Conjecture on the absence of limit cycles in second-order digital filters with minimum L2-sensitivity subject to L2-scaling constraints

This paper gives conjecture on the absence of limit cycles of second-order digital filters with minimum L₂-sensitivity subject to L₂-scaling constraints. We evaluate a sufficient condition for the absence of limit cycles of second-order digital filters with minimum L₂-sensitivity subject to L₂-scaling constraints from their filter coefficient matrices. We next design second-order digital filters for various pole-zero configurations, synthesize the minimum L₂-sensitivity realizations, and confirm if their coefficient matrices satisfy the sufficient condition for the absence of limit cycles. Experimental results show 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 in the range of practical pole radii.