Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 16

Another drawback of the HLSVD is the fact that from the onset it is designed
to work only with Lorentzian lineshapes, despite the abundant appearance
of nonLorentzians in all spectra encountered in MRS with encoded FIDs. In
practice, the HLSVD is implemented for an overdetermined system of linear
equations. In such a system, the number of equations is larger than the num
ber of the unknowns. Consequently, the redundant input information from
the data matrix leads to the socalled singular eigenvalues. These are the
nearly zero eigenvalues that cause the inverse of the diagonalized Hankel ma
trix to become almost singular. Removing such singular values reduces the
rank of the investigated matrix, and this occurrence is an integral part of the
procedure called the singular value decomposition (SVD).
The fast Pade transform is a powerful and versatile quantummechanical
processor for parametric estimations which simultaneously lifts both of the
mentioned restrictions of the HLSVD. The FPT can be set up in both the
time and frequency domain of estimations. This is achievable through several
numerical algorithms and algebraic methods that invariably give the spectrum
in the form of the quotient of two uniquely determined frequencydependent
polynomials. Such rational polynomials can be in several different forms,
e.g., the most general nondiagonal P L /Q K (K≥L), diagonal P K /Q K , para
diagonal P K−1 /Q K , etc. For example, in the case of the diagonal quotient
where both polynomials have the same degree K, the algebraic condition for
a strictly determined system is given by 2K = N. In other words, here, at
least 2K FID points are needed to obtain K frequencies and K amplitudes,
since in a determined system, the number of equations is equal to the number
of unknown quantities.
While staying in the same computational framework, the FPT comprises
both the usual Pade approximant and the causal Pade z−transform (PzT).
The PA is a wellknown method from numerical analysis and the theory of
approximations whose most important branch is the class of rational functions.
Similarly, the PzT is also wellknown from signal processing and statistical
mathematics [84, 85]. The PA and the PzT variants of the FPT are defined
with their initial convergence regions lying outside and inside the unit circle
in the complex harmonic variable plane, or the z−plane. The acronyms for
these two versions of the FPT are FPT (−) and FPT (+) , respectively.
It is important to emphasize that the FPT (+) does not diverge outside the
unit circle and neither does the FPT (−) inside the unit circle. This crucial
feature is secured by the universal Cauchy principle of analytical continuation.
Such a principle applied to the FPT (+) prolongs its initial convergence region,
which is inside the unit circle, to the complementary domain outside the unit
circle. Likewise, the same Cauchy principle when used in connection with the
FPT (−) , extends its initial convergence region, which is outside the unit circle,
to the complementary domain inside the unit circle. Thus, in the case of both
the FPT (+) and FPT (−) , their respective initial and extended convergence
regions cover the entire complex frequency plane, with the exception of poles
as points of singularity.