On semi-classical questions related to signal analysis

Abstract

This study explores the reconstruction of a signal using spectral quantities associated with some self-adjoint realization of an h-dependent Schrödinger operator −h 2 d 2 dx 2 − y(x), h > 0, when the parameter h tends to 0. Theoretical results in semi-classical analysis are proved. Some numerical results are also presented. We first consider as a toy model the sech 2 function. Then we study a real signal given by arterial blood pressure measurements. This approach seems to be very promising in signal analysis. Indeed it provides new spectral quantities that can give relevant information on some signals as it is the case for arterial blood pressure signal.

Spectral data de‐noising using semi‐classical signal analysis: application to localized MRS

“…This is achieved by choosing a large value of the parameter h to minimize N h , and iteratively lowering it to increase N h to include more eigenfunctions until the noiseless spectrum s h is accurately reconstructed. This is in agreement with the properties of the eigenfunctions ψ ( f ), where it has been shown that the n th squared eigenfunction has n wells, implying that the first squared eigenfunction is represented by a single well function localized at its maximum, the second squared eigenfunction has two wells, and so on . By analogy, one can see that the squared eigenfunctions with low n h values represent, in a broad manner, the profiles of the peaks of the spectrum, whereas the functions with high n h values characterize more the fine details of these profiles.…”

“…The SCSA method, pioneered as a classification method to discriminate between pathological and control patients in arterial blood pressure data analysis, [17][18][19] has been theoretically evaluated as a de-noising technique in biomedical signal processing. 20,21 The method employs the discrete spectrum of the Schrödinger operator to decompose the input signal into a set of squared eigenfunctions, with shapes derived from the potential function of the operator.…”

“…It is found that, when h tends to zero, the reconstructed spectrum s h converges towards the true spectrum s. This is in compliance with the semi-classical properties of the Schrödinger operator where the number N h of negative eigenvalues and consequently the number of eigenfunctions increases when h decreases. [17][18][19][20][21] Since the main goal of the technique is to separate the useful information from noise, it is preferable to represent the input signal with the minimum number possible of eigenfunctions (SL functions). This is achieved by choosing a large value of the parameter h to minimize N h , and iteratively lowering it to increase N h to include more eigenfunctions until the noiseless spectrum s h is accurately reconstructed.…”

“…This is in agreement with the properties of the eigenfunctions ψ(f), where it has been shown that the nth squared eigenfunction ψ 2 nh f ð Þ has n wells, implying that the first squared eigenfunction ψ 2 1h f ð Þ is represented by a single well function localized at its maximum, the second squared eigenfunction ψ 2 2h f ð Þ has two wells, and so on. [17][18][19][20][21] By analogy, one can see that the squared eigenfunctions with low nh values represent, in a broad manner, the profiles of the peaks of the spectrum, whereas the functions with high nh values characterize more the fine details of these profiles. Therefore, by incorporating squared eigenfunctions with higher nh values, the peaks are accurately represented into the reconstructed spectrum s h .…”

“…This is achieved using the iterative SCSA algorithm described below. [17][18][19][20][21] 1. Compute the eigenvalues and the eigenfunctions of the discrete Schrödinger operator using Equation (2) represents the differentiation matrix of the Laplacian, computed using a Fourier pseudo-spectral method, 17 and Y is a diagonal matrix whose entries are the values of the MR spectrum s.…”