Closed-Form Approximations to the High-order Statistics of the -μ Extreme Fading

This paper presents two closed-form approximations for the high-order statistics of theκ-μ Extreme fading model. The rationale for the approximations arose from the observation of the empirical behavior of the level crossing statistics for data whose first order statistics closely follow theκ-μ Extreme model. Field measurements are then used to validate the formulations. Keywords—κ-μ Extreme distribution, level crossing rate, average fade duration, field measurements, validation. Resumo— Este artigo apresenta duas aproximaç̃ oes em fórmula fechada para as estat́ ısticas de ordem superior do modelo de desvanecimentoκ-μ Extremo. A razão para as aproximaç̃oes surge da observaç̃ ao das estat́ ısticas emṕıricas de cruzamento de nı́vel de dados nas quais as estat ı́s icas de primeira ordem seguem de maneira muito próxima o modeloκ-μ Extreme. Medidas de campo s̃ao usadas para validar a formulaç̃ao. Palavras-Chave— distribuiç ão κ-μ Extrema, taxa de cruzamento de ńıvel, tempo ḿedio de desvanecimento, medidas de campo, validaç̃ao. I. I NTRODUCTION I N wireless communications, a well investigated propagation phenomenon that increasingly and continuously raises the interest of the researchers is the fading. The occurrenc e of fading depends on several factors, including the environme nt and the propagation frequency. Its harshness ranges from ve ry mild to extremely severe, and a number of models appear in the literature that reasonably well describe such a phenome non in its various aspects. Among the fading distributions, Rayleigh, Hoyt, Weibull, Rice, and Nakagami, are the best known. Recently, more general fading models, namely α-μ [1], κ-μ [2], and η-μ [2], have been proposed that comprise the previous ones as special cases and that better fit experiment al field data. Because wireless communications applications h ave been progressively diversified, not only outdoor and indoor environments have experienced their surge, but enclosed sc enarios too. Enclosed and some indoor environments are characterized by very severe fading conditions. More specifica lly, due to the harsh variation of the received signal, a great dea l of reception points may be found which are well below the receiver sensitivity. Therefore, even though the signal ma y not necessarily be nil, it may be sufficiently low with such an eve nt occurring a sufficient number of times that the probability of finding it at such a condition may not be negligible. In E. M. Ferdinando and M. D. Yacoub are with the Wireless Techno logy Laboratory (WissTek), Department of Communications, School of E lectrical and Computer Engineering, State University of Campinas, 13083-8 52, Campinas, SP, Brazil (e-mail: [erickmf,michel]@decom.fee.unicamp.br). U. S. Dias is with the Microwave and Wireless System Lab (MWSL) , Department of Electrical Engineering, University of Brasil ia, 70910-900, Braśılia, DF, Brazil (e-mail: udias@unb.br). addition, unlike the traditional environments, where diff erent combinations of large number of multipath components lead to known fading channels, enclosed environments may presen t only a few radio paths, therefore rendering the utilization of the Central Limit Theorem inappropriate. Furthermore, kno w and useful propagation mechanisms (e.g., Plane Earth [3]) predict that direct and reflected waves may be combined to yield nulls at some reception points. In [4], the severe fadi ng conditions – worse than that predicted by the Rayleigh case – in enclosed environments was named hyper-Rayleigh fading [4]–[6]. In [2], an extreme condition of fading was found for the κ-μ distribution. Such a condition, namely κ-μ Extreme, was later explored in [7]. In particular, field measurements conducted in a large transport helicopter, as reported in [4 ], and others collected in a university parking lot with moving cars and within a sports gymnasium were used to validate the κ-μ Extreme high-order statistics obtained here. Currently, theκ-μ Extreme model is limited to its first order statistics. Exploring high-order statistics is certainly of interest to fully characterize the fading channel. It is anticipated , however, that, specifically for the κ-μ Extreme case, this is a tough problem whose exact solution is still open. The aim of this paper is to find approximate expressions for the level crossing rate (LCR) and for the average fade duration (AFD) of this new channel. The remainder of this work is structured as follows. Section II revisits the physical model and some expressions for the κμ Extreme fading. Section III develops the joint probability density function of the envelope and its time derivative. Section IV proposes two approaches to approximate the LCR and AFD statistics. Section V describes the conditions with which the field measurements have been conducted. Section VI compares the proposed approximations and the statistics obtained from field measurements. Section VII concludes the paper. II. T HE κ-μ EXTREME MODEL REVISITED The κ-μ distribution is a general fading distribution that can be used to represent the small variation of the fading signal under LOS conditions. It includes as special cases important other distributions such as Rice (Nakagamin) and Nakagamim [2]. Therefore, One-Sided Gaussian and Rayleigh also constitute special cases of it. As its name implies, it i s written in terms of two physical parameters, namely κ and μ. The parameter κ > 0 concerns the ratio between the total power of the dominant components and the total power of the scattered waves, whereas the parameter μ > 0 is related to the XXX SIMPÓSIO BRASILEIRO DE TELECOMUNICAÇ̃OES SBrT’12, 13-16 DE SETEMBRO DE 2012, BRAS ÍLIA, DF multipath clustering. Theκ-μ Extreme distribution was also originally proposed in [2] and arises as a particular case of κ-μ distribution, for which the fading parameters assume extre me values, i.e.,κ → ∞ (very strong LOS) andμ → 0 (very few multipaths), denoting very high severe fading conditio ns. According to [2], the PDF of theκ-μ Extreme envelope can be written in terms of the Nakagamim fading parameter, m. Such result shows that for a given m, an infinite number of curves of κ-μ distribution can be obtained for appropriate values of κ andμ, rendering it well suited to field measurements in LOS conditions with very severe fading scenarios [2], [7]. For a fading signal with envelope R, with r̂ = √ E(R) being therms value ofR, theκ-μ Extreme PDFfP(ρ) of the normalized envelopeP = R/r̂ is given as [7] fP(ρ) = 4mI1(4mρ) exp[2m(1 + ρ)] + exp(−2m)δ(ρ), (1) in which Iν(·) is the modified Bessel function of the first kind and orderν [8, Eq. 9.6. 20], andδ(·) is the Dirac delta function. For convenience, Equation (1) is now rewritten as fP(ρ) = g(ρ) + exp(−2m)δ(ρ), (2) in which g(ρ) is the continuous part of the PDF. The corresponding CDF is given as [7]