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

摘要

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( omega )= mod F( omega ) mod exp (j Delta ( omega )) of a real argument omega , the extrema of its magnitude mod F( omega ) mod and its argument (or phase) Delta ( omega ), as functions of omega , are determined by finding the roots of one common equation, Im (G( omega ))=0, where G=(F'/F)/sup 2/ and F'= delta F/ delta omega . The extrema of mod F mod and Delta 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.