THE GENERAL THEORY OF DIGITAL FILTERS WITH APPLICATIONS TO SPECTRAL ANALYSIS

The filtering theories for both continuous-time and discrete-time signals are formulated in terms of abstract Hilbert space, with the notion of a stable filter defined as a bounded linear operator. A specific isomorphism is then constructed which connects the filtering theories for continuous-time and discrete-time signals, and in the linear time-invariant case the two theories are shown to be essentially identical. This means that many optimization problems can be solved simultaneously for continuoustime and digital systems. The isomorphism developed above is used to reduce the approximation problem for digital filters to that for continuous-time filters. The problem of estimating the power-spectral-density of a signal from equally spaced samples is discussed. It is shown that bandpass digital filters generate a class of spectral windows which produce always positive estimates of the power-spectral-density. The optimum bandwidth and shape of such a filter are then derived. A method for identifying unknown parameters in the power-spectraldensity of a digital signal is presented.