# Optimized Deployment of Generalized OCDM in Deep-Sea Shadow-Zone Underwater Acoustic Channels
## 3 GOCDM Underwater Acoustic Communication System
### Properties of GOCDM  
Building upon the methodology of [13], the complete receiver system workflow employed in this paper consists of the following: Doppler factor estimation-resampling-carrier frequency offset estimation-carrier frequency offset equalization-channel estimation-channel equalization. After completing the aforementioned four steps, the channel matrix can be approximately treated as a circulant matrix. We then analyze the properties of the GOCDM modulation and demodulation method based on the circulant matrices. Assuming that the cyclic prefix of the signal is sufficiently long, the ideal multipath channel matrix can be represented as a circulant matrix \(\mathbf{H}_{K}\). Neglecting the impact of noise, the signal modulated by GOCDM and transmitted through the ideal multipath channel can be expressed as follows:  
\[ \begin{array}{ll}\mathbf{y}&=(\mathbf{\Phi}_{N}\otimes\mathbf{I}_{M}) \mathbf{H}_{K}\big{(}\mathbf{\Phi}_{N}^{\mathrm{H}}\otimes\mathbf{I}_{M}\big{)} \mathbf{d}\\ &=\mathbf{\Psi}_{K}\mathbf{H}_{K}\mathbf{\Psi}_{K}^{\mathrm{H}}\mathbf{d}. \end{array}  \tag{10}\]  
Figure 2: GOCDM underwater acoustic communication system diagram. To analyze the properties of the GOCDM modulation matrix \(\mathbf{\Psi}_{K}\) with respect to the circulant matrix, consider the matrix product \(\mathbf{\hat{H}}_{K}=\mathbf{\Psi}_{K}\mathbf{\Psi}_{K}^{\mathrm{H}}\). For simplicity, in the following proof, all matrix and vector indices will start from 1. To obtain the value in the \(m\)-th row and \(n\)-th column of matrix \(\mathbf{\hat{H}}_{K}(m,n)\), the \(m\)-th row of \(\mathbf{\Psi}_{K}\mathbf{H}_{K}\) must be multiplied by the \(n\)-th column of \(\mathbf{\Psi}_{K}^{\mathrm{H}}\). The \(n\)-th column of \(\mathbf{\Psi}_{K}\) has a non-zero value only at the row index, \([\mathrm{mod}(n,M),\mathrm{mod}(n,M)+M,\ldots,\mathrm{mod}(n,M)+(N-1)M]\), which corresponds to the \(\mathrm{ceil}(n/M)\)-th column of \(\mathbf{\Phi}_{N}^{\mathrm{H}}\). Let matrix \(\mathbf{\Psi}_{K}\mathbf{H}_{K}\) be represented as \(\mathbf{\hat{H}}_{K}\); then, \(\mathbf{\hat{H}}_{K}(m,n)\) can be expressed as follows:  
\[\mathbf{\hat{H}}_{K}(m,n)=\sum\limits_{k=0}^{N-1}\mathbf{\hat{H}}_{K}(m, \mathrm{mod}(n,M)+kM)\mathbf{\Phi}_{N}^{\mathrm{H}}\Big{(}k+1,\mathrm{ceil} \Big{(}\frac{n}{M}\Big{)}\Big{)}, \tag{11}\]  
To obtain \(\mathbf{\hat{H}}_{K}(m,\mathrm{mod}(n,M)+(N-1)M)\), similar to the previous step, the \(m\)-th row of \(\mathbf{\Psi}_{K}\) needs to be multiplied by the \((\mathrm{mod}(n,M)+(N-1)M)\)-th column of \(\mathbf{H}_{K}\). Since the \(m\)-th row of \(\mathbf{\Psi}_{K}\) has a non-zero value only at the column index, \([\mathrm{mod}(m,M),\mathrm{mod}(m,M)+M,\ldots,\mathrm{mod}(m,M)+(N-1)M]\), which corresponds to the \(\mathrm{ceil}(m/M)\)-th row of \(\mathbf{\Phi}_{N}\), \(\mathbf{\hat{H}}_{K}(m,\mathrm{mod}(n,M)+kM)\) can be expressed as follows:  
\[ \begin{array}{ll}\mathbf{\tilde{H}}_{K}(m,\mathrm{mod}(n,M)+kM)=\sum \limits_{v=0}^{N-1}\mathbf{\Phi}_{N}\Big{(}\mathrm{ceil}\Big{(}\frac{m}{M} \Big{)},v+1\Big{)}\\ \times\mathbf{H}_{K}(\mathrm{mod}(m,M)+vM,\mathrm{mod}(n,M)+kM).\end{array}  \tag{12}\]  
Substituting (12) into (11), we obtain the following:  
\[ \begin{array}{ll}\mathbf{\hat{H}}_{K}(m,n)&=\sum\limits_{k=0}^{N-1}\sum \limits_{v=0}^{N-1}\mathbf{\Phi}_{N}\Big{(}\mathrm{ceil}\Big{(}\frac{m}{M} \Big{)},v+1\Big{)}\\ &\times\mathbf{H}_{K}(\mathrm{mod}(m,M)+vM,\mathrm{mod}(n,M)+kM)\\ &\times\mathbf{\Phi}_{N}^{\mathrm{H}}\Big{(}k+1,\mathrm{ceil}\Big{(}\frac{n} {M}\Big{)}\Big{)}.\end{array}