Interrelations between the coefficients of FIR digital differentiators and other FIR filters and a versatile multifunction configuration

Abstract

Interrelationships between seven types of digital FIR filters, viz. the differentiators (DD), Hilbert transformers, half-band lowpass/highpass filters, bandpass/bandstop filters and frequency discriminators, have been established. It has been shown that all the aforementioned filters can be derived from the design of a DD. If we use the optimality criteria of maximal linearity of the frequency response of a differentiator, the proposed relations yield maximally flat/linear frequency responses for the remaining six filters. Similarly, an equiripple relative error design of a DD gives equiripple filters through the use of the suggested interrelations. A versatile, multipurpose, FIR configuration has also been proposed which yeilds optimal frequency response for all the seven types of digital filters. The suggested design is shown to be particularly useful for operation over the frequency ranges 0 ≤ ω ≤ 0.50π(0.5π ≤ ω ≤ π) for lowpass (highpass) filters and 0.25π ≤ ω ≤ 0.75π for the remaining ones. Mathematical formulas for computing the weights of aforementioned filters have also been derived. Possible uses of the suggested design have been given.