The concept of the
Fourier transform is involved in two very important instrumental methods in
chemistry. In Fourier transform infrared
spectroscopy (FTIR), the Fourier
transform of the spectrum is measured directly by the instrument, as the
interferogram formed by plotting the detector signal vs mirror displacement in
a scanning Michaelson interferometer. In Fourier Transform Nuclear Magnetic
Resonance spectroscopy (FTNMR),
excitation of the sample by an intense, short pulse of radio frequency energy
produces a free induction decay signal that is the Fourier transform of the
resonance spectrum. In both cases the spectrum is recovered by inverse Fourier
transformation of the measured signal.
The power spectrum or frequency spectrum is a simple way of
showing the total amplitude at each of these frequencies; it is calculated as
the square root of the sum of the squares of the coefficients of the sine and
cosine components.
A signal with n points gives a power
spectrum with only (n/2)+1 points. The x-axis is the harmonic number.
The first point (x=0) is the zero-frequency (constant) component. The second
point (x=1) corresponds to the fundamental frequency, the next point (x=2) to
twice the fundamental frequency, the next point (x=3) to three times the
fundamental frequency, etc. An example of a practical application of the use of
the power spectrum as a diagnostic tool is shown in Figure 10.
In the example
shown here, the signal is a time-series signal with time as the independent
variable. More generally, it is also possible to compute the Fourier transform
and power spectrum of any signal, such as an optical spectrum, where the
independent variable might be wavelength or wavenumber, or an electrochemical
signal, where the independent variable might be volts. In such cases the units
of the x-axis of the power spectrum are simply the reciprocal of the units of
the x-axis of the original signal (e.g. nm-1
for a signal whose x-axis is in nm).
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