On the eigenstructure of linear quasi-time-invariant systems

In this paper, the eigenstructure of a class of linear time-varying systems, termed linear quasi-time-invariant (LQTI) systems, is investigated. A system composed of dynamic devices such as linear time-varying capacitors and resistors can be an example of the class. To describe and analyze the LQTI systems effectively, a differential operator G composed of the derivative operator D and some time functions is adopted. Then, the dynamic systems described by the operator G are studied in terms of eigenvalue, frequency characterisics, stability and an extended convolution. Some basic attributes of the operator G are compared with those of the differential operator D. The corresponding generalized Laplace transform pair is defined and relevant properties are derived for frequency-domain analysis and design of the filters. It is also noted that the stability is determined by the position of poles in the G frequency domain, where the stable region in the complex plane is different from the classical left half s plane. A point in the extended Laplace transform space represents a complex exponential function modulated by some functions. As an application example, an LQTI filter is examined and designed by using the concept of eigenstructure of LQTI system. Also, a new type of modulation or frequency shift property is examined as in the linear time-invariant filter theory.