Design techniques for orthogonal modulated filterbanks based on a compact representation

This paper presents a method to design perfect reconstruction orthogonal modulated filterbanks having a very large number of subbands. At first, using the angular parameterization of the underlying lattice structure, the problem is formulated as an unconstrained optimization problem. Then, it is shown for two different optimization criteria (the minimization of the out-of-band energy and the maximization of the time-frequency localization) that the optimized lattice coefficients can be interpolated in two different ways. A first characterization takes advantage of the fact that the angles related to successive polyphase components of the prototype filter have a smooth behavior. A new parameterization, named compact representation, is therefore introduced, which shows that a linear increase of the number of parameters with respect to the number of subbands can be avoided with practically no loss of accuracy. A second observation concerns the also smooth behavior of each angular parameter when considering an increasing number of subbands. The efficiency of the two approaches is illustrated by the design of filterbanks with a large number of subbands going from 32 up to 4096, including prototype filters up to a length of 32 768.