Second-Order Statistics for the Envelope and Phaseof η-µ Generalized Fading ChannelsDaniel Benevides da Costa, Michel Daoud Yacoub, and Gustavo Fraidenraich

Abstract— This paper proposes an analysis of the second-orderstatistics concerning the envelope and phase of η-µ generalizedfading channels. So, a simple and closed-form expression forthe phase crossing rate of the signal operating in such channelswill be obtained. Furthermore, exact expressions for the levelcrossing rate and average fade duration of η-µ channels will alsobe attained. The analytical results are thoroughly validated byreducing the general case to some particular cases (Nakagami-m,Hoyt, and Rayleigh), for which the solutions are known, and bymeans of Monte Carlo simulation, for the more general case.

Index Terms— Average fade duration, generalized fading dis-tribution, level crossing rate, phase crossing rate

I. INTRODUCTION

In wireless communications, an accurate modeling of thepropagation channel is of extreme importance. Over the years,a great number of channel models have been proposed thatwell describe the statistics of the mobile radio signal. The shortterm signal variation is described by several models such asHoyt (Nakagami-q), Rayleigh, Rice, Nakagami-m and Weibull.Among the short term fading models, Nakagami-m has beengiven a special attention for its ease of manipulation and widerange of applicability. Although, in general, it has been foundthat the fading statistics of the mobile radio channel maywell be characterized by Nakagami-m, situations are easilyfound for which other models such as Hoyt, Rice, and Weibullyield better results [1]–[4]. More importantly, situations areencountered for which no models seem to adequately fitexperimental data, though one or another may yield a moderatefitting. Recently [5], [6], one new fading model (the η-µfading model) has been proposed that includes or closelyapproximates the most common fading models.

The η-µ distribution, whose associated fading model will berevisited in the next section, is a general fading distribution thatcomprises as special cases important other distributions suchas One-Sided Gaussian, Hoyt, Rayleigh, and Nakagami-m. Itsflexibility renders it suitable to better adjust to field measure-ment data, as shown in several field measurement campaigns[5], [6]. In particular, its tail closely follows the true statisticswhere other distributions fail to yield good fit [3]. In addition,the η-µ distribution can be used to approximate the distributionof the sum of independent, non-identical Hoyt variates havingarbitrary mean powers and arbitrary fading degrees. In thiscase, the results show that the differences between exact and

The authors are with the Wireless Technology Laboratory (WissTek),Department of Communications, School of Electrical and Computation Engi-neering, State University of Campinas, DECOM/FEEC/UNICAMP, PO Box6101, 13083-852 Campinas, SP, Brazil (Tel: +55 (19) 3788 5106, Fax: +55(19) 3289 1395, E-mail:[daniel,michel,gf]@wisstek.org). This work was partlysupported by FAPESP (05/59259-7).

approximate statistical curves are almost imperceptible [7].Since it is a recently proposed distribution, few works in theliterature deal with the statistical characterization of η-µ fadingchannels. The aim of this paper is to provide a detailed studyof the second-order statistics of the envelope and phase of suchchannels.

It is well known that in wireless communications, the enve-lope and phase of the received signal vary in a random mannerbecause of multipath fading. Thus, a good characterizationof the envelope and phase variations of the received signalrenders the design of any communication technique very moreaccurate. Statistics concerned with the envelope variation are,for instance, the level crossing rate (LCR) and the average fadeduration (AFD). These statistics have already been derivedfor the more known fading channels [8]–[12]. Related to thesignal phase variation, a statistics commonly used to quantifyits behavior is the phase crossing rate (PCR). But, differ-ently from other distributions (Hoyt, Rice, Rayleigh), whenthe η-µ distribution, as well as the Nakagami-m one, wereproposed, no information on the signal phase was provided.Only recently, a simple and closed-form expression for thephase-envelope joint probability density function (PDF) of theNakagami-m and η-µ distributions was derived in [13] and[14], respectively. Based on [13], a study of the second-orderstatistics of the phase process for Nakagami-m fading channelswas exhibited in [15]. However, as far as the authors are aware,such statistics remains unexplored for the η-µ fading channels.

This paper derives a simple and closed-form expression forthe PCR of η-µ fading channels, based on the results obtainedin [14]. Furthermore, exact expressions for the LCR and AFDover η-µ fading channels are also attained. The analyticalresults are thoroughly validated by reducing the general case tosome particular cases (Nakagami-m, Hoyt, and Rayleigh), forwhich the solutions are known, and by means of simulation,for the more general case.

This paper is organized as follows. Section II establishes themodel for η-µ fading channels and introduces some importantstatistics used throughout of this paper. Section III derives thePCR for η-µ fading channels. Section IV derives the LCRand AFD for the same fading scenario. Section V showssome numerical plots, and finally Section VI draws someconclusions. Appendix shows the joint PDF of the phase andits respective time derivative necessary to attain the PCR.

II. THE η-µ FADING MODEL

In this Section, we revisit the η-µ fading model as proposedin [5], [6]. In addition, based on this model and in theresults of [14], we present some higher order statistics of the

85-89748-04-9/06/$25.00 © 2006 SBrT ITS2006923

envelope and phase necessary for the calculations that follow.The fading model for the η-µ distribution considers a signalcomposed of clusters of multipath waves propagating in annon-homogeneous environment. Within any one cluster, thephases of the scattered waves are random and have similardelay times with delay-time spreads of different clusters beingrelatively large. The in-phase and quadrature components ofthe fading signal within each cluster are assumed to havedifferent powers. The envelope R of the η-µ fading model canbe written in terms of the in-phase and quadrature componentsof each one of the n clusters of the fading signal as

R2 =

n∑

i=1

(

X2i + Y 2

i

)

(1)

where Xi and Yi are mutually independent Gaussian processeswith zero-mean, i.e, E(Xi) = E(Yi) = 0, and non-identicalvariances so that E(X2

i ) = σ2X and E(Y 2

i ) = σ2Y . (E(·)

denotes statistical average) The η-µ envelope PDF is givenby [5], [6]

fR(r) =4√

πµµ+ 1

2 hµ

Γ(µ)Hµ− 1

2 r

(

r

r

)2µ

exp

[

−2µh

(

r

r

)2]

× Iµ− 1

2

[

2µH

(

r

r

)2]

(2)

where r =√

Ω =√

E(R2) , η = σ2X/σ2

Y (0 ≤ η ≤ 1, orequivalently, 0 ≤ η−1 ≤ 1), h = 2+η−1

+η4

, H = η−1−η

4, Γ(·)

is the Gamma function [16, Eq. 6.1.1], Iν [·] is the modifiedBessel function of the first kind and arbitrary order ν [16, Eq.9.6.20], and µ > 0 is the real extension of n/2 so that

µ =E2(R2)

V (R2)× 1 + η2

(1 + η)2∴ µ =

Ω2

E(R4) − Ω2× 1 + η2

(1 + η)2

(3)

with V (·) denoting variance. By setting µ = 0.5 in (2), theHoyt envelope PDF with Hoyt parameter b = (η − 1)/(η + 1)can be obtained in an exact manner. In addition, for µ = mand η → 0, or in the same way, for µ = m/2 and η → 1, theη-µ distribution reduces to the Nakagami-m one. Defining [6]

Yυ(λ, β) =2−υ+ 3

2

√π

Γ(υ)

(1 − λ2)υ

λυ− 1

2

∫

∞

β

x2υ exp(

−x2)

Iυ− 1

2

[

λx2]

dx (4)

the cumulative distribution function (CDF) of the η-µ envelopeR is given by

FR(r) = 1 − Yµ

(

1 − η

1 + η,

√

µ

2η

(1 + η) r

r

)

(5)

In addition, it can be shown that the in-phase X andquadrature Y components of the η − µ fading signal areindependent of each other and have marginal PDFs [14]

fZ(z) =µµ|z|2µ−1

Ωµz Γ(µ)

exp

(

−µz2

Ωz

)

,−∞ < z < ∞ (6)

where Z ≡ X and Ωz ≡ Ωx , E(X2) = 2µσ2X = ηr2/(1 +

η), or Z ≡ Y and Ωz = Ωy , E(X2) = 2µσ2Y = r2/(1+ η),

as required.Next, we shall derive the PCR for η-µ fading channels.

III. PHASE CROSSING RATE

The PCR, denoted as NΘ(θ), is defined as the averagenumber of upward (or downward) crossings per second of afading signal at a specific phase level θ. A general expressionfor this statistic can be formulated as

NΘ(θ) =

∫

∞

0

θfΘ,Θ(θ, θ)dθ (7)

where fΘ,Θ(·, ·) stands for the joint PDF of the phase Θ and

its time derivative Θ. Making the appropriate substitutions andcarrying out the necessary algebraic manipulation, we obtaina simple and closed-form expression for the PCR operating inη-µ fading channels

NΘ(θ) =fm

√π ηµ− 1

2 | sin(2θ)|2µ−1 Γ(2µ − 1

2)

23

2 Γ2(µ)

× (1 + η + (1 − η) cos(2θ))1−2µ (8)

where fm is the maximum Doppler shift in Hz. Someproperties related to (8) will be posteriorly discussed. How-ever, it is noteworthy now that (8) is valid only for µ >1/4, which is consistent with the Nakagami-m fading model(m > 1/2) presented in [15]. To our best knowledge, thereare no works in the literature reporting this statistics for η-µ generalized fading channels. In the following, a detailedexplanation concerning the achievement of (8) is depicted.

A. Derivation of the PCR

From [13], the component Z , Z = X or Z = Y , can bewritten as Z = S|Z|, where S represents sgn(Z) (sign of Z)and |Z| is Nakagami-m distributed. For mathematical simpli-city, we write Z = SN , in which N denotes a Nakagami-mvariate.

Let Z be the time derivative of Z . From the above, Z =SN +SN . Because S assumes the constant values ±1, exceptfor the transition instants (−1 → +1 and +1 → −1), its timederivative S is nil. In addition, because Z is continuous, thetransition instants occur exactly and only at the zero crossinginstants of Z , when N = |Z| is nil. Therefore, SN = 0always and Z = SN . It has been shown in [10] that N isindependent of N and Gaussian distributed. Knowing thatZ = SN , Z is also Gaussian distributed conditioned onZ = SN , having the same distribution parameters as N . Morespecifically, X and Y obey a Gaussian distribution with zeromean and standard deviation σx = πfmr

√

η/(µ(1 + η)) andσy = πfmr

√

1/(µ(1 + η)), respectively. Consequently, X isindependent of X , and Y is independent of Y . As X andY are independent processes, follows that X is independentof Y and Y is independent of X . Then, noting that X andY are Gaussian distributed with the cited parameters, andthat the PDF of X or Y is expressed in (6), the joint PDF

924

fX,X,Y,Y (x, x, y, y) is given by

fX,X,Y,Y (x, x, y, y) = fX(x) fX(x) fY (y) fY (y) =

=|x|2µ−1|y|2µ−1µ2µ+1(1 + η)2µ+1

2 f2m π3 r4µ+2 ηµ+ 1

2 Γ2(µ)

× exp

(

−µ(1 + η)

2 r2

(

x2

f2m π2η

+y2

f2m π2

+2x2

η+ 2y2

))

(9)

From [14], it is possible to write X and Y as X = R cos(Θ)and Y = R sin(Θ). Therefore, X = R cos(Θ) − RΘ sin(Θ)and Y = R sin(Θ) + RΘ cos(Θ). Following the standardstatistical procedure for the transformation of variates and afteralgebraic manipulations, the joint PDF fR,R,Θ,Θ(r, r, θ, θ) ofthe envelope, the phase, and their time derivatives, is obtainedas

fR,R,Θ,Θ(r, r, θ, θ) =r4µ µ2µ+1(1 + η)2µ+1| sin(2θ)|2µ−1

22µ f2m π3 r4µ+2 ηµ+ 1

2 Γ2(µ)

× exp

(

−µ(1 + η)

2 r2

(

(r cos(θ) − r θ sin(θ))2

f2m π2η

+(r sin(θ) + r θ cos(θ))2

f2m π2

+2r2 cos2(θ)

η+ 2r2 sin2(θ)

))

(10)

For η = 1 and µ = m/2, (10) reduces in an exact mannerto the Nakagami-m case given in [15, Eq. 4], as expected.For µ = 0.5 in (10), the Hoyt joint PDF with parameter b =(η − 1)/(η + 1) [11, Eq. 10] is also obtained. In order toderive the PCR, it is necessary now to attain f

Θ,Θ(θ, θ) (seeAppendix). For this purpose, we integrate (10) with respectto r, from 0 to ∞, and with respect to r, from −∞ to ∞.After obtaining f

Θ,Θ(θ, θ) and substituting it in (7), the PCRfor η-µ fading channels is achieved subsequent to perform theintegration respecting to θ.

B. Special Cases

One of the means to verify the correctness of the PCRobtained in (8) is reducing it for some particular cases whosesolutions are known. These special cases will be reproducedhere for convenience. By substituting η = 1 and µ = m/2 in(8), the PCR for Nakagami-m fading channels is obtained as[15, Eq. 12]

NΘ(θ) =

√π fm Γ(m − 1

2) | sin(2θ)|m−1

2m+ 1

2 Γ2(m2

)(11)

In the same way, for η arbitrary and µ = 0.5, (8) reduces tothe Hoyt case presented in [17]

NΘ(θ) =fm

2√

2(12)

Note that (12) is independent of the phase level θ, which iscoherent with the result obtained in [17]. Moreover, it givesthe same result of the Rayleigh case (η = 1).

IV. LEVEL CROSSING RATE AND AVERAGE FADE

DURATION

The LCR is defined as the average number of times a fadingsignal crosses a given signal level in the negative (or positive)direction within a certain period of time. Denoting the crossinglevel as r, the LCR is estimated as

NR(r) =

∫

∞

0

rfR,R(r, r)dr (13)

where fR,R(·, ·) is the joint PDF of R and R. The AFD isdefined as the mean time the received envelope is found belowthe given threshold r and is formulated as

TR(r) =FR(r)

NR(r)(14)

in which FR(r) is given in (5). The LCR for η − µ fadingchannels can be expressed as

NR(r) =fm

√π r4µ−1(1 + η)2µ− 1

2 µ2µ− 1

2

ηµ r4µ−1 22µ−2 Γ2(µ)

×∫ π

2

0

(sin(2θ))2µ−1√

1 + η + (−1 + η) cos(2θ)

× exp

(

−r2(1 + η)µ

2r2η(1 + η − (−1 + η) cos(2θ))

)

dθ (15)

The AFD follows directly from (15) and (5). As far as weknow, these results are new.

A. Derivation of the LCR

The derivation of the LCR follows the same steps used toobtain the PCR in the previous section. The difference nowis that we have to find the joint PDF fR,R(r, r) instead offΘ,Θ(θ, θ). As an intermediate step, let us first derive the the

following

NR,Θ(r, θ) =

∫

∞

0

rfR,R,Θ(r, r, θ)dr (16)

Integrating (10) with respect to θ and applying this result in(16), we have

NR,Θ(r, θ) =fm

√π r4µ−1(1 + η)2µ− 1

2 µ2µ− 1

2 | sin(2θ)|2µ−1

ηµ r4µ−1 22µ Γ2(µ)

×√

1 + η + (−1 + η) cos(2θ)

× exp

(

−r2(1 + η)µ

2r2η(1 + η − (−1 + η) cos(2θ))

)

(17)

Integrating (17) with respect to θ from 0 to 2π, the LCR, asgiven in (15), is finally attained.

B. Special Cases

For illustration purposes, some special cases will be re-produced here. The LCR for the Nakagami-m fading case isobtained from (15) for η = 1 and µ = m/2 as

NR(r) =

√2πfm mm−

1

2 r2m−1

r2m−1Γ(m)exp

(

−mr2

r2

)

(18)

925

which is in accordance with [10, Eq. 17]. By setting µ = 0.5in (15), the LCR for the Hoyt case with parameter b = (η −1)/(η +1) is derived, being once more coherent with [11, Eq.13]. However, it remains in a integral form. For η = 1, theLCR for the Rayleigh case [18, Eq. 4.64] is obtained from theHoyt one as

NR(r) =√

2πfm

(

r

r

)

exp

(

−r2

r2

)

(19)

V. NUMERICAL RESULTS AND DISCUSSIONS

In this Section, some plots illustrate the expressions ob-tained here. In addition, the validity of these expressions ischecked by comparing the theoretical curves against the MonteCarlo simulation results. As will be observed, an excellentagreement is achieved between the simulation results and theanalytical data. Fig. 1 depicts the normalized PCR, NΘ(θ)/fm,as a function of the phase, for arbitrary values of µ and η =0.5. For µ = 0.5 (Hoyt case), this statistics is independent ofthe phase level, assuming a constant value equal to 1/(2

√2).

For values of µ higher than 0.5, the curves assume null valueat integer multiples of π/2. On the other hand, for values ofµ lower than 0.5, the curves are convex tending to infinity atintegers multiples of π/2. It is worth noting that, except for thecase µ = 0.5 in which it is uniform, the PCR is periodic withperiod π. This result is indeed expected since the distributionof the phase [14] is also periodic with period π. The simulatedcurves contrasting with the theoretical ones are presented inFig. 2 for several fading conditions. A very good adjustmentis noticeable.

Figs. 3 and 4 show the normalized LCR (left axis),NR(r)/fm, and AFD (right axis), TR(r)fm, as a function ofthe normalized envelope ρ = r/r. Some implications can besaid concerning these figures. By increasing µ and keepingη constant, note that lower levels are crossed at lower rateswhereas higher levels are crossed at higher rates. In otherwords, when the parameter µ increases, the signal has asubstantial improvement in its level. The same can be said forthe case of η increasing and µ constant. Comparing Figs. 3 and4, we note that both LCR and AFD are more influenced by thevariation of the parameter µ than the one of the parameter η. Infact, the shape of the AFD curves is loosely dependent of theparameter η. For several fading conditions, Figs. 5 and 6 plotthe simulation curves of the normalized LCR as a functionof the normalized envelope. Note the excellent agreementbetween the simulated and theoretical curves. For other fadingconditions, exhaustive simulations have been performed and,in all the cases, a very good adjustment has been achieved.

VI. CONCLUSIONS

In this paper, a simple and closed-form expression forthe PCR operating in η-µ generalized fading channels wasobtained. An interesting feature related to the shape of thecurves is that, apart from µ = 0.5, the PCR is periodic withperiod π. Also, exact formulas for the LCR and AFD wereattained for channels undergoing the η-µ fading statistics. Forseveral fading conditions, the formulas have been validatedby reducing the general case to some special cases, for which

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

/20- /2-

= 3, 2, 1.5, 1, 0.75, 0.5, 0.45, 0.425, 0.4, 0.375, 0.35

N(

) / f m

Fig. 1. PCR for η-µ fading channels (η = 0.5, µ varying).

0,0

0,1

0,2

0,3

0,4

0,5

= 0.5

= 2

= 1

= 0.5

0 /2- /2

N

() /

f m

-

Fig. 2. Comparison between the theoretical and simulated curves of the PCRfor η-µ fading channels (solid → theoretical curves; squared → Monte Carlosimulated curves).

the solutions are known, and, more generally, by means ofsimulation. For other fading conditions, exhaustive simulationshave been performed and, in all the cases, a very goodadjustment has been achieved.

REFERENCES

[1] C.-X. Wang, N. Youssef, and M. Patzold, “Level crossing rate andaverage fade duration of fades of deterministic simulation models forNakagami-Hoyt fading channels ,” in The 5th International Symposiumon Wireless Personal Multimedia Communications, vol. 1, Oct. 2002,pp. 272–276.

[2] J. D. Parsons, The Mobile Radio Channel, 2nd ed. Chichester, U.K.:John Wiley Sons, 2000, vol. 1.

[3] S. Stein, “Fading channel issues in system engineering,” IEEE JournalSelected Areas in Commun., vol. 5, no. 2, pp. 68–69, Feb. 1987.

926

10-5

10-4

10-3

10-2

10-1

100

101

102

103

-30 -25 -20 -15 -10 -5 0 5 1010-4

10-3

10-2

10-1

100

101

TR (r) fm

= 0.5, 0.75, 1,..., 2.5

= 2.5, 2.25, 2,..., 0.5

NR(r

) / f m

20Log( )

Fig. 3. LCR and AFD for η-µ fading channels (η = 0.5, µ varying).

10-3

10-2

10-1

100

101

102

103

104

-30 -25 -20 -15 -10 -5 0 5 1010-4

10-3

10-2

10-1

100

101

TR (r) fm

= 0.01, 0.02, 0.03, 0.05, 0.1, 0.2, 0.3, 0.4, 1

= 1, 0.4, 0.3, 0.2, 0.1, 0.05, 0.03, 0.02, 0.01

NR(r

) / f m

20Log( )

Fig. 4. LCR and AFD for η-µ fading channels (µ = 0.6, η varying).

[4] G. Tzeremes and C. G. Christodoulou, “Use of Weibull distributionfor describing outdoor multipath fading,” in Proc. IEEE Antennas andPropagation Society Int. Symp., San Antonio, TX, 2002, pp. 232–235.

[5] M. D. Yacoub, “The η−µ distribution: A general fading distribution,” inIEEE Atlantic City Fall Vehicular Technology Conference 2000, Boston,USA, Sep. 2000.

[6] ——, “The κ−µ and the η −µ distribution,” Accepted for publicationin IEEE Ant. and Prop. Magazine, 2006.

[7] J. C. S. S. Filho and M. D. Yacoub, “Highly accurate η−µ approxima-tion to the sum of M independent non-identical Hoyt variates ,” IEEEAnt. and Wireless Prop. Lett., vol. 4, pp. 436–438, 2005.

[8] W. C. Jakes, Microwave Mobile Communications. New York: Wiley,1974.

[9] N. C. Sagias, D. A. Zogas, G. K. Karagiannidis, and G. S. Tombras,“Channel capacity and second-order statistics in Weibull fading,” IEEECommun. Lett., vol. 8, no. 6, pp. 377–379, Jun. 2004.

[10] M. D. Yacoub, J. E. Vargas B. and L. G. R. Guedes, “On high orderstatistics of the Nakagami-m distribution,” IEEE Trans. Veh. Technol.,vol. 48, no. 3, pp. 790–793, May 1999.

-30 -25 -20 -15 -10 -5 0 5 1010-4

10-3

10-2

10-1

100 = 0.5

= 2, 1.5, 1.25, 1, 0.75

NR(r

) / f m

20Log( )

Fig. 5. Comparison between the theoretical and Monte Carlo simulatedcurves of the LCR for η-µ fading channels (solid → theoretical curves;squared → Monte Carlo simulated curves).

-30 -25 -20 -15 -10 -5 0 5 1010-4

10-3

10-2

10-1

100 = 1 = 0.9, 0.1, 0.03, 0.01, 0.001

NR(r

) / f m

20Log( )

Fig. 6. Comparison between the theoretical and Monte Carlo simulatedcurves of the LCR for η-µ fading channels (solid → theoretical curves;squared → Monte Carlo simulated curves).

[11] N. Youssed, C-X. Wang, and M. Patzold, “A Study on the second-orderstatistics of Nakagami-Hoyt mobile fading channels,” IEEE Trans. Veh.Technol., vol. 54, no. 4, pp. 1259–1265, Jul. 2005.

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[13] M. D. Yacoub, G. Fraidenraich, J. C. S. Santos Filho, “Nakagami-mphase-envelope joint distribution,” Electron. Lett., vol. 41, no. 5, Mar.2005.

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APPENDIX

The goal of this Appendix is to show the final expressionof the joint PDF f

Θ,Θ(θ, θ) necessary to the achievement ofthe PCR. After integrate (10) with respect to r, from 0 to ∞,and with respect to r, from −∞ to ∞, follows

fΘ,Θ(θ, θ) =

24µ−1 f2m π4µ− 1

2 (1 + η)2µ−1 µ2µ−1| sin(2θ)|2µ−1 Γ(

2µ + 1

2

)

ηµ− 3

2 Γ2(µ)[

(2f2mπ2 + η θ2) cos4(θ) + 2η (2f2

mπ2 + θ2) cos2(θ) sin2(θ) + η (2f2mπ2η + θ2) sin4(θ)

]

×(

(1 + η)µ (f2m π2(3 + 2η + 3η2) + 4 η θ2 + f2

mπ2(−4(−1 + η2) cos(2θ) + (−1 + η)2 cos(4θ)))

f2m η (cos2(θ) + η sin2(θ))

)1

2−2µ

×√

(1 + η)µ (cos2(θ) + η sin2(θ))

f2m η

(20)

928