New design of FIR digital differentiators having maximally linearity at middle of the frequency band

Maximal linear FIR digital differentiators are preferred over others, like minimax designs, for narrow band applications. Most of the existing maximally linear designs achieve maximal linearity at zero frequency, and relatively lower attention has been given to the designs accurate in mid and higher frequency bands. We present a simple design accurate for midband frequencies, by forcing the maximal linearity constraints at half of the Nyquist frequency. The design problem is formulated as the solution of a system of linear equations, obtained by imposing maximal linearity constraints to the general frequency response of the filter. Certain special characteristics of the determinant of the coefficients matrix of these equations are explored and used in derivation of the explicit formulas for the impulse response coefficients.