The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 65

∞

n=0 c n u −n−1 ≈
P K−1 (u)/Q K (u) can be considered as a generating fraction, which upon in
version yields the geometric progression model c n =

P K−1 (u)/Q K (u) from the stated representation G(u) =

K
k=1 d k u k

for the time

signal.
Moreover, if spectral analysis is performed directly in the time domain via,
e.g., the Shanks algorithm, the FPT is found to be the exact filter for a gen
eral time signal represented by the most conventional form of a sum of com
plex damped exponentials with stationary and/or timedependent amplitudes.
Overall, the discussed arguments for modeling in the time and frequency do
main highlight the optimal suitability of the FPT for MRS.

2.5 Pade acceleration and analytical continuation of time

series

Considering all the currently available processors, the FPT is especially im
portant because of its selfcontained crossvalidation. Specifically, the FPT,
as a system function for modeling an optimal response to general external
perturbations, has two complementary variants. As noted earlier, these rep
resent the equivalent versions FPT (+) and FPT (−) that are initially defined in
two totally different convergence regions located inside (|z|< 1) and outside
(|z|> 1) the unit circle, respectively [18, 19].
By design, the fast Pade transform performs equally well in the time and
frequency domains. For instance, in the frequency domain, the FPT (+) and
FPT (−) possess a common starting platform given by the same input spectrum
in the form of the following truncated power series representation of the finite
rank Green function

N−1

1
N

G N (z −1 ) = c n z −n z = e iωτ . (2.162)

n=0

Here,{c n }(0≤n≤N−1), are the expansion coe cients that represent

the signal points, where N is the total length. As discussed, the time signal
from the Green function in (2.162) is equivalent to the quantummechanical
autocorrelation function. By reference to the theory of the z−transform, the
Green function (2.162) is introduced by the expansion in powers of the inverse
z −1 of the harmonic variable z.
The superscripts±in FPT (±) indicate the power±1 of the expansion
harmonic variable z ±1 . Also in use are the usual conventions z +1 ≡z + = z
and z −1 ≡z − = 1/z. In the limit N−→∞, the series (2.162) converges
outside and diverges inside the unit circle,|z|> 1 and|z|< 1, respectively.
The initial convergence regions of the FPT (+) and the FPT (−) are|z|< 1 and