IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008
819
Second-Order Statistics of η -μ Fading Channels: Theory and Applications Daniel Benevides da Costa, Student Member, IEEE, Jos´e Cˆandido Silveira Santos Filho, Member, IEEE, Michel Daoud Yacoub, Member, IEEE, and Gustavo Fraidenraich, Member, IEEE
Abstract— In this work, a number of new closed-form expressions for the η-μ fading channels envolving the joint statistics of the envelope, phase, and their time derivatives are obtained. Level crossing rate (LCR), average fade duration (AFD), and phase crossing rate (PCR) are also derived. The expressions are thoroughly validated by reducing them to some particular known cases and, more generally, by means of Monte Carlo simulation. We then provide alternative (i) singlefold integral exact formulations and (ii) highly-accurate approximations to the level-crossing statistics of multibranch maximal-ratio combining (MRC) and equal-gain combining (EGC) systems, respectively, operating over independent Hoyt fading channels, for which the exact solutions appear in the literature in multifold integral forms. Index Terms— Average fade duration, generalized fading channels, level crossing rate, phase crossing rate.
I. I NTRODUCTION VER the years, a great number of channel models have been proposed that closely follow the behaviour of the mobile radio signal. The short term signal variation is described by several distributions such as Rayleigh, Hoyt (Nakagami-q), Rice, Nakagami-m, and Weibull. Among the short term fading scenarios, Nakagami-m has been given a special attention for its ease of manipulation and wide range of applicability. On the other hand, in many situations, its tail does not seem to follow the true statistics [1], [2]. Efforts have been made to extend existing fading models along different directions so as to obtain more flexible scenarios (e.g., [3], [4]). Recently [5], [6], the η-μ fading model has been investigated that includes as special cases Nakagami-m and Hoyt (Nakagami-q). Its flexibility renders it more adaptable to situations in which neither of these two distributions yield good fit [5], [6], particularly at the tail portion, where several distributions fail to follow the true statistics. Moreover, it has been shown that the η-μ distribution can be used to accurately
O
Manuscript received October 3, 2006; revised December 21, 2006; accepted February 19, 2007. The associate editor coordinating the review of this paper and approving it for publication was R. Mallik. This work was partially supported by FAPESP (05/59259-7). The material in this correspondence was presented in part at the IEEE International Telecommunication Symposium, Fortaleza, Brazil, September, 2006. D. B. da Costa, J. C. S. Santos Filho, and M. D. Yacoub are with the Wireless Technology Laboratory (WissTek), Department of Communications, School of Electrical and Computer Engineering, University of Campinas, PO Box 6101, 13083-852 Campinas, SP, Brazil (e-mail: {daniel, candido, michel}@wisstek.org). G. Fraidenraich was with the the Wireless Technology Laboratory (WissTek), Department of Communications, School of Electrical and Computer Engineering, University of Campinas, Brazil. He is now with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305-9515 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2008.060774.
approximate the distribution of the sum of independent, nonidentical Hoyt variates having arbitrary mean powers and arbitrary fading degrees [7]. The gain in flexibility of the η-μ distribution is obtained with the inclusion of one additional parameter. With respect to Nakagami-m, the additional parameter is η, which accounts for the unequal power of the in-phase and quadrature components of the fading signal or, equivalently, for the correlation between these components. With respect to Hoyt (Nakagamiq), the additional parameter is μ, which accounts for the clustering of the multipath signals. All this comes with an attempt to better describe the fading signal of an environment within which the diffuse scattering field results from spatially correlated surfaces, characterizing a real non-homogeneous environment [8]. The aim of this paper is to explore the second-order statistics for the η-μ fading channels. With such a target, a reasonable number of original, exact, closed-form expressions concerning the η-μ statistics are derived that include Hoyt and Nakagami-m as particular cases. Additionally, an application is proposed in which the exact expressions of LCR and AFD for the η-μ distribution is used to very closely approximate those of multibranch equal-gain combining (EGC) for which complex, multifold integrals yield the exact solution [9, Eq. 14]. Moreover, the derived expressions constitute exact solutions for the multibranch maximal-ratio combining (MRC), and replace the complex, multifold integrals for the exact solution [9, Eq. 18]. The analytical results are thoroughly validated by reducing them to some known particular cases and, more generally, by means of Monte Carlo simulation. II. T HE η-μ FADING M ODEL In this Section, we revisit the η-μ fading model [5], [6]. The η-μ fading model considers a signal composed of clusters of multipath waves, so that within any one cluster the phases of the scattered waves are random and have similar delay times with delay-time spreads of different clusters being relatively large. The envelope R of the η-μ fading signal can be written as 2μ 2 Xi + Yi2 (1) R2 = i=1
where Xi and Yi are Gaussian in-phase and quadrature zeromean processes, i.e, E(Xi ) = E(Yi ) = 0, and E(·) denotes statistical average. The η-μ envelope distribution may be found in two formats, namely Format 1 and Format 2. In Format 1, the variances of the independent in-phase and quadrature processes are arbitrary with their ratio defined as η = E(Xi2 )/E(Yi2 ), η > 0. In Format 2, the variances of the
c 2008 IEEE 1536-1276/08$25.00
820
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008
dependent in-phase and quadrature processes are identical with the correlation coefficient defined as η = E(Xi Yi )/E(Xi2 ) (or, equivalently, η = E(Xi Yi )/E(Yi2 )), −1 < η < 1. The η-μ envelope probability density function (PDF) is given by [5], [6] 2μ 2 √ 1 r 4 πμμ+ 2 hμ r exp −2μh fR (r) = 1 μ− rˆ Γ(μ)H 2 rˆ rˆ 2 r × Iμ− 12 2μH (2) rˆ √ where rˆ = Ω = E(R2 ), Γ(·) is the Gamma function [10, Eq. 6.1.1], Iν [·] is the modified Bessel function of the first kind and arbitrary order ν [10, Eq. 9.6.20], and μ > 0 is related to the number of multipath clusters so that
2 H E 2 (R2 ) × 1+ (3) μ= 2V (R2 ) h with V (·) denoting variance. In Format 1, h = (2+η −1 +η)/4 and H = (η −1 − η)/4, whereas in Format 2, h = 1/(1 − η 2 ) and H = η/(1 − η 2 ). One Format can be obtained from the other by the bilinear transformation η1 = (1 − η2 )/(1 + η2 ), where η1 denotes the parameter η in Format 1 and η2 denotes the parameter η in Format 2. Although derived for 2μ integer, the formulations are used for any μ > 0. The Hoyt (or Nakagami-q) distribution can be obtained from the η-μ distribution in an exact manner by setting μ = 0.5. In this case, the Hoyt (or Nakagami-q) parameter is given by b = − (1 − η) / (1 + η) (or q 2 = η) in Format 1 or b = −η (or q 2 = (1 − η) / (1 + η)) in Format 2. The Nakagami-m distribution can be obtained from the η-μ distribution in an exact manner by setting μ = m and η → 0 or η → ∞ in Format 1, or η → ±1 in Format 2. In the same way, it can be obtained by setting μ = m/2 and η → 1 in Format 1 or η → 0 in Format 2. From (2), the kth moment of R is calculated as k Γ 2μ + k2 Ω 2 E(Rk ) = k k hμ+ 2 (2μ) 2 Γ(2μ) 2 k 1 H k 1 × 2 F1 μ + + , μ + ; μ + ; (4) 4 2 4 2 h where 2 F1 (·, ·; ·; ·) is the hypergeometric function [10, Eq. 15.1.1]. The cumulative distribution function (CDF) of the η-μ envelope R is written as H r , 2hμ (5) FR (r) = 1 − Yμ h rˆ where [6]
3 √ 2−υ+ 2 π (1 − λ2 )υ Yυ (λ, β) 1 Γ(υ) λυ− 2 ∞ x2υ exp −x2 Iυ− 12 λx2 dx
(6)
β
The derivations that follow shall are detailed for the Format 1 case only. As already detailed, one format can be converted into another by a simple bilinear transformation.
III. J OINT E NVELOPE AND P HASE D ISTRIBUTION In this Section, the joint probability density function ˙ of the envelope R, phase Θ, and their ˙ θ, θ) fR,R,Θ, ˙ ˙ (r, r, Θ ˙ is derived. Following respective time derivatives R˙ and Θ [11], the in-phase X and quadrature Y components of the η-μ fading signal are independent of each other and have marginal PDFs μz 2 μμ |z|2μ−1 exp − fZ (z) = μ , −∞ < z < ∞ (7) ΩZ Γ(μ) ΩZ where Z ≡ X or Z ≡ Y , as required. In addition, ΩZ = E(Z 2 ). Of course, because η = ΩX /ΩY , it can be shown that ΩX = rˆ2 η/(1 + η) and ΩY = rˆ2 /(1 + η). Each quadrature component Z can be written as Z = S|Z|, where S = sgn(Z) (sign of Z) and |Z| is Nakagami-m distributed [11]. For convenience, we write Z = SN , in which N denotes a Nakagami-m variate. Differentiating Z with respect ˙ + S N˙ . Because S assumes to time, it follows that Z˙ = SN the constant values ±1, except for the transition instants (−1 → +1 and +1 → −1), its time derivative S˙ is nil. In addition, because Z is continuous, the transition instants occur exactly and only at the zero crossing instants of Z, ˙ when N = |Z| is nil. Therefore, SN = 0 always and Z˙ = S N˙ . It has been shown in [12] that N˙ is Gaussian and independent of N . Knowing that Z˙ = S N˙ , then Z˙ is also Gaussian distributed conditioned on Z = SN , having the same distribution parameters as those of N˙ . Thus, Z˙ is indeed independent of Z. More specifically, X˙ and Y˙ are zero mean Gaussian distributed with standard deviations πfm rˆ η/(μ(1 + η)) and πfm rˆ 1/(μ(1 + η)), respectively, where fm is the maximum Doppler shift in Hz. As already ˙ and Y is independent of mentioned, X is independent of X, Y˙ . Because X and Y are independent processes, it follows that ˙ Y , and Y˙ are mutually independent. Finally, noting that X, X, X˙ and Y˙ are Gaussian distributed with the cited parameters, and that the PDF of X or Y are expressed in (7), the joint PDF fX,X,Y, ˙ y, y) ˙ is given by fX,X,Y, ˙ y, y) ˙ = ˙ ˙ Y˙ (x, x, Y˙ (x, x, ˙ fY (y) fY˙ (y). ˙ Now, the quadrature components fX (x) fX˙ (x) X and Y can be written in terms of the envelope R and phase Θ as X = R cos Θ and Y = R sin Θ. Accordingly, ˙ sin Θ and Y˙ = R˙ sin Θ+RΘ ˙ cos Θ. Then, X˙ = R˙ cos Θ−RΘ following the standard statistical procedure of transformation of variables, and after some algebraic manipulations, the joint ˙ is obtained as PDF fR,R,Θ, ˙ θ, θ) ˙ ˙ (r, r, Θ 2μ+1 | sin(2θ)|2μ−1 r4μ ˙ = [(1 + η) μ] fR,R,Θ, ˙ θ, θ) ˙ ˙ (r, r, Θ 2 η μ+ 12 Γ2 (μ) r 22μ π 3 fm ˆ4μ+2
(1 + η)μ [r˙ cos(θ) − r θ˙ sin(θ)]2 + × exp − 2 π2 η 2 rˆ2 fm 2 [r˙ sin(θ) + r θ˙ cos(θ)]2 cos (θ) + + sin2 (θ) + 2r2 2 π2 fm η (8)
For the appropriate conditions, as detailed before, (8) reduces in an exact manner to the Nakagami-m case [13, Eq. 4] and Hoyt one [14, Eq. 10]. From (8), by performing the appropriate integrations, several exact, closed-form joint PDFs can be found. These are listed in Appendix.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008
IV. C ROSSING R ATES AND AVERAGE FADE D URATION Let P be a process, P˙ its time derivative and p a certain level. The crossing rate of the process P , NP (p), is defined as ∞ NP (p) = p˙ fP,P˙ (p, p)d ˙ p˙ (9)
821
A. Maximal-Ratio Combining ˜ is given as In MRC, the combiner output envelope R ˜2 = R
0
where fP,P˙ (·, ·) is the joint PDF of P and P˙ . The AFD TP (p) is obtained as FP (p) (10) TP (p) = NP (p) where FP (p) is the CDF of P . In the sequel, the LCR, AFD, and PCR shall be derived for η-μ fading channels.
M
Ri2
(13)
i=1
where Ri is the envelope at the ith branch, i = 1, . . . , M . Here, Ri is Hoyt distributed with mean power Ωi = E(Ri2 ) and Hoyt fading parameter bi . Indeed, each Hoyt envelope Ri can be modeled as a sum of independent unbalanced squared Gaussian variates, so that (13) coincides with the η-μ envelope ˜ ≡ R) for the η-μ distribution parameters model (1) (i.e., R chosen as
A. LCR and AFD
Ω = M Ωi
(14a)
μ = M/2 1 + bi η= 1 − bi
(14b)
LCR and AFD concern the envelope statistics. In this case, in (9) and in (10) P ≡ R and p ≡ r. The joint PDF fR,R˙ (r, r) ˙ is derived from (8), after the appropriate integrations. Then √ 2μ− 12 4μ−1 r fm π [(1 + η)μ] NR (r) = 2μ−2 μ 2 4μ−1 2 η Γ (μ) rˆ π2 [sin(2θ)]2μ−1 1 + η − (1 − η) cos(2θ) 0 [(1 + η)2 + (1 − η 2 ) cos(2θ)]μ r2 × exp − dθ 2 η rˆ2 (11)
Using (14) into (11) and (5), we obtain alternative singlefold integral exact formulations to the LCR and AFD of MRC systems over Hoyt fading channels, currently available in the literature in a multifold integral form [9, Eq. 18].
The AFD follows directly from (11) and (5).
B. Equal-Gain Combining
B. PCR
˜ is given as For EGC, the combiner output envelope R M i=1 Ri ˜= √ R (15) M In this case, as before, the exact formulations for LCR and AFD are available in the literature in a multifold integral form [9, Eq. 14]. Here we propose highly-accurate approximations to the output LCR and AFD by means of (11) and (5). In order to render these expressions good approximations (i.e., ˜ ≈ R), we match the parameters of R, namely Ω, η, and R μ, to those calculated from the exact moments of the EGC ˜ In such a case [5]–[7] combiner output R.
PCR concerns the phase statistics. In this case, in (9) P ≡ Θ ˙ is derived from (8), after and p ≡ θ. The joint PDF fΘ,Θ˙ (θ, θ) the required integrations. Then √ 1 fm π η μ− 2 Γ(2μ − 12 )| sin(2θ)|2μ−1 NΘ (θ) = 3 (12) 2μ−1 2 2 Γ2 (μ) [1 + η + (1 − η) cos(2θ)] For the appropriate fading conditions, as described previously, (11) and (12) reduce exactly and respectively to the LCR and PCR of the Nakagami-m signal derived in [12, Eq. 17] and [13, Eq. 12]. Analogously, the LCR as well as the PCR of the Hoyt fading channel given in [14, Eq. 13] and [15], respectively, can be obtained exactly from (11) and (12). V. A N A PPLICATION The LCR and AFD of multibranch MRC and EGC systems operating over independent Hoyt fading channels are currently known in a multifold integral form [9, Eqs. 14 and 18]. In this Section, we use the above-derived LCR and AFD of the η-μ fading channels as simple alternative exact solutions to the LCR and AFD of multibranch MRC systems over Hoyt channels. In the same way, they are used as highly-accurate approximations to the LCR and AFD of EGC systems in the same scenario. In each case, the η-μ distribution parameters Ω, η, and μ must be properly calculated in terms of the Hoyt branch parameters. The procedure is detailed next. The exactly same approach can be used for the Nakagami-m case, in which case the exact solution is given in [16], [17]. For brevity, only the Hoyt case is explored in this paper.
ηa,b
˜2) Ω = E(R √ √ 2c − 3 − 2c ± 9 − 8c =√ √ 2c + 3 − 2c ± 9 − 8c c
μa,b =
˜6) E(R Ω3
2
−
˜4) 3E(R Ω2
˜4) E(R Ω2
+2 2 −1
2 1 + ηa,b Ω2 × ˜ 4 ) − Ω2 (1 + ηa,b )2 E(R
(14c)
(16) (17)
(18)
(19)
By carrying out this, two pairs of estimators for η and μ are found, namely (ηa ,μa ) and (ηb ,μb ). The correct pair is the one ˜ − E(R)|, for E(R) estimated that leads to the smallest |E(R) from (4) at (Ω, η, μ) = (Ω, ηa , μa ) and (Ω, η, μ) = (Ω, ηb , μb ). ˜ k ) (k = 1, 2, 4, 6) It remains to find the exact moments E(R from the EGC conditions as required above. These can be
822
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008
Fig. 1.
LCR and AFD for η-μ fading channels (η = 0.5, μ varying).
Fig. 2.
LCR and AFD for η-μ fading channels (μ = 0.6, η varying).
calculated from (15) as
Fig. 3. Comparison between the analytical and simulated curves of the LCR for η-μ fading channels (solid → analytical curves; squared → simulated curves).
Fig. 4. Comparison between the analytical and simulated curves of the LCR for η-μ fading channels (solid → analytical curves; squared → simulated curves).
kM −2 k1 k k k1 kM −2 1 validated in Figs. 3 and 4 through Monte Carlo simulation. ... ... k1 k2 kM −1 The Monte Carlo simulation was performed by generating M k/2 k =0 k =0 kM −1 =0 1 2 k the autocorrelated variates Xi and Yi in accordance with the k−k1 E(R1 )E(R2k1 −k2 ) . . . E(RMM −1 ) well-established Jakes/Clark model [18]. The parameter η was (20) adjusted by changing the variance ratio between Xi and Yi , j and the parameter μ was introduced by varying the number of where E Ri of each Hoyt summand Ri in (20) is obtained Gaussian variates as in (1). From the illustrations, an excellent from (4) with R = Ri , μ = 0.5, η = (1 + bi )/(1 − bi ). Of agreement between the simulated and theoretical curves is course, Ωi = E(Ri2 ). observed. A myriad of other examples have been investigated by the authors, and an excellent agreement has always been VI. N UMERICAL R ESULTS AND D ISCUSSIONS attained. Fig. 5 depicts the simulated and theoretical curves Figs. 1 and 2 show the normalized LCR (left axis), of the PCR for η-μ fading channels and, once more, a very NR (r)/fm , and AFD (right axis), TR (r)fm , as functions of good adjustment between the curves occurs. For μ = 0.5 of the phase, assuming the normalized envelope ρ = r/ˆ r. By increasing μ for η (Hoyt case), the PCR is independent √ constant, lower levels are crossed at lower rates, whereas a constant value equal to 1/2 2, which agrees with [15]. In Fig. 6, our singlefold integral LCR and AFD approximahigher levels are crossed at higher rates. A similar behaviour is observed for increasing η and μ constant. By comparing tions to EGC systems operating in Hoyt fading are compared Figs. 1 and 2, we note that the variation of the parameter μ to the multifold integral exact formulations presented in [9, has a stronger influence on the LCR and AFD performances Eq. 14], for M = 2, 3, 4, bi = 0.1, 0.6, and Ωi = 1. Note than that of η. The LCR and AFD analytical expressions are how the proposed simple approximations yield results that ˜k ) = E(R
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008
823
1
(1 + η)1+2μ μ2μ+ 2 | sin(2θ)|2μ−1 r4μ 3 η μ 22μ−1 fm π 2 Γ2 (μ) (1 + η)2 + (1 − η 2 ) cos(2θ) rˆ4μ+1 ⎫ ⎧ 2 2 2 2 ⎨ μ(1 + η) fm π [3 + η(2 + 3η)] + 4 η θ˙2 + fm π [4(1 − η 2 ) cos(2θ) + (1 − η)2 cos(4θ)] r2 ⎬ × exp − 2 π 2 η[cos2 (θ) + η sin2 (θ)] r ⎩ ⎭ 8 fm ˆ2
˙ = fR,Θ,Θ˙ (r, θ, θ)
(21)
1
[μ(1 + η)] 2 +2μ | sin(2θ)|2μ−1 r4μ−1 3 η μ 22μ−1 fm π 2 Γ2 (μ) 1 + η − (1 − η) cos(2θ) rˆ4μ+1 " 2 2 # 2 2 μ(1 + η) fm π [1 + η(6 + η)]r2 + 4 η r˙ 2 − fm π (1 − η)2 cos(4θ) r2 × exp − 2 π 2 η[η cos2 (θ) + sin2 (θ)] r 8 fm ˆ2
˙ θ) = fR,R,Θ ˙ (r, r,
˙ = fΘ,Θ˙ (θ, θ)
(22)
2 4μ− 12 24μ−1 fm π (1 + η)2μ−1 μ2μ−1 | sin(2θ)|2μ−1 Γ 2μ + 12
3 2 π 2 + η θ˙ 2 ) cos4 (θ) + 2η (2f 2 π 2 + θ˙ 2 ) cos2 (θ) sin2 (θ) + η (2f 2 π 2 η + θ˙ 2 ) sin4 (θ) η μ− 2 Γ2 (μ) (2fm m m 12 −2μ 2 2 2 2 π (3 + 2η + 3η 2 ) + 4 η θ˙2 + fm π [4(1 − η 2 ) cos(2θ) + (1 − η)2 cos(4θ)]} (1 + η) μ {fm × 2 η [cos2 (θ) + η sin2 (θ)] fm $ (1 + η) μ [cos2 (θ) + η sin2 (θ)] (23) × 2 η fm
Fig. 5. Comparison between the theoretical and simulated curves of the PCR for η-μ fading channels (solid → theoretical curves; squared → simulated curves).
are almost indistinguishable from the exact multifold integral formulations. As mentioned before, for MRC, our alternative singlefold integral expressions are indeed exact and coincide with the multifold formulations given in [9, Eq. 18]. Of course, in this case, there is no need to compare both exact solutions. VII. C ONCLUSIONS In this work, a number of new exact second order statistics for the η-μ fading channels has been derived. The joint statistics for the envelope, the phase, their time derivatives, combined in different forms, and the PCR are obtained in closed-form formulas. The envelope LCR is obtained in a singlefold integral form. The expressions have been fully validated by reducing them to some particular cases for which the results are already known, and, more generally, by means of Monte Carlo simulation. Furthermore, based on
Fig. 6. Approximation of the LCR and AFD of multibranch EGC systems operating in a Hoyt fading environment by the LCR and AFD of the η-μ envelope.
the new results, we provided alternative singlefold integral exact formulations for the LCR and AFD of multibranch MRC and highly-accurate approximations to the LCR and AFD of multibranch EGC operating over independent Hoyt fading channels, for which multifold integral solutions appear in the literature.
A PPENDIX In this Appendix, several new closed-form joint PDFs [(21), (22), and (23)] are shown that can be obtained from the ˙ by carrying out an appropriate ˙ θ, θ) joint PDF fR,R,Θ, ˙ ˙ (r, r, Θ integration. The joint PDF fR,Θ (r, θ) is also obtained in a simple closed-form manner and has been the focus at [19].
824
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008
R EFERENCES [1] C. -X. Wang, N. Youssef, and M. P¨atzold, “Level crossing rate and average fade duration of fades of deterministic simulation models for Nakagami-Hoyt fading channels,” in Proc. 5th Intern. Symp. on Wirel. Pers. Multimedia Commun., vol. 1, Oct. 2002, pp. 272–276. [2] S. Stein, “Fading channel issues in system engineering,” IEEE J. Select. Areas Commun., vol. 5, no. 2, pp. 68–69, Feb. 1987. [3] Q. T. Zhang, “A generic correlated Nakagami fading model for wireless communications,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1745– 1748, Nov. 2003. [4] G. D. Durgin, T. S. Rappaport, and D. A. de Wolf, “New analytical models and probability density functions for fading in wireless communications,” IEEE Trans. Commun., vol. 50, no. 6, pp. 1005–1015, June 2002. [5] M. D. Yacoub, “The η-μ distribution: A general fading distribution,” in Proc. IEEE Fall Veh. Technol. Conf., Boston, MA, Sept. 2000. [6] ——, “The κ-μ and the η-μ distribution,” in IEEE Antennas Propag. Mag., vol. 49, no. 1, pp.68–81, Feb. 2007. [7] J. C. S. Santos Filho and M. D. Yacoub, “Highly accurate η-μ approximation to the sum of M independent non-identical Hoyt variates,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 436–438, 2005. [8] W. R. Braun and U. Dersch, “A physical mobile radio channel model,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 472–482, May 1991. [9] G. Fraidenraich, J. C. S. Santos Filho, and M. D. Yacoub, “Second-order statistics of maximal-ratio and equal-gain combining in Hoyt fading,” IEEE Commun. Lett., vol. 9, no. 1, pp. 19–21, Jan. 2005. [10] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.
[11] M. D. Yacoub, G. Fraidenraich, and J. C. S. Santos Filho, “Nakagami-m phase-envelope joint distribution,” Electron. Lett., vol. 41, no. 5, Mar. 2005. [12] M. D. Yacoub, J. E. V. Bautista, and L. G. R. Guedes, “On higher order statistics of the Nakagami-m distribution,” IEEE Trans. Veh. Technol., vol. 48, no. 3, pp. 790–793, May 1999. [13] D. B. da Costa, M. D. Yacoub, J. C. S. Santos Filho, G. Fraindenraich, and J. R. Mendes, “Generalized Nakagami-m phase crossing rate,” IEEE Commun. Lett., vol. 10, no. 1, pp. 13–15, Jan. 2006. [14] N. Youssef, C. -X. Wang, and M. Patzold, “A study on the second-order statistics of Nakagami-Hoyt mobile fading channels,” IEEE Trans. Veh. Technol., vol. 54, no. 4, pp. 1259–1265, July 2005. [15] N. Youssef, W. Elbahri, M. P¨atzold, and S. Elasmi, “On the crossing statistics of phase processes and random FM noise in Nakagami-q mobile fading channels,” IEEE Trans. Wireless Commun., vol. 4, no. 1, pp. 24–29, Jan. 2005. [16] M. D. Yacoub, C. R. C. M. da Silva, and J. E. V. Bautista, “Secondorder statistics for diversity-combining techniques in Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1464–1470, Nov. 2001. [17] C. D. Iskander and P. T. Mathiopoulus, “Analytical level crossing rates and average fade durations for diversity techniques in nakagami fading channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1301–1309, Aug. 2002. [18] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [19] D. B. da Costa and M. D. Yacoub, “The η-μ joint phase-envelope distribution,” in IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 195– 198, 2007.
