State modelling of the land mobile propagation channel for dualsatellite systems
 Daniel Arndt^{1}Email author,
 Alexander Ihlow^{1},
 Thomas Heyn^{2},
 Albert Heuberger^{2},
 Roberto PrietoCerdeira^{3} and
 Ernst Eberlein^{2}
DOI: 10.1186/168714992012228
© Arndt et al.; licensee Springer. 2012
Received: 29 November 2011
Accepted: 18 May 2012
Published: 23 July 2012
Abstract
The quality of service of mobile satellite reception can be improved by using multisatellite diversity (angle diversity). The recently finalised MiLADY project targeted therefore on the evaluation and modelling of the multisatellite propagation channel for land mobile users with focus on broadcasting applications. The narrowband model combines the parameters from two measurement campaigns: In the U.S. the power levels of the Satellite Digital Audio Radio Services were recorded with a high sample rate to analyse fast and slow fading effects in great detail. In a complementary campaign signals of Global Navigation Satellite Systems (GNSS) were analysed to obtain information about the slow fading correlation for almost any satellite constellation. The new channel model can be used to generate time series for various satellite constellations in different environments. This article focuses on realistic state sequence modelling for angle diversity, confining on two satellites. For this purpose, different state modelling methods providing a joint generation of the states ‘good good’, ‘good bad’, ‘bad good’ and ‘bad bad’ are compared. Measurements and resimulated data are analysed for various elevation combinations and azimuth separations in terms of the state probabilities, state duration statistics, and correlation coefficients. The finally proposed state model is based on semiMarkov chains assuming a lognormal state duration distribution.
Keywords
Land mobile satellite Statistical propagation model Satellite diversity Markov chain SemiMarkov chain1 Introduction
Satellites play an important role in today’s commercial broadcasting systems. In cooperation with terrestrial repeaters they can ensure uninterrupted service of multimedia content (e.g. audio and video streaming) to stationary, portable, and mobile receivers. However, in case of mobile reception fading regularly disrupts the signal transmission due to shadowing or blocking objects between satellite and receiver. To mitigate these fading effects, diversity techniques such as angle diversity (multiple satellites) and time diversity (interleaving) are attractive. For linklevel studies of the land mobile satellite (LMS) channel, statistical channel models are frequently used that are able to generate timeseries of the received fading signal. Statistical LMS channel models describe several fading processes of the received signal: slow variations of the signal are caused by obstacles between the satellite and the receiver, which induce varying shadowing conditions of the direct signal component. Fast signal variations are caused by multipath effects due to static or moving scatterers in the vicinity of the mobile terminal. For short time periods these two components (slow and fast variations) are usually modelled by a stationary stochastic processes, e.g., as a Loodistributed fading signal[1]. For longer time periods the received signal can not assumed to be stationary. Therefore, statistical LMS channel models describe different receive states to assess the large dynamic range of the received signal. The states correspond to slow varying environmental conditions (e.g. line of trees, buildings, lineofsight (LOS)) in the transmission path. Traditional LMS channel models simulate series of three states (‘lineofsight’, ‘shadowed’, and ‘blocked’) or two states (‘good’ state and ‘bad’ state) by using Markov or semiMarkov concepts.
While several LMS channel models for singlesatellite systems are already available and consolidated[2–4], models for multisatellites systems are of ongoing interest for modern transmission standards, e.g. DVBSH[5]. Early studies on multisatellite transmission were carried out in 1992. Based on circular scans of fisheyecamera pictures in different environments an empirical model was developed describing the correlation coefficient between two satellite signals depending on their azimuth separation[6]. In 1996 a statistical channel model for two correlated satellites based on firstorder Markov chains was developed[7]. The state sequence generation is based on state transition probabilities of two independent satellites. Both satellites are combined by a state correlation parameter which can be taken from empirical models. Due to its simplicity this modelling approach is frequently used today. However, firstorder Markov chains have limitations in state duration modelling, as their state durations follow an exponential distribution. Studies in[8–10] found that this condition does not hold for the LMS channel. Nevertheless, a correct state duration modelling is of high interest for the optimal configuration of physical layer and link layer parameters for modern broadcasting standards with long time interleaving (e.g. for physical layer interleaver size, link layer protection time). Therefore, different concepts improving the state duration modelling were introduced: semiMarkov chains[10] and dynamic Markov chains[9]. For these approaches some exemplary parameters for the singlesatellite reception are published. However, an intense study for multisatellite state duration modelling and a corresponding channel model including parameters for different environments and orbital positions does not exist so far. In this sense, a new channel model for two or more satellites was developed in the context of the project MiLADY (M obile satelli te channeL with A ngle D iversitY )[11]. This project covered two measurement campaigns in the U.S. and in Europe: In the first campaign the power levels of the Satellite Digital Audio Radio Services (SDARS) satellites (SBand) were recorded synchronously with a sample rate of 2.1 kHz. The signals allow to study slow and fast fading effects in combination with angle diversity for a limited set of elevation and azimuth angle combinations. A second measurement campaign was carried out in the area of Erlangen in Germany to record the carriertonoise spectral density ratio (C/N_{0}) from Global Navigation Satellite System (GNSS) satellites in the LBand. The C/N_{0} allows a comprehensive analysis of the state correlation (line of sight, shadowed/blocked) for multiple satellites for a wide range of elevation and azimuth angle combinations.
This article focuses on the state sequence generation for a dualsatellite channel model. The parameters are derived from the measurements for different modelling approaches assuming two states per satellite (‘good’, ‘bad’). Chosen models are: firstorder Markov approach[7], semiMarkov approach[10], and dynamic Markov approach[9]. To assess the performance of these models, correlation coefficients, state probabilities and state duration statistics are gained from resimulated state sequences and compared with the measurement data. As state sequence modelling is only a part of a complete LMS channel model, we describe an overall LMS channel model and give the complete set of parameters.
The article is organised as follows: In Section 3 basics of the LMS channel and of different state modelling methods for single and dualsatellite reception are explained. Further on, these methods are compared on an exemplary scenario for two satellites. Section 3 gives an overview on the GNSS and SDARS measurements and the data processing to derive the channel states. In Section 3 the state models are compared on a high number of receive scenarios with the measurements. The evaluation criteria are state probabilities and state duration statistics. Finally, in Section 3 the conclusions are drawn.
2 Statistical channel modelling for singlesatellite and dualsatellite systems
The statistical approach of generating time series for the LMS propagation channel includes two processes: First, the very slow fading components of the channel due to varying shadowing conditions between the satellite and the receiver are modelled in terms of channel states. LMS models with three states[3], namely ‘lineofsight’, ‘shadowed’, and ‘blocked’, or two states[2, 4] ‘good’ and ‘bad’ are available in the literature. Once the channel states are modelled, in a second process the amplitudes of direct and indirect signal components are generated. They depend on the current state and the receive environment. In common narrowband LMS propagation models the fading is described as a combination of lognormal, Rice and Rayleigh models.
Focus of this article is the state sequence modelling for single and dualsatellite systems, assuming two states per satellite: ‘good’ state and ‘bad’ state. For this purpose, different state modelling methods are compared with the measurement data in terms of the state probability, state duration probability, and correlation coefficient. Moreover the practicability of various state generation methods in terms of generating a database, e.g. for different environments and elevation angles, is assessed.
2.1 Channel state models for singlesatellite systems
Three types of state modelling approaches for singlesatellite systems are found in the literature:
In the following the main characteristics of these models are described.
2.1.1 Firstorder Markov model
where p_{ ii } is the state transition probability between two equal states, and Δd denotes the sampling distance (frame length).
In this article the function P(D) will be further denoted as state duration probability density function (SDPDF). The SDPDF of the firstorder Markov chain follows an exponential distribution.
2.1.2 Dynamic Markov model
For this purpose, the twodimensional STPM is extended to a threedimensional state transition probability tensor (STPT)${\mathcal{P}}_{\text{trans}}\in {\mathbb{R}}_{0+}^{N\times N\times {n}_{\text{max}}}$, where n_{max} corresponds to the maximum state length obtained from the measurements with D_{max} = n_{max}Δd.
If the values for the STPT are directly derived from the measured state sequence (assuming a sample length of e.g. Δ d = 1m), the dynamic Markov model enables an exact reproduction of the state probabilities as well as an exact remodelling of the measured SDPDF. A significant disadvantage is the high number of parameters required to describe the STPT.
In[9] some model approximations are proposed to reduce the number of required parameters of the STPT:

partial dynamic Markov model: From Equation (5) it is derived that an exact state duration modelling requires only a subset of the STPT. Only the transition coefficients p_{ ii }(n) need to change as a function of the current state duration. For a twostate model the remaining values p_{ ij }(n) can be recalculated easily with p_{ ij }(n) + p_{ ii }(n) = 1 . For a multistate model some additional coefficients S_{ iZ } are required to calculate the relative ratio between the state transition probabilities p_{ ij }(n) . The coefficients are derived from the STPT${\mathcal{P}}_{\text{trans}}$ at position n = 1 for each state i$\begin{array}{l}{p}_{\mathit{\text{ii}}}\left(1\right)+{S}_{i1}\left(1\right)+{S}_{i2}\left(1\right)+\cdots +{S}_{\mathit{\text{iZ}}}\left(1\right)=1\phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{2em}{0ex}}\\ {S}_{\mathit{\text{iZ}}}\left(1\right)={p}_{\mathit{\text{ij}}}\left(1\right),i\ne j.\phantom{\rule{2em}{0ex}}\end{array}$(6)

approximated partial dynamic Markov model: For a further reduction of the model parameters, the function p_{ ii }(n) can be approximated by a curve fit. In[9], a piecewise linear approximation at 8 predefined values of the state duration D = nΔd is proposed. Assuming this, a twostate model would require 8·2 parameters for state sequence generation. A multistate model would need 8·N parameters to describe the functions p_{ ii }(n) (with N being the number of states) and further N(N − 1) parameters to describe the relative ratios S_{ iZ } between the coefficients p_{ ij }(n),i ≠ j (Equation (6)).
2.1.3 SemiMarkov model
Another Markov approach is the semiMarkov model introduced in[10] to enable a correct state duration modelling. In contrast to the firstorder and dynamic Markov model, the state transitions do not occur at concrete time intervals. In fact, the time interval of the model staying in state i depends directly on its SDPDF. As with the Markov models, the state transitions are described with the state transition probability p_{ ij }, but with i ≠ j. Assuming a singlesatellite model of only two states, the state transition probability is p_{ ij } = 1 (cf. Figure2c). The equilibrium probability of the states can be calculated as the product of the mean state duration$\stackrel{\u0304}{D}$ and the probability of entering a state (which is described with the STPM).
The semiMarkov model offers some options to describe the SDPDF of each state:

The measured state duration statistic is used without any approximation for remodelling, i.e., the state duration is a random realisation of the measured SDPDF.$P\left(D\right)=P\left({D}_{\text{measured}}\right)$(8)

The measured SDPDF is approximated with a lognormal distribution, as proposed in[4, 10] individually for the singlesatellite state ‘good’ and ‘bad’. The lognormal probability density function describing the state duration probability P(D) is given by$P\left(D\right)=\frac{1}{D\sigma \sqrt{2\Pi}}exp\left[\frac{{\left(\text{ln}\right(D)\mu )}^{2}}{2{\sigma}^{2}}\right]\phantom{\rule{0.3em}{0ex}},$(9)

In[9], a piecewise exponential curve fit of the SDPDF using four segments is proposed. Clearly, this requires more parameters than the lognormal curve fit, but it enables a more flexible remodelling of the state duration statistic.$P\left(D\right)=\left\{\begin{array}{ll}{a}_{1}\phantom{\rule{0.3em}{0ex}}{\mathrm{e}}^{{b}_{1}D},& \phantom{\rule{1em}{0ex}}{D}_{0}\le D\le {D}_{1}\\ {a}_{2}\phantom{\rule{0.3em}{0ex}}{\mathrm{e}}^{{b}_{2}D},& \phantom{\rule{1em}{0ex}}{D}_{1}\le D\le {D}_{2}\\ {a}_{3}\phantom{\rule{0.3em}{0ex}}{\mathrm{e}}^{{b}_{3}D},& \phantom{\rule{1em}{0ex}}{D}_{2}\le D\le {D}_{3}\\ {a}_{4}\phantom{\rule{0.3em}{0ex}}{\mathrm{e}}^{{b}_{4}D},& \phantom{\rule{1em}{0ex}}{D}_{3}\le D\le {D}_{4}\end{array}\right.$(11)
In any case, the STPM has to be derived from measurements, which is independent from the used SDPDF approximation.
2.2 Channel state models for dualsatellite systems
2.2.1 Straightforward method
2.2.2 Approximation of joint state duration statistics for semiMarkov chains
The lognormal distribution is accepted for singlesatellite state duration modelling in the literature[4, 10]. In the sequel, we use it for the dualsatellite case, too. To illustrate some limitations, we discuss two border cases:
Example 2: Assuming two satellites having the same elevation and a marginal azimuth separation. The state sequence from satellite 2 is exactly the same as the state sequence from satellite 1, but with a delay of 5 m. As a result, the state duration distribution for ‘good bad’ or ‘bad good’ is limited to the range of [0 m; 5 m] and has a peak at 5 m (cf. Figure4 (right)).
Both examples show that a curvefit of the joint state duration requires some degree of flexibility. A good fit would be a piecewise approximation. Nevertheless, when analysing real measurement data it is obtained that already a (simple) lognormal fit provides good approximations of the state durations, as shown later in this article.
2.2.3 Lutz model
Besides the above straightforward method of extending multiple singlesatellite states to joint states, a very effective approach for two correlated satellites was introduced in[7]. This algorithm is based on firstorder Markov chains and generates a joint STPM (4 × 4 elements) from two independent singlesatellite STPMs (each with 2 × 2 elements). Using the joint STPM, a joint sequence of four states can be generated as mentioned above. The high flexibility of this algorithm becomes clear, since it requires only singlesatellite parameter sets (in form of 2 × 2 STPMs), that are easy to parametrise and are already available in literature for different elevation angles and a high number of environments. Databases for correlation coefficients are available for different environments, elevation angles, and angular separations of the azimuth and elevation angle[6] as well.
In contrast to this Lutz model, the above mentioned ‘straightforward methods’ need complete datasets for any combination of elevation angles, azimuth angle separations and environments to achieve the same variability.
2.3 Comparison of state models for dualsatellite systems
Measured and resimulated state probabilities and state correlation coefficients using different dualsatellite state models for an exemplary scenario: urban, elevation1 = 45°, elevation2 = 25°, azimuth separation = 45° (g… ‘good’state, b… ‘bad’state, gb … ‘good bad’state, etc.)
Algorithm  Corr.  Joint state prob. (Sat1 & Sat2)  Sat1  Sat2  Parameters  

coef.  P _{gg}  P _{gb}  P _{bg}  P _{bb}  P _{g}  P _{b}  P _{g}  P _{b}  
measured (Reference)  0.15  0.32  0.46  0.05  0.16  0.78  0.22  0.37  0.63  
1st order Markov (Lutz)  0.15  0.32  0.46  0.05  0.16  0.78  0.22  0.37  0.63  9 
dynamic Markov  0.15  0.32  0.46  0.05  0.16  0.78  0.22  0.37  0.63  11120 
partial dynamic Markov  0.10  0.35  0.37  0.10  0.17  0.72  0.28  0.45  0.55  2792 
SemiMarkov, no fit  0.15  0.32  0.46  0.05  0.16  0.78  0.22  0.37  0.63  2796 
SemiMarkov, lognormal fit  0.15  0.33  0.46  0.05  0.16  0.79  0.21  0.38  0.62  20 
SemiM., logn.fit + correction  0.15  0.32  0.46  0.05  0.16  0.78  0.22  0.37  0.63  20 
SemiMarkov, piecew. exp. fit  0.15  0.33  0.45  0.05  0.17  0.78  0.22  0.38  0.62  64 
Mean Squared Error (MSE) of the resimulated state duration PDF to the measured state duration PDF using different dualsatellite state models for an exemplary scenario (cf. Figure 5 )
Algorithm  MSE in[10^{−5}] for Sat1 & Sat2  MSE in[10^{−5}] for Sat1  MSE in[10^{−5}] for Sat2  Parameters  

gg  gb  bg  bb  g  b  g  b  
1st order Markov (Lutz)  0.41  0.99  1.19  0.72  0.72  1.10  0.67  1.36  9 
Dynamic Markov  0  0  0  0  0.31  0.15  0.23  0.47  11120 
Partial dynamic Markov  0  0  0  0  0.28  0.28  0.53  0.57  2792 
SemiMarkov, no fit  0  0  0  0  0.33  0.25  0.34  0.58  2796 
SemiMarkov, lognormal fit  0.23  0.17  0.19  0.13  0.39  0.38  0.45  0.67  20 
SemiMarkov, logn.fit + correction  0.24  0.17  0.21  0.17  0.40  0.36  0.47  0.67  20 
SemiMarkov, piecewise exp. fit  0.24  0.19  0.21  0.23  0.42  0.45  0.54  0.64  64 
From Figure5, Table1, and Table2 the following results are obtained:

firstorder Markov model (Lutz model)

perfect match of P_{state} and ϱ_{states} according to theory, furthermore P_{jointState} is estimated correctly

some deviations between the state duration statistics and the measurements. The MSE of the SDPDF is high compared to other algorithms.

dynamic Markov model

perfect match of P_{jointState}, P_{state}, and ϱ_{states}

according to theory, joint state durations are modelled accurate (MSE = 0 )

good (but not perfect) match of the singlesatellite state durations

partial dynamic Markov model

perfect match of joint state durations (MSE = 0 )

good match of the singlesatellite state durations

despite the high model complexity, P_{jointState} and P_{state} can not be remodelled accurately

semiMarkov with measured SDPDF

perfect match of P_{jointState}, P_{state}, ϱ_{states}, and joint state duration statistics

good match of singlesatellite duration statistics

same results as with the dynamic Markov model, but with less parameters

semiMarkov with lognormalfit

marginal differences of P_{jointState}, P_{state}, and ϱ_{states} from measurements

good match of the of joint states and singlesatellite state durations: the MSE is much lower compared to firstorder Markov models, but higher than dynamic Markov models

semiMarkov with lognormalfit and correction of mean state duration (cf. Equation (12))

due to the modification, perfect match of P_{jointState}, P_{state}, and ϱ_{states}

since the correction of the state duration is only marginal, the MSE is marginal greater than the initial lognormalfit

semiMarkov with piecewise exponential approximated SDPDF

marginal differences of P_{jointState}, P_{state}, and ϱ_{states} from measurements

good match of duration statistics for joint states and singlesatellite states
Performance comparison of dualsatellite state models
Algorithm  State probability modelling  State duration modelling  Model complexity 

1st orderMarkov (Lutz)  perfect  bad  low 
Dynamic Markov  perfect  perfect  high 
Partial dynamicMarkov  bad  perfect  high 
SemiMarkov,no fit  perfect  perfect  high 
SemiMarkov, lognormal fit  good  good  medium 
SemiMarkov, logn.fit+correction  perfect  good  medium 
SemiMarkov,piecewise exp. fit  good  good  medium 
With respect to the required number of parameters, it can be concluded that the ‘dynamic Markov model’ (no approximations) and the ‘semiMarkov model with measured SDPDF’ are not feasible to generate a dualsatellite model database representing arbitrary receive situations, although they achieve a perfect remodelling of state probabilities and state duration statistics. The ‘partial dynamic Markov model’ (and consequently its approximated versions) is not able to reproduce the state probabilities correctly. As a consequence, its applicability for state modelling is limited. The Lutz model, the semiMarkov model with a lognormal SDPDF, and the semiMarkov model with a piecewise exponential approximated SDPDF achieve good modelling results by using an acceptable number of parameters. These three models are compared in Section 3 using a large number of receive scenarios.
2.4 Channel state models for multisatellite systems
Some preliminary investigations of this concept for dualsatellite reception yields that correlation coefficients, state probabilities of single satellites, and joint state probabilities can be modelled accurately, whereas a correct state duration modelling of the Slaves has limitations.
To validate the Master–Slave concept for multisatellite reception, a statistical analysis for constellations with at least three satellites needs to be performed. This is addressed in near future activities.
2.5 Dualsatellite LMS model with correlated state sequences
In the context of the project MiLADY, extensive satellite signal measurements were analysed to develop a new multisatellite channel model. The basis is the twostate model from PrietoCerdeira et al.[4]. It includes a versatile selection of Looparameter triplets after each state transition to describe slow and fast signal variations.
However, the analysis of the measurement data captured in the MiLADY project indicate some changes to the initial twostate model[4] in terms of describing slow and fast fading characteristics. Further on, correlation effects for slow and fast variations between the satellites were obtained from the data. A comprehensive study for slow and fast fading effects is topic of ongoing work and out of scope of this article.
3 Satellite signal measurements and data processing for angle diversity analysis
To derive the parameters for a new multisatellite LMS model, two measurement campaigns were carried out in the context of the MiLADY project.
3.1 SDARS measurements, U.S. east coast
3.2 GNSS measurements, Erlangen, Germany
A second measurement campaign was carried out around Erlangen (Germany) to record the carriertonoise spectral density ratio (C/N_{0}) from GNSS satellites in the LBand. Due to a permanent availability of at least eight satellites on the hemisphere, a comprehensive analysis of fading effects for a wide range of elevation and azimuth angle combinations of multiple satellites is possible. Because of the low C/N_{0} resolution in time (20 Hz) and in amplitude (1 dB quantisation), only parameters for slow variations can be derived.
The GNSS campaign was split in two parts:

The first part of the campaign has been carried out in July 2009. The GNSS antenna was mounted on a measurement van at a height of 2 m. A measurement roundtrip of 38 km length was driven ten times, covering several environments (suburban, forest, open, commercial) in and around Erlangen (cf. Figure8, red line).

The second part of the measurements was done in late September and early October 2010 by mounting the setup onto two city buses, driving on different routes. The GNSS antenna was mounted at a height of 3.1 m. The city buses drove an identical route for 3 days. The covered environments were urban, suburban and partly open rural areas. The individual routes of the two buses spanned 7 km and 6 km in North–South direction and 6 km and 5 km in West–East direction, respectively (cf. Figure8, green and yellow line).
The trials were carried out in summer and in autumn months, where leaves were on the trees.
For the measurements a professional GNSS receiver (built by Fraunhofer IIS by using a Javad receiver core) was used. Beside information of vehicle speed and positioning data, the GNSS signal includes a C/N_{0} estimation of the GPS L1 carrier at 1575.420 MHz. It has a dynamic range of 20 dB and a quantisation of 1 dB. The time resolution of the C/N_{0} estimation is 20 Hz. At lower signal levels of a specific satellite (e.g. during deep blockages), the GNSS receiver loses the satellites signal synchronisation and the C/N_{0} estimation of this satellite is no more available. In terms of state detection it will be defined as ‘bad’ state. Additional information of azimuth and elevation of the individual satellites were captured with 1° resolution.
Overview on two measurement campaigns for the parameter extraction of a multisatellite LMS model
SDARS measurements(USA, East Coast)  GNSS measurements(Germany, Erlangen) 

High sample rate (2.1 kHz)  Low sample rate (20 Hz) 
→ Reliable for Loo parameterextraction and state parameterextraction  → Reliable for state parameterextraction 
4 satellites (2 GEO from XM, 3 HEO from Sirius)  > 20 satellites (MEO from GPS,GLONASS) 
→ Limited combinations of orbital positions  → Many combinations of orbital positions 
Environments: urban, suburban,treeshadowed, forest, commercial, highway (open)  Environments: urban, suburban,forest, commercial, open 
Model validation for a limited set oforbital positions  Preliminary state parameters formany orbital positions whichneed refinement and validation 
3.3 Data processing and state identification
The analysis pipeline for both the SDARS and GNSS data is similar. First, the signal is normalised to LOS level[13]. Afterwards, the timeseries is resampled into travelled distance units. As resolution, 1 cm and 10 cm is chosen for the SDARS and GNSS data, respectively. State identification is performed by global thresholding (threshold 5 dB below LOS) of the lowpass filtered signal (sliding window over 5 m), similar to[10].
3.4 Separation of data into environments and satellite positions
3.4.1 Singlesatellite analysis
Singlesatellite reception depends on the kind of environment, the elevation angle of the satellite, and the azimuth of the satellite relative to the driving direction. For a detailed analysis of the singlesatellite state characteristics, the SDARS and GNSS measurement data were divided into:

different environment types ‘Urban’, ‘Suburban’, ‘Forest’, ‘Commercial’, and ‘Open’. The environment classification for the SDARS measurement data was performed by visual inspection of the image material from two cameras. For the GNSS measurement data the LandUsage data from the European Corine project[14] was used.

different elevation angles from 10° to 90° in segments of 10°. The mean elevation angles represented by these datasets therefore are 15°, 25°, …, 85°, respectively.

four classes of driving directions (for GNSS data, only) with the intervals 0°…10°, 10°…30°, 30°…60°, and 60°…90°.
3.4.2 Dualsatellite analysis
The dualsatellite (and multisatellite) reception depends on the kind of environment, the elevation angle of each satellite, and the azimuth of each satellite relative to the driving direction. Especially the angular separation of the satellites of elevation and azimuth are crucial, since it affects the correlation of the received signals. For a detailed analysis of state characteristics for dualsatellite reception, the measurement data from SDARS and GNSS measurements were divided into:

five environments (the same as for singlesatellite case).

combinations of eight elevation angles as used for singlesatellite analysis. Thus, the elevation angle separation is included as well.

seven intervals of the azimuth angle separation between the satellites (0°…10°, 10°…30°, 30°…60°, 60°…90°, 90°…120°, 0°…150°, and 150°…180°).
4 Measurement results and state modelling statistics
In this section the results of the GNSS and SDARS analyses are presented in terms of state probabilities and state duration statistics for different environments and orbital positions of the satellites. In two subsections the characteristics for singlesatellite reception and dualsatellite reception are addressed. Furthermore, for each dualsatellite receive scenario the parameters for different channel models are derived. Thus, the remodelling results can be compared with the measurements.
4.1 Results for the singlesatellite channel
The following observations can be made in Figure13:

The ‘bad’state probability increases with increasing angle between satellite azimuth and driving direction within the interval 0°…90° . Except, for high elevation angles (>70° ) the influence of the driving direction is low. For system planning and the application of effective fading mitigation techniques it is important to consider the worst case (≈90° azimuth).

In general, the ‘bad’state probability decreases with increasing satellite elevation. The slope of the curve depends on the driving direction.

Comparing different environments, the ‘bad’state probability in urban and forest areas is on average higher than in other environments. Also the variance between the worst (90° azimuth) and best (0° azimuth) reception case is higher than in suburban, commercial and forest.

In case of GNSS, for elevation angles above 70° the ‘bad’state probability in urban environments is lower than in suburban and forest environments. A reason could be that trees reach above the streets in suburban environments and forests, whereas in urban the probability of trees is low.

Comparing SDARS and GNSS measurements similar results are obtained for urban areas only. In suburban, forest, commercial, and open environments a lower signal availability is obtained for SDARS measurements. A reason could be mainly wider streets in the U.S. than in Europe. Furthermore, for interurban measurement sections during SDARS trials the streets were mainly oriented towards the satellites. Also different methods for environment classification may have an influence to the results.
4.2 Results for the dualsatellite channel
A dualsatellite analysis was done for five different environments, the combination of eight different elevation angles from two satellites, and seven intervals of the azimuth angle separation between the satellites (cf. Figure11). A total of 5·8·8·7=2,240 segments of measurement data were defined^{b}. Due to that fact, displaying the results of dualsatellite modelling is much more complex than in the singlesatellite case. Therefore, in this article only the urban results are presented for only a subset of receive scenarios. With regard to an acceptable number of model parameters, four state modelling approaches are selected and compared in this section: the Lutz model based on firstorder Markov chains, the semiMarkov model assuming a lognormal SDPDF fit and its modification according to Equation (12), and the semiMarkov model assuming a piecewise exponential SDPDF fit.
4.2.1 State correlation
The following observations can be made in Figure14:

In case of small azimuth separations, both state sequences are highly correlated (up to ϱ_{states} = 0.9). (Note: For two exactly colocated satellites ϱ_{states} = 1 is expected, this special case is not covered in the results.) The correlation has a minimum for azimuth separations between 60° and 120°. It can reach values ϱ_{states} < 0 . Towards 180° azimuth separation, the correlation coefficient slightly increases.

In case of small azimuth separation, the correlation coefficient further depends on the elevation angle separation between two satellites.

The Lutz model perfectly resimulates the correlation coefficient. This is not surprising, as the correlation coefficient is one parameter of the Lutz model. It combines two independent state sequences of a singlesatellite simulator into a dualsatellite variant.

A good fit of the correlation coefficients is also achieved with the semiMarkov approaches assuming a lognormal fit or a piecewise exponential fit of the SDPDF. For only a small number of receive scenarios the correlation coefficient deviates by ±0.1 from the measurements. It is acceptable, since the variation of the correlation coefficients between different elevation angles and azimuth separations is much higher.

After a modification of the lognormal fit according to Equation (12), the correlation ϱ_{states} generated with the semiMarkov model matches exactly with the measurements.
Similar to Figure14, the Additional files2,3,4 and5 show the correlation coefficients between two satellites for the environments Suburban, Forest, Commercial, and Open.
4.2.2 State probability
The following can be concluded from Figure15:

The ‘bad bad’state probability strongly depends on the elevation angles of the single satellites (cf. Figure13). Assuming 15° elevation of two satellites, P_{bad bad} is between 0.5 and 0.7 with respect to the azimuth angle separation. For the combination elevation1=45° and elevation2=15°P_{bad bad} is between 0.15 and 0.25. When both satellites have 45° elevation, P_{bad bad} is only between 0.05 and 0.20.

The ‘bad bad’state probability depends further on the azimuth angle separation. It is related to the state correlation between the satellites (cf. Figure14), whereas a low correlation coefficient results in a low ‘bad bad’state probability and provides therefore a high signal availability. A large variance due to the azimuth angle separation is seen for low elevation angles. Here, between 5° azimuth separation and about 90° azimuth separation a reduction of P_{bad bad} of 20% is obtained for elevations < 30° .

The Lutz model perfectly fits the state probabilities. This is one of the great advantages of firstorder Markov chains.

Both semiMarkov models reproduce the ‘bad bad’state probability with an accuracy of ±0.03 in general. An exception is obtained for 15° elevation of both satellites, where the lognormal fit is 0.10 higher, and the piecewise exponential fit is 0.07 lower than the measured state probability.

A correction of the lognormal fit provides an exact reproduction of the state probabilities.
4.2.3 Mean state duration
From Figure16 it is obtained that the Lutz model and the piecewise exponential curvefit accurate describe the mean state length. For both algorithms, only a small difference between resimulation and measurements is obtained for 15° elevation. In case of the lognormal curve fit (without correction) the mean duration is clearly overestimated for 15° elevation. The modified lognormal fit reproduces exactly the mean state duration.
4.2.4 Validation of state duration modelling
Figure17 shows the state duration statistics for the ‘bad bad’state for the measurements and resimulations with four state models for different elevation angles and azimuth angle separations. The quality of the curvefits is given in Figure18 in terms of MSEs between the measured and remodelled SDPDF^{c}. From Figures17 and18 it can be concluded that:

As already stated, the state durations simulated with the Lutz model (based on firstorder Markov chains) follow always an exponential distribution. It is seen that the exponential distribution doesn’t fit the measured state durations in an accurate manner. As seen in Figure17, the probability of the ‘bad bad’state durations between 1 and 100 m are mainly overestimated, whereas the probability of long states above 100 m is too low. A deviation of the state duration statistic (complementary state duration CDF) up to 20% is seen between the measurements and the resimulation with the Lutz model.

SemiMarkov chains have high flexibility in state duration modelling. It is seen that the piecewise exponential fit as well as the lognormal fit accurately approximate the measured state durations. The MSE values Figure18 underline this fact.

Although a piecewise exponential function has higher flexibility in curvefitting, the lognormal fit provides the same good results for state duration modelling. Due to the less number of lognormal parameters per SDPDF (2 lognormal parameters instead of 11 parameters for piecewise exp. fit with 4 segments, cf. Equation (11)), it is the preferred semiMarkov approximation.
4.2.5 Conclusions on dualsatellite state modelling
For dualsatellite propagation modelling, the Lutz model is able to describe the state probabilities and average state durations exactly. It has a low complexity, since only singlesatellite parameters and a correlation coefficient is needed to generate correlated state series for two satellites. A weakness is the capability of correct state duration modelling. For the optimisation of physical layer and link layer parameters for satellite broadcasting systems with high qualityofservice (QoS) requirements it has some limitations, when e.g. system blockage lengths or LOS durations are insufficiently described.
SemiMarkov models accurately describe the state durations of the dualsatellite LMS propagation channel. To reduce the complexity, two variants of SDPDF approximations were analysed: a piecewise exponential fit and a lognormal fit. The curve fits can be modified such that the state probabilities and the correlation coefficients of the measurements are exactly resimulated. It has been shown that both the piecewise exponential fit as well as the lognormal fit describe the state duration with high accuracy. Due to the low number of required parameters, the lognormal distribution is the preferred curve fit of the SDPDF.
5 Conclusions
In this article, we compared different approaches for dualsatellite state modelling based on experimental data from different environment types, various combinations of two satellite elevation angles and different azimuth angle separations.
To find an appropriate state model architecture, the first part of this article gives a closed overview on existing state modelling approaches for singlesatellite and dualsatellite reception. Three categories of models are presented: firstorder Markov models, dynamic Markov models, and semiMarkov models. Based on one measurement example, a detailed analysis of diverse variants of these approaches for dualsatellite state modelling is performed. As evaluation criterion, the single and dualsatellite state probabilities and the state duration statistics are compared with the measurements. Further on, the practicability of the state modelling approaches in terms of dualsatellite channel models is discussed. It was concluded that, due to the high number of required parameters and the high complexity, dynamic Markov models are not feasible in terms of dualsatellite state modelling.
In the second part of this article, three dualsatellite modelling approaches with low complexity are analysed on a large set of receive scenarios: a firstorder Markov model for correlated satellites (Lutz model), a semiMarkov approach assuming a lognormal SDPDF fit, and a semiMarkov approach assuming a piecewise exponential SDPDF fit. For this purpose, the GNSS measurements are separated into various receive scenarios including five environments (urban, suburban, forest, commercial, open), 8 × 8 sections with constant elevation angles of two satellites between 10° and 90°, and seven different intervals of the azimuth angle separation. Parameters for these receive scenarios has been derived for the three selected modelling approaches. Afterwards, the remodelling results are compared with the measurements in terms of the correlation coefficient, the state probability and the state durations of the critical system state ‘bad bad’ in dependency on the azimuth angle separation and the elevation angles of two satellites for the urban environment. It was shown that the Lutz model accurately resimulates the correlation coefficient and the state probability, whereas the state duration statistics are insufficiently described. The semiMarkov models describe accurately the state probabilities, the correlation coefficients, and also the state duration statistics. With respect to the number of parameters, the semiMarkov approach using a lognormal fit of the SDPDF is the preferred model for the dualsatellite state modelling and is proposed therefore for the a new dualsatellite channel model for broadcasting applications.
State parameters for the semiMarkov model as well as for the Lutz model for single and dualsatellite reception are found in the Additional files6,7,8 and9. For the sake of completeness, we also derived Loo parameters from SDARS measurement data describing slow and fast fading effects (cf. Additional file1). By using the twostate model according to PrietoCerdeira et al.[4], the parameters enable a simulation of LMS timeseries for singlesatellite and dualsatellite reception for different environments, elevation angles, and azimuth separations.
In the near future we will also focus on modelling of the smallscale fading. By analysing the extensive SDARS measurement data in terms of slow and fast signal variations within MiLADY, some modifications of the twostate model from[4] are indicated. Recently, modifications were proposed in[15]. A validation of these new concepts for a multisatellite model is topic of ongoing work.
A further task is the state analysis of multisatellite constellations with three or more satellites. Due to the exponential growth of the model complexity with the number of satellites, new concepts must be investigated. A promising approach would be a Master–Slave concept, where several ‘Slave’ satellites are modelled according to their correlation with one ‘Master’ satellite, while neglecting the correlation between the ‘Slave’ satellites (cf. Section 3). Based on statistical parameters derived from measurement data (such as the joint state probability for ‘bad bad bad’ for three satellites), the Master–Slave concept will be evaluated.
To improve the consistency of a state parameter database, activities are planned to extract state parameters with alternative methods, such as the analysis of environmental images from fisheye cameras.
6 Endnotes
^{a} The state duration statistic of a combined state include ≥ 50 elements.
^{b} For singlesatellite analysis only 160 segments (8 elevation angles, 5 environments and 4 driving directions) were required.
^{c} Note: The MSE is estimated between measured and resimulated state duration PDF, but the figures show the complementary CDF of the state duration for better visibility.
Declarations
Acknowledgements
The GNSS and SDARS measurements were carried out in the context of the project MiLADY. This project is funded by the ARTES 5.1 Programme of the Telecommunications and Integrated Applications Directorate of the European Space Agency.
Authors’ Affiliations
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