new physical-mathematical model for predicting slant-path rain attenuation statistics based on...

www.ietdl.org

9©

Published in IET Microwaves, Antennas & PropagationReceived on 30th January 2013Revised on 10th June 2013Accepted on 13th June 2013doi: 10.1049/iet-map.2013.0206

70The Institution of Engineering and Technology 2013

ISSN 1751-8725

New physical-mathematical model for predictingslant-path rain attenuation statistics based on inverseGaussian distributionCharilaos Kourogiorgas, Athanasios D. Panagopoulos

School of Electrical and Computer Engineering, National technical University of Athens, Athens, Greece

E-mail: [email protected]

Abstract: A new physical-mathematical model for predicting first-order rain attenuation statistics for slant-path links is presentedin this study. It is assumed that point rain rate and slant-path rain attenuation follow the inverse Gaussian (IG) distribution. Thestatistical parameters of the IG distribution of the induced rain attenuation on a slant path are calculated in terms of the rain ratestatistical parameters by adopting a spatial inhomogeneity model for specific rain attenuation. The method is validated usingexperimental data taken from Database of Study Group 3 (radiowave propagation) of International Telecommunication Union-Radio (ITU-R), hereafter named DBSG3 and with simulated rain attenuation data using the Synthetic Storm Technique onrecent rain rate measurements in Athens, Greece. Its performance is also compared to the ITU-R P. 618-10 model. Theproposed model shows better performance comparing to the ITU-R model and the results are very encouraging.

1 Introduction

The demand for high-data transmission rates in moderncommunication systems and services increases nowadays.Moreover, the spectrum congestion is another cruciallimitation towards the wireless vision. Consequently, thefuture satellite communication systems migrate to higherfrequency bands such as Ka and Q/V bands, in order toprovide users with large capacities. At these frequency bands,the satellite links are susceptible to a series of physicalphenomena related to the propagation of radiowaves throughthe troposphere thus leading to significant signal degradation[1]. Among these tropospheric phenomena, rain is the maindominant fading mechanism for satellite links operatingabove 10 GHz and is the only atmospheric phenomenon thatexhibits significant spatial inhomogeneity within the distancesof interest [1, 2].Owing to the high values of rain attenuation, Fade

Mitigation Techniques (FMTs) must be integrated into thesatellite communication systems in order to improve itsperformance in terms of outage probability, capacity, packetand bit-error rate, throughput etc. [1]. For the evaluationand the reliable implementation of the FMTs for systemoperating above 10 GHz, rain attenuation and rain ratestatistics are required. The values of rain attenuation that areexceeded for time percentages from 0.001 to 0.1%,considering the desired availability, should be predicted forthe design of satellite links in order to define the margins ofthe communication system.Numerous propagation models have been developed since

the decade of 1970 for the prediction of the long-term (e.g. fora period of at least a year) exceedance probability of rain

attenuation at Earth-space links. In the InternationalTelecommunication Union-Radio (ITU-R) Recommendationfor propagation data and prediction methods required fordesign of Earth-space communication systems (ITU-RP.618-10 [3]) an empirical model is suggested forprediction of the rain attenuation exceedance probability.This ITU-R model is using as inputs the R0.01 value ofrainfall rate which is the rainfall rate that is exceeded forthe 0.01% of total time.A model which can be included in ‘semi-physical’ models

category is the SC-EXCELL model [4]. In this category ofprediction models, rain attenuation is evaluated throughmodelling of rain rate distribution with a population of raincells. The SC-EXCELL model discriminates stratiform andconvective rain and it is based on the EXCELL model[5, 6]. In [7], the Synthetic Storm Technique (SST) forcalculating rain attenuation simulated time series usingexperimental rain rate time series is presented. For the lattermodel, in [8], the long-term exceedance probability derivedfrom the SST has been validated with rain attenuationmeasurements, and it has been found that it shows a goodaccuracy of the prediction. Furthermore, physical-mathematical models for the prediction of rain attenuationdistribution based on lognormal and gamma distributionassumptions with inputs of the statistical distribution ofrainfall rate, have been proposed in [9, 10]. In [11], a globalmethod for the prediction of the slant-path rain attenuationstatistics based on Weibull distribution has also beenproposed. Moreover, an easy calculated formula for thecalculation of exceedance probability based on regressionfitting analysis of the Weibull model [11] has beenpresented in [12].

IET Microw. Antennas Propag., 2013, Vol. 7, Iss. 12, pp. 970–975doi: 10.1049/iet-map.2013.0206

www.ietdl.org
In this paper, a new physical-mathematical method for the
prediction of rain attenuation statistics on Earth-space linksusing as inputs the rain rate statistics, assuming that rainattenuation and rain rate can be approximated by theinverse Gaussian (IG) distribution [13], is proposed. IGdistribution has been recently used to describe shadowingphenomena [14] and turbulence-induced phenomena infree-space optical systems operating in weak turbulenceconditions [15]. Furthermore, the IG distribution has beenalso used to model the conditional distribution of rain ratedue to the fact that it is a skewed distribution [16]. Owingto this skewness property, we propose the modelling of rainrate and rain attenuation using IG distribution. Themodelling of the first-order statistics of rain rate and rainattenuation with the IG distribution through a fitting of thetheoretical distribution to rain rate and rain attenuation datahas been presented in [17], whereas in [18], the bivariate IGdistribution is used for the prediction of outage probabilityin dual-site Earth-space diversity systems.The rest of the paper is organised as follows: in Section 2,

the method for modelling of rain attenuation statistics isdescribed. Section 3 includes the validation tests of theproposed model. The model is compared with the ITU-RP. 618-10 model using the Databank of ITU-R Study Group3 (DBSG3) [19]. Considering the validation tests, we alsocompare our model with 1 year simulated rain attenuationby applying SST [7] in rain rate data measured in two sitesinside the campus of National Technical University ofAthens (NTUA), Athens, Greece with very good results.Finally, some useful conclusions derived from this paperare given in the final Section 4.

2 Outage prediction method

A fundamental assumption of the proposed model is that bothpoint rainfall rate and rain attenuation follow the IGdistribution. Since, the statistical parameters of the IGdistribution of rain rate are needed as an input to theproposed model they can be computed through a non-linearregression to fit the theoretical complementary cumulativedistribution function (CCDF) of IG distribution to theexperimental or predicted exceedance probability of rain rate.Since, it is also assumed that rain attenuation follows the IG

distribution, the exceedance probability of slant-path rainattenuation (As), that is, the probability that rain attenuationexceeds a certain threshold value (Ath) is given from the IGdistribution [13, 20, 21]

P As ≥ Ath

[ ] = 1− Q

��l

Ath

√1− Ath

m

( )( )

− e2(l/m)Q

��l

Ath

√1+ Ath

m

( )( ) (1)

where Q()is the Gaussian Q-function [22], λ and μ are the twopositive parameters of slant-path rain attenuation. The meanvalue and the variance of a random variable which followsthe IG distribution are [13]

E As

[ ] = m (2)

E As − m( )2[ ]

= m3

l(3)


respectively, with E[] being the expectance operator. For anIG random variable, its moment for a non-integer order b is(by using the integral 9, in page 368 of [22])

mb = 2

��l

2p

√e(l/m) m2( ) (b−1/2)/2( )

Kb−1/2l

m

( )(4)

where Kn(x) is the modified Bessel function of second kind oforder n.Following Crane’s hypothesis, that the vertical profile of

rain rate is uniform [23], rain attenuation along the slantpath (AS) with average effective path length LS is related tothe rain attenuation along the projection of the slant path onthe ground (A) with average effective path length (L)according to

A = AS cosw (5)

where j is the elevation angle of the link and the relationshipbetween the two effective path lengths is L = LS cosj.Considering the CCDFs of A and AS, both derived from (1)with different IG parameters (λ and μ for slant path rainattenuation and λA and μA for the projected rainattenuation), it holds that

P As ≥ xs[ ] = P A ≥ xD

[ ](6)

where xD = xScosj. It is well-known that [24]

A =∫L0aR(x)b dx (7)

where R(x) is the rain rate at point x, a and b are the specificattenuation coefficients which depend on the elevation angle,the polarisation and the frequency of the link. The twocoefficients a, b can be calculated from ITU-R. P. 838-3[25]. Hereafter it is assumed that the statistical parametersof the IG distribution of rain rate are notated as λR and μRand are assumed constant with space, which follows fromthe assumption that the mean value and variance of rainrate are constants.Therefore the mean value of A can be computed from (7) as

mA, 1 =∫L0E aR(x)b[ ]

dx = amR, bL (8)

where E[] is the expectance operator and mR, b is the momentof rain rate of order b calculated from (4). As for the varianceof rain attenuation it holds that

s2A =

∫L0

∫L0E G(x)G(x′)[ ]

dx dx′ −∫L0E[G(x)] dx

( )2

(9)

where Γ(x) = aR(x)b, is the specific rain attenuation. In orderto solve (9) a spatial correlation coefficient (ρ) of specificrain attenuation is introduced

r(x, x′) = E G(x, t)G(x′, t)[ ]− m2

G, 1

s2G

(10)

where mΓ, 1 = a ·mR, b is the mean value of specific rainattenuation and s2

G = a2 mR, 2b − m2R, b

( )its variance with

mR, 2b the moment of rain rate of order 2b. From (9), the

971© The Institution of Engineering and Technology 2013

Fig. 2 Standard deviation of the error for the IG model and theITU-R. P. 618 model

Fig. 1 Mean value of the error for the IG model and theITU-R. P. 618 model

Fig. 3 RMS value of the error for the IG model and theITU-R. P. 618 model

www.ietdl.org
expressions of the mean value and variance of specific rainattenuation and the assumption that the statisticalparameters of rain rate are constant it follows that
s2A = s2

GH1 = a2 mR, 2b − m2R, b

( )H1 (11)

with

H1 =∫L0

∫L0r(x, x′) dx dx′ (12)

For the computation of the variance of rain attenuation aspatial coefficient must be adopted. Although, anycoefficient can be used in (12), in this paper the onepresented in [9] is recommended

r = G��G2 + d = x− x′| |( )2

√ (13)

where d is the distance between points x and x′, G is aconstant which depends on the local climatic characteristics,with a typical value of 1.5 km for Europe and USA. Thedouble integral of (12) is calculated for the specificcorrelation coefficient shown in (13) as

H1 = 2LG sinh−1 L

G

( )+ 2G2 1−

��L

G

( )2

+1

√⎡⎣

⎤⎦ (14)

Finally, from the expressions of the mean value and varianceof an IG random variable with the parameters μ and λ, (2) and(3), the IG statistical parameters of the projection of slant-pathrain attenuation on the ground (A) are

mA = amR, bL (15)

lA = amR, bL( )3

a2 mR, 2b − m2R, b

( )H1

(16)

The outage probability can be calculated using (1) and thestatistical parameters of the expressions (15) and (16).

3 Numerical results and discussion

3.1 Earth-space experiments

The proposed method is first tested against the DBSG3database of ITU-R [19]. From the database 83 experimentswere chosen taking into account the existence of concurrentrain rate and rain attenuation exceedance probabilities andthe comments given by ITU-R propagation experts. Thefrequency range of these experiments is from 11.198 to28.6 GHz, the elevation angle range is from 10.7° to 52°.Considering the geographical coordinates of the Earthstations of experiments the absolute value of latitude rangesfrom 0.2° (equatorial area) up to 67.4° and the longitudefrom 1.3° up to 358.57°, taking into account as manydifferent climatology as possible. From the tests, the meanvalue the standard deviation and the root-mean square(RMS) errors are computed for two cases: for every timepercentage, in Figs. 1–3, respectively, and the statistics oftotal error, shown in Table 1. The error is computedaccording to the ITU-R Recommendation P. 311-13 [26].



Table 1 Mean value, standard deviation and RMS value of theerror for IG model and ITU-R. P. 618-10 model, considering allthe errors

ɛmean, % ɛstd, % ɛrms, %

IG model − 1.55 21.53 21.59ITU-R. P.618 − 7.68 20.42 21.81

www.ietdl.org

The same approach is used for the computation of the error ofITU-R. P.618-10 [3]. As it is observed from the three figuresthe performance of the model is better comparing to the onefrom ITU-R. P.618-10. More particularly, as shown inTable 1, the RMS error of the proposed model is less thanthis of ITU-R P.618-10. Consequently, the proposed modelis flexible and can be used for any location of world withvery good accuracy. Here, it must be noted that if we haveused a different experimental database, it is expected thatthe new model will have better performance comparing toITU-R, since the parameters of the ITU-R model have beenrefined based on fitting process of the DBSG3 data.

Fig. 4 CCDF of rain attenuation derived from the application ofSST (solid line) on measured rain rate times series in NTUA, site1 (Ku band), from the IG model (dashed line) andITU-R. P. 618-10 (dash-dotted line)

Fig. 5 CCDF of rain attenuation derived from the application ofSST (solid line) on measured rain rate times series in NTUA, site2 (Ka band), from the IG model (dashed line) andITU-R. P. 618-10 (dash-dotted line)

3.2 Validation in simulated Earth-space rainattenuation time series

In this subsection, the proposed model and the ITU-RRecommendation P. 618-10 are compared to the CCDFcomputed from synthesised rain attenuation time series. Thetime series are derived after the application of the SST [7]on rain rate time series measured inside the NTUA campusin Athens, Greece. One reason why these time series areused is to show the performance of the proposed model andits comparison to ITU-R. P. 618 with data apart from theseof the DBSG3 databank, as the latter data have alreadybeen used for the derivation of the parameters ofITU-R. P. 618-10. A second one is that the exceedanceprobability calculated by the SST applied on measured rainrate time series has been validated with variousexperimental data [7, 8]. A great advantage of the SST isthat time series of rain attenuation can be generated for thewhole period in which measured time series of rain rate areavailable, taking into account all the events with high rainrate values or low ones.For the case presented in this paper, two rain gauges were

placed in the campus of NTUA with a separation distance of387 m. The rain gauges have an aerodynamic shape tominimise the effect of the wind on the measurements andthey measure the rainfall using tipping buckets with a tipsensitivity of 0.2 mm of rainfall per tip. Although there aredata measured for a period of 15 months, presented recentlyin [27], only 1 year measured rain rate time series wereused, started from 18 June 2010, as the random variable ofrain rate seems to be cycloperiodic. We use the 12 monthdata in order not to boost a certain period of the year. Thedata were processed in the same way as presented in [27].In order to apply the SST on rain rate measurements, there

was a change in the formulation of the SST. In [7], it refersthat the rain height is 5 km and the depth of the meltinglayer was assumed to be 0.4 km. To be consistent with theITU-Recommendation and the description of the proposedmodel made in Section 3, the rain height is derived fromthe map of ITU-R. P. 839 [28] and the depth of the meltinglayer was considered equal to 0.36 km, as this isrecommended in [28]. For comparing the two models twolinks were considered, one for every measured rain rate


time series. For the first site the elevation angle of the linkhas been set equal to 30° and the operating frequency equalto 12 GHz (Ku band), whereas for the second site theelevation angle of the link was chosen 43.2° and thefrequency 20 GHz (Ka band). In Fig. 4, three curves ofthe CCDF of rain attenuation for site 1 are shown derivedfrom the SST, ITU-R. P. 618-10 and the proposed model,whereas in Fig. 5, the same curves are shown for site 2.As in the case of the validation with DBSG3 data, the mean

error, the standard deviation and the RMS value of the errorhave been calculated. For site 1 the results are shown inTable 2 and in Table 3 for site 2. These errors werecalculated for the time percentages from 10− 3 to 10− 1% tobe consistent with the comparison made in Section 3.1. The


Table 2 Mean, standard deviation and RMS error (%) for site 1

ɛmean ɛstd ɛrms

IG model 11.91 20.5 23.71ITU-R. P.618-10 − 1.10 47.40 47.41

Table 3 Mean, standard deviation and RMS error (%) for site 2

ɛmean ɛstd ɛrms

IG model 8.91 25.35 26.87ITU-R. P.618-10 26.01 42.73 50.02

www.ietdl.org

same figures of merit employed in the previous subsectionhave been also used in this subsection.From Figs. 4 and 5, it can be observed that the prediction

made by the proposed model based on the IG distribution isvery close to the one derived from the time series, whereasthe recommendation of ITU-R. P. 618 behaves much worsethan the proposed model. This can be also pointed from thetwo tables (Tables 2 and 3) which present the statistics ofthe error. For site 1 the mean error of the proposed model isgreater than this of ITU-R P.618-10. However, the standarddeviation of the IG model is much less than this of ITU andso for the RMS value of the error (actually the half of thisof ITU). In site 2, all the statistics of the error of theproposed model are less than these of ITU-R. P. 618, with afinal RMS error of 26.85 and 50.01% for the IG model andthe ITU model, respectively.The poor prediction made by the ITU-R. P. 618 and the

great difference between the statistics of the error atSections 3.1 and 3.2 are due to the fact that its parameterswere derived from the data of DBSG3, as this has been alsopointed before and this gives a credit to the fact that for anynew data ITU-R model may not have the same behaviour,probably worse, comparing with the DBSG3 data. Theproposed model which is based on a very few basicassumptions for the physical process of rain rate and hasinputs the long-term statistics of rain rate is expected tohave generally very good performance since it is aphysical-mathematical one.

4 Conclusions

A new simple physical-mathematical method has beenproposed for predicting rain attenuation distribution onsatellite radio paths. Assuming that the rainfall rate and rainattenuation are following IG distribution and adopting asemi-empirical spatial correlation coefficient for the specificrain attenuation horizontal variation the statisticalparameters of rain attenuation can be calculated fromrainfall rate’s distribution parameters. The model shows avery good performance, it is based on physical andstatistical assumptions and still remains simple.The proposed model behaves very well for Earth-space

links having approximately the same overall RMS errorcomparing with experimental databanks. Comparing it withITU-R P.618-10 for satellite links, the proposed model isslightly better than this of ITU. Moreover, the proposedmodel has been compared to the CCDF computed from rainattenuation time series generated by the application of theSST on measured at Athens, Greece rain rate time series.From the last comparison, the error of the IG model is


much less than the prediction made by ITU-R. P. 618-10.As it has been discussed, the much better performance ofthe IG model than the ITU-R Recommendation in the lastcomparison, as well as the great difference between thestatistics of the error for the ITU-R model at Earth-spacelinks database (DBSG3) and the one derived from the SSTmay comes from the fact that the ITU model has beendeveloped by calculating its parameters from fitting theITU-R’s model to data from DBSG3 database.The advantages of the proposed model are its simplicity

and the direct derivation of the main long-term statisticalparameters of rain attenuation employing the expressions(15) and (16), using just a few input parameters.Furthermore, because of its simple and global physicalassumptions, the model will have approximately the sameor better behaviour than ITU-R models, which are empiricalalgorithms, as this was shown in the extended comparativetests. Moreover, the assumption of a single distribution formodelling rain attenuation has many applicationsconsidering the derivation of basic telecommunicationquantities such as the capacity with analytical expressions.In [29], analytical expressions for computing the outagecapacity for high frequency MIMO satellite systems aregiven assuming that rain attenuation follows lognormaldistribution. Similarly with this paper, if rain attenuation ismodelled using another distribution different thanlognormal, closed forms can be also derived for the outagecapacity. On the same point, as IG distribution gives goodresults for the outage prediction because if rain attenuation,the proposed model can be used for calculating the capacityfor diversity combining schemes. On the other hand,ITU-R. P. 618-10 model does not provide a specificdistribution for rain attenuation. Therefore a regressionfitting process must be used in order to obtain the statisticalparameters of the predicted rain attenuation according tothe chosen distribution. This can have negative effects, asthe fitted distribution may not reproduce accurately thepredicted rain attenuation.Finally, in the ITU-R Recommendation P. 1853-1 [30] a

rain attenuation time series synthesiser is proposedassuming that rain attenuation follows lognormaldistribution. To generate time series of rain attenuation,again a regression process must take place in order tocalculate the statistical parameters of rain attenuation byfitting the unconditional lognormal distribution to theprediction made by the model of ITU-R [3]. However,using the proposed model and developing a new rainattenuation time series synthesiser based on IG distribution,the first-order statistics of the time series will be calculateddirectly and will have only the innate errors of theassumptions of the proposed model.

5 Acknowledgment

This work has been carried out under the framework ofCOST IC0802 and NTUA-THALES-MIMOSA. Thisresearch has been co-financed by the European Union(European Social Fund – ESF) and Greek national fundsthrough the Operational Program ‘Education and LifelongLearning’ of the National Strategic Reference Framework(NSRF) – Research Funding Program: THALES. Investingin knowledge society through the European Social Fund.The authors are grateful to all the Propagation Expertsmade the experiments that have been included in theITU-R Databanks.


www.ietdl.org
6 References
1 Panagopoulos, A.D., Arapoglou, P.-D.M., Cottis, P.G.: ‘Satellitecommunications at Ku, Ka and V bands: propagation impairments andmitigation techniques’, IEEE Commun. Surv. Tutorials, 2004, 6, (3),pp. 2–14

2 Crane, R.K.: ‘Propagation handbook for wireless communication systemdesign’ (CRC Press LLC, 2003)

3 ITU-R P.618-10: ‘Propagation data and prediction methods required forthe design of Earth-space communication systems’ (Geneva, 2009)

4 Capsoni, C., Luini, L., Paraboni, A., Riva, C., Martelucci, A.: ‘A newprediction model of rain attenuation that separately accounts forstratiform and convective rain’, IEEE Trans. Antennas Propag., 2009,57, (1), pp. 196–204

5 Capsoni, C., Fedi, F., Paraboni, A.: ‘A comprehensive meteorologicallyoriented methodology for the prediction of wave propagation parametersin telecommunication applications beyond 10 GHz’, Radio Sci., 1987,22, (3), pp. 387–393

6 Capsoni, C., Fedi, F., Magistroni, C., Paraboni, A., Pawlina, A.: ‘Dataand theory for a new model of the horizontal structure of rain cells forpropagation applications’, Radio Sci., 1987, 22, (3), pp. 395–404

7 Matricciani, E.: ‘Physical-mathematical model of the dynamics of rainattenuation based on rain intensity time series and two layer verticalstructure of precipitation’, Radio Sci., 1996, 31, pp. 281–295

8 Matricciani, E., Riva, C.: ‘The search for the most reliable long-term rainattenuation CDF of a slant path and the impact on propagation models’,IEEE Trans. Antennas Propag., 2005, 53, (9), pp. 3075–3079

9 Lin, S.H.: ‘A method for calculating rain attenuation distributions onmicrowave paths’, Bell Syst. Tech. J., 1975, 54, (6), pp. 1051–1083

10 Morita, K., Higuti, I.: ‘Prediction methods for rain attenuationdistributions of micro and millimeter waves’, Rev. Electron. Commun.Labs, 1976, 24, pp. 651–668

11 Livieratos, S., Katsambas, V., Kanellopoulos, J.D.: ‘A Global methodfor the prediction of the slant path rain attenuation statistics’,J. Electromagn. Waves Appl., 2000, 14, pp. 713–714

12 Panagopoulos, A.D., Arapoglou, P.-D.M., Kanellopoulos, J.D., Cottis, P.G.:‘Long-term rain attenuation probability and site diversity gain predictionformulas’, IEEE Trans. Antennas Propag., 2005, 53, (7), pp. 2307–2313

13 Chikkara, R.S., Folks, J.L.: ‘The inverse Gaussian distribution: theory,methodology and applications’ (M. Dekker, New York, 1989)

14 Chatzidiamantis, N.D., Sandalidis, H.G., Karagiannidis, G.K.,Kotsopoulos, S.A.: ‘On the inverse Gaussian shadowing’. Proc. 2011Int. Conf. Communications and Information Technology, 2011

15 Chatzidiamantis, N.D., Sandalidis, H.G., Karagiannidis, G.K.,Matthaiou, M.: ‘Inverse Gaussian modeling of turbulence-induced


fading in free-space optical systems’, J. Lightwave Technol., 2011, 29,(10), pp. 1590–1596

16 Kedem, B., Chiu, L., Karni, Z.: ‘An analysis of the threshold methodfor measuring area-average rainfall’, J. Appl. Meteorol., 1990, 29,pp. 3–20

17 Kourogiorgas, C.I., Panagopoulos, A.D., Kanellopoulos, J.D.,Karagiannidis, G.K.: ‘On the inverse Gaussian modeling of rainfallrate and slant path and terrestrial links rain attenuation’. IEEE Proc.Sixth European Conf. Antennas and Propagation (EuCAP), Prague,Czech Republic, 26–30 March 2012

18 Kourogiorgas, C.I., Panagopoulos, A.D., Kanellopoulos, J.D.: ‘On theearth-space site diversity modeling: a novel physical-mathematicaloutage prediction model’, IEEE Trans. Antennas Propag., 2012, 60,(9), pp. 4391–4397

19 (2010) ITU-R. Databank DBSG3, available at http://www.itu.int/publ/R-SOFT-SG3/en

20 Seshardi, B.: ‘The inverse Gaussian distribution’ (Clarendon Press,Oxford, 1993)

21 Al-Hussaini, E.K., Abd-El-Hakim, N.S.: ‘Bivariate inverse Gaussiandistribution’, Ann. Institute Stat. Math., 1981, 33, (1), pp. 57–66

22 Gradshteyn, I.S., Ryzhik, I.M.: ‘Table of integrals, series and products’(Elsevier academic press, 2007)

23 Crane, R.K.: ‘Prediction of attenuation by rain’, IEEE Trans. Commun.,1980, 28, (9), pp. 1717–1733

24 Olsen, R., Rogers, D., Hodge, D.: ‘The aRbrelation in the calculation ofrain attenuation’, IEEE Trans. Antennas Propag., 1978, 26, (2),pp. 318–329

25 ITU-R P.838-3: ‘Specific attenuation model for rain for use in predictionmodels’ (Geneva, 2005)

26 ITU-R 311-13: ‘Acquisition, presentation and analysis of data in studiesof tropospheric propagation’ (Geneva, 2009)

27 Kourogiorgas, C.I., Panagopoulos, A.D., Moraitis, N., Kanellopoulos,J.D., Livieratos, S.N., Chatzarakis, G.E.: ‘Analysis of 15-months rainrate measurements at NTUA campus’. IEEE Proc. Sixth EuropeanConf. Antennas and Propagation (EuCAP), Prague, Czech Republic,26–30 March 2012

28 ITU-R P.839-3: ‘Rain height for prediction methods’ (Geneva, 2001)29 Liolis, K.P., Panagopoulos, A.D., Cottis, P.G.: ‘Multi-satellite MIMO

communications at Ku-band and above: investigation on spatialmultiplexing for capacity improvement and selection diversity forinterference mitigation’, EURASIP J. Wireless Commun. Netw., 2007,2007, pp. 11

30 ITU-R. P. 1853-1: ‘Tropospheric attenuation time series synthesis’(Geneva, 2012)


new physical-mathematical model for predicting slant-path rain attenuation statistics based on...

Documents