Harmonic analysis of linear sampled-data systems

Abstract

This chapter has presented a group theoretic approach to the concept of frequency response of sampled-data systems. Specifically, we have considered sampled-data systems as operators on the space of finite-energy signals defined on the real line where a specific group of transformations acts. Under the action of this group the signal space decomposes into certain infinitesimal invariant submanifolds. The frequency response is defined as the restriction of the sampled-data operator to these invariant submanifolds. This decomposition yields a unitary transformation which is a generalized Fourier transform matched to sampled-data signal and system analysis and its relation with the lifting technique has been pointed out. The approach proposed in this chapter provides a unified framework for the concept of frequency response of sampled-data systems and shows that recently introduced approaches for frequency domain analysis of sampled-data systems follow actually from one mathematical core, i.e. a group structure describing a time domain symmetry.