Matrix-Based Algorithms for Constrained Least-Squares and Minimax Designs of 2-D Linear-Phase FIR Filters

The impulse response coefficients of a two-dimensional (2-D) finite impulse response (FIR) filter are in a matrix form in nature. Conventional optimal design algorithms rearrange the filter's coefficient matrix into a vector and then solve for the coefficient vector using design algorithms for one-dimensional (1-D) FIR filters. Some recent design algorithms have exploited the matrix nature of the 2-D filter's coefficients but not incorporated with any constraints, and thus are not applicable to the design of 2-D filters with explicit magnitude constraints. In this paper, we develop some efficient algorithms exploiting the coefficients' matrix nature for the constrained least-squares (CLS) and minimax designs of quadrantally symmetric 2-D linear-phase FIR filters, both of which can be formulated as an optimization problem or converted into a sequence of subproblems with a positive-definite quadratic cost and a finite number of linear constraints expressed in terms of the filter's coefficient matrix. Design examples and comparisons with several existing algorithms demonstrate the effectiveness and efficiency of the proposed algorithms.