outage probability over composite η−μ fading–shadowing radio channels

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Published in IET CommunicationsReceived on 24th November 2011Revised on 23rd March 2012doi: 10.1049/iet-com.2011.0873

ISSN 1751-8628

Outage probability over composite h 2 m fading–shadowing radio channelsN.Y. Ermolova O. TirkkonenDepartment of Communications and Networking, Aalto University, FI-00076, Aalto, FinlandE-mail: [email protected]

Abstract: The authors analyse the outage probability over compound h 2 m fading–log-normal shadowing radio channels. Thegamma distribution is used as a substitute to the log-normal shadowing model, and new finite-integral expressions for theprobability density and cumulative distribution functions of the composite h 2 m – gamma distribution. Approximateestimates obtained by reducing the considered problem to that over generalised K-fading are also presented.

1 Introduction

Recently, the h 2 m fading distribution was proposed formodelling small-scale fading in non-line-of-sight scenarios[1]. It was shown in [1] that the h 2 m distribution fitsbetter to experimental data than other frequently usedfading models (e.g. the Nakagami-m distribution), and itinvolves some of them as particular cases. The h 2 mmodel considers the fading signal as the composition ofmultipath clusters where either the in-phase and quadratureGaussian components within each cluster have differentpowers, or these components are correlated. It was shownin [1] that under both scenarios (formats), the probabilitydensity function (PDF) is expressed by the same formula,but with different definition of the parameter h. Analysis ofthe outage probability over h 2 m fading is given in [1, 2].

In a wireless environment, large-scale fading (shadowing)is often superimposed on small-scale fading, and the log-normal distribution based on experimental data iscommonly used for modelling shadowing effects [3]. Thecompound fading–shadowing PDF can be obtained byaveraging the small-scale fading PDF conditioned on themean signal power over the log-normal distribution [3].Such an approach does not result, however, in closed-formPDF expressions even for the Rayleigh small-scale fadingmodel. This fact makes any analytical evaluations over thecompound fading–shadowing channels difficult. Differentsubstitutes to the log-normal distributions have beenproposed to overcome this problem. Such areapproximations by the gamma and inverse Gaussiandistributions [4, 5]. These approximations lead to closed-form expressions for the composite PDF for Rayleigh andNakagami-m small-scale fading models [4–7]. Thecompound gamma–gamma distribution is the generalisedK-distribution [6–9], and the gamma-inverse Gaussiandistribution is the G-distribution [5]. The outage probabilityover G-and K-fading is evaluated in [5, 7–9], respectively.

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The composite Weibull-gamma distribution is introducedand analysed in [10]. In [11], a generalised fadingcomposite model, termed the extended generalised K-fadingdistribution, is proposed and studied.

The compound h 2 m – gamma distribution is consideredin [12]. However, the analysis in [12] is restricted byobtaining formulas for the PDF only. Those are anapproximate formula for arbitrary values of the fadingparameters and a closed-form expression for integer valuesof m. Meanwhile, shadowing channels are often not ergodicbecause the mean signal power varies slowly. In this case,the probability of the outage is a very importantcharacteristic of such channels. The problem of thedefinition of the outage probability is not addressed in [12]at all. Additionally, no numerical estimates were given toshow accuracy of the applied approximation andapplicability of the derived results to the log-normalshadowing model.

In this paper, in contrast with [12], we analyse the outageprobability Pout2comp(V0) over the compound h 2 mfading–gamma shadowing radio channels. We obtain a newfinite-integral expression for Pout2comp(V0) for arbitraryvalues of the fading parameters and a novel simplifiedformula for integer values of the fading parameter m. Thelatter result corresponds to the case of an even number ofthe multipath clusters in the small-scale fading model [1].Both formulas are derived on the basis of new PDFformulas of the compound h 2 m – gamma distribution forthe arbitrary fading parameters and for integer values of thefading parameter m. In addition to the exact formulas, wepresent approximate estimates obtained by reducingcomposite h 2 m – gamma fading to generalised K-fading.We present numerical estimates confirming applicability ofthe obtained results for analysing the outage probabilityover h 2 m fading–log-normal shadowing radio channels.To the best of our knowledge, no results on Pout2comp(V0)have been reported yet.

IET Commun., 2012, Vol. 6, Iss. 13, pp. 1898–1902doi: 10.1049/iet-com.2011.0873

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2 Fading and shadowing models

2.1 h 2 m fading distribution

The PDF of the h 2 m power variable g with the averageE{g} ¼ V is [1]

fgh−m(x, V) = 2

��p

√mm+(1/2)hmxm−(1/2)

G(m)Hm−(1/2)Vm+(1/2)

× exp − 2mh

Vx

( )Im−(1/2)

2mH

Vx

( )(1)

where 2m is the number of multipath clusters, m ¼ V2/(2var{g})[1 + (H/h)2] (where var{} denotes the variance),G(.) is the gamma function and Ia(.) is the modified Besselfunction of the first kind of the order a. In Format 1,0 , h , 1 is the power ratio of the in-phase andquadrature scattered waves in each multipath cluster;H ¼ (h21 2 h)/4 and h ¼ (2 + h21 + h)/4. In Format 2,21 , h , 1 is the correlation coefficient between the in-phase and quadrature scattered waves; H ¼ h/(1 2 h2) andh ¼ 1/(1 2 h2).

2.2 Gamma approximation to log-normaldistribution

Under shadowing (large-scale fading), the average power V isa random variable that is commonly modelled by the log-normal distribution with the PDF [3]

flog -norm(V) = 1��2p

√Vs

exp − ( lnV− n)2

2s2

( )(2)

where n and s are the respective mean and standard shadowdeviations expressed in nepers. The corresponding values indB are: n(s) [dB] ≃ 8.686n(s) [Np].

The gamma distribution was proposed in [4] as anapproximation to (2):

fgamma(V) = 1

G(M )

M

Vs

( )M

VM−1 exp −VM

Vs

( )(3)

where Vs is a measure of the mean power, and M . 0 is theshape parameter of the gamma distribution. Differentapproaches for specifying the parameters of theapproximating distribution (3) have been proposed. In ournumerical evaluations, we follow a widely used methodpresented in [4] that is based on an approximate equality ofall raw moments of the distributions (2) and (3):

C′(M ) = s2 and Vs = M exp(n−C(M )) (4)

where C(x) ¼ d/dx(lnG(x)) is the digamma function [11,vol. 3, Section II. 4]. The value of the parameter M iscalculated numerically from the first equation in (4), andthen it is used for evaluation of Vs on the basis of thesecond equation.


3 Outage probability over compoundh 2 m – gamma channels

3.1 PDF of the composite h 2 m – gammadistribution

The compound h 2 m – gamma PDF is defined by averagingthe distribution (1) over the gamma PDF (3), that is

fcomp(x) =∫1

0

fgh−m(x, V)fgamma(V) dV (5)

Proposition 1: The compound h 2 m – gamma PDF can beexpressed as

fcomp(x) = 2(a1a2)m

[G(m)]2G(M )

M

Vs

( )M/2+m

xm+M/2−1

×∫1

0

tm−1(1 − t)m−1[a2 − t(a2 − a1)]M/2−m

× K2m−M 2

��M

Vs

x[a2 − t(a2 − a1)]

√( )dt (6)

where a1 , a2, a1(2) ¼ 2m[h + (2)H ] and Kl() is themodified Bessel function of the second kind of the order l[13, vol. 2, Section 2.16].

Proof: We use a representation of the h 2 m power variableas the sum of two gamma variables with the same shapeparameter m and different scale parameters b1 ¼ V/[2m(h + H )] and b2 ¼ V/[2m(h + H )] [14]. Then the PDF(1) can be represented as the convolution of two gammaPDFs [15]. Thus the compound PDF is

fcomp(x) = (a1a2)m

[G(m)]2G(M )

M

Vs

( )M

×∫1

0

∫x

0

tm−1(x − t)m−1

yM−2m−1

× exp − (x − t)a2 + ta1

y

[ ]{ }exp −y

M

Vs

( )dt dy

(7)

Changing the order of integration in (7) and applying [13, vol.1, eq. (2.3.16.1)] (valid for a1 , a2) we obtain (6). A

Equation (6) is given in the form of a finite-range integral.This is a proper integral that can be easily evaluated via anymodern software.

3.2 Outage probability over h 2 m – gamma fading

We use the derived formula (6) for evaluation of the outageprobability.

Proposition 2: The outage probability Pout−comp(V0) = V0

0fcomp(x) dx over the compound h 2 m fading–gamma

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shadowing radio channel is

Pout−comp(V0) = (a1a2)m

[G(m)]2G(M )

{VM

0

G(2m− M )

M

M

Vs

( )M

×∫1

0

[a2 − t(a2 − a1)]M−2m · tm−1(1 − t)m−1

× 1F2

(M ; 1 + M − 2m, M + 1; V0

× M

Vs

[a2 − t(a2 − a1)]

)dt +V

2m0

× G(M − 2m)

m

M

Vs

( )2m∫1

0

tm−1(1 − t)m−1

× 1F22m; 1 + 2m− M , 2m+ 1; V0

× M

Vs

[a2 − t(a2 − a1)]

)dt

}(8)

where 1F2(.) is a hypergeometric function [13, vol. 3, Section7.2.3].

Proof: Equation (8) is obtained by application of anintegration formula [13, vol. 2, eq. (1.12.1.2)] to (6) afterchanging the order of integration in the double integral. A

The integrals in (8) are proper too, and they can be easilyevaluated via any modern software.

3.3 PDF of composite h 2 m – gamma distributionfor integer values of m

Simplified formulas for fcomp(x) and Pout2comp(x) can beobtained for integer values of the fading parameter m, thatis, for the small-scale fading model with the even numberof the multipath clusters [1].

Proposition 3: For integer values of m, the PDF of thecompound h 2 m – gamma distribution can be expressedas (for x . 0)

fcomp(x) = hm ·m22m−3 · [(m− 1)!]2H2m−1 ·G(M )

· M

Vs

( )M

×∑2m−2k−2

m=0

(2x)k+(M+m−1)/2 · (mH)2k+m

m!

×∑2m−2k−2

m=0

(2x)k+(M+m−1/2) · (mH)2k+m

m!

× (−1)m m(h−H)Vs

M

( )(M−m−1)/2−k[

×KM−2k−m−1 2

��2m ·M (h−H)x

Vs

√( )

− m(h+H)Vs

M

( )(M−m−1)/2−k

·

×KM−2k−m−1 2

��2m ·M (h+H)x

Vs

√( )]; fcomp(0) = 0

(9)

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Proof: First we derive a new PDF formula for the h 2 mdistribution with integer values of m. By using an integration

formula [13, vol. 1, eq. (2.3.5.1)], (2mH/V)

−(2mH/V)((4m2H2 /V2)−

t2)m−1 exp(−xt)dt = 2m−1/2 ��p

√G(m)((x/2mH)V)1/2−mIm−1/2

((2mH/V)x), we express (1) in the integral form

fgh−m(x,V)= m ·hm

[G(m)]222m−2H2m−1Vexp −2mh

Vx

( )︸��︷︷��︸

C

·x2m−1

×∫(2mH/V)

−(2mH/V)

4m2H2

V2 − t2

( )m−1

exp(−xt)dt

(10)

Taking into account the definition of the lower incompletegamma-function g(a, x)=

x

0ta−1 × exp(− t)dt [13, vol. 1,

eq. (1.3.2.3)], we obtain that the integral in (10) is

fgh−m(x,V)=C

∑m−1

k=0

m−1

k

( )(−1)m−1−k 2mH

Vx

( )2k

× g 2m−2k −1,2mH

Vx

( )[

− g 2m−2k −1, −2mH

Vx

( )](11)

Using the representation of g(a, x)= (a−1)! 1− exp(−x)[∑a−1

n=0 (xn/n!)] for integer values of a [13, vol. 1, eq.(4.1.7.10)], we obtain that

fgh−m(x,V)= hm ·m

22m−2 · [(m−1)!]2H2m−1V

×∑m−1

k=0

m−1

k

( )(−1)m−1−k

× 2mH

Vx

( )2k

(2m−2k −2)!

×∑2m−2k−2

m=0

1

m!

2mH

Vx

( )m

× (−1)m exp −2m(h−H)

Vx

( )[

− exp −2m(h+H)

Vx

( )](12)

Equation (12) is a new PDF formula of the h 2 m distributionwith integer values of m.

Applying an integration formula [13, vol. 1, eq. (2.3.16. 1)]that is valid for x . 0, we obtain the expression for x . 0given in (9). By direct evaluation of (5) we find thatfcomp(0) ¼ 0. A

3.4 Outage probability for integer values of m

From (9), we obtain a simplified expression for the outageprobability.


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Proposition 4: The outage probability for integer values of m is

Pout−comp(V0) = hm

(2H)2m−1 · [(m− 1)!]2 · G(M )

×∑m−1

k=0

m− 1

k

( )(−1)m−1−k(2m− 2k − 2)

×∑2m−2k−2

m=0

H2k+m

m!· [(−1)m

× G(h − H) − G(h + H)] (13)

where

G(z) = 1F2

(2k + m + 1; 2k + m − M

+ 2, 2k + m + 2;2m · M · z

Vs

V0

)

× G(M − 2k − m − 1)

2k + m + 1

2m · M

Vs

V0

( )2k+m+1

+ G(2k + m + 1 − M )

M

2m · M

Vs

V0

( )M

zM−2k−m−1

× 1F2 M ; M − 2k − m, M + 1;2m · M · z

Vs

V0

( )(14)

Proof: Equation (13) is obtained by the application of theintegration formula [13, vol. 2, Eq. (1.12.1.2)] to (9). A

We note that (8) is not defined for integer values of(2m 2 M ), and (13) is not defined for integer values of theshadowing parameter M. However, the outage probability iscontinuous with respect to these parameters. ThusPout2comp(V0) can be found numerically from (8) or (13) bysetting the values of the above parameters close to theintegers.

3.5 Approximate expression for the outageprobability

Along with the exact expressions for the outage probability(8) and (13), we present also approximate formulas basedon reducing the compound h 2 m – gamma fading to thegeneralised K-fading.

Proposition 5: The outage probability Pout2comp(V0) isexpressed approximately as

Pout−comp(V0) ≃ G(M − m)

G(M )G(1 + m)(JV0)m

×1 F2(m; 1 − M + m, 1 + m; JV0)

+ G(m − M )

G(m)G(1 + M )(JV0)M

×1 F2(M ; 1 − m + M , 1 + M ; JV0) (15)

where m ¼ {(2m+ 1)/(2m . hm+2) .2F1(m+ 1.5,m+ 1;m+

0.5;(H/h)2) 2 1}21, 2F1(.) is the Gauss hypergeometricfunction [13, vol. 3, Section 7.2.1] and J ¼ m . M/Vs.


Proof: We use the representation of the h 2 m power variablein the form of the sum of two independent gamma variables(see the proof of Proposition 1) and then approximate thePDF of the sum of two independent gamma variables bythe PDF of the gamma distribution with properly chosenparameters [16]. In such a way we obtain thatfgh−m

(z, V) ≃ mm/(G(m)Vm) · zm−1 exp (−(m/V)z), wherem ¼ E2{g}/(E{g2} 2 E2{g}) [16]. Using [1, eq. (21)] weobtain an expression for the parameter m in (15). Thus thegeneralised K (gamma–gamma)–distribution can be viewedas an approximation to the compound h 2 m fading–gamma distribution. The PDF of the generalisedK-distribution is given by [6–9]

fK (x) = 2J(M+m)/2

G(M )G(m)x(M+m)/2−1KM−m(2

��Jx

√) (16)

and (15) represents the outage probability over the generalisedK-distribution [7, eq. (13)]. A

Numerical aspects of the proposed approximation arediscussed in the next section.

4 Numerical results

We emphasise that all the derived expressions show verygood agreement with the corresponding integrals evaluatednumerically for various combinations of the parameters of(1) and (3).

In Fig. 1, we present estimates of the outage probabilityagainst the shadowing severity s for Format 1 of small-scale fading, and in Fig. 2, we show estimates of the outageprobability against the parameter n for Format 2. Theseresults were obtained via (8) and (13) and via numericalaveraging over composite h 2 m–log-normal fading. Weshow also estimates of the outage probability obtained viacomputer simulations, where the log-normal and gammavariables are generated by applying the standard software,and the sum of two independent gamma variables withproperly chosen parameters (see Section 3.1) is used togenerate the h 2 m variables. In both figures, analyticalresults agree well with the estimates obtained via thenumerical integration and Monte-Carlo simulations. It isalso seen that the h 2 m – gamma fading model provides

Fig. 1 Outage probability against shadow deviation; Format 1 ofsmall-scale fading; n ¼ 0 dB, V0 ¼ 27 dB

Lines represent estimates for h 2 m – log-normal fading obtained vianumerical integration of composite PDF, squares report analytical results(8), (13), and (15), and black points show simulation results

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an acceptable accuracy of the approximation of h 2 m–log-normal fading in all cases considered.

We also check the accuracy of the approximate formula(15). Under all scenarios tested, we observe very slightdifferences between the estimates obtained on the basis of(8) and (15), which are not distinguished in Figs. 1 and 2.This is because of a good accuracy of the approximation ofthe compound h 2 m – gamma distribution by thegeneralised K-distribution. In order to confirm this fact, wepresent in Fig. 3 graphs of the PDF (6) of the compositeh 2 m fading–log-normal distribution for a few sets of thefading parameters and their approximations by the PDF(16) of the generalised K-distribution. The good accuracy ofthe approximation is observed.

5 Conclusions

In this paper, we analyse the outage probability over thecompound h 2 m fading–log-normal shadowing radiochannels. Our approach is based on applying the gammadistribution as a substitute to the log-normal distribution[4]. We present finite-integral expressions for the PDF andcumulative distribution function of the composite h 2 m –gamma distribution. The obtained formulas contain properfinite-range integrals that can be easily evaluated via anymodern software. We also derive simplified formulas for

Fig. 2 Outage probability against shadowing parameter n;Format 2 of small-scale fading; s ¼ 7.77 dB, V0 ¼ 27 dB

Lines represent estimates for h 2 m – log-normal fading obtained vianumerical integration of composite PDF, squares report analytical results(8), (13) and (15), and black points show simulation results

Fig. 3 Comparison of PDFs of h 2 m – gamma (lines) andgeneralised K-distributions (circles)

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the case of integer values of the small-scale fadingparameter m. The PDF formulas obtained in this paper aregiven in the form of linear transforms of the products of themodified Bessel functions Kl() and power functions. Thus,many theoretical results derived for the generalised K-fading via integral transforms of the PDF of the K-distribution (e.g. those given in [6–9]) can be applied toperformance evaluation of communication systemsoperating over h 2 m fading–shadowing radio channels. Asa by-product, we derive a new formula for the PDF of theh 2 m distribution with integer values of m.

Along with the exact formulas for the outage probability,we present approximate expressions obtained by applicationof the generalised K-distribution as a substitute to thecomposite h 2 m – gamma distribution. Our numericalestimates confirm good accuracy of the used approximationand applicability of all presented results to the analysis ofthe outage probability over the h 2 m fading – log-normalshadowing radio channels.

6 Acknowledgment

The authors would like to thank the reviewers for theirconstructive comments and suggestions. This work wassupported by the Academy of Finland.

7 References

1 Yacoub, M.D.: ‘The k 2 m distribution and the h 2 m distribution’,IEEE Antennas Propag. Mag., 2007, 49, (2), pp. 68–81

2 Morales-Jimenez, D., Paris, J.F.: ‘Outage probability analysis for h 2 mfading channels’, IEEE Commun. Lett., 2010, 14, (6), pp. 521–523

3 Simon, M.K., Alouini, M.-S.: ‘Digital communication over fadingchannels’ (Wiley, New York, 2005)

4 Abdi, A., Kaveh, M.: ‘K distribution: an appropriate substitute forRayleigh – lognormal distribution in fading-shadowing wirelesschannels’, IEE Electron. Lett., 1998, 34, (4), pp. 851–852

5 Laourine, A., Alouini, M.-S., Affes, S., Stephenne, A.: ‘On theperformance analysis of composite multipath/shadowing channelsusing the G-distribution’, IEEE Trans. Commun., 2009, 57, (4),pp. 1162–1170

6 Shankar, P.M.: ‘Error rates in generalized shadowed fading channels’,Wirel. Person. Commun., 2004, 28, pp. 233–238

7 Shankar, P.M.: ‘Outage probabilities in shadowed fading channels usinga compound statistical model’, IEE Proc. Commun., 2005, 152, (6),pp. 828–832

8 Bithas, P.S., Sagias, N.C., Mathiopoulos, P.T., Karagiannidis, G.K.,Rontogiannis, A.A.: ‘On the performance analysis of digitalcommunications over generalized-K fading channels’, IEEE Commun.Lett., 2006, 10, (5), pp. 353–355

9 Efthymoglou, G.P., Ermolova, N.Y., Aalo, V.A.: ‘Channel capacity andaverage error rates in generalized –K fading channels’, IET Commun.,2010, 4, (11), pp. 1364–1372

10 Bithas, P.S.: ‘Weibull-gamma composite distribution: alternativemultipath/shadowing fading model’, IET Electron. Lett., 2009, 45,(14), pp. 749–751

11 Yilmaz, F., Alouini, M.-S.: ‘A new simple model for composite fadingchannels: second order statistics and channel capacity’. Proc. SeventhInt. Symp. Wireless Communication Systems, ISWCS 2010, Jork,UK, September 2010, pp. 676–680

12 Sofotasios, P.S., Freear, S.: ‘The h 2 m/gamma composite fading model’.Proc. IEEE Int. Conf. Wireless Information Technology and Systems,ICWITS 2010, Honolulu, USA, August–September 2010, pp. 1–4

13 Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: ‘Integrals and series’(Gordon and Breach Science Publishers, 1986)

14 Peppas, K., Lazarakis, F., Alexadridis, A., Dangakis, K.: ‘Errorperformance of digital modulation schemes with MRC diversityreception over h 2 m fading channels’, IEEE Trans. Wirel. Commun.,2009, 8, (10), pp. 4974–4980

15 Papoulis, A.: ‘Probability, random variables, and stochastic processes’(McGrow Hill, 1984)

16 Da Costa, D.B., Yacoub, M.D.: ‘Outage performance of two hop AFrelaying systems with co-channel interferes over Nakagami-m fading’,IEEE Commun. Lett., 2011, 15, (9), pp. 980–982


outage probability over composite η−μ fading–shadowing radio channels

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