Digital Signal Processing Reference
In-Depth Information
2.6 Description of the background contribution by the
off-diagonal fast Pade transform
Mathematically, the values of the degrees L and K of the polynomials P
±
L
(z
±1
)
and Q
±
K
(z
±1
) in the general GreenPade functions G
L,K
(z
±1
) from (2.164)
can be any positive integer. Of course, the sum L + K cannot exceed the
total number of available signal points, as indicated in (2.164). Physically,
the input FID has a definite number of harmonics, and this number is equal
precisely to K which is not known prior to spectral analysis. However, this
does not mean at all that K should be guessed as regularly done in all fitting
recipes throughout MRS, and beyond, as well as in the HLSVD. Quite the
opposite practice is recommended, by which the unknown true value K is
unambiguously reconstructed from the input FID together with retrieval of the
unknown fundamental frequencies and amplitudes, as accomplished exactly in
the FPT.
Two cases are particularly important in versatile applications of the FPT.
These are the diagonal (L = K) and paradiagonal (L = K−1) variants of
the fast Pade transform as denoted by
G
±
K
(z
±1
)≡G
±
K−1,K
(z
±1
)
G
±
K
(z
±1
)≡G
±
K,K
(z
±1
)
(2.168)
P
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
.
G
±
K
(z
±1
) =
P
±
K
(z
±1
)
G
±
K
(z
±1
) =
(2.169)
Q
±
K
(z
±1
)
Here, the diagonal GreenPade functions G
±
K
(z
±1
) can be given by a sum of a
constant term and the corresponding paradiagonal form, labeled as G
±
K
(z
±1
)
P
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
G
±
K
(z
±1
)
G
±
K
(z
±1
) =
G
±
K
(z
±1
) = b
±
K
+
(2.170)
K−1
=
p
±
K
q
±
K
P
±
K−1
(z
±1
) =
p
±
r
z
±r
p
±
r
= p
±
r
−b
±
K
q
±
r
b
±
K
.
(2.171)
r=r
±
All absorption spectra constructed from FIDs encoded via MRS are invariably
seen to exhibit quite pronounced backgrounds beneath the main physical res
onances. These backgrounds are usually interpreted as stemming from large
macromolecules that all the fitting techniques from MRS treat roughly by
adjusting some 3-4 expansion coe
cients of ad hoc 3rd or 4th degree least
square polynomials. Such background polynomials are subsequently patched
in an artificial way to the employed basis set for the remaining resonances
in the fitted spectra.
As opposed to this empirical procedure, the FPT is
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