Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 12

coded FID to the frequency domain with a spectrum of resonances. However,
such a transformation is merely a qualitative description of metabolites of the
studied tissue. Each metabolite has a molecular structure which might contain
more than one resonance. This often leads to a number of congested peaks in a
typical spectrum of metabolites. Among such spectral structures, unambigu
ous quantification of overlapping resonances presents a major challenge for
adequate interpretation of encoded data. Attractiveness for processing FIDs
by the FFT within MRS is largely due to the stability, e ciency and robust
ness of automatic computations via the fast Nln 2 N algorithm for any fixed
signal length N whenever N = 2 s (s = 0, 1, 2,...). For a given FID, the FFT
spectrum has a simple stick structure and, as such, it exists only at the Fourier
grid frequencies 2πk/T (0≤k≤N−1). Here, T as the total acquisition time
is given by T = Nτ, whereas τ > 0 is the sampling rate, or equivalently, dwell
time, which is equal to the inverse of the selected bandwidth. Clearly, for a
theoretically synthesized FID, T has the same significance as the total acqui
sition time in measurement, but actually it is called the total duration of the
FID. Naturally, in a larger interval, no encoded FID is zero for any realistically
selected T. Nevertheless, the common practice with the FFT is to artificially
double the original FID length by adding zeros, and this is called zero fill
ing or zero padding. As a result, the length of the ensuing FFT spectrum is
also doubled. The supplementary N frequencies in the FFT spectrum give
a simple sincinterpolation characterized by wiggling side lobes around each
genuine resonance. A drawback of such an outcome, especially for any two
closely spaced resonances, is that the sinc side lobes coupled with truncation
artefacts might interfere constructively or destructively to yield extraneous
peaks or dips. Any extraneous or spurious spectral peak is a false and un
physical structure which is not a part of the true information contained in the
investigated FID. This sincinterpolation cannot be systematically improved,
as empirical practice shows that no enhanced spectral quality is achieved by
introducing additional zeros into the FID beyond the first doubling of N.
The mentioned truncation artefacts in the FFT are spectral deformations
known as Gibbs oscillations that stem from the experimental impossibility
to encode infinitely long FIDs. On the other hand, only FIDs of infinite
length can give the exact Fourier coe cients. Therefore, the FFT itself can
give merely some approximate Fourier coe cients for any encoded FID. In
encoding an FID, the first selected quantity is the bandwidth. The second
selected quantity is the signal length N, so that T is automatically fixed by
T = Nτ. However, the unavoidable truncation in the time domain is critically
determined by τ itself. This occurs because τ is the only quantity determining
the Nyquist frequency 1/(2τ) as the largest frequency which can be sampled
for a fixed bandwidth. The Nyquist frequency sets the upper limit to the
frequency content of the FID prior to encoding. Hence, it is such a limit which
leads to unavoidable truncation errors in the time domain, since no FID can
be limited simultaneously in the two conjugate domains - time and frequency.
The role of a concrete value of N appears on the level of indicating the actual