The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 84

whole from the Pade quotient P K+m /Q K+m .
As discussed, if we limit ourselves to estimation of total shape spectra
alone for a given FID with the (unknown) number K, we would use the
nonparametric FPT and compute P K ′ /Q K ′ at a selected frequency grid by
gradually increasing the polynomial degree K ′ until a saturation occurs at
K ′ = K + m (m = 1, 2, 3,...), as in (2.226). The found constancy in the total
shape spectra determines the exact number K of the components in the con
sidered FID. Thus, remarkably, even the nonparametric FPT can determine
with certainty at least one parameter: the total number K of the FID's com
ponents. Of course, to extract all the components themselves from the FID,
the parametric FPT is needed when staying within the Pade methodology.
Given a time signal with K harmonics, we have 2K unknowns, K fundamen
tal frequencies{ω k }and K corresponding amplitudes d k (k = 1, 2, 3,...,K).

If we apply the parametric FPT, we need two systems of linear equations,
each yielding K solutions for the expansion coe cients of P K and Q K . In this
case, the problem of spectral analysis is said to be “determined” by having
the 2K unknown frequencies and amplitudes to be computed by means of the
extracted 2K Pade polynomial coe cients that are themselves obtained from
their linearly independent equations.
In practice, with the unknown K, we can pick up a running order K ′ of the
FPT. For example, with K ′ > K, we would have an “overdetermined” spec
tral analysis with more equations than unknowns. For, e.g., K ′ = K+m (m >
1), the system of equations for the Pade polynomial coe cients will possess m
linearly dependent solutions that would lead to m Froissart doublets. Thus, in
the parametric FPT, linear dependence of the system of equations for the Pade
polynomial coe cients can be connected with the associated situations where
the problem of spectral analysis is overcomplete. However, as stated, Frois
sart doublets regularly also appear in the underdetermined case (K ′ < K)
with fewer equations for the polynomial expansion coe cients than the un
known spectral parameters.
Linear dependence of the systems of equations for the expansion coe cients
of the Pade polynomials, as a mechanism by which Froissart doublets are
produced, also remains in place for noisecorrupted FIDs, as stated. Namely,
no estimator is able to model noise in a fully adequate manner, but the FPT
attempts to do this by solving a larger system of equations (K + m ′ ,m ′ > 1)
than is actually needed for the Pade polynomial coe cients.
Such “larger than needed” systems possess m ′ linearly dependent equations
that lead to Froissarttype spuriousness. In practice, we have m ′ > m where
m is associated with noiseless FIDs. As mentioned, the striking difference be
tween the appearances of Froissart doublets for noisefree and noisecorrupted
FIDs is that the former spurious features are distributed in a relatively orderly,
regular manner in the complex plane of the harmonic variables (z), whereas
the locations of the latter polezero pairs are irregular and random, but still
near the circumference|z|= 1 of the unit circle (for detailed illustrations, see
Refs. [11, 34, 35] and the present chapters 3 and 6).