Ξ1γ,⋯,ΞLγ∣∣∣(−Ω;{1}i=1:L)(1−Ω;{1}i=1:L)∣∣∣ [(1−mi−msi,1),(1−mi,1)]i=1:Li=1:L] (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 built-in 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]

 Fγ(γ)=L∏i=1Fγi(γ) (4)

Substituting (2) into (4), yielding

 Fγ(γ) =L∏i=1ΞmiiγmiΓ(mi)Γ(msi) ×H1,22,2[Ξiγ∣∣∣(1−mi−msi,1),(1−mi,1)(0,1),(−mi,1)] (5)

After using the definition of the single variable Fox’s -function [11, eq. (1.2)], (5) can be expressed in multiple Barnes-type closed contours as

 Fγ(γ)=(L∏i=1ΞmiiΓ(mi)Γ(msi))1(2πj)L∫U1⋯∫UL {L∏i=1Γ(ui)Γ(mi+msi−ui)Γ(mi−ui)Γ(1+mi−ui)}Ξ−u11⋯Ξ−uLL γ∑Li=1mi−uidu1⋯duL (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

 fγ(γ)=(L∏i=1ΞmiiΓ(mi)Γ(msi))1(2πj)L∫U1⋯∫UL {L∏i=1Γ(ui)Γ(mi+msi−ui)Γ(mi−ui)Γ(1+mi−ui)}Ξ−u11⋯Ξ−uLL Γ(1+∑Li=1mi−ui)Γ(∑Li=1mi−ui)γ∑Li=1mi−ui−1du1⋯duL (7)

With the help of [11, eq. (A.1)], (7) can be written in exact closed-form expression as in (3), which completes the proof.

###### Proposition 2

The MGF of , , is given as

 Mγ(s)=(L∏i=1ΞmiiΓ(mi)Γ(msi))1sΩ ×H0,1:[1,2]i=1:L1,0:[2,2]i=1:L[Ξ1s,⋯,ΞLsγ∣∣∣(−Ω;{1}i=1:L)−∣∣∣ [1−mi−msi,1−mi]i=1:Li=1:L] (8)
###### Proof:

The MGF can be calculated by plugging (6) into where denotes the Laplace transform. Hence, we have

 Mγ(s)=(L∏i=1ΞmiiΓ(mi)Γ(msi))1(2πj)L∫U1⋯∫UL {L∏i=1Γ(ui)Γ(mi+msi−ui)Γ(mi−ui)Γ(1+mi−ui)}Ξ−u11⋯Ξ−uLL sL{γ∑Li=1mi−ui;−s}du1⋯duL (9)

The Laplace transform in (8) is recoded in [10, eq. (3.381.4)]; thus, can be derived as

 Mγ(s)=(L∏i=1ΞmiiΓ(mi)Γ(msi))1(2πj)L∫U1⋯∫UL {L∏i=1Γ(ui)Γ(mi+msi−ui)Γ(mi−ui)Γ(1+mi−ui)}Ξ−u11⋯Ξ−uLL Γ(1+∑Li=1mi−ui)s∑Li=1mi−uidu1⋯duL (10)

Again, with the aid of [11, eq. (A.1)], (8) is deduced and the proof is accomplished.

## Iii Performance of SC over Non-Identically Distributed Fisher-Snedecor F Fading Channels

Due to the space limitations, the following unified framework can be utilised

 P=∫∞0P(γ)fγ(γ)dγ (11)

where and are the average and the conditional of the performance metric, respectively.

Substituting (7) into (11), we have

 P=(L∏i=1ΞmiiΓ(mi)Γ(msi)) 1(2πj)L∫U1⋯∫ULΓ(1+Ω−∑Li=1ui)Γ(Ω−∑Li=1ui) {L∏i=1Γ(ui)Γ(mi+msi−ui)Γ(mi−ui)Γ(1+mi−ui)}Ξ−u11⋯Ξ−uLL ∫∞0γΩ−∑Li=1ui−1P(γ)dγIdu1⋯duL (12)

### Iii-a Average Bit Error Probability

The ABEP can be evaluated by [1]

 Pe=∫∞0Q(√2ργ)fγ(γ)dγ (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

 I =12√π∫∞0γΩ−∑Li=1ui−1H2,01,2[ργ∣∣∣(1,1)(0,1),(0.5,1)]dγ (a)=ρ−Ω+∑Li=1uiΓ(Ω−∑Li=1ui)Γ(0.5+Ω−∑Li=1ui)Γ(1+Ω−∑Li=1ui) (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

 Pe=(L∏i=1ΞmiiΓ(mi)Γ(msi))12√πρΩ ×H0,1:[1,2]i=1:L1,0:[2,2]i=1:L[Ξ1ρ,⋯,ΞLρ∣∣∣(0.5−Ω;{1}i=1:L)−∣∣∣ [(1−mi−msi,1),(1−mi,1)]i=1:Li=1:L] (15)

### Iii-B Average Channel Capacity

According to Shannon theory, the ACC, , can be computed by

 ¯C=Bln2∫∞0ln(1+γ)fγ(γ)dγ (16)

where is the bandwidth of the channel.

Inserting (7) in (16), of (12) for becomes

 I =Bln2∫∞0γΩ−∑Li=1ui−1ln(1+γ)dγ (b)=Bln2Γ(1−Ω+∑Li=1ui)[Γ(Ω−∑Li=1ui)]2Γ(1+Ω−∑Li=1ui) (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

 ¯C=(L∏i=1ΞmiiΓ(mi)Γ(msi))Bln2H1,1:[1,2]i=1:L1,1:[2,2]i=1:L[Ξ1,⋯,ΞL∣∣∣ (1−Ω;{1}i=1:L)(1−Ω;{1}i=1:L)∣∣∣[(1−mi−msi,1),(1−mi,1)]i=1:Li=1:L] (18)

It can be noted that (18) reduces to [15, eq. (18)] for .

### Iii-C ED with SC over Fisher-Snedecor F fading conditions

#### Iii-C1 Average Detection Probability

The ADP can be evaluated by [9, eq. (9)/eq. (4)]

 ¯Pd=∫∞0Qu(√2γ,√λ)fγ(γ)dγ (19)

where is the threshold value, stands for the time-bandwidth product and is the generalized Marcum -function.

It can be observed that the generalized Marcum -function can be expressed as

 Qu(√2γ,√λ) (c1)=1−e−γ2u+12γu−12∫λ0xu−12e−x2Iu−1(√2γx)dx (c3)=1−π(λ2)ue−γ1(2πj)2∫R1∫R2Γ(u−r1−r2)Γ(1+u−r1−r2) Γ(r1)Γ(r2)Γ(0.5+r2)Γ(u−r2)Γ(0.5−r2)(λ2)−r1(λγ2)−r2dr1dr2 (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 th-order. 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

 I=1−π(λ2)ue−γ1(2πj)2∫R1∫R2Γ(u−r1−r2)Γ(1+u−r1−r2) Γ(r1)Γ(r2)Γ(0.5+r2)Γ(u−r2)Γ(0.5−r2)(λ2)−r1(λγ2)−r2 ∫∞0γΩ−∑Li=1ui−r2−1e−γdr1dr2 (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 closed-from computationally tractable expression in terms of a single variable Fox’s -function.

#### Iii-C2 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)]

 ¯A=∫∞0A(γ)fγ(γ)dγ (23)

where is the AUC at the instantaneous SNR.

The is given as [9, eq. (35)]

 A(γ)=1−u−1∑k=0k∑l=0(k+u−1k−l)12k+l+ul!γle−l2 (24)

where denotes the binomial coefficient.

Substituting (24) and (7) into (23) and invoking , we have of (12) as

 I=1−u−1∑k=0k∑l=0(k+u−1k−l)12k+l+ul! ×∫∞0γl+Ω−∑Li=1ui−r2−1e−l2dγ (25)

Utilising [10, eq. (3.381.4)] to evaluate the integral of (25) and plugging the result in (12), we have a closed-form 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. Fisher-Snedecor 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. Fisher-Snedecor 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 missed-detection () versus the probability of false alarm for and dB111Here, 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 no-diversity 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 Fisher-Snedecor 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 non-identically distributed branches. To be specific, the ABEP, the ACC, the ADP, and the AUC of ED technique were obtained in exact mathematically tractable closed-form 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.

## References

• [1] M. K. Simon and M.-S. Alouini, Digital Communications over Fading Channels. New York: Wiley, 2005.
• [2] P. S. Bithas, P. T. Mathiopoulos, and S. A. Kotsopoulos, Diversity reception over generalized-K (KG) fading channels, IEEE Trans. Wireless Commun., vol. 6, no. 12, pp. 4238-4243, Dec. 2007.
• [3] J. Paris, Statistical characterization of shadowed fading, IEEE Trans. Veh. Technol., vol. 63, no. 2, pp. 518-526, Feb. 2014.
• [4] H. Al-Hmood, and H. S. Al-Raweshidy, On the sum and the maximum of non-identically distributed composite

/gamma variates using a mixture gamma distribution with applications to diversity receivers,

IEEE Trans. Veh. Technol., vol. 65, no. 12, pp. 10048-10052, Dec. 2016.
• [5] H. Al-Hmood, Performance Analysis of Energy Detector over Generalised Wireless Channels in Cognitive Radio. PhD Thesis, Brunel University London, 2015.
• [6] S. K. Yoo, S. L. Cotton, P. C. Sofotasios, M. Matthaiou, M. Valkama, and G. K. Karagiannidis, The Fisher-Snedecor distribution: A simple and accurate composite fading model, IEEE Commun. Lett., vol. 21, no. 7, pp. 1661-1664, March 2017.
• [7] O. S. Badarneh, D. B. Da Costa, P. C. Sofotasios, S. Muhaidat, and S. L. Cotton, On the sum of Fisher-Snedecor variates and its application to maximal-ratio combining, IEEE Commun. Lett., vol. 7, no. 6, pp. 966-969, Dec. 2018.
• [8] O. S. Badarneh, S. Muhaidat, P. C. Sofotasios, S. L. Cotton, K. Rabie and D. B. da Costa, The NFisher-Snedecor cascaded fading model, in Proc. IEEE WiMob, Oct. 2018, pp. 1-7.
• [9] S. K. Yoo et al., Entropy and energy detection-based spectrum sensing over composite fading channels, IEEE Trans. Commun., 2019, pp. 1-1.
• [10] I. S. Gradshteyn, and I. M. Ryzhik, Table of Integrals, Series and Products, 7th edition. Academic Press Inc., 2007.
• [11] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications. Springer, 2009.
• [12] H. Chergui, M. Benjillali, and M.-S. Alouini, (2018) Rician -factor-based analysis of XLOS service probability in 5G outdoor ultra-dense networks, [Online]. Available: https://arxiv.org/abs/1804.08101
• [13] O. S. Badarneh and F. S. Almehmadi, Performance analysis of -branch maximal ratio combining over generalised

fading channels with imperfect channel estimation,

IET Commun., vol. 10, no. 10, pp. 1175-1182, July 2016.
• [14] The Wolfram Functions Website. (Last accessed April 2019).
• [15] S. K. Yoo et al., A comprehensive analysis of the achievable channel capacity in composite fading channels, IEEE Access, vol. 7, pp. 34078-34094, March 2019.
• [16] H. Al-Hmood, Performance of cognitive radio systems over shadowed with integer and Fisher-Snedecor fading channels, in Proc. IEEE IICETA, May 2018, pp. 130-135.