Abstract

The formula for the noise-equivalent change in radiance (NEdN) for sampling noise [Appl. Opt. 38, 139 (1999)] can work well when applied to the double-sided interferograms of radiance spectra dominated by isolated emission lines, but it does not work well when applied to broad, slowly varying radiance spectra such as a Planck blackbody curve. The modified formula for the sampling-noise NEdN works well when applied to the double-sided interferograms of both types of radiance spectra.

© 2003 Optical Society of America

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  1. D. Cohen, “Performance degradation of a Michelson interferometer due to random sampling errors,” Appl. Opt. 38, 139–151 (1999).
    [CrossRef]
  2. D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).
  3. D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).
  4. W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).
  5. P. Haschberger, “Impact of the sinusoidal drive on the instrument line shape function of a Michelson interferometer with rotating retroreflector,” Appl. Spectrosc. 48, 307–315 (1994).
    [CrossRef]
  6. A. S. Zachor, “Drive nonlinearities: their effects in Fourier spectroscopy,” Appl. Opt. 16, 1412–1424 (1977).
    [CrossRef] [PubMed]
  7. A. S. Zachor, S. M. Aaronson, “Delay compensation: its effect in reducing sampling errors in Fourier transform spectroscopy,” Appl. Opt. 18, 68–75 (1979).
    [CrossRef] [PubMed]
  8. E. E. Bell, R. B. Sanderson, “Spectral errors resulting from random sampling-position errors in Fourier transform spectroscopy,” Appl. Opt. 11, 688–689 (1972).
    [CrossRef] [PubMed]
  9. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
    [CrossRef]
  10. D. Cohen, “Performance degradation of a Michelson interferometer when its misalignment angle is a rapidly varying random time series,” Appl. Opt. 36, 4034–4042 (1997).
    [CrossRef] [PubMed]

1999 (1)

1997 (1)

1994 (1)

1988 (1)

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

1979 (1)

1977 (1)

1972 (1)

Aaronson, S. M.

Bell, E. E.

Bicknell, W. E.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Bold, D. R.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Buijs, H.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

Burnside, J. W.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Cafferty, M. S.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Candell, L. M.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Cohen, D.

Cohen, D. L.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Colao, A. A.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Feinstein, H. W.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Filip, A. E.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Forman, S. E.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Haschberger, P.

Hearn, D. R.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Howell, H. B.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

Jimenez, H. J.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Kerekes, J. P.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

LaPorte, D. D.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

Malyak, P. H.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Miller, R. W.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Mooney, D. L.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Persky, M. J.

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Pillsbury, A. D.

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Plaut, A. B.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Revercomb, H. E.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

Ryan-Howard, D. P.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Sanderson, R. B.

Scouler, W. J.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

Smith, W. L.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

Sromovsky, L. A.

H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer sounder,” Appl. Opt. 27, 3201–3218 (1988).
[CrossRef]

Weidler, D. E.

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

Zachor, A. S.

Appl. Opt. (6)

Appl. Spectrosc. (1)

Other (3)

D. L. Mooney, D. R. Bold, A. A. Colao, A. E. Filip, S. E. Forman, J. P. Kerekes, P. H. Malyak, R. W. Miller, M. J. Persky, A. D. Pillsbury, D. E. Weidler, “POES high-resolution sounder study final report,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

D. L. Mooney, D. R. Bold, M. S. Cafferty, D. L. Cohen, H. J. Jimenez, J. P. Kerekes, R. W. Miller, M. J. Persky, D. P. Ryan-Howard, “POES advanced sounder study (Phase II),” (Lincoln Laboratory, MIT, Cambridge, Mass., 1994).

W. E. Bicknell, J. W. Burnside, L. M. Candell, H. W. Feinstein, D. R. Hearn, J. P. Kerekes, A. B. Plaut, D. P. Ryan-Howard, W. J. Scouler, D. E. Weidler, “GOES high-resolution interferometer study,” (Lincoln Laboratory, MIT, Cambridge, Mass., 1995).

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Figures (6)

Fig. 1
Fig. 1

Signal diagram showing the flow of information from the input scene radiance that enters the Michelson interferometer to the digital signal that leaves the Michelson interferometer. Although the Michelson interferometer is drawn with flat return mirrors, the equations derived still hold true when corner cubes are used to return the signal to the beam splitter.

Fig. 2
Fig. 2

Scaled so that the x axes cover the same spread of wave numbers. The zero wave-number position of (b) is aligned just below the peak value of the emission line in (a) to show how the power spectrum in (b) brackets the emission line in (a). Each peak in (b) has a wave-number width of σM.

Fig. 3
Fig. 3

Noise-power spectrum S 4(σ) used to generate random errors in the sampling position for the simulated interferometer discussed. The width of each block is 10 cm-1, and the top of each block corresponds to a spectral power of [(10-11/4)cm2]/(2 × 10 cm-1). Consequently the rms error in the sampling position is (10-11/2/2) cm ≅ 1.58 × 10-6 cm.

Fig. 4
Fig. 4

Graph of ten simulated measurements of a 400 K blackbody spectrum contaminated by the sampling noise in Fig. 3. The noise was increased by a factor of 20 over the size specified by the spectrum in Fig. 3 to make it easier to see.

Fig. 5
Fig. 5

Top curve, sampling-noise NEdN from the formula in Ref. 1; bottom curve, sampling-noise NEdN from Eq. (8e). The crosses that mark the calculated standard deviations of the spectral error follow the bottom curve. The standard deviations are from 30 noise-contaminated measurements of a 400 K Planck curve, with the sampling noise obeying the noise-power spectrum in Fig. 3.

Fig. 6
Fig. 6

Plot of the T(σ) curve showing where it crosses zero on the wave-number axis.

Equations (104)

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Zx=-Zσexp2πiσxdσ,
Zσ=Zsσ+Zbσ,
Zsσ=1/4AdΩscη|σ|R|σ|τa|σ|τf|σ|Bsc|σ|,
Zbσ=1/4Adη|σ|R|σ|τa|σ|ΩfBf|σ|-ΩaBa|σ|.
Gx=-dxuhx-xuZx,
Πx, LGx,
Πx, L=1 for |x|L0 for |x|>L.
-LxL.
Gx+rxGx+rxdGdx,
Erx=0.
Erxrx=Rrx-x.
Rrx-x=Rrx-x.
Rr-x=Rrx,
ImRrx=0.
Srσ=-Rrxexp-2πiσxdx,
Rrx=-Srσexp2πiσxdσ.
Sr-σ=Srσ,
ImSrσ=0.
Gmσ=- Πx, LGxexp-2πiσxdx,
GmσGσ=Huσ·Zσ.
Gmσ=-Πx, LGxexp-2πiσxdx=-LLGxexp-2πiσxdx,
Gσ=-Gxexp-2πiσxdx.
nsσ=-Πx, LrxdGdxexp-2πiσxdx.
Gmσ+nsσHuσ·Zσ+nsσ.
Ensσ=-Πx, LErxdGdxexp-2πiσxdx=0.
dGdx=ddx-Gσexp2πiσxdσ=2πi-σGσexp2πiσxdσ.
dGdx=2πi - σHuσZσexp2πiσxdσ.
spectral radiance contaminated by sampling noise=Gσ+nsσ-G1σG2σ-G1σ·B2σ-B1σ+B1σ,
G1σ=Huσ·Zs1σ+Zbσ,
G2σ=Huσ·Zs2σ+Zbσ,
Zs1,2σ=1/4AdΩscη|σ|R|σ|×τa|σ|τf|σ|B1,2|σ|.
spectral radiance contaminated by sampling noise=Bscσ+4nsσAdΩscHuση|σ|R|σ|τa|σ|τf|σ|.
4nsσAdΩscHuση|σ|R|σ|τa|σ|τf|σ|
sampling noise in measurement=4Rensσ/HuσAdΩscη|σ|R|σ|τa|σ|τf|σ|.
Huσ=|Huσ|expiψσ.
sampling noise in measurement=4Reexp-iψσnsσAdΩscη|σ|R|σ|τa|σ|τf|σ||Huσ|.
NEdNsamp=4{E[(Reexp-iψσnsσ)2]}1/2AdΩsc|Huσ|η|σ|R|σ|τa|σ|τf|σ|.
E[(Reexp-iψσnsσ)2].
E|exp-iψσnsσ|2=E|nsσ|2
E[(Reexp-iψσnsσ)2]12E|nsσ|2.
EReexp-iψσnsσ2
EReexp-iψσnsσ2=Jσ,
Jσ=π2-Srσ|σ+σexp-iψσ×Huσ+σZσ+σ-σ-σexpiψσ×Huσ-σ*Zσ-σ|2dσ.
NEdNsamp=4JσAdΩsc|Huσ|η|σ|R|σ|τa|σ|τf|σ|.
EReexp-iψσnsσ212E|nsσ|2Jσ.
Jσ=12E|nsσ|2-2π2 Reexp-2iψσ×-Srσσ-σHuσ-σ×Zσ-σσ+σHuσ+σ×Zσ+σdσ.
σ=-Srσσ-σHuσ-σZσ-σ×σ+σHuσ+σZσ+σdσ
Baσ=Bfσ=0
σ=-σC+σM-σCSrσσ-σHuσ-σZσ-σσ+σHuσ+σZσ+σdσ+σCσC+σMSrσσ-σHuσ-σZσ-σσ+σHuσ+σZσ+σdσ.
τaσ=τfσ=ησ=1,
Rσ=1 amp s/s.
AdΩsc=3.07×10-3 cm2 s.
ψσKσ
Jσ=π2-Srσ|σ+σ|Huσ+σ|×Zσ+σ-σ-σ|Huσ-σ|×Zσ-σ|2dσ.
Ba|σ|=Bf|σ|=0 and τa|σ|=τf|σ|=η|σ|=1 with R|σ|=1 amp s/s,
NEdNsamp=π|Huσ|-Srσ|σ+σ×|Huσ+σ|Zσ+σ-σ-σ×|Huσ-σ|Zσ-σ|2dσ1/2
gσ=|Huσ|Bscσ
gσ±σgσ±σdgdσat σ
-Srσσ+σgσ+σ-σ-σ×gσ-σ2dσ4gσ+σdgdσat σ2-σ2Srσdσ.
NEdNsamp2π|Huσ|·gσ+σdgdσat σ·-σ2Srσdσ1/2.
σ=gσ+σdgdσat σ
Huσ+σHuσ,
R|σ+σ|R|σ|,
τa,f|σ+σ|τa,f|σ|,
η|σ+σ|η|σ|
Baσ=Bfσ=0.
Jσπ2AdΩsc42|Huσ|2R|σ|2τa|σ|2τf|σ|2η|σ|2·-Srσσ+σBsc|σ+σ|-σ-σBsc|σ-σ|2dσ.
NEdNsampπ-Srσσ+σBscσ+σ-σ-σBscσ-σ2dσ1/2.
-Srσσ+σσ-σBscσ+σ×Bscσ-σdσ0,
NEdNsampπ-Srσσ+σBscσ+σ2×dσ+-Srσσ-σ×Bscσ-σ2dσ1/2.
-Srσσ+σBscσ+σ2dσ=-Srσσ-σBscσ-σ2dσ.
NEdNsampghostπ2-Srσσ-σ×Bscσ-σ2dσ1/2=π2Srσ * σ2Bscσ21/2,
F±iσxfx=- fxexp±2πiσxdx.
Wx=dGdx=2πi - σHuσZσexp2πiσxdσ=Fiσx2πiσHuσZσ.
T1=EF-iσxΠx, LrxWx2,
T2=EFiσxΠx, LrxWx2,
T3=EF-iσxΠx, LrxWx·FiσxΠx, LrxWx.
T2=T1*.
T1=E-dxΠx, LWxexp-2πiσx×-dxΠx, LWx×exp-2πiσxrxrx=-dxΠx, LWxexp-2πiσx×-dxΠx, LWx×exp-2πiσxErxrx=-dxΠx, LWxexp-2πiσx×-dxΠx, LWx×exp-2πiσxRrx-x.
-Πx, Lexp±2πiσxdx=F±iσxΠx, L=2L sinc2πσL
-Wxexp-2πiσxdx=F-iσxWx=2πiσHuσZσ.
-Πx, LWxexp-2πiσxdx=2L sinc2πσL * 2πiσHuσZσ.
-Πx, LWxexp-2πiσxdx2πiσHuσZσ.
T1=-4π2·-Srσσ+σHuσ+σZσ+σσ-σHuσ-σZσ-σdσ.
T2=-4π2·-Srσσ+σHuσ+σ*Zσ+σσ-σHuσ-σ*Zσ-σdσ.
T3=-dxΠx, LWxexp-2πiσx-dxΠx, LWxexp2πiσxRrx-x.
Rrx-x=12Rrx-x+12Rrx-x.
T3=12-dxΠx, LWxexp-2πiσx-dxΠx, LWxexp2πiσxRrx-x+12-dxΠx, LWxexp-2πiσx-dxΠx, LWxexp2πiσxRrx-x.
-Πx, LWxexp±2πiσxdx*=-Πx, LWxexp2πiσxdx,
T3=2π2·-Srσσ+σHuσ+σ×Zσ+σσ+σHuσ+σ*×Zσ+σdσ+-Srσσ-σ×Huσ-σZσ-σ×σ-σ×Huσ-σ*Zσ-σdσ.
Jσ=EReexp-iψσnsσ2.
Jσ=E12exp-iψσnsσ+12expiψσnsσ*2=14exp-2iψσEnsσ2+14exp2iψσEnsσ*2+12E|nsσ|2.
nsσ=F-iσxΠx, LrxWx.
nsσ*=FiσxΠx, LrxWx.
Jσ=T14exp-2iψσ+T24exp2iψσ+12E|nsσ|2.
Jσ=12ReT1 exp-2iψσ+12E|nsσ|2,
Jσ=-2π2Reexp-2iψσ·-Srσσ+σHuσ+σZσ+σσ-σHuσ-σZσ-σdσ+12E|nsσ|2.
Jσ=T14exp-2iψσ+T24exp2iψσ+T32.
Jσ=π2-Srσ·-exp-iψσσ+σ×Huσ+σZσ+σ·exp-iψσσ-σHuσ-σ×Zσ-σ-expiψσσ+σ×Huσ+σ*Zσ+σ·expiψσσ-σHuσ-σ*×Zσ-σ+exp-iψσσ+σ×Huσ+σZσ+σ·expiψσσ+σHuσ+σ*×Zσ+σ+exp-iψσσ-σ×Huσ-σZσ-σ·expiψσσ-σHuσ-σ*×Zσ-σdσ.
a=exp-iψσσ+σHuσ+σZσ+σ, b=expiψσσ-σHuσ-σ*Zσ-σ,
-ab*-ba*+aa*+bb*=a-ba*-b*=|a-b|2.
Jσ=π2-Srσ|exp-iψσσ+σ×Huσ+σZσ+σ-expiψσσ-σ×Huσ-σ*Zσ-σ|2dσ,
EReexp-iψσnsσ2=π2-Srσ|exp-iψσσ+σHuσ+σZσ+σ-expiψσσ-σHuσ-σ*Zσ-σ|2dσ.
EReexp-iψσnsσ2

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