Triangular function Fourier transform

Fig. 26 Fourier transform spectrum of v2 of ammonia. Trace (a) is a section of the infrared absorption spectrum of ammonia recorded on a Digilab Fourier transform spectrometer at a nominal resolution of 0.125 cm-1. In this section of the spectrum near 848 cm-1 the sidelobes of the sine response function partially cancel, but the spectrum exhibits negative absorption and some sidelobes. Trace (b) is the same section of the ammonia spectrum using triangular apodiza-tion to produce a sine-squared transfer function. Trace (c) is the deconvolution of the sine-squared data using a Jansson-type weight constraint.

It is found that multiplication of the Fourier transform of the data by a carefully chosen window function is very effective in removing the artifacts around peaked functions. This process is called apodization. Apodization with the triangular window function is often applied to Fourier transform spectroscopy interferograms to remove the ringing around the infrared... 

An additional consequence of finite retardation is the appearance of secondary extrema or "wings" on either side of the primary features. The presence of these features is disadvantageous, especially when it is desired to observe a weak absorbance in proximity to a strong one. To diminish this problem the interferogram is usually multiplied by a triangular apodization function which forces the product to approach zero continuously for s = + Fourier transformation of the... 

In the transformation the physical units are inverted. When the interferogram is expressed in optical path difference units (cm), the spectrum is obtained in wave-numbers (cm-1) and when the interferogram is expressed in time units (s) the spectrum is in frequency units (s 1). Apart from sine and cosine functions, box-car and triangular, etc. functions are also known, for which the Fourier transformation can be calculated. When applying the Fourier transformation over the whole area + oo, the arm of the interferometer also would have to be moved from — co to +co. When making a displacement over a distance of +L only, the interferogram has to be multiplied by a block function, which has the value of 1 between + and —I and the value 0 outside. I then influences the resolution that can be obtained. 

The Fourier transform necessary to convert I (s) into /(l ) may be executed either in an analogue or in a digital way. In this section, we shall concentrate on the first possibility. One potential way is illustrated in Fig. 19 the movable mirror of the Michelson interferometer is moved according to a triangular wave function (Fig. 19 upper left). This can be done by means of an appropiate mechanical cam system. Problems arise only at the points of reversal, but usually these difficulties are bypassed by the apodization. During half a period the mirror moves with constant velocity, say uq- Hence s=2vat and, for monochromatic radiation, there is an a.c. component in the signal recorded by the detector ... 

Another way of looking at it starts from the periodic nature of the basis functions. As far as the sines and cosines are concerned, the signal could be just one period of a cyclic phenomenon. When we plot the concatenated signal (Fig. 22(a)), we see that it is dominated by a triangular oscillation, a saw-tooth as it is called. The Fourier transform of the signal will be equally dominated by the transform of that saw-tooth, which, due to the sharp edges in the saw-tooth, contains a lot of high frequencies. To illustrate this, the saw-tooth and its power spectrum have been plotted in Fig. 22(b) and (c). 

An additional problem in FT deconvolution results from the finite number of data points. Back transformation frequently leads to wave-like curves, which cannot be attributed to real periodicities of the Fourier transform. To suppress these undesired side effects, an apodization function in the form of a triangular or parabolic function is applied. In analogy to Eq. (3.27), the decon-voluted signal is calculated by using an apodization function, Z)(v), by... 

Figure 5.3. Various apodization functions (left) and the instrumental lineshape produced by them (right) (a) boxcar truncation (b) triangular (c) trapezoidal (d) Norton-Beer weak, medium, and strong (e) Happ-Gen-zel (f) Blackman-Harris 3-term and 4-term. The maximum retardation is set to / = 1. In the Fourier transform the FWHH of the main lobe is indicated.

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

Figure 4. A triangular apodization function and its Fourier transform, the sinc function.

To compute the convolution of these two functions, Eq. 2.19 requires that/(v) be reversed left to right [which is trivial in this case, since/(v) is an even function], after which the two functions are multiplied point by point along the wavenumber axis. The resulting points are then integrated, and the process is repeated for all possible displacements, v, of/( relative to B v). One particular example of convolution may be familiar to spectroscopists who use grating instruments (see Chapter 8). When a low-resolution spectrum is measured on a monochromator, the true spectrum is convolved with the triangular slit function of the monochromator. The situation with Fourier transform spectrometry is equivalent, except that the true spectrum is convolved with the sine function/(v). Since the Fourier transform spectrometer does not have any slits,/(v) has been variously called the instrument line shape (ILS) Junction, the instrument function, or the apparatus function, of which we prefer the term ILS function. 

The function Ai(8), called a triangular apodization function, is a popular apodization function used in Fourier transform infrared spectrometry, which is unfortunate since the triangular apodization function has deleterious effects on the photometric accuracy in a spectrum (see Section 8.3). Nonetheless, for lines separated by 1 /A, a 20% dip is found, as shown in Figure 2.6a. If the lines were separated by 2/A, they would be fully resolved (i.e., resolved to baseline). The FWHH for the function /] (v) is 0.88/A, and the lines separated by this amount are just resolved however, the dip is extremely small, that is, on the order of 1%. 

The relation between the exponential decay and the Lorentz profile and that between the cosine function and the delta function as the Fourier transform pairs are described, respectively, in Sections D.3.4 and D.3.5. The Fourier transform of the triangular function shown in Figure 6.6f is a sine function squared as described in Section D.3.6. 

Fig. 5.8.5 Fourier transforms of several apodization functions shown in the insert. (1) refers to the box-car function, (2) to triangular apodization, and (3) to the Hamming function. (Hanel, 1983)...

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