## 4 SNR Analysis of a QADA Receiver  
To analyze the SNR performance of a QADA receiver, two different receiver designs with different FOVs and two different transmitter radiation patterns are used. We also show how the performance can be improved by combining the outputs from different quadrants. Figure 5 shows the receiver and the transmitter coordinate system \(xyz\). This is different from the coordinate system used within the QADA front-end so that the effect of different receiver orientations can be studied. In this section, a single transmitter with fixed optical power is considered. It is placed directly above the origin, \(O\). To make the results in this section consistent with the next section, the transmitter and the receiver are placed at \(z=2.5\,\mathrm{m}\) and \(z=0.7\,\mathrm{m}\), which represent typical ceiling and table heights in an indoor scenario. The block diagram of the main components in the transmitter and the receiver are shown in Figure 6. An electrical signal that carries the data is generated at the transmitter, which is then used to modulate the intensity of the LED. At the receiver, the optical signal is converted back to an electrical signal by the quadrant PD. At the output of the optical front-end, four signals are generated, each corresponding to one of the quadrants. After the analog-to-digital converter (ADC), the received signal from each quadrant \(k\) is given by  
\[v_{k}\left(t\right)=h_{k}s\left(t\right)+w_{k}\left(t\right) \tag{9}\]  
where \(s\left(t\right)\) is the transmitted signal, \(w_{k}\left(t\right)\) is the noise, and \(h_{k}\) is the channel gain. The electrical SNR of the signal received from the \(k\)-th quadrant is expressed as  
Figure 5: Communication system with a single transmitter at the center (**a**) a 3-D view of the configuration with a single transmitter and a QADA receiver; (**b**) a side view of the configuration; (**c**) a top view of the configuration. \[\text{SNR}_{k}=\frac{h_{k}^{2}E\Big{[}|s(t)|^{2}\Big{]}}{E\Big{[}|w_{k}\left(t \right)|^{2}\Big{]}} \tag{10}\]  
where \(E\left\{\cdot\right\}\) is the expected value. The transmitted data can be recovered directly from one of the quadrants; however, this does not take advantage of the potential benefit of the four quadrants. The total received signal from all quadrants is expressed as  
\[\textbf{v}\left(t\right)=\textbf{h}s\left(t\right)+\textbf{w}\left(t\right) \tag{11}\]  
where \(\textbf{v}=[v_{1}\left(t\right),v_{2}\left(t\right),v_{3}\left(t\right),v_{4 }\left(t\right)]^{\text{T}}\), \(\textbf{w}=[w_{1}\left(t\right),w_{2}\left(t\right),w_{3}\left(t\right),w_{4 }\left(t\right)]^{\text{T}}\), and \(\textbf{h}=[h_{1},h_{2},h_{3},h_{4}]^{\text{T}}\). Combining can be used to benefit from the diversity in the four signals. In general, the SNR after combining is given by  
\[\text{SNR}=\frac{|\alpha^{\text{T}}\textbf{h}|^{2}E\Big{[}|s(t)|^{2}\Big{]}}{ E\Big{[}\|\textbf{w}(t)\|^{2}\Big{]}} \tag{12}\]  
where \(\alpha=[\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}]^{\text{T}}\) and \(\alpha_{k}\) is the weighting factor for the signal from the \(k\)-th quadrant. There are multiple combining methods that can be applied and thus improve the performance. The first is the selection combining where the quadrant with the highest power is chosen. The weighting factor, in this case, is given by  
\[\alpha_{k}=\left\{ \begin{array}{ll}1,&\text{if}\,E\left[|v_{k}\left(t\right) |^{2}\right]=\max_{n\in[1,\cdots,A]}\left\{E\left[|v_{n}\left(t\right)|^{2} \right]\right\}\\ 0,&\text{otherwise}\end{array} \right. \tag{13}\]  
The SNR can be improved if the four signals are added together. In its simplest form, the signals from all quadrants are combined using EGC. The weighting factor using EGC is \(\alpha_{k}=0.5\) for all \(k\). EGC is only optimum when the signals have the same channel gain, which is not usually the case for a QADA receiver. To fully use the diversity of the signals, the combining should be proportional  
Figure 6: System block diagram of the transmitter and the QADA receiver.