A general transform theory of rational orthonormal basis function expansions

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 the 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 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