Chen K.

Traditionally, frequency mapping must be carried before curve-fitting to render a better numerical condition of the normal matrix. With borrowing the complex orthogonal polynomial, however, the normal matrix condition of the rational fractional polynomial method (RFPM) is insensitive to frequency scaling. The other two concerns are transforming coefficients between bases of orthogonal polynomials and monomials, and computing polynomials. The leading term coefficient of the orthogonal polynomial generated over the frequency scale [0,1] is approximately exponential, so is the diagonal element of transfer matrix. By examining the recursion of Legendre polynomials, which the asymptotic of the Forsythe polynomials, a scale [0,2] can efficiently avoid the aforementioned exponential trend. Concerning the sub-band fitting, a numerically empirical formula of frequency mapping was proposed. This formula is a function of the ratio of the upper bound to the low band of the frequency band. Numerically results demonstrate this formula, not only work well in the case of a constant weight vector, but also 3 typical cases of weight vectors. By the aid of the Horner scheme for a general polynomial and recursion algorithm for an orthogonal polynomial, computing a polynomial is also insensitive the frequency scale. In conclusion, the mapping scale proposed here, not [0, 1] can give better numerical condition.