## 2 System Description
### Analog CAP Demodulator  
At the receiver side, independent devices need to demodulate a single band bearing in mind the IoT requirements of low cost and low power consumption. The receiver diagram is shown in Figure 4: the optical signal is translated to a current by a photodiode having a responsivity of \(\Re\), and a transimpedance gain, \(A\), is applied. A noise source, \(n(t)\), is added in order to include the system noise. The signal is then multiplied in each branch by a sinusoidal wave, with a phase shift of 90\({}^{\circ}\) between them, with angular frequency \(\omega_{LO}\), plus a constant phase, \(\phi\). The mixing stage shifts the received signal to its baseband form. Following the mixing stage, a filtering stage with low-pass filters is used to remove both the undesired and imaging bands. The impulse response of the filters is denoted by \(g(t)\). Analog-to-digital conversion occurs to sample the demodulated symbols at the rate \(R\), which after a QAM de-mapping process, results, ideally, in the original data stream. When compared to its digital equivalent, the analog demodulator uses less active components, thus requiring less power and reducing its overall cost. A digital CAP demodulator would require complex Digital Signal Processors (DSPs), demanding hundreds of logical gates implemented by thousands of transistors. Instead, the analog demodulator requires a Local Oscillator (LO), two mixers (implemented with few transistors), and analog filtering stages, severely reducing its cost and power consumption. Another advantage of using an analog frontend is the required sample rate of the receiver ADCs, which are much slower when compared with the digital demodulator. In other words, in digital demodulation, the signal must be sampled according to the Nyquist sampling theorem, i.e., more than twice the signal bandwidth (of all bands). Instead, using the analog frontend, the ADCs' sampling frequency can be lowered to the symbol rate. Therefore, the sampling frequency can be decreased by a factor proportional to the number of bands. At the input of the quadrature mixer, the signal is given by:  
\[rx(t)=\Re A[y_{d}(t)*h_{t}(t))]+\eta(t) \tag{6}\]  
where \(\eta(t)\) is considered to be Additive White Gaussian Noise (AWGN) resulting from the photodiode shot noise and the Transimpedance Amplifier (TIA) noise [50]. The mixers multiply the \(rx(t)\) signal by a sinusoidal wave, with the central frequency of the desired band, \(\omega_{LO}\), having a \(90^{\circ}\) phase shift between each branch. Assuming \(\phi=0\), i.e., the LO is in phase with \(rx(t)\), the mixers' signal output is given by,  
\[rx_{I}(t)=rx(t)cos(\omega_{LO}t) \tag{7}\]  
\[rx_{Q}(t)=rx(t)cos(\omega_{LO}t-\frac{\pi}{2})=-rx(t)sin(\omega_{LO}t) \tag{8}\]  
In order to simplify the mathematical analysis of the demodulation, \(h_{t}(t)\) will be assumed unity in the forgoing analysis. By combining Equations (6)-(8), as well as the usage of the convolution definition in the Equation (5) convolution operations, the in-phase and quadrature signals are demodulated as follows:  
\[rx_{I}(t)=\Bigg{[}\Re A\sqrt{2}\sum_{m=1}^{M}\int_{-\infty}^{ \infty}\Big{(}s_{I}^{m}(\tau)F_{I}^{m}(t-\tau)-s_{Q}^{m}(\tau)F_{Q}^{m}(t-\tau )\Big{)}d\tau+\eta(t)\Bigg{]}cos(\omega_{LO}t) \tag{9}\] \[rx_{Q}(t)=-\Bigg{[}\Re A\sqrt{2}\sum_{m=1}^{M}\int_{-\infty}^{ \infty}\Big{(}s_{I}^{m}(\tau)F_{I}^{m}(t-\tau)-s_{Q}^{m}(\tau)F_{Q}^{m}(t-\tau )\Big{)}d\tau+\eta(t)\Bigg{]}sin(\omega_{LO}t) \tag{10}\]  
Figure 4: Analog CAP receiver. TIA, Transimpedance Amplifier.