# Optical OFDM for SiPM-Based Underwater Optical Wireless Communication Links
## 5 Description of the Considered Signaling Schemes
### Dco-Ofdm  
Figure 1 shows the block diagram of DCO-OFDM signaling for a typical SiPM-based UWOC link. First, blocks of input data bits are mapped into \(M\)-QAM (complex) symbols \(X_{k},k=0,1...,N-1\), which are then passed through an inverse fast Fourier transform (IFFT) block generating the \"time-domain\" OFDM signal \(x_{n}\):  
\[x_{n}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}X_{k}\exp\left(j\frac{2\pi}{N}nk\right),\quad n=0,...,N-1. \tag{6}\]  
As explained previously, to ensure that \(x_{n}\) is real, Hermitian symmetry is imposed on \(X_{k}\) symbols before IFFT such that [26]  
\[\left\{\begin{array}{rcl}X_{0}&=&X_{N/2}=0,\\ X_{k}&=&X_{N-k}^{*},\quad\quad 0<k<N/2,\end{array}\right.\]  
where \(.^{*}\) denotes complex conjugate. After IFFT, a CP is added to each block to avoid inter-symbol interference (ISI) as a result of the delay spread \(\tau_{0}\) of the aggregate channel. The length of the CP, \(N_{\text{CP}}\), is set as to be larger than \(\tau_{0}/T_{s}\), where \(T_{s}\) is the OFDM symbol duration, hence allowing to restore the signal at the Rx using a one-tap equalizer. Afterwards, a scaling factor \(\alpha\) is applied to the signal in order to adequately fit it to the LED DR (this will be further clarified later in Section 5.5); the resulting signal is denoted by \(\tilde{x}_{n}=\alpha x_{n}\). Next, to obtain a positive signal, a DC bias is added to \(\tilde{x}_{n}\) before upper and lower clipping due to the limited LED DR, which gives rise to the so called \"clipping noise\". The resulting double-side clipped signal is denoted by \(\tilde{x}_{n}\).  
Without loss of generality, we consider driving the LED with a voltage, i.e., \(x_{n}\), \(\tilde{x}_{n}\), and \(\tilde{x}_{n}\) are all in units of Volts. Note that this is not restricting, as there is a bijection relationship between the input current and voltage of the LED, as described in Section 6.1.  
After being transmitted through the aquatic channel, the received optical intensity at the Rx is converted to an electrical signal. \"Photon to amplitude conversion\" performs the conversion of the number of generated photo-electrons at the SiPM output to an electrical signal amplitude [7]. Here, it is obtained by multiplying \(C_{\text{ph}}\) by the photon energy \(E_{p}\). Afterwards, after removing the CP, the recovered time-domain OFDM symbols \(r_{n}\) are passed through an FFT block whose output is given by  
\[R_{k}=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}r_{n}\exp\Big{(}-j\frac{2\pi}{N}nk\Big{)},\quad k=0,...,N-1. \tag{7}\]  
The obtained signals \(\hat{R}_{k}\) after equalization of the aggregate channel are then passed to the QAM-demapping block to recover the transmitted bits.  
An important point here concerns QAM signal demodulation since the Rx is clearly shot noise limited due to the use of an SiPM. Indeed, for a multi-level modulation scheme, the corresponding signal-dependent noise should be processed carefully, e.g., by performing the so-called square-root transformation [9; 39] to avoid a degradation of the Rx performance. However, it has been shown in [39] that for the case of O-OFDM signaling, although the time-domain signal \(r_{n}\) at the Rx (before the FFT block in Figure 1) is affected by signal-dependent noise, for the frequency-domain signal (i.e., \(R_{k}\) at the output of the FFT block) the noise is in practice nearly independent of the signal [39]. As a result, conventional QAM demodulation can be used in our case, that is, by assuming an effectively signal-independent noise.  
Assuming \(M\)-QAM constellation for \(X_{k}\), the spectral efficiency of DCO-OFDM is given by  
\[\gamma_{\text{DCO}}=\frac{\log_{2}M\;(N-2)}{2\,(N+N_{\text{CP}})}\quad\text{( bps/Hz)}. \tag{8}\]