A Closed Form Solution to L2-Sensitivity Minimization of Second-Order Digital Filters Subject to L2-Scaling Constraints

L₂-sensitivity is one of the functions that evaluates the coefficient quantization effects of state-space digital filters. This paper proposes a closed form solution to L₂-sensitivity minimization of 2nd-order state-space digital filters subject to L₂ -scaling constraints. The proposed solution reduces a constrained optimization problem to an unconstrained optimization problem by appropriate variable transformation. Futhermore, by restricting ourselves to the case of 2nd-order state-space digital filters, we can formulate the L₂-sensitivity minimization problem via hyperbolic functions. As a result, we can express the L₂-sensitivity in closed form, whose minimization subject to L₂-scaling constraints is achieved without iterative calculations