# Characterization of the Body-to-Body Propagation Channel for Subjects during Sports Activities
## 4 The Channel Model  
It has been observed in Section 3 that the received signal power for BBNNs oscillates between intervals where it is above a certain threshold, and an interval where it is below it (see Figure 3). In addition, the oscillations can be approximated to have a constant period depending on the activity involved. These kinds of channels can be represented using the two-state semi-Markov model (TSSMM) as done in land mobile satellite links [17; 18]. In this section, a TSSMM for BBWN channels under various running and cycling scenarios is developed. We start the modeling process by obtaining the appropriate type of amplitude distribution for the 'good' state \(D_{g}\), representing an interval in which the received signal is above the local mean, and for the 'bad' states \(D_{b}\), representing an interval in which the received signal is below the local mean. For simplicity, the choice was restricted to the use of same type of distribution for both states. The distribution was determined by obtaining the maximum likelihood (ML) estimates of the measured received signal amplitudes of the two states for Gamma, Lognormal, Nakagami-m, Rayleigh, Rician, and Weibull distributions, and comparing their total negative-log-likelihood. The distribution which gave the largest value of the total negative-log-likelihood was chosen as the best representation  
Figure 6: The power spectral density (PSD) of running activity. (**a**) Scenario 1; (**b**) Scenario 2. Figure 7: The PSD of cycling activity. (**a**) Scenario 1; (**b**) Scenario 2.  
of the fading distributions of the two states [19]. It should be noted though that only Scenario 1 (subject behind the other) of cycling activity experience regular line of sight and hence Nakagami-m given by  
\[f(x|m,\omega)=\frac{2m^{m}}{\omega^{m}\Gamma(m)}x^{2m-1}e^{-\frac{m\omega^{2}}{ \omega}}, \tag{2}\]  
gave the best representation of the fading distributions for the two states in that scenario. As the LOS path was subjected to shadowing in the other scenarios, Lognormal distribution given by  
\[f(x|\mu,\sigma)=\frac{1}{x\sigma\sqrt{2\pi}}e^{-\frac{(\ln x-\mu)^{2}}{2\
u^{ 2}}}, \tag{3}\]  
gave the best representation of the fading distributions for the two states instead. To obtain the time period of each state, we use the obtained ACF shown in Figure 5. From the ACF, one can determine the total period of the two states, which is equal to the oscillation period of the ACF. The ratio of the time period of each state can then be obtained by using the ratio between the samples present in each state. The resulting state periods obtained, \(T_{g}\) and \(T_{b}\), control the time spent in each state as the state machine alternates between the two states. The simulation model also included the shadowing and fading effects caused by the change in the surrounding environment as well as the changes in the relative position between the subjects (see Figure 4). Since these changes are relatively slow, they are normally considered as large scale fading effects and are modelled using Lognormal distribution [20; 21]. The parameters of the distribution were determined by performing ML from the samples of the local mean. Lastly, to ensure the presence of the right frequency composition, the simulated envelope has to pass through an appropriate filter. The PSD of the measured data shown in Figures 6 and 7 can in general be modeled using a lowpass filter. For the case of running activity, where the PSD decreases exponentially in logarithmic scale (see Figure 6), a first-order Butterworth filter with cutoff frequency around 10 Hz gives a good approximation. For the case of cycling activity, the PSD tend to settle between 20 and 50 Hz before it start decreasing again (see Figure 7). This motivated the use of a third-order Butterworth filter instead, with cutoff frequency around the turning point (50 Hz). Figure 8 summarizes the simulation model for BBWN channels during running and cycling, and the corresponding parameters used in the simulation are given in Tables 2 and 3.