# Optimized Deployment of Generalized OCDM in Deep-Sea Shadow-Zone Underwater Acoustic Channels
## 3 GOCDM Underwater Acoustic Communication System
### Underwater Acoustic Communication GOCDM System  
The first part of this section introduced the modulation and demodulation methods of GOCDM. To apply GOCDM in an underwater acoustic communication system, channel estimation and equalization are also necessary. After performing essential synchronization, Doppler scaling factor estimation, nonuniform Doppler compensation via resampling, carrier frequency offset estimation, and carrier frequency offset compensation at the receiver [13], the effects of other noise in the marine environment can be approximated as additive white Gaussian noise. Under this assumption, the received signal can be expressed as follows:  
\[ \begin{array}{ll}\mathbf{y}&=\mathbf{\Psi}_{K}\mathbf{H}_{K}\mathbf{\Psi}_{ K}^{\mathrm{H}}\mathbf{d}+\mathbf{\Psi}_{K}\mathbf{w}\\ &=\mathbf{H}_{K}\mathbf{d}+\mathbf{v},\end{array}  \tag{19}\]  
where \(\mathbf{w}\) represents Gaussian white noise. Since \(\mathbf{\Psi}\) is a unitary matrix, \(\mathbf{v}\) can also be regarded as Gaussian white noise. When using block pilots for channel estimation, the received signal can be expressed as \(\mathbf{r}=\mathbf{P}_{K}\mathbf{h}+\mathbf{z}\), where \(\mathbf{P}_{K}\) is a circulant matrix with the pilot sequence \(\mathbf{p}\) as its first column, and \(\mathbf{h}\) is the time-domain response vector of the channel. Least Squares (LS) and Minimum Mean Square Error (MMSE) methods can be applied for channel estimation; however, both methods suffer from a high computational complexity. Alternatively, a sparse channel estimation method can be employed by appropriately designing the pilot sequence, though this approach requires prior knowledge of the exact number of channel multipaths. Another efficient approach leverages the flexibility of GOCDM by setting the modulation parameter \(M\) of the estimation sequence to 1, assigning a value to only one subcarrier in the pilot sequence, thereby transforming the pilot sequence into a chirp signal and enabling the use of the channel estimation method described in [37]. Similar to OCDM, GOCDM supports low-complexity equalization algorithms. By performing a direct FFT on the received signal, we obtain the following:  
\[\mathbf{x}=\mathbf{F}_{K}\mathbf{H}_{K}\mathbf{\Psi}_{K}^{\mathrm{H}}\mathbf{ d}+\mathbf{F}_{K}\mathbf{w}. \tag{20}\]By utilizing the property whereby a circulant matrix can be diagonalized by the DFT matrix, two IFFT and FFT operations are inserted in the middle of the above equation:  
\[ \begin{array}{ll}\mathbf{x}&=\mathbf{F}_{K}\mathbf{H}_{K}\mathbf{F}_{K}^{\rm H }\mathbf{F}_{K}\mathbf{F}_{K}^{\rm H}\mathbf{F}_{K}^{\rm H}\mathbf{F}_{K} \mathbf{d}+\mathbf{v}\\ &=\mathbf{\Lambda}_{K}\mathbf{T}_{K}^{\rm H}\mathbf{F}_{K}\mathbf{d}+\mathbf{v},\end{array}  \tag{21}\]  
where \(\mathbf{\Lambda}_{K}=\mathbf{F}_{K}\mathbf{H}_{K}\mathbf{F}_{K}^{\rm H}\) is the channel frequency response, and \(\mathbf{T}_{K}^{\rm H}=\mathbf{F}_{K}\mathbf{F}_{K}^{\rm H}\mathbf{F}_{K}^{\rm H}\) is the coefficient matrix. It follows from the above equation that each subcarrier in the GOCDM modulation scheme achieves full diversity across the frequency band. Given that, for the DFnT matrix [19], \(\mathbf{T}_{K}^{\rm H}=\mathbf{F}_{K}\mathbf{\Phi}_{K}^{\rm H}\mathbf{F}_{K}^{ \rm H}\), the diagonal matrix \(\mathbf{\Gamma}_{K}\) contains the eigenvalues of the DFnT matrix relative to the IDFT matrix, given by the following:  
\[\mathbf{\Gamma}_{K}(k)=e^{-\frac{i}{N}k^{2}}. \tag{22}\]  
To determine the coefficient matrix, the eigenvalues of \(\mathbf{\Psi}_{K}\) relative to the IDFT matrix must be known. Since \(\mathbf{\Phi}_{N}\) is a symmetric matrix, the following property holds for symmetric matrices:  
\[\text{eig}(\mathbf{\Phi}_{N}\otimes\mathbf{I}_{M})=\text{eig}(\mathbf{\Phi}_{ N})\text{eig}(\mathbf{I}_{M})^{\rm T} \tag{23}\]  
Thus, the coefficient matrix \(\mathbf{T}_{K}\) is given by the following:  
\[\mathbf{T}_{K}(k)= \begin{cases}e^{-j\frac{\pi}{N}k^{2}}&k=0,1,\ldots,N-1\\ e^{-j\frac{\pi}{N}\text{mod}(k,N)^{2}}&k=N,N+1,\ldots,NM.\end{cases}