МЕТОДИКА И АЛГОРИТМ СИНТЕЗА УПРАВЛЯЕМЫХ ЦИФРОВЫХ ФИЛЬТРОВ ЧЕБЫШЕВА I РОДА НИЖНИХ ЧАСТОТ НА БАЗЕ МЕТОДА БИЛИНЕЙНОГО ПРЕОБРАЗОВАНИЯ

Abstract

The article presents a technique for the synthesis of controlled digital recursive Chebyshev lowpassfilters of the first kind with an infinite impulse response. The frequency response of such filters hasripples in the passband and is as flat as possible in the stopband. Controllability is understood as anexplicit dependence of the filter coefficients on the cutoff frequency. The technique is based on the bilineartransformation of the transfer function of the analog low-pass filter prototype and the frequencytransformation 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 ofthe transfer function of biquadratic or bilinear links, which have the dimension of frequency, will benumerically equal to the correction factors for similar parameters of a controlled filter with an arbitrarycutoff frequency. As an example, the synthesis of a digital Chebyshev filter of the first kind of the fifthorder is considered. In this article, the transfer function of an arbitrary order filter is represented as acascade connection of II order links if the filter is of an even order. In the case of an odd order greaterthan one, one cascaded link of the first order is added. Despite the relative simplicity of the frequencyconversion, 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 approximationsof the frequency response of an ideal low-pass filter), a series arises non-trivial specific momentsthat complicate the engineering use of this method of synthesizing controlled digital filters. Therefore, inaddition to the technique, a step-by-step algorithm has been developed that allows one to synthesize afilter without knowing these moments. The algorithm is implemented in the Mathcad environment; asan 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 explicitlydepend on the cutoff frequency, the amplitude-frequency characteristics of this filter and its lowfrequencyprototype converted into a filter with the same cutoff frequency, the amplitude-frequencycharacteristics are given in the same coordinates. Due to the good formalization of the algorithm, thelatter is suitable for the implementation of computer-aided design systems for controlled digitalChebyshev low-pass filters of the first kind.