基于Nyquist采样的椭圆球面波函数数值解法

Numerical Solution of Prolate Spheroidal Wave Functions Based on Nyquist's Sampling

为了解决椭圆球面波函数数值解法时积分方程该如何离散化以及采用何种方法求取实对称矩阵的特征值和特征向量问题,提出了基于奈奎斯特采样的椭圆球面波函数数值解法。该数值解法利用奈奎斯特采样定理确定的采样频率对积分方程进行离散化,利用Jacobi方法求取实对称矩阵的全部特征值和相应的特征向量,求得的特征向量就是椭圆球面波函数的近似数值解。对基于奈奎斯特采样的椭圆球面波函数数值解法进行了理论推导、性能分析和仿真。理论分析和仿真结果表明,该数值解法方法简单,实用性强,求得的椭圆球面波函数精度高,椭圆球面波函数之间正交性好。

In order to solve the problems that how to discretize integral equation,and with which methods to obtain eigenvalues and eigenvectors of real symmetric matrix during numerical solution process of Prolate Spheroidal Wave Functions（PSWF）,a numerical solution of PSWF based on Nyquist＇ sampling is proposed.The method discretizes integral equation by making use of sampling frequency which is defined by Nyquist＇ sampling theorem.All the eigenvalues and corresponding eigenvectors of real symmetric matrix are obtained by use of Jacobi＇s method.The eigenvector is just the approximate numerical solution of PSWF.Theoretical deduction,performance analysis and simulation are conducted for numerical solution of PSWF.Theoretical analysis and simulation results show that the method is simple and applicable,the achieved PSWF are with high precision and the orthogonality between PSWF is good.