Stability-constrained variable mathematical model for odd-order recursive variable filters

This paper presents a stability-triangle-constrained mathematical model (transfer function) for modeling the odd-order stable recursive variable digital filter (recursive variable filter). In variable-filter (VF) designs, a few spectral parameters are usually used for tuning the frequency-domain characteristics, and the VF coefficients are expressed as the functions of those parameters. In this paper, we only consider the single parameter case. To guarantee the stability of an odd-order recursive VF, this paper expresses the coefficients in the denominator of the odd-order transfer function as composite functions of the single spectral parameter to constrain the values of the composite functions within the stability triangle of the second-order sections. As a result, the stability of the odd-order recursive VF is always guaranteed whenever the VF coefficients are changed. The proposed stability-triangle-constrained odd-order transfer function lays the foundation of stable odd-order recursive VF design.