On the frequency mapping of modal parameters identification

With borrowing the complex orthogonal polynomial (OP), the numerical condition of the normal matrix of the rational fractional polynomial method is insensitive to frequency scaling. The other two concerns are, the first, transforming coefficients based on OPs into coefficients based on monomials, and the second, computing polynomials. The leading term coefficients of OPs usually blast approximately exponentially as the order increases, so are the diagonal entries of the transitional matrix. This can be contributed to that the orthogonal relationship among the selected OPs is over the frequency band [0,1]. By examining the recursion relation of the Legendre polynomials, we found that mapping the actual frequency vector into [0,2] can efficiently avoid the aforementioned exponential trend. Moreover, in the case of sub-band fitting, an empirical formula for frequency mapping was proposed. Numerical simulation demonstrates that this formula, not only works well in the case of a uniform weight function, but also three typical cases of non-uniform weight cases. The second concern, overflow in computing polynomials with new mapping, can be overcome by the Horner's scheme for a general polynomial and three-term recurrent algorithm for an OP.