Pad´e processing for MR spectra from in vivo time signals - Signal Processing in MRS with Biomedical Applications - page 228

Such extrapolation properties of the FPT (±)

lengths. lead directly to the

resolution enhancement relative to the FFT.
As known, the FFT is not an extrapolator, since the Fourier resolving
power is limited by the total acquisition time T. Moreover, the FFT is not an
interpolator either, since the frequency mesh in the Fourier spectra is fixed in
advance by the basic limitation of the Fourier grid 2πk/T(0≤k≤N−1).
Another related disadvantage of the FFT, especially when compared with the
corresponding flexibility in the Pade grid of the FPT (±) , is that the number
of frequency points in the Fourier spectrum is rigorously restricted to the
number N/M(M≥1) of employed time signal points. In computation for
M > 1, the FFT employs N−N/M zeros for completion to N = 2048, but
the FPT (+) and FPT (−) do not.
We have noted that due to its nonlinearity, as is apparent from the poly
nomial quotient for the spectrum, the FPT can effectively reduce noise. Noise
reduction by the FPT is also due to the fact that short, i.e., truncated sig
nals can be used with better SNR to extract the full spectral information.
This is opposed to the FFT which, as a linear processor, brings the entire
noise content from the time to the frequency domain. Furthermore, reduc
tion of noise by the FFT is not possible by truncation of the encoded data,
since the Fourier method must use the full signal length to achieve reasonable
resolution, as will now be illustrated.

7.2 The FIDs, convergence regions and absorption spec-
tra at full signal length encoded at high magnetic
field strengths
In this chapter, the FPT (+) and FPT (−) are applied to two complexvalued
time signals{c n }with the common bandwidth of 6001.5 Hz as encoded via
MRS at 4T and 7T from the brain of a healthy volunteer [141]. These FIDs
are long (N = 2048) with very good SNR, so that the shape spectra from the
FFT using all N points are excellent and can be used as a gold standard.
The investigated FIDs and the corresponding absorption spectra in the FFT
and FPT (±) computed with the full signal length (N = 2048) are displayed
in Figs. 7.1 and 7.2 for 4T and 7T, respectively. The real and the imaginary
parts of the FIDs are displayed on the top and the middle panels of the left
column, respectively. On the bottom left panels of Figs. 7.1 and 7.2, we show
the “initial convergence regions” of the FPT (+) and FPT (−) that are located
inside (|z|< 1) and outside (|z|> 1) the unit circle, respectively.
Recall as well that by the Cauchy concept of analytical continuation, the
FPT (+) and FPT (−) are valid in the whole complex frequency plane with the
exception of the poles. Both variants, the FPT (+) and FPT (−)

of the fast