Analytical continuation by numerical means in spectral analysis using the Fast Padé Transform (FPT)

A numerical method is used to assess the practical usefulness of the Cauchy concept of analytical continuation of a formal power series T(z−1)=∑n cnz−n for a frequency spectrum, which is originally divergent i.e. undefined for |z|<1. Here, z is a dependent variable, z=exp(iωτ), where ω is any real or complex angular frequency and τ is a fixed parameter in the role of a real time lag. The present implementation of analytical continuation is accomplished for a class of response rational functions given as a ratio of two polynomials. For a given Taylor or Laurent series T(z−1) in the expansion variable z−1, we introduce a rational polynomial A+(z)/B+(z), not in the original, but rather in the reciprocal variable z. Nevertheless, the ansatz A+(z)/B+(z) is still recognised as a variant of the Padé approximant defined in the complementary convergence or stability region inside the unit circle (|z|<1) relative to the input series which does not exist for |z|<1. For Iω>0, the rational polynomial A+(z)/B+(z) is equivalent to the causal z-transform whose inverse Fourier integral contains only the exponentially decaying components as the building blocks of generic time signals. Therefore, the response function A+(z)/B+(z) is very well suited for adequate physical representations of both Lorentzian and non-Lorentzian spectra. For the purpose of illustration, we presently perform parametric estimations by generating magnitude spectra |A+(z)/B+(z)| using experimentally measured in vivo time signals from Magnetic Resonance Spectroscopy. An equivalent parametric analysis is also done in the time domain. Many experimentally measured data stem from mechanisms that intrinsically describe the time evolution of the studied system. Such physical time functions have their customary meaning as probability amplitudes and, therefore, are expected to decay exponentially with the passage of time. The most prominent examples are auto-correlation functions in signal and image processing or activity curves in decays of radionuclides encountered in e.g. Positron Emission Tomography (PET), etc. For the given experimental data, the task is to retrieve the exact number of the constituent components of the measured time functions as well as the individual parameters of each exponential, i.e. the amplitudes and the time decaying constants. The unique answer to this difficult inverse problem is shown to be the Padé approximant which, as such, can be used in very diverse applications ranging from spectral decomposition of time signals to spectroscopic analysis of PET data leading to a novel medical tool called spectroscopic PET as presently acronymed by sPET.