Synthesis of optimum RC phase-shift network

This paper is concerned with the following problem. Given two terminating resistances R, and R0, design a low-pass ladder RC network of n sections such that the magnitude of the transfer-voltage ratio
|V_{1}/V_{2} (j\omega_{o})|
is minimized at a given phaseshift
\angle V_{1}/V_{2}(j\omega_{0})) = \phi_{0}
and at a given frequency
\omega
. This problem is solved explicitly. It was found that the optimum
RC
phase-shift network is a geometrical progression network properly terminated at both ends. In the distributed case (
n
approaches infinity), the optimum
\overline{RC}
network is an exponentially tapered line, terminated by four lumped impedances.