Spectral transformations for digital filters

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^jθπⁿ_i=1z⁻¹−α_i/1−α^*_iz⁻¹known as unit functions.