Theory, Design and Application of Arbitrary Order Arbitrary Delay Filterbanks

Classical design of filterbanks results in equal order inverses, thus resulting in filterbanks with equal analysis and synthesis filter length. While it is known that filterbanks may have unequal orders, there does not exist a simple, systematic method to design a pth-order analysis and qth-order synthesis filterbank where p not equal q. Also, the design criteria of the th-order analysis having th-order synthesis filters (p not equal q) with a flexibility to control the system delay has never been addressed concomitantly. In this work, we propose the use of unit non-diagonal matrix polynomials to design such filterbanks for given orders p, q with arbitrary delay. We demonstrate how these orders depend on the sequence of unit matrices. Not only the achievable ranges of p, q are found, but construction methods for all such cases are also suggested. The proposed design has several desirable properties such as structural perfect reconstruction in its factorized form, completeness, and ability to control the resulting system delay of the filterbank. Several design examples and an application example for audio illustrate the flexibility and usefulness of the proposed design.