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消趾函数

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消趾函数（也称为锥削函数或窗函数）是一种用于在采样区域边缘将采样信号平滑降至零的函数。这抑制了在执行离散傅里叶变换时可能产生的泄漏旁瓣，但这种抑制是以加宽谱线为代价的，从而导致分辨率降低。

下面总结了一些用于对称（双边）干涉图的消趾函数，以及它们产生的仪器函数（或设备函数）和仪器函数旁瓣的放大图。给定消趾函数A(x)对应的仪器函数 I(k) 可以通过取有限傅里叶余弦变换来计算，

I(k)=int_(-a)^acos(2pikx)A(x)dx.
(1)
InstrumentFunctions
类型消趾函数仪器函数
Bartlett1-(|x|)/aasinc^2(pika)
BlackmanB_A(x)B_I(k)
Connes(1-(x^2)/(a^2))^28asqrt(2pi)(J_(5/2)(2pika))/((2pika)^(5/2))
余弦cos((pix)/(2a))(4acos(2piak))/(pi(1-16a^2k^2))
高斯e^(-x^2/(2sigma^2))2int_0^acos(2pikx)e^(-x^2/(2sigma^2))dx
HammingHm_A(x)Hm_I(k)
HanningHn_A(x)Hn_I(k)
均匀12asinc(2pika)
Welch1-(x^2)/(a^2)W_I(k)

其中

B_A(x)=(21)/(50)+1/2cos((pix)/a)+2/(25)cos((2pix)/a)
(2)
B_I(k)=(a((21)/(25)-9/(25)a^2k^2)sinc(2piak))/((1-a^2k^2)(1-4a^2k^2))
(3)
Hm_A(x)=(27)/(50)+(23)/(50)cos((pix)/a)
(4)
Hm_I(k)=(a((27)/(25)-(16)/(25)a^2k^2)sinc(2piak))/(1-4a^2k^2)
(5)
Hn_A(x)=cos^2((pix)/(2a))
(6)
=1/2[1+cos((pix)/a)]
(7)
Hn_I(k)=(asinc(2piak))/(1-4a^2k^2)
(8)
=a[sinc(2pika)+1/2sinc(2pika-pi)+1/2sinc(2pika+pi)]
(9)
W_I(k)=a2sqrt(2pi)(J_(3/2)(2pika))/((2pika)^(3/2))
(10)
=a(sin(2pika)-2piakcos(2piak))/(2a^3k^3pi^3).
(11)

下表总结了常用消趾函数的宽度、峰值和峰值旁瓣峰值（负和正）。

类型FWHM 仪器函数IF 峰值(peak (-) sidelobe)/(peak)(peak (+) sidelobe)/(peak)
Bartlett1.7717910.000000000.0471904
Blackman2.29880(21)/(25)-0.001067240.00124325
Connes1.90416(16)/(15)-0.04110490.0128926
余弦1.639414/pi-0.07080480.0292720
高斯--1----
Hamming1.81522(27)/(25)-0.006891320.00734934
Hanning2.000001-0.02670760.00843441
均匀1.206712-0.2172340.128375
Welch1.590444/3-0.08617130.0356044

一个通用的对称消趾函数 A(x) 可以写成傅里叶级数

A(x)=a_0+2sum_(n=1)^inftya_ncos((npix)/b),
(12)

其中系数满足

a_0+2sum_(n=1)^inftya_n=1.
(13)

相应的仪器函数是

I(t)=int_(-b)^bA(x)e^(-2piikx)dx
(14)
=2b{a_0sinc(2pikb)+sum_(n=1)^(infty)[sinc(2pikb+npi)+sinc(2pikb-npi)]}.
(15)

要获得在 ka=3/4 处为零的消趾函数，请使用

a_0sinc(3/2pi)+a_1[sinc(5/2pi)+sinc(1/2pi)]=0.
(16)

代入 (14),

-(1-2a_1)2/(3pi)+a_1(2/(5pi)+2/pi)=-1/3(1-2a_1)+a_1(1/5+1)=0
(17)
a_1(6/5+2/3)=1/3
(18)
a_1=(1/3)/(6/5+2/3)=5/(6·3+2·5)=5/(28)
(19)
a_0=1-2a_1=(28-2·5)/(28)=(18)/(28)=9/(14).
(20)

Hamming 函数接近于仪器函数在 ka=5/4 处变为 0 的要求，得到

a_0=(25)/(46) approx 0.5435
(21)
a_1=(21)/(92) approx 0.2283.
(22)

Blackman 函数的选择使得仪器函数在 ka=5/4 和 ka=9/4 处变为 0，得到

a_0=(3969)/(9304) approx 0.42659
(23)
a_1=(1155)/(4652) approx 0.24828
(24)
a_2=(715)/(18608) approx 0.38424,
(25)

另请参阅

Bartlett 函数, Blackman 函数, Connes 函数, 余弦消趾函数, 半峰全宽, 高斯函数, Hamming 函数, Hanning 函数, 泄漏, Mertz 消趾函数, Parzen 消趾函数, 均匀消趾函数, Welch 消趾函数

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参考文献

Ball, J. A. "The Spectral Resolution in a Correlator System" §4.3.5 in Astrophysics, Part C: Radio Observations (Ed. M. L. Meeks). New York: Academic Press, pp. 55-57, 1976.Blackman, R. B. and Tukey, J. W. "Particular Pairs of Windows." In The Measurement of Power Spectra, From the Point of View of Communications Engineering. New York: Dover, pp. 95-101, 1959.Brault, J. W. "Fourier Transform Spectrometry." In High Resolution in Astronomy: 15th Advanced Course of the Swiss Society of Astronomy and Astrophysics (Ed. A. Benz, M. Huber, and M. Mayor). Geneva Observatory, Sauverny, Switzerland, pp. 31-32, 1985.Harris, F. J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proc. IEEE 66, 51-83, 1978.Norton, R. H. and Beer, R. "New Apodizing Functions for Fourier Spectroscopy." J. Opt. Soc. Amer. 66, 259-264, 1976.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 547-548, 1992.Schnopper, H. W. and Thompson, R. I. "Fourier Spectrometers." In Astrophysics, Part A: Optical and Infrared (Ed. N. P. Carleton). New York: Academic Press, pp. 491-529, 1974.

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消趾函数

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Weisstein, Eric W. "消趾函数。" 来自 —— 资源。 https://mathworld.net.cn/ApodizationFunction.html

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