## 3 System and Channel Model Development
### Channel Model  
Let \(\alpha_{l}\) be the complex amplitude of the \(l^{th}\) multipath component and \(\tau_{l}\) be the path delay for that component. The complex envelope model for the multipath channel impulse response is given by:  
\[h(t)=\sum_{l=1}^{L}\alpha_{l}\delta(t-\tau_{l}) \tag{3}\]  
where L is the number of the multipath components and is assumed to be the same for all the sensors. Our objective is to determine the values of the amplitude \(\alpha_{l}\), path delay \(\tau_{l}\), DOA \(\theta_{l}\), the power for the multipath components. We start by determining the distribution of the DOA for a particular multipath component as a function of time-of-arrival. To simplify the notation, it is convenient to introduce the normalized multipath delay, \(\tau_{l}=\frac{c\tau_{l}}{d_{0}}=\frac{\tau_{l}}{\tau_{0}}\), where the distribution of \(\tau_{l}\) is given by:  
\[f_{r}(r)=\frac{2r^{2}-1}{\beta\sqrt{r^{2}-1}},1\leq r\leq\tau_{m} \tag{4}\]  
where \(\beta=\tau_{m}\sqrt{r_{m}^{2}-1}\) and \(\tau_{m}=\frac{\tau_{m}}{\tau_{0}}\) is the maximum value of the normalized path delay. Several techniques for selecting \(\tau_{m}\) are outlined in [2]. A detailed analysis on the pdf of multipath delays, AOA and power spectrum of the elliptical channel model can be found in [14]. Figure 2: Geometry of the GBSBEM. The idea is first to define an ellipse corresponding to the maximum multipath delay, \(\tau_{m}\) and uniformly placed scatterers inside the ellipse. The relevant signal parameters can then be calculated from the coordinates of the scatterers. It is assumed that the number of multipaths, L and the separation distance between the cluster head and the receiver, D is known. A value of the maximum multipath propagation delay, \(\tau_{m}\) is chosen and samples of two uniformly distributed random variables, \(x_{l}\) and \(y_{l}\), \(l=1,2,...\,L\) are generated over the interval [\(-1\),\(1\)]. These L samples of a random variable are described by the polar coordinates \((r_{l},\varphi_{l})\)according to the following relationships \(r_{l}=\sqrt{x_{l}^{2}+y_{l}^{2}}\) and \(\varphi_{l}=tan^{-1}\left(\frac{y_{l}}{x_{l}}\right)\). These samples are translated so that they are uniformly distributed in an ellipse; the following two transformations are performed:  
\[x_{l}=a_{m}r_{l}\cos(\theta_{l})+\frac{D}{2},y_{l}=b_{m}r_{l}\sin(\theta_{l}) \tag{5}\]  
Thus, the multipath propagation distance, \(d_{l}\), and, the propagation delays, \(\tau_{l}\), can be calculated as \(d_{l}=\sqrt{x_{l}^{2}+y_{l}^{2}+\sqrt{(D-x_{l})^{2}+y_{l}^{2}}}\), and \(\tau_{l}=\frac{d_{l}}{c}\), respectively. Following that the receiver system is located at the origin of the coordinate system, the angle of arrivals (AOA) of the multipaths at the receiver are given by \(\theta_{l}=tan^{-1}\left(\frac{y_{l}}{x_{l}}\right)\). The power of the direct path component (LOS) can be calculated as below:  
\[P_{0}(dBm)=P_{ref}\left(dBm\right)-10 nlog\left(\frac{D/c}{d_{ref}}\right)+G_{ t}(\theta_{d})+G_{r}(\theta_{a}) \tag{6}\]  
where \(P_{ref}\) is the reference power measured at a distance \(d_{ref}\) from the transmitter using omni-directional antennas at the transmitter and the receiver. \(P_{ref}\) can be calculated using [PERSON] free space propagation model given by:  
\[P_{ref}\left(dBm\right)=P_{T}(dBm)-20 log\left(\frac{4\pi d_{ref}}{\lambda}\right) \tag{7}\]  
where \(P_{T}\) is the transmitted power and \(\lambda=c/f\) is the wavelength for a particular carrier frequency, \(f\). The path loss exponent, n typically ranges from 3 to 4 in a microcell environment. \(G_{t}(\theta_{d})\)_and_\(G_{r}(\theta_{a})\) are the gains of the transmit and the receive antennas as functions of the angle of departure, \(\theta_{d}\) and the angle of arrival, \(\theta_{a}\) respectively. For the LOS component, \(\theta_{d}\) and \(\theta_{a}\)are both zero.