Optimal free parameters in orthonormal approximations

Abstract

Abstract-We consider orthonormal expansions where the basis functions are governed by some free parameters. If the basis functions adhere to a certain differential or difference equation, then an expression can be given for a specific enforced convergence rate criterion as well as an upper bound for the quadratic truncation error. This expression is a function of the free parameters and some simple signal measurements. Restrictions on the differential or difference equation that make this possible are given. Minimization of either the upper bound or the enforced convergence criterion as a function of the free parameters yields the same optimal parameters, which are of a simple form. This method is applied to several continuous-time and discrete-time orthonormal expansions that are all related to classical orthogonal polynomials.

Optimum waist of localized basis functions in truncated series employed in some optical applications

“…Optimal free parameter selection in orthonormal bases can be accomplished by minimizing the upper bound of the quadratic truncation error for a class of functions. Den Brinker et al in [3] used this approach to find the optimal waist parameter for several bases, including HG and LG. This suggestion, although interesting for signal processing applications, has several major shortcomings.…”

“…Once the set of basis functions is chosen, there is usually a set of free parameters left to be determined. In particular, the waist parameter for localized basis functions can play an important role in the accuracy of the expansion for nonsmooth functions, and its optimum determination has attracted much attention in different applications [1][2][3][4][5][6][7]. To determine the optimum waist for projecting a specific function on a specific set of localized basis functions, different suggestions have been proposed in the literature.…”

“…normalized frequency of an optical waveguide, to determine the basis function parameters [8][9][10][11]. Some others have further contemplated on the problem for a class of signals and have minimized an upper bound for the squared error (e.g., see [3] and references therein).…”

Conditions for generation of stable and accurate hybrid TD-FD MoM solutions

“…They can be expressed as (12) where is the Hermite polynomial. The Hermite polynomial can be computed recursively through (13) Using (12), associate Hermite functions can be calculated easily by the recursive relationship (14) Since the associate Hermite functions are the eigen functions of the Fourier transform operator, we have their Fourier transforms as (15) Thus, if we can expand the function by the orthogonal basis , its Fourier transform can be expressed by adding up the terms with the same coefficients.…”

“…This is due to the fact that the basis functions are high-order polynomials with windowed functions. The optimal parameters can be obtained if is small or if we know the original data in the whole domain [8]- [12]. Searching for optimal , and analytically in this application of EM computation is a nonlinear problem and, in most practical cases, it is not feasible.…”