## 2 Channel Model and Signal-to-Noise Ratio
### Effective Signal-Collection Area Model for Non-Imaging Angle-diversity Receivers
#### 2.2.1 Conventional Angle-Diversity Receiver  
A conventional angle-diversity receiver uses multiple receiving elements or branches that are oriented in different directions, where each element employs its own filter and non-imaging concentrator, such as a compound parabolic concentrator or hemispherical lens (see Figure 3a). By adding a filter and concentrator to a bare detector, the effective signal-collection area of the receiver becomes:  
\[A_{eff}^{cf}(\psi)=A_{R}T_{S}(\psi)g(\psi)cos(\psi)rect\left(\frac{\psi}{\pi/2}\right) \tag{10}\]  
where \(T_{S}(\psi)\) is the filter transmission and \(g(\psi)\) the concentrator gain. Non-imaging concentrators exhibit a trade-off between gain and FOV. An idealized non-imaging concentrator [4] having an internal refractive index \(n\) achieves a constant gain expressed as:  
\[g(\psi)=\frac{n^{2}}{sin^{2}\psi_{c}}rect\left(\frac{\psi}{\psi_{c}}\right) \tag{11}\]  
where \(\psi_{c}\) is the concentrator FOV, which is usually less than or equal to \(\pi/2\). In the model used, the concentrator gain is affected by the optical efficiency \(\eta(\psi)\), which represents the reflection losses of the concentrator. The propagation delay introduced by the concentrator is also considered [18]:  
\[g(\psi)=\frac{n^{2}}{sin^{2}\psi_{c}}\eta(\psi)rect\left(\frac{\psi}{\pi/2} \right),\ t(\psi)\
eq 0 \tag{12}\]  
Replacing \(g(\psi)\) in the expression that defines the effective signal-collection area of the receiver yields:  
\[A_{eff}^{cf}(\psi)=\frac{n^{2}A_{R}T_{S}(\psi)cos(\psi)}{sin^{2}\psi_{c}}\eta (\psi)rect\left(\frac{\psi}{\pi/2}\right) \tag{13}\]  
In addition, in a wireless IR communications system, an optical bandpass filter can be used to limit the ambient radiation reaching the detector. A common form of bandpass filter consists of a stack of dielectric thin-film layers. By properly choosing the number of layers, their thicknesses, and their refractive indexes, it is possible to control the surface reflectance and thus the filter transmittance. The filter transmission \(T_{S}(\psi)\) can be described fairly accurately by a simple five-parameter model [23]. In this model, for radiation of wavelength \(\lambda_{o}\) incident at angle \(\psi\), the filter transmission is given by:  
\[T(\psi;\Delta\lambda,\psi^{\prime})=\frac{T_{0}}{1+\left[\frac{\lambda_{0}- \lambda^{\prime}(\psi;\psi^{\prime})}{\Delta\lambda/2}\right]^{2m}} \tag{14}\]  
Figure 3: Non-imaging angle-diversity receiver geometries: (a) Conventional angle-diversity receiver; (b) Sectored angle-diversity receiver; (c) Self-orienting receiver. where \(\psi^{\prime}\) is the filter orientation, \(T_{0}\) is the peak transmission at \(\psi^{\prime}\), \(\Delta\lambda\) is the spectral half-power bandwidth, \(m\) is the filter order and \(\lambda^{\prime}\)(\(\psi\);\(\psi^{\prime}\)) represents the shifting to shorter wavelengths at non-normal incidences, which is described by:  
\[\lambda^{\prime}(\psi;\psi^{\prime})=\lambda_{0}\left(\frac{n_{s}^{2}-n_{1}^{2 }sin^{2}\psi}{n_{s}^{2}-n_{1}^{2}sin^{2}\psi^{\prime}}\right)^{\frac{1}{2}} \tag{15}\]  
where \(n_{1}\) is the index of the input layer and \(\mathrm{n_{s}}\) is the effective index for the spacer layer. The design of the optical filter thus boils down to specifying the two parameters \(\Delta\lambda\) and \(\psi^{\prime}\). The remaining three parameters (\(n_{s}\), \(m\) and \(T_{0}\)) should be chosen to be as large possible, while considering the constraints imposed by technology. In keeping with reference [23], we used \(n_{s}=2.293\), \(m=3\), and \(T_{0}=0.92\). Furthermore, to provide the best utilization of the CPC and filter, the angular bandwidth \(\Delta\psi\) should be equal to the concentrator FOV. Figure 3a shows a conventional angle-diversity receiver composed of seven elements, one of them oriented vertically towards the ceiling and six angled at a \(56^{\circ}\) elevation with a \(60^{\circ}\) separation in azimuth. Each element uses a bandpass optical filter and a CPC with a \(50^{\circ}\) field of view.