Once an interferogram is collected, it needs to be translated into a spectrum (emission, absorption, transmission, etc.). The process of conversion is through the Fast Fourier Transform algorithm. The discovery of this method by J.W. Cooley and J.W. Tukey in 1965, followed by an explosive growth of computational power at affordable prices, has been the driving force behind the market penetration of FTIR instruments.
A number of steps are involved in calculating the spectrum. Instrumental imperfections and basic scan limitations need to be accommodated by performing phase correction and apodization steps. These electronic and optical imperfections can cause erroneous readings due to different time or phase delays of various spectral components. Apodization is used to correct for spectral leakage, artificial creation of spectral features due to the truncation of the scan at its limits (a Fourier transform of sudden transition will have a very broad spectral content).
FTIRs are capable of high resolution because the resolution limit is simply an inverse of the achievable optical path difference, OPD. Therefore, a 2 cm OPD capable instrument, such as our FTIR scanner model 80351, can reach 0.5 cm^{1} resolution. Table 2 shows the relationship between resolution expressed in wavenumbers and in nanometers, as is customary in dispersive spectroscopy.
Wavelength
(µm) 
Resolution
(cm^{1}) 
Resolution
(nm) 
0.2 
1 
0.004 
0.5 
1 
0.025 
1 
1 
0.1 
2 
1 
0.4 
5 
1 
1.0 
10 
1 
10 
20 
1 
40 
Following, we talk about three significant advantages that FTIR instruments hold over dispersive spectrometers, but first we compare the two instruments.

MIR 8035™ FTIR Scanner 
Cornerstone™ 260 1/4 m Grating Monochromator 
Wavelength Range 
700 nm  25 µm 
180 nm  24 µm 
Max. Resolution 
0.024 nm @ 700 nm 
0.25 nm with 10 um slit and 1200 line/mm grating @ blaze wavelength 
Etendue @ 1 µm, 0.15 nm resolution 
0.38 @ 1 µm, 0.15 nm resolution 
0.002 
In a dispersive spectrometer, wavenumbers are observed sequentially, as the grating is scanned. In an FTIR spectrometer, all the wavenumbers of light are observed at once. When spectra are collected under identical conditions (spectra collected in the same measurement time, at the same resolution, and with the same source, detector, optical throughput, and optical efficiency) on dispersive and FTIR spectrometers, the signaltonoise ratio of the FTIR spectrum will be greater than that of the dispersive IR spectrum by a factor of √M, where √M is the number of resolution elements. This means that a 2 cm^{1} resolution, 800  8000 cm^{1} spectrum measured in 30 minutes on a dispersive spectrometer would be collected at equal S/N on an FTIR spectrometer in 1 second, provided all other parameters are equal.
The multiplex advantage is also shared by Array Detectors (PDAs and CCDs) attached to spectrographs. However, the optimum spectral ranges for these kinds of systems tend to be much shorter than FTIRs and therefore the two techniques are mostly complementary to each other.
FTIR instruments do not require slits (in the traditional sense) to achieve resolution. Therefore, much higher throughput with an FTIR can be achieved than with a dispersive instrument. This is called the Jacquinot Advantage. In reality there are some slitlike limits in the system, due to the fact that one needs to achieve a minimum level of collimation of the beams in the two arms of the interferometer for any particular level of resolution. This translates into a maximum useable detector diameter and, through the laws of imaging optics, it defines a useful input aperture.
Spectral resolution is a measure of how well a spectrometer can distinguish closely spaced spectral features. In a 2 cm^{1} resolution spectrum, spectral features only 2 cm^{1} apart can be distinguished. In FTIR, the maximum achievable value of OPD determines spectral resolution. The interferograms of light at 2000 cm^{1} and 2002 cm^{1} can be distinguished from each other at values of 0.5 cm or longer.
A collimated, monochromatic light source will produce an interferogram, in the form of a sinusoid, at the detector. When the light intensity of the interferogram changes from one maximum to the next, the optical path difference between the two legs in the interferometer will change by exactly 1 wavelength of the incoming radiation.
To determine the wavelength of the incoming radiation, we can measure the frequency f_{i} or period t_{i} = 1/f_{i} of the interferogram with an oscilloscope. Then we can find the wavelength through the formula:
λ_{i} = V_{o}*t_{i} = V_{o}/f_{i}................................(1)
Where:
V_{o} = the speed of change of the optical path difference (V_{o} is directly related to the speed of the scanning mirror. For MIR 8035™ FTIR Scanners, V_{o} is 4 times the optical speed of the scanning mirror: V_{o} = 4nV_{m})
There is, however, an important practical difficulty. We need to keep the velocity Vm constant at all times, and we need to know what this velocity is with a high degree of accuracy. An error in the velocity value will shift the wavelength scale according to (1). Fluctuations in Vm have a different effect; they manifest themselves as deviations of the interferogram from a pure sine wave that in turn will be considered as a mix of sinusoids. In other words, we will think that there is more than one wavelength in the incoming radiation. This behavior produces what are called “spectral artifacts”.
Since the manufacture of an interferometricallyaccurate drive is extremely expensive, FTIR designers added an internal reference source into the interferometer to solve the drive performance problem. A HeNe laser emits light with a wavelength which is known to a very high degree of accuracy and which does not significantly change under any circumstance. The laser beam parallels the signal path through the interferometer and produces its own interferogram at a separate detector. This signal is used as an extremely accurate measure of the interferometer displacement (optical path difference).
We can, therefore, write the following equation for a HeNe based FTIR:
λ_{i} = λ_{r}*(f_{r}/f_{i})................................(2)
Where subscript r denotes HeNe reference.
We can now calculate the spectrum without extremely tight tolerances on the velocity.
This was just a theoretical example. Now let us see how the reference interferogram is actually used in the MIR 8035™ FTIR Scanners. The signal from the interfering beams of the HeNe is monitored by a detector. What is observed is a sinusoidal signal. The average value is the same as we would see if the beam was not divided and interference produced. The sinusoid oscillates about this value. The average signal level is called zero level. A high precision electronic circuit produces a voltage pulse when the HeNe reference sinusoid crosses zero level. By use of only positive zero crossings, the circuitry can output one pulse per cycle of the reference interferogram, or use all zero crossings for two pulses per cycle of this interferogram. The latter case is often called oversampling. These pulses trigger the A/D converter which immediately samples the main interferogram.
There is a fundamental rule called the Nyquist Theorem which can be paraphrased to state that a sinusoid can be restored exactly from its discrete representation if it has been sampled at a frequency at least twice as high as its own frequency. If we apply this rule to the above formula we find immediately that since the minimum value of (f_{r}/f_{i}) is 2, the minimum value of λ_{i} is twice the wavelength of the reference laser:
λ_{min} = 633 nm*2 = 1.266 µm
With oversampling, the reference laser wavelength is effectively halved. So in this case:
λ_{min} = (633 nm/2)*2 = 633 nm
In practice, the FFT math runs into difficulties close to the theoretical limit. That is why we say 1.4 µm is the limiting wavelength without oversampling, and 700 nm is the limiting wavelength with oversampling