二维FIR数字滤波器优化设计的二维优化算法研究

项目名称： 二维FIR数字滤波器优化设计的二维优化算法研究

项目编号： No.61304142

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 自动化技术、计算机技术

项目作者： 赵瑞杰

作者单位： 山东大学

项目金额： 24万元

中文摘要： 随着多维数字信号处理技术的快速发展，二维FIR数字滤波器被广泛用于图像处理，声纳信号处理，雷达信号处理，地震波信号处理等诸多方面。二维FIR数字滤波器优化设计的主要挑战是其高度的计算复杂性。针对此问题，本项目将充分利用二维FIR滤波器参数的矩阵形式，建立基于矩阵变量的二维FIR滤波器优化设计模型，推导出最优性条件，并以矩阵方程组形式表达。根据最优性条件，提出求解加权最小二乘优化设计的一种矩阵迭代算法和一种广义共轭梯度算法，和求解最小lp范数优化设计的一种迭代重加权最小二乘算法。所提出的算法中保持二维滤波器参数的原始矩阵形式，故称之为二维优化算法。这种结构将大大减小设计所需计算机内存，提高算法的计算效率，降低计算复杂性。所提出的算法将在收敛速度，计算精度及数值稳定性方面较现有算法有很大提高。最后，运用矩阵分析理论、线性算子谱理论和Hilbert内积空间理论证明所得算法的收敛性。

中文关键词： 二维FIR滤波器；线性相位；加权最小二乘设计；最小lp范数设计；二维算法

英文摘要： With the rapid development of multidimensional signal processing techniques, two-dimensional (2-D) FIR digital filters have been widely applied in image processing, sonar and radar signal processing, geophysical signal processing and so on. High compuational complexity is the major challenge encountered in the optimal design of 2-D FIR filters. For overcoming this difficult, taking advantage of the matrix form of the coefficients of 2-D filters, we would establish the optimization design models of 2-D FIR filters in matrix varibles and deduce their optimality conditions formed as matrix equations. Based on the optimality conditions, a matrix iterative algorithm and a generalized conjugate gradient algorihtm would be proposed for the weighted least squares design, and an iterative reweighted least squares algorithm would also be developed for the least lp-norm design. Those proposed algorithms that retain the natural matrix form of the 2-D fitler's coefficients are called two-dimension based algorithms and would achieve considerable savings in the memory space and the amount of computation required. Compared with existing methods, the proposed algorithms could provide a significant improvement in terms of the convergence rate, computational precision and numerical stability. Finally, convergences of those propos

英文关键词： Two-dimensional FIR filter；linear phase；weighted least squares design；least lp-norm design；two-dimension based algorithm