In vivo magnetic resonance spectroscopy by the fast Padé transform.

The fast Padé transform (FPT) is thoroughly illustrated on two in vivo time signals encoded at 4 T and 7 T via magnetic resonance spectroscopy (MRS). The exact quantum-mechanical spectrum as the Green function series truncated at any partial sum reduces to the unique quotient of two polynomials, which is the FPT. In this Green function as a Maclaurin series in powers of the harmonic variable, the expansion coefficients are the time signal values as damped complex-exponentials with stationary and non-stationary amplitudes for non-degenerate (Lorentzian) and degenerate (non-Lorentzian) spectra. This is automatically shared by the FPT to represent an enormous advantage over the Hankel-Lanczos singular value decomposition (HLSVD) which works only for Lorentzian spectra. Moreover, the resonance amplitudes in the FPT are obtained analytically, rather than solving a system of linear equations as done in the HLSVD. We use two variants of the FPT, initially defined inside and outside the unit circle, but extended automatically to the whole complex frequency plane by the Cauchy analytical continuation. The converged spectra from these two variants of the FPT are found to give the same results, within the experimental background noise level, and this represents an intrinsic cross-validation of the findings and the error analysis.