3.7 Numerical simulations
3.7.4 Fade duration statistics
Finally, we discuss fade duration statistics to prove that the generated series are reliable in terms of attenuation fades intervals. Fades duration statistics are of concern for the evaluation of parameters associated with the risk of failure of a variety of
0 50 100 150 200 250 300 10−2 10−1 100 Distance [km] Normalized χ index
Average χ from NIMROD dataset
Correlated TS: average 10 realizations Uncorrelated TS: average 10 realizations
Figure 3.14: χ index obtained from the proposed model compared to the average χ
index obtained by the NIMROD network. Theχ index obtained from the conditioning process (red dots) is compared with the χindex resulting from the complete model (circles). The effect of the conditioned process, which imposes rain
attenuation correlation, is evident for small distances (up to 50 km) where the impact of rain attenuation correlation is higher. It is also interesting the more accurate decorrelation achieved for large distances: the result of the two processes of
the model better represent the expected rain decorrelation than the single conditioning process.
telecommunication systems. Fade duration is defined as the time interval between two crossings above the same attenuation threshold. In the context of availability criteria, of particular interest is the distinction between fades of shorter or longer duration than 10 s. Knowledge of the distribution of fade duration as function of fade depth is also a prerequisite for the application of risk concepts in the provision of telecommunication services [67].
• P rob(d > D|a > A), the probability of occurrence of fades duration d longer thanD(s), given that the attenuationais greater thanA(dB). This probability can be estimated from the ratio of the number of fades of duration longer than
Dto the total number of fades observed, given that the thresholdAis exceeded;
• F(d > D|a > A), the cumulative exceedance probability, or equivalently, the total fraction (between 0 and 1) of fade time due to fades of duration d longer thanD(s), given that the attenuationais greater thana(dB). This probability can be estimated from the ratio of the total fading time due to fades of duration longer than D given that the thresholdA is exceeded, to the total exceedance time of threshold.
The latter statistics has been considered to test the proposed model. Fade duration statistics are evaluated from the obtained time series according to:
F(A|D) = NS(D)
NT OT (3.23)
where NS(D) is the total time the attenuation A (dB) is exceeded composed of intervals longer than D (s) and NT OT the total time the attenuation A is exceeded. The ITU-R model for fades duration [67] is taken as reference in terms of prediction accuracy. The ITU-R model [67] is expected to be valid for durations longer than 1 s and requires the following parameters:
• f: frequency (GHz): 10−50 GHz;
• ϕ: elevation angle (degrees): 5−60o
Table 3.3: List of the DBSG3 considered experiments
Name Country Lat.
[deg] Lon. [deg] Alt. [m] Duration [dd] Freq. [GHz] Elev. [deg.] Ottawa CAN 45.34 284.11 81 362.74 20.19 27.32 Chibolton UK 51.13 358.57 89 365 20.7 28 Sparsholt UK 51.08 358.61 119 365 20.7 28 Dundee UK 56.45 357.02 49 365 20.7 23.29
The proposed model was tested against a set of experimental data, results have been compared with the predicted statistics of the ITU-R model for the same experiments. Measured statistics of fade duration are obtained from the DBSG3 database [37], a set of 9 experimental statistics for 4 different sites, listed in Tab. 3.3, have been selected. The experiments were chosen based on the availability of statistical data at the frequencies close to 20 GHz.
Time series of rain attenuation for the considered links were generated by the proposed model. Statistics were obtained for different duration (60 120 180 300 600 900 1200 1500 1800 2400 s) and for different attenuation thresholds (3 5 10 15 20 25 dB). The statistics obtained by the time series and the ITU-R model are compared with those of the experimental data, the overall results are presented in terms of average error and RMSE error of the figure of merit ε(D, A) [68]:
ε(D, A) = ln 100%−Fp(D|A) 100%−Fm(D|A) (3.24)
Tab. 3.4 shows the overall error obtained by averaging over all the attenuation values and thresholds, for all the stations considered. The proposed model exhibits a mean error and RMSE very similar to the ITU-R statistical prediction, proving that fades duration statistics obtained from the correlated series are reliable.
Model Mean error RMS error
ITU-R -0.45 0.66
TS -0.48 0.69
Table 3.4: Fade Durations Test Results
3.8
Conclusions
This chapter describes a model to generate time series of rain attenuation values (1 sample/s) for multiple receiving stations in a large geographical area by adapting a single set of experimental rain attenuation measurements. The reference rain atten- uation measurements are obtained from a large database collected at Spino d’Adda, Italy, during ITALSAT propagation campaign (from 1994 to 2000). The task espe- cially addressed in this work is to reproduce concurrent attenuation time series for a large number of stations with the desired rain statistics and correlation, both in terms of rain occurrence and rain attenuation.
The proposed model consists of two independent processes, one assigning the rainy time to the stations and the other assigning concurrent rain attenuation values for the identified “rainy” stations. A mathematical description based on a multidimen- sional Gaussian model allows to suit the measured values to the different locations, arranging the attenuation time series with the desired spatial correlation. The ob- tained time series are composed of basic periods of fixed length (1 hour in this case, which was found to be a good trade off between the correlation interval of events). The generated time series is composed of discrete periods either of real rain measure- ments or no rain intervals, with a time resolution equal to 1 sample/sec. An important advantage of the proposed model is that, within the basic period duration, the char- acteristics of the rain process are defined by the real measurements. The basic period duration defines the update timing of the model: at every period (“epoch”) rainy
stations are properly identified and series of ran attenuation measurements assigned. The temporal evolution of the rainy events is driven by an exponential correlation model and the spatial correlation of rain events and attenuation is correctly imposed. The model has been tested on its ability to reproduce the long-term rain statis- tics of single stations with different climatological characteristics and for different simulation durations. In addition to the first-order statistics of rain attenuation, the model is able to reproduce the spatial distribution of rain attenuation: concurrent rain conditions over multiple sites have been verified by the evaluation of joint at- tenuation CDF and by measuring the decorrelation index of rain among the stations (χ index). From a temporal point of view, a validation in terms of fade duration statistics is proposed.
Results obtained by simulations are compared with current models, showing a good agreement and proving the model validity. Especially, long-term statistics are well reproduced for a wide range of probability levels with a limited number of simulated year-long time series. In terms of space diversity, joint attenuation CDF are reproduced with good accuracy compared to ITU-R statistical model. Looking at multiple sites, the rain correlation among multiple stations well matches independent reference measurements, validating the mutual correlation processes implemented. Finally, fade duration statistics show the same accuracy of current statistical model for a set of radio links with available measurements. In conclusion, the model here proposed is suitable for simulating accurate concurrent rain attenuation conditions for multiple sites, taking into consideration the spatial correlation of rain events and attenuation.
The knowledge of concurrent attenuation values over a large geographical area can be used to verify the effectiveness of proper Fade Mitigation Techniques FMTs for current and new SatCom systems. In this perspective, the joint use of the gen-
erated time series with other FMT, such as Adaptive Modulation and Coding and Reconfigurable Antenna [69] is introduced and discussed in the next Chapters.