A.S. Korotkov1, A. Kavruk2

1,2 Peter the Great St. Petersburg Polytechnic University (St. Petersburg, Russia)

1 korotkov@spbstu.ru; 2 kavruk.a@edu.spbstu.ru

The paper presents a solution to the problem of transforming the transfer function of an analogue prototype in the p-domain to the transfer function of a digital/time-discrete filter in the z-domain. The known methods of p-z transform are considered: inverse difference, bilinear, based on interpolation and reduction of high-order numerical integration rules, conversion with instability elimination. The developed method of transforming the transfer function based on integration by Simpson's rule and by Tic's rule is presented, the hodographs of transformations are analyzed. As a result of comparison of the developed methods with the known ones it is established that based on the criterion of the mean modulus of the error vector on the complex plane in the z-domain the proposed methods of transformation allow to significantly reduce the error in the filter bandwidth. Thus, in the given examples the error is reduced by 27% in comparison with bilinear transformation when applying the method based on Simpson's rule, and by 40% - by Tick's rule.

Korotkov A.S., Kavruk A. Application of high-order integration methods for p-z-transform. Radiotekhnika. 2025. V. 89. № 3. P. 5−17. DOI: https://doi.org/10.18127/j00338486-202503-01 (In Russian)

1. Korotkov A.S. Switched-capacitor filter designs: tutorials, St. Petersburg: Polytechnic University Publishing House. 2014. 193 р.
2. Tustin A. A method of analysing the behavior of linear systems in terms of time series. J. Inst. Electrical Eng. Part IIA: Automat Regul Servo Mech. 1947. V. 94. № 1. Р. 130-142.
3. Bruton L. Low-sensitivity digital ladder filters. IEEE Transactions on Circuits and Systems. March 1975. V. 22. № 3. Р. 168-176. DOI: 10.1109/TCS.1975.1084021.
4. Dostal T., Mikula Ja. Preobrazovanie p-z dlja umen'shenija jeffekta deformacii harakteristik i ego ispol'zovanie v SC-fil'trah. Izvestija vuzov. Ser. Radiojelektronika. 1994. T. 37. № 6. S. 70-73 (in Russian).
5. Korotkov A.S. Mikrojelektronnye analogovye fil'try na preobrazovateljah impedansa. SPb: Nauka. 1999/2000. 416 s. (in Russian).
6. Bahvalov N.S.  Chislennye metody: uchebnik. Izd. 8-e. M.: MGU im. Lomonosova. 2015. 637 s. (in Russian).
7. Muljavka Ja. Shemy na operacionnyh usiliteljah s perekljuchaemymi kondensatorami: Per. s pol'sk. M: Mir. 1992. 416 s. (in Russian).
8. N'juton I. Matematicheskie nachala natural'noj filosofii: Per. s lat. A.N. Krylova. M.: Nauka. 1989. 706 s. (in Russian).
9. Al-Alaoui M. Novel class of digital integrators and differentiators. IEEE Transactions on Signal Processing. August 2008. V. 5.
   № 99. Р. 1-10. DOI: 10.1109/TSP.2008.929668.
10. Yadav R., Gupta M. Design of fractional order differentiators and integrators using indirect discretization approach. 2010 International Conference on Advances in Recent Technologies in Communication and Computing. Kottayam. India. 2010. Р. 126-130. DOI: 10.1109/ARTCom.2010.67.
11. Steiglitz K. Computer-aided design of recursive digital filters. IEEE Transactions on Audio and Electroacoustics. June 1970. V. 18. № 2. Р. 123-129. DOI: 10.1109/TAU.1970.1162099.
12. Rajasekhar K., Krishna B.T. Design of novel fractional order differintegrators at low frequency region. 2020 7th International Conference on Signal Processing and Integrated Networks (SPIN). Noida. India. 2020. Р. 322-327. DOI: 10.1109/SPIN48934.2020.9070879.
13. Rajasekhar K., Krishna B.T. Design and implementation of fractional order differintegrators using reduced s- to z-transforms. Journal of Communications Technology and Electronics. 2018. V. 63. Р. 1406-1417. DOI: 10.1134/S1064226918120185.