Abstract: Non-contact measurement of rotating blades using blade tip timing technology is recognized as an effective method for blade vibration monitoring. However, due to the limitation on the number of sensors that can be arranged in the casing, the vibration signal acquired by the blade tip timing system has a serious under-sampling problem. The current convex relaxation and non-convex relaxation methods do not take into account the important scale invariance of $\ell_0$ norm, which limits the reconstruction accuracy and makes it impossible to obtain the sparsest solution. To solve this problem, a scale-invariant norm ratio function is first introduced to approximate $\ell_0$ norm. For its non-convex and non-smooth properties, a smooth approximation proxy function(Smooth $\ell_p$ over $\ell_q$,SPOQ) is constructed to avoid falling into the local minimum. Then, a sparse representation model of blade tip timing signals is constructed based on the SPOQ function, and the model is solved using a variable metric forward–backward algorithm. The results of blade tip timing simulation and test show that, by comparing with $\ell_1$ regularization and GMC regularization, this method can not only accurately reconstruct the signal amplitude, but also effectively filter out the interference frequency components, making the solution sparser.

Key words: blade tip timing, undersampling, sparse reconstruction, scale-invariant