Unity-Bounded Functions for Designing Stable Variable Digital Filters

The system function of a recursive variable filter (VF) has both numerator and denominator that are parameterized as the polynomial functions of a parameter called tuning parameter. That is, the VF coefficients are the polynomial functions of the tuning parameter. However, changing the values of the denominator polynomials may drive the filter to instability. A wise way to avoid the instability problem is to replace the original denominator polynomial with other functions such that the stability condition of a recursive VF is always met. This paper first presents various novel coefficient-replacement (CR) functions for replacing the original denominator polynomial with the CR-functions, and then rigorously proves that replacing the original denominator polynomial with the proposed CR-functions can avoid the instability. That is, varying the values of the CR-functions will never incur instability. As a result, the proposed CR-functions are useful in the design of recursive VFs with theoretically ensured stability.