The rate of change of rain attenuation, or ‘fade slope,’ observed on a microwave link, is dependant on system parameters of the receiver. This paper assesses its dependence on two data processing parameters: the time interval over which the fade slope is defined, and the low-pass filter bandwidth used to remove tropospheric scintillation. A theoretical expression of this dependence is derived. The expression is compared with measured results, which show a good agreement. The results show that the time interval has a strong influence when it is larger than the inverse of the filter bandwidth, and the filter bandwidth has a strong influence when its inverse is larger than the time interval. In general, the fade slope always depends strongly on at least one of the two parameters.
 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  and Sweeney and Bostian .
 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 fB of the LPF, and different bandwidths are used in different experiments.
 In the definition of the fade slope ζ, usually two attenuation sample values A are subtracted and divided by the time Δt between them:
The time interval Δt is most conveniently a multiple of the sampling time of the measured data. Values of Δt used in practice usually vary between 2 s [van de Kamp, 1999] and 10 s [Feil et al., 1997].
 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 σζ is approximately proportional to attenuation [Lin et al., 1980; van de Kamp, 1999].
 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.
 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 σζ of the distribution of fade slope values ζ, coinciding with attenuation A. As mentioned above, σζ is approximately proportional to A.
2. Theoretical Dependence on Filter Bandwidth and Time Interval
Matricciani  performed experiments with fade slope statistics, measured from the satellite SIRIO at the ground station of Gera Lario, Italy. Scintillation effects were removed by low-pass filtering the data with two moving-average filters with different time periods: 1 s and 4 s, in order to compare the results. From these two analyses, he found distributions of similar shapes, apart from a multiplication factor: the fade slope exceeded for the same probability was significantly larger for the larger filter bandwidth. He explained this by the smoothing and resampling which causes longer rising and sinking times over a given attenuation interval. This indicates that fade slope results are dependent on the bandwidth fB of the filter used.
Maseng and Bakken  derived a theoretical model for the dynamic behavior of rain attenuation. From the result it can be seen that for time intervals Δt much shorter than 500 s between two attenuation samples, the standard deviation of the second one, given the first one is known, increases proportionally to . If the fade slope ζ is calculated as the difference between these two values divided by the time interval Δt between them, as in equation (1), it follows directly that its standard deviation σζ for a given attenuation value A(t−Δt) is proportional to .
 However, neither of these dependencies is likely to be uniformly valid. If the fade slope is calculated between two samples with a time interval Δt between them much larger than the inverse of the filter bandwidth fB, the LPF will not affect attenuation changes over the time Δt, so the fade slope standard deviation σζ will not depend on fB.
 On the other hand, if Δt is much smaller than 1/fB, then on a time scale of the order of Δt, the filtered signal will show no sudden changes. In this situation the calculated fade slope is the time derivative of attenuation, regardless of the exact length of Δt. The instantaneous fade slope will then be independent of Δt, and so will its standard deviation σζ.
 The theoretical dependence of the fade slope standard deviation on time interval and filter bandwidth can be estimated using spectral analysis [Clérivet, 2001]. The square of the standard deviation, i.e., the variance, of any signal is equal to the integral over its entire power spectrum. For the fade slope this can be expressed as
where S(f) is the frequency spectrum of (filtered) fade slope fluctuations, H(f) is the LPF characteristic, A(f) is the spectrum of rain attenuation fluctuations, and Z(f) is the transfer function of the calculation of the fade slope from the attenuation according to equation (1) (i.e., the Fourier transform of equation (1) is equal to Z(f)A(f)).
 The LPF is approximated as an ideal filter, with
This can be expressed as a limitation of the integration range. The transfer function Z(f) is found through the Fourier transform of equation (1):
Matricciani  found that signal fluctuations due to rain attenuation are approximately inversely proportional to the Fourier frequency, for frequencies larger than a value fL, below which the spectrum is flat. The value of fL depends on wind speed and rain height, and is of the order of 10−4 Hz. Therefore, in this calculation A(f) is approximated as
where a is a constant dependent on the attenuation level and its fluctuations during the period under consideration. The above expressions, substituted in equation (2), fL< fB):
The integral in the second term can be written in two parts:
If fL ≪ 1/(2Δt), this can be approximated by
The integral in the first of the final two terms of equation (8) is difficult to solve, but some special cases of it can easily be recognized. In the case where fBΔt ≫ 1, it can be approximated by an integral from 0 to ∞, the result of which is π/2, so the term becomes 4a2π2/Δt. On the other hand, in the case where fBΔt ≪ 1, the integrand can be approximated by ‘1,’ so in this case the term becomes 8a2π2fB.
 It can be seen that if fL ≪ fB, and fBΔt ≪ 1, the second of the final two terms of equation (8) is negligible with respect to the first (equal to 8a2π2fB). But since the integrand in the first term is never negative, the result of the integral increases monotonously with fBΔt. Therefore, as long as fL ≪ fB, the first term will be much larger than the second term for any value of fBΔt, and the second term can be neglected completely.
 Concluding, the fade slope standard deviation can be approximated by
for fB ≫ fL and Δt ≪ (fL being around 10−4 Hz). The following special cases are found, based on the observations above:
The results that σζ is dependent on fB if fBΔt ≪ 1, and dependent on Δt if fBΔt ≫ 1, agree with the expectations mentioned at the beginning of this section.
 The result of equation (9) as a function of Δt and fB is plotted in Figure 1 (where a is taken equal to 1). In the next section, this result will be verified using measurements.
3. Experimental Verification
 In Eindhoven, Netherlands, the three beacon signals of the satellite Olympus were received and processed at Eindhoven University of Technology. The signal level was measured with a time resolution of 1 Hz. All rain attenuation event data measured between January 1991 and July 1992 from the 20 GHz beacon were analysed to verify the dependence of σζ on Δt and fB.
 Scintillation fluctuations were removed from the data using a low-pass filter. The filter calculated the Fourier transform of the data of each event, removed all contributions for larger Fourier frequencies than the bandwidth fB, and transformed the result back to the time domain. This filter is equivalent to the filter characteristic of equation (3); such a filter will hereafter be referred to as a ‘sharp filter.’
 The fade slope was calculated from the attenuation data according to equation (1), and collected in conditional distributions for different attenuation classes, with attenuation bin size 1 dB. The standard deviation σζ of every conditional distribution was calculated. This was all done for various filter bandwidths fB from 1 mHz to 1 Hz, and for various time intervals Δt from 2 s to 100 s. It was found that in all of these cases, σζ was approximately proportional to A, as found before [Lin et al., 1980; van de Kamp, 1999]. Furthermore, σζ appeared to depend significantly on both fB and Δt.
Figure 2 shows the result for the attenuation bin of (arbitrarily chosen) 6 dB, obtained using various values of fB and Δt. The conditional distribution for 6 dB attenuation contained about 3600 data points, depending slightly on the filter bandwidth. Comparing Figure 2 with Figure 1, it can be seen that the variation of σζ with Δt and fB agrees with the theoretically predicted behavior. In particular, σζ is proportional to and independent of Δt when Δt ≪ 1/fB, and proportional to and independent of fB when Δt ≫ 1/fB. These observations are in agreement with equation (10).
 The two results of Figures 1 and 2 differ only by a constant factor, which is due to the unknown value of the parameter a for A = 6 dB. Study of this parameter is outside the scope of this paper. The measured results confirm that equation (9) is a good estimate for the variation of the standard deviation of rain fade slope with filter bandwidth and time interval.
4. Dependence on Filter Type
 The filtering method in the theoretical analysis, and in the analysis of the measurements, was to sharply cut off all frequency components above fB. However, in many fade slope analysis procedures, moving-average (or ‘running-average’) filters are used [e.g., Stutzman et al., 1995]. These filters replace each sample by the average of the samples in the block of length ta around it. In order to assess the effects of these filters, a comparative test was performed. The fade slope standard deviation was calculated from the same measured data as above, using the same procedures, with Δt = 10 s, but using moving-average filters with ta from 1 to 500 s. The results of this, again for the attenuation bin of 6 dB, are shown in Figure 3.
 The same test was performed for so-called cos2-filters, which are also often used in fade slope analysis [e.g., Feil et al., 1997]. This type of filter is similar to a moving-average filter, but while averaging it weighs the samples in the block by a cos2-shaped function, with the nulls at the edges and the maximum in the center (this function is also known as a ‘Hanning window’). This action suppresses discontinuities in the signal. The cos2-filter was tested in the same way as the moving-average filter, under the same conditions (Δt = 10 s; ta = 1 s to 500 s). The results are included in Figure 3.
 It appears that the results of these experiments are very similar to the results using the sharp filter when an effective filter bandwidth for moving-average filters is defined as
and for cos2-filters as
This is shown in Figure 4, where the results are compared with those for a sharp filter, with the filter bandwidths of the other two filters estimated from the averaging times as in equations (11) and (12).
 It can be concluded that the dependence of σζ on filter bandwidth for any LPF filter can be expressed by equation (9). For moving-average filters and cos2-filters, the effective signal bandwidth can be calculated from equations (11) and (12).
5. An Approximate Expression
 The integral in equation (9) makes it a relatively complicated expression to use in practice. Because of this, an expression was derived which approximates the theoretical expression reasonably, and is easier to use. After numerically calculating equation (9) for many values of fB and Δt, analytic functions were fitted to the results by trial and error. A good fit was found with the following expression:
with b = 2.3. This expression approximates equation (9) with a relative error smaller than 2.3% for any value of fB or Δt. The expression is plotted in Figure 5 for a = 1, as well as its relative error with equation (9).
 The standard deviation of the distribution of fade slope, conditional for attenuation values, is strongly dependent on the low-pass filter bandwidth used to remove scintillation, and on the time interval used to calculate the fade slope. The dependence on these parameters is as expressed in equation (9). For moving-average or cos2-filters, the effective filter bandwidth is given by equations (11) and (12). The expression of equation (9) agrees with measurement results. An easier-to-use approximate expression is given by equation (13).
 The dependence on filter bandwidth and time interval should be taken into account when fade slope results from different sources are compared, or when results are analysed to study the dynamic behavior of rain attenuation. A modeled expression based on the results obtained in this paper, has been submitted to and adopted by the ITU-R (International Telecommunication Union; Radiocommunication Sector), as part of a prediction model of rain fade slope.