Apodization Function

An apodization function (also called a tapering function or window function) is a function used to smoothly bring a sampled signal down to zero at the edges of the sampled region. This suppresses leakage sidelobes which would otherwise be produced upon performing a discrete Fourier transform, but the suppression is at the expense of widening the lines, resulting in a decrease in the resolution.

A number of apodization functions for symmetrical (two-sided) interferograms are summarized below, together with the instrument functions (or apparatus functions) they produce and a blowup of the instrument function sidelobes. The instrument function corresponding to a given apodization function can be computed by taking the finite Fourier cosine transform,

(1)

type	apodization function	instrument function
Bartlett
Blackman
Connes
cosine
Gaussian
Hamming
Hanning
uniform	1
Welch

where

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

The following table summarizes the widths, peaks, and peak-sidelobe-to-peak (negative and positive) for common apodization functions.

type	instrument function FWHM	IF peak
Bartlett	1.77179	1	0.00000000
Blackman	2.29880			0.00124325
Connes	1.90416
cosine	1.63941
Gaussian	--	1	--	--
Hamming	1.81522			0.00734934
Hanning	2.00000	1		0.00843441
uniform	1.20671	2
Welch	1.59044

A general symmetric apodization function can be written as a Fourier series

(12)

where the coefficients satisfy

(13)

The corresponding instrument function is

			(14)
			(15)

To obtain an apodization function with zero at , use

(16)

Plugging in (14),

(17)

(18)

			(19)
			(20)

The Hamming function is close to the requirement that the instrument function goes to 0 at , giving

			(21)
			(22)

The Blackman function is chosen so that the instrument function goes to 0 at and , giving

			(23)
			(24)
			(25)

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