Synthesis filterbank design for multi-channel systems by the theory of fusion frames

Filterbanks of multi-channel systems consist of analysis filters and synthesis filters. Rigorous and effective design of synthesis filters of an arbitrary multi-channel system is a challenging topic. In this thesis, a fusion frame theoretical approach to synthesis filter design is studied. Each channel is described by a subspace spanned by the channel (local) frame that corresponds to the analysis filter. The class of all synthesis filters is then constructed through dual frames and fusion frame principles. One major goal is to design synthesis filters that are finite in length or decaying fast. Techniques such as L1 minimizations and L1 tail minimizations are examined to solve the associated L0 minimization problem. Algorithms are derived. Error bound analyses are obtained for truncated and approximate synthesis filters. Furthermore, an algebraic approach designing finite length synthesis filters is also presented using the general dual frame formula. Algorithms are also derived and implemented.