Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 97

are included beyond N/4. This is shown on panels (v) and (vi) for N/2 = 512
and N = 1024, respectively. This unique feature of the Pade polynomial
quotient, e.g., P − K /Q − K and the like, is related to polezero cancellations or
Froissart doublets, as analyzed in section 3.6 . A veritable signature of re
construction of the true number K of resonances is provided by the attained
convergence of the spectral parameters. If one continues to increase K in the
ratio P − K /Q − K even after convergence has been achieved, the converged results
for the Pade quotient in the FPT (−) will not change. This is the case be
cause the new poles from Q − K+m will be exactly the same as the new zeros of
P − K+m . In such a case, the polezero cancellation occurs in the Pade quotient
yielding P − K+m /Q − K+m = P − K /Q − K , as per (2.226), where m is any positive
integer. We have verified that this holds true in the present computation, as
seen on panels (iv)-(vi) of Table 3.2 . Synergistically, the same computation
demonstrates that all the amplitudes{d − k }associated with the poles from

the Froissart doublets are identical to zero, as in (2.192). Theoretically, the
strict algebraic condition 2K = N implies that only 100 FID points should
be su cient for the FPT (−) to exactly reconstruct all the 25 unknown com
plex frequencies and 25 complex amplitudes. However, panel (iv) of Table 3.2
reveals that full convergence is attained with the first 256 time signal points.
This occurs because the FPT (−) produces genuine as well as spurious reso
nances. In other words, in the polynomial quotient P − K /Q − K , spurious poles
and spurious zeros from the denominator and the numerator, respectively,
come in pairs as Froissart doublets. Therefore, they cancel each other. Thus,
each addition of more time signal points yields new Froissart doublets. How
ever, another process takes place at the same time, namely stabilization of
the values of the reconstructed physical spectral parameters. Ultimately, sat
uration occurs when the total number of genuine resonances stop fluctuating
as described in (2.226). At that point, all the spectral parameters become
constant for varying partial signal length. This process of stabilization illus
trates how the FPT (−) determines, with certainty, the true total number K
of genuine resonances. For the time signal which is currently under study,
this stabilization actually takes place by using less than a quarter N/4 = 256
of the full FID. In fact, this occurs by exhausting the first 210 signal points
(not shown in Table 3.2 where all the signal lengths are of the form 2 s with s
being a positive integer). This would give K = 105, but the elimination of all
the Froissart doublets and other extraneous poles finally yields the exact total
number K = 25 of the genuine resonances in a Froissartcleaned spectrum.

3.1.3 Numerical values of the reconstructed spectral param-

eters near full convergence for 3 partial signal lengths
N P = 180, 220, 260

Table 3.3 for the reconstructed spectral parameters from the FPT (+) (left
column) and FPT (−) (right column) focuses upon a narrow convergence range