Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 25

any other external disturbance. Importantly, a convolution (folding) of two
or more Voigtians gives a single Voigt frequency profile. This is similar to the
wellknown fact that a convolution of two or more Gaussians (Lorentzians)
gives a single Gaussian (Lorentzian). In plasma physics, the Doppler broaden
ing of spectral lines originates from statistical velocity distributions of emitting
atoms. By reference to the Boltzmann thermodynamical theory of gases, it
follows that the average velocity and plasma temperature are directly propor
tional to each other. When gas temperatures are far from the BoseEinstein
condensation, the velocity distribution of atoms is Maxwellian and, therefore,
the said Doppler broadening will appear as a Gaussian profile. Alternatively,
in some other applications, the Voigt function can be viewed as a primary
Gaussian profile, which is narrowed through a convolution with a Lorentzian
and the basis for this is the occurrence of collisions among atoms.
In MRS, the basic forms of lineshapes are also rooted in pure Lorentzians
that stem from absorption of radiofrequency photons. However, whenever
there are inhomogeneities of static magnetic fields, the original Lorentzian
profiles could appropriately be modified by Gaussian broadening of the basic
Lorentzian profiles. Such a physically motivated convolution of the Lorentzian
leads to the Voigt profile. In practical applications, the Voigt folding em
ployed in MRS has often been simplified by a distribution in the form of a
linear combination of Lorentzians and Gaussians with some constant fitting
coe cients [98, 99]. Alternatively, the Voigt convolution integral can be com
puted directly. This might be especially necessary if this convolution is used
many times in the course of an iterative analysis, e.g., in various least square
estimations of spectra. In particular, it is wellknown that the Voigt convo
lution integral represents the real part of the complexvalued errorfunction
or the probability function. The errorfunction of complex variables can be
generated e ciently to within machine accuracy by the continued fraction
(CF) algorithm proposed by Gautschi [100]. This latter algorithm enables
the Voigt frequency profile to be used extensively without any simplification
as a sum of a Gaussian and a Lorentzian. Continued fractions are the special
cases (diagonal and paradiagonal) of the general Pade approximant.
It should be noted that there exists also the Voigt time profile. This func
tion is defined by the inverse Fourier integral of the Voigt frequency profile.
Explicitly, the result of such an inversion is represented by the twofold expo
nential function exp (−γt−σ 2 t 2 /4) where t is time (t≥0). This stems from
the convolution theorem according to which the Fourier integral of two func
tions convolved in the frequency domain is effectively reduced (in the time
domain) to the ordinary product of the two inverse Fourier transforms. In
particular, the onesided inverse Fourier integral over frequencies (from zero
to infinity) of a Lorentzian∝1/(γ 2 + ω 2 ) is given by∝exp (−γt), whereas
that of a Gaussian∝exp (−ω 2 /σ 2 ) is∝exp (−σ 2 t 2 /4). This yields the time
domain Voigt time profile in the abovequoted form,∝exp (−γt−σ 2 t 2 /4).
Hence, in the case of static field inhomogeneities, for an MR spectrum with
more than one peak, the corresponding time signal c(t) could be described by a