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Article

Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis

1
SpaceForest Ltd., 81-451 Gdynia, Poland
2
Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, 80-233 Gdansk, Poland
*
Author to whom correspondence should be addressed.
Academic Editor: Alessandro Mauro
Energies 2021, 14(5), 1488; https://doi.org/10.3390/en14051488
Received: 18 January 2021 / Revised: 2 March 2021 / Accepted: 4 March 2021 / Published: 9 March 2021
In this paper, novel methods for the evaluation of digital-filter stability are investigated. The methods are based on phase analysis of a complex function in the characteristic equation of a digital filter. It allows for evaluating stability when a characteristic equation is not based on a polynomial. The operation of these methods relies on sampling the unit circle on the complex plane and extracting the phase quadrant of a function value for each sample. By calculating function-phase quadrants, regions in the immediate vicinity of unstable roots (i.e., zeros), called candidate regions, are determined. In these regions, both real and imaginary parts of complex-function values change signs. Then, the candidate regions are explored. When the sizes of the candidate regions are reduced below an assumed accuracy, then filter instability is verified with the use of discrete Cauchy’s argument principle. Three different algorithms of the unit-circle sampling are benchmarked, i.e., global complex roots and poles finding (GRPF) algorithm, multimodal genetic algorithm with phase analysis (MGA-WPA), and multimodal particle swarm optimization with phase analysis (MPSO-WPA). The algorithms are compared in four benchmarks for integer- and fractional-order digital filters and systems. Each algorithm demonstrates slightly different properties. GRPF is very fast and efficient; however, it requires an initial number of nodes large enough to detect all the roots. MPSO-WPA prevents missing roots due to the usage of stochastic space exploration by subsequent swarms. MGA-WPA converges very effectively by generating a small number of individuals and by limiting the final population size. The conducted research leads to the conclusion that stochastic methods such as MGA-WPA and MPSO-WPA are more likely to detect system instability, especially when they are run multiple times. If the computing time is not vitally important for a user, MPSO-WPA is the right choice, because it significantly prevents missing roots. View Full-Text
Keywords: digital filters; discrete-time systems; stability analysis; digital signal processing digital filters; discrete-time systems; stability analysis; digital signal processing
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MDPI and ACS Style

Trofimowicz, D.; Stefański, T.P. Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis. Energies 2021, 14, 1488. https://doi.org/10.3390/en14051488

AMA Style

Trofimowicz D, Stefański TP. Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis. Energies. 2021; 14(5):1488. https://doi.org/10.3390/en14051488

Chicago/Turabian Style

Trofimowicz, Damian, and Tomasz P. Stefański 2021. "Testing Stability of Digital Filters Using Optimization Methods with Phase Analysis" Energies 14, no. 5: 1488. https://doi.org/10.3390/en14051488

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