Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 133

makes the total shape spectrum converge, on panel (iv) in Fig. 3.11 , even in
the absence of the peak k = 11 and the overestimation of the peak k = 12.
This also clearly shows that it is tenuous to use an envelope spectrum in
attempts to describe the corresponding component spectra, as is customarily
attempted in every fitting routine used in MRS and beyond.

3.6 Preview of illustrations for the concept of Froissart

doublets

A critically important feature of the FPT is its ability to reconstruct exactly
the total true number of physical harmonics in the given FID. Each such FID
is a sum of damped complex exponentials with stationary and nonstationary
(polynomial type) amplitudes, associated with nondegenerate (Lorentzian)
and degenerate (nonLorentzian) spectra. This type of FID is ubiquitous
across the interdisciplinary research fields, including MRS [5]. In practice,
determination of the exact number of resonances can be accomplished via
Froissart doublets [44] or polezero cancellations [45]. The total number of
genuine resonances is given by the degree K of the denominator polynomials
Q ± K . The only known information about this degree K is that it must obey
the inequality 2K≤N.
Algebraically, the 2K unknown spectral parameters (frequencies and am
plitudes) require at least 2K signal points from the whole set of N available
FID entries, as mentioned. To determine K unequivocally, we compute a
short sequence of the FPTs by varying the degree K ′ of the polynomials in
the Pade spectra{P ± K ′ /Q ± K ′ }until all the results stabilize/saturate. When
this happens, e.g., at some K ′ = K ′′ , we are sure that the true number K is
obtained as K = K ′′ . If we keep increasing the running order K ′ of the FPT
beyond the stabilized value K, we would always obtain the same results for
K ′ = K +m and for K using any positive integer m according to (2.226). The
mechanism by which this is achieved (i.e., the maintenance of the overall sta
bility, including the constancy of the value of the true number of resonances)
is provided by polezero cancellations or Froissart doublets [44].
By not knowing the exact number K in advance (as in encoding via, e.g.,
MRS), we would keep increasing the order K ′ = K + m, and this would lead
to extra zeros from P ± K+m and Q ± K+m . All the zeros of P ± K+m and Q ± K+m
are the respective zeros and poles in the spectra P ± K+m /Q ± K+m because these
latter rational polynomials are meromorphic functions. These extra zeros
and poles are spurious, as they cannot be found in the input FID, which is
built from K true harmonics alone. However, such spurious poles and zeros
in the spectra P ± K+m /Q ± K+m for m > 1, beyond the stabilized number K of
resonances, will automatically cancel each other, because of the special form