## 5 LMS Channel Models
### Single-State Channel Model  
A statistical model for LMS systems is represented by the Loo model [10; 57]. It assumes that the amplitude of the LoS component under foliage attenuation (shadowing) is lognormally distributed and that the received multipath interference has a Rayleigh distribution. In the following, a brief mathematical description of the model is provided. This model previews that the signal is modeled using the sum of the lognormal and Rayleigh random variables with independent phases, as shown:  
\[a\cdot exp(j\theta)=l\cdot exp(j\phi_{0})+r\cdot exp(j\phi), \tag{10}\]  
where \(a\) represents the signal amplitude, \(\phi_{0}\) and \(\phi\) are the phases uniformly distributed in \((0,2\pi)\), and \(l\) and \(r\) represent the lognormal and the Rayleigh distribution, respectively. The PDF of \(a\) is given by:  
\[p_{Loo}(a)=\frac{a}{b_{0}}\int_{0}^{\infty}exp[\frac{-(a^{2}+l^{2})}{2b_{0}}] I_{0}(\frac{al}{b_{0}})p(z)dz, \tag{11}\]  
where \(b_{0}\) represents the average scattered power due to the multipath and \(I_{0}(\cdot)\) is the modified zero-order Bessel function. Considering \(a\) as the lognormal for large values and distributed as Rayleigh for small values, the \(p(a)\) can be represented by:  
\[p_{Loo}(a)= \begin{cases}\frac{a\sqrt{2\pi}d_{0}}{exp[\frac{-(ln-\mu)^{2}}{2d_ {0}}]},&for\ a\gg\sqrt{b_{0}}\\ \frac{a}{b_{0}}exp[\frac{-a^{2}}{2b_{0}}],&for\ a\ll\sqrt{b_{0}},\end{cases}  \tag{12}\]  
with \(\sqrt{d_{0}}\) and \(\mu\) being the standard deviation and the mean, respectively. Moreover, in order to allow integration over \((0,R)\) instead of \((R,\infty)\), it is possible to use the following relation:  
\[p(a>R)=\int_{R}^{\infty}p_{Loo}(a)da=1-\int_{0}^{R}p_{Loo}(a)da \tag{13}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Markov Model** & **Urban** & **Suburban** & **Rural** \\ \hline
**Single-State** & Rice/Lognormal & Rice/Lognormal & Rice/Lognormal \\
**Two-State** & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh \\
**Three-State** & Lognormal/Rayleigh & Lognormal/Rayleigh & Lognormal/Rayleigh \\
**Rice/Lognormal/Rayleigh** & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh \\
**Four-State** & Nakagami-Rice/Lognormal/Rayleigh & Nakagami-Rice/Lognormal/Rayleigh & Nakagami-Rice/Lognormal/Rayleigh \\
**Five-State** & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh \\
**Six-State** & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh \\
**Multi-State** & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh & Rice/Lognormal/Rayleigh \\ \hline PDF expression & & & \\ \hline
**Rice** & & \(p_{Kur}(a)=\frac{a}{a^{2}}exp[-\frac{a^{2}+l^{2}}{2}]\left[\frac{a}{a^{2}}\right]\) \\
**Lognormal** & & \(p_{Lognormal}(a)=\frac{a}{a^{2}}exp[-\frac{a^{2}}{2a^{2}}]\) \\
**Rayleigh** & & \(p_{Lognormal}(a)=\frac{a}{a^{2}}exp[-\frac{a^{2}}{2a^{2}}]\) \\
**Nakagami–Rice** & & \(p_{Lognormal}(a)=\frac{a}{a^{2}}exp[-\frac{a^{2}}{2a^{2}}]\) \\
**Nakagami–Rice** & & \(p_{Lognormal}(a)=\frac{a}{a^{2}}exp[-(a^{2}+l^{2})/2a^{2}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Urban, suburban, and rural environment statistic comparison. It is worth noting that, if the attenuation due to shadowing (lognormal distribution) is maintained constant, the PDF in Equation (12) just provides a Rician distribution. Experiments in rural regions with elevation angles of up to 30 degrees have been used to validate this model [31]. Other important contribution to this type of channel model exist in the literature, one of which is represented by the Corazza model [33]. In this paper, the authors introduce a statistical channel model that uses the combination of two important distributions, Rice and lognormal, to model the effects of the LoS and shadowing components. The principle provided in the work can be suitable for all types of environments, such as rural, urban, and suburban, thanks to a set of parameters that can be tuned opportunely.