# Optimized Deployment of Generalized OCDM in Deep-Sea Shadow-Zone Underwater Acoustic Channels
## 3 GOCDM Underwater Acoustic Communication System
### GOCDM Modulation and Demodulation  
Assume that a symbol block of length \(K\), denoted as \(\mathbf{d}=\left[d(0),d(1),\ldots,d(K-1)\right]^{\mathrm{T}}\), is transmitted. This block is divided into \(N\) symbol vectors, each of length \(M\), and arranged into an \(N\times M\) matrix following a row-wise allocation method. Each column of this matrix is then subjected to an Inverse Discrete Fresnel Transform (IDFnT), after which the transformed data is read out row by row to obtain the transmitted signal, denoted as \(\mathbf{s}\). This entire process can be mathematically expressed using a formula. As shown in [33], a \(K\times K\) permutation matrix is defined to represent this operation.  
\[\mathbf{P}_{N,M}=\left[\begin{array}{c}\mathbf{I}_{N}\otimes\mathbf{e}_{M} ^{\mathrm{T}}(0)\\ \mathbf{I}_{N}\otimes\mathbf{e}_{M}^{\mathrm{T}}(1)\\ \vdots\\ \mathbf{I}_{N}\otimes\mathbf{e}_{M}^{\mathrm{T}}(M-1)\\ \end{array}\right], \tag{6}\]  
The modulation scheme for GOCDM can thus be expressed as follows:  
\[\begin{array}{ll}\mathbf{s}&=\mathbf{P}_{N,M}^{\mathrm{H}}(\mathbf{I}_{M} \otimes\mathbf{\Phi}_{N}^{\mathrm{H}})\mathbf{P}_{N,M}\mathbf{d}\\ &=\left(\mathbf{\Phi}_{N}^{\mathrm{H}}\otimes\mathbf{I}_{M}\right)\mathbf{d}. \end{array} \tag{7}\]  
Similar to OSDM, GOCDM allows for adjustments to the parameters \(M\) and \(N\). When \(M=1\), the system reduces to OCDM, and, when \(M=K\), the system becomes Single-Carrier Frequency-Domain Equalization (SC-FDE). As a multicarrier modulation technique evolved from OCDM, GOCDM also requires the addition of a cyclic prefix. After removing the cyclic prefix at the receiver, the received signal \(\mathbf{r}\) is obtained. The demodulation process is the inverse of the modulation process. Specifically, the signal \(\mathbf{r}\) is first arranged row by row into an \(N\times M\) matrix; then, DFnT is applied to each column of this matrix. The matrix is then read out row by row to obtain the demodulated signal \(\mathbf{y}\). The demodulation process can also be mathematically expressed as follows:  
\[\begin{array}{ll}\mathbf{y}&=\mathbf{P}_{N,M}^{\mathrm{H}}(\mathbf{I}_{M} \otimes\mathbf{\Phi}_{N})\mathbf{P}_{N,M}\mathbf{r}\\ &=(\mathbf{\Phi}_{N}\otimes\mathbf{I}_{M})\mathbf{r}.\end{array} \tag{8}\]  
Let the demodulation matrix of GOCDM be denoted as \(\mathbf{\Psi}\). Since the DFnT matrix is a unitary matrix, it follows that  
\[\begin{array}{ll}\mathbf{\Psi}_{K}^{-1}&=(\mathbf{\Phi}_{N}\otimes\mathbf{I }_{M})^{-1}=\mathbf{\Phi}_{N}^{-1}\otimes\mathbf{I}_{M}\\ &=\mathbf{\Phi}_{N}^{\mathrm{H}}\otimes\mathbf{I}_{M}=\left(\mathbf{\Phi}_{N} \otimes\mathbf{I}_{M}\right)^{\mathrm{H}}=\mathbf{\Psi}_{K}^{\mathrm{H}}, \end{array} \tag{9}\]  
The above equation utilizes the property that the inverse of the Kronecker product of two matrices equals the Kronecker product of their respective inverses. From this equation, it is evident that \(\mathbf{\Psi}_{K}\) is also a unitary matrix, with its inverse given by \(\mathbf{\Phi}_{N}^{\mathrm{H}}\otimes\mathbf{I}_{M}\), which corresponds to the modulation matrix \(\mathbf{\Psi}_{K}^{\mathrm{H}}\). In this paper, GOCDM is applied to the underwater acoustic communication system, and the flowchart is shown in Figure 2.