Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 13

extent of the truncation error. A su ciently large N means that the total
truncation error will be dominated by noise. Gibbs oscillations are reduced for
a long FID, but the resulting signaltonoise ratio is diminished. A better SNR
is obtained for a shorter FID, but this enhances Gibbs oscillations. Thus, in
the FFT, a longer FID reduces spectral deformations, but also worsens SNR.
This trouble is unsolvable using only the FFT because of its fundamental
drawback - the lack of extrapolation features.
As noted, the resolution in the FFT is defined by the minimal separation
ω min between any two adjacent frequencies, ω min = 2π/T. Thus, all the FIDs
with the same T, irrespective of their internal structure, will have the same
resolution which is fixed prior to processing by the FFT. In other words,
the resolving power in the FFT is predetermined by the separation between
any two adjacent Fourier grid points 2πk/N (0≤k≤N−1). The minimal
frequency separation ω min is also called the Rayleigh bound or the Fourier
uncertainty principle.
In signal processing, two kinds of spectral shapes are encountered. These
are the component shape spectra and the corresponding total shape spectrum.
The component shape spectra are the spectra for every separate resonance,
and they can be generated only via parametric estimators. The total shape
spectrum represents the sum of all the separate component shape spectra.
The total shape spectrum is also the envelope spectrum. This stems from
the fact that a constructive and destructive interference of all the individual
resonances yields the total spectral shape.
As stated, the FFT as a nonparametric processor can give only envelope
spectra. For this reason, the FFT and other nonparametric processors are
called envelope estimators. By contrast, parametric estimators are able to
yield both component and total shape spectra, since they can obtain the peak
positions, widths, heights and phases of individual physical resonances. The
main difference between these two categories of processors is in the kind of
information extracted from the input FID. Nonparametric processors can
give merely qualitative information, which is an apparent information seen
on graphs of spectral shapes. However, more important quantitative informa
tion can be obtained by parametric estimators through unfolding the hidden
spectral structure of the envelope. Crucially, parametric methods can recon
struct the quantitative features of resonances (peak position, width, height,
phase) as the essential ingredient for reliable estimates of concentrations and
relaxation times of the clinically most relevant metabolites of the examined
tissue.
The spectral parameters of resonances that determine chemical shifts, re
laxation times and concentrations of metabolites are the complex frequencies
and the corresponding complex residues (amplitudes) as the main constituents
of the damped harmonics from the associated FID. Surprisingly, with all its
drawbacks, fitting is still most frequently employed in MRS for estimation of
these critically important spectral parameters. Here, envelope spectra from
the FFT are fitted by least square (LS) adjustments of some assumed reso