Biomedical Engineering Reference
In-Depth Information
of general FIDs as well as those in MRS. Time-frequency duality implies that the
inverse fast Pade transform (IFPT) computed from the Pade spectrum from (
25.1
)
will yield the
damped complex exponentials. Similar to the
inverse FFT, the ability of the IFPT to exactly retrieve the input FID irrespective of
the level of noise corruption is precisely the feature which justifies the use of the
term “transform” in the FPT. This can be seen by casting the Pade spectrum from
(
25.1
) into its equivalent form of continued fractions (CF) [
1
]. Namely, every signal
point
fc
n
g
as a sum of
K
can be exactly reconstructed for any noise level from the
general analytical expression for the expansion coefficients
fc
n
g .0 n N 1/
in the CF [
1
]. This
determines that the optimal mathematical model for the frequency spectrum of these
time signals is prescribed quantum-mechanically as the finite-rank response Green
function in the form of the unique ratio of two polynomials, i.e., the FPT. Similarly
to the time domain, where the Schr odinger time evolution operator predicts the
FID as the sum of damped exponentials, the same quantum physics automatically
prescribes that the frequency spectrum is given by the Green function via the Pade
quotient of two polynomials. This is the true origin of the unprecedented algorithmic
success of the FPT, via its demonstrable, exact reconstructions [
1
,
2
].
To cross-validate its finding, the FPT uses two equivalent, but conceptually
different versions denoted by FPT
.C/
fa
n
g
and FPT
./
:
Their diagonal forms have
the following representations for the exact infinite-rank Green function, which is
defined as the Maclaurin series with the time signal points
fc
n
g
as the expansion
coefficients:
X
z
1
/
c
n
z
n1
Exact
W G.
(25.2)
nD0
P
r D1
p
r
P
K
.
/
z
r
z
G
.C/
K
FPT
.C/
;
.
z
/
D
W
(25.3)
P
sD0
q
r
Q
K
.
/
z
z
s
P
r D0
p
r
z
1
/
Q
K
.
P
K
.
z
r
./
K
.
z
1
/
FPT
./
:
G
D
W
(25.4)
P
sD0
q
r
z
1
/
z
s
P
K
.
fp
r
;q
s
g
z
˙1
/
The expansion coefficients
of the numerator
and denominator
Q
K
.
z
˙1
/
polynomials can be extracted exactly and uniquely from the given signal
fc
n
g
points
by solving only one system of linear equations from the definitions
(
25.3
)and(
25.4
), after truncation of the Maclaurin series for
G.
z
1
/
n D N 1:
at
P
K
.
z
˙1
/=Q
K
.
z
˙1
/
Spectra
can equivalently be written in their canonical forms:
Y
z
˙1
z
k;P
P
K
.
z
˙1
/
p
K
q
K
D
z
k;Q
:
(25.5)
Q
K
.
z
˙1
/
z
˙1
kD1
Q
K
.
z
˙1
/ D 0;
have the solutions
z
˙1
k
Roots of the characteristic equations,
z
k;Q
.1 k K/
that represent one constituent part of the fundamental harmonics
