Sparse Optimization of Vibration Signal by ADMM

Abstract

In this paper, the alternating direction method of multipliers (ADMM) algorithm is applied to the compressed sensing theory to realize the sparse optimization of vibration signal. Solving the basis pursuit problem for minimizing the $L1$ norm minimization under the equality constraints, the sparse matrix obtained by the ADMM algorithm can be reconstructed by inverse sparse orthogonal matrix inversion. This paper analyzes common sparse orthogonal basis on the reconstruction results, that is, discrete Fourier orthogonal basis, discrete cosine orthogonal basis, and discrete wavelet orthogonal basis. In particular, we will show that, from the point of view of central tendency, the discrete cosine orthogonal basis is more suitable, for instance, at the vibration signal data because its error is close to zero. Moreover, using the discrete wavelet transform in signal reconstruction there still are some outliers but the error is unstable. We also use the time complex degree and validity, for the analysis of the advantages and disadvantages of the ADMM algorithm applied to sparse signal optimization. The advantage of this method is that these abnormal values are limited in the control range.