Pad´e processing for MR spectra from in vivo time signals - Signal Processing in MRS with Biomedical Applications - page 227

7

Pade processing for MR spectra from in vivo
time signals

We now will turn our attention to Pade processing of in vivo time signals.
These FIDs were encoded using highfield MR scanners, and have excellent
SNR, so that the converged shape spectra from the FFT with all N points
(N =2048) can be used as reliable envelopes for comparisons.

7.1 Relative performance of the FPT and FFT for total
shape spectra for encoded FIDs
In this chapter we shall use both the truncated N/M(M > 1) and the full
length N of the time signal{c n }to perform detailed comparisons of the

FPT (±) with the FFT. Our primary goal is to assess the resolving power,
convergence rate and robustness of the Pade and the Fourier estimators rela
tive to each other for encoded in vivo time signals.
To ensure complete fairness through strictly valid comparisons, all the FFT
and FPT (±) spectra are consistently computed by employing exactly the same
number N/M(1≤M≤32) of time signal points from either a truncated
(M > 1) or full (M = 1) set of the originally encoded time signal points{c n

}.
In the frequency domain, the Pade grid differs conceptually from the more
customary Fourier grid.
Thus, for example, the absorption spectra in the FPT (±) given by the real
parts of the complexvalued polynomial quotients P ± K (z ±1 )/Q ± K (z ±1 ), where
z ±1 = e ±iωτ , can be evaluated at any frequency ω. Therefore, the Pade grid
is simply an arbitrary plotting mesh for frequency ω chosen independently of
the employed number of time signal points{c n }. The ability to possess such

flexibility in the Pade grid is rooted deeply in the interpolation feature of the
FPT (±) as achieved via the rational functions P ± K (z ±1 )/Q ± K (z ±1 ).
Recall that these polynomial quotients are extrapolators, since they stem
from the explicit summation of the Maclaurin series (i.e., an infinite sum) with
the signal points{c n }as the expansion coe cients. Thus, the Pade spectra
inherently contain inferences on the otherwise unmeasurable infinitely long
time signals by relying solely upon the corresponding encoded data of finite

227