Abstract: Hilbert transform is an important tool for signal analysis and processing. Due to the singularity of the Cauchy kernel at the origin, the calculation of the Hilbert transform becomes very difficult. Recently, researchers firstly propose the AFD (Adaptive Fourier Decomposition) method which uses complex analysis to calculate the Hilbert transform. The AFD method adaptively approximates the analytical signal through the linear combination of the parameterized Szegö kernel to obtain original signals' Hilbert transform. Compared with the traditional method, the AFD method can give approximate analytical expressions and thus has a wider application. However, when using the principle of maximum selection to select parameters, the AFD method needs to exhaust all points of the unit disk which is time-consuming. Sparse Bayesian learning has been a hot spot in machine learning research in recent years. The Szegö kernel-based complex sparse Bayesian learning algorithm can provide a sparse rational approximation. A complex sparse Bayesian learning algorithm will be proposed based on the Szegö kernel to calculate the Hilbert transform. This method has the same advantages as the AFD method, and an iterative optimization can be performed without parameter control. The calculation speed of the proposed method is fast. Experimental results show that the proposed method is effective.

Key words: complex sparse Bayesian learning, Szeg? kernel, Hilbert transform, analytic function