A method of determining the transfer function of a linear system in terms of its pol es and zeros from the frequency response

A procedure is developed for obtaining the transfer function of a linear system from the frequency-response data in terms of its poles and zeros. The method is based upon the fact that the transfer function is an analytic function of the complex variable s at all points in the s plane except the isolated points which are the poles of the function. Hence if the function is specified along the j¿ axis, it is possible to extend the function in the left-half or right-half s plane by means of conformai mapping techniques. The curvilinear squares in the G(s) plane corresponding to a grid of constant ¿ and constant ¿ lines in the s plane can be constructed with fair accuracy after a little practice. The particular values of ¿ and ¿, for which the lines pass through the origin of the s plane, can thus be determined by locating the zeros of the transfer function. The poles are similarly obtained by working with the inverse function. Since there are several values of s satisfying G(s) = 0, the mapping from the s plane to the G(s) plane is not entirely a one-to-one mapping, and several regions in the s plane are mapped on to the same region in the G(s) plane. A method is suggested to overcome this difficulty and to obtain, step-by-step, all the zeros and poles of the transfer function corresponding to the given frequency-response locus. The method is illustrated by means of three numerical examples.