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消趾函式

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消趾函式（也稱為錐削函式或窗函式）是一種用於在取樣區域邊緣將取樣訊號平滑降至零的函式。這抑制了在執行離散傅立葉變換時可能產生的洩漏旁瓣，但這種抑制是以加寬譜線為代價的，從而導致解析度降低。

下面總結了一些用於對稱（雙邊）干涉圖的消趾函式，以及它們產生的儀器函式（或裝置函式）和儀器函式旁瓣的放大圖。給定消趾函式A(x)對應的儀器函式 I(k) 可以透過取有限傅立葉餘弦變換來計算，

I(k)=int_(-a)^acos(2pikx)A(x)dx.
(1)
InstrumentFunctions
型別消趾函式儀器函式
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))
餘弦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)
均勻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)

下表總結了常用消趾函式的寬度、峰值和峰值旁瓣峰值（負和正）。

型別FWHM 儀器函式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
餘弦1.639414/pi-0.07080480.0292720
高斯--1----
Hamming1.81522(27)/(25)-0.006891320.00734934
Hanning2.000001-0.02670760.00843441
均勻1.206712-0.2172340.128375
Welch1.590444/3-0.08617130.0356044

一個通用的對稱消趾函式 A(x) 可以寫成傅立葉級數

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

其中係數滿足

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

相應的儀器函式是

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)

要獲得在 ka=3/4 處為零的消趾函式，請使用

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 函式接近於儀器函式在 ka=5/4 處變為 0 的要求，得到

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

Blackman 函式的選擇使得儀器函式在 ka=5/4 和 ka=9/4 處變為 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)

另請參閱

Bartlett 函式, Blackman 函式, Connes 函式, 餘弦消趾函式, 半峰全寬, 高斯函式, Hamming 函式, Hanning 函式, 洩漏, Mertz 消趾函式, Parzen 消趾函式, 均勻消趾函式, Welch 消趾函式

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參考文獻

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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消趾函式

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Weisstein, Eric W. "消趾函式。" 來自 —— 資源。 https://mathworld.tw/ApodizationFunction.html

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