Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 149

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Harmonic transients in time signals

A large number of parametric estimators [146]-[205] can be used for tackling
spectral analysis of time signals. In so doing, longtime practice across inter
disciplinary fields firmly established that the most adequate strategy for this
purpose is the powerful methodology of rational approximations. Moreover,
this same practice has also found that the most frequently employed ratio
nal function is the Pade approximant to which many of the existing methods
[146]-[205] can either be reduced or become equivalent. This is a good moti
vation for devoting this chapter to the theory of rational approximations with
the highlight on the leading role of the Pade approximant in this field.

4.1 Rational response function to generic external per-

turbations

From the outset, the quantification problem in MRS is the same as the har
monic inversion from quantum theory of resonances and spectroscopy [6, 92].
Thus, the general quantummechanical relaxation formalism can advanta
geously be used to solve the quantification problem in MRS. To this end,
we shall use the quantummechanical parametric signal processing via the
frequencydependent Green function G(ω). With this strategy, upon its algo
rithmic implementation through the FPT, it becomes feasible to reconstruct
exactly all the spectral parameters [10, 11, 24, 34]. The FPT provides the
frequency spectrum as the unique ratio of two polynomials

G L,K (ω)∝ P L (ω)

Q K (ω) .

(4.1)

The two subscripts L and K are written in the label for the Green function
G L,K to indicate the degrees of the numerator and denominator polynomial,
respectively. As discussed, for the given degrees L and K, these polynomial
quotients can yield the offdiagonal (L = K) and the diagonal (L = K) vari
ant of the FPT. Here, the particular case L = K−1 of the offdiagonal FPT is
called the paradiagonal fast Pade transform. In practice, the most frequently
used versions are the paradiagonal and the diagonal FPT because of their

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