Transfer functions of second-order digital filters with two equal second-order modes

This paper clarifies the class of second-order digital filters with two second-order modes equal. We consider three cases for second-order digital filters: complex conjugate poles, distinct real poles, and multiple real poles. We derive a general expression of the transfer function of second-order digital filters with two second-order modes equal. Furthermore, we show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter.