## 2 Related Works and Problem Formulation  
### OCDM System Model  
The OCDM system model is shown in Figure 1. The bitstream data is transformed from serial to parallel at the transmitter, followed by encoding and mapping. Equation (1) shows the OCDM time domain signal is obtained through inverse discrete Fresnel transform (IDFnT) from multiplexing \(N\) orthogonal chirp waves. \[s(n)=\sum_{k=0}^{N-1}x(k)\psi_{k}\left(n\frac{T}{N}\right)=e^{i\frac{T}{4}}\sum _{k=0}^{N-1}x(k)e^{-i\frac{T}{N}(n-k)^{2}}. \tag{1}\]  
The chirp waves as shown in Figure 1. Solid lines represent in-phase components and dashed lines represent orthogonal components. Transform above expression in a concise matrix form, shown in Equation (2). \[\mathbf{s}=\mathbf{\Phi}^{H}\mathbf{x}. \tag{2}\]  
Figure 1: OCDM chirp waveforms for \(N\). In Equation (2), \(\mathbf{x}=[x(0),x(1),\ldots,x(N-1)]\) and DFnT unitary matrix \(\mathbf{\Phi}\in C^{N*N}\) is defined as Equation (3),  
\[\left\{\mathbf{\Phi}\right\}_{m,n}=\frac{1}{\sqrt{N}}e^{-j\frac{\pi}{8}}exp\Big{(}j \frac{\pi}{N}(m-n)^{2}\Big{)}. \tag{3}\]  
After IDFnT, \(N\) parallel subcarriers are converted into post-serial bit streams. After that, a cyclic prefix sequence is added at the head of data streams to avoid ISI. The transmitting signal after the UWA channel can be concluded as Equation (4). \[\mathbf{r}=\mathbf{H}\mathbf{s}+\mathbf{n}=\mathbf{H}\mathbf{\Phi}^{H}\mathbf{x}+\mathbf{w} \tag{4}\]  
where \(\mathbf{w}\) is the Gaussian white noise vector in the channel, the circulant matrix \(\mathbf{H}\) is the channel impulse response (CIR), and the first column is:  
\[\mathbf{h}=[h(0),h(1),\ldots,h(L-1),0,\ldots,0]. \tag{5}\]  
where \(h(l),l=0,\ldots,L-1\) is the first moment's CIR tap and \(L\) is the maximum delay spread of the channel. To reduce computational load, based on the eigen-decomposition properties of the DFnT matrix [22], the discrete FFT transform is employed for DFnT implementation, which is shown in Equation (6). \[\mathbf{y}=\mathbf{F}\cdot\mathbf{r}=\mathbf{\Gamma}^{\mathbf{H}}\mathbf{\Lambda}\mathbf{F}\mathbf{x}+\mathbf{w}. \tag{6}\]  
where \(\mathbf{F}\) is a Fourier matrix of size \(N\), and \(\mathbf{\Gamma}\) and \(\mathbf{\Lambda}\) are both diagonal matrices of size \(N*N\). The k-th diagonal term of \(\mathbf{\Lambda}\) is the k-th frequency point of channel frequency response (CFR). When the size of \(N\) is even: \(\mathbf{\Gamma}_{n,n}=e^{-j\frac{\pi}{8}n^{2}}\). Therefore, both ZF and MMSE channel equalization matrix G in the above expressions can be obtained from Equations (7) and (8):  
\[\mathbf{G}_{\mathbf{ZF}}=\mathbf{\Lambda}^{\mathbf{H}}/\mathbf{\Lambda}^{\mathbf{H}}\mathbf{\Lambda}. \tag{7}\]  
\[s(n)=\sum_{k=0}^{N-1}x(k)\psi_{k}\left(n\frac{T}{N}\right)=e^{i\frac{T}{4}}\sum _{k=0}^{N-1}x(k)e^{-i\frac{T}{N}(n-k)^{2}}. \tag{1}\]0  
The OCDM system model as shown in Figure 2, the transmitter transforms bitstream data from serial to parallel and generally utilizes modulation to achieve higher data transmission rates. Each modulated subcarrier can carry more information, thereby achieving higher spectral efficiency. A discrete time-domain signal is obtained by IDFnT. Due to inter-symbol interference caused by multipath effect, additional guard intervals are inserted into adjacent blocks. Typically, cyclic prefixes are used to fill guard intervals. The length of the guard interval depends on the specific channel environment, it should be greater than the maximum delay of the channel. After serial-to-parallel conversion, the UWA transducer only needs to convert the digital signal into its analog counterpart for transmission through seawater. Upon successful synchronization at the receiver, the subsequent procedure entails the digitization of the captured signal through quantization and sampling. After removing the cyclic prefix from the signal, DFnT operations are applied to demodulate data from each subcarrier. Due to channel interference, the receiver conducts channel estimation to obtain accurate channel information to recover the signal.