Bandlimited Signal Processing Methods Research

Numerical approximation is a fundamental part in modern numerical analysis.Using the signal sampling{f(xj)}to reconstruct f is essentially a numerical approximation problem. The e?ectiveness of the reconstruction process is determined by what approximation model is selected or namely di?erent models(such as compact support generator function B spline(frame) FFT(orthonormal basis) wavelet etc.)would lead to di?erent reconstruction results. In this paper we conduct a comprehensive theoretical analysis about the reconstruction of bandlimited signals. In the analysis we discuss B spline,cubic spline interpolation multiresolution analysis Fourier transform and present some specific instances. The main contributions of this paper can be summarized as follows.(1) We select the B spline function which has compact support property as the generator of translation invariant space and use the algorithm proposed by Section3.1.3 to reconstruct signal and analyze its error level.We analyze the advantages and the ill-conditioned problem of B-spline as the generator of translation invariant space.There is no special literature indicates that B spline as the generator of translation invariant space can cause ill-conditioned problems.This paper not only points out the ill-conditioned problems and analyzes the specific reasons the e?ective solution is given.Some numerical examples are studied and analyzed.(2) We use the cubic spline interpolation to approximate the signal and present a comparable analysis between the algorithm and the B spline fitting method.(3) We use FFT and DWT for compression and denoising on audio signal respectively and analyze their e?ciency and e?ectiveness.