# LDPC-Coded CAP with Spatial Diversity for UVLC Systems over Generalized-Gamma Fading Channel
## 2 CAP-Based UVLC System  
### System Model  
Figure 1 shows a simplified block diagram of the LDPC-coded CAP modulation scheme with spatial diversity at the receiver. The system consisted of one transmitter and \(N\) receivers with the same aperture size. The pseudorandom binary streams were parsed into groups of \(k\)-bits and encoded using LDPC code, which resulted in corresponding groups of codewords with \(n\)-bits, where \(k\) and \(n\) were the information length and code length of LDPC, respectively. The \(n\)-bits in each codeword were then mapped into an \(M\)-QAM signal using a Gray mapping rule. The QAM signal was upsampled by a factor of \(N_{s}\) followed by separation of inphase (I) and quadrature-phase (Q) sequences, and application of two orthogonal pulse-shaping filters. The impulse responses of the orthogonal filters were given by the multiplication of cosine and sine with a root-raised cosine filter (RRCF) for I and Q signals, respectively, i.e., the impulse responses of pulse shaping filters were given by [21; 22]. \[f_{I}(t)=g(t)\cos(2\pi f_{c}t) \tag{1}\]  
\[f_{Q}(t)=g(t)\sin(2\pi f_{c}t), \tag{2}\]  
where \(f_{c}\) is the carrier frequency, and \(g(t)\) is the RRCF, given by  
\[g(t)=\frac{\sin\left[\frac{\pi t}{T}(1-\beta)\right]+\frac{4 gt}{T}\cos\left[ \frac{\pi(1+\beta)t}{T}\right]}{\frac{\pi t}{T}\left[1-\left(\frac{4 gt}{T} \right)^{2}\right]} \tag{3}\]  
where \(T\) is the symbol duration, and \(\beta\) is the roll-off factor of RRCF. The filter outputs were then summed, i.e.,  
\[S(t)=A_{I}(t)\otimes f_{I}(t)-A_{Q}(t)\otimes f_{Q}(t) \tag{4}\]  
where \(\otimes\) represents time-domain convolution operation, and \(A_{I}(t)\) and \(A_{Q}(t)\) denote in-phase and quadrature \(M\)-QAM symbols. The signal was then converted to an analogue signal by a digital-to-analogue converter (DAC). Since the IM/DD requires a positive signal, DC bias \(I_{\text{DC}}\) was added to bipolar CAP signals. The DC bias was given by [23]  
\[I_{\text{DC}}=g\sqrt{E_{s}} \tag{5}\]  
Where \(g\) is the normalised bias, and \(E_{s}\) is the energy per symbol. Figure 1: Block diagram of spatial-diversity underwater visible-light communication (UVLC) system employing carrierless amplitude and phase (CAP) modulation scheme and low-density parity-check (LDPC) codes. The modulated beam was expanded and collimated to ease alignment and sufficiently illuminate all receiver apertures. After going through the turbulence channel (details of turbulence channel are given in the following section), the beam was received by spatial-diversity receivers. We assumed that the spatial-correlation width was larger than the aperture diameter of each receiver, and the receiver separations were sufficiently larger than coherence length, whose typical value is on the order of millimeters [24; 25]. Thus, the signal at each receiver branch was independent, and the received instantaneous electrical signal with EGC was given by  
\[y(t)=\eta x(t)Ps\sum_{i=1}^{N}\alpha_{i}h_{i}+\sum_{i=1}^{N}n_{i}(t) \tag{6}\]  
where \(\eta\) denotes photodetector responsivity, \(x(t)\) represents the modulated symbol, \(Ps\) is the irradiance at the receiver, and \(\alpha_{i}\), \(h_{i}\), and \(n_{i}(t)\) represent the fading coefficient, path loss, and additive white Gaussian noise (AWGN) of the \(i^{\text{th}}\) branch, respectively. We assumed that path loss \(h_{i}\) was independent of the fading process and normalised to unity. Meanwhile, all receivers had the same noise variance of \(\sigma^{2}\). Thus, the electrical signal-to-noise ratio (SNR) of EGC was given by  
\[\gamma_{\text{EGC}}=\frac{(\eta Ps)^{2}\text{E}\Big{[}x^{2}(t)\Big{]}\Big{(} \sum_{i=1}^{N}\alpha_{i}\Big{)}^{2}}{N\sigma^{2}} \tag{7}\]  
where E[.] is the expectation operation.