## 2 Channel Model and Signal-to-Noise Ratio
### Effective Signal-Collection Area Model for Non-Imaging Angle-diversity Receivers
#### 2.2.2 Sectored Angle-Diversity Receiver  
A sectored angle-diversity receiver consists of a set of photodetectors located on a hemisphere. The space of sectored receivers enclosed between two parallels is called a crown, and a sector is the region of the crown enclosed between two equally spaced meridians. Every sector in each crown has an equal azimuth aperture, and therefore the same limiting elevation angles. In summary, the sectored receiver is defined by a set of parameters [11], \(\Psi\), which specifies, for each crown, its number of sectors, \(N_{S}\), the azimuth offset of its first sector, \(\varepsilon\), and its limiting elevation angles, \(\theta_{l}\) and \(\theta_{h}\). As an example, bottom illustration in Figure 3b shows the configuration of a sectored receiver with three crowns, which is defined by \(\Psi=\{(4,\,0^{\circ},\,0^{\circ},\,24^{\circ}),\,(4,\,20^{\circ},\,24^{\circ},\,5 3^{\circ}),\,(8,\,0^{\circ},\,53^{\circ},\,9^{\circ})\}\). As we can see, the first crown in the sectored receiver consists of 4 sectors (\(N_{S}=4\)). The azimuth offset of its first sector is \(\varepsilon=0^{\circ}\), and its limiting elevation angles are \(0^{\circ}\) and \(24^{\circ}\) (\(\theta_{l}=0^{\circ}\) and \(\theta_{h}=24^{\circ}\)). Analogously to the conventional angle-diversity receiver shown in Figure 3a, the configuration of the sectored receiver was the result of a study focused on achieving the receiver structure that exhibits the minimum simultaneous rms delay spread and path loss [12]. As with a single-element detector, each sector of the sectored receiver is defined by its position, orientation, vertical and horizontal apertures, and effective signal-collection area. The region of a sector is specified by the two limiting elevation angles, \(\theta_{l}\) and \(\theta_{h}\), and the two limiting azimuth angles, \(\phi_{l}\) and \(\phi_{h}\), where \(\theta_{l}<\theta_{h}\) and \(\phi_{l}<\phi h\) (see top illustration in Figure 3b). Consequently, the vertical aperture is given by \((\theta_{h}-\phi_{l})/2\), whereas the horizontal aperture is set by \((\phi_{h}-\phi_{l})/2\). The orientation of each sector is defined by its azimuth and elevation angles \((\phi_{S},\theta_{S})\), which are given by:  
\[\phi_{S}=\left(\phi_{l}+\phi_{h}\right)/2\quad\text{and}\quad\theta_{S}=\left( \theta_{l}+\theta_{h}\right)/2 \tag{16}\]  
except in the case of a polar crown with a single sector where \(\theta_{S}=0^{\circ}\) (sector aimed vertically upward, \(\phi_{h}-\phi_{l}=2\pi\), \(\theta_{h}\) any, \(\theta_{l}=0\)). Each sector in a sectored receiver has an effective signal-collection area given by:  
\[A_{eff}(\psi)=A_{R}T_{S}\left(\psi\right)cos(\psi)rect\left(\frac{\psi}{\pi/2}\right) \tag{17}\]  
where \(T_{S}(\psi)\) is the transmission characteristic of the filter used to limit the ambient radiation reaching each photodetector, \(\psi\) is the incidence angle of radiation with respect to the sector orientation, and \(A_{R}\) represents the physical area of the detector, which is described by:  
\[A_{R}=r^{2}\cdot(\phi_{h}-\phi_{l})\cdot(cos\ \theta_{l}-cos\ \theta_{h}) \tag{18}\]where \(r\) is the radius of the hemisphere. In a sectored receiver, the optical filter should be deposited or bonded onto the outer surface of the sectored receiver, i.e., onto all the sectors or photodetectors that comprise the receiver. In this sense, a bandpass or interference optical filter cannot be used because the radiation that reaches each photodetector is incident upon the filter at any value of \(\psi\) between \(0^{\circ}\) and \(90^{\circ}\), shifting the filter transmission to shorter wavelengths and minimizing its transmission. Therefore, the ambient light radiation and the desired signal striking the filter at wide incidence angles will not be adequately filtered. However, in order to reject ambient light, longpass absorption optical filters could be used [4], which pass light at every wavelength beyond the cutoff wavelength.