# Software-Defined Visible Light Communication for Internet of Things: A Low-Complexity Approach
## 3 Modeling of BBP Unit in SD-VLC System
### Performance Analysis of CAP Modulation  
Figure 3 illustrates the CAP modulation and demodulation process in a VLC system. The Tx path begins with QAM mapping, followed by an interpolation stage (\(k\uparrow\)), which increases the sampling rate. The complex signal is then separated into real and imaginary components, which are passed through orthogonal waveform filters: the in-phase filter (\(h_{1}(t)\)) and the quadrature filter (\(h_{Q}(T-t)\)). The outputs are combined to form the transmitted CAP signal. At the Rx path, the received signal undergoes matched filtering using the in-phase (\(h_{1}(t)\)) and quadrature (\(h_{Q}(Q-t)\)) filters, ensuring optimal signal recovery. The filtered signals are recombined into a complex signal, which is then downsampled (\(k\downarrow\)) and QAM-demapped to reconstruct the original bitstream \(\{b_{n}\}\). This structure ensures efficient spectral utilization and robustness against channel impairments, making CAP a suitable modulation scheme for VLC systems. CAP modulation offers an efficient way to encode data in two-dimensional signal space without relying on a conventional local oscillator. Instead, it employs two orthogonal signature waveform filters, \(h_{1}(t)\) and \(h_{Q}(t)\), which modulate the signal in separate quadrature components. This design not only improves spectral efficiency but also eliminates IQ imbalance issues commonly encountered in traditional IQ modulation schemes, making CAP particularly suitable for VLC systems operating under IM/DD constraints. At the transmitter, these orthogonal filters shape the signal before transmission; at the receiver, matched complementary filters recover the original data with minimal inter-component interference. The impulse responses of the two orthogonal filters are expressed as follows:  
Figure 3: Block diagram of the CAP modulation and demodulation process. Figure 2: Block diagram of the BBP unit for the proposed SD-VLC system. \[ \begin{cases}h_{I}(t)=g(t)\cos(2\pi f_{c}t)\\ h_{Q}(t)=g(t)\sin(2\pi f_{c}t).\end{cases}  \tag{1}\]  
Here, \(g(t)\) represents the raised cosine filter (RCF), which is used to shape the waveform of the baseband signal and is crucial for minimizing ISI. The modulation frequency \(f_{c}\) must be higher than the highest baseband frequency, ensuring effective separation of the orthogonal components in the passband signal. By carefully tuning the rolloff factor (\(R\)) and symbol period (\(T\)) of \(g(t)\), CAP modulation balances bandwidth efficiency and robustness to channel impairments. \[g(t)=\frac{\sin(\pi t/T)}{\pi t/T}\cdot\frac{\cos(\pi Rt/T)}{1-4R^{2}t^{2}/T^{ 2}}. \tag{2}\]  
Both the transmitter and the receiver are made in the digital domain. The transmitter's signature filters are designed as fixed finite-impulse response (FIR) filters. CAP differs from QAM in utilizing a square-root RCF for pulse shaping and performing frequency up-conversion in two consecutive steps to create a passband signal centered around the designated carrier frequency. CAP is executed using two FIR filters in a single step, as shown in Figure 4. In this process, the input symbol sequences are up-sampled to align with the FIR filter's sampling rate, whereas the signal received by the detector is down-sampled to match the filters \(h_{I}(T-t)\) and \(h_{Q}(T-t)\) in accordance with the original symbol rate. The receiver employs inverse filtering, followed by the application of a digital FIR filter in the baseband domain to mitigate ISI in frequency-selective channels. In SD-VLC systems, CAP modulation was proposed as an alternative to QAM modulation [20; 21]. One of the key advantages of CAP over QAM is that it inherently avoids IQ imbalance, which is a common issue in analog QAM systems due to the nonlinearity of analog components. IQ imbalance introduces distortions, such as DC offset, amplitude mismatch, and phase shift, leading to constellation stretching, translation, or rotation. These distortions degrade signal integrity and require additional IQ imbalance compensation in QAM-based systems.