Pad´e processing for MR spectra from in vivo time signals - Signal Processing in MRS with Biomedical Applications - page 242

7.4.1 Residual spectra as the difference between the fully

converged Fourier and Pade spectra at various partial
signal lengths

Figures 7.11 and 7.12 corresponding to 4T and 7T, respectively, show the
residual absorption spectra Re(F)−Re(P − K /Q − K ) computed at the Fourier
grid points with the FPT (−) being taken at the fractions N/32 = 64,N/16 =
128,N/8 = 256,N/4 = 512,N/2 = 1024 as well as at the full signal length
N = 2048, whereas the FFT is obtained at N = 2048 throughout.
The extremely stable and fast convergence pattern of the FPT (−) seen in
Figs. 7.4 and 7.8 i s also reflected on the residual or error spectra depicted in
Figs.7.11 and 7.12 at 4T and 7T, respectively. At N/2 = 1024 (the right mid
dle panels in Figs. 7.11 and 7.12), the residual spectra are indistinguishable
from the background noise. This demonstrates that the total shape spectra
in the FPT (−) has fully converged by exhausting at most one half of the full
signal length, as discussed. In the residual spectra displayed in Figs. 7.11 and
7.12 as the difference between the two absorption spectra (“Fourier minus
Pade”), the FFT is always taken at the full signal length (N = 2048). This
is because the fully converged FFTs for the studied signals are of excellent
SNR and, as such, can rightly be considered as a gold standard for the cor
responding total shape spectra, as emphasized before. On the other hand,
the Pade absorption spectra are taken at N/M(M = 1−32) with no zero
filling. Therefore, the mentioned difference “Fourier minus Pade” represents
the error spectra that ideally should be indistinguishable from the background
noise after the FPT (−) has reached full convergence. This is indeed observed
on Figs. 7.11 and 7.12 already at N/2 = 1024.
At this point, it is pertinent to return to Figs. 7.1 and 7.2 to recall that
the second half of the time signal contains mainly noise. In such a case, it is
natural to expect that only the first half of the signal should su ce to extract
the entire physical information. Such an expectation is not fulfilled in the
FFT as seen on the middle right panel in Figs. 7.3 and 7.7 . Namely, the
triplet of glutamineglutamate around 2.4 ppm is not fully resolved by the
FFT at N/2 = 1024, as opposed to the FPT (−) . The FFT can completely
resolve this triplet only by exhausting the full signal length N = 2048 as seen
on the bottom right panels in Figs. 7.3 and 7.7.
This illustrates the severity of the Rayleigh bound 2π/(Nτ) imposed on
the resolution by the linearity feature of the Fourier transform. The FPT
rescues the situation by means of its nonlinearity to surpass the Rayleigh
bound and, therefore, improves the Fourier resolution. Both nonlinearity and
extrapolation of the FPT are due to the very definition of the Pade spectrum
as a ratio of two polynomials P/Q. The inverse Q −1 of the denominator
polynomial Q is a series, i.e., an infinite expansion. Given a finite Riemann
sum

N−1
n=0 c n z −n , the FPT obtains the unique polynomial quotient P/Q,
which by means of an implicit series in Q −1 , effectively extrapolates the input
time signal{c n }(0≤n≤N−1) beyond the original acquisition time T = Nτ.