Apodization and Interpolation in Fourier-Transform Spectroscopy*

A. S. Filler

doi:10.1364/JOSA.54.000762

Apodization and Interpolation in Fourier-Transform Spectroscopy

A. S. Filler

Journal of the Optical Society of America
Vol. 54,
Issue 6,
pp. 762-767
(1964)
•https://doi.org/10.1364/JOSA.54.000762

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A. S. Filler, "Apodization and Interpolation in Fourier-Transform Spectroscopy*," J. Opt. Soc. Am. 54, 762-767 (1964)

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- Original Manuscript: December 14, 1963
- Published: June 1, 1964

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Abstract

An interpolation formula is derived which gives an apodized spectrum as the convolution of the unapodized spectrum (sampled at suitable points) with the apodized apparatus function. This allows many apodizations to be applied to a single interferogram with the performance of only a single Fourier transformation. A further saving in computation effort is possible if the apodized apparatus function decays rapidly away from its center. Examples of such cases are presented, where the interferogram is weighted by a function which is a cosine series of only a few terms.

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Interferogram	Spectrum
Unterminated interferogram = I (l)	↔ True spectrum = S(k)
Termination function = $T (l) = {\begin{array}{l} 1 & if & \| l \| < L \\ 0 & if & \| l \| > L \end{array}$	↔ Primary apparatus function (normalized) = A′(k) = sin2πkL/2πkL
An even function = V (l)	↔ U(k)
Weighting function W(l) = T(l) · V(l)	↔ Apodized apparatus function = A″(k) = A′(k)*U(k)
Terminated interferogram = I′(l) = I(l) · T(l)	↔ Unapodized measured spectrum = S′(k) = S(k)*A′(k)
Weighted interferogram = I″(l) = I(l) · W(l) = I′(l) · V(l)	↔ Apodized measured spectrum = S″(k) = S(k)A″(k) = S′(k)U(k)

Case	2Lw	\|h\|	ν(0.01)	ν(0.001)
A	1.22	0.22	32	∼320
B	1.76	0.047	32	∼320
C	2.24	0.010	5	12
D	1.64	0.071	5	15
E	2.00	0.027	3	7
F	2.60	0.0046	3	4
D_0.12	1.81	0.018	3	12
D_0.18	1.92	0.0092	2	10
D_0.24	2.05	0.0031	2	8
E_0.06	2.08	0.012	3	6
E_0.13	2.20	0.0035	2	6
E_0.20	2.32	0.00093	3	4

Interferogram	Spectrum
Unterminated interferogram = I (l)	↔ True spectrum = S(k)
Termination function = $T (l) = {\begin{array}{l} 1 & if & \| l \| < L \\ 0 & if & \| l \| > L \end{array}$	↔ Primary apparatus function (normalized) = A′(k) = sin2πkL/2πkL
An even function = V (l)	↔ U(k)
Weighting function W(l) = T(l) · V(l)	↔ Apodized apparatus function = A″(k) = A′(k)*U(k)
Terminated interferogram = I′(l) = I(l) · T(l)	↔ Unapodized measured spectrum = S′(k) = S(k)*A′(k)
Weighted interferogram = I″(l) = I(l) · W(l) = I′(l) · V(l)	↔ Apodized measured spectrum = S″(k) = S(k)A″(k) = S′(k)U(k)

Case	2Lw	\|h\|	ν(0.01)	ν(0.001)
A	1.22	0.22	32	∼320
B	1.76	0.047	32	∼320
C	2.24	0.010	5	12
D	1.64	0.071	5	15
E	2.00	0.027	3	7
F	2.60	0.0046	3	4
D_0.12	1.81	0.018	3	12
D_0.18	1.92	0.0092	2	10
D_0.24	2.05	0.0031	2	8
E_0.06	2.08	0.012	3	6
E_0.13	2.20	0.0035	2	6
E_0.20	2.32	0.00093	3	4

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Equations (18)

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