# Optical Camera Communication as an Enabling Technology for Microalgae Cultivation +
## 3 Communications Modelling
### Plant Distribution
These approximations would greatly simplify the practical comparison of two cases without incurring harsh penalties. First, it is important to remark that the camera's hardware and configuration would not change. Consequently, the bandwidth, which depends on the sampling frequency and the image sensor's exposure time, remains constant (\(BW^{\prime}=BW_{\text{ideal}}\)). On the other hand, the maximum capacity will be achieved when the transmitting source occupies the image entirely in the scanning dimension (usually from top to bottom). In this way, the link availability is kept while capturing a frame (\(\tau_{\text{rx,ideal}}=1\)). With these considerations, Equation (23) could be reduced to Equation (24). \[Q=\frac{log_{2}(1+S/N)}{log_{2}(1+(S/N)_{\text{ideal}})}\cdot\tau_{\text{rx}} \tag{24}\]  
Now, using the first order Taylor approximation of the logarithmic function (at x = 0), Equation (25), results in Q as given by Equation (26). \[\lim_{x\to 0}log_{2}(1+x)=\frac{x}{ln(2)} \tag{25}\]  
\[Q\approx\frac{S/N}{(S/N)_{\text{ideal}}}\ \frac{h_{\text{roi}}}{h_{\text{res}}} \tag{26}\]  
In this way, the measurement of the capacity of a placement can be approximated to the comparison of the SNR with the best case and the size of its projection in the image. However, assumptions regarding the relationship between the SNR under study and the best-case SNR can simplify the analysis. These assumptions are detailed below. First, the container radiance varies smoothly through the entire surface, or at least in the area were the signal is recovered. Not the whole surface of the container is utilised for signal detection, but a fraction of it. The radiance at any given direction is expressed with respect to the maximum radiance \(L_{\text{max}}\) (Equation (27)). \[L(\vartheta,\varphi)=L_{\text{max}}\hat{L}(\vartheta,\varphi) \tag{27}\]  
The pixel FoV is small enough to assume that the pixel irradiance's contributions come from the same emitter radiance's output angles. Furthermore, the use of image-forming optics compensates the power loss due to spherical propagation with the projected size of the optical source on the image sensor [31]. Therefore, the power received by a pixel (Equation (28)) in a given direction can be approximated as the ratio of the total received power in the direction of maximum radiation, and it can be related to the ratio of the emitted radiance. \[\text{where}:\quad\text{SUR}=\frac{\text{V}_{\text{cont}}}{\text{ V}_{\text{room}}} \tag{19}\]0  
Furthermore, assuming that the region of interest (ROI) where the signal will be recovered is constant in width and height, then the total received power would be the aggregation of all contributions from the ROI's pixels. Therefore, the optical power received within the ROI is approximated by Equation (29). \[\text{where}:\quad\text{SUR}=\frac{\text{V}_{\text{cont}}}{\text{ V}_{\text{room}}} \tag{19}\]1  
Regarding the noise power, no external interfering optical sources were considered for simplicity. The primary noise sources are thermal noise (\(\sigma_{\text{th}}\)), shot noise (\(\sigma_{\text{shot}}\)) and quantisation noise (\(\sigma_{\text{add}}\)), which is generated by the ADC of the camera. Besides, thermal noise does not depend on the signal power, whilst shot noise is affected by the received optical power. Nonetheless, considering the application scenarios of this work, it is expected that the shot noise contribution could be neglected with respect to thermal noise. Hence, the noise power is the same regardless of the receiver's position. Therefore, comparing the radiation at different viewing angles can give a good approximated idea about the difference in the SNR of different container's arrangements. \[\text{where}:\quad\text{SUR}=\frac{\text{V}_{\text{cont}}}{\text{ V}_{\text{room}}} \tag{19}\]2  
\[\text{where}:\quad\text{SUR}=\frac{\text{V}_{\text{cont}}}{\text{ V}_{\text{room}}} \tag{19}\]3  
In conclusion, the final approximation of Q (Equation (32)) is obtained by combining Equations (24) and (30).