Abstract
Sole lineshape estimation of non-parametrically computed higher-order derivatives of spectral envelopes in different modes (complex, real, imaginary, magnitude) is investigated. The processed time signals are sums of complex attenuated exponentials (harmonics). The peak parameters of the derivative spectra are directly connected to those of the customary (non-derivative) absorption lineshapes. Crucially, this permits the reconstruction of the latter from the former parameters (the latter are sought since they are unknown, whereas the former are extractable from the derivative envelopes). The cross-checking relationships of the lineshapes for the magnitude modes with the real and imaginary parts of the complex-valued envelopes (total shape spectra) are established. The explicit procedure and the necessary analytical expressions are presented for reconstruction of the exact locations, widths and heights of all the retrieved physical resonances (spectral peaks). These facets are illuminated in the derivative fast Padé transform (dFPT) using its non-parametric version, i.e. without solving the quantification problem (no polynomial rooting, no tackling of eigen-value problems, etc.). Two kinds of illustrations for derivative spectra are reported. One deals with the general Breit-Wigner resonance formula and its first three derivatives. The other is concerned with the dFPT in clinical diagnostics of relevance to breast cancer detection by magnetic resonance spectroscopy. A systematic parallel between these two examples is drawn to highlight, in a stepwise manner, the role of paramount importance played by derivative lineshapes, especially for disentangling overlapping resonances that invariably plague all quantitative analyses of spectra.
Highlights
A very common stumbling block in signal processing across inter-disciplinary fields of scientific, as well as technological/industrial research and development, is the occurrence of unresolvent components in compound spectral peaks [1]
An example of spectral crowding is the abundance of closely overlapped peaks generated from time signals encoded by conventional magnetic resonance spectroscopy (MRS) used, e.g. in medicine through cancer diagnostics [2]
Subsections 4.1 and 4.2 are on the para-diagonal and diagonal rational polynomials, respectively. These polynomial quotients are of the Padé forms that belong to the Lorentzian-type frequency distributions
Summary
A very common stumbling block in signal processing across inter-disciplinary fields of scientific, as well as technological/industrial research and development, is the occurrence of unresolvent components in compound spectral peaks [1]. Irrespective of their origin, are given in their ubiquitous forms of linear combinations of exponentials These can be either oscillatory (i.e. containing sine and cosine functions of time) or non-oscillatory (i.e. without such trigonometric functions). These, invariably exhibit unresolved components that the FFT cannot autonomously quantify to determine the peak positions, widths, heights and phases. To estimate the latter parameters, most techniques resort to post-processing of the envelopes from the FFT by fitting the compound peaks with several Lorentzians or Gaussians. Linear combinations of these two lineshapes are used in fitting as surrogates of the Voigt profile, which is a convolution via integration of the product of a Lorentzian and a Gaussian. There is no way to separate the physical lineshapes from the non-physical component profiles that are used to merely minimize the errors of fitting by way of some, more or less, arbitrary constraints [4,5,6]
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