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Level crossing rate and average fade duration for the Beaulieu-Xie fading model Olutayo, Adebola; Ma, Hui; Cheng, Julian; Holzman, Jonathan F. Mar 19, 2018

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1Level Crossing Rate and Average Fade Duration forthe Beaulieu-Xie Fading ModelAdebola Olutayo, Student Member, IEEE, Hui Ma, Julian Cheng, Senior Member, IEEE,and Jonathan F. Holzman, Member, IEEEAbstract—Level crossing rate (LCR) and average fade duration(AFD) of a new (Beaulieu-Xie) fading model are analyzed. Thecharacteristic function method is used to derive the LCR fora diversity scheme using maximal ratio combining. The LCRand AFD of the Beaulieu-Xie fading model show improvementbeyond the performance levels of the Ricean and Nakagami-mfading models.I. INTRODUCTIONTHE Ricean fading model has gained acceptance dueto its ability to characterize wireless systems with adominant line-of-sight (LOS) channel and multiple non-line-of-sight (NLOS) channels. However, the Ricean fading modelhas limited flexibility in characterizing the broad range ofsignal fades from reflections and shadowing. In contrast, theNakagami-m distribution has been found to be an effectivefading model because it includes a flexible fading parameter,allowing it to be adapted for varying levels of severity in signalfades. However, attempts to apply the Nakagami-m fadingmodel to wireless systems with both LOS and NLOS channelshave been unsuccessful. This is because the Nakagami-mfading model is a normalized form of the chi-distribution,which cannot define a LOS channel [1]-[3].Given the benefits and challenges of the Ricean andNakagami-m fading models, these models were recentlymerged into a new fading model by Beaulieu and Xie [4].The Beaulieu-Xie fading model acquires the benefits of theNakagami-m fading model, in that it has a flexible fadingparameter, and the benefits of the Ricean fading model, in thatits use of a non-central chi-distribution allows it to characterizeboth LOS and NLOS channels.In this work, the level crossing rate (LCR) and averagefade duration (AFD) [5] are computed to evaluate the dynamicperformance of the Beaulieu-Xie model. The LCR and AFDare key second-order statistics that define the quality of thereceived signal. The AFD is especially critical in understand-ing the statistics of error bursts [6]. The characteristic function(CF) method is used to analyze diversity reception via maximalratio combining (MRC) [7]. Such a method overcomes theanalytical complexities of derivations employing the jointprobability density function (pdf). We conclude that the LCRand AFD obtained for this model are improved beyond theperformance levels of the Ricean and Nakagami-m fadingmodels. To the authors’ knowledge, this is the first work thatevaluates the dynamic performance of this new fading modelwith diversity combining. A simple formula is ultimatelyThe authors are with the School of Engineering, The University of BritishColumbia, Kelowna, BC, Canada V1V 1V7 (e-mail: julian.cheng@ubc.ca).This research is supported by Natural Sciences Engineering Research Councilof Canada (NSERC).presented to show the effect of the LOS component(s), in termsof the signal mean and fading parameter.II. PROBABILITY DENSITY FUNCTION OF THEBEAULIEU-XIE FADING MODELFor the Beaulieu-Xie fading model, it is assumed that wehave a non-central chi-distributed random variable (RV), d =√k∑i=1(Xii)2, where Xi is a non-zero Gaussian distributedRV and Xi ∼ N(iF 2i)F t = 1F 2F · · · F k; with mean iand variance 2i , and the k parameter denotes the degree offreedom. The pdf of d can be obtained byfY (y; kF ) =exp(−y2+22)yk(y)k2T k2−1 (y) (1)where Ty (·) is the yth order modified Bessel function ofthe first kind and  is the non-centrality parameter, which isrelated to the mean and variance values of the Gaussian RVs by =√k∑i=1(ii)2. A new RV can be obtained by normalizingd by√2mΩ with the parameters m =k2 and Ω. The Beaulieu-Xie fading pdf is given by (2) at the top of the next page[4]. Using the in-phase and quadrature approach [8], one canderive the phase distribution of the Beaulieu-Xie fading model.The parameter m quantifies the severity of the fading, i.e., thehigher the value of m, the less severe the fading conditions.Ultimately, m controls the shape, Ω controls the spread, and impacts the location and height of the mode of the pdf [4].The performance of this new fading model in different fadingconditions is illustrated with symbol error rate curves in [4].III. LEVEL CROSSING RATEThe LCR and AFD are key metrics to dynamic performancein mobile communication systems [5]. The rate of change ofthe received signal in mobile communication can be easilyrelated to the crossing (signal) level1 and velocity [9]. TheLCR, YR(r), is the average number of times a fading signalcrosses a given signal level, r, within a certain period of time,t. Here, the envelope of the received signal, R (t), is subjectedto Beaulieu-Xie fading with the pdf in (2). The LCR in thefading environment can be expressed asYR(r) =∫ ∞0r˙fR;R˙(rF r˙)dr˙ =∫ ∞0r˙fR (r) fR˙ (r˙) dr˙ r ≥ 0(3)1Crossing level and signal level are used interchangeably in this work.(c) 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. 2fR (r;mFFΩ) =exp(−mΩ (r2 + 2)) rm ( 2mΩ )m−1Tm−1(2mΩr)(2)where fR;R˙ (rF r˙) is the joint pdf of R (t) and its timederivative, R˙ (t), and the second equality follows when R(t) isassumed to be independent of R˙(t). Hereafter, for simplicitywe will omit t. For single branch reception, the pdf of thetime derivative of the received signal envelope, R˙ (t), is [10,eq. (8)]fR˙ (r˙) =√12˙2r˙exp{− r˙22˙2r˙}(4)where ˙2r˙ is the time derivative of the variance of R˙ (t). Thevariance of R˙ (t) for the Ricean fading model is [10]˙2r˙ = 22f2m2 (5)where 22 is the Ricean power of the NLOS component(s) andfm is the maximum Doppler frequency. The transformationthat unifies the Ricean fading model to the Beaulieu-Xie fadingmodel is s = √2mΩ  [4], where s2 is the Ricean LOS power,2 is the Beaulieu-Xie LOS power, and Ω is the Beaulieu-XieNLOS power. Then, ˙2r˙ for the Beaulieu-Xie fading model is˙2r˙ =2f2mΩmE (6)We insert (6) into (4), and (2) into (3), and apply mathematicalmanipulations to give LCR in (7) at the top of next page.IV. AVERAGE FADE DURATIONThe AFD, TR (r), is the average time that the receivedenvelope, R (t), remains below a given crossing (signal) level,r, after crossing it in the negative direction [10]. The AFD isTR (r) =FR (r)YR (r)=∫ r0fR(t)dtYR (r)(8)where FR (r) is the cumulative distribution function (CDF) ofR (t) and the integral in (8) is expressed as [4, eq. (5)]∫ r0fR(t)dt =∫ r0exp(−mΩ (t2 + 2)) tm ( 2mΩ )m−1× Tm−1(2mΩt)dtE (9)V. SIMULATION RESULTSThe Beaulieu-Xie fading model is simulated using MAT-LAB to show the effects of various parameters and comparethe observations to those of the existing Ricean fading model.Figure 1 shows the normalized LCR curves of the Beaulieu-Xie, Ricean, and Nakagami-m fading models as a functionof the crossing level, r. It is seen that the normalized LCRof the Beaulieu-Xie fading model approaches that of theRicean fading model as m is reduced to 1, corresponding toincreased fading severity, at the same LOS and NLOS powers.The LCR values of the Beaulieu-Xie fading model are lowerthan those of the Ricean and Nakagami-m fading model. TheCrossing level, r-15 -10 -5 0 5 10NR/fm10-1010-810-610-410-2100RiceanNakagami m = 1.5Nakagami m = 2Beaulieu-Xie m = 1.5Beaulieu-Xie m = 2Crossing level, r-15 -10 -5 0 5 10NR10-1010-5100BX - K-factor = 0 dBBX - K-factor = 6.99 dBBX - K-factor = 10 dBFig. 1. The normalized LCR curves of the Beaulieu-Xie, Ricean andNakagami-m fading models versus crossing level, r. The Beaulieu-Xie,Ricean and Nakagami-m fading model curves have a total power of 4.88 dB.The Beaulieu-Xie and Nakagami-m curves are shown for fading parametersof m = 1.5 and 2. Inset: The LCR curves of the Beaulieu-Xie fading modelversus crossing level, r, with a K-factor of 0, 5 and 10 dB.Crossing level, r-15 -10 -5 0 5 10N R10-1610-1410-1210-1010-810-6Speed,v = 5 km/hSpeed,v = 30 km/hSpeed,v = 80km/hSpeed,v = 100 km/hSpeed,v = 140 km/hSpeed,v = 200 km/hFig. 2. The LCR curves of the Beaulieu-Xie fading model versus crossinglevel, r, for vehicular speeds of v = 5, 30, 80, 100, 140 and 200 km/h.inset figure shows the effect of the K-factor on the LCR ofBeaulieu-Xie fading. An increase in K-factor corresponds toan increase to the LOS power, which leads to lower LCRvalues. Figures 2 and 3 show the LCR and AFD as a functionof the crossing level, r, for vehicular speeds up to 100 km/h.The LCR and AFD are seen to be dependent on the speed ofthe mobile device, i.e., dependent on the maximum Dopplershift frequency. The LCR increases as fm becomes large whileAFD decreases as the fm becomes large. Figure 4 shows theAFD as a function of the crossing level, r, for the fadingparameter, m.VI. LEVEL CROSSING RATE EXPRESSION FOR MAXIMALRATIO COMBININGDiversity combining techniques can counter variations inthe envelope of a received signal in wireless systems. TheLCR and AFD are two important statistics that can be used toanalyze such diversity schemes.In [7], the LCR was calculated using a CF method, as analternative to the standard pdf method. Using this approach,we define the LCR as the expected number of occurrences per3YR(r) =fm√Ω√2mexp(−mΩ (r2 + 2)) rm ( 2mΩ )m−1Tm−1(2mΩr)(7)Crossing level, r-10 -8 -6 -4 -2 0 2 4 6 8 10TR10-5100105Speed, v = 0 km/hSpeed, v = 5 km/hSpeed, v = 30 km/hSpeed, v = 80 km/hSpeed, v = 100 km/hFig. 3. The AFD curves of the Beaulieu-Xie fading model versus crossinglevel, r, for vehicular speeds of v = 0, 5, 30, 80, and 100 km/h.unit time that a stationary random process R (t) crosses thesignal level r. This is expressed using Rice’s formula as [12]E sYr {R (t)}] =∫ ∞−∞|r˙|fR;R˙ (rF r˙) dr˙ (10)where E s·] denotes the expectation operation and |·| denotesthe absolute value operator, which is obtained from its jointCF. Note that |r˙| = r˙sgn (r˙), where sgn (·) is the signumfunction. The joint CF of R (t) and R˙ (t) is defined asΦR;R˙(,1F ,2) = E[e(j,1R+j,2R˙)], where j2 = −1. Werewrite (10) using Fourier transforms as (11) at the top ofthe next page, where F s·] is the Fourier transform operator,∗ is the complex conjugate operator, andF ssgn (r˙)] =2j,2(12a)F[fR;R˙ (rF r˙)]=12∫ ∞−∞ΦR;R˙(,1F ,2)e−j,1rd,1E (12b)Substituting (12) into (11) gives (13) on the next page [7],[14]. We define the total instantaneous signal-to-noise ratio(SNR) at the output of MRC diversity system asR =L∑l=1Rl (14)where L is the number of branches and Rl is the instantaneousSNR of the lth branch. Here, Rl is expressed asRl =k∑i=1X 2l;i (15)assuming unit noise variance, where k = 2m signifies thedegrees of freedom for the diversity system andm is the fadingparameter. For the lth branch, Xl;i’s are independent andidentically distributed (i.i.d.) Gaussian RVs with equal meanl and equal variance ll. Thus, the joint CF for independentbranches can be written as (16) at the top of the next page.For each l, X˙l;i’s are Gaussian RVs with zero mean and equalvariance ml. Equation (16) can also be expressed with thefollowing vectors and matrices:Ml =[l;1 0 l;2 · · · 0]T1×2kD =,1 0 0 · · · 0 ,20 0 0 · · · ,2 00 0 ,1 . ..0 00 0 ,2. . . 0 00 ,2 0 · · · ,1 0,2 0 0 · · · 0 02k×2kCl =ll 0 0 0 0 00 ml 0 0 0 00 0 ll 0 0 00 0 0. . . 0 00 0 0 0 ll 00 0 0 0 0 ml2k×2kwhere the superscript T denotes the transpose, Ml is amean vector, and Cl is a covariance matrix. We continue byexpressing (16) as [7], [14]ΦRl;R˙l(,1F ,2) =exp(−MTl C−1l sI−(I−j2ClD)−1]Ml2)√det(I− j2ClD)(17)where I is a 2k-by-2k identity matrix and det(·) denotes thedeterminant. Substituting the vectors and matrices into (17)and applying mathematical manipulations, we obtainΦRl;R˙l(,1F ,2) =exp(−42 (2ml,22−j,1)(1+4llml,22−j2ll,1))(1 + 4llml,22 − j2ll,1)m(18)for the Beaulieu-Xie fading model. We assume that the instan-taneous SNR of each branch is independent and writeΦR;R˙(,1F ,2) =L∏l=1ΦRl;R˙l(,1F ,2)E (19)The LCR is then obtained by substituting (19) into (13).Figure 5 shows the LCR as a function of the crossinglevel, r, at the output of a three-branch MRC combiner,using CF methods for Ricean, Nakagami-m and Beaulieu-Xie fading models. The inset figures show LCR curves forfading parameters of m = 1.5, 2, 3, and 4 and identicallyand independently distributed (i.i.d.) and non-identically andindependently distributed (non-i.i.d.) branches. Here, the i.i.d.branches all have Ricean-distributed envelopes and the non-i.i.d. have a Beaulieu-Xie distributed envelope in the firstbranch and Ricean-distributed envelopes in the second andthird branches. The average power per branch is 7.64 dBand the K-factor is 10 dB for the i.i.d. case. The averagepowers are 7.64, 7.58 and 6.71 dB for the first, second andthird branches in the non-i.i.d., respectively. The Beaulieu-Xie4E sYr {R (t)}] = 12∫ ∞−∞F sr˙sgn (r˙)]{F[fR;R˙ (rF r˙)]}∗d,2 (11)E sYr {R(t)}] = − 122∫ ∞−∞∫ ∞−∞ΦR;R˙(,1F ,2)− ΦR(,1),22e−j,1rd,1d,2 (13)ΦRl;R˙l(,1F ,2) = Esexp(j,1(X 2l;1 + X 2l;2 + · · ·+ X 2l;k) +j,2(2Xl;1X˙l;1 + 2Xl;2X˙l;2 + · · ·+ 2Xl;kX˙l;k))](16)Crossing level, r-10 -8 -6 -4 -2 0 2 4 6 8TR*fm10-310-210-1100101102103m = 1m = 1.5m = 2m = 3m = 4Fig. 4. The AFD curves of the Beaulieu-Xie fading model versus crossinglevel, r, for fading parameters of m = 1, 1.5, 2, 3, and 4.fading model clearly outperforms the Ricean and Nakagami-m fading models. Also, it is seen that m impacts the LCR ofthe system; as m is reduced, the rate of fading in the systemincreases. A notable difference between the non-i.i.d. and i.i.d.cases is apparent here. This can be a key consideration insystem design [7].VII. CONCLUSIONSWe studied the LCR for the Beaulieu-Xie fading model andused the CF method to develop an analytical expression forthe LCR in an MRC diversity system. The performance of theBeaulieu-Xie fading model was analyzed for various crossinglevels, LOS powers, and maximum Doppler frequencies. Thisfading model can be potentially used to describe high speedtrain scenario where the LCR becomes relatively insensitiveto large Doppler shift. While more evidence needs to beestablished and show the Beaulieu-Xie model is a better fitto certain realistic communication scenarios, this task furthercalls for the need to develop efficient parameter estimation ofthe Beaulieu-Xie fading model.REFERENCES[1] M. D. Yacoub, “Nakagami-m phase-envelope joint distribution: A newmodel,” IEEE Trans. Veh. Technol., vol. 59, no. 3, pp. 1552-1557, Mar.2010.[2] N. C. Beaulieu and S. A. Saberali, “A generalized diffuse scatter plusline of-sight fading channel model,” in Proc. IEEE ICC 2014, Sydney,NSW, Jun. 2014, pp. 5849-5853.[3] S. Wyne, A. P. Singh, F. Tufvesson, and A. F. Molisch, “A statisticalmodel for indoor office wireless sensor channels,” IEEE Trans. WirelessCommun., vol. 8, no. 8, pp. 4154-4164, Aug. 2009.[4] N. C. Beaulieu and J. Xie, “A novel fading model in channel with multipledominant specular components,” IEEE Wireless Commun. Lett., vol. 4, no.1, pp. 54-57, Feb. 2015.Crossing level, r0 2 4 6 8 10 12 14 16 18 20NR10-210-1100Ricean, L=3Nakagami-m, m = 2, L= 3Beaulieu-Xie, m = 2, L = 3Crossing level, r5 10 15 20NR10-1100Beaulieu-Xie,m = 1.5Beaulieu-Xie,m = 2Beaulieu-Xie,m = 3Beaulieu-Xie,m = 4Crossing level, r 0 5 10 15 20NR10-210-1100IIDNIIDFig. 5. The LCR curves of the Beaulieu-Xie fading model versus crossinglevel, r, at the output of a 3-branch MRC combiner using the CF methodfor Ricean, Nakagami-m and Beaulieu-Xie fading models, with an averagepower of 7.64 dB per branch. Inset: The LCR curves of Beaulieu-Xie fadingversus crossing level, r, with fading parameters of m = 1.5, 2, 3 and 4. TheLCR curves for i.i.d. and non-i.i.d. branches have a total average power of12.41 dB and a K-factor of 10 dB per branch.[5] D. B. Da Costa, J. C. S. S. Filho, M. D. Yacoub, and G. Fraidenraich,“Crossing rates and fade duration for diversity-combining schemes over−  fading channels,” IEEE Commun. Lett., vol. 6, no. 12, pp. 4263-4267, Dec. 2007.[6] Z. Cao and Y. D. Yao, “Definition and derivation of level crossing rateand average fade duration in an interference-limited environment, ” inProc. IEEE Vehicle Technology Conf. (VTCFall 01), Atlantic City, NJ,Oct. 2001, pp. 16081611.[7] A. Abdi and M. Kaveh, “Level of crossing rate in terms of characteristicsfunction: New approach for calculating the fading rate in diversitysystems,” IEEE Trans. on Commun., vol. 50, no. 9, pp. 1394-1400, Sept.2002.[8] M. D. Yacoub, G. Fraidenraich, and J.C.S. Santos Filho, “Nakagami-mphase-envelope joint distribution,” Electronic Letters, vol. 41, issue 5, pp.250-261, Mar. 2005.[9] T. S. Rappaport, Wireless Communication: Principles and Practise, 2ndEd. Upper Saddle River, NJ, USA: Prentice Hall, 2001.[10] X. Dong and N. C. Beaulieu, “Average level of crossing rate and averagefade duration of selection diversity,” IEEE Commun. Lett., vol. 5, no. 10,pp. 396-398, Oct. 2001.[11] Z. Hadzi-Velkov, “Level crossing rate and average fade duration of selec-tion diversity with Ricean faded co-channel interferers,” IEEE Commun.Lett., vol. 10, no. 9, pp. 649-651, Sept. 2006[12] A. Abdi and S. Nader-Esfahani, “Expected number of maxima in theenvelope of a spherically invariant random process,” IEEE Trans. Inf.Theory, vol. 49, no. 5, pp. 1369-1375, May 2003.[13] S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech.J., vol. 23, pp. 282-332, July 1944.[14] G. L. Turin, “The characteristics function of Hermitian quadratic formsin complex normal variables,” Biometrika, vol. 47, pp. 199-201, June1960.[15] K. H. Biyari and W. C. Lindsey, “Statistical distributions of Hermitianquadractic forms in complex Guassian variables,” IEEE Trans. Inf.Theory, vol. 39, no. 3, pp. 1076-1082, May 1993.[16] J. G. Proakis and M. Salehi, Digital Communication, 5th Ed. New York,NY, USA: McGraw-Hill, 2007.

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