A Unified Framework to Design Orthonormal Wavelets Packet Bases for Multi-Carrier Modulation

Wavelet Packet Modulation (WPM) is a novel multicarrier communication technique where information carrying symbols are placed on orthogonal wavelet packet bases. The WPM system is realized by a tree structure obtained by cascading a pair of half-band Quadrature Mirror Filter (QMF) duo. By altering the properties of these filters the transmission characteristics of the WPM system can be customized to a given design specification. In this article we present a general, unified approach to design and develop orthogonal wavelet packet bases according to a requirement. To this end, the design criterion and the wavelet constraints are first listed. The problem which is originally non-linear and non-convex in nature is then converted into a tractable convex optimization problem and finally solved using suitable Semi Definite Programming (SDP) tools. The proposed mechanism is demonstrated through two toy examples where families of wavelets which are i) maximally frequency selective and ii) have the lowest cross correlation energy, respectively, are developed.