# An Analytical Prediction Model of Time Diversity Performance for Earth-Space Fade Mitigation

Authors: Pantelis-Daniel M. Arapoglou; Athanasios D. Panagopoulos; Panayotis G. Cottis
Journal: International Journal of Antennas and Propagation
 (2008-02-11)

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## Abstract

Time diversity (TD) has recently attracted attention as a promising and cost-efficient solution
for high-frequency broadcast satellite applications. The present work proposes a general
prediction model for the application of TD by approximating the time dynamics of rain attenuation
through the use of the joint lognormal distribution. The proposed method is tested against experimental
data and its performance is investigated with respect to the basic parameters of a satellite link.


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## Body

## 1. Introduction

Several diversity reception schemes have been thoroughly investigated in
the past to mitigate rain fading in fixed/broadcast satellite communication
systems operating above 10 GHz. These schemes offer a significant performance
improvement, especially considering the recent trend of commercial satellite
networks to migrate to higher-frequency bands due to spectral and/or orbital
congestion: first, from C to the Ku frequency band and, more
recently, from Ku to the Ka, or even to the EHF frequency
band. The established diversity techniques include [1](i)
site diversity (SD), which takes advantage
of the spatial characteristics of the rainfall medium by engaging two or more
jointly receiving earth terminals to combat strong rain fades [2];(ii)
orbital (or satellite) diversity (OD), which takes advantage of the
spatial characteristics of the rainfall medium by allowing earth stations to
choose the best signal from two or more satellites [3];(iii)
frequency
diversity (FD), which takes advantage of the spectral
characteristics of the rainfall medium by employing a high-frequency band (e.g.,
Ka or EHF) during normal operation, and switching to a lower-frequency band
(e.g., Ku or C) when the attenuation due to rain exceeds a certain threshold
[4].All the above fade mitigation techniques
(FMT) produce significant diversity gains but their use is limited due to
specific technical and other factors. For SD, the main limitation comes from
cost considerations, since additional earth stations and terrestrial connections
enabling the processing of the jointly received signals are required. In OD,
limitations are imposed by the switching procedure between the satellites, as
well as by the waste of satellite bandwidth.
Finally, FD is rather expensive, requiring dual reception at the earth terminals.Time diversity (TD), is a less
studied diversity alternative for satellite systems [5]. Basically, it consists
of retransmitting the corrupted information at times when the channel is
expected to be more favourable, that is at time spacings exceeding the channel coherence
time. The basic principle of the TD technique is depicted in Figure 1. Actually,
it resembles the ARQ (automatic repeat request) FMT implemented in layer 2,
which falls under the general category of error correction. The difference
between the two techniques is that ARQ is characterized by a fixed or random
retransmission period, while TD retransmits the information after having estimated
the duration of the unfavorable propagation phenomenon. Clearly, the
performance of the TD technique is closely related to the time period selected
for retransmission, which ranges from a few seconds to several hours. One of
its main advantages when compared to other diversity techniques is that TD does
not require additional equipment or complicated synchronization procedures,
since it involves only a single satellite link and a single reception unit.
This renders the TD option cost efficient. However, the application of the TD
technique is restricted to delay-tolerant applications, such as video-on-demand
or multimedia and data transfer, and seems particularly attractive for
broadcasting services.Figure 1
Application of
the TD technique.Given that only a few similar models
exist in the open literature (briefly outlined in Section2), usually derived
from measurements in specific regions, the present paper proposes a physical
prediction model for the performance estimation of TD. The model, detailed in
Section 3, relies on the assumption that the joint statistics of rain
attenuation at both the initial and the retransmission instants follow the
lognormal distribution. In Section 4, the prediction model is tested against TD
experimental data and the main parameters influencing its performance are discussed.


## 2. Related Measurement and Modeling Activities

One of the first TD measurement campaigns was carried
out for the equatorial region of Malaysia using a satellite beacon
receiver at 12 GHz to record data from a MEASAT geostationary satellite. Apart from
presenting the analysis of the 20-month data, an empirical exponential model to
fit the cumulative distribution of TD was developed [6]. This distribution is defined by means of the exceedance probability
(1)Pr{ATD(Δt)>a}=1−Pr{min(A(t),A(t+Δt))≤a}, where A(t) is the sample of the rain attenuation
stochastic process at time t, ATD(Δt) denotes the attenuation that a TD system with
a retransmission time of Δt is subjected to, and a is the fade
margin. All attenuation quantities are expressed in dB. Furthermore, TD
measurements from the South of England in the V frequencies bawd were
reported in [7] using ITALSAT beacon data. The duration of this measurement
campaign has been three years. Valuable insight on TD both through experimental
data and empirical modeling for the area of Japan is given in [8], which includes
eight years of attenuation data at 20 GHz. The relationship between TD gain,
time delay, and time percentage was approximated by a function similar to the
one proposed by the ITU-R Recommendation [9, page 618] for single site rain
attenuation prediction. Note that the TD gain for the p% time percentage
is given by the difference (2)GTD(Δt;p%)=ATD(Δt=0;p%)−ATD(Δt≠0;p%). Results of the data processing for TD from the ITALSAT geostationary
satellite were presented in [10]. The data were collected at 19 GHz in Italy and at 12, 20, 30, 40, and 50 GHz in Belgium over an
observation period of six years. The authors in [10] assume the bivariate
lognormal distribution to model the joint statistics of two attenuation random
variables A(t), A(t+Δt) separated by a time delay of Δt.
The correlation between these random variables in the time domain is assumed to
follow the same distribution as the corresponding spatial correlation function
employed in SD. This is taken into account by simply replacing the site
separation distance with the TD delay. Finally in [11], a statistical
assessment of TD is presented by simulating rain attenuation time series from
rain rate data with the synthetic storm technique (SST) in the frequency range
10–100 GHz. The rain
rate databank spans over a period of seven years.


## 3. Proposed TD Model

To simplify the notation, the rain attenuation random variables corresponding
to the first and the second transmission of the same signal are here after denoted
byA1 and A2, respectively. The starting point of the proposed approach is the
assumption that the two samples A1=A(t) and A2=A(t+Δt) of the rain attenuation random
process follow the lognormal distribution. Based on this hypothesis, it can
easily be shown that the exceedance probability given by (1) may also be
determined by (3)PTD=Pr(A1>a,A2>a)=∬a∞pA1A2(A1,A2)dA1dA2, where pA1A2(A1,A2) is the joint probability density
function (pdf) of A(t) and A(t+Δt). The above joint pdf is
assumed to follow the bivariate lognormal distribution. The exceedance
probability of a single transmission is (setting for simplicity A=A(t)) (4)Pr(A1>a)=∫a∞pA(A)dA=12erfc(lna−lnAm2Sa), where pA(A) is the lognormal pdf, Am and Sa are the mean value and the standard
deviation of the corresponding normal random variable lnA, and erfc(x) is the
complementary error function (4) [3]. Based on the transformation (5)U=lnA−lnAmSa, the exceedance probability given by (4) is expressed as a function of
the normalized attenuation U(6)Pr(U>u)=12erfc(u2). Assuming that long-term rain attenuation measurements
are available in the form of attenuation probability pairs, (Ai,Pi), i=1,…,N,
where N is the number of available data pairs, the statistical parameters Am and Sa may be evaluated by first reversing (6) to
obtain (7)Ui=2erfc−1(2Pi),i=1,…,N. Since Ui also satisfy (5), the following set of
equations is obtained: (8)Ui=C1Zi+C2,i=1,…,N, where (9)Zi=lnAi,C1=Sa−1,C2=−lnAmSa.   Since Ui and Zi are related through the linear relationship (8),
a least-squares regression procedure may easily lead to the evaluation of the
coefficients C1 and C2 and, then, to Am and Sa.
Since to obtain (4) and (6) the unconditional lognormal distribution is adopted,
in the calculations only values of Ai higher than a minimum 0.5 dB threshold are
taken into account. Also, in case local measurements are not available, the
statistical parameters Am and Sa may be calculated applying the procedure,
summarized in (6)–(9), to the rain
attenuation model of the ITU-R Recommendation [9, page 618].Returning to the bivariate case and adopting for both
random variablesA1 and A2 the transformation (5), two
normal random variables come up, namely, U1 and U2,
each with zero mean and standard deviation equal to 1. Their joint pdf is given by
(10)PU1U2(U1,U2)=12π1−ρn2(Δt)exp{−12(1−ρn2(Δt))⋅(U12+U22−2ρn(Δt)U1U2)}, where the normalized time correlation coefficient ρn is defined by [12, 13] (11)ρn(Δt)=exp(Sa2exp(−dA|Δt|))−1exp(Sa2)−1. It should be emphasized at this point that the critical dependence of pU1U2(U1,U2) and of the subsequent results on Δt is expressed by the correlation
coefficient, which is a function of Δt.The parameterdA [s-1] is the dynamic parameter
of rain attenuation (also referred to as beta in the literature)
describing the temporal variation of the rain attenuation process. A
comprehensive investigation on how to calculate dA based on electrical, geometrical, and climatic
characteristics of the link is carried out in [14].Based on Bayes’ theorem, (10) may be expressed in
terms of its marginal and conditional pdfs (12)pU1U2(U1,U2)=pU2(U2)⋅pU1|U2(U1,U2), where (13)pU1|U2(U1,U2)=12π(1−ρn2(Δt))exp{−(U1−ρn(Δt)⋅U2)22(1−ρn2(Δt))}, and pU2(U2) is the normal pdf. Upon replacing
(12), (13) into (3), the final expression for the joint exceedance probability
of the two rain attenuation random variables separated in time by Δt comes
up (14)PTD=12∫u∞pU2(U)erfc(u−ρn(Δt)U2(1−ρn2(Δt)))dU.


## 4. Performance Evaluation

Next, a first attempt to validate the TD statistical
model based on the bivariate lognormal distribution is carried out for a
variety of measured data related to various climatic zones and for links of various
electrical and geometrical characteristics. The measured results for TD are
simply reproduced from the open literature, as briefly outlined in Section2.The statistical parameters of the lognormal
distributionAm and Sa required for the application of
the proposed model are obtained by fitting the experimental data of the exceedance
probability concerning the single transmission. The dynamic parameter dA is determined following the methology presented in [14]. The parameters
employed in the following figures are given in Table 1.Table 1
Parameters employed.
ExperimentFrequencyElevation angleAmSadASparsholt, UK49.5 GHz29.9°1.18450.93592.1⋅10−4 s-1Kuala
Lumpur, MLA12 GHz77.43°0.032191.87911.4⋅10−4 s-1Spino d’ Adda19 GHz30.6°0.114561.33521.0⋅10−4 s-1In Figure2, TD curves for data originating from
measurements reported in [7] for Sparsholt, UK, together with curves obtained
applying the proposed model are
depicted. A very good convergence is observed between the two sets of
curves demonstrating the validity of the proposed model.Figure 2
Performance
comparison of the proposed model and measured data from a TD system operating
at 50 GHz in Sparsholt, UK.In Figure3, the results of
the proposed model are compared to single transmission TD experimental results
from Kuala Lumpur,
MLA [6]. The deviation observed in Figure 3 between the appropriately
modified lognormal pdf and the curves depicting the measurement is mainly due
to the failure of the lognormal model to represent rain attenuation in
subtropical and tropical regions of the Earth, where the gamma distribution
exhibits an improved prediction performance [15].Figure 3
Performance
comparison of the proposed model and measured data from a TD system operating
at 12 GHz in Kuala Lumpur,
MLA.On the other hand, the overall performance of the proposed TD model
depends critically on the value selected for the dynamic parameterdA.
Although in Figures 2 and 3 the evaluation of dA was based on the analytical methodology
described in [14], there are several suggestions for a global value of this
parameter in the range 3.16⋅10−4s−1≤dA≤3.16⋅10−3s−1. Furthermore,
after testing experimental data from the OLYMPUS and ITALSAT campaigns, the authors in [16] suggest that a rough estimate of dA=10−4s−1 can be used in North western European regions
for elevation angles between 25° and 38° and frequencies
between 12 GHz and 50 GHz. This value has been adopted to determine the TD gain
as a function of the retransmission delay Δt in the area of Spino d' Adda, IT. TD gain
curves adopting and
applying the proposed model have been
plotted in Figure 4. A very satisfactory convergence between the experimental values and those of the
proposed model is observed, demonstrating the validity of the proposed model.Figure 4
Proposed model and measured data in terms of time diversity gain for a TD
system operating at 19 GHz in Spino d' Adda, IT.


## 5. Conclusions

The present paper proposes
a general TD prediction method based on the joint correlated bivariate
lognormal distribution in the time domain. The correlation function proposed
depends critically on the dynamic parameter of rain attenuation. The proposed
model exhibits a very satisfactory convergence to experimental data originating
from temperate climatic regions.



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*Source: 142497-2008-03-23.xml*