ORGANIC AND INORGANIC GASES by FTIR Spectrometry: METHOD 3800, Issue 1, dated 15 March 2003 - Page 21 of 47
C2. The Fast Fourier Transform (FFT)
Every data point in the interferogram contains intensity information about every infrared wavelength transmitted from the source to the detector. It is possible to recover the intensity information as a function of wavelength through application of a Fast Fourier Transform (FFT), from which the FTIR technique's name is derived. This digital transformation of the interferogram can be thought of as the mathematical inverse of the optical modulation applied to the infrared beam as it passes through the interferometer. Its function is similar to that of the human brain and ear, which provide intensity information (loudness) versus wavelength (pitch) for complex signals (sound waves) incident on the eardrum. (Note that, as for an interferogram, each point in a complex sound wave contains intensity information about every pitch contained in the wave. Yet the ear and brain allow a symphony audience to immediately perceive, for instance, that the piccolo is playing very loudly while the tuba is playing very quietly.) Reference 15 (Chapter 3) provides a complete mathematical description of the FFT.
C3. Instrument Resolution, Apodization Function, and Minimum Instrumental Linewidth (MIL)
Most software packages supplied with FTIR systems provide several options associated with the collection of data and application of the FFT. These typically include – at least – the nominal "instrument resolution" (specified in cm-1) and the "apodization function" (e.g., "Boxcar" and "Triangular"). These parameters are very important in quantitative spectroscopy, and are addressed in turn below.
The instrument resolution is the most fundamental and important instrument parameter. It specifies the nominal minimum full-width-at-half-maximum (FWHM, in cm-1) of any spectral "peak" (or "line") in the final instrument output. Every FTIR instrument has a minimum FWHM determined by the maxim um distance traversed by the interferometer's moving element during a single scan. (For the basic Michelson interferometer, the FWHM in cm-1 is equal to (2d)-1, where d is the distance in cm traversed by a moving mirror during a scan.) Clearly, instruments with low FWHM provide more spectral information than instruments with higher FWHM capability. However, this additional information comes at high costs associated with the design, construction, size, mechanical stability, portability, speed, and residual noise area (RSA) of the instruments.
It is important to recognize the two uses of the word “resolution” in the nomenclature used to describe FTIR spectrometers: Instruments of high resolving power, or “high resolution,” provide spectral features of low FWHM; when the nominal resolution is specified in units of cm-1, a lower cm-1 specification corresponds to higher resolving power, or “higher resolution”. Most commercially available FTIR spectrometers suitable for field use provide FWHM values greater than or equal to 0.5 cm-1—that is, they are systems whose nominal spectral resolution is specified as a number higher than 0.5 cm-1. Most of the instruments capable of higher resolution (lower FWHM) are suitable for use only in very stable laboratory environments.
Standard FTIR operating software always provides options for recording spectra with FW HM values higher than the instrument's actual lower FWHM limit. These options simply move the mirror (or other optical element) through only some fraction of its maximum possible travel. Operating the instrument in this manner results in larger FWHM values ("lower" resolution, and shorter interferograms) than the instrument is mechanically capable of providing. Spectra of lower resolution (higher FWHM) provide less information, but can be generated more quickly and, in most cases, with lower RSA than spectra of higher resolution.
The instrument operator can also choose the apodization function to be used in the generation of FTIR spectra. Apodization is a mathematical alteration of the interferogram that can be performed before application of the FFT. Several standard alteration functions have been devised, and each affects the final absorption spectrum of the sample gas in a different way. As with the selection of instrument resolution, each choice has its advantages and drawbacks. The simplest choice, known as the "boxcar apodization" function, results in the lowest FWHM but also in relatively low S/N ratio. (Spectra generated with the boxcar function are often referred to as “unapodized” spectra.) Other choices (triangular, Norton-Beer, and several other apodization functions) provide higher S/N ratio at the cost of higher FWHM values and other tradeoffs in quantitative spectroscopy. Reference 15 provides a more thorough description of the characteristics of various apodization functions.
For a given instrument configuration—which includes the nominal spectral resolution and the choice of apodization function—every FTIR system is capable of generating absorption bands with a minimum instrumental linewidth (MIL). Unlike the actual spectral resolution (which has several accepted physical
NIOSH Manual of Analytical Methods, Fourth Edition
