Low-sensitivity active filter realization using a complex all-pass filter

A wide class of continuous-time transfer functions may be implemented as the parallel combination of two all-pass filters, including Butterworth, Chebyshev, and elliptic low-pass approximations of odd order. Here, we consider the realization of even-order low-pass classical approximations and show that they may be decomposed in terms of complex all-pass functions. A systematic realization approach, based on scattering domain simulation (i.e., wave active filters), allows for a low-sensitivity active filter implementation. Further insight into the low-sensitivity property is gained by connecting the insertion loss of doubly terminated antimetric networks with the imaginary return loss of complex lossless networks.