Least squares FIR filter design using frequency domain piecewise polynomial approximations

A general framework for specifying and designing one-dimensional FIR digital filters is presented. The system uses piecewise polynomials to guarantee that the ideal impulse response maintains continuity up to a given number of derivatives (given as parameter p). In this system, it is possible to efficiently design optimal (in the least-squares sense) digital filters that converge pointwise to any (continuous) desired frequency response, while the filters' coefficients decay in the order of 1/n
^p+1
. It is shown experimentally that the maximum absolute error also decays as 1/n
^p+1
. Results on (i) bandpass filter design and (ii) bandpass differentiating filter design are shown.