Extrema of the absolute value and argument of a complex function of a real variable: application to the spectral amplitude and phase response of analog filter circuits

The application of a sample theorem that provides an efficient means of locating jointly the extrema of the amplitude and phase responses of linear systems is discussed. Given a complex function F (ω)=|F(ω)| exp [jΔ(ω)] of a real argument ω, the extrema of its magnitude |F(ω)| and its argument (or phase) Δ(ω), as functions of ω, are determined by finding the roots of one common equation, Im [G(ω)]=0, where G=(F´/F)² and F´=∂ F/∂ω. The extrema of |F| and Δ are associated with Re G<0 and Re G>0, respectively. This easy-to-prove theorem is used to determine the extrema of the spectral amplitude and phase response of analog filter circuits. A variant of the theorem, in which F is expressed in terms of its real and imaginary parts. is developed. Examples confirm the usefulness of the theorem in locating the extrema of the amplitude and phase frequency response of linear circuits