Abstract

Fourier transform spectroscopy (FTS) has been widely used in a variety of fields due to its high signal-to-noise ratio, simultaneous acquisition of a broad spectrum, and versatility for different radiation sources. Further improvement of the spectroscopic performance will widen its scope of applications. Here, we demonstrate improved spectral resolution by overcoming the time-window size limitation using a mode-locked terahertz (THz) pulse train as precisely periodic pulsed radiation in discrete Fourier transform spectroscopy (dFTS). Since infinitesimal resolution can be achieved at harmonic components of its repetition frequency 1/T when the time-window size is exactly matched to the repetition period T, a combination of dFTS with a spectral interleaving technique achieves a spectral resolution limited only by the spectral interleaving interval. Linewidths narrower than 1/(50T) are fully resolved by THz-dFTS, allowing rotational-transition absorption lines of low-pressure molecular gases to be attributed within a 1.25 MHz band.

© 2015 Optical Society of America

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References

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2014 (1)

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

2013 (1)

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

2012 (1)

2011 (2)

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

2010 (1)

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

2008 (1)

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

2006 (1)

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

2005 (2)

T. Yasui, E. Saneyoshi, T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

C. Janke, M. Först, M. Nagel, H. Kurz, A. Bartels, “Asynchronous optical sampling for high-speed characterization of integrated resonant terahertz sensors,” Opt. Lett. 30, 1405–1407 (2005).
[Crossref]

2004 (1)

2002 (1)

1998 (3)

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

1996 (1)

J. C. Pearson, H. S. P. Müller, “The submillimeter wave spectrum of isotopic methyl cyanide,” Astrophys. J. 471, 1067–1072 (1996).
[Crossref]

Abe, H.

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

Araki, T.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

T. Yasui, E. Saneyoshi, T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

Baraniuk, R. G.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

Bartels, A.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

C. Janke, M. Först, M. Nagel, H. Kurz, A. Bartels, “Asynchronous optical sampling for high-speed characterization of integrated resonant terahertz sensors,” Opt. Lett. 30, 1405–1407 (2005).
[Crossref]

Baumann, E.

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Becker, E. D.

T. C. Farrar, E. D. Becker, Pulse and Fourier Transform NMR: Introduction to Theory and Methods (Academic, 1971).

Bernhardt, B.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Bracewell, R. N.

R. N. Bracewell, “Sampling and series,” in Fourier Transform and its Applications, R. N. Bracewell, ed. (McGraw-Hill, 1999).

Bruchhausen, A.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Cassady, C. J.

G. H. Kruppa, C. J. Cassady, Fourier Transform Mass Spectrometry (CRC Press, 2006).

Coddington, I.

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Cohen, E. A.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

Dekorsy, T.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Delitsky, M. L.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

Deschênes, J.-D.

Ekström, M.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Erbe, A.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Eriksson, P.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Farrar, T. C.

T. C. Farrar, E. D. Becker, Pulse and Fourier Transform NMR: Introduction to Theory and Methods (Academic, 1971).

Först, M.

Gebs, R.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Genest, J.

Giorgetta, F. R.

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Gohle, C.

Griffiths, P. R.

P. R. Griffiths, J. A. D. Haseth, Fourier Transform Infrared Spectrometry (Wiley-Interscience, 2007).

Guelachvili, G.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Hänsch, T. W.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Haseth, J. A. D.

P. R. Griffiths, J. A. D. Haseth, Fourier Transform Infrared Spectrometry (Wiley-Interscience, 2007).

Hindle, F.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Holzwarth, R.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

F. Keilmann, C. Gohle, R. Holzwarth, “Time-domain mid-infrared frequency-comb spectrometer,” Opt. Lett. 29, 1542–1544 (2004).
[Crossref]

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Hosako, I.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Hoshina, H.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Hsieh, Y.-D.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Hudert, F.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Huntzinger, J.-R.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Inaba, H.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Issenmann, D.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Iyonaga, Y.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Jacobsen, R. H.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

Jacquet, P.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Jacquey, M.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

Janke, C.

Kabetani, Y.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

Kasai, Y.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Keilmann, F.

Klatt, G.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Kobayashi, Y.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

Kruppa, G. H.

G. H. Kruppa, C. J. Cassady, Fourier Transform Mass Spectrometry (CRC Press, 2006).

Kurz, H.

Loßow, S.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Mandon, J.

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Matsuura, S.

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

Minoshima, K.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Mittleman, D. M.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

D. M. Mittleman, Sensing with THz Radiation (Springer-Verlag, 2003).

Mlayah, A.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Mori, Y.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Muller, H. S. P.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

Müller, H. S. P.

J. C. Pearson, H. S. P. Müller, “The submillimeter wave spectrum of isotopic methyl cyanide,” Astrophys. J. 471, 1067–1072 (1996).
[Crossref]

Murtagh, D.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Nagel, M.

Neelamani, R.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

Newbury, N. R.

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Nuss, M. C.

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

Otani, C.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Ozawa, A.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

Ozeki, H.

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

Pearson, J. C.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

J. C. Pearson, H. S. P. Müller, “The submillimeter wave spectrum of isotopic methyl cyanide,” Astrophys. J. 471, 1067–1072 (1996).
[Crossref]

Pickett, H. M.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

Picqué, N.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

Potvin, S.

Poynter, R. L.

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

Roy, J.

Saito, S.

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

Sakaguchi, Y.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Sakai, K.

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

Saneyoshi, E.

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

T. Yasui, E. Saneyoshi, T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

Schecker, O.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Scheer, E.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Schiller, S.

Seta, T.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Sigrist, M. W.

M. W. Sigrist, Air Monitoring by Spectroscopic Techniques (Wiley, 1994).

Swann, W. C.

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Takahashi, Y.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Tani, M.

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

Udem, T.

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

Urban, J.

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Waitz, R.

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Yasui, T.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

T. Yasui, E. Saneyoshi, T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

Yokoyama, S.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

Yoshimura, M.

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

Zolot, A. M.

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Appl. Phys. B (1)

D. M. Mittleman, R. H. Jacobsen, R. Neelamani, R. G. Baraniuk, M. C. Nuss, “Gas sensing using terahertz time-domain spectroscopy,” Appl. Phys. B 67, 379–390 (1998).
[Crossref]

Appl. Phys. Lett. (2)

T. Yasui, E. Saneyoshi, T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
[Crossref]

T. Yasui, Y. Kabetani, E. Saneyoshi, S. Yokoyama, T. Araki, “Terahertz frequency comb by multifrequency-heterodyning photoconductive detection for high-accuracy, high-resolution terahertz spectroscopy,” Appl. Phys. Lett. 88, 241104 (2006).
[Crossref]

Astrophys. J. (1)

J. C. Pearson, H. S. P. Müller, “The submillimeter wave spectrum of isotopic methyl cyanide,” Astrophys. J. 471, 1067–1072 (1996).
[Crossref]

IEEE Trans. Terahertz Sci. Technol. (1)

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, Y. Takahashi, M. Yoshimura, Y. Mori, T. Araki, T. Yasui, “Terahertz comb spectroscopy traceable to microwave frequency standard,” IEEE Trans. Terahertz Sci. Technol. 3, 322–330 (2013).

J. Mol. Spectrosc. (1)

S. Matsuura, M. Tani, H. Abe, K. Sakai, H. Ozeki, S. Saito, “High-resolution terahertz spectroscopy by a compact radiation source based on photomixing with diode lasers in a photoconductive antenna,” J. Mol. Spectrosc. 187, 97–101 (1998).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (2)

H. M. Pickett, R. L. Poynter, E. A. Cohen, M. L. Delitsky, J. C. Pearson, H. S. P. Muller, “Submillimeter, millimeter, and microwave spectral line catalog, J. Quant. Spectrosc. Radiat. Transfer 60, 883–890 (1998).
[Crossref]

T. Seta, H. Hoshina, Y. Kasai, I. Hosako, C. Otani, S. Loßow, J. Urban, M. Ekström, P. Eriksson, D. Murtagh, “Pressure broadening coefficients of the water vapour lines at 556.936 and 752.033  GHz,” J. Quant. Spectrosc. Radiat. Transfer 109, 144–150 (2008).
[Crossref]

Nat. Photonics (1)

B. Bernhardt, A. Ozawa, P. Jacquet, M. Jacquey, Y. Kobayashi, T. Udem, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Cavity-enhanced dual-comb spectroscopy,” Nat. Photonics 4, 55–57 (2010).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

E. Baumann, F. R. Giorgetta, W. C. Swann, A. M. Zolot, I. Coddington, N. R. Newbury, “Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer,” Phys. Rev. A 84, 062513 (2011).
[Crossref]

Phys. Rev. Lett. (1)

A. Bruchhausen, R. Gebs, F. Hudert, D. Issenmann, G. Klatt, A. Bartels, O. Schecker, R. Waitz, A. Erbe, E. Scheer, J.-R. Huntzinger, A. Mlayah, T. Dekorsy, “Subharmonic resonant optical excitation of confined acoustic modes in a free-standing semiconductor membrane at GHz frequencies with a high-repetition-rate femtosecond laser,” Phys. Rev. Lett. 106, 077401 (2011).
[Crossref]

Sci. Rep. (1)

Y.-D. Hsieh, Y. Iyonaga, Y. Sakaguchi, S. Yokoyama, H. Inaba, K. Minoshima, F. Hindle, T. Araki, T. Yasui, “Spectrally interleaved, comb-mode-resolved spectroscopy using swept dual terahertz combs,” Sci. Rep. 4, 3816 (2014).

Other (7)

R. N. Bracewell, “Sampling and series,” in Fourier Transform and its Applications, R. N. Bracewell, ed. (McGraw-Hill, 1999).

M. W. Sigrist, Air Monitoring by Spectroscopic Techniques (Wiley, 1994).

P. Jacquet, J. Mandon, B. Bernhardt, R. Holzwarth, G. Guelachvili, T. W. Hänsch, N. Picqué, “Frequency comb Fourier transform spectroscopy with kHz optical resolution,” in Fourier Transform Spectroscopy 2009, OSA Technical Digest (Optical Society of America, 2009), paper FMB2.

P. R. Griffiths, J. A. D. Haseth, Fourier Transform Infrared Spectrometry (Wiley-Interscience, 2007).

D. M. Mittleman, Sensing with THz Radiation (Springer-Verlag, 2003).

T. C. Farrar, E. D. Becker, Pulse and Fourier Transform NMR: Introduction to Theory and Methods (Academic, 1971).

G. H. Kruppa, C. J. Cassady, Fourier Transform Mass Spectrometry (CRC Press, 2006).

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Figures (5)

Fig. 1.
Fig. 1.

Principle of operation. (a) FT for a measured temporal waveform h ( t ) of a phenomenon. (b) Temporal overlapping of multiple repeating phenomena using precisely periodic radiation with a repetition period T . (c) Signal acquisition of portions of h ( t ) with different timings with a time-window size τ . (d) FT of portions of h ( t ) with different timings. (e) Temporal connection of portions of h ( t ) with different timings and FT of the temporally connected h ( t ) without limitation of the time-window size. (f) Discrete Fourier transform spectrum for the temporally connected h ( t ) and the spectral interleaving by changing T .

Fig. 2.
Fig. 2.

Experimental setup. Rb-FS, rubidium frequency standard; SFG-XC, sum-frequency-generation cross correlator; SHG, second-harmonic-generation crystals; L, lenses; EDFA, erbium-doped fiber amplifier; OSC, erbium-doped fiber oscillator; PCA1, dipole-shaped low-temperature GaAs photoconductive antenna for THz emitter; PCA2, dipole-shaped low-temperature-GaAs photoconductive antenna for THz detector; Si-L, silicon lenses; AMP, current preamplifier.

Fig. 3.
Fig. 3.

Absorbance spectrum at the water rotational-transition ( 1 10 1 01 ) absorption line with the expected pressure broadening linewidth of 10.6 MHz for (a)  τ = T , (b)  τ = 0.9995 T , and (c)  τ = T achieved by connecting the temporal data for 0.9995 T and the null data for 0.0005 T (the spectral interleaving interval is 25 MHz). Absorbance spectrum at the water absorption line 1 10 1 01 with the expected pressure broadening linewidth of 504 MHz for (d)  τ = T and (e)  τ = 0.9995 T (the spectral interleaving interval is 25 MHz). (f) Pressure broadening characteristic of the 1 10 1 01 water absorption line at 0.557 THz.

Fig. 4.
Fig. 4.

Absorbance spectrum of low-pressure CH 3 CN gas within a frequency range (a) from 0.2 to 1 THz, (b) from 0.6 to 0.7 THz, and (c) around 0.6428 THz with a spectral interleaving interval of 25 MHz. Red asterisks in Fig. 3(b) show the spectral features caused by the rotational transition of CH 3 CN molecules in the vibrationally excited state. (d) Expanded absorbance spectrum for two adjacent absorption peaks around 0.643265 THz with a spectral interleaving interval of 1.25 MHz.

Fig. 5.
Fig. 5.

Comparison of the spectral behavior between (a) mode-resolved DCS and (b) dFTS with and without spectral interleaving.

Equations (9)

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

H ( f ) = h ( t ) exp ( 2 π i f t ) d t .
f n = n T ( n = 0,1 , 2 , , N 1 ) ,
H ( f ) = h ( t ) exp ( 2 π i f t ) d t = + T 0 h ( t ) exp ( 2 π i f t ) d t + 0 T h ( t ) exp ( 2 π i f t ) d t + T 2 T h ( t ) exp ( 2 π i f t ) d t + = + 0 T h ( t T ) exp { 2 π i f ( t T ) } d t + 0 T h ( t ) exp ( 2 π i f t ) d t + 0 T h ( t + T ) exp { 2 π i f ( t + T ) } d t + = k = 0 T h ( t k T ) exp { 2 π i f ( t k T ) } d t ,
g ( t ) = k = h ( t k T ) .
G ( f ) = 0 T g ( t ) exp ( 2 π i f t ) d t = k = 0 T h ( t k T ) exp { 2 π i f t + 2 π i n k } d t ,
G ( f n ) = k = 0 T h ( t k T ) exp { 2 π i f n ( t k T ) } d t = H ( f n ) ,
g s ( t ) = m = 0 N 1 g ( t ) δ ( t m T N ) ,
G s ( f n ) = m = 0 N 1 g ( t ) δ ( t m T N ) exp ( 2 π i f n t ) d t = m = 0 N 1 g ( t m ) exp ( 2 π i f n t m ) ,
ν = 2 B ( J + 1 ) 2 D J K K 2 ( J + 1 ) ,

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