Simplified parameter-transformation for guaranteeing variable-filter stability

In designing a variable recursive digital filter (variable-recursive-filter), one of the most important problems to be solved is to guarantee the stability of the designed variable recursive digital filter. This paper simplifies the existing parameter-transformation presented in the literature, which is used for transforming the denominator-coefficients of the transfer function of a variable-recursive-filter into another set of new parameters. Such a parameter-transformation enables all the new parameters to take arbitrary values while without affecting the stability of the variable-recursive-filter. That is, the variable-recursive-filter is guaranteed to be stable for arbitrary parameter values. Based on this parameter-transformation, we can simplify the variable-recursive-filter design problem subject to stability-guarantee condition (constrained non-linear minimization problem) as an unconstrained non-linear minimization problem, and any non-linear minimization solver can be applied to the design of a stable variable-recursive-filter. Most importantly, when this parameter-transformation is applied to the variable-recursive-filter design where all the filter coefficients are frequently updated, based on the proposed parameter-transformation, each variable-recursive-filter coefficient is represented as a different function of the spectral parameter, the stability of the obtained variable-recursive-filter is theoretically guaranteed.