The Application of Spectral Representations in Coordinates of Complex Frequency for Digital Filter Analysis and Synthesis

Abstract

The suitability of using one or another spectral representation depends on the type of signal to be analysed and problem to be solved, etc. (Kharkevich, 1960, Jenkins, 1969 ). Thus, the spectral representations, based on Fourier transform, are widely applied for linear circuit and frequency filter analysis for sinusoidal and periodical input signals (Siebert, 1986, Atabekov, 1978). However, using these spectral representations for a filter analysis of nonstationary signals would not be so simple and visually advantageous (Kharkevich, 1960). In the majority of cases input signals of automation and measurement devices have an analogue nature, and can be represented as a set of semi-infinite or finite damped oscillatory components. In the case of IIR filter impulse functions the representation uses this set of damped oscillatory components. Impulse functions of FIR filters representation are also based on this set of damped oscillatory components, but with the difference of a finite duration of the impulse functions. Thus, the generalized signal and impulse function of analog filters have similar mathematical expressions. In this case it is reasonable to use the Laplace transform instead of the Fourier transform, because the Laplace transform operates with complex frequency, and its damped oscillatory component is a base function of the transform (Mokeev, 2006, 2007, 2009a). The application of the spectral representations based on Laplace transform, or in other words, the spectral representations in complex frequency coordinates, enables to simplify significantly calculations of stationary and non-stationary modes and get efficient methods of filter synthesis (Mokeev, 2006). It also extends the application area of the complex amplitude method, including use of this method for analysis of stationary and nonstationary modes of analog and digital filters (Mokeev, 2007, 2008b, 2009a).