An efficient closed-form approach to the design of linear-phase FIR digital filters with variable-bandwidth characteristics

This paper deals with the design of variable-bandwidth linear-phase FIR digital filters. Such filters are implemented as a linear combination of fixed-coefficient linear-phase filters and the variable bandwidth characteristics are provided by a tuning parameter embedded in the filter structure. These filters are designed in a least-square sense by formulating an error function reflecting the difference between the desired variable bandwidth filter and the practical filter represented as a linear combination of fixed-coefficient filters in a quadratic form. The filter coefficients are obtained by solving a system of linear equations comprising of a block-symmetric positive-definite matrix in which each block is a Toeplitz-plus-Hankel matrix. Consequently, a significant reduction in computational complexity can be achieved in obtaining the entries of this matrix. Moreover, closed-form expressions are provided for both the block-symmetric matrix as well as the vector involved in the system of linear equations.