Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising
Weighted singular value decomposition basis of Szegő kernel and its applications to signal reconstruction and denoising
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DOI：
10.1016/j.cam.2023.115067
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发表时间：
2023-01
期刊：
影响因子：
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通讯作者：
Wen Xu;L. Tan;Rongrong Lin
中科院分区：
文献类型：
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作者：
Wen Xu;L. Tan;Rongrong Lin
对于任意循环解析信号f+(E I T)∈H2(T)，如果单位圆盘D中的序列{a k}k=0∞满足双曲不可分条件∑k=0∞(1−|a k|)=∞，则已知基于竹中-Malmquist系统(S N+1f+)(E I T)=∑k=0 N<；f+(E I T)，Bk(E I T)>的第一个N+1项近似函数；B k(E It)，可以逼近L 2范数意义下的解析信号f+(E It)，其中B k(E It)=1−|a k|2 1−a k‘e t∏L=0 k−1 e t−a L 1−a L e it，k∈N是Takenaka-Malmquist系统，可视为傅立叶级数的推广.我们发现(S N+1f+)(Ett)也可以用标准基、拉格朗日基和加权奇异值分解基来表示。不同的碱基有不同的优点。标准基数很简单。拉格朗日基形式简洁，便于理论分析。对于Takenaka-Malmquist系统，第一个N+1项近似函数(S N+1f+)(e，t)能够以迭代形式写出。Takenaka-Malmquist系统具有继承性的优点，并且可以通过贪婪原则容易地逐步选择点ak∈D。在本文中，我们主要关注加权奇异值分解基。给出了加权奇异值分解基的一些正交性。证明了任意实数信号f(e，i，t)∈L 2(T)都可以在给定的误差范围内，通过加权奇异值分解得到精确的重构。实验结果表明，基于加权SVD基的算法具有更高的稳定性和准确性。实验结果表明，与经典傅立叶分解和自适应傅立叶分解算法相比，加权SVD基分解对噪声的敏感性更高。这将为信号去噪提供一种新的方法。
For any circular analytic signal f+(e i t)∈ H 2 (T), if the sequence {a k} k= 0∞ in the unit disc D satisfies the hyperbolic non-separability condition∑ k= 0∞(1−| a k|)=∞, it is known that the first N+ 1 term approximation function based on the Takenaka–Malmquist system (S N+ 1 f+)(e i t)=∑ k= 0 N< f+(e i t), B k (e i t)> B k (e i t), can approximate the analytic signal f+(e i t) in the L 2-norm sense, where B k (e i t)= 1−| a k| 2 1− a k¯ e i t∏ l= 0 k− 1 e i t− a l 1− a l¯ e i t, k∈ N is the Takenaka–Malmquist system which can be seen as a generalization of Fourier series. We find that (S N+ 1 f+)(e i t) 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 (S N+ 1 f+)(e i t) is able to write in an iterative form. The Takenaka–Malmquist system has the advantage of inheritance, and the point a k∈ 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 (e i t)∈ L 2 (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.