Al-Hmood, H

Al-Raweshidy, HS

IEEE Wireless Communications Letters

Brunel University Research Archive

IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, JUNE 2020 1Selection Combining Scheme over Non-identicallyDistributed Fisher-Snedecor F Fading ChannelsHussien Al-Hmood, Member, IEEE, and H. S. Al-Raweshidy, Senior Member, IEEEAbstract—In this paper, the performance of the selectioncombining (SC) scheme over independent and non-identicallydistributed (i.n.i.d.) Fisher-Snedecor F fading channels is anal-ysed. Accordingly, the probability density function (PDF) and themoment generating function (MGF) of the maximum of i.n.i.d.Fisher-Snedecor F variates are derived first. Based on thesestatistics, the exact expression and the asymptotic behaviour athigh average signal-to-noise ratio value of the average bit errorprobability (ABEP), the normalised average channel capacity(ACC), and the average area under the receiver operating char-acteristics curve (AUC) of the energy detection based spectrumsensing with i.n.i.d. SC diversity receivers are provided. Tovalidate our analysis, the numerical results are compared withthe Monte Carlo simulations.Index Terms—Selection combining, Fisher-Snedecor F fading,average error probability, channel capacity, energy detection.I. INTRODUCTIONSELECTION combining (SC) diversity reception has beenwidely used in the literature to improve the performance ofwireless communications systems. This is because it has lowimplementation complexity in comparison with the maximalratio combining (MRC) where the branch with a high signal-to-noise ratio (SNR) is selected among many branches [1].The probability density function (PDF), the cumulative dis-tribution function (CDF), and the moment generating function(MGF), of the maximum of random variables (RVs) are em-ployed to analyse the performance of the SC scheme [2]-[5].For instance, the behaviour of SC receivers over independentand non-identically distributed (i.n.i.d.) generalized-K (KG)fading was investigated in [2]. The average bit error probability(ABEP) of SC with i.n.i.d. branches over κ − µ shadowedfading was derived in [3]. In [4], the statistics of the maximumof η − µ/gamma RVs were provided with applications tothe average channel capacity (ACC). The average area underthe receiver operating characteristics (ROC) curve (AUC) ofenergy detection (ED) with SC branches was presented in [5].Recently, the Fisher-Snedecor F fading has been proposedas a composite of Nakagami-m and inverse Nakagami-m dis-tributions to model the device-to-device (D2D) fading channelat 5.8 GHz in both indoor and outdoor environments [6], [7]. Incontrast to the KG fading, the statistics of the Fisher-SnedecorF distribution are expressed in simple elementary functions.Manuscript received June 6, 2020; xxxxxx xxxxx xxxxx xxxxx.Hussien Al-Hmood is with the Electrical and Electronics EngineeringDepartment, College of Engineering, University of Thi-Qar, Iraq, e-mails:Hussien.Al-Hmood@brunel.ac.uk, Hussien.Al-Hmood@utq.edu.iq.H. S. Al-Raweshidy is with the Electronic and Computer EngineeringDepartment, College of Engineering, Design and Physical Sciences, BrunelUniversity London, UK, e-mail: Hamed.Al-Raweshidy@brunel.ac.uk.Furthermore, it includes Nakagami-m, Rayleigh, and one-sided Gaussian as special cases. The Fisher-Snedecor F fadingcan be utilised for both the line-of-sight (LoS) and the non-LoS (NLoS) communications scenarios with better fitting tothe empirical measurements than the KG. Consequently, theABEP of the MRC was analysed in [8] using the PDF of thesum of i.n.i.d. Fisher-Snedecor F variates. The behaviour of anED with square law selection (SLS) in i.n.i.d. Fisher-SnedecorF fading was studied in [9]. The distribution of the ratio ofproducts of Fisher-Snedecor F RVs was presented in [10].To the best of authors’ knowledge, no work has been yetdevoted in the literature to derive the statistical properties ofthe maximum of i.n.i.d. Fisher-Snedecor F variates. Motivatedby this, our main contributions in this paper are twofold.• We derive the exact and the asymptotic expressions athigh average SNR value of the PDF and the MGF of themaximum of i.n.i.d. Fisher-Snedecor F RVs.• Capitalising on the above, the performance of the ABEP,the ACC, and the average AUC of ED with i.n.i.d. SCdiversity receivers are analysed. To this end, mathemati-cally tractable closed-form expressions are obtained.II. FISHER-SNEDECOR F FADING CHANNELThe CDF of the instantaneous SNR at ith receiver, γi overFisher-Snedecor F fading is expressed as [7, eq. (12)]Fγi(γ) =Ξmii γmimiB(mi,msi)2F1(mi +msi ,mi; 1 +mi;−Ξiγ)(a1)= ΦiγmiH1,22,2[Ξiγ∣∣∣∣(1−mi −msi , 1), (1−mi, 1)(0, 1), (−mi, 1)](1)where Ξi = mi(msi−1)γ̄i, Φi =ΞmiiΓ(mi)Γ(msi ), mi is thereal extension of the number of multipath clusters, msi > 1stands for the shadowing severity parameter, B(., .) is thebeta function [11, eq. (8.380.1)], 2F1(., .; .; .) is the Gausshypergeometric function [11, eq. (9.14.1)], and Hm,np,q [.] is theFox’s H-function (FHF) defined in [12, eq. (1.2)]. Step (a1)is developed by applying the identities [12, eq. (1.132)] and[11, eq. (8.384.1)/ eq. (8.331.1)].III. MAXIMUM I.N.I.D. FISHER-SNEDECOR F VARIATESProposition 1: Let all RVs, γi ∼ F(mi,msi , γ̄i) fori ∈ {1, · · · , L} where L is the number of the variates,follow i.n.i.d. Fisher-Snedecor F distribution. Thus, the PDFof γ = max{γ1, · · · , γL} is derived as in (2) shown at the topof the next page. In (2), Φ =∏Li=1 Φi, Ω =∑Li=1mi andIEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, JUNE 2020 2fγ(γ) = ΦγΩ−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:L[(0, 1), (−mi, 1)]i=1:L](2)Mγ(s) =ΦsΩH0,1:[1,2]i=1:L1,0:[2,2]i=1:L[Ξ1s, · · · , ΞLs∣∣∣∣(−Ω; {1}i=1:L)−∣∣∣∣[(1−mi −msi , 1), (1−mi, 1)]i=1:L[(0, 1), (−mi, 1)]i=1:L](9)Pe =Φ2√πρΩH0,1:[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:L[(0, 1), (−mi, 1)]i=1:L](14)Hm,n:[mi,ni]i=1:Lp,q:[pi,qi]i=1:L[.] is the extended generalised bivariate Fox’sH-function (EGBFHF) [12, eq. (A.1)] that can be evaluatedby using an efficient MATLAB code of [13].Proof: The CDF of the maximum i.n.i.d. variates can becomputed via inserting (1) in Fγ(γ) =∏Li=1 Fγi(γ) [4, eq.(12)]. Accordingly, we haveFγ(γ) =ΦL∏i=1γmiH1,22,2[Ξiγ∣∣∣∣(1−mi −msi , 1), (1−mi, 1)(0, 1), (−mi, 1)](4)Recalling the definition of the FHF [12, eq. (1.2)], (4) canbe expressed in multiple Barnes-type closed contours asFγ(γ) =Φ(2πj)L∫U1· · ·∫UL{ L∏i=1Γ(ui)Γ(mi +msi − ui)Γ(mi − ui)Γ(1 +mi − ui)Ξ−uii}γΩ−∑Li=1 uidu1 · · · duL (5)where j =√−1 and Ui is the ith suitable contour in theu-plane from σi− j∞ to σi+ j∞ with σi is a constant value.Differentiating (5) with respect to γ to obtain fγ(γ), i.e.,fγ(γ) = dFγ(γ)/dγ and then invoking the identity Γ(1+x) =xΓ(x) [11, eq. (8.331.1)]. Thus, this yieldsfγ(γ) =Φ(2πj)L∫U1· · ·∫UL{ L∏i=1Γ(ui)Γ(mi +msi − ui)Γ(mi − ui)Γ(1 +mi − ui)Ξ−uii}Γ(1 + Ω−∑Li=1 ui)Γ(Ω−∑Li=1 ui)γΩ−∑Li=1 ui−1du1 · · · duL (6)With the help of [12, eq. (A.1)], (6) can be written in exactclosed-form expression as in (2) which finishes the proof.Since 2F1(., .; .; .) of (1) tends to unity when γ̄i →∞, theasymptotic of the CDF, FAsyγ (γ), can be expressed asFAsyγ (γ) ≈L∏i=1ΞmiimiB(mi,msi)γmi (7)Making use of fAsyγ (γ) = dFAsyγ (γ)/dγ, the PDF at highaverage SNR value is obtained asfAsyγ (γ) ≈ Ω( L∏i=1ΞmiimiB(mi,msi))γΩ−1 (8)Proposition 2: The MGF of γ, Mγ(s), is provided in (9)as shown at the top of this page.Proof: The MGF can be calculated by plugging (5) inMγ(s) = sL{Fγ(γ);−s} where L{.} represents the Laplacetransform operation. Hence, we haveMγ(s) =Φ(2πj)L∫U1· · ·∫UL{ L∏i=1Γ(ui)Γ(mi +msi − ui)Γ(mi − ui)Γ(1 +mi − ui)Ξ−uii}sL{γΩ−∑Li=1 ui ;−s}du1 · · · duL(10)Invoking [11, eq. (3.381.4)] to compute the Laplace trans-form of (10) and recalling [12, eq. (A.1)], (9) is deduced.Using (7) and following the same steps of the Proposition2, the asymptotic of the MGF, MAsyγ (s), is given byMAsyγ (s) ≈( L∏i=1ΞmiimiB(mi,msi))Γ(1 + Ω)sΩ(11)IV. PERFORMANCE ANALYSIS OF SC SCHEMEA. Average Bit Error ProbabilityThe ABEP can be evaluated by [1, eq. (9.11)]Pe =1π∫ π20Mγ(ρsin2 θ)dθ. (12)where ρ = 0.5, ρ = 1, and ρ = 0.715 for coherent binaryfrequency shift keying (BFSK), binary phase shift keying(BPSK), and BFSK with minimum correlation, respectively.Substituting (9) into (12) and employing [12, eq. (A.1)] andx = sin2 θ, the following inner integral is obtained∫ 10xΩ−∑Li=1 ui−12√1− xdx(a2)=Γ( 12 )Γ(12 + Ω−∑Li=1 ui)Γ(1 + Ω−∑Li=1 ui) (13)where (a2) follows [11, eq. (3.191.3)/ eq. (8.384.1)].Next, plugging the result of (13) and the remaining partsof (9) in (12) and invoking [12, eq. (A.1)], Pe is yielded asgiven in (14) at the top of this page.Inserting (11) in (12) and using x = sin2 θ as well asfollowing the same procedure of (13), the asymptotic of theABEP at high average SNR regime, PAsye , is expressed asPAsye ≈( L∏i=1ΞmiimiB(mi,msi))Γ(0.5 + Ω)2√πρΩ(15)After substituting Ξi = mi(msi−1)γ̄iinto (15) and approxi-mating the result to be PAsye ≈ γ̄−Gd where Gd denotes theIEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, JUNE 2020 3C̄ =Φln2H1,1:[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:L[(0, 1), (−mi, 1)]i=1:L](18)Ā =1− Φu−1∑k=0k∑l=0(k + u− 1k − l)12k−Ω+ul!×H0,2:[1,2]i=1:L2,1:[2,2]i=1:L[2Ξ1, · · · , 2ΞL∣∣∣∣(−Ω, {1}i=1:L), (1− l − Ω, {1}i=1:L)(1− Ω, {1}i=1:L)∣∣∣∣[(1−mi −msi , 1), (1−mi, 1)]i=1:L[(0, 1), (−m1, 1)]i=1:L](23)diversity order, one can observe that Gd is proportional to Land mi, i.e., PAsye ≈∏Li=1 γ̄−mii , whereas for independentand identically distributed (i.i.d.) branches, Gd = Lm. Theseobservations are consistent with [6] in which Gd = m forL = 1.B. Average Channel CapacityThe normalised ACC, C̄, can be computed by [7, eq. (20)]C̄ =1ln2∫ ∞0ln(1 + γ)fγ(γ)dγ (16)Plugging (6) in (16), we have the following inner integral∫ ∞0γΩ−∑Li=1 ui−1ln(1 + γ)dγ(a3)=Γ(1− Ω +∑Li=1 ui)[Γ(Ω−∑Li=1 ui)]2Γ(1 + Ω−∑Li=1 ui)(17)where (a3) obtains after using [11, eq. (4.293.10)] and recall-ing [11, eq. (8.334.3)/ eq. (8.331.1)].Substituting the result of (17) and the remaining terms of(6) into (16), C̄ is yielded as in (18) at the top of this page.Inserting (11) in [4, eq. (27)], the asymptotic of the nor-malised ACC at γ̄i →∞, C̄Asy, is derived asC̄Asy ≈M∑l=1wlxl ln 2[1−( L∏i=1ΞmiimiB(mi,msi))Γ(1 + Ω)xΩl](19)where M is the number of terms for the Gaussian-Laguerreintegration whereas xl and wl are the abscissas and the weightfactors, respectively [16].It is obvious that Gd of (19) is proportional to L and mifor i.n.i.d. case, whereas Gd = Lm for identical receivers.C. Average AUC of Energy Detection Based-Spectrum SensingThe AUC is a single figure of merit that evaluates the areaunder the ROC curve. Hence, the AUC is used in the analysisof the ED when the ROC curve doesn’t give a clear insightinto the behaviour of the system.The average AUC, Ā, can be calculated by [9, eq. (36)]Ā =∫ ∞0A(γ)fγ(γ)dγ (20)where A(γ) is given as [9, eq. (35)]A(γ) = 1−u−1∑k=0k∑l=0(k + u− 1k − l)12k+l+ul!γle−l2 (21)where u = TB is the time (T )-bandwidth (B) product and(ba)denotes the binomial coefficient.After plugging (21) and (6) in (20) and invoking∫∞0fγ(γ)dγ , 1, the following inner integral is obtained∫ ∞0γl+Ω−∑Li=1 ui−1e−l2 dγ(a4)= 2l+Ω−∑Li=1 uiΓ(l + Ω−L∑i=1ui)(22)where (a4) arises after utilising [11, eq. (3.381.4)].Substituting the result of (22) as well as the remaining partsof (6) and (21) into (20), we have a closed-form expressionof Ā as shown in (23) at the top of this page.Inserting (8) and (21) in (20) and with the aid of [11, eq.(3.381.4)], the asymptotic analysis of the AUC at high averageSNR regime, ĀAsy, is derived asĀAsy ≈( L∏i=1ΞmiimiB(mi,msi))×[1−u−1∑k=0k∑l=0(k + u− 1k − l)Γ(l + Ω)2k−Ω+ul!](24)Similar to (15) and (19), the diversity order, Gd, of (24)which is for arbitrary distributed branches, is proportional toL and mi, whereas Gd = Lm for i.i.d. case.V. ANALYTICAL AND SIMULATION RESULTSIn this section, to validate our analysis, the Monte Carlosimulations that are obtained via generating 107 realizationsfor each RV are compared with the analytical results. TheFisher-Snedecor F fading channel of SC scheme was gen-erated by the product of independent variates that were ob-tained via the inverse CDF of the inverse Nakagami-m andNakagami-m distributions. Three scenarios of the shadowingimpact, which are heavy, moderate, and light are studied byusing ms = 1.5, ms = 5 and ms = 50, respectively.Figs. 1 and 2 illustrate the ABEP for BPSK and the nor-malised ACC with M = 15 for single receiver, dual, and tripleSC branches over i.n.i.d. Fisher-Snedecor F fading channels.As anticipated, the performance becomes better when the SCscheme is employed and monotonically improves with theincreasing in the number of the diversity branches. This isbecause the received average SNR of SC with L = 3 ishigher than the no-diversity, i.e. L = 1, and L = 2 cases.For example, in Fig. 1, at fixed γ̄ = 7 dB and ms = 1.5, theIEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 00, NO. 00, JUNE 2020 40 5 10 15 20 2510−710−610−510−410−310−210−1100Average SNR per Branch γ̄ (dB)AverageBitErrorProbabilityPe  L=1L=2L=3MRCm = 3.5, ms = 1.5m = [3.5, 3.5], ms = 1.5m = [3.5, 4.5], ms = 1.5m = [3.5, 4.5, 5.5], ms = 1.5m = [3.5, 4.5, 5.5], ms = 5m = [3.5, 4.5, 5.5], ms = 50SimulationAsymptoticFig. 1. ABEP for BPSK of SC scheme with i.n.i.d. branches.0 5 10 15 20 250246810121416Average SNR per Branch γ̄ (dB)NormalisedAverageChannelCapacity(bits/sec/Hz)  L=1L=2L=3(m, ms) = (0.5, 1.1), [7]m = 3.5, ms = 1.5m = [3.5, 3.5], ms = 1.5m = [3.5, 4.5], ms = 1.5m = [3.5, 4.5, 5.5], ms = 1.5m = [3.5, 4.5, 5.5], ms = 5m = [3.5, 4.5, 5.5], ms = 50SimulationAsymptoticFig. 2. Normalised ACC of SC scheme with i.n.i.d. branches.ABEP for L = 3 is approximately 82% and 60% less thanthat for L = 1 and L = 2 with m = [3.5, 4.5], respectively.Furthermore, for further validations, a comparison between theABEP of SC and MRC of [8], has been shown in Fig. 1.As expected, the MRC provides less ABEP than the SC butwith high implementation complexity. In addition, the resultfor L = 1 of [7] in Fig 2 is obtained by utilising (18). Inboth figures, the perfect matching between the numerical andthe simulation results as well as their asymptotic counterpartsat high average SNR can be observed, which confirms thecorrectness of our derived expressions.Fig. 3 shows the behaviour of an AUC versus m and msand different values of u and L for γ̄ = 5 dB. In this figure, asubstantial enhancement in the AUC is occurred when m, ms,or/and L become large. This is because the increasing in m,ms, and L would lead to a large number of multipath clusters,less effect of the shadowing, and high received average SNR,respectively. Also, when the parameter u reduces, the value ofthe AUC increases due to the increasing in both the detectionand the false alarm probabilities simultaneously.VI. CONCLUSIONSIn this paper, both the exact and the asymptotic of thePDF and the MGF of the maximum of i.n.i.d. Fisher-SnedecorF RVs were derived in mathematically tractable closed-formexpressions. These statistics were then used to study the01020304050123450.830.840.850.860.870.880.890.90.910.92 msm(u, L) = (3, 2), (2, 2), (2, 3) AverageAreaundertheROCA0.830.840.850.860.870.880.890.90.910.92Fig. 3. AUC versus ms and m and different values of u and L.performance of the SC diversity reception over i.n.i.d. Fisher-Snedecor F fading channels. To this end, the exact andthe asymptotic expressions of the ABEP, the ACC, and theAUC of an ED were analysed. The diversity order which isproportional to the number of the diversity branches and theparameter m is also provided to validate of our results.REFERENCES[1] M. K. Simon and M.-S. Alouini, Digital Communications over FadingChannels. New York: Wiley, 2005.[2] P. S. Bithas, P. T. Mathiopoulos, and S. A. Kotsopoulos, “Diversity recep-tion over generalized-K (KG) fading channels,” IEEE Trans. WirelessCommun., vol. 6, no. 12, pp. 4238-4243, Dec. 2007.[3] J. Paris, “Statistical characterization of κ − µ shadowed fading,” IEEETrans. 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 maximumof non-identically distributed composite η − µ/gamma variates usinga 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 GeneralisedWireless Channels in Cognitive Radio. 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Academic Press Inc., 2007.[12] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theoryand Applications. Springer, 2009.[13] H. Chergui et al., “Rician K-factor-based analysis of XLOS serviceprobability in 5G outdoor ultra-dense networks,” IEEE Wireless Commun.Lett., vol. 8, no. 2, pp. 428-431, Apr. 2018.[14] O. S. Badarneh and F. S. Almehmadi, “Performance analysis of L-branch maximal ratio combining over generalised η−µ fading channelswith imperfect channel estimation,” IET Commun., vol. 10, no. 10, pp.1175-1182, Jul. 2016.[15] “The Wolfram Functions Website.” (Last accessed April 2020).[16] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions:With Formulas, Graphs, and Mathematical Tables. New York, NY, USA:Dover, 1965.

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

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

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