This paper considers the minimax design of two-dimensional (2D) finite impulse response (FIR) half-band filters. First, the design problem is formulated in a matrix form, where the half-band constraints are expressed as a pair of matrix equations. By matrix transformations, the constrained minimax problem is transformed into an unconstrained one. Then, we propose an efficient iterative reweighted least squares (IRLS) algorithm to solve this problem. The weighted least squares (WLS) subproblems arising from the IRLS algorithm are solved using a generalized conjugate gradient (GCG) algorithm. Moreover, the GCG algorithm is guaranteed to converge in a finite number of iterations. In the proposed algorithm, the design coefficients of filters are solved in their matrix form, leading to a great saving in computations and memory space. Design examples and comparisons with existing methods are provided to demonstrate the effectiveness and efficiency of the proposed algorithm.

中文翻译：

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使用基于矩阵的高效 IRLS 算法的 2D FIR 半带滤波器的 Minimax 设计

本文考虑了二维 (2D) 有限脉冲响应 (FIR) 半带滤波器的极小极大设计。首先，设计问题以矩阵形式表示，其中半带约束表示为一对矩阵方程。通过矩阵变换，将受约束的极大极小问题转换为无约束问题。然后，我们提出了一种有效的迭代重加权最小二乘（IRLS）算法来解决这个问题。由 IRLS 算法产生的加权最小二乘 (WLS) 子问题使用广义共轭梯度 (GCG) 算法求解。此外，保证 GCG 算法在有限次数的迭代中收敛。在所提出的算法中，滤波器的设计系数以矩阵形式求解，从而大大节省了计算量和存储空间。