Rational Orthonormal Functions on the Unit Circle and on the Imaginary Axis, with Applications in System Identification

In this report we present a collection of results concerning two families of rational orthonormal functions, one on the unit circle, and another on the imaginary axis. We also describe in detail an interesting link between the two families. Special cases of these rational orthonormal functions include the Laguerre and Kautz orthonormal functions, as well as the orthonormal functions recently introduced by Heuberger, Van den Hof and Bosgra. Among the results presented herein are completeness and uniform boundedness conditions, and their respective proofs, for the above mentioned families of rational orthonormal functions, as well as some interpolatory properties of truncated orthonormal expansions based on these functions. We will only discuss the case of (rational) functions orthonormal with respect to a unit weighting function. The general case of a non constant (non-negative) weighting function is much more involved and will be described in a future report. We derive the rational orthonormal functions by two methods. The first one is essentially the classical approach, and is based on the Gram-Schmidt orthonormalization of a set of exponentials. This method was the one used by Lee and Wiener, Kautz, and others, and is based on certain elementary properties of contour integrals of rational functions. The second one, which may be called the modern approach, is based on balanced realizations of rational all-pass transfer functions. To the best of our knowledge this method was developed by Roberts and Mullis and is at the root of the Heuberger et al. family of rational orthonormal functions. Some interesting elementary properties of linear models constructed from rational orthonormal functions are also presented. Special cases of these models include the very well known FIR model and the recently studied Laguerre and “two-parameter” Kautz models, as well as the Heuberger et al. model. Curiously, these models have many properties in common with the FIR model, such as the theoretical upper bound of the condition number of the auto-correlation matrix of their internal signals. Together with the inherent structure of these models this leads to a simple LMS-like algorithm to adapt its weights that can use the same adaptation step size as the standard LMS algorithm for transversal filters.