A general transform theory of rational orthonormal basis function expansions∗

In this paper a general transform theory is presented that underlies expansions of stable discrete-time transfer functions in terms of rational orthonormal bases. The types of bases considered are generated by cascade connections of stable all-pass functions. If the all-pass sections in such a network are all equal, this gives rise to the Hambo basis construction. In this paper a more general construction is studied in which the all-pass functions are allowed to be different, in terms of choice and number of poles that are incorporated in the all-pass functions. It is shown that many of the interesting properties of the so-called Hambo transform that underlies the Hambo basis expansion carry over to the general case. Especially the recently developed expressions for the computation of the Hambo transform on the basis of state-space expressions can be extended to the general basis case. This insight can for instance be applied for the derivation of a recursive algorithm for the computation of the expansion coefficients, which are then obtained as the impulse response coefficients of a linear time-varying system.