Explicit expressions of the balanced realizations for second-order filters

This paper proposes explicit expressions of the balanced realizations of second-order digital and analog filters. We first consider the case of digital filters. Second-order digital filters are categorized into the following three cases: complex conjugate poles, distinct real poles, and multiple real poles. For each case, simple formulas are derived for the synthesis of the balanced realizations. As a result, we obtain explicit expressions of the balanced realizations of all types of second-order digital filters. Furthermore, we show that explicit expressions for second-order digital filters can be applied for also second-order analog filters by simple modifications of definitions of some parameters.