## 5 LMS Channel Models
### Four-State Markov Channel Model  
A first four-state Markov channel model was proposed by [PERSON] in [36] as a model for two correlated land mobile satellite channels, as shown in Figure 5. However, the four-state model for a single channel is proposed in [46]. A semi-Markov model for the low-elevation satellite-Earth propagation channel is presented in [46], where semi-Markov means no \(P_{ii}\) arches. The model provided by the authors describes long-term variations of the received signal by a chain of four distinct states. This represents a novel satellite channel model that takes into account the classical LoS, shadowing and blockage states, and a new state in order to model all those periods of time when the receiver is unable to carry out the acquisition and tracking of the satellite. In particular, the authors propose a propagation model in which the LoS signal and multipath (state #1), due to surrounding elements, is modeled by a Nakagami-Rice distribution with a cumulative distribution function (CDF) so defined [60]:  
\[f_{1}(x\leq x_{0})=\int_{0}^{x_{0}}\frac{2x}{M_{r,1}}exp\bigg{(}-\frac{1+x^{2 }}{M_{r,1}}\bigg{)}I_{0}\bigg{(}\frac{2x}{M_{r,1}}\bigg{)}dx, \tag{18}\]  
where \(I_{0}\) is a modified Bessel function of the first kind of zero order, \(M_{r,1}=2\sigma^{2}\) is the mean multipath power described in [60], and \(x\) is the received voltage. The shadowing (state #2) also consists of LoS and multipath components, but in this case, the LoS component is attenuated by trees and/or small obstacles. A good representation for this state is the model presented by [PERSON] in [10], where the LoS signal follows a lognormal distribution and the reflected component follows a Rayleigh distribution:  
\[f_{2}(x\leq x_{0})=\frac{6.390}{\sigma M_{r,2}}\int_{0}^{x_{0}}x\int_{\epsilon} ^{\infty}\frac{1}{z}exp\bigg{(}-\frac{[20 log(z)-m]^{2}}{2\sigma^{2}}-\frac{x^ {2}+z^{2}}{M_{r,2}}\bigg{)}I_{0}\bigg{(}\frac{2 xz}{M_{r,2}}\bigg{)}dzdx, \tag{19}\]  
where \(\epsilon\) is a very small value (\(\epsilon=0.001\) is suggested), \(m\) and \(\sigma\) are the mean and standard deviation of the signal fading in decibels for the direct wave component, and \(M_{r,2}\) is the mean multipath power. The third state (state #3) is represented by blockage, that is, the state in which the LoS component is blocked by large obstacles, so no LoS contribution, only the multipath, is present. The fading distribution for this state is described in the ITU-R model [58] by a Rayleigh distribution with CDF:  
Figure 5: Four-state Markov channel model. \[f_{3}(x\leq x_{0})=1-exp\left(-\frac{x_{0}^{2}}{M_{r,3}}\right), \tag{20}\]  
where \(M_{r,3}\) is the mean multipath power of the diffuse reflections. Taking into account the above formulations for the three cases, the cumulative distribution function of the received signal envelope can be expressed as:  
\[CDF(x\leq x_{0})=P_{1}\cdot f_{1}+P_{2}\cdot f_{2}+P_{3}\cdot f_{3}, \tag{21}\]  
where \(P_{1}\), \(P_{2}\), and \(P_{3}\) are the occurrence probability of the three states that fulfill the equation \(P_{1}+P_{2}+P_{3}=1\). The authors have added a new state to the three-state model for modeling those periods when the fade caused by a large obstacle leads the signal to values lower than the sensitivity of the receiver, and neither direct nor multipath components can be recovered, so the receiver is not able to track the satellite. This new state, indicated with #4, will have an initial probability of \(P_{4}\). The #4 state #4 scope is mainly included to provide better knowledge of the duration of fades deeper than the sensitivity of the receiver that cause the interruption of the channel. The CDF of the state #4 is expressed by:  
\[CDF(x\leq d)=\frac{1}{2}\bigg{[}1+erf\bigg{(}\frac{ln(d)-m}{\sigma\sqrt{2}} \bigg{)}\bigg{]}, \tag{22}\]  
where \(m\) and \(\sigma\) are the mean and standard deviation of the logarithm of the distance \(d\), and _erf_ is the _error function_[60].