A second order s-to-z transform and its implementation to IIR filter design

Abstract

Purpose -The purpose of this paper is to design a tool for IIR digital filters obtained from analog prototypes, which preserves simultaneously the amplitude and the group delay response. Design/methodology/approach -A new s-to-z transform is developed based on a second order formula used for numerical integration of differential equations. Stability of the newly obtained transfer functions in the z-domain is proved to be preserved. Distortions introduced by the new transform into the original amplitude and group delay responses are studied. Findings -The new formula, when implemented to all-pole prototypes, exhibits lower selectivity than the original while reducing the pass-band group delay distortions. In the same time its structure is importantly simpler than the functions obtained by the well-known bilinear transform. When implemented to a prototype having "all kinds" of transmission zeros the resulting filter has almost ideally the same characteristic as the prototype.Research limitations/implications -The new transform may be used exclusively to synthesize even order filters. The new function is twice the order of the analog prototype. This kind of transformations are used to design IIR digital filters only. Low-pass transfer functions were studied being prototypes for all other cases. Originality/value -This is a new result never mentioned in the literature. Its effectiveness is confined to a niche problem when simultaneous sharp selectivity and low group delay distortions are sought.

IIR digital filters with critical monotonic pass-band amplitude characteristic

“…the maximal migration of the real pole from 0.7578207520 to 2.2161297265) while not as clear for the parallel (the same pole is migrating to 0.75). As the best confirmation of the theory developed in Section 5 and expressed by (15) one may use this maximal migration of the real pole since it represents the minimal radius of the disc encompassing all poles of the system. Therefore, the new radius of the circle encompassing the poles is now R c 7 bits = 2.2161297265.…”

“…Note, the worst case increase of the radius (with reference to Table 7), as predicted with (15), for the canonical realization using 7-bit arithmetic is…”

“…In that way, one has the Matched-z Transform method, discussed in [22], and three methods obtained by approximation of the analog integrator by a digital one. These are known as the Backward Euler (backward difference); the Trapezoidal method or the Bilinear Transform method, discussed in [23], and the second order formula introduced in [15].…”

“…As listed in [15], these methods may be categorized in two groups. In the first group are put the ones which implement a specific criterion of approximation such as: the Impulse Response Invariant method, first introduced in [16] and discussed in [17]; the Modified Impulse Response Invariant method, discussed in [18], the Step Response Invariant method (or Zero Order Hold), discussed in [19]; the Magnitude-invariance method, first introduced in [20]; and the Phase-invariance method, first introduced in [21].…”

Difference equation−based transient-state and steady-state analysis of flyback converter circuit

“…Difference equations (Brand, 1966;Miller, 1968;Bainov and Stamova, 1997) allow a nonlinear system to transform into a linear one. Therefore, this technique is used in many applications of electrical engineering from signal processing to power system tools (Shoihet and Slonim, 2010;Babiarz, 2015;Li et al, 2017;Ngoc, 2018;Shen et al, 2016;Bujosa and Ferrer, 2015;Mirkovic et al, 2014). The nonlinear converter topologies (Flyback, Forward, Push-Pull, halfbridge and full-bridge topologies) are solved by using genetic algorithms in Versele et al (2014).…”