Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 134

of the Pade spectrum given by the polynomial quotients. Hence stability of
the Pade estimation, as per (2.226).
This stabilization condition is the signature of the determination of the
exact total number K of resonances. If the quantification problem is solved
first, then the ensuing stability in (2.226) will be due to the constancy of
all the spectral parameters that are reconstructed exactly by the FPT from
the given FID. Of course, once all the parameters are reconstructed, it is not
mandatory to search for saturation of the corresponding spectra in the FPT.
In such a parametric signal processing within the FPT, the exact number of
genuine resonances is determined by monitoring solely the constancy of all
the found spectral parameters.
In the nonparametric version of the FPT, the exact number of genuine
resonances is determined without spectral analysis, i.e., with no recourse
to reconstruction of peak parameters. In such a case, the Pade spectra
P ± K ′ /Q ± K ′ are explicitly computed. This is done in a given frequency window
or in the whole Nyquist range by systematically increasing the running or
der K ′ . When saturation of these computed spectra occurs at a certain value
of K ′ , the true number K of genuine resonances is extracted via K ′ = K.
Here, the constancy of spectra is not achieved through Froissart doublets,
because poles and zeros are not extracted at all. Rather, the polynomi
als P ± K+m and Q ± K+m de facto contain certain factored polynomials S ± m , i.e.,
P ± K+m = P ± K S ± m and Q ± K+m = Q ± K S ± m . Thus, the sought saturation of spectra
occurs by cancellation of the common factors

S ± m (z ±1 ), in the Pade quotient

P ± K+m /Q ± K+m ={P ± K S ± m }= P ± K / Q ± K .

Even in the nonparametric FPT, we may think of Froissart doublets being
implicitly present in this cancellation of S ± m as a whole. Of course, whether
or not we solve the characteristic equations

}/{Q ± K S ± m

S ± m (z ±1 ) = 0, polynomials S ± m do

inherently contain their roots z ±1
s

(1≤s≤m) such that their canonical forms

S ± m

s=1 (z ±1 −z ± s ) implicitly exist. Therefore, given that S ± m disappear al
together as the common factors in the quotients P ± K+m (z ±1 )/Q ± K+m (z ±1 ), this
occurrence may be viewed as implicit cancellations of the hidden/latent joint
terms z ±1 −z ± s (1≤s≤m) in the numerators P ± K+m (z ±1 ) and denominators
Q ± K+m (z ±1 ). Here, the same zeros{z ±1

m

∼

}that satisfy the characteristic equa
tion S ± m (z ± s ) = 0, act simultaneously as zeros and poles depending whether
they appear in the numerator or denominator of the quotients P ± K S ± m / Q ± K S ± m .
In this way, one may say that irrespective of whether poles or zeros{z ±1
s

s

}
are explicitly available or not, Froissart doublets implicitly act by effectively
reducing P ± K+m /Q ± K+m to P ± K / Q ± K . Such a lowering of the degrees of the char
acteristic polynomials from K +m to K, with parametric and nonparametric
estimations within the FPT (±) promotes the concept of Froissart doublets
to the status of an e cient method for reduction of the dimensionality of
the problem. Note that the problem of dimensionality reduction in itself is
a very important issue in the field of system theory, especially when dealing
with large degrees of freedom [5]. Large systems are di cult to handle in any