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Alternative TitleWavelet Filter Banks Construction and Approximation
Thesis Advisor彭思龙
Degree Grantor中国科学院研究生院
Place of Conferral中国科学院自动化研究所
Degree Discipline控制理论与控制工程
Keyword小波滤波器 Cayley变换 仿酉矩阵 线性相位 Wavelet Filter Cayley Transform Paraunitary Matrix Linear Phase
Abstract不可分离二维小波(滤波器)由于有设计上的更多自由度和更好的频率可选择性,成为当前小波理论及应用领域的热点。尽管目前已有了一些二维不可分离小波滤波器构造的方法,但在实际应用中,我们仍需要构造小波滤波器以满足不同情形下的需要以提高滤波的效果。 本文在第二章首先研究了含参变量的仿酉矩阵。证明了一类对称的仿酉矩阵一定能够块对角化,给出了一类任意阶紧支撑二元正交小波滤波器组的构造问题;同时我们也刻画了 二元一次中心对称仿酉矩阵的通解,构造了具有线性相位的低通滤波器。本文第三章提出了用Cayley变换的方法构造正交滤波器组的方法,从而将仿酉矩阵的构造简化为斜的仿厄米特矩阵的构造。通过对斜的仿厄米特矩阵的研究,给出了滤波器组的构造及 滤波器组的通解。在第四章本文提出了一种小波滤波器逼近的方法,我们采用分步优化的方法来降低滤波器逼近的复杂度,我们能保证每一步是最优的。第五章给出了本文的总结和展望。
Other AbstractNonseparable bivariate wavelet (filter bank) are designed in more freedom and better frequency selectivity. Therefore, nonseparable bivariate wavelet is the focus of today’s wavelet theory and application field. Although there are some way of constructing bivariate nonseparable wavelet filter bank, In application, to improve the effect of filtering, we still need to construct wavelet filter to meet some special requirements. In the second chapter of paper, we study paraunitary matrices with bivariate at first, we prove that a class of bivariate symmetric paraunitary matrices are able to characterized by block diagonal matrix. Using this property, we construct non-separable bivariate compactly orthogonal wavelet filter banks. At the same time, the general solution of central-symmetric paraunitary matrix (each element is a bivariate polynomial of order one) with support being 4 times 4 is given and and also construct lowpass filters with linear phase. In the third chapter of paper, we propose a novel design method for orthogonal filter banks using the cayley transform, so constructing a paraunitary matrix is simplified constructing a para-skew-hermitian matrix. In the chapter, A para-skew-hermitian matrices are studied at first, we present a class of orthogonal filter banks and give complete solutions of filter banks with support being 2 times 2. In the forth chapter of paper, we present a orthogonal wavelet filters approximating method. to decrease complexity , our optimal method is step by step. We can prove that each step is optimal. The last chapter summarizes the previous four chapters.At the same time it is pointed out that there are many attentive questions in the construction for wavelets and approximation.
Document Type学位论文
Recommended Citation
GB/T 7714
李林杉. 小波滤波器的构造与逼近[D]. 中国科学院自动化研究所. 中国科学院研究生院,2006.
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