Optimization of Synthesis Oversampled Complex Filter Banks

TL;DR

This paper develops algorithms to determine and optimize inverse synthesis filter banks for oversampled FIR analysis filter banks, ensuring properties like Hermitian symmetry and simplifying design via unconstrained optimization.

Contribution

It introduces a method to verify the existence of inverse FIR synthesis filter banks and provides a parameterization for their entire family, simplifying the design process.

Findings

Algorithm to determine inverse FIR synthesis filter banks.

Parameterization of all inverse solutions via null space projection.

Gradient-based optimization for time and frequency localization.

Abstract

An important issue with oversampled FIR analysis filter banks (FBs) is to determine inverse synthesis FBs, when they exist. Given any complex oversampled FIR analysis FB, we first provide an algorithm to determine whether there exists an inverse FIR synthesis system. We also provide a method to ensure the Hermitian symmetry property on the synthesis side, which is serviceable to processing real-valued signals. As an invertible analysis scheme corresponds to a redundant decomposition, there is no unique inverse FB. Given a particular solution, we parameterize the whole family of inverses through a null space projection. The resulting reduced parameter set simplifies design procedures, since the perfect reconstruction constrained optimization problem is recast as an unconstrained optimization problem. The design of optimized synthesis FBs based on time or frequency localization criteria is then investigated, using a simple yet efficient gradient algorithm.