# Orthogonal Chirp Division Multiplexing for Underwater Acoustic Communication
## 3 System Model
### Decoding Structure  
The decoding process structure considered in this study is mainly based on the decoder architecture introduced in [14] from which the UWAC Doppler processing and channel decoding block are added. The overall schematic flowchart of the decoder is illustrated in Figure 5. After base-band conversion, frame synchronization is performed by cross-correlation between the received stream and the PN sequence, then motion-induced Doppler shifts are estimated and compensated by resampling and phase correction [12]. After serial-to-parallel conversion and CP removal, for each block \(k\), a vector noted \(\mathbf{r}_{k}\) is obtained. In order to remove the multipath effect brought by the UWAC channel, Frequency Domain Equalization (FDE) is invoked by applying Fast Fourier Transform (FFT) to \(\mathbf{r}_{k}\), leading to the vector \(\mathbf{y}_{k}\), which can be expressed as [14]:  
\[\mathbf{y}_{k}=\mathbf{F}\cdot\mathbf{r}_{k}=\mathbf{\Gamma}^{H}\mathbf{\Lambda F }\mathbf{x}_{k}+\mathbf{w}_{k} \tag{11}\]  
where \(\mathbf{F}\) is the Fourier matrix of size \(N\), \(\mathbf{\Gamma}\) and \(\mathbf{\Lambda}\) are both diagonal matrices of size \(N\times N\) with:  
\[\{\mathbf{\Gamma}\}_{n,n}=e^{-i\frac{\pi}{N}n^{2}} \tag{12}\]  
in the case of \(N\) being even and:  
\[\{\mathbf{\Lambda}\}_{n,n}=H_{n} \tag{13}\]  
where \(H_{n}\) is the Channel Frequency Response (CFR) of the channel at the \(n\)-th frequency bin. Additionally, \(\mathbf{w}_{k}\) is the resulting noise vector assumed Gaussian with zero-mean and variance noted \(\sigma_{w}^{2}\).  
Under the assumption that the CFR is constant during the frame, an estimation of the transmitted vector \(\mathbf{\tilde{x}}_{k}\) can be formed by compensating the CFR and the phase rotation induced by matrix \(\mathbf{\Gamma}\) and then transferred to the time domain via inverse FFT as follows:  
\[\mathbf{\tilde{x}}_{k}=\mathbf{F}^{H}\mathbf{G}\mathbf{I}\mathbf{y}_{k} \tag{14}\]where \(\mathbf{G}\) is a diagonal equalization matrix optimized under the Minimum Mean Square Error (MMSE) criterion such that:  
\[\mathbf{G}=\boldsymbol{\Lambda}^{H}\big{(}\boldsymbol{\Lambda}^{H}\boldsymbol{ \Lambda}+\sigma_{w}^{2}\mathbf{I}\big{)}^{-1} \tag{15}\]  
In practice, the equalization matrix is computed from an estimation of CFR obtained from the pilot symbol with the conventional Least Square (LS) algorithm.  
Then, in order to compensate the residual Doppler shift within the equalized signal, a fine phase correction stage is applied as:  
\[\hat{x}_{n}^{k}=e^{-\hat{\psi}_{d}}\hat{x}_{n}^{k} \tag{16}\]  
The estimation of the Doppler phase shift \(\hat{\phi}_{d}\) is initialized to zero at the beginning of each frame and updated for each OCDM block by measuring the average phase rotation between previous estimates and the closest constellation points:  
\[\hat{\phi}_{d}=\hat{\phi}_{d}-\frac{1}{N}\sum_{n=1}^{N}\arg\left((\hat{x}_{n}^ {k-1})^{*}\cdot\text{Dec}\left(\hat{x}_{n}^{k-1}\right)\right) \tag{17}\]  
where \(\text{Dec}(.)\) refers to the decision process choosing the closest complex cell from the QAM constellation. Finally, from estimated QAM data, Logarithm Likelihood Ratio (LLR) is formed for each bit, then the data are de-interleaved and finally fed to a Viterbi-based channel decoder providing an estimation of the data bits.  
When considering system complexity, it can be shown that IDFnT, in the transmission side, can be efficiently performed by using an Inverse Fast Fourier Transform (IFFT) yielding a complexity increase of only two phase multiplications per symbol with respect to traditional OFDM [14]. At the reception side, an additional IFFT is required compared to OFDM that brings an additional complexity of \(0.5\log_{2}N\) per chirp/sub-carrier. By combining the transmitter and the receiver, the total complexity increase is \(2+0.5\log_{2}N\) multiplications per symbol, representing a slight increase with respect to OFDM.  
Figure 5: Decoder structure.