Sub-Sampling Rate Acquisition Based on Hilbert Semi-Tensor Compressed Sensing

‍ ‍To address the contradiction between data redundancy and limited wireless resources caused by high sampling rates in wireless distributed transient pressure testing， a compressive sensing method is proposed to realize compressed sampling of redundant data at the encoding end. However， storing high-dimensional compressed sensing observation matrices at each node poses new challenges for the limited resources of wireless nodes. Compressive sensing technology based on the semi-tensor product utilizes semi-tensor theory at the encoding end to overcome the limitation of matrix multiplication dimensions， drastically reducing the dimension of observation matrices. However， some essential signal information may be lost， and the multiple priority may be reduced. This study proposes a compressive sensing method based on the Hilbert semi-tensor， utilizing the orthogonal space of Hilbert and Fourier transform to approximate the energy of shock wave signals， enhancing the incoherence between sparse representation and observation matrices， thus reducing observation loss due to semi-tensor product operations. Furthermore， an optimal atom selection strategy without prior information is proposed in the reconstruction algorithm， penalizing the “energy” of transformed data using energy regularization to improve the accuracy of atom support set selection. Finally， a variable step update strategy is proposed to dynamically adjust the step size when updating the support set in the reconstruction algorithm， reducing the time for atom selection and improving operational efficiency. The analysis of the simulation results of multi-range measured muzzle shock wave signals revealed that the proposed method can achieve a lower sampling rate compared to Nyquist sampling， reducing the total amount of data and ensuring real-time communication. Furthermore， compared to traditional compressive sensing technology， high-precision reconstruction at the decoding end can still be guaranteed， even when the dimension of the observation matrix is reduced by half， with reconstruction errors below 1e-6 and a reduced reconstruction time of approximately 87%. Additionally， the proposed method can be applied to high-dimensional signal acquisition in distributed wireless transmission systems， effectively addressing the contradiction between redundant data and limited network resources.