Indirect derivative fast Padé transform applied to magnetic resonance spectroscopy

This article is on lineshape profiles by indirect derivatives of spectra reconstructed from time signals or free induction decay (FID) data encoded by magnetic resonance spectroscopy (MRS). Only the nonparametric reconstructions (shape estimations) are performed using for the first time the indirect version of the derivative fast Padé transform (dFPT). Here, the frequency-dependent derivative operator is not applied straight to the spectrum of the precomputed nonderivative fast Padé transform (FPT). Rather, the derivative operator is applied to the frequency-dependent complex exponential harmonic variable in the starting MacLaurin expansion, generating therein a time power function (monomial). From this derivative-modified MacLaurin expansion, the FPT is constructed. Thus, the indirect dFPT can equivalently be conceived as the FPT which processes the so-called FID moments defined as the products of the time monomial and the FID. The time monomial leads to ill-conditioning by enhancing noise from encoded FIDs. It is shown that this problem is automatically solvable in the indirect dFPT without any regularizing optimization. Moreover, high concordance is recorded in the results from the indirect dFPT with a constant (static) or derivative-adapted (dynamic) apodizations or by ignoring both attenuations of the FID moments.