Compressed sensing techniques for arbitrary frequency-sparse signals in structural health monitoring

Structural health monitoring requires collection of large number sample data and sometimes high frequent vibration data for detecting the damage of structures. The expensive cost for collecting the data is a big challenge. The recent proposed Compressive Sensing method enables a potentially large reduction in the sampling, and it is a way to meet the challenge. The Compressed Sensing theory requires sparse signal, meaning that the signals can be well-approximated as a linear combination of just a few elements from a known discrete basis or dictionary. The signal of structure vibration can be decomposed into a few sinusoid linear combinations in the DFT domain. Unfortunately, in most cases, the frequencies of decomposed sinusoid are arbitrary in that domain, which may not lie precisely on the discrete DFT basis or dictionary. In this case, the signal will lost its sparsity, and that makes recovery performance degrades significantly. One way to improve the sparsity of the signal is to increase the size of the dictionary, but there exists a tradeoff: the closely-spaced DFT dictionary will increase the coherence between the elements in the dictionary, which in turn decreases recovery performance. In this work we introduce three approaches for arbitrary frequency signals recovery. The first approach is the continuous basis pursuit (CBP), which reconstructs a continuous basis by introducing interpolation steps. The second approach is a semidefinite programming (SDP), which searches the sparest signal on continuous basis without establish any dictionary, enabling a very high recovery precision. The third approach is spectral iterative hard threshold (SIHT), which is based on redundant DFT dictionary and a restricted union-of-subspaces signal model, inhibiting closely spaced sinusoids. The three approaches are studied by numerical simulation. Structure vibration signal is simulated by a finite element model, and compressed measurements of the signal are taken to perform signal recovery. Comparison of the performance of the three approaches is made, and future work on design of compressive sampling testing system for vibration signal is proposed.