三角样条小波(TSWn)构造方法

CONSTRUCTION METHOD OF TRIGONOMETRIC SPLINE WAVELETS

该文首先分析了三角样条函数(TSn)的性质,得到了三角样条函数具有与B样条函数(nM)相同的性质,如相同的图像、同阶的光滑性、相同的局部支撑区间、对称性以及全正性等。在此分析基础上,提出了一种新的样条小波—三角样条小波(TSWn,n=1,2,3,?构造方法。三角样条小波对偶数n是对称的;对于奇数n是反对称的,因而具有线性相位或广义线性相位。它具有良好的时频局部性,时频窗口面积接近“测不准定理”中的下限0.5与B样条小波一样具有最 小支撑。由于三角样条小波由三角函数张成,基于它的小波变换又具有Fourier变换的特点。另外,文章的构造方法对分段样条函数构造小波函数具有通用性。

Analyzing the properties of trigonometric spline functions (TSn), it is revealed that some features properties of TSn is similar to that in B spline functions, e.g., the sameimages, the smoothness for same order, same local compactsupport interval, symmetry, etc. On this basis a new method to construct a new spline wavelet, i.e., the trigonometric spline wavelets (TSWn, n=1,2,3,...), is put forward. The TSWn will be symmetric when n is even and be anti-symmetric when n is odd, so the trigonometric spline wavelets possess linear phases or generalized linear phases, good time-frequency locality.When n increases, its area of time-frequency windows isclose to 0.5 that is the optimal value in uncertainty principle and the trigonometric spline wavelets exist minimum compact support as well as B spline wavelets. Because the trigonometric spline wavelets are spanned by trigonometric functions, so the wavelet transform based on TSWn contains some characters of Fourier Transformation (FT). In addition, the proposedconstruction method is a versatile one for constructing thewavelet functions by piecewise spline functions.