Closed form expressions of the minimum L2-sensitivity realizations for all-pass digital filters

This paper presents closed form expressions of the minimum L₂-sensitivity realizations of all-pass digital filters. We first prove that the L₂-sensitivity minimization problem can be solved analytically if the all second-order modes of a digital filter are equal. We focus on the fact that the second-order modes of all-pass digital filters are all unity. As a result, it is clarified that the minimum L₂-sensitivity realization of an all-pass digital filter is equal to the balanced realization, of which closed form expression can be derived analytically.