Abstract. A method for studying complex filter with the use of the transition characteristics is developed. Presentation of input signal as a sum of step functions shifted relative to each other leads to a new form of complex superposition integral allowing to determine the filter output signal for a given complex-valued input and a known complex transition characteristic. Application of complex filters is shown to enable variation of their frequency settings without a change in shape of the amplitude and phase-frequency characteristics. The concepts of complex delta function and complex transition characteristics are introduced; the concepts are employed in the study of dynamic properties of the filters and transmission of various signals. In contrast to the known method of analysis of linear systems based on the pulse characteristic, the proposed method may be used to synthesize complex discrete filter with dynamic properties coinciding with those of a continuous filter-analog. The developed theoretical principles are confirmed by presented examples.

Keywords: complex signal, complex filter, complex pulse characteristic, complex transition characteristic, superposition integral

References:

1. Vorob'ev S. N. Tsifrovaya obrabotka signalov (Digital Signal Processing), Moscow, 2013, 317 р. (in Russ.)
2. Gonorovskiy I.S. Radiotekhnicheskie tsepi i signaly (Radio Engineering Chains and Signals), Moscow, 1986, 512 p. (in Russ.)
3. Oppenheim A.V., Schafer R.W., Buck J.R. Discrete-Time Signal Processing, 2 Ed. Prentice Hall, 1999, 870 p. ISBN: 0137549202. (in Russ.)
4. Ziatdinov S.I. Informatsionno-upravliaiushchie sistemy (Information and Control Systems), 2016, no. 5, рp. 98–101. (in Russ.)
5. Ziatdinov S.I. Radioelectronics and Communications Systems, 2016, no. 3, рp. 3–6. (in Russ.)
6. Ziatdinov S.I. Informatsionno-upravliaiushchie sistemy (Information and Control Systems), 2016, no. 2, рp. 104–106. (in Russ.)
7. Ziatdinov S.I. Journal of Instrument Engineering, 2015, no. 8(58), рp. 167–172. (in Russ.)
8. Ziatdinov S.I. Journal of Instrument Engineering, 2016, no. 7(59), рp. 253–259. (in Russ.)
9. Besekerskiy V.A., ed., Mikroprotsessornye sistemy avtomaticheskogo upravleniya (Microprocessor Systems of Automatic Control), Leningrad, 1988, 364 р. (in Russ.)
10. Besekerskiy V.A. Tsifrovye avtomaticheskie sistemy (Digital Automatic Systems), 1976, 576 p. (in Russ.)
11. Brychkov Yu.A., Marichev O.I., Prudnikov A.P. Tablitsy neopredelennykh integralov (Tables of Indefinite Integrals), Moscow, 1986. 192 р. (in Russ.)