Hyperbolic Wavelets and Multiresolution in H 2 (T)
Margit Pap · 2011
In signal processing and system identification for H 2 (T) and H 2 (D) the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are con- structed from rational Blaschke functions, which form a group with respect to func- tion composition that is isomorphic to the Blaschke group, respectively to the hy- perbolic matrix group. Consequently, the background theory uses tools from non- commutative harmonic analysis over groups and the generalization of Fourier trans- form uses concepts from the theory of the voice transform. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer func- tion that may cause a drawback of the method. In this paper we give a set of poles and using them we will generate a multiresolution in H 2 (T) and H 2 (D). The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the space H 2 (T). The constructed discretization scheme gives opportunity of prac- tical realization of hyperbolic wavelet representation of signals belonging to H 2 (T) and H 2 (D) if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet repre- sentation will be studied.