Symmetric Unity-Bounded Functions for Stabilizing Recursive Variable Digital Filters

A variable digital filter (VDF) is a highly flexible reconfigurable signal processing system that is able to change its frequency-domain characteristic (frequency response) during filtering operations. Although a recursive-type VDF requires lower computational complexity than its non-recursive counterpart, it may become unstable, depending on the transfer function’s denominator coefficients, where transfer function means the mathematical model in the frequency domain. That is, the stability of a recursive-type VDF is affected by the transfer function’s denominator coefficients. If the denominator coefficients do not satisfy the stability condition, a recursive VDF becomes an unstable one. To prevent a recursive VDF from becoming unstable, a wise strategy is to transform the denominator coefficients into the functions of other new parameters by employing a special function named as unity-bounded (UB) function. In this paper, a pair of UB functions are presented for carrying out the parameter transformations. One of the two UB functions has odd symmetry, and the other has even symmetry. Since both the two functions meet a particular condition required for a UB function, both the two UB functions are applicable to the stability guarantee of a recursive-type VDF.