Improved filter sharpening

We propose a natural extension to Kaiser-Hamming (1977) filter sharpening methods to allow for a piecewise linear desired amplitude change function (ACF). The primary advantages of the proposed ACF over piecewise constant ACFs is that we obtain better control of selective improvement (or degradation) in either the passband or stopband or both, and we are not restricted to applying our methods to filters with piecewise constant pass and stopbands, since linear segments of slope 1 can be used to retain existing filter performance in either passband or stopband. The proposed ACF approximating polynomial (AP) is easy to compute, may be constrained to have simple (or integer) coefficients, and may be expressed as the AP of Kaiser and Hamming plus a correction polynomial. We also provide applications for motivation.