Arbitrarily precise equalization of the phase of linear analogue systems

A procedure is introduced, to develop the transfer function of an analogue all-pass system for the equalization of a given phase contour. The approximation error between the total phase of the cascade of the given system with this all-pass and the ideal linear phase can be prescribed arbitrarily small in the desired frequency range. A smaller equalization error will naturally cause a larger group delay of the total system, and an increase in the number of required building elements for the realization of the all-pass system. As a first step it is shown that the above described problem can be reduced to the equalization of the phase of a first order real all-pass section. In the second part of this paper, it is proved that this simplified problem can be solved arbitrarily precisely in a prescribed frequency interval