A generalized orthonormal basis for linear dynamical systems

In many areas of signal, system and control theory orthogonal functions play an important role in issues of analysis and design. In this paper, it is shown that there exist orthogonal functions that, in a natural way, are generated by stable linear dynamical systems, and that compose an orthonormal basis for the signal space l₂ⁿ . To this end use is made of balanced realizations of inner transfer functions. The orthogonal functions can be considered as generalizations of, e.g., the Laguerre functions and the pulse functions, related to the use of the delay operator, and give rise to an alternative series expansion of rational transfer functions. It is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion, i.e. to obtain a good approximation by retaining only a finite number of expansion coefficients