## 2 System Description
### m-CAP Modulator  
Figure 3 presents the emitter's block diagram, i.e., the digital m-CAP modulator. It represents several \(m\) data streams, intended for different users, each modulated in Quadrature Amplitude Modulation (QAM), formatted by two pass band filters (Hilbert pair) and added in the time domain to generate a digital m-CAP modulated signal, \(y_{d}(t)\). This signal is then converted to the analog domain and transmitted as light with an LED. The impulse response \(h(t)\) represents the combined impulse responses of the Digital-to-Analog Converter (DAC), the LED driver circuit, the LED, and the VLC channel.  
The following analysis establishes the mathematical model of the m-CAP digital modulator. Considering \(m\) as the index of independent data sources, the \(m^{th}\) data signal is denoted by \(x^{m}(t)\), where \(t\) is the time. Prior to QAM modulation, the given QAM symbols are:  
\[x^{m}_{QAM}(t)=A^{m}(t)+jB^{m}(t) \tag{1}\]  
where \(A^{m}\) and \(B^{m}\) are the in-phase and quadrature components for the \(m^{th}\) data source, respectively. The QAM symbols are then upsampled by a factor of \(L\) by means of zero padding. The upsampled signals are denoted by \(s^{m}_{l}(t)\) and \(s^{m}_{Q}(t)\). The upsampled QAM symbols are filtered by digital filters, \(F^{m}_{l}(t)\) and \(F^{m}_{Q}(t)\), the impulse responses of which form a Hilbert pair, given by, respectively:  
Figure 3: m-CAP modulator in the VLC base station.  
Figure 2: Modulated m-CAP signal frequency spectrum: gray/dashed bands may be suppressed to create guard bands.  
\[F_{I}^{m}(t)=p(t)cos(\omega_{c}^{m}t) \tag{2}\]  
\[F_{Q}^{m}(t)=p(t)sin(\omega_{c}^{m}t) \tag{3}\]  
where \(p(t)\) is a pulse shaping filter and \(\omega_{c}^{m}=2\pi f_{c}^{m}\), where \(f_{c}^{m}\) is the central frequency on the \(m^{th}\) band. Equation (4) gives the values for \(f_{c}^{m}\):  
\[f_{c}^{m}=\frac{B}{2}(2m-1) \tag{4}\]  
where \(B\) is the bandwidth of each band. The pulse shaping filter design will be addressed in Section 2.3.2.  
The filter outputs are added at the modulator output, resulting in the following digital signal:  
\[y_{d}(t)=\sqrt{2}\sum_{m=1}^{M}\left(s_{I}^{m}(t)*F_{I}^{m}(t)-s_{Q}^{m}(t)*F_ {Q}^{m}(t)\right) \tag{5}\]  
Note that, if only the odd bands are modulated, the \(s_{I}^{m}(t)\) and \(s_{Q}^{m}(t)\) contribution is zero for \(m=2i-1,\forall i\in\mathbb{N}\).  
In order to transmit the \(y_{d}(t)\) signal, the LED current must be modulated by an analog signal. This means that the digital voltage signal must be converted to an analog current, using a DAC and LED driver, which do not have unitary impulse responses. The emitter optical frontend usually has a low-pass filter characteristic due to the ADC anti-imaging filter, the LED driver, and the LED itself. DACs require a low-pass frequency response to remove the image bands resulting from the sampling process, which by the Nyquist sampling theorem must have its cut-off frequency less than half of the sampling frequency. Regarding LED drivers, parasitics in the driver components result in bandwidth limitations for high frequencies. Typical high-brightness white LEDs used in lighting systems have a cut-off frequency of a few MHz due to the phosphor coating [46]. Finally, the indoor VLC channel has an impulse response, the contributions of which usually arise from signal attenuation and signal reflections, resulting in Signal-to-Noise Ratio (SNR) degradation and interference phenomena such as intersymbol interference [47; 48]. Note that the VLC channel often presents a non-flat frequency response, which may degrade the SNR, in particular for the higher frequency bands [49]. The combined impulse responses of the described process will be denominated as \(h(t)\). However, due to the narrowband characteristic of each carrier in the proposed architecture, \(h(t)\) was considered to have a low-pass response with a constant attenuation in the pass band of each user, i.e., the frequency response is considered to be flat for a particular band.