A new closed form method for design of variable bandwidth linear phase FIR filter using different polynomials

In this paper, a new method for the design of variable bandwidth linear-phase finite impulse response (FIR) filters using different polynomials such as shifted Chebyshev polynomials, Bernstein polynomials and shifted Legendre polynomials is proposed. For this purpose, the transfer function of a variable bandwidth filter, which is a linear combination of fixed-coefficient linear-phase filters and the above polynomials are separately exploited as tuning parameters to control bandwidth of the filter. In order to determine the filter coefficients, mean squared difference between the desired variable bandwidth filter and the practical filter is minimized by differentiating it with respect to its coefficients leading to a system of linear equations. The matrix elements can be expressed in form of Toeplitz-plus-Hankel matrix, which reduces the computational complexity. Several examples are included to demonstrate effectiveness of the proposed method in terms of passband error (ep), stopband error (es) and stopband attenuation (As).