## 2 Dynamic Lognormal Shadowing Framework
### Fade Durations and Level Crossings  
Of great importance to several aspects of wireless communications is the average outage or fade duration (AOD). AOD is a second order statistical property, which significantly affects packet length, channel coding schemes, length interleaver, etc. The dynamic model proposed here permits the calculation of AOD in a multidimensional setting by considering correlated shadowing and without the adoption of asymptotic results of level crossings as in [40]. The shadowing process of (18) is a vector stochastic process of Ornstein-Uhlenbeck type, i.e., a Gaussian and Markov process. It is a well-known fact already from the one dimensional case that the power carried by the high frequencies of such a process is so large that the AOD is zero and the average level crossing rate (LCR) is non-finite [41]. However the inductances and capacitances always present in the circuits used in radio engineering act to smooth the process [42], so analysis of fade durations may be done through appropriate filtering of the process (18). Similar approaches can be found in [21] for a general Markov process, and in [16; 24; 43] for the shadowing process. The filtering approach finds a very strong reasoning based on the time scale of the physical process of shadowing. As shadowing refers to the local mean of the received signal power, this is calculated by measuring and averaging the power received over a spatial distance of 20-30 wavelengths in order to filter out the multipath fading [44]. The shadowing then experiences slow variations over distances of tens of wavelengths due to the presence of obstructions depending on their relative size. Translating to time scale by considering the effect of movement this gives rise to a coarser time scale of shadowing and an upper frequency limit of the shadowing process. However the limits are not strict, especially in dense environments like in indoor propagation channels, where the large and small scale phenomena tend to get mixed. Thus, averaging for finding the local mean by filtering out the multipath effect has the drawback that the resultant shadowing process is unable to reflect shadowing variations inside the averaging window [45]. We nevertheless assume through the whole study that the multipath effect has been eliminated, an assumption common in the shadowing literature. We also do not confine ourselves to a specific filtering method as in [16; 24], where moving averages (temporal or local) have been considered. Instead, we consider any appropriate linear filtering. However, filtering has a major impact on second order statistics and the selection of the filter characteristics must correspond to the specific analysis scenario. Now as filtering is a linear operation and the shadowing process is a Gaussian process, the same is true for the smoothed process, thus permitting the use of well-known facts for LCR and AOD of Gaussian processes. By use of the generalized Rice method for vector stochastic processes [46], we can calculate the out-crossing rate \(v^{+}\) of a limit surface \(\partial S\) bounding a domain \(D\) in the space of values of the filtered process \(\mathbf{Z}_{t}\) as  
\[v^{+}=\int\limits_{\partial S}\!\left\{\int\limits_{0}^{\infty}\dot{z}_{n}f_{ \mathbf{Z}\dot{z}_{n}}\left(\mathbf{z},\dot{z}_{n}\right)\dot{d}\dot{z}_{n} \right\}\!d\mathbf{z} \tag{34}\]  
where \(\dot{z}_{n}=\mathbf{n}^{T}(\mathbf{z})\cdot\dot{\mathbf{z}}\) is the normal velocity of vector process \(\mathbf{Z}_{t}\) at the limit surface \(\partial S\) and \(f_{\dot{z}\dot{z}_{n}}\left(\mathbf{z},\dot{z}_{n}\right)\) is the joint probability density function (PDF). As \(\mathbf{Z}_{t}\) is a Gaussian vector process, its derivatives are independent of the process simplifying the expressions for the joint PDF.