# Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics
## 2 Materials and Methods
### Relevance of the Elevation Angle for the Satellite Channel
Most of channels for NGEO systems have been developed based on previous models for GEO satellites, specially on well-known models, such as those by [PERSON] [21] and [PERSON] [22]. Even though the [PERSON] and Lutz models were not originally intended for LEO satellites, those were later used as a base for much of the non-geosynchronous (NGSO) channel models. The Loo channel assumes that the complex envelope of the received signal, \(r_{T}\), contains a LOS, \(r_{D}\), and a multipath component, \(r_{M}\), which in its phasor notation can be represented as  
\[r_{T}\exp(j\phi_{T})=r_{D}\exp(j\phi_{D})+r_{M}\exp(j\phi_{M}) \tag{4}\]  
where \(\phi_{T}\), \(\phi_{D}\), and \(\phi_{M}\) indicate the phase of the total received signal, of the direct or LOS component, and of the multipath component, respectively. The channel developed by [PERSON] [22] models a varying received signal affected by different levels of shadowing. This channel is characterized by a Markov chain with a state corresponding to light shadowing, \(G\), and another corresponding to deep shadowing conditions \(B\). Figure 4 illustrates the Lutz channel. The Loo channel [21] was adapted by [PERSON] and [PERSON] [3] to formulate one of the best-known channel models for LEO satellites; similarly, it was modified by [PERSON] et al. [23], to achieve straightforward analytical expressions for the first and second order statistics of the received signal based on the Nakagami distribution. However, current models for GEO and NGEO systems have recognized the limitations of a single distribution and a single state to describe the received signal, and multistate models with different distributions at each state have became popular. Channel models for LEO with multiple states as in [22] have been developed containing the same kind of distribution at each state. For example, a combination of the Lutz and Loo models was implemented for NGSO by [PERSON] et al. [24], focusing on time series generation. This model used a three-state Markov chain, as shown in Figure 5, to indicate different levels of shadowing; the first state, \(S_{1}\), indicates deep shadow; the second, \(S_{2}\), indicates moderate shadowing; and the third state, \(S_{3}\), indicates line-of-sight (LOS) conditions. Each state describes the received signal according to a Loo distribution with a different Loo triplet, \(\alpha\), \(\psi\), \(MP\); and a different Markov chain for distinct elevation angle values. The so-called Loo triplet, which includes the mean of the direct signal amplitude, \(\alpha\), the standard deviation of the direct signal amplitude, \(\psi\), and the multipath power, \(MP\), is the set of parameters required by the Loo distribution. Figure 6 shows two Loo distributions to characterize the satellite channel at two elevation angles; the CDF's were computed assuming presence of a LOS component, and a multipath power component very close to the LOS level. The Loo distributions in Figure 6 can be understood as two of the states of Figure 5 at different elevation angles; e.g., a state transition from \(S_{3,\beta_{1}}\) to \(S_{3,\beta_{2}}\). In addition to the shadowing level at a receiver environment, which is mainly determined by natural and human made objects at the surroundings of the ES, the received  
Figure 4: First order Markov chain illustrating the channel model developed by [PERSON] [22]. signal level will be changing according to the elevation angle value. For LEO satellites, the elevation angle will change rapidly, and with a different rate at every contact. As observed in Figure 6, the Loo CDF curves of the received signal rely on the elevation angle value since those depend on the LOS component of the received signal, \(\alpha\). Then, in order to predict different curves we need to know the expected value and measures of dispersion of the elevation angle.