# Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications
## 2 System Model
After passing through the channel, the received signal is given by  
\[\hat{g}(t)=\sum_{l=1}^{L}\alpha_{l}\bar{x}((1+a)t-\tau_{l})+\tilde{\sigma}(t), \tag{6}\]  
where \(\tilde{\sigma}(t)\) is the passband additive white Gaussian noise (AWGN). Due to the Doppler effects, the received signal suffers from compressing and broadening effects. The scalar coefficient \(a\) is estimated by the resampling method as \(\hat{a}=\frac{\hat{T}_{\text{rx}}}{\hat{T}_{\text{rx}}}-1\), where \(\hat{T}_{\text{rx}}\) is the length of the transmitted signal and the length of the received signal \(\hat{T}_{\text{rx}}\) is calculated by cross-correlating the linear frequency modulated (LFM) preambles of neighboring frames. After the resampling and downshifting operation, the baseband received signal is given by  
Figure 1: OFDM-IM system for UWA communications. \[y(t)=LP\Big{[}\mathcal{Y}(t/(1+\hat{a}))e^{-j2\pi f_{\ell}t}\Big{]} \tag{7}\] \[\approx e^{j2\pi\epsilon t}\sum_{m}\Big{\{}x[m]e^{j2\pi m\Delta f _{t}}\Big{[}\sum_{l}\alpha_{l}e^{-j2\pi f_{m}\eta}g(t-\tau_{l})\Big{]}\Big{\}}+v (t),\]  
where \(v(t)\) denotes the baseband AWGN and \(\epsilon=\frac{q-\hat{a}}{1+\hat{a}}f_{\ell}\) is the residual carrier frequency offset (CFO). To discretize the baseband signal \(y(t)\) at the baseband rate \(1/B\), then the \(p\)th received sample is given by \(y[p]=e^{\frac{2\pi\epsilon p}{B}}\sum_{m}\Big{\{}x[m]e^{\frac{2\pi\epsilon p }{N}}\sum_{l}\alpha_{l}e^{-j2\pi f_{m}\eta}g(p-l)\Big{\}}+v[p]\). Through expansion and union simplification operation, the received samples are denoted as  
\[y[p]=e^{\frac{2\pi\epsilon p}{B}}\sum_{l}h_{l}\sum_{m}x[m]e^{\frac{2\pi(p-l)}{ N}}g[p-l]+v[p], \tag{8}\]  
where \(h_{l}=\sum_{m=-N/2}^{K/2-1}C(f_{m})e^{\frac{2 ml}{N}}\) and \(C(f)=\sum_{p}\alpha_{p}e^{-j2\pi f/\tau_{p}}\) is the channel frequency response. Based on Formula (8), CFO matrix is defined as \(\mathbf{\
abla}(\epsilon)=\text{diag}\Big{(}1,e^{\frac{2\pi\epsilon}{B}},\cdots,e ^{\frac{(2\pi\epsilon(N-1)}{B}}\Big{)}\). Part of inactive subcarriers is used for CFO mitigation. The process is operated in the frequency domain based on the goal of minimizing the leaked energy of the zero subcarriers after the signal passes through the channel. After CP removal and FFT transform, the received signal in the frequency domain is denoted as \(\mathbf{Y}=\mathbf{Fy}\), where \(\mathbf{F}\) is the Discrete Fourier Transform (DFT) matrix and \(\mathbf{y}\) is the received signal in the time domain. The selection matrix \(\mathbf{\Xi}\) extracts zero subcarriers from \(\mathbf{Y}\), and the energy is expressed as \(\|\mathbf{\Xi}\mathbf{Fy}\|_{2}^{2}\). Assuming CFO is perfectly known and compensated, the optimization problem is formulated as  
\[\hat{\epsilon}=\arg\min_{\epsilon}\|\mathbf{\Xi}\mathbf{F}\mathbf{Y}(\epsilon) \mathbf{y}\|_{2}^{2}. \tag{9}\]  
One dimensional (1-D) search method is used to solve the problem (9) [17]. The detailed effects of CFOs are analyzed in the following experiments. With \(\hat{\epsilon}\), the received signal components are corrected through phase reversal. Then the inter-carrier interference (ICI) is mitigated and the ICI-free signal is obtained. Channel state information (CSI) \(\mathbf{h}\) is obtained through the channel estimation module. \(N_{p}\) pilot subcarriers are utilized as input, and the received pilot subcarriers are the output, then the channel estimation problem can be formed as a sparse signal recovery (SSR) problem. To design efficient channel estimation, representative CS-based methods can be involved. Then frequency-domain channel \(\mathbf{H}\) is fed to the data detection module. For the detection of OFDM-IM, symbol detection includes index detection and constellation symbol detection.