Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 145

are both formulated as the unique polynomial quotients in their respective in
dependent variables z and 1/z. Thus, as mentioned, the initial convergence
regions of the FPT (+) and FPT (−) are inside (|z|< 1) and outside (|z|> 1)
the unit circle, respectively. As rational analytical functions, both polynomial
quotients expand their initial validity to the entire complex plane of the har
monic variables according to the Cauchy principle of analytical continuation.
Thereby, the FPT (+) and FPT (−) can be computed everywhere with the ex
ception of the singular points given by the poles that represent the locations
of fundamental frequencies from the examined FID.
Herein, the original Maclaurin series is defined in the variable 1/z. Con
sequently, this exact Green function expansion is convergent for|z|> 1 and
divergent for|z|< 1, respectively. Thus, the FPT (+) analytically continues a
divergent series forcing it to converge for|z|< 1. In contrast, the FPT (−) has
the relatively easier task of accelerating the convergence rate of an already
convergent series for|z|> 1.
It should be pointed out, however, that analytical continuation entails more
than transforming an initial series in terms of 1/z into another series in powers
of z. Analytical continuation also denotes that a function originally defined,
e.g., only for real frequencies, can be mapped into another more general func
tion of complex frequencies. Both the FPT (+) and FPT (−) accomplish this
task. Therefore, in fact, they both perform analytical continuation: FPT (+)
for|z|> 1 and FPT (−) for|z|< 1. Insofar as the defining polynomial quo
tients from the FPT (+) and FPT (−) are expanded in their own Maclaurin
series, they coincide with the original Maclaurin expansion at a fixed order
depending on the degrees of the numerator and denominator polynomials.
In other words, all the converged results from the FPT (+) and FPT (−) are

expected to be identical. This is the basis of the uniqueness of the FPT.
This internal crossvalidation of the FPT is confirmed with numerical il
lustrations by carrying out the exact reconstructions of all the input spectral
parameters. Furthermore, as discussed, the FPT accomplishes this task by
using a quarter of the full MR time signal.
For a given MR time signal, the spectrum in the FPT can be formulated,
for example, in the diagonal form as the quotient of two unique polynomials
P K /Q K . Then, the sought exact number of resonances due to the fundamental
harmonics from the given FID can be determined unequivocally in the FPT
by the degree K of the denominator polynomial Q K . In sharp contrast, as said
earlier, all the other estimators used in MRS and especially fitting algorithms
can only guess the true number K. We have demonstrated that the FPT
takes a completely different approach by viewing the exact number K of
resonances as yet another spectral parameter to be recovered together with the
complex frequencies and amplitudes in the process of solving the quantification
problem. We should herein emphasize that the entire strategy of the FPT is
completely orthogonal to the usual practice with attempted quantifications in
MRS. Therefore, the Pade methodology actually represents a paradigm shift
in the field of signal processing.