A Digital Computer Program for the Design of Phase Correctors

A digital computer program for the design of an all-pass function, with n natural modes, to correct the phase of a prescribed transfer function is discussed. The program is based on the expansion of both the desired phase and that of the all-pass function as a Tchebycheff polynomial series, together with the selection of the all-pass natural modes, so that the first n coefficients in the two series agree. If the over-all delay is unimportant, it may be chosen either to increase the number of matched coefficients by one, or to seek a minimum peak error over the approximating bandwidth. The design process is based on a continued fraction expansion, and a detailed technique for adjusting the over-all delay is discussed. Recurrence relationships are established for expanding the phase of any transfer function as a Tchebycheff polynomial series. The dependence of peak error on approximating bandwidth is discussed, and a routine for the automatic adjustment of the bandwith to meet a specified peak error is described. The design accuracy depends primarily on the root extraction technique, and on the loss of significant figures during continued fraction expansion. Adequate results have been obtained with 8 or 10 natural modes. The speed of the program depends on the number of basic iterations of the design process, which varies from one to ten. On a Ferranti Mercury Computer with , the basic iteration time is 20 seconds.