Exact quantum-mechanical, Pad´e-based recovery of spectral parameters - Signal Processing in MRS with Biomedical Applications - page 87

and z − k ≈z − k , respectively. Using the Padecomputed signal poles z ± k and

the corresponding amplitudes d ± k , one can build the two Pade time signals,
one in the FPT (+)

K

= e iω k τ

d k (z k ) n z k Re(ω k ) > 0 |z k c n = |< 1 (3.5)

k=1

and the other in the FPT (−)

K

= e −iω − k c − n = d − k (z − k ) n z − k

τ

Re(ω − k ) > 0 |z − k

|> 1 (3.6)

k=1

where|c ± n

|<∞. A joint expression for (3.2) and (3.5) was given earlier in
(2.193). The representations (3.5) and (3.6) are the Pade approximations to
the same input c n from the two formally different, but otherwise identical
expressions (3.2) and (3.4), respectively

c n c − n ≈c n ≈c n . (3.7)

If the input time signal c n has precisely K harmonics, then for the noiseless
case, the two variants of the FPT will exactly reconstruct the input FID via

c n = c − n

= c n (noiseless FID). (3.8)

On the other hand, when the input signal, containing exactly K resonances,
is noisecorrupted, the timedomain residuals r ± n obtained by the FPT (+) and

FPT (−) via

r n ≡c n r − n ≡c − n −c n = 0 −c n = 0 (noisy FID) (3.9)

will be different (r n = r − n ) and shall represent an approximate reconstruction

of the input noise.
In order to formulate the proper theoretical standards, it is vital to re
construct the spectral parameters from noiseless FIDs, as we shall do in this
chapter. These allow validity assessments to be made concerning the designed
estimator, such as the FPT in the present case. We reemphasize that the
quantification problem in MRS can be kept under complete control throughout
the analysis exclusively via synthesized noisefree FIDs. Thereby, everything
that is expected from the theory can be known exactly, and then the remain
ing part of the investigation can be geared to testing the appropriateness of
the employed algorithms. In later steps, the idealized FIDs can be corrupted
with random Gaussian noise which can be controlled. Solving the correspond
ing quantification problem in this latter case can then be used to get a handle
on actual data from in vivo time signals encoded via MRS.
In noisefree as well as noisecorrupted synthesized FIDs, the spectral pa
rameters are recognized by detecting their stability or constancy with gradu
ally increasing partial signal length N/M (M > 1). This expected constancy