Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising

Abstract

For any circular analytic signal f+(eit)∈H2(T), if the sequence {ak}k=0∞ in the unit disc D satisfies the hyperbolic non-separability condition ∑k=0∞(1−|ak|)=∞, it is known that the first N+1 term approximation function based on the Takenaka–Malmquist system (SN+1f+)(eit)=∑k=0N〈f+(eit),Bk(eit)〉Bk(eit),can approximate the analytic signal f+(eit) in the L2-norm sense, where Bk(eit)=1−|ak|21−ak¯eit∏l=0k−1eit−al1−al¯eit, k∈N is the Takenaka–Malmquist system which can be seen as a generalization of Fourier series. We find that (SN+1f+)(eit) can also be represented by the standard basis, Lagrange basis and weighted SVD basis. Different bases have different merits. The standard basis is simple. The Lagrange basis has a neat and compact form, which is very convenient for theoretical analysis. With the Takenaka–Malmquist system, the first N+1 term approximation function (SN+1f+)(eit) is able to write in an iterative form. The Takenaka–Malmquist system has the advantage of inheritance, and the point ak∈D can be easily selected step by step through the greedy principle. In this paper, we focus our attention on the weighted SVD basis. We give some orthogonal properties of the weighted SVD basis. It demonstrates that any real signal f(eit)∈L2(T) can be precisely reconstructed within the given error range by the weighted SVD basis decomposition. The experimental results reveal that the algorithm based on weighted SVD basis is more stable and accurate. Furthermore, compared with classical Fourier decomposition and adaptive Fourier decomposition algorithms, the experimental results show that the weighted SVD basis decomposition is more insensitive to noise. This will provide a new method for signal denoising.