Non-linear signal processing using index calculus DBNS arithmetic

This paper discusses the use of a recently introduced index calculus Double-Base Number System (IDBNS) for representing and processing numbers for non-linear digital signal processing; the target application is a digital hearing aid processor. The IDBNS representation uses 2 orthogonal bases (2 and 3) to represent real numbers with arbitrary precision. By restricting the number of digits to one or two, it is possible to efficiently represent the real number using the indices of the bases rather than the distribution of the digits. In this paper we discuss the use of the two-digit form of this representation (2-IDBNS) to efficiently perform arithmetic associated with the non-linear processing required to correct the usual forms of hearing loss in a digital hearing aid. The non-linear processing takes the form of dynamic range compression as a function of frequency band. Currently developed digital hearing instrument processors require large dynamic range representations (20-24 bits) in order to accurately generate the dynamic range compression associated with typical hearing loss. We show that the natural non-linear representation afforded by the IDBNS provides both a more efficient signal representation and a more efficient technique for processing the dynamic range compression. We pay particular attention to a novel technique of converting from a linear binary input directly to the 2-IDBNS representation using an observation of partial cyclic repetition in the indices along with near unity approximants.