# Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications
###### Abstract
The work [8] proposes to use the AMP framework to realize iterative channel estimation and data detection for OFDM-IM. Approximate Message Passing (AMP) algorithm belongs to the Bayesian estimation framework [9; 10]. The simplifications of the Gaussian message according to the central limit theorem and Taylor expansions make the IM detector efficient depending on several posterior parameters. One constraint of the AMP method is the sensitivity to the non-Gaussianity of the dictionary matrix. The generalized AMP (GAMP) is incorporated as a step for joint phase-noise Estimation and decoding in OFDM-IM [11]. The performance of these OFDM-IM detectors for UWA communications cannot be concluded and is to be verified and analyzed. OFDM-IM was first introduced to UWA communications in [12]. The work verifies the reduced PAPR effects due to power averaging by the inactivated subcarriers and better BER performance compared with OFDM. Since then, OFDM-IM-based UWA communications have been developing. Combining the IM with advanced OFDM schemes, many UWA OFDM-IM transceivers are proposed such as the fully quadrature subcarrier-index shift keying OFDM (RQSISK-OFDM) modulation scheme [13], which put their emphasis on the design of activation strategy. The work [14] combines the IM concept with Orthogonal Time Frequency Space (OTFS) and proposes a Hamming distance optimized model to modify the index combinations. Currently, there is a lack of a design scheme for a low-complexity OFDM-IM detector for UWA communications and a detailed analysis of the effects from the UWA channel for UWA OFDM-IM systems. To fill this gap, this paper considers the OFDM-IM receiver design and especially focuses on data detection, which is a major challenge for real-time and high SE data transmission schemes in the UWA physical layer. In this paper, we involve the data detection for OFDM-IM in the Bayesian estimation framework. Instead of a loopy factor graph (FG), a vector approximate message passing (VAMP) detector based on a scalar FG is proposed. An appropriate statistical prior model is beneficial for the achievement of the optimal Bayesian solution. By exploiting the sparsity of symbols in the frequency domain, the SAP constraint is considered using the statistical prior and a novel minimum mean square error (MMSE)-optimal shrinkage function is derived. The data reconstruction performance is improved through the forward-back message passing scheduling, where the involved statistical parameters are learned automatically. Inherently from the robustness of VAMP for the deviations of Gaussian matrices [15], the proposed detector is less sensitive to the non-Gaussianity of the measurement matrix which is composed of unknown channel components and presents good generality and convergence. Aiming to mitigate the possible invalid SAPs, a modified trick using the criterion of minimal Euclidean distance with the space of a look-up table is used to replace the possible non-existent results. The VAMP-based detection has relatively low complexity, which is advantageous for real-time data transmission. Simulation results verify the proposed receiver outperforms the benchmarks in terms of complexity, BER as well as robustness to the time-varying UWA channel. The rest of this paper is organized as follows. Section 2 introduces the UWA OFDM-IM communication systems. The proposed VAMP-based detector is presented in Section 3. The computation complexity and numerical experiments are analyzed in Section 4. Section 5 makes the conclusive remarks. Notations: Lower case boldface letter \(\mathbf{x}\) and upper case boldface letter \(\mathbf{X}\) denote vector and matrix, respectively. \((\cdot)^{T}\) and \((\cdot)^{H}\) stand for the transpose and conjugate transpose operation. \(\text{diag}\{\mathbf{x}\}\) is the diagonalized operation with \(\mathbf{x}\) as the diagonal element. \(\mathbb{E}\{\cdot\}\) and \(\text{var}\{\cdot\}\) denote the expectation and variance operation. \(\lfloor\cdot\rfloor\) denotes the floor operation.