Conjecture on the Absence of Limit Cycles in Second-Order Digital Filters with Minimum L₂-Sensitivity Subject to L₂-Scaling Constraints

Abstract

This paper gives a conjecture on the absence of limit cycles in second-order digital filters with minimum L₂-sensitivity subject to L₂-scaling constraints. We design second-order digital filters for various pole-zero configurations. For given second-order digital filters, we synthesize the minimum L₂-sensitivity realizations and examine whether their coefficient matrices satisfy a sufficient condition for the absence of limit cycles. Experimental results show that the minimum L₂-sensitivity realizations subject to L₂-scaling constraints of second-order digital filters satisfy a sufficient condition for the absence of limit cycles for all given practical pole-zero configurations.