ISSN 0021-3454 (print version)
ISSN 2500-0381 (online version)

12
Issue
vol 63 / December, 2020
Article

DOI 10.17586/0021-3454-2019-62-11-951-957

UDC 621.396:681.323

## DIFFERENTIAL EQUATIONS OF CONTINUOUS COMPLEX FILTERS

S. I. Ziatdinov
St. Petersburg State University of Aerospace Instrumentation, Department of Information Network Technologies; Professor

Abstract. A method to derive differential equation for complex continuous filter using specified frequency transfer function is considered. The method may be applied to low-pass and high-pass filters, band-pass and band-reject continuous complex filters. An equation is presented that relates the spectral densities of the complex input and output signals of the filter with its frequency transfer function. To obtain a differential equation describing the complex filter operation, the inverse Fourier transform is applied to both parts of the equation, so that the input and output signals of the filter are converted from the frequency domain to the time domain. As examples, continuous complex filters of various orders are considered, which at the zero tuning frequency are either low-pass filters or high-pass filters, and at a nonzero tuning frequency, they are either band-pass or notch filters, depending on the specified frequency transfer function type. The differen-tial equations of continuous complex filters are shown to be complex functions of time determined by two quadrature components. To find the transient and impulse characteristics of continuous complex filters, so-lution of the differential equation of low-pass filter (band-pass filter) for a single step input signal is ana-lyzed. The general and particular solutions of the complex differential equation are obtained that determine two quadrature components of the complex transition characteristic of the filter under consideration. By dif-ferentiating the complex transient response, an expression is found for the impulse response of the filter, which is also complex and is represented by two quadrature components.
Keywords: complex filter, differential equation, frequency transfer function

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