Weighted least squares design of 2-D FIR filters using a matrix-based generalized conjugate gradient method

One of the important issues in the weighted least-squares (WLS) design of two-dimensional (2-D) finite impulse response (FIR) filters is the computational complexity of the design algorithms. This paper presents a matrix formulation of the design problem and derives a matrix-based generalized conjugate gradient (GCG) algorithm. By defining a linear operator acting on the coefficient matrix of the filter, the optimality condition of the design problem is expressed as a linear operator equation. By proving the boundedness, self-adjointness and positive-definiteness of the linear operator in the Hilbert space of the coefficient matrix, a GCG algorithm designated for the Hilbert matrix space is used to solve the linear operator equation. The convergence of the proposed algorithm in finite steps is established and its high computational efficiency is demonstrated by design examples and comparison with some existing methods.