Optimization of digital filters performances simultaneously in frequency and time domains

Wide used method of digital filters design consists in transformation of analog filter-prototype with required performances into digital filter. This method is applicable if the transformation preserves optimality of filter performances under specified set of quality indexes (QI). It was denoted earlier that such situation is possible when gain-frequency response (GFR) and phase-frequency response are optimized simultaneously. The task of simultaneous optimization of digital filters GFR and step response (SR) is also important but yet a little explored. Alternative method of this problem solving consists in search of digital filter transfer function (TF) which is optimal under GFR and SR QI’s. To investigate capabilities of the first method we found examples of analog filters Pareto-optimal under rise time and transient duration. Other QI’s of these filters fulfilled specified constraints. Then these filters were transformed into digital filters. Bilinear transformation and transformation with invariant impulse response were applied. Further we did the search of digital filters optimal under the same set of QI’s. In either method the hybrid heuristic algorithm was applied for search optimal solutions in the space of TF poles and zeroes coordinates. The results of investigation demonstrated that digital filters developed via search are superiorly under specified set of QI’s then digital filters developed via transformation of analog filters. Accordingly Pareto-optimality for QI of GFR and SR is not preserved during such transformation and direct search must be applied to optimized digital filters simultaneously in frequency and time domains. Further in some cases analog filters developed via reverse bilinear transformation of the found optimal digital filters are superiorly under the same set of QI’s then analog filters developed using search. In such cases using of digital filter-prototypes for design of analog filters is practical.

digital filter, filter-prototype, bilinear transformation, impulse response invariance, gain-frequency response, step response, Pareto-optimality

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