METHOD AND ALGORITHM FOR SYNTHESIS OF CONTROLLED CHEBYSHEV DIGITAL FILTERS OF THE FIRST KIND OF LOW FREQUENCIES BASED ON THE BILINEAR TRANSFORMATION METHOD

Abstract

The article presents a technique for the synthesis of controlled digital recursive Chebyshev lowpass
filters of the first kind with an infinite impulse response. The frequency response of such filters has
ripples in the passband and is as flat as possible in the stopband. Controllability is understood as an
explicit dependence of the filter coefficients on the cutoff frequency. The technique is based on the bilinear
transformation of the transfer function of the analog low-pass filter prototype and the frequency
transformation of the amplitude-frequency characteristics of the obtained digital filter. The main idea ofthe technique is that for an analog prototype filter with a cutoff frequency of 1 rad/s, the parameters of
the transfer function of biquadratic or bilinear links, which have the dimension of frequency, will be
numerically equal to the correction factors for similar parameters of a controlled filter with an arbitrary
cutoff frequency. As an example, the synthesis of a digital Chebyshev filter of the first kind of the fifth
order is considered. In this article, the transfer function of an arbitrary order filter is represented as a
cascade connection of II order links if the filter is of an even order. In the case of an odd order greater
than one, one cascaded link of the first order is added. Despite the relative simplicity of the frequency
conversion, in its practical use for digital filters synthesized using computer-aided design of digital filters
(or using reference books containing calculated prototype low-pass filters for various approximations
of the frequency response of an ideal low-pass filter), a series arises non-trivial specific moments
that complicate the engineering use of this method of synthesizing controlled digital filters. Therefore, in
addition to the technique, a step-by-step algorithm has been developed that allows one to synthesize a
filter without knowing these moments. The algorithm is implemented in the Mathcad environment; as
an example, a digital recursive Chebyshev filter of the 1st kind of the 5th order is calculated.
The example shows the calculated coefficients of a digital controlled low-pass filter, which explicitly
depend on the cutoff frequency, the amplitude-frequency characteristics of this filter and its lowfrequency
prototype converted into a filter with the same cutoff frequency, the amplitude-frequency
characteristics are given in the same coordinates. Due to the good formalization of the algorithm, the
latter is suitable for the implementation of computer-aided design systems for controlled digital
Chebyshev low-pass filters of the first kind.