A Note on the Frequency Characteristics of Discrete Systems in the Complex Plane

Frequency characteristics of discrete-time systems in the complex plane and in linear coordinates are used to design controllers (correction terms) by frequency methods, to determine stability by frequency stability criteria, to assess the dynamic properties and behavior of the discrete system with respect to the harmonic components of the input signal, and to design various filters. Therefore, their determination is of considerable importance and their mapping and compilation into comprehensive summaries from which their shape and waveform can be determined is not yet done for discrete systems. This part is the first aim of the paper. Furthermore, a comparison of the construction of frequency characteristics of continuous and discrete linear time-invariant (LTI) systems is made and their differences are highlighted. Also, a comparison is made between the shape and waveform of the frequency characteristics of continuous and discrete systems with formally similar transfer functions. This comparison showed a relatively high similarity, which could be used to determine the dynamical properties of these systems. From the comparison of the constructions can also be seen how difficult it is to construct discrete versus continuous characteristics.