Frame-theory-based analysis and design of oversampled filter banks: Direct computational method

This paper studies the frames corresponding to oversampled filter banks (FBs). For this class of FB frames, we present a state-space parameterization of perfect reconstruction FB frames and explicit and numerically efficient formulas to compute the tightest frame bounds, to obtain the dual FB frame, and to construct a tight (paraunitary) FB frame from a given untight (nonparaunitary) FB frame. The derivation uses well-developed techniques from modern control theory, which results in the unified formulas for generic infinite-impulse-response (IIR) and finite-impulse-response (FIR) FBs. These formulas involve only algebraic manipulations of real matrices and can be computed efficiently, reliably, and exactly without the approximation required in the existing methods for generic FBs. The results provide a unified framework for frame-theory-based analysis and systematic design of oversampled filter banks.