## 1. Introduction

[2] Information on the rate of change of rain attenuation or ‘fade slope’ on microwave links is important for the design of adaptive fade countermeasures, determining the required response speed of rain attenuation compensation systems [e.g., *COST255*, 2002]. It is also essential for the development of the modeling of dynamic rain attenuation behavior, such as undertaken by *Maseng and Bakken* [1981] and *Sweeney and Bostian* [1992].

[3] In order to provide information on rain fade slope, various experiments have been carried out to analyse its statistical properties, using, e.g., terrestrial links [e.g., *Lin et al.*, 1980], and the satellites SIRIO [e.g., *Matricciani et al.*, 1986], Olympus [e.g., *Stutzman et al.*, 1995; *van de Kamp*, 1999], ACTS [e.g., *Feil et al.*, 1997] and Italsat [e.g., *Schnell and Fiebig*, 1997].

[4] In all of these studies, the measured time series also contain signal fluctuations due to tropospheric scintillation. Since these are typically faster (containing higher spectral components) than the fluctuations of rain attenuation, they are removed from the signal by a low-pass filter (‘LPF’), in order to study just one effect at a time. However, the spectra of rain attenuation and scintillation overlap, and vary with time and climate, which means that there is no overall optimum value for the bandwidth *f*_{B} of the LPF, and different bandwidths are used in different experiments.

[5] In the definition of the fade slope ζ, usually two attenuation sample values *A* are subtracted and divided by the time Δ*t* between them:

The time interval Δ*t* is most conveniently a multiple of the sampling time of the measured data. Values of Δ*t* used in practice usually vary between 2 s [*van de Kamp*, 1999] and 10 s [*Feil et al.*, 1997].

[6] It has been found that the rain fade slope is stochastic, and depends on the rain attenuation level [*Lin et al.*, 1980; *Matricciani et al.*, 1986; *Feil et al.*, 1997; *van de Kamp*, 1999]. There are indications that fade slope also depends on the elevation angle [*Feil et al.*, 1997], and on meteorological parameters, such as the wind speed, the path length through rain, and the type of rain (widespread or convective) [*Buné et al.*, 1988].

[7] Because the fade slope ζ is stochastic, no deterministic relation between ζ and other parameters has been found, and measured results are usually studied statistically: measured fade slope values are collected in distributions and the characteristics of these distributions are assessed. The distributions obtained are generally found to be symmetrical around a zero mean. In many studies, the distribution of fade slope ζ(*t*) is evaluated conditional to the coinciding attenuation value *A*(*t*). It is generally found that this conditional distribution is also symmetrical around a zero mean, and has a standard deviation which depends on attenuation. Some studies showed that this standard deviation σ_{ζ} is approximately proportional to attenuation [*Lin et al.*, 1980; *van de Kamp*, 1999].

[8] The dependencies of fade slope on various system and meteorological parameters, although they have been noted in various experiments, have not yet been thoroughly assessed. In order to study and quantify these dependencies, it will be necessary to compare measurement results from different sites. However, when comparing these results, it is important to note the dependence of the fade slope statistics on data processing parameters, such as the LPF bandwidth, and the time interval in the fade slope definition. This is also important when using statistical fade slope information in the design of fade countermeasures. This dependence of fade slope on time interval and filter bandwidth is not mentioned in most publications of fade slope results.

[9] This paper theoretically estimates the dependence of fade slope statistics on time interval and filter bandwidth, and verifies the obtained results by analysing measured statistics using various time intervals and filter bandwidths. This evaluation is performed in terms of a characteristic parameter of fade slope statistics: the standard deviation σ_{ζ} of the distribution of fade slope values ζ, coinciding with attenuation *A*. As mentioned above, σ_{ζ} is approximately proportional to *A*.