Influence of time interval and filter bandwidth on measured rain fade slope

Transcription

1 RADIO SCIENCE, VOL. 39, RS5, doi:1.19/rs837, 4 Influence of time interval and filter bandwidth on measured rain fade slope M. M. J. L. van de Kamp 1 ONERA-DEMR, Toulouse, France P. Clérivet IN-SNEC, les Ulis, France Received 18 November ; revised 18 June 3; accepted 4 December 3; published 3 April 4. [1] 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. INDEX TERMS: 6964 Radio Science: Radio wave propagation; 1854 Hydrology: Precipitation (3354); 674 Electromagnetics: Signal processing and adaptive antennas; 694 Radio Science: Atmospheric propagation; KEYWORDS: fade slope, LPF filtering, propagation modeling, rain attenuation Citation: van de Kamp, M. M. J. L., and P. Clérivet (4), Influence of time interval and filter bandwidth on measured rain fade slope, Radio Sci., 39, RS5, doi:1.19/rs Introduction [] 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., COST55, ]. 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 [199]. [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., 198], 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]. 1 Now at Department of Electronic and Electrical Engineering, University of Bath, Bath, UK. Copyright 4 by the American Geophysical Union /4/RS837 [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 z, usually two attenuation sample values A are subtracted and divided by the time between them: RS5 zðþ¼ t Atþ ð =Þ At ð = Þ ð1þ The time interval is most conveniently a multiple of the sampling time of the measured data. Values of used in practice usually vary between s [van de Kamp, 1999] and 1 s [Feil et al., 1997]. [6] It has been found that the rain fade slope is stochastic, and depends on the rain attenuation level 1of8

2 RS5 RS5 [Lin et al., 198; 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 z is stochastic, no deterministic relation between z 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 z(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 s z is approximately proportional to attenuation [Lin et al., 198; 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 s z of the distribution of fade slope values z, coinciding with attenuation A. As mentioned above, s z is approximately proportional to A.. Theoretical Dependence on Filter Bandwidth and Time Interval [1] Matricciani [198] 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 f B of the filter used. [11] Maseng and Bakken [1981] derived a theoretical model for the dynamic behavior of rain attenuation. From the result it can be seen that for time intervals much shorter than 5 s between two attenuation samples, the standard deviation of the second one, p given ffiffiffiffiffi the first one is known, increases proportionally to.ifthe fade slope z is calculated as the difference between these two values divided by the time interval between them, as in equation (1), it follows directly that its standard deviation s z for ap given attenuation value A(t ) is proportional to 1= ffiffiffiffiffi. [1] However, neither of these dependencies is likely to be uniformly valid. If the fade slope is calculated between two samples with a time interval between them much larger than the inverse of the filter bandwidth f B, the LPF will not affect attenuation changes over the time, so the fade slope standard deviation s z will not depend on f B. [13] On the other hand, if is much smaller than 1/f B, then on a time scale of the order of, 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. The instantaneous fade slope will then be independent of, and so will its standard deviation s z. [14] The theoretical dependence of the fade slope standard deviation on time interval and filter bandwidth can be estimated using spectral analysis [Clérivet, 1]. 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 of8 s z ¼ Z 1 1 jsðf Þj df ¼ Z 1 1 jhðf ÞZðf ÞAðf Þj df ðþ 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

3 RS5 RS5 Figure 1. Theoretical fade slope standard deviation s z versus filter bandwidth f B, for different time intervals. a =1. fade slope from the attenuation according to equation (1) (i.e., the Fourier transform of equation (1) is equal to Z( f )A( f )). [15] The LPF is approximated as an ideal filter, with Hðf Þ ¼ 1 jf j f B elsewhere ð3þ 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): Ffeqn ð1þ g ¼ e jpf jpf e j sin pf ¼ ð Þ Aðf Þ Aðf Þ ¼ Zðf ÞAðf Þ ð4þ Matricciani [1994] found that signal fluctuations due to rain attenuation are approximately inversely proportional to the Fourier frequency, for frequencies larger than a value f L, below which the spectrum is flat. The value of f L depends on wind speed and rain height, and is of the order of 1 4 Hz. Therefore, in this calculation A( f )is approximated as Aðf Þ ¼ a=f jfj f L ð5þ a=f L jf j < f L where a is a constant dependent on the attenuation level and its fluctuations during the period under consideration. The above expressions, substituted in equation (), give (assuming f L < f B ): Z fl s z ¼ a sinðpf Þ df þz f L a sinðpf Þ df f L f 3of8 f L þ Z f B f L ¼ 8a fl ¼ a f Z f L 8a pf L 3 sinðpfþ df f B Z fb sin ðpf Þdf þ 8a p sinðpf Þ df pf Zpf L sin xdx þ 8a p f L Zpf B pf L dx ð6þ x

4 RS5 RS5 The integral in the second term can be written in two parts: Zpf L Zpf B s z ¼ 8a pfl sin xdx þ 8a p dx 3 x 8a p Zpf L dx ð7þ x If f L 1/(), this can be approximated by Zpf L Zpf B s z 8a pfl x dx þ 8a p dx 3 x 8a p Zpf L dx ¼ 8 3 a p f L þ 8a p ¼ 8a p Figure. Fade slope standard deviation s z measured in Eindhoven from Olympus at GHz, versus f B, for different. Attenuation A = 6 db. Zpf B Zpf B dx 8a p f L x dx 16 x 3 a p f L ð8þ 4of8 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 f B 1, it can be approximated by an integral from to1, the result of which is p/, so the term becomes 4a p /. On the other hand, in the case where f B 1, the integrand can be approximated by 1, so in this case the term becomes 8a p f B. [16] It can be seen that if f L 3 f B, and f B 1, the second of the final two terms of equation (8) is negligible with respect to the first (equal to 8a p f B ). But since the integrand in the first term is never negative, the result of the integral increases monotonously with f B. Therefore, as long as f L 3 f B,the first term will be much larger than the second term for any value of f B, and the second term can be neglected completely. [17] Concluding, the fade slope standard deviation can be approximated by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Zpf B up s z at dx x ð9þ

5 RS5 RS5 Figure 3. The standard deviation of fade slope using moving-average- and cos -filters, as a function of the averaging time. =1s. for f B 3 f L and 1=ðf L Þ (f L being around 1 4 Hz). The following special cases are found, based on the observations above: p s z ap= ffiffiffiffiffi if f B 1 p s z ap ffiffiffiffiffiffi ð1þ f B if f B 1 The results that s z is dependent on f B if f B 1, and dependent on if f B 1, agree with the expectations mentioned at the beginning of this section. [18] The result of equation (9) as a function of and f B 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 [19] 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 199 from the GHz beacon 5of8 were analysed to verify the dependence of s z on and f B. [] 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 f B, 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. [1] 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 s z of every conditional distribution was calculated. This was all done for various filter bandwidths f B from 1 mhz to 1 Hz, and for various time intervals from s to 1 s. It was found that in all of these cases, s z was approximately proportional to A, as found before [Lin et al., 198; van de Kamp, 1999]. Furthermore, s z appeared to depend significantly on both f B and. [] Figure shows the result for the attenuation bin of (arbitrarily chosen) 6 db, obtained using various values of f B and. The conditional distribution for 6 db

6 RS5 RS5 Figure 4. The results of Figure 3 and those using a sharp filter, as a function of the filter bandwidth, where the averaging times are converted to filter bandwidths using equations (11) and (1). =1s. attenuation contained about 36 data points, depending slightly on the filter bandwidth. Comparing Figure with Figure 1, it can be seen that the variation of s z with and f B agrees with the theoreticallyppredicted ffiffiffiffi behavior. In particular, s z is proportional to f B and independent of when 1/f B, and proportional to 1= ffiffiffiffiffi p and independent of f B when 1/f B. These observations are in agreement with equation (1). [3] The two results of Figures 1 and 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 [4] The filtering method in the theoretical analysis, and in the analysis of the measurements, was to sharply cut off all frequency components above f B. However, in many fade slope analysis procedures, moving-average (or running-average ) filters are used [e.g., Stutzman et 6of8 al., 1995]. These filters replace each sample by the average of the samples in the block of length t a 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 = 1 s, but using moving-average filters with t a from 1 to 5 s. The results of this, again for the attenuation bin of 6 db, are shown in Figure 3. [5] The same test was performed for so-called cos - 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 cos -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 cos -filter was tested in the same way as the moving-average filter, under the same conditions ( =1s;t a = 1 s to 5 s). The results are included in Figure 3. [6] It appears that the results of these experiments are very similar to the results using the sharp filter when an

7 RS5 RS5 Figure 5. (top) The expression of equation (13) versus f B. (bottom) The relative error of this expression with respect to equation (9). a = 1. 7of8

8 RS5 RS5 effective filter bandwidth for moving-average filters is defined as and for cos -filters as f B ¼ :445=t a f B ¼ :719=t a ð11þ ð1þ 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 (1). [7] It can be concluded that the dependence of s z on filter bandwidth for any LPF filter can be expressed by equation (9). For moving-average filters and cos -filters, the effective signal bandwidth can be calculated from equations (11) and (1). 5. An Approximate Expression [8] 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 f B and, analytic functions were fitted to the results by trial and error. A good fit was found with the following expression: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s z ap u t 1=b ð13þ 1=fB b þ ðþb with b =.3. This expression approximates equation (9) with a relative error smaller than.3% for any value of f B or. The expression is plotted in Figure 5 for a =1,as well as its relative error with equation (9). 6. Conclusions [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 cos -filters, the effective filter bandwidth is given by equations (11) and (1). The expression of equation (9) agrees with measurement results. An easierto-use approximate expression is given by equation (13). [3] 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. References Buné, P. A. M., M. H. A. J. Herben, and J. Dijk (1988), Rain rate profiles obtained from the dynamic behavior of the attenuation of microwave radio signals, Radio Sci., 3(1), 13. Clérivet, P. (1), Fade slope modeling for the satellite communication in the Ka- and V-band, graduation report, INSA Rennes/ONERA Toulouse, France. COST55 (), Radio wave propagation modelling for SatCom services at Ku-band and above, final report of COST Action 55, chap. 5.3, SP-15, Eur. Space Agency, Paris. Feil, J., L. J. Ippolito, Jr., H. Helmken, C. E. Mayer, S. Horan, and R. E. Henning (1997), Fade slope analysis for Alaska, Florida, and New Mexico ACTS propagation data at and 7.5 GHz, Proc. IEEE, 85(6), Lin, S. H., H. J. Bergmann, and M. V. Pursley (198), Rain attenuation on Earth-satellite paths Summary of 1-year experiments and studies, Bell Syst. Tech. J., 59(), Maseng, T., and P. M. Bakken (1981), A stochastic dynamic model of rain attenuation, IEEE Trans. Commun., 9(5), Matricciani, E. (198), Effects of filtering on rate of change of rain-induced attenuation, Electron. Lett., 18(11), Matricciani, E. (1994), Physical-mathematical model of dynamics of rain attenuation with application to power spectrum, Electron. Lett., 3(6), Matricciani, E., M. Mauri, and A. Paraboni (1986), Joint analysis of duration and rate of change of rain attenuation at 11.6 GHz, Electron. Lett., (17), Schnell, M., and U.-C. Fiebig (1997), Fade slope statistics of 4 GHz beacon signals, Electron. Lett., 33(1), Stutzman, W. L., T. Pratt, A. Safaai-Jazi, P. W. Remaklus, J. Laster, B. Nelson, and H. Ajaz (1995), Results from the Virginia Tech Propagation Experiment using the Olympus satellite 1, and 3 GHz beacons, IEEE Trans. Antennas Propag., 43(1), Sweeney, D. G., and C. W. Bostian (199), The dynamics of rain-induced fades, IEEE Trans. Antennas Propag., 4(3), van de Kamp, M. M. J. L. (1999), Climatic radiowave propagation models for the design of satellite communication systems, Ph.D. thesis, Eindhoven Univ. of Technol., Eindhoven, Netherlands. P. Clérivet, IN-SNEC, 9194 les Ulis, France. M. M. J. L. van de Kamp, Department of Electronic and Electrical Engineering, University of Bath, Bath BA 7AY, UK. (eexmvdk@bath.ac.uk) 8of8