Multiscale filter diagonalization method for spectral analysis of noisy data with nonlocalized features

Abstract

Stability and performance of the filter diagonalization method (FDM) for harmonic inversion [i.e., fitting a time signal by C(t)=∑k dke−itωk] of noisy data are examined. Although FDM is capable to extract accurately the parameters of narrow spectral peaks, in the presence of broad peaks (or strong background spectrum) and noise, the FDM ersatz spectrum, i.e., I(ω)=∑kdk/(ωk−ω), maybe distorted in some regions and be sensitive to the FDM parameters, such as window size, window position, etc. Some simple hybrid methods, that can correct the ersatz spectrum, are discussed. However, a more consistent approach, the multiscale FDM, is introduced to solve the instability problem, in which some coarse basis vectors describing (in low resolution) the global behavior of the spectrum are added to the narrow band Fourier basis. The multiscale FDM is both stable and accurate, even when the total size of the basis (i.e., the number of coarse plus narrow band basis vectors) used is much smaller than what would previously be considered as necessary for FDM. This, in turn, significantly reduces the computation cost. Extension of the 1D multiscale FDM to a multidimensional case is also presented.