Digital Signal Processing Reference
In-Depth Information
capable of describing these spectral backgrounds from MRS with full accu
racy and does so quite naturally by reliance upon the offdiagonal GreenPade
functions G
L,K
(z
±1
) for L > K. This is achieved by using the relationships
P
±
L
(z
±1
)
Q
±
K
(z
±1
)
= B
±
L−K
(z
±1
) +
A
±
K
(z
±1
)
Q
±
K
(z
±1
)
.
(2.172)
Functions B
±
L−K
(z
±1
) are the polynomials of degree L−K > 0 and they
can describe the background contributions to typical spectra from MRS. The
remaining genuine resonances are described by the diagonal GreenPade func
tions A
±
K
(z
±1
)/Q
±
K
(z
±1
) where A
±
K
(z
±1
) are the numerator polynomials of the
same degree K as the denominators Q
±
K
(z
±1
). Of course, this latter ratio of
polynomials could also be replaced by the paradiagonal GreenPade functions
A
±
K−1
(z
±1
)/Q
±
K
(z
±1
). In such a case, the degree of the new polynomial for
the background should be increased accordingly by 1
A
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
.
P
±
L
(z
±1
)
Q
±
K
(z
±1
)
= B
±
L−K+1
(z
±1
) +
(2.173)
The sole purpose of writing the expressions (2.172) and (2.173) is to indicate
that a modeling of the background due to macromolecules is achieved natu
rally by a simple division in the Pade quotients. This division can factor out
a background polynomial and the remaining quotient is responsible for the
description of the physical resonances. When a given spectrum to be modeled
possesses a rolling and smoothly varying background, as encountered in MRS,
one would not first extract P
±
L
(z
±1
)/Q
±
K
(z
±1
) for L > K and then perform
divisions as done in, e.g., (2.172) to single out macromolecules from the con
tributions due to other resonances. Rather, from the outset, one should set up
the Pade approximants with the distinct contributions from the background
and the remaining spectrum via the model
G
N
(z
−1
)≈B
±
L−K
(z
±1
) +
A
±
K
(z
±1
)
(2.174)
Q
±
K
(z
±1
)
with the diagonal polynomial ratio, or with the corresponding paradiagonal
quotient
A
±
K−1
(z
±1
)
Q
±
K
(z
±1
)
.
G
N
(z
−1
)≈B
±
L−K+1
(z
±1
) +
(2.175)
In such a procedure, the expansion coe
cients of the background polynomi
als B
±
L−K
(z
±1
) and those of the remaining two polynomials in the quotients
A
±
K
(z
±1
)/Q
±
K
(z
±1
) can be extracted simultaneously from the system of lin
ear equations that are implicitly present, e.g., in (2.174). By this design of
the FPT, the background for large macromolecules and the main physical
resonances are adequately treated on the same footing without any artificial
patching of one contribution to the other.
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