Optimization of Synthesis Oversampled Complex Filter Banks
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.
Summary
This paper presents algorithms for deciding when a complex oversampled FIR analysis filter bank admits an FIR synthesis inverse and for computing such inverses. It shows how to enforce Hermitian symmetry on the synthesis side for real signals and parameterizes the full family of inverses via a null-space projection, enabling constrained perfect-reconstruction design to be turned into an unconstrained optimization over a reduced parameter set.
Key Takeaways
- Determine whether a given complex oversampled FIR analysis filter bank has an FIR synthesis inverse using the provided algorithm.
- Enforce Hermitian symmetry on synthesis filters to guarantee compatibility with real-valued signals.
- Parameterize the entire family of synthesis inverses via null-space projection from a particular solution.
- Recast perfect-reconstruction constrained optimization into an unconstrained optimization over a reduced parameter set to simplify design.
Who Should Read This
DSP engineers and researchers experienced in multirate filter-bank and wavelet design who need practical methods to test invertibility and optimize synthesis filters for oversampled complex filter banks.
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