The α μ - Joint Envelope-Phase Fading Distribution Anastasios K. Papazafeiropoulos, and Stavros A. Kotsopoulos Wireless Telecommunications Laboratory Dept. of Electrical and Computer Engineering University of Patras, Greece Abstract—In this paper, a new, simple, and exact closed- form expression for the envelope-phase joint distribution of a signal, operating in - α μ generalized fading channels, has been derived. The marginal distribution of the phase is also obtained in an exact closed-form expression and studied in depth. The analytical results are general and after suitable simplifications, they are verified mathematically, since they coincide to the respective ones of the models included in the - α μ distribution (i.e., the Nakagami-m, the Weibull, and the Rayleigh distributions for which the corresponding solutions are known). Finally, a sample of numerical results, obtained by the Monte Carlo simulation, is presented in order to validate the formulations developed in this paper. Index Terms—Fading channels; Nakagami-m distribution; Weibull distribution; α μ - distribution. I. INTRODUCTION Since the need for higher data rates and spectral efficiency in mobile communications has increased, the wireless communication channel has attracted a lot of attention over the years. As a result, many statistical distributions have been proposed to model the fading characteristics of the received signal in different geographical environments. The short-term signal variation is described by several distributions such as Nakagami-m [1], Hoyt [2], Rayleigh, Rice, and Weibull [3], which was originally derived for reliability study purposes. However, the flexibility of these models is too limited and often not adequate for a sufficient adaptation to the statistics of real-world channels, even if Rayleigh and Nakagami-m are characterized by their ease of manipulation [4]. The key for a more comprehensive description is to study the assumptions during the derivation of the above distributions. For example, in most cases, a homogeneous diffuse scattering field is considered as a result of randomly distributed point scatterers. In [5], the nonlinearity of the propagation medium is being explored in terms of the α parameter. There, Yacoub proposed the α μ - distribution, which is a rewritten version of the generalized Gamma distribution [6]. This model assumes that the signal is composed of many clusters of multipath waves and not only one. As a result, a better and more thorough characterization of the physical nature of the channel is achieved. It must be mentioned that beyond the fact that α μ - can describe directly the physical properties of the propagation medium through its various fading conditions, its tail closely follows the true statistics, where other distributions fail to yield a good fit. The distribution of the phase has a wide variety of applications in communications systems [7], [8], mainly when the information is transmitted in the phase of a sinusoid. For instance, the probability density function (PDF) of the phase may be useful in determining probabilities of error for M-phase signalling over fading channels using diversity [9]. Also, the characterization of the phase behavior is useful in the design of optimal carrier schemes needed in the synchronization subsystem of coherent receivers [10]. Contrary to Rayleigh, Rice, and Hoyt models where the phase PDF was provided when these distributions were proposed, for α μ - and Nakagami-m fading models, as well as for η μ - [11], no information corresponding on the signal phase was presented originally. Recently, in [12], a model for the envelope and phase of the Nakagami-m signal was presented that led to a simple joint distribution written in a closed-form manner. The same approach was followed in [13], in order to obtain the PDF of the phase of the η μ - fading model. The purpose of this paper is to derive the α μ - joint envelope-phase PDF and next the marginal PDF of the phase, in order to explore the nonlinearity of the channel. Its structure is as follows. In Section II, the physical and mathematical description of the α μ - fading model is revisited. In Section III, the envelope-phase joint PDF and the PDF of the phase of this general model are derived in simple, exact, and closed- form expressions. The most important contribution of the paper is that the nonlinearity affects interestingly not only the envelope, but also the phase of the fading signal. This fact assigns interesting properties to the phase. After suitable simplifications, the compatibility between the PDF of the phase of the α μ - distribution with the existed known phase PDFs is shown by specializing into formulations already found in the literature. Some plots, described in Section IV, illustrate the behavior of the phase PDF for several fading conditions and the Monte Carlo simulation is performed, in order to validate the phase PDF. Finally, Section V concludes the paper. II. GENERAL STATISTICAL MODELING The α μ - is a general fading distribution that describes better the small scale variation of the fading signal in an environment, where there is no line of sight (nLOS) component. Its physical background is hidden behind its name. More specifically, each letter implies a physical parameter. Thus, α declares a nonlinearity in terms of an exponent and μ is associated with the number of the multipath clusters. As a consequence, the α μ - model considers a signal composed of clusters of multipath waves propagated in a nonlinear environment, where the in-phase and quadrature components of the fading signal within each cluster are independent from each other and have equal powers. Let R and Θ be random variables representing the envelope and phase, respectively, of the complex signal 978-1-4244-5213-4/09/ $26.00 ©2009 IEEE 919