Digital Signal Processing Reference
In-Depth Information
concept of the approximate SNS: appearance of quasi Froissart doublets and
very small amplitudes, and this was set to explain. The smallness of Froissart
amplitudes implies their sensitivity to even the slightest perturbations.
Therefore, these small amplitudes are the cause for great instability of
spurious resonances, i.e., Froissart doublets. This is in sharp contrast to
the stability of genuine resonances. Such a diametrically opposite behavior
of physical and unphysical resonances greatly facilitates the task of distin
guishing one from the other. In practical computations with, e.g., para
metric signal processing, this is easily accomplished by merely monitoring
the Pade table when passing from one Pade approximant [K/K] to another
[(K + m)/(K + m)] (m = 1, 2, 3,...). In so doing, as the order of the PA
changes, we would observe that some resonances are robustly stable, whereas
the others exhibit great instability. Then the former resonances are identi
fied as genuine and the latter as spurious. A more detailed exposition on the
theory and illustrations of Froissart doublets is given in chapters 5 and 6,
respectively (for our previous studies, see Refs. [11, 24, 30, 31, 34]).
3.7
The importance of exact quantification for MRS
Single voxel in vivo MRS uses encoded time signals to extract biochemical
information about concentrations of metabolites of the examined tissue. This
can be critically important for cancer diagnostics, because malignant tumors,
benign processes and healthy tissue usually differ markedly in their spectral
information. These important differences are potentially detectable by MRS
which can yield the initial information for the assignment of a particular
disease to the identified spectral pattern of the scanned tissue.
In vitro MRS, which employs strong external magnetic fields, can resolve
some fifty resonances. With this high resolution and excellent SNR, valuable
diagnostic information is provided about tissue metabolites. However, such
an excellent performance of in vitro MRS is not matched by the corresponding
in vivo MRS on clinical scanners that operate at considerably weaker fields
and with poorer SNR. Moreover, the usual information from in vivo MRS is
limited to a relatively small number of metabolites that can be resolved and
eventually quantified for the purpose of clinical diagnostics.
The main reason for this limitation is that signal processing for in vivo MRS
relies predominantly upon the FFT. The diagnostic value of in vivo MRS could
certainly be enhanced by employing highresolution signal processors, such as
the FPT. Especially because of its capability to resolve closely overlapped
resonances, the FPT can be advantageously applied to clinical FIDs encoded
by in vivo MRS at short echo times. Most frequently, FIDs encoded via MRS
use long TE, because a larger number of shortlived tightly packed resonances
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