On digital differentiators, Hilbert transformers, and half-band low-pass filters

Abstract

Interrelationships between the digital differentiator (DD), the digital Hilbert transformer (DHT), and the half-band low-pass (½-LPF) are brought out. A number of important properties confirming the close proximity of these filters are highlighted. Theoretical results are substantiated by transforming minimax relative error DDs to equiripple DHTs and equiripple ½-LPFs. Relations connecting their impulse responses and their frequency responses are brought out. The precise frequencies of ripple extrema and the magnitudes of their peaks are shown to be simple related to the corresponding values for the minimax relative error differentiators.