A Rain Attenuation Time-Series Synthesizer Based ... - Semantic Scholar

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A Rain Attenuation Time-Series Synthesizer Based on a Dirac and Lognormal Distribution Xavier Boulanger, Laurent Féral, Laurent Castanet, Nicolas Jeannin, Guillaume Carrie, and Frederic Lacoste

Abstract—In Recommendation ITU-R P.1853-1, a stochastic approach is proposed to generate long-term rain attenuation time series , including rain and no rain periods anywhere in the world. Nevertheless, its dynamic properties have been validated so far from experimental rain attenuation time series collected at mid-latitudes only. In the present paper, an effort is conducted to derive analytically the first- and second-order statistical properties of the ITU rain attenuation time-series synthesizer. It is then shown that the ITU synthesizer does not reproduce the first-order statistics (particularly the rain attenuation cumulative distribution function CDF), however, given as input parameters. It also prevents any rain attenuation correlation function other than exponential to be reproduced, which could be penalizing if a worldwide synthesizer that accounts for the local climatology has to be defined. Therefore, a new rain attenuation time-series synthesizer is proposed. It assumes a mixed Dirac-lognormal modeling of the absolute rain attenuation CDF and relies on a stochastic generation in the Fourier plane. It is then shown analytically that the new synthesizer reproduces much better the first-order statistics given as input parameters and enables any rain attenuation correlation function to be reproduced. The ability of each synthesizer to reproduce absolute rain attenuation CDFs given by Recommendation ITU-R P.618 is finally compared on a worldwide basis. It is then concluded that the new rain attenuation time-series synthesizer reproduces the rain attenuation CDF much better, preserves the rain attenuation dynamics of the current ITU synthesizer for simulations at mid-latitudes, and, if it proves to be necessary for worldwide applications, is able to reproduce any rain attenuation correlation function. Index Terms—Rain attenuation time-series, satellite communication systems, stochastic processes.

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I. INTRODUCTION

HE conventional frequency bands (C, Ku, i.e., 4–15 GHz) used in mid-latitudes for fixed satellite telecommunication systems are nearly to be saturated. Nevertheless, fixed satelManuscript received June 26, 2012; revised November 16, 2012; accepted November 19, 2012. Date of publication February 07, 2013; date of current version February 27, 2013. This study has been partly carried out in the framework of the European action COST IC0802. X. Boulanger is with the French Aerospace Lab (ONERA), Département Electromagnétisme et Radar (DEMR), Toulouse 31055, France, and also with the French Space Agency (CNES), Toulouse 31055, France (e-mail: Xavier. [email protected]). L. Féral is with the Laboratoire LAPLACE, Groupe de Recherche en Electromagnétisme (GRE), Université de Toulouse (Paul Sabatier, Toulouse III), Toulouse 31400, France (e-mail: [email protected]). L. Castanet, N. Jeannin, and G. Carrie are with the French Aerospace Lab (ONERA), Département Electromagnétisme et Radar (DEMR), Toulouse 31055, France (e-mail: [email protected]; [email protected]; [email protected]). F. Lacoste is with the French Space Agency (CNES), Toulouse 31055, France (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2012.2230237

lite services require higher and higher data rates. As a consequence, systems operating at higher frequency bands (such as Ka or Q/V, i.e., 20–50 GHz) are envisaged to reach higher performances in terms of spectrum availability, equipment size, or bandwidth. Besides, in tropical and equatorial areas, where the terrestrial telecommunication infrastructures are less developed than in temperate regions, the satellite is an interesting alternative as it allows faster and cheaper service deployment. Nevertheless, satellite telecommunication systems at frequencies greater than 5 GHz suffer from tropospheric attenuation. The latter increases with the radio link frequency and can reach some tens of decibels at Ka or Q/V bands [1]. Among the various tropospheric components, rain is the major contributor to total tropospheric attenuation. Moreover, depending on the local climatology, rain attenuation is expected to be more severe in tropical and equatorial areas than in mid-latitudes, both in terms of occurrence and in attenuation level. To counteract tropospheric impairments, a fixed power margin that depends on the required service availability is commonly introduced in the link budget. Nevertheless, this extra resource allocation is not optimal for systems above 20 GHz because the required margin becomes too high for continuous operation, resulting in costly user Earth stations and causing excessive interferences. Therefore, to insure a favorable link budget, the fade mitigation techniques (FMT) (such as adaptive coding or modulation and frequency diversity [2], [3]) have to be implemented to counteract tropospheric impairments. That is the reason why, for satellite communication systems in centimetric or millimetric bands, adaptive real-time algorithms are being developed [4]. Clearly, tropospheric attenuation time series are necessary for their development, their evaluation, and their optimization [4]. As the local climatology changes from one location to another, experimental attenuation time series are thus needed all around the world, for various radio wave configurations in terms of frequency, polarization, and elevation angle. Unfortunately, propagation experiments above 20 GHz are not numerous, so have to be that the synthetic rain attenuation time series considered. During the 1980s, a stochastic approach was proposed in [5] to simulate conditional rain attenuation time series (i.e., only for rainy periods) as a first-order Markov process. This approach has been used in [6]–[8] and has then been extended in [9]–[11] to generate long-term rain attenuation time series (i.e., including rain and no rain periods). It is important to note that [9]–[11] and [12] have shown that the dynamics of their long-term synthetic rain attenuation time series compares satisfactorily (from a statistical point of view) with experimental propagation time series collected at mid-latitudes for frequencies between 10 and 50 GHz and elevation angles between 25

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and 40 . Although recently adopted by ITU in Recommendation ITU-R P.1853-1 [13], the stochastic approach proposed in [9], [10], and [11] has two shortcomings. First, in [9]–[11] or [13], it is assumed that the long-term (or absolute) rain attenuation complementary cumulative distribution function (CCDF) is lognormal. Yet, the analysis of experimental rain attenuation time series clearly shows that the long-term rain attenuation process is not rigorously lognormal as its CCDF falls to 0 for time percentages between 1% and 20 of an average year depending on the local probability of rain. To overcome this limitation, [11] and [13] have introduced a posteriori an empirical offset parameter . Therefore, from a conceptual point of view, a contradiction appears: the synthetic time series does not follow given as an input paramthe lognormal CCDF eter as will be shown in Section II. Besides, from rain rate measurements collected with disdrometers at various sites located from middle to tropical and equatorial latitudes, [14] has shown that the conditional rain rate CCDF is lognormal. Disregarding the spatial correlation of rainy events and recalling the power relationship between rain intensity and rain specific attenuation [15], it follows that only the conditional CCDF of rain attenuation should be considered as a lognormal process in compliance with the first approach developed in [5] to synthesize conditional rain attenuation time series (i.e., in presence of rain only). Second, in [9]–[11] or [13], the long-term rain attenuation time series are modeled as a first-order Markov process, which implies that the rain attenuation correlation function is asymptotically exponential for small time lags whatever the location (i.e., whatever the local climatology), as will be illustrated in Section II. On the one hand, this assumption has been validated in [9], [11], and [12] from experimental rain attenuation time series collected at mid-latitudes. On the other hand, its validity in other climatic areas (in tropical or equatorial areas for instance where the convective nature of rainy events is much more pronounced than in mid-latitudes) has not been demonstrated so far in the literature and might be viewed skeptically. Therefore, for worldwide applications, a rain attenuation time series synthesizer that allows any correlation function to be reproduced would represent a generalization of the current ITU synthesizer described in [13]. To overcome the above limitations, a new rain attenuation time series synthesizer including rain and no-rain periods is proposed. It relies on a mixed Dirac-lognormal modeling of the absolute rain attenuation CCDF and on a stochastic generation in the Fourier plane. Contrary to the current ITU synthesizer, the new model reproduces very accurately the long term rain attenuation CCDF given as an input parameter and, for worldwide applications, allows any rain attenuation correlation function to be reproduced. The paper is organized as follows. First, the rain attenuation time series synthesizer [13] adopted by ITU-R Study Group 3 is recalled in Section II. Its conceptual shortcomings are demonstrated from the analytical derivation of first- and second-order statistics. Then, a new rain attenuation time series synthesizer is presented in Section III. Its first- and second-order statistical properties are derived and a methodology that allows any correlation function to be reproduced is presented. Particularly, as

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the dynamics of the current ITU synthesizer has been intensively tested and validated at mid-latitudes in [9], [11], and [12] from experimental rain attenuation time series, the capability of the new synthesizer to reproduce the rain attenuation correlation function of the ITU synthesizer is demonstrated. Lastly, the ability of each synthesizer to reproduce absolute rain attenuation CCDFs given by Recommendation ITU-R P.618 is compared on a worldwide basis in Section IV. II. RAIN ATTENUATION TIME SERIES SYNTHESIS: RECOMMENDATION ITU-R P.1853-1 A. Principle As mentioned in Section I, Recommendation ITU-R P.1853-1 [13] relies on a stochastic modeling of rain attenuation time series. First, a centred reduced Gaussian process is generated. Second, is low-pass filtered with a cutoff frequency to define a correlated Gaussian process . As detailed in [5], generated that way is a centred, reduced, first-order stationary Markov process of which the correlation function is exponential and depends only on the lag time so that (1) is the correlation time. Note that in [5] and where 5000 s in Recommendation ITU-R P.1853-1 [13]. that Third, [13] assumes that the absolute rain attenuation CCDF given as input parameter is well represented by and stana lognormal distribution with average dard deviation or, equivalently in terms of natural logarithm, and standard deviation . In such condiand the correlated tions, Gaussian process is turned into a correlated lognormal process [dB] through (2) From classical statistical results, (2) implies that the first-order statistics given as input parameters are (3a) (3b) (3c) (3d) where is the CCDF of the process and the inverse complementary error function. Fourth, to be representative of rain and no-rain periods, an empirical offset parameter is introduced in [13] with (4) where is the probability to have rain attenuation on the link. The long-term rain attenuation time series finally given by [10] is if otherwise.

(5)

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In compliance with (2) and (4), the rain attenuation time series synthesizer driven by (5) requires as input parameters , and . Ideally, the latter two should be regressed from experimental rain attenuation CCDF . In practice, as experimental rain attenuation time series are not available worldwide and as recommended by [10], Recommendation ITU-R P.618-10 [16] is used to derive worldwide for probabilities in the range 10 % to 5% while the probability to have is approximated by the probability of rain rain attenuation given by Recommendation ITU-R P.837 [17] anywhere in the world. B. Derivation of First Order Statistics From (4) and (5), it follows that the statistical averages and of the process are given by

(6a)

the introduction of an offset parameter. In particular, comparing (7d) and (3d), the introduction a posteriori of makes the current ITU synthesizer unable to reproduce the lognormal absolute rain attenuation CCDF yet given as input parameter. This point will be quantitatively assessed in Section IV where the abilities of the ITU synthesizer and the new model presented in Section III to reproduce rain attenuation CCDFs given by Recommendation ITU-R P.618 [16] are compared on a worldwide basis. C. Derivation of Second-Order Statistics The dynamics of a process such as (5) or (11) is fully driven by its correlation function. Obviously, the correlation function —(1)—of the underlying Gaussian process drives the correlation function of the ITU rain attenuation process defined by (5). Our objective here is to analytically assess from (5), (7a), and (7c). In particular, defining , , , , it follows from (4) and (5) that the covariance function of the ITU process driven by (5) is given by

(6b)

is the centred reduced where normal probability density function (PDF) and in compliance with (2). After some manipulations, (6) finally leads to

(7a) (8)

, , is the joint PDF of the rain attenuation process and is the bivariate normal PDF (see equation (A2) in Appendix A for the definition). Considering an elementary approximation for [18], the covariance (8) is derived analytically in Appendix A from equations (A4), (A13), and (A15). Consequently, using (7a) and (7c), an analytical dependency between and is finally obtained as a function of , , and . As an illustration, Fig. 1 shows as a function of for a typical range of values of varying from 0 to 1. 5.48 , 1.69, and 1.59 in compliance with Section IV, where a hypothetical radio link at 50 GHz between an Earth station situated at Toulouse 43.60 1.44 and a geostationary satellite located at longitude 0 is considered. where

(7b) (7c) Moreover, the derivation of the CCDF process driven by (5) is straightforward:

of the ITU

(7d) Therefore, (7a)–(7d) compare satisfactorily with (3a)–(3d) only if , which, from a conceptual point of view, prevents

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Fig. 1. Dependency between synthesizer driven by (5). compliance with Section IV.

and for the ITU rain attenuation time series 5.48 , 1.69, and 1.59 in

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Now, and as mentioned in Section I, more flexibility on the shape of might be required if a worldwide rain attenuation time series synthesizer that accounts for the local climatology has to be defined. Therefore, the exponential definition of that is a basic assumption of the current ITU rain attenuation synthesizer and that defines the shape of is a strong constraint that would be relaxed to accept any analytical definition, particularly the one that best reproduces the local dynamics of experimental rain attenuation time series. This flexibility is one of the advantages of the new rain attenuation time series synthesizer detailed in Section III. In addition, note that the analytical framework developed in Appendix A can be used to assess the dynamic parameter in (1) from rain attenuation measurements. Indeed, once , , , and have been derived from experimental rain attenuation time series, an optimization routine based on equations (A13), (A15), (A4), (7a), and (7c) can be used to find that minimizes the error between of model [10] and the experimental correlation . Consequently, the analytical derivation of conducted in Appendix A offers an alternative to the method of the second-order conditional moment that has been previously used in [9] or [11] or to the methodology based on the hitting time statistics developed in [19] to infer . III. NEW RAIN ATTENUATION TIME SERIES SYNTHESIZER A. Definition

Fig. 2. Analytical correlation function of the ITU process comestimated from 100 random yearly realizations of or pared with 5.48 , 1.69, derived from a numerical computation of (8). 1.59 in compliance with Section IV. is given by (1) with 5000 s in compliance with Recommendation ITU-R P.1853-1 [13].

On the other hand, Fig. 2 shows the correlation function derived analytically from given by (1) with 5000 s (i.e., s ) in compliance with [9] or [13]. For comparison, the average correlation function derived from 100 random yearly realizations of is plotted on Fig. 2. Moreover, a numerical computation of (8) has also been conducted to finally give a numerical evaluation of . The result is also reported on Fig. 2. The three curves match satisfactorily, confirming the validity of the analytical framework laid in Appendix A for the analytical derivation of . Fig. 2 shows that once the rain attenuation process is defined in compliance with (5) with given by the exponential formulation (1), then the correlation function of has an exponential asymptotic behavior for . This result is confirmed by first asymptotic results derived from equations (A13), (A15), (A4), (7a), and (7c) that are not developed here for the sake of conciseness.

To overcome the limitations listed in Section II, a new synthesizer is proposed. The latter has to generate rain attenuation time including rain and no-rain periods. It must reproduce series the first-order statistics—i.e., average , variance , CCDF —given as input parameters and, for worldwide applications, must be able to reproduce any correlation function . First, in compliance with Section I, it is now assumed that only the rain attenuation conditional PDF is lognormal with mean and standard deviation :

(9) In such conditions, the absolute rain attenuation CCDF given as input parameter is now supposed to be well represented by a mixed Dirac-lognormal distribution:

(10) is still the probability to have rain attenuation on where the link. Errors potentially introduced by the mixed Diraclognormal modeling (10) will be quantitatively assessed on a worldwide basis in Section IV. Second, a stationary, centred, reduced, correlated Gaussian process with normal PDF and arbitrary correlation

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function is generated in the Fourier domain in compliance with the methodology presented in Section III-D. is then turned into a rain attenuation process according to (11), shown at the bottom of the page, where is given by (4). B. Derivation of First-Order Statistics From (11) and recalling equation (A6), it is clear that and

. There-

fore, the new model (11) allows rain and no-rain periods to be reproduced without any additional offset parameter. Moreover, for , it can be verified that the random variable in (11) is normal, with zero average and unit standard deviation. Therefore, the absolute CCDF of the stochastic process defined by (11) is given by

Fig. 3. Dependency between and for the new rain attenuation time series 5.48 , 0.96, and 1.10 synthesizer driven by (11). and for Recin compliance with Section IV. The relationship between ommendation ITU-R P.1853-1 driven by (5)—see Fig. 1—is also plotted for comparison (dashed line).

(12a) Contrary to the current ITU model [13], (12a) shows that the new rain attenuation time series synthesizer driven by (11) reproduces the (mixed Dirac-lognormal) rain attenuation absolute CCDF —(10)—given as input parameter. Recalling the classical statistical results (3), it follows from (11) and (12a) that (12b) (12c)

In (13), , now refers to the joint PDF of the rain attenuation process defined by (11), and is still the bivariate normal PDF [see equation (A2) in Appendix A for the definition]. Unfortunately, due to the complexity of , (13) cannot be solved analytically so that a numerical computation is required. It is then convenient to change the variables and to and according to and . In such conditions, (13) becomes

(12d) The full parameterization of (11) now requires the definition of the second-order statistics.

(14)

C. Derivation of Second-Order Statistics Similarly to Section II-C, defining , the covariance function uation process (11) is now given by

, of the rain atten-

(13)

Equation (14) is evaluated numerically. The correlation funcof the rain attenuation process (11) follows from (12b) tion and (12d) as a function of , , , and . As an example, Fig. 3 shows as a function of for a typical range of values of varying from 0 to 1. 5.48 , 0.96 and 1.10 in compliance with Section IV where a hypothetical radio link at 50 GHz between an Earth station situated at Toulouse 43.60 1.44 and a geostationary satellite located at longitude 0 is considered. Clearly, from the explicit dependency between and illustrated in Fig. 3, the correlation function of the correlated Gaussian process in (11) can be derived once is

if otherwise

(11)

BOULANGER et al.: A RAIN ATTENUATION TIME-SERIES SYNTHESIZER

Fig. 4. Correlation function that must be given to the Gaussian in (11) to insure that in (11) has the same correlation funcprocess (i.e., the same dynamics) as Recommendation ITU-R P.1853-1 tion 5.48 , 1.69, 1.59, 0.96, driven by (5). 1.10 in compliance with Section IV. The correlation function and of the Gaussian process of Recommendation ITU-R P.1853-1—(1) 5000 s (dashed line)—is added for comparison. with

known. Particularly, if experimental rain attenuation time series are available, then can be derived from the experimental values , , , and obtained from measured time series in compliance with (14), (12b), and (12d). On the other hand, as experimental data are not available worldwide and as the dynamics of the current ITU synthesizer has been intensively tested and validated at mid-latitudes in previous studies from experimental rain attenuation time series [9], [11], [12], the capability of the new synthesizer to reproduce the rain attenuation correlation function of the ITU synthesizer is demonstrated. To do it, must be defined so that the random process (11) reproduces the correlation function of the ITU rain attenuation time series synthesizer driven by (5). This can be done very easily from the analytical derivations conducted in Section II. Indeed, as an illustration, consider the Recommendation ITU-R P.1853-1 input parameters 5.48 , 1.69, 1.59 in compliance with Section IV. The rain attenuation correlation function of the ITU synthesizer driven by (5) is then given in Fig. 2. The associated conditional parameters also given in Section IV are 0.96 and 1.10. According to Fig. 3, the one to one correspondence between and of the new time series synthesizer driven by (14), (12b), and (12d) is then used to interpolate at . Finally, the correlation function that must be given to the underlying Gaussian process in (11) to insure that the rain attenuation process in (11) has the same correlation function —i.e., the same dynamics—as Recommendation ITU-R P.1853-1 driven by (5) is shown in Fig. 4. In accordance with Fig. 4, the logarithm of the correlation function of in (11) shows a linear dependency with respect to . Therefore, the correlation function of in (11) is exponential, i.e., accepts the analytical formulation (1), but now with 4340 s. Obviously, other departures from the ITU parameter 5000 s have to be

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Fig. 5. Worldwide map of the correlation time that has to be given to the —supposed to be exponential—of in (11) to correlation function in (11) reproduces the dynamics of the current ITU synthesizer insure that driven by (5). The frequency is 40 GHz and the radio link geometrical configurations are defined in Section IV.

expected depending on the local values of the parameters , , , , and . This point is highlighted worldwide in Fig. 5 where that has to be given to in (11) to mimic the dynamics of the ITU synthesizer is regressed on a worldwide basis considering the satellite radio links operating at 40 GHz defined in Section IV. It is important to note that, as the dynamics of the ITU synthesizer have been validated so far only from experimental rain attenuation time series collected at mid-latitudes, the validity of Fig. 5 should be limited to mid-latitudes areas. At this stage, an algorithmic scheme to generate stationary correlated Gaussian processes with arbitrary correlation function (the one derived in Fig. 4 for instance) is still required to make effectual the new time series synthesizer driven by (11). This point is addressed in Section III-D. D. Generating of a One-Dimensional Correlated Gaussian Process in the Fourier Domain Our objective is to generate a one-dimensional stationary real Gaussian process with zero mean, variance one, and arbitrary correlation function . The methodology lies on the algorithmic approach defined by [20] to simulate bidimensional Gaussian processes with arbitrary spatial covariance function. Here, its adaptation to the one-dimensional case is conducted and is fully demonstrated in Appendix B. For numerical implementation, define , where are the points (or instants) where has to be specified and is the length of the random process (or duration so that refers to the sampling rate). is constructed using a Fourier series: (15a) and (15b) where and are the direct and inverse Fourier transforms, respectively. The correlated Gaussian random process

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is then constructed in the Fourier domain according to the following algorithm: 1) Generate uncorrelated random complex numbers which real and imaginary parts are normal, with 0 mean and variance 1. For and , put the imaginary part of to 0. 2) Complete the definition of so that for . 3) Define except for and where . 4) From the analytical (if available) or the numerical definition of , compute the Fourier transform according to (15b). 5) Define (16) from (15a). 6) Derive Justifications of the algorithm described above are given in Appendix B. In particular, it is demonstrated that the correlated Gaussian process constructed that way has 0 mean, variance 1, and correlation function . It is important to note that any correlation function can be introduced: as required, the exponential definition of that was a basic assumption of the current ITU synthesizer is now relaxed. IV. CAPABILITY OF THE MODELS TO REPRODUCE ABSOLUTE RAIN ATTENUATION CCDFS WORLDWIDE To quantitatively assess on a worldwide basis the ability of both rain attenuation synthesizer to reproduce absolute rain attenuation CCDFs , satellite radio links operating at 20, 30, 40, and 50 GHz with three geostationary satellites arbitrary located at longitudes 80 , 0 , and 80 are considered. As experimental data are not available worldwide, Recommendation ITU-R P.837 [17] and ITU-R P.618-10 [16] are used to predict worldwide (resolution 1.125 ) the probability of rain and the absolute rain attenuation CCDF , respectively. It is important to note that we recall that the validity range of the worldwide predictions of given by Recommendation ITU-R P.618-10 is limited to 5% in compliance with [16]. Moreover, as noted by Recommendation ITU-R P.1853-1 [13], the probability to have rain attenuation is approximated by . On the one hand, for each frequency and each location, the input parameters and required by the ITU synthesizer are derived worldwide from the lognormal regression of the absolute CCDFs given by Recommendation ITU-R P.618-10. The regression is conducted in compliance with the methodology recommended in [13], for probabilities between 10 % and the minimum value between % and 5%. To make the regression reliable, areas where is lower than 10 % are disregarded. Moreover, to reduce the computation time, only links with elevation angle greater than 25 are considered. For pixels in the coverage intersection of two satellites, the parameters corresponding to the highest elevation link are kept. On the other hand, the absolute rain attenuation CCDFs given by Recommendation ITU-R P.618-10 are regressed by a mixed Dirac-lognormal distribution as required

Fig. 6. Absolute rain attenuation CCDF given by Recommendation ITU-R P.618-10, reproduced by the current ITU synthesizer, and reproduced by the new synthesizer for an hypothetical radio link at 50 GHz between 1.44 and a geostationary an Earth station situated at Toulouse 43.60 satellite located at longitude 0 .

by the new rain attenuation synthesizer. Worldwide maps of and are then obtained. the conditional parameters Fig. 6 is an example of the absolute rain attenuation CCDF given by Recommendation ITU-R P.618 for an hypothetical radio link at 50 GHz between an Earth station situated at Toulouse 43.60 1.44 and the geostationary satellite located at longitude 0 . On the one hand, the lognormal regresconducted in compliance with [13] leads sion on to 1.69 and 1.59 while Recommendation ITU-R P.837 gives 5.48% . It follows that the in (5) is equal to 2.35 dB in compliance offset parameter with (4). On the other hand, the Dirac-lognormal regression of leads to 0.96 and 1.10. The rain attenuation CCDFs finally reproduced by the current ITU synthesizer (7d) and by the new synthesizer (12a) are plotted for comparison. For time percentages between 10 % and 5% (i.e., the highest probability value given by Recommendation ITU-R P.618-10), Fig. 6 underlines the greater ability of the new model to reproduce the input CCDF given 5.48%, by Recommendation ITU-R P.618-10. Beyond both model assumes clear sky conditions and the rain attenuation CCDF of both synthesizer falls to 0 as expected. In compliance with Fig. 6, note that the probability to have rain attenuation suggested by Recommendation ITU-R P.618-10 (i.e., derived from an extrapolation) would differ from the probability of rain 5.48%. Of course, the probability to have rain attenuation on a slant path that may extend to a few kilometers depending on the radio link elevation and on the local rain altitude is expected to be greater than the local probability of rain but probably not in the proportion suggested in Fig. 6. Such a difference is clearly acknowledged in Recommendation ITU-R P.1853-1 and refers to the difficulty to measure and a fortiori to model the probability to have rain attenuation on an Earth–space satellite link. Discussions on this difficult topic are out the scope of the present paper. To quantitatively assess the greater ability of the new model to reproduce given by Recommendation ITU-R of the error recommended in P.618-10, the rms value

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percent) computed worldwide for the radio links at 20, 30, 40, and 50 GHz, respectively. In compliance with the definition of , note that the new synthesizer is all the better with respect to the current ITU model as tends to 100%. 0% means no improvement and 0% means that the new model reproduces less accurately the input rain attenuation CCDF given by Recommendation ITU-R P.618-10 than the current ITU synthesizer. For the satellite link at Toulouse considered in Fig. 6, 50.6%. First, according to Fig. 7(a)–(d), 0% whatever the frequency and whatever the location. Therefore, the new synthesizer reproduces better the absolute rain attenuation CCDF given as input parameter than the current ITU synthesizer. The improvement is all the better as the frequency increases in compliance with the analytical derivations conducted in Section II-B. Indeed, recalling that the rain attenuation increases with frequency, the same behavior applies to and , and it can be concluded from (4) that the offset parameter increases with the frequency. Therefore, in accordance with equation (7d), the current ITU synthesizer departs all the more from given by Recommendation ITU-R P.618-10 as the frequency increases. Lastly, the improvement strongly depends on the location. Indeed, varies from about 30% in mid-latitudes and reaches 100% in tropical or equatorial areas. V. CONCLUSION

Fig. 7. Ratio (%) computed worldwide for the radio links at (a) 20, (b) 30, (c) 40, and (d) 50 GHz. The location of the three satellites considered (SAT 1, SAT 2, SAT 3) are recalled.

Recommendation ITU.R P.311 [21] is computed for each model on a worldwide basis. In particular, Fig. 7(a)–(d) shows the ratio (in

From the analytical derivation of the first- and second-order statistical properties of the rain attenuation time series synthesizer proposed in Recommendation ITU-R P.1853-1, two shortcomings have been demonstrated. First, due to the offset parameter, it has been shown analytically that the ITU synthesizer does not reproduce the absolute rain attenuation CCDF yet given as an input parameter. Second, in Recommendation ITU-R P.1853-1, it is necessary that the correlation function used to generate the underlying stationary Gaussian process is exponential. Clearly, this assumption does not allow any rain attenuation correlation function to be reproduced. More flexibility is required, especially if a worldwide synthesizer able to account for the local climatology has to be defined and if better performances are expected for any climatic area. Therefore, a new rain attenuation time series synthesizer has been proposed. It relies on a mixed Dirac-lognormal modeling of the absolute rain attenuation CCDF. It has then been shown analytically that the new synthesizer allows the 1st order statistics given as input parameters to be reproduced. Second, for worldwide applications, the new synthesizer is able to reproduce any rain attenuation correlation function . Particularly, as the dynamics of the current ITU synthesizer has been intensively tested and validated in previous studies from experimental rain attenuation time series collected at mid-latitudes, the capability of the new synthesizer to reproduce the rain attenuation correlation function of the ITU synthesizer has been demonstrated. To make the approach effectual and offer a framework allowing potential future improvements, a methodology to simulate one- dimensional Gaussian processes with arbitrary correlation function has been presented and thoroughly justified.

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Then, the ability of each synthesizer to reproduce absolute rain attenuation CCDFs given by Recommendation ITU-R P.618 has been compared on a worldwide basis. The new synthesizer then shows is greater ability to reproduce worldwide. Therefore, the new rain attenuation time series synthesizer reproduces better the rain attenuation CCDF given as input parameter, preserves the rain attenuation dynamics of the current ITU synthesizer for simulations at mid-latitudes, and, if it proves to be necessary for worldwide applications, is able to reproduce any rain attenuation correlation function. Consequently, the new synthesizer might be considered as an improvement and a generalization of the current ITU rain attenuation time series synthesizer. Now, the ability of the new synthesizer to reproduce statistics derived from experimental rain attenuation time series collected worldwide must be intensively tested. In particular, the mixed Dirac-lognormal modeling of the absolute CCDF must be carefully investigated on a worldwide basis from experimental CCDFs. Moreover, at mid-latitudes, the capability of the new synthesizer to reproduce any rain attenuation correlation function should be used to confirm (or not) that an exponential is definitively the analytical formulation that best restitutes the experimental rain attenuation dynamics as reported by [9]–[11] or [12]. Its flexibility should also be used to investigate seasonal effects. The same exercise must be conducted from equatorial and tropical experimental propagation data to finally define a worldwide parameterization that optimally reproduces the rain attenuation experimental statistics.

Let

and define (A3)

so that (A4) In such conditions

(A5) As

, (A5) implies that . Recalling that (A6)

the integration with respect to leads to

in (A5) is straightforward and

APPENDIX A ANALYTICAL DERIVATION OF THE CORRELATION FUNCTION OF THE ITU RAIN ATTENUATION PROCESS In compliance with Section II, the analytical derivation of requires the computation of the covariance function:

(A7) Now, let (A8) After some manipulations, (A7) becomes (A9) where (A10)

(A1)

where and

and

, , is the bivariate normal PDF given by (A11)

(A2)

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Depending on , two cases have to be considered. If : we use the approximation

and designating by gate, (15) leads to

the complex conju-

(A12) for [18]. which holds with an error less than , , , In (A11), , and . In such conditions, (A12) in (A9) and recalling (A6) finally leads to

(B1) is real with zero average and variance one and As of the stationary recalling that the correlation function process is an even function, it follows that . Defining ’, (B1) becomes

(A13) where erfc is the complementary error function, , and If : we use the approximation

,

(B2) Consequently

(A14) for [18] and which holds with an error less than have been defined in (A12). Therefore, where a, , , , (A14) in (A9) and (A6) finally lead to

(B3) It follows from (B3) that (A15) where as before . Depending on the sign of

but now with

and

defined by (A10), can be derived analytically from (A13) or (A15) with good accuracy. The correlation function of the ITU rain atdefined by (5) is finally obtained using tenuation process (7a) and (7c). It is important to note that is a function of , , , and . APPENDIX B STATISTICAL PROPERTIES OF THE GAUSSIAN PROCESS DEFINED BY (15) AND (16) AND JUSTIFICATIONS OF THE ALGORITHMIC APPROACH TO GENERATE 1) Link between the correlation function and the Fourier coefficients driven by (15a) and (15b). Defining

if otherwise 2) Statistical properties of the random process from (16). It follows from (15) that

(B4) derived

(B5) In compliance with (16), since is normal with mean 0 so that average. Moreover

has zero

(B6)

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With

’, (B6) leads to

Using (B4), (B7) finally leads to

(B7) (B8)

In compliance with (B8), the random process constructed from (16) complies with the requirements, i.e., is Gaussian with correlation function , has 0 average and variance by definition of the correlation function at lag 0. 3) Details and justifications of the algorithm to construct the correlated Gaussian field in the Fourier domain. By using expansions (15a) and (15b), we are implicitly assuming that is periodic since . Moreover, as is real, it follows that . In such conditions, the Fourier coefficients in (15b) must verify

(B9) Moreover, as

is an even function, it follows that from (B4) is real. Therefore, is real and in (16) implies that and must be real for and in compliance with step 1 and 2 of the algorithm given in Section III-D From (16) and (B4): (B10) . As the real which holds whatever only if and imaginary parts of are uncorrelated with variance 1 (except for and where the imaginary part of is 0), it follows that except for and , where . It follows that except for and where in compliance with step 3 of the algorithm given in Section III-D. REFERENCES [1] E. Salonen, S. Karhu, P. Jokela, W. Zhang, S. Uppala, H. Aulamo, S. Sarkkula, and P. P. Baptista, “Modelling and calculation of atmospheric attenuation for low-fade margin satellite communications,” ESA J., vol. 16, no. 3, pp. 299–316, 1992. [2] L. Castanet, M. Bousquet, M. Filip, P. Gallois, B. Gremont, L. D. Haro, J. Lemorton, A. Paraboni, and M. Schnell, “Impairment mitigation and performance restoration,” ESA Publications Division, SP-1252, COST 255 Final Rep. Ch. 5.3, Mar. 2002. [3] L. Castanet, A. Bolea-Alamañac, and M. Bousquet, “Interference and fade mitigation techniques for Ka and Q/V band satellite communication systems,” presented at the COST 272-280 Int. Workshop Satellite Commun. from Fade Mitigation to Service Provision, Noordwijk, The Netherlands, May 2003. [4] L. Castanet, D. Mertens, and M. Bousquet, “Simulation of the performance of a Ka-band VSAT videoconferencing system with uplink power control and data rate reduction to mitigate atmospheric propagation effects,” Int. J. Satell. Commun., vol. 20, no. 4, pp. 231–249, Jul./Aug. 2002. [5] T. Maseng and P. M. Bakken, “A stochastic dynamic model of rain attenuation,” IEEE Trans. Commun., vol. 29, no. 5, pp. 660–669, May 1981.

[6] F. J. A. Andrade and L. A. R. da Silva Mello, “Rain attenuation time series synthesizer based on the gamma distribution,” IEEE Antennas Wireless Propag. Lett., vol. 10, pp. 1381–1384, 2011. [7] G. Karagiannis, A. D. Panagopoulos, and J. D. Kanellopoulos, “Multidimensional rain attenuation stochastic dynamic modeling: Application to earth-space diversity systems,” IEEE Trans. Antennas Propag., vol. 60, no. 11, pp. 5400–5411, Oct. 2012. [8] M. Cheffena, L. E. Braten, and T. Ekman, “On the space-time variations of rain attenuation,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1771–1782, Jun. 2009. [9] F. Lacoste, M. Bousquet, L. Castanet, F. Cornet, and J. Lemorton, “Improvement of the ONERA-CNES rain attenuation time series synthesizer and validation of the dynamic characteristics of the generated fade events,” Space Commun. J., vol. 20, no. 1–2, 2005. [10] F. Lacoste, M. Bousquet, L. Castanet, F. Cornet, and J. Lemorton, “Event-based analysis of rain attenuation time series synthesizers for the Ka-band satellite propagation channel,” presented at the ClimDiff Conf., Cleveland, OH, USA, Sep. 2005. [11] G. Carrie, F. Lacoste, and L. Castanet, “A new ‘event-on-demand’ synthesizer of rain attenuation time series at Ku-, Ka- and Q/V bands,” Int. J. Satell. Commun. Netw., vol. 29, no. 1, pp. 47–60, Jan./Feb. 2009. [12] Influence of the Variability of the Propagation Channel on Mobile, Fixed Multimedia and Optical Satellite Communications, L. Castanet, Ed. et al. Aachen: Shaker Verlag, 2008, SatNEx JA-2310 book, ISBN 978-3-8322-6904-3. [13] Tropospheric Attenuation Time Series Synthesis, ITU-R Recommendation P.1853-1, ITU, Geneva, Switzerland, 2009. [14] H. Sauvageot, “The probability density function of rain rate and the estimation of rainfall by area integrals,” J. Appl. Meteor., vol. 33, no. 11, pp. 1255–1262, 1994. [15] Specific Attenuation Model for Rain for Use in Prediction Methods, ITU-R Recommendation P.838-3, ITU, Geneva, Switzerland, 2005. [16] Propagation Data and Prediction Methods Required for the Design of Earth-Space Telecommunication Systems, ITU-R Recommendation P.618-10, ITU, Geneva, Switzerland, 2009. [17] Characteristics of Precipitation for Propagation Modelling, ITU-R Recommendation P.837-5, ITU, Geneva, Switzerland, 2007. [18] F. G. Lether, “Elementary approximation for erf(x),” J. Quant. Spectrosc. Radiat. Transfer, vol. 49, no. 5, pp. 573–577, 1993. [19] S. A. Kanellopoulos, A. D. Panagopoulos, and J. D. Kanellopoulos, “Calculation of the dynamic input parameter for a stochastic model simulating rain attenuation: A novel mathematical approach,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3257–3264, 2007. [20] T. L. Bell, “A space-time stochastic model of rainfall for satellite remote-sensing studies,” J. Geophys. Res., vol. 92, pp. 9631–9643, 1987. [21] Acquisition, Presentation and Analysis of Data in Studies of Tropospheric Propagation, ITU-R Recommendation P.311-13, ITU, Geneva, Switzerland, 2007. Xavier Boulanger was born in Montpellier, France, in 1986. He received the Diploma of Engineering degree in electronics and digital communications from Ecole Nationale de l’Aviation Civile (ENAC), Toulouse, France, in 2010. In 2009, he conducted his End of Study Internship in ONERA, France, on the improvement of the modeling of the propagation channel for fixed Satcom systems over temperate climates. Since 2010, he has been working towards the Ph.D. degree in collaboration between the French Space Agency (CNES), France, and ONERA, France. He was also a contributor in the European action COST IC-0802. Laurent Féral, photograph and biography not available at the time of publication. Laurent Castanet, photograph and biography not available at the time of publication. Nicolas Jeannin, photograph and biography not available at the time of publication. Guillaume Carrie, photograph and biography not available at the time of publication. Frederic Lacoste, photograph and biography not available at the time of publication.