On determining the unknown band-parameter and truncated sinc series coefficients from a time sampled band-limited function

摘要

In this paper, we address the problem of either approximating or estimating the band-parameter B^ and a coefficient vector C^=[c^−(m−1)/2,c^−(m−1)/2+1,…,c^(m−1)/2]T of specified odd dimension m so that X(t)≈∑k=−(m−1)/2(m−1)/2c^ksinc(2B^t−k), in which the band-limited real-valued sampled function X(t)∈C2(−∞,∞) with true band-parameter B† ∈ [r1,r2]⊂R + , the positive reals.We formulate the problem as the minimization of the over-determined nonlinear least squares function: f:R×Rm→R+, f∈C2([r1,r2]×Rm), f(B,C)=(1/2)∥H(B)C−X∥22, H(B) an n × m matrix of sinc function values at an odd number, n, of equally spaced sampling points such that n > m + 1.We show that this nonlinear least squares problem in m + 1 variables reduces to minimizing a function of a single real variable, f*(B), f* ∈ C2([r1,r2]). In both noise-free and Gaussian error-perturbed data cases, an FFT is first applied to a vector X of equally spaced time samples. The FFT is followed by a simple threshold search on the transform spectrum. This search yields an accurate first guess of B^=argminB∈[r1,r2](f*(B)) that initiates a Gauss–Newton iterative algorithm.Monte Carlo performance statistics for one hundred sample problems in both the approximation and estimation cases are tabulated. These statistics are the mean, standard deviation, the median of the relative band error, the root mean square coefficient error, the relative energy error, an approximate Cramer–Rao error bound in B^ as an estimate of B†, and the mean and standard deviation of the approximate Cramer–Rao normalized error of the estimate in the estimation cases.