Machine accurate quantification and illustrated signal-noise separation - Signal Processing in MRS with Biomedical Applications - page 224

of absolute values of amplitudes of genuine resonances than in the discussed
case of harmonic variables. This is confirmed in Figs. 6.11 and 6.14 for the
FPT (+) and FPT (−) , respectively.

6.6 Practical significance of the Froissart filter for exact

signal-noise separation

In chapter 4, we presented the theory of quantummechanical spectral anal
ysis based upon the Pade approximant, the Lanczos algorithm as well as on
their combination called the PadeLanczos approximant and Lanczos contin
ued fractions. These are all equivalent. Lanczos continued fractions are of the
type of contracted continued fractions that contain two times as many expan
sion terms as the ordinary continued fractions for the same order or rank. In
signal processing, the Pade approximant is called the fast Pade transform. It
has its two equivalent versions, the FPT (+) and FPT (−) , that are initially de
fined to lie inside and outside the unit circle for complex harmonic variables.
By the Cauchy analytical continuation, these two versions are also defined
everywhere in the complex plane with the exception of the poles. This could
be considered analogous to the typical outgoing and incoming boundary con
ditions ingrained in the standard Green function.
The FPT (+) and FPT (−) are equivalent to Lanczos continued fractions of
the even and odd order, respectively. These two variants of the FPT can ex
actly reconstruct the input spectral parameters of every resonance from noise
less and noisy time signals. When convergence is achieved in both variants,
the results of the FPT (+) and FPT (−) are identical. This provides an indis
pensable intrinsic check of the validity of the FPT. We have proven through
illustrations that the FPT is an extremely reliable method for quantifying
noisecorrupted FIDs reminiscent of those encoded by in vivo MRS.
A critical hurdle for spectral analysis is how to unambiguously separate
genuine from spurious information in FIDs. We have shown that this exceed
ingly di cult problem can indeed be solved via the powerful concept of exact
signalnoiseseparation using Froissart doublets [44] or polezero cancellations
[45]. This separation is unique to the FPT, due to its polynomial quotient
form P K /Q K of the frequencydependent response function, which is the total
Green function of the investigated system. The true number K G of genuine
resonances, as the exact order or rank K ex of the FPT with K G = K ex , is re
constructed by reaching the constancy of P K /Q K when the polynomial degree
K is systematically increased. By augmenting the 'running order' K above
the plateau attained at K = K ex , the same values of P K /Q K are obtained
via P K ex +m /Q K ex +m = P K ex /Q K ex (m = 1, 2,...) as per (2.226). This can
only be possible when, for K > K ex , all the new poles and zeros coincide