## 2 Channel Model and Signal-to-Noise Ratio
### Signal to Noise Ratio for Angle-Diversity Receivers  
Assuming no co-channel interference, a \(J\)-element angle-diversity receiver yields \(J\) receptions of the form:  
\[I_{j}(t)=R\;x(t)\otimes h_{j}(t)+n_{j}(t),\;\;\;j=1,\ldots,J \tag{26}\]  
where \(I_{j}(t)\) and \(n_{j}(t)\) represent, respectively, the received instantaneous current and the total noise at the output of the \(j\)-th receiving element, \(t\) is the time, \(x(t)\) is the transmitted instantaneous optical power, \(h_{j}(t)\) is the channel between the transmitter and the \(j\)-th receiver, \(\bigotimes\) denotes convolution, and \(R\) is the responsivity of the \(j\)-th receiver, which are considered equal for all \(J\) receiving elements. In general, much of the performance of an angle diversity receiver depends on how the signals received in the different elements \(I\)/\((t)\) are processed and detected. There are several possible diversity schemes that can be considered. The most common techniques include MLC, MRC, SB, and (EGC) [4]. When multipath distortion is significant, the optimum reception technique is MLC, also known as matched-filter combining (MFC). In this technique, each \(I_{j}(t)\) is processed by a separate continuous-time matched filter \(h_{j}(\)-\(t)\) of the \(j\)-th receiving element. The output from each matched-filter is sampled and adjusted in magnitude by a weight factor \(w_{j}\), which is chosen to be inversely proportional to thepower-spectral density (PSD) of the noise \(n_{j}(t)\) at the output of the \(j\)-th receiving element. The weighted output signals are summed in order to obtain the combined signal. The main problem of MLC is its highly complex implementation, since this technique requires estimating each of the \(J\) channel impulse responses and noise PSD's separately. For this reason, MLC is not suited to many applications. There are simple alternatives to MLC that are practical to implement, such as MRC, SB and EGC, which differ depending on how the signals are weighed and combined. Independent of the combined method and of the weight factor, and according to Equation (21), the signal to noise ratio at the output of the \(j\)-th receiving element \(SNR_{j}\), and the SNR of the combined channel are defined as:  
\[SNR_{j}=\left(R\;P_{S,j}\right)^{2}/\sigma_{j}^{2} \tag{27}\]  
\[SNR_{combined}=\frac{\left(\sum\limits_{j=1}^{J}w_{j}RP_{S,j}\right)^{2}}{ \sum\limits_{j=1}^{J}w_{j}^{2}\sigma_{j}^{2}} \tag{28}\]  
where \(P_{S,j}\) and \(\sigma^{2}_{j}\) represent the received optical average power and the total noise variance of the \(j\)-th receiving element, respectively. In MRC, the \(I_{j}(t)\), \(j=1\),..., \(J\), are summed together with weights \(w_{j}\) proportional to the signal current to noise-variance ratios, thereby maximizing the SNR of the weighted sum. Simulations have shown that MRC can reduce transmitter optical power requirements by 4-6 dB in diffuse links at low bit rates [26]. Moreover, in situations where the ambient noise and the strong signal components arrive from different directions, MRC can decrease the multipath distortion in comparison to a single, wide FOV receiver. When multipath distortion is not significant, the optimum MLC reduces to MRC. MRC requires estimating the SNR's at the each of the receiving elements, thus increasing the complexity over non-diversity reception. In order to obtain the weight factor \(w_{j}\) at the \(j\)-th receiving element, the \(J\) partial-derivatives of the combined SNR with respect to each of the \(w_{j}\), \(j=1\),..., \(J\) are calculated and set equal to zero, so \(w_{j}\) becomes:  
\[w_{j}=\frac{\sum\limits_{j=1}^{J}w_{j}^{2}\sigma_{j}^{2}}{\sum\limits_{j=1}^{J }w_{j}RP_{S,j}}\cdot\frac{RP_{S,j}}{\sigma_{j}^{2}}=k_{0}\cdot\frac{RP_{S,j}}{ \sigma_{j}^{2}} \tag{29}\]  
where \(k_{0}\) is a constant independent of \(j\), and \(w_{j}\) is proportional to the signal current to noise-variance ratio, as expected.