Abstract

We describe a nonuniform spectral resampling transform (NUSRT) that resamples a frequency-scaled spectrum that has been measured by a Fourier-transform spectrometer (FTS). Frequency scaling of a spectrum can arise from measurements made with off-axis detectors and Doppler shifts induced by motion of a spaceborne FTS relative to an input radiation source. In addition, a spectrum may need to be rescaled in frequency to match spectral lines for applications such as the retrieval of atmospheric state parameters. The NUSRT is cast as a linear algebraic expression that relates a nonuniformly sampled interferogram to an input spectrum. A polynomial approximation is applied to this expression that reduces the inverse of the NUSRT to a series of Fourier transforms that can be implemented as fast Fourier transforms (FFTs). We show that this NUSRT algorithm requires on the order of 6N log N flops, which reduces the computational cost of rescaling by more than 1 order of magnitude compared with conventional FFT-based Shannon interpolation techniques while comparable accuracy is maintained.

© 2003 Optical Society of America

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References

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  1. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).
  2. S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).
  3. J. Genest, P. Tremblay, “Instrument line shape of Fourier transform spectrometers: analytic solutions for nonuniformly illuminated off-axis detectors,” Appl. Opt. 38, 5438–5446 (1999).
    [CrossRef]
  4. K. Bowman, H. Worden, R. Beer, “Instrument line shape modeling and correction for off-axis detectors in Fourier transform spectrometry,” Appl. Opt. 39, 3765–3773 (2000).
    [CrossRef]
  5. J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
    [CrossRef]
  6. K. Silk, E. R. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
    [CrossRef]
  7. R. Beer, T. Glavich, D. Rider, “Tropospheric emission spectrometer for the Earth Observing System’s Aura satellite,” Appl. Opt. 40, 2356–2367 (2001).
    [CrossRef]
  8. D. Siméoni, C. Singer, G. Chalon, “Infrared atmospheric sounding interferometer,” Acta Astronaut. 40, 113–118 (1997).
    [CrossRef]
  9. J. J. Puschell, P. Tompkins, “Imaging spectrometers for future Earth observing systems,” in Earth Observing Systems II, W. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
    [CrossRef]
  10. C. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practise (World Scientific, London, 2000).
  11. A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
    [CrossRef]
  12. R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
    [CrossRef]
  13. K. Gröchenig, “A discrete theory of irregular sampling,” Linear Algebra Its Appl. 193, 129–150 (1993).
    [CrossRef]
  14. Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
    [CrossRef]
  15. A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838–856 (1998).
    [CrossRef]
  16. H. G. Feichtinger, K. Gröchenig, T. Strohmer, “Efficient numerical methods in nonuniform sampling theory,” Numer. Math. 69, 423–440 (1995).
    [CrossRef]
  17. R. Beer, Remote Sensing by Fourier Transform Spectrometry (Wiley, New York, 1992).
  18. A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

2001 (1)

2000 (1)

1999 (1)

1998 (2)

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838–856 (1998).
[CrossRef]

1997 (1)

D. Siméoni, C. Singer, G. Chalon, “Infrared atmospheric sounding interferometer,” Acta Astronaut. 40, 113–118 (1997).
[CrossRef]

1995 (1)

H. G. Feichtinger, K. Gröchenig, T. Strohmer, “Efficient numerical methods in nonuniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

1993 (1)

K. Gröchenig, “A discrete theory of irregular sampling,” Linear Algebra Its Appl. 193, 129–150 (1993).
[CrossRef]

1977 (1)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Abrams, M. C.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

Beer, R.

Bowman, K.

Brault, J. W.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

Chalon, G.

D. Siméoni, C. Singer, G. Chalon, “Infrared atmospheric sounding interferometer,” Acta Astronaut. 40, 113–118 (1997).
[CrossRef]

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

Clark, C. C.

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Davis, S. P.

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

Feichtinger, H. G.

H. G. Feichtinger, K. Gröchenig, T. Strohmer, “Efficient numerical methods in nonuniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

Genest, J.

Glavich, T.

Gröchenig, K.

H. G. Feichtinger, K. Gröchenig, T. Strohmer, “Efficient numerical methods in nonuniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

K. Gröchenig, “A discrete theory of irregular sampling,” Linear Algebra Its Appl. 193, 129–150 (1993).
[CrossRef]

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Kerr, J.

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Liu, Q. H.

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

Maillard, J. P.

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Marks, R. J.

R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[CrossRef]

Massey, S.

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Nguyen, N.

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Puschell, J. J.

J. J. Puschell, P. Tompkins, “Imaging spectrometers for future Earth observing systems,” in Earth Observing Systems II, W. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
[CrossRef]

Rider, D.

Rodgers, C.

C. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practise (World Scientific, London, 2000).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Schildkraut, E. R.

K. Silk, E. R. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
[CrossRef]

Silk, K.

K. Silk, E. R. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
[CrossRef]

Siméoni, D.

D. Siméoni, C. Singer, G. Chalon, “Infrared atmospheric sounding interferometer,” Acta Astronaut. 40, 113–118 (1997).
[CrossRef]

Simons, D. A.

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Singer, C.

D. Siméoni, C. Singer, G. Chalon, “Infrared atmospheric sounding interferometer,” Acta Astronaut. 40, 113–118 (1997).
[CrossRef]

Smith, S.

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

Strohmer, T.

H. G. Feichtinger, K. Gröchenig, T. Strohmer, “Efficient numerical methods in nonuniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

Tompkins, P.

J. J. Puschell, P. Tompkins, “Imaging spectrometers for future Earth observing systems,” in Earth Observing Systems II, W. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
[CrossRef]

Tremblay, P.

Ware, A. F.

A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838–856 (1998).
[CrossRef]

Worden, H.

Acta Astronaut. (1)

D. Siméoni, C. Singer, G. Chalon, “Infrared atmospheric sounding interferometer,” Acta Astronaut. 40, 113–118 (1997).
[CrossRef]

Appl. Opt. (3)

IEEE Microwave Guided Wave Lett. (1)

Q. H. Liu, N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFT’s),” IEEE Microwave Guided Wave Lett. 8, 18–20 (1998).
[CrossRef]

Linear Algebra Its Appl. (1)

K. Gröchenig, “A discrete theory of irregular sampling,” Linear Algebra Its Appl. 193, 129–150 (1993).
[CrossRef]

Numer. Math. (1)

H. G. Feichtinger, K. Gröchenig, T. Strohmer, “Efficient numerical methods in nonuniform sampling theory,” Numer. Math. 69, 423–440 (1995).
[CrossRef]

Proc. IEEE (1)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

SIAM Rev. (1)

A. F. Ware, “Fast approximate Fourier transforms for irregularly spaced data,” SIAM Rev. 40, 838–856 (1998).
[CrossRef]

Other (9)

R. Beer, Remote Sensing by Fourier Transform Spectrometry (Wiley, New York, 1992).

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

R. J. Marks, Introduction to Shannon Sampling and Interpolation Theory (Springer-Verlag, New York, 1991).
[CrossRef]

J. P. Maillard, D. A. Simons, C. C. Clark, S. Smith, J. Kerr, S. Massey, “CFHT’s imaging Fourier transform spectrometer,” in Instrumentation in Astronomy VIII, D. Crawford, E. Craine, eds., Proc. SPIE2198, 185–193 (1994).
[CrossRef]

K. Silk, E. R. Schildkraut, “Imaging Fourier transform spectroscopy for remote chemical sensing,” in Electro-Optical Technology for Remote Chemical Detection and Identification, M. Fallahi, E. Howden, eds., Proc. SPIE2763, 169–177 (1996).
[CrossRef]

J. J. Puschell, P. Tompkins, “Imaging spectrometers for future Earth observing systems,” in Earth Observing Systems II, W. Barnes, ed., Proc. SPIE3117, 36–48 (1997).
[CrossRef]

C. Rodgers, Inverse Methods for Atmospheric Sounding: Theory and Practise (World Scientific, London, 2000).

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

S. P. Davis, M. C. Abrams, J. W. Brault, Fourier Transform Spectrometry (Academic, New York, 2001).

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Figures (12)

Fig. 1
Fig. 1

Examples of band-limited spectra residing in different alias bands. Note that the even alias band is frequency reversed relative to the odd alias band.

Fig. 2
Fig. 2

Nonuniform sampling of the ideal interferogram. The distance between the uniformly sampled and the nonuniformly sampled points in the ideal interferogram increases with increasing n.

Fig. 3
Fig. 3

Relationship between a frequency axis and a scaled frequency axis. The points on the scaled frequency axis are compressed and shifted to the origin relative to the points on the natural frequency axis. Note that the end points of the positive frequency set are sampled, whereas end points are not sampled for the negative frequency set. This sampling always occurs for an even number of samples.

Fig. 4
Fig. 4

Unit circle, which represents the magnitude of the complex exponential used in a DFT. The dashed lines along the unit circle correspond to the angles at which the complex exponential is sampled in a DFT. The circles correspond to the angles at which the complex exponential is sampled for the NUSRT.

Fig. 5
Fig. 5

Spectral resampling of a monochromatic input spectrum with Shannon interpolation. The uncorrected signal is treated as an off-axis spectrum for which the frequency axis has been scaled by 1.818 × 10-5 such that the apparent peak frequency is 1049.981 cm-1. This spectrum is then spectrally resampled with Shannon interpolation to recover the input spectrum with a peak frequency at 1050.0 cm-1.

Fig. 6
Fig. 6

Error caused by spectral resampling of the monochromatic input spectrum with Shannon interpolation.

Fig. 7
Fig. 7

Spectral resampling of a monochromatic input spectrum with the NUSRT. The uncorrected signal is treated as an off-axis spectrum for which the frequency axis has been scaled by 1.818 × 10-5 such that the apparent peak frequency is 1049.981 cm-1. This spectrum is then spectrally resampled with the NUSRT to recover the input spectrum with peak frequency at 1050.0 cm-1.

Fig. 8
Fig. 8

Error caused by spectral resampling of the monochromatic input spectrum with the NUSRT.

Fig. 9
Fig. 9

ILS correction and spectral resampling of off-axis convolved spectral radiances with the Shannon interpolation representing the ozone band viewed from the TES limb mode.

Fig. 10
Fig. 10

Error caused by the ILS correction and spectral resampling of the off-axis convolved spectral radiances with the Shannon interpolation.

Fig. 11
Fig. 11

ILS correction and spectral resampling of off-axis convolved spectral radiances with the NUSRT for the ozone band viewed from the TES limb mode.

Fig. 12
Fig. 12

Error caused by the ILS correction and spectral resampling of the off-axis convolved spectral radiances with the NUSRT.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

fx= Sνexp2πiνxdν,
fn=fnδx,
Al=l-1ν0, lν0,
VAl,
sk=Sζkνmin+kδν,k=-N-1/2,, -1, 0, 1,, N/2,
ζk=1k0-1k<0,
-1l-1nfn=k=-N-1/2N/2 skexpi2πkn/N.
sk=1Nn=-N-1/2N/2-1l-1nfnexp-i2πkn/N.
Fs=f,
fin=-1l-1nfn,
sik=sk,
Finik=expi2πkn/N.
1NFHf=s,
f˜x= Sνexp2πi1-ρνxdν,
f˜nδx=fn1-ρδx.
S˜ν=Sν1-ρ.
νmin+kδν1-ρνmin+kδν.
-1l-1nf˜n=k=-N-1/2N/2 sk×expi2π1-ρk-ζkβn/N,
Eρ, βs=f˜,
Eρ, βinik=expi2π1-ρk-ζkβn/N
E-11NEH
e=I-1/NEHEss
e<λ, ρ, βT<,
e=I-1/NEHEssI-1/NEHE ss=I-1NEHE.
Eρ, βinik =expi2π1-ρk-ζkβn/N=expi2π1-ρkn/N×exp-i2πζkβn/N=cos2πN βnexpi2π1-ρkn/N-ζki sin2πN βnexpi2π1-ρkn/N,
E=CE0ρ+iSE0ρĨ,
E0ρ=Eρ, 0
diagĨT=-1, -1, -1,, -1N/2+1, 1,, 1, 1N-1/2.
diagCβ=cos2πN β diagD,
diagSβ=sin2πN β diagD.
diagDT=0, 1, 2,, N/2,-N-1/2,, -2, -1.
E0ρinik=expi2π1-ρkn/N.
E0ρinikp=0m1p! αpkp expi2πkn/Nnpρp,α=-i2πN.
E0ρp=0m1p! αpDpFDpρp=F+αDFDρ+12! α2D2FD2ρ2++1m! αmDmFDmρm.
E0mρ=p=0m1p! αpDpFDpρp.
Eρ, βCβE0mρ+iSβE0mρĨ.
EHρ, βE0mHρCβ-iĨE0mHρSβ,
E0mHρ=p=0m1p! α¯pDpFHDpρp,α¯=i2π/N.
s1NEHρ, βf˜.
s1Np=0m1p! α¯pDpFHDpρpCβf˜-i 1NĨ×p=0m1p! α¯pDpFHDpρpSβf˜.

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