Spectral transformations for digital filters

TLDR: In this paper, the authors describe general transformations for digital filters in the frequency domain, which operate on a lowpass-digital-filter prototype to give either another lowpass or a highpass, bandpass or band-elimination characteristic.

Abstract: The paper describes certain general transformations for digital filters in the frequency domain. The term digital filter is used to denote a processing unit operating on a sampled waveform, so that the input, output and intermediate signals are only defined at discrete intervals of time; the signals may be either p.a.m. or p.c.m. The transformations discussed operate on a lowpass-digital-filter prototype to give either another lowpass or a highpass, bandpass or band-elimination characteristic. The transformations are carried out by mapping the lowpass complex variable Z -1 [where Z -1 = exp ( -jωT) and T is the time interval between samples] by functions of the form e jθ Πni=1z -1 -α I /1-α * I z -1 known as unit functions.