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– The issue of constructing discrete band-pass and notch filters using differential equations of continuous analog filters of low and high frequencies is considered. Purpose – Development of a technique for the synthesis of tunable band-pass and notch filters with a constant width of the amplitude-frequency characteristics, regardless of the filter tuning frequency. Setting the task – Based on the invariant differential equations of continuous analog filters of low and high frequencies, find the weight coefficients of the difference equations that determine the operation of discrete bandpass and notch filters that have a constant width of amplitude-frequency characteristics under conditions of variable filter tuning frequency. Method – The synthesis is based on the presentation of the mathematical frequency transfer function of the filter separately in the region of positive and negative frequencies, for each of which a bilinear z-transform is applied shifted to the tuning frequency of the selected region. Results – Difference equations of discrete filters are obtained from the given differential equations of continuous analog filters of low and high frequencies. Frequency-shifted bilinear z-transformations are presented. Transfer functions in the z-plane are obtained separately for the regions of the frequency transfer function of the filter located in the ranges of positive and negative frequencies. By summing the transfer functions in the z-plane for the regions of positive and negative frequencies and multiplying these transfer functions, the resulting transfer functions in the z-plane are obtained for discrete bandpass and notch filters. By passing from the z-plane to the jω, expressions for the frequency transfer functions of the synthesized band-pass and notch filters are obtained. The calculation of their amplitude-frequency characteristics has been made. It is shown that the width of the amplitude-frequency characteristics of the synthesized filters remains constant when the filter tuning frequency is changed. Practical relevance – The developed technique for the synthesis of tunable real discrete band-pass and notch filters with a constant frequency response width based on invariant differential equations of continuous analog filters will be very useful in the construction of adaptive systems and signal processing devices, such as signal detection and filtering systems, Doppler velocity meters, selection systems moving targets.

Synthesis of discrete band and notch filters by the method of invariant differential equations.