Identification Of Rational Transfer Function From Frequency Response Samples

A method for identifying a transfer function, H(z) = A(z)/B(z), from its frequency response values is presented. The method has its roots in the nineteenth century work of Cauchy and Jacobi. Identifying the transfer function involves determining the unknown degrees and coefficients of the polynomials A(z) and B(z), given the frequency response samples. Although the problem is seemingly nonlinear, the method for finding the parameters of the transfer function, involves solving linear simultaneous equations only. An important aspect of the method is the decoupled manner in which the polynomials A(z) and B(z) are determined. We present two slightly different derivations of the linear equations involved, one based on the properties of divided-differences and the other using Vandermonde matrices or equivalently, Lagrange interpolation. A certain matrix synthesized from the given frequency response samples is shown to have a rank equal to the number of poles in the system. These results are strikingly similar to the Prony\u27s method which is used to identify a linear system from its impulse response samples. The method has many applications in filter design, system identification, and signal modeling. © 1990 IEE