Derivation of the Class of Digital Filters With All Second-Order Modes Equal

This paper reveals the class of digital filters with all second-order modes equal. We first prove that if the second-order modes of a digital filter are all equal, the
L
₂
-sensitivity minimization problem of the digital filter can be solved analytically. We derive a general expression of the transfer function of digital filters with all 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.