Apodization

Apodization, sometimes also called tapering, is a mathematical technique used to reduce the Gibbs phenomenon ``ringing'' which is produced in a spectrum obtained from a truncated interferogram. Since interferograms can only be measured out to some finite distance, all laboratory interferograms are truncated. As shown by (), the observed spectrum is related to the true spectrum by a convolution with the ``instrument function'' (or ``apparatus function'') obtained by Fourier transforming the apodization function. The instrument function corresponding to the simplest apodization--the rectangle function produced by a finite-length interferogram--is a sinc function. Because of the large sidelobes of this function, it is sometimes desirable to multiply the original interferogram by some other function which goes smoothly to zero at the end of the interferogram (e.g., Schnopper and Thompson 1974). Table lists the most commonly used apodization functions and their transforms, both of which are plotted in Fig. .

**Figure 4.4:** Apodization functions, their instrument functions, and blowups of the first few sidelobes.
$\begin{figure} \begin{center} \BoxedEPSF{InstrumentFunctions.epsf scaled 500}\\ \end{center}\end{figure}$

While apodization suppresses sidelobes, it also results in a broadening of spectral features (Thompson et al. 1991, p. 239). Table lists the widths, peak, and peak sidelobes of the apodization functions in Table . For a given application, these two factors must be balanced when selecting an appropriate apodization function. Spectra obtained with the CSO FTS have been processed using a variety of apodization functions. For planetary interferograms, apodization made no discernible difference. This is true because the broad width of planetary features smears out the ringing of the instrument functions, averaging out their effect. Furthermore, because the signal levels in our interferograms are very weak near the maximum optical path difference, the interferograms are effectively ``self-apodized'' by noise, making additional apodization unnecessary. Even for fairly narrow lines such as those in the Orion Molecular Cloud core (Serabyn and Weisstein 1995), only a single sidelobe of ringing was evident for the strongest CO lines. In this case, the desire for the highest possible resolution precluded the use of apodization.

Table 4.1: Various commonly used apodization functions and their corresponding instrument functions as illustrated in Fig. . L is the length of the one-sided portion of an interferogram (in sample numbers).

Type	Apodization Function	Instrument Function
Bartlett	$1-{\vert x\vert\over L}$	$L\mathop{\rm sinc}\nolimits ^2(\pi kL)$
Blackman	$0.42+0.5\cos\left({\pi x\over L}\right)+0.08\cos\left({2\pi x\over L}\right)$	${L(0.84-0.36 L^2k^2-2.17\times 10^{-19} L^4k^4)\mathop{\rm sinc}\nolimits (2\pi Lk)\over (1-L^2k^2)(1-4L^2k^2)}$
Connes	$\left({1-{x^2\over L^2}}\right)^2$	$8L\sqrt{2\pi}\, {J_{5/2}(2\pi kL)\over (2\pi kL)^{5/2}}$
Cosine	$\cos\left({\pi x\over 2L}\right)$	${4L\cos(2\pi Lk)\over \pi(1-16L^2 k^2)}$
Gaussian^*	$e^{-x^2/(2\sigma^2)}$	2 $\int_0^L \cos(2\pi kx)e^{-x^2/(2\sigma^2)}\,dx$
Hamming	$0.54+0.46\cos\left({\pi x\over L}\right)$	${L(1.08-0.64L^2k^2)\mathop{\rm sinc}\nolimits (2\pi Lk)\over 1-4L^2k^2}$
Hanning ${}^\dagger$	$\cos^2\left({\pi x\over 2L}\right)$	${L\mathop{\rm sinc}\nolimits \,(2\pi L k)\over 1-4L^2k^2}$
Uniform	1	$2L\mathop{\rm sinc}\nolimits \,(2\pi kL)$

^*For Gaussian apodization, $\sigma^2$ is the variance of the Gaussian function, which can be chosen independently of L. ${}^\dagger$ The instrument function for Hanning apodization can also be written

$\begin{displaymath}a[\mathop{\rm sinc}\nolimits (2\pi kL)+{\textstyle{1\over 2}}... ...\textstyle{1\over 2}}\mathop{\rm sinc}\nolimits (2\pi kL+\pi)].\end{displaymath}$

Table 4.2: Width, peak value, and peak positive and negative sidelobes for the instrument functions illustrated in Fig. . The values for Gaussian apodization depend on the choice of the Gaussian variance $\sigma^2$ .

Type	FWHM	PSF Peak	${\hbox{Peak $(-)$\ Sidelobe}\over \hbox{Peak}}$	${\hbox{Peak $(+)$\ Sidelobe}\over \hbox{Peak}}$
Bartlett	1.77179	1	$\phantom{-}0.00000000$	$0.0471904\phantom{0}$
Blackman	2.29880	0.84	-0.00106724	0.00124325
Connes	1.90416	${\textstyle{16\over 15}}$	$-0.0411049\phantom{0}$	$0.0128926\phantom{0}$
Cosine	1.63941	${\textstyle{4\over \pi}}$	$-0.0708048\phantom{0}$	$0.0292720\phantom{0}$
Gaussian	--	1	--	--
Hamming	1.81522	1.08	-0.00689132	0.00734934
Hanning	2.00000	1	$-0.0267076\phantom{0}$	0.00843441
Uniform	1.20671	2	$-0.217234\phantom{00}$	$0.128375\phantom{00}$