6 - Working with digital interferograms, fourier transforms, and spectra

Abstract

The precision required in calculating the Fourier transform, that is at the heart of high-resolution Fourier transform spectrometer (FTS), is far too great for any kind of analog technique. The development of the modem digital computer and algorithms, such as the fast Fourier transform (FFT), is absolutely essential for high-resolution Fourier spectroscopy. But digital invariably implies discrete, and so one must understand the constraints imposed by the operation of sampling the interferogram. To avoid aliasing in an extended spectrum, the spectrum often must be limited with an anti-aliasing low-pass filter that attenuates the undesirable frequencies. If a higher order alias is used, it is essential that both the optical bandwidth for the spectrum and the electronic bandwidth for the noise be limited to that single order, because all aliases contribute to the final spectrum, including the noise. The best way to start interpolation is to increase the number of points in the spectrum. One can do this by increasing the number of points in the interferogram before transforming it. When finding line positions, the application of simple peak-finding algorithms to relatively complex spectra usually benefits from some lighter apodization, so that only the strongest lines are accompanied by a few sidelobes. An apodized spectrum should not be used for finding line. One should always fit the data with a model, even if one needs only line positions.