Triangular apodization function

Although a variety of apodization functions have been examined 34,35,36), triangular apodization has the following form... 

Fig. 4. The functions a) I(Vt) and b) S(Vk), which are the instrument line shape functions for spectra computed using no apodization and triangular apodization, respectively.

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... 

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.

When a spectrum is measured on a dispersive instrument, the true spectrum is convolved with the instrumental line shape (ILS) of the monochromator, which is the triangular slit function. The situation with the FT technique is equivalent, except that the true spectrum is convolved with the (sinx)/x function (no apodization) or with the FT of an appropriate apodization function. Hence, FT instruments offer a free choice of ILS according to the apodization selected and thus make it possible to optimise the sampling condition for a particular application. 

Figure 2.5 shows three examples of apodization functions and the corresponding ILS. It can be observed that the triangular and squared triangular apodization functions (green and red, respectively) present a reduced side lobe intensity. However,... 

Fig. 2.5 Apodization functions (left) and corresponding Instrumental Line Shape (right) boxcar (blue), triangular (green) and squared triangular (red)...

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.

On the other hand, it can be shown that if the Rayleigh criterion is obeyed and a triangular apodization function is used the resolution becomes... 

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 series of curves shown in Figure 8.3 is quite similar to the corresponding series of curves calculated by Ramsay for a monochromator with a triangular slit function. However, with the triangular apodization function, the slope of the plot of log A pg versus log Ap j can be as small as indicating that Ap can vary as (Ap j ) when Apg is very large. 

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)...

For FTIR instrumentation, this absorbance limitation arises from the apodization process. For the common triangular (sinc ) apodization function, a linear relationship is found between the true absorbance and the apparent (observed) absorbance up to about 0.7 absorbance units. Above 0.7 absorbance units, the apparent absorbance becomes nonlinear, with the amount of decrease depending on the true absorbance and the resolution parameter, r, where r is the ratio of the instrument nominal resolution to the full width at half height. For absorbance values greater than 1, resolution parameters ofO. 1 or greater produce nonlinear behavior [12]. 

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... 

Fig. 15 Apodization, or the reduction of artifacts in the spectral line by the multiplication of the interferogram by a window function that tapers to zero at the end point of the interferogram. (a) Cosine interferogram of Fig. 13(a) multiplied by the triangular window function of Fig. 14(d). (b) Resulting spectral line, the sine-squared function.

Fig. 10. Interferogram I (s) of a continuous spectrum versus path difference s (upper part), the corresponding spectrum I (f) versus wave number v, and the instrument line-shape function (for triangular apodization)...

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 ... 

Figure 2.7. The sinc instrument lineshape function computed for triangularly apodized interferograms note that its full width at half-height is greater than that of the sine function resulting ftom boxcar truncation of the same interferogram.

In such cases it is better to apodize with a triangular rather than box-car function. This situation may be compared with the rule-of-thumb for conventional infrared spectrometers that the peak width at half height should be at least five times the instrumental spectral slit width if accurate results are to be obtained. 

On the left is the mean value of 7(v), within the interval Av, weighted by the instrument function, which may have a nonrectangular shape in this case. Narrow-band interference filters are often bell-shaped grating spectrometers and Michelson interferometers with triangular apodization have a (sin a /a ) response function. If one defines I(v) as the spectrum weighted by the narrow instrument function one obtains... 

Boxcar truncation of the interferogram results in a sine function which has side lobes. The reduction in the side lobes on the spectral lines observed can be accomplished by apodization. Triangular apodization gives a sine function with the side lobes considerably reduced. The reduction in side lobes is accomplished at the expense of a some loss in spectral resolution. 

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