# Vector Approximate Message Passing Based OFDM-IM Detection for Underwater Acoustic Communications
###### Abstract  
Orthogonal frequency division multiplexing with index modulation (OFDM-IM) has great potential for the implementation of high spectral-efficiency underwater acoustic (UWA) communications. However, general receivers consisting of the optimal maximum likelihood detection suffer from high computational load, which prohibits real-time data transmissions in underwater scenarios. In this paper, we propose a detection based on a vector approximate message passing (VAMP) algorithm for UWA OFDM-IM communications. Firstly, a VAMP framework with a non-loopy factor graph for index detection is formulated. Secondly, by utilizing the sparsity inherently existing in OFDM-IM symbols, a novel shrinkage function is derived based on the minimum mean square error criterion, which guarantees better posterior estimation. To reduce the errors from estimated non-existing indices, one trick is utilized to search the elements from the look-up table with the minimal Euclidean distance for the replacement of erroneously estimated indices. Experiments verify the advantages of the proposed detector in terms of low complexity, robustness and effectiveness compared with the state-of-art benchmarks. underwater acoustic communications; OFDM-IM; data detection; vector approximate message passing +
Footnote †: journal: _Entropy_  
0000-0001-999-4000-0000-0000-0000-0000-0000-0000-0000-0000-0  
activation patterns (SAPs), which determine the activated or inactivated subcarriers. In OFDM-IM, additional information requires special handling because the conventional detection for constellation symbols is only part of data detection modules. The detection of SAPs is the prerequisite for constellation data detection, which is the key to avoiding decoding error propagation. For OFDM-IM detection, maximum likelihood (ML) detection is the best, which searches the optimal indices and constellation points among all combinations. The complexity of ML detection increases exponentially with the number of index combinations as well as the order of constellation modulations. Even the searches have been employed for each subblock, the complexity is heavy, especially for the multi-carrier systems with large-number subcarriers and high-order constellation modulations. A log-likelihood (LLR) detector can achieve near-optimal ML performance with reduced complexity [1]. If the ratio of activated subcarriers is low, the LLR detector tends to find nonexistent index combinations with high probability. Subsequently, many sub-optimal detectors are developed [2; 3; 4]. The work in [3] proposes a low-complexity detector that conveys data using all possible SAPs to avoid errors from invalid SAPs. The mapping of non-fixed length bits makes the communication system complicated. The work in [4] proposes a modified k-largest-value (klv) detector, which chooses \(k\) active indices with the largest values according to the active likelihood metrics. This method provides a method for dealing with illegal SAPs and operates on each subcarrier. As research deepens, the inherent characteristics of OFDM-IM symbols from IM concept and SAPs are further exploited. SAPs are constructed by zero subcarriers and activated subcarriers, which makes OFDM-IM symbols present inherent sparsity characteristics. The sparsity leverages powerful tools to solve the detection problem. The paper [5] proposes to interpret data detection as a problem of convex optimization, on which the AP constraints are imposed, and then a semi-definite relaxation method is utilized. The convergence speed of this method is severely limited and the error floor exists in high SNR ranges. Exploiting the sparsity of symbols, compressed sensing (CS) based methods have recently emerged. The work [6] proposes a CS-assisted signaling strategy, based on which an iterative residual check (IRC)-based detector is formulated. An AMP-based detector is proposed for OFDM-IM by exploiting the statistic of OFDM-IM symbols in the frequency domain [7].