Basic tasks of signal processing in spectroscopy - Signal Processing in MRS with Biomedical Applications - page 24

approximants, as the simplest rational functions in the form of the unique
quotient of two polynomials from (1.11). The overall goal of this succinct re
minder is to illuminate the key feature of the Pade approximant, i.e., the fact
that this method forms a natural integral part of the leading mathematical
formulations and implementations of quantum mechanics.

1.3 Resonance profiles
As stated, collision theory predicts that on physical grounds the most natural
spectral lineshapes are given by Lorentzian profiles. The term Lorentzian for
a bellshaped profile of a spectral line stems from optics where Lorentz [94]
used a dispersiontype function of the form (1.3). Alternatively, in mathe
matics, a Lorentzian functional form is called the Cauchy distribution.
There exist other lineshapes that are frequently used, e.g., the Gauss and
Voigt profiles [95]. In particular, the Voigt profile f V is a convolution integral
of a Lorentzian (γ/π)/[γ 2 + (y−ω) 2 ] by a Gaussian [1/(σ

√

π )] exp (−y 2 /σ 2 )

∞

e −y 2 /σ 2
γ 2 + (y−ω) 2 dy.

f V (ω) = γ
π

1

√ (1.17)

σ π

−∞

As such, the Voigt function can be perceived as a broadened Lorentzian, or
equivalently, a narrowed Gaussian depending upon which of the two con
stituents (Gaussian or Lorentzian) is taken to be the primary profile. Origi
nally, the Voigt profile was introduced in optics with the purpose of accounting
for the Doppler broadening (via Gaussians) of the primary lineshapes given
by dispersion functions (Lorentzians) to describe absorption spectra [95]-[97].
After this initial appearance, many applications of the Voigt profile emerged
in diverse fields, including MRS [98, 99]. Being a symmetric function, the
Voigt frequency profile is intermediate to a Gaussian and a Lorentzian. More
specifically, the Voigt profile is reduced to a Gaussian or a Lorentzian function
in the pertinent limiting cases of the two widths. For instance, not too far
from the center ω≈ω 1 , the Lorentz and Gauss distributions look very much
alike. Nevertheless, the discrepancy between these two distributions becomes
appreciable away from the center where a Lorentzian decreases more slowly
than a Gaussian when the value of the frequency variable ω is augmented.
From the mathematical viewpoint, a Lorentz distribution represents the
onesided Fourier integral of an exponentially damped oscillatory wave func
tion. Likewise, from the physics standpoint, exposing an oscillator (e.g., a
pendulum, an atom, a molecule, etc.) to a given external influence will pro
duce damped harmonic oscillations of a type of forced excitations as a response
to the applied perturbation. This response function due to exponentially at
tenuated harmonic oscillations represents a pure Lorentzian in the absence of