An Orthogonal View of the Polyreference Least-Squares Complex Frequency Modal Parameter Estimation Algorithm

The polyreference least-squares complex frequency (PLSCF) modal parameter estimation algorithm has gained some popularity since the introduction of its single-reference predecessor shortly after the turn of this millennium. It is a z-domain (i.e., discrete time) method that uses a complex exponential frequency mapping from the imaginary frequency axis to the unit circle on the complex plane. While it operates directly on frequency response functions, this method has been interpreted to be essentially equivalent to the polyreference time-domain algorithm, with the application of the discrete Fourier transform implicit in its formulation. Another way to view this algorithm is that its basis functions are a set of orthogonal polynomials evaluated around the unit circle. This paper shows that the PLSCF method can be implemented as an orthogonal polynomial algorithm by a simple substitution of the basis functions. Furthermore, the PLSCF method is extended for applicability to uneven frequency spacing by generating the z-domain basis functions with the same procedure that is used for the traditional Laplace-domain orthogonal polynomials. The paper also illustrates how PLSCF, the orthogonal polynomial algorithm, and their ancestor the rational fraction polynomial method all start from the same place but move to different neighborhoods to do their work.