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对长度为L =MK的正交插值尺度滤波器进行了特征化 ,然后给出设计高阶正交插值尺度滤波器的实现方案 ,特别对长度为L =4M的三阶正交插值尺度滤波器进行了参数化 ,并具体构造出二阶和三阶的 3带正交紧支插值尺度函数 .并讨论了M带正交紧支插值小波的性质 ,并由此定性和定量分析Mallat投影的逼近性能 :对光滑信号 ,获得了Mallat投影逼近误差的渐近公式 ;对带限信号 ,给出了误差上界较精确的定量估计 .结果表明 ,Mallat投影与正交投影具有相同的逼近阶 ,特别对于偶数阶的M带正交紧支插值尺度函数 ,两者渐近逼近性能完全相同 .定量结果还表明 ,对给定的尺度函数 ,初始尺度的选取依赖于信号频谱的分布和尺度函数的正则阶 .利用这些结果 ,对给定的信号和尺度函数 ,可根据误差精度的要求确定出初始尺度 ,同时较准确地估计出初始的WST系数而无需进行预滤波 .
The orthogonal interpolation scale filter with length L = MK is characterized, and then the design of high order orthogonal interpolation scale filter is given. Especially for the third order orthogonal interpolation scale filter with length L = 4M The parameters of second-order and third-order orthogonal interpolation with 3-band compensation are constructed, and the properties of orthogonal interpolation with M-band interpolation are discussed, and the approximation of Mallat projection is qualitatively and quantitatively analyzed Performance: The asymptotic formula of the approximation error of the Mallat projection is obtained for the smooth signal, and a more accurate quantitative estimation of the error bound is given for the band-limited signal.The results show that the Mallat projection has the same approximation order as the orthogonal projection, For the even order M-band orthogonal compactly interpolated scaling function, the asymptotic approximation performance of the two is exactly the same.The quantitative results also show that for a given scale function, the selection of the initial scale depends on the distribution of the signal spectrum and the regularity of the scale function Using these results, for the given signal and scale functions, the initial scale can be determined according to the requirements of the error precision, while the initial WST coefficients can be more accurately estimated without pre-filtering .