## 2 Dynamic Lognormal Shadowing Framework  
### Random Field of Shadowing  
Shadowing refers to the random variations of the received signal power around a mean value that is noticed when a user is located at a given distance from a base station. This variation is mainly caused because of different objects located in the radio-path and not taken into account by path gain models that are used for the calculation of the mean power based only on distance. Thus, these fluctuations are taken into account in a statistical manner and the received signal to noise ratio \(SNR_{R}\) for a specific location is expressed in logarithmic scale as:  
\[SNR_{R}=SNR_{T}-\overline{PL}(d)+X(\mathrm{dB}) \tag{1}\]  
where \(SNR_{T}\) is the transmitted signal to noise ratio and \(\overline{PL}(d)\) is the mean propagation loss, as calculated by an appropriate path loss model dependent only on the distance \(d\) between the base station and the receiver. The term \(X\) in (1) is the shadowing component, a random variable (r.v.) location specific, commonly accepted to be zero mean Gaussian [7]. Denoting by \(\gamma_{R}\) the received signal to noise ratio in linear scale, then from (1) it is evident that it is a lognormal r.v. with a probability density function (pdf) given by:  
\[p(\gamma)=\frac{\xi}{\gamma\sqrt{2\pi}\sigma}e^{-\frac{(10\log\gamma-\mu)}{2 \sigma^{2}}} \tag{2}\]where \(\xi=10/\ln 10\) is a normalization constant and \(\mu\) (dB) and \(\sigma\) (dB) are the mean and the standard deviation of \(10\log\gamma_{R}\) respectively. The standard deviation \(\sigma\) of \(10\log\gamma_{R}\) is actually that of \(X\), whereas the mean \(\mu\) expressed in terms of the median \(\gamma_{m}\) of \(\gamma_{R}\) and the transmission characteristics is given by:  
\[\mu=\ln(\gamma_{m})=\ln\gamma_{T}-\frac{1}{\xi}\overline{PL} \tag{3}\]  
where \(\gamma_{T}\) is the transmitted signal to noise in linear scale. In (1) and what follows we assume that any multipath losses have been filtered out and we focus next on the shadowing r. v. \(X\). This is modeled for a given base station placement as a random field \(X(\mathbf{x})\) with correlated values in different locations, what is known as serial correlation or autocorrelation of shadowing. Though being traditionally modeled as a time-invariant spatial process, it is actually a spatio-temporal process \(S(\mathbf{x},t)\). The temporal evolution is caused by variabilities in the propagation environment caused by moving obstructions. As wireless networking becomes a critical requirement even in complex and unfavorable environments, a novel dynamic shadowing model is presented here able to deal with the temporal evolution of shadowing. In this section we present the proposed model mostly informally through the covariance structure in order to explain the motivation and to show the similarity to well accepted models in the literature. The covariance function permits not only the second order description of the shadowing random field, but its complete characterization, as it is a Gaussian random field. A commonly accepted autocorrelation model for shadowing is a simple exponential characterized by the shadowing correlation distance \(r_{c}\)[34]. \[\rho(\mathbf{x}_{1},\mathbf{x}_{2})=e^{-\frac{|\mathbf{x}_{2}-\mathbf{x}_{1}| }{r_{c}}} \tag{4}\]  
The variance of shadowing is referred to as location variability \(\sigma\) and depends on frequency and the propagation environment (urban, suburban, indoors, etc.). It is usually assumed common for all locations belonging to the same general category of environment. Thus the spatial covariance of shadowing is described as:  
\[C(\mathbf{x}_{1},\mathbf{x}_{2})=e^{-\frac{|\mathbf{x}_{2}-\mathbf{x}_{1}|}{r _{c}}}\sigma^{2} \tag{5}\]  
It is a common practice when the user is moving this spatial variability to be translated to a temporal variability scaled according to mobile's speed. This is a standard scenario for measuring the spatial variability, however careful data averaging is used to exclude the effects of multipath fading.