Nyquist plots under frequency transformations

The relevance of odd, and in particular loss-less, frequency transformations is well established in analog filter design. A loss-less frequency transformation s←F(s), in fact, allows to design multi-bandpass multi-bandstop filters as transformed systems G̃(s)=G(F(s)) from an original prototype lowpass filter G(s), by properly selecting F(s). In this paper the invariance of the shape of the Nyquist plot for linear time-invariant continuous-time systems under a class of odd frequency transformations is proved. This result allows to determine closed-loop stability conditions for this class of transformed systems on the basis of the Nyquist's criterion referred to the original, lower order, system.