Explicit extraction of absorption peak positions, widths and heights using higher order derivatives of total shape spectra by nonparametric processing of time signals as complex damped multi-exponentials

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.