Harmonic transients in time signals - Signal Processing in MRS with Biomedical Applications - page 156

required optimality and, therefore, the Pade approximant has no competitor
for investigating those input functions that are themselves defined as ratio
nal polynomials. In signal processing, the input function is not a quotient of
two polynomials, but rather a single series

n c n+s u −n−1 for which the exact
processor does not exist. Nevertheless, this situation can be salvaged, and an
exact theory could still be formulated for the spectrum

n c n+s u −n−1 by in
voking the appropriate prior information, such as the existence of a harmonic
structure of the signal via (4.7). In this case, with no additional approxima
tion, the single sum

∞

n=0 c n+s u −n−1 from the original input data coincides
exactly with the rational polynomial (4.14) or (4.15), for which the Pade
approximation is again the exact theory.
Regarding the general issue of prior information, the minimal knowledge
needed in advance could be the assumption that the time signal possesses a
structure. For instance, by merely plotting a given signal, it might become pos
sible for a trained eye to qualitatively discern certain oscillatory patterns with
a harmonictype structure. These structures often become more pronounced
by considering the associated derivative of the time signal. Alternatively, the
Fourier shape spectrum in the frequency domain would give a more definitive
indication of an underlying structure via a clearer emergence of a number
of peaks. Here, the otherwise generic structure would become more specific
through the appearance of the resonant nature of the spectrum.
Despite such a favorable circumstance, the Fourier method does not exploit
this key finding from its own analysis, since only the envelope spectrum is ob
tained as the final result of signal processing. Such an occurrence is reflected
in the socalled 'Fourier uncertainty principle'. This principle states that for
a fixed total acquisition time T, or equivalently, a given full signal length N
at a fixed bandwidth, the corresponding Fourier spectrum cannot predict an
angular frequency resolution better than the Fourier bound 2π/T. The obvi
ous drawback of this principle is in the implication that, regardless of their
intrinsic nature, all the signals of the identical acquisition time T will pos
sess the same frequency resolution 2π/T. In other words, all Fourier spectra
stemming from time signals of the same length N will have the same resolu
tion. As such, in advance of both measurement and processing, the Fourier
method preassigns the frequencies at which all the spectra should exist and
these are the frequencies from the Fourier grid 2πk/T, (k = 0, 1,...,N−1).
Such a severe limitation of the Fourier analysis rules out the chance for in
terpolation or extrapolation. However, without an interpolation property,
the Fourier method is restricted to the predetermined minimal separation
ω min = 2π/T between any two adjacent frequencies. Further, without an ex
trapolation feature, the Fourier processing has no predictive power. In lieu of
extrapolation, the Fourier method employs zero filling or zero padding beyond
N or uses signal's periodic extensions c n+N = c n . However, in most circum
stance encountered in practice, time signals are nonperiodic and, therefore,
no extrapolation or any other new information can be obtained by periodic
extensions, as mentioned in chapter 1.