Orthogonal-like fractional-octave-band filters

Abstract

This paper addresses the design of digital fractional-octave-band filters with energy conservation and perfect reconstruction—i.e., whose outputs to each fractional-octave-band correctly sum up to the original signal and whose partial energies at each output correctly sum up to the overall signal energy—a combination of properties that cannot be met by any current design despite its considerable importance in many applications. A solution is devised based on the introduction of complex basis functions that span the outputs of the fractional-octave bands and whose real and imaginary parts form two individually—but not mutually—orthogonal bases. This imposes a “partition-of-unity” condition on the design of the filter frequency gains such that they exactly sum up to one over the frequency axis. The practical implementation of the proposed solution uses the discrete Fourier transform, and a fast algorithm is implemented using the fast Fourier transform. The proposed filters are well suited to any application involving the post-processing of finite-energy signals. They closely match the international standard templates, except for a small departure at the bandedge frequencies which can be made arbitrarily small.