Abstract

A system using a nonlinear multiswept sine wave is described, which employs multichannel multiswept orthogonal waves, to separate channels and make multiple, simultaneous online/offline CO2 measurements. An analytic expression and systematic method for determining the orthogonal frequencies for the unswept, linear swept, and nonlinear swept cases is presented. It is shown that one may reduce sidelobes of the autocorrelation function while preserving cross channel orthogonality, for thin cloud rejection.

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  2. G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
    [CrossRef]
  3. J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
    [CrossRef]
  4. J. F. Campbell, N. S. Prasad, and M. A. Flood, “Pseudorandom noise code–based technique for thin-cloud discrimination with CO2 and O2 absorption measurements,” Opt. Eng. 50, 126002 (2011).
    [CrossRef]
  5. J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
    [CrossRef]
  6. R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
    [CrossRef]
  7. O. Batet, F. Dios, A. Comeron, and R. Agishev, “Intensity-modulated linear-frequency-modulated continuous-wave lidar for distributed media: fundamentals of technique,” Appl. Opt. 49, 3369–3379 (2010).
    [CrossRef]
  8. M. Imaki, S. Kameyama, Y. Hirano, S. Ueno, D. Sakaizawa, S. Kawakami, and M. Nakajima, “Laser absorption spectrometer using frequency chirped intensity modulation at 1.57 μm wavelength for CO2 measurement,” Opt. Lett. 37, 2688–2690 (2012).
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  10. E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).
  11. S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).
  12. S. Kameyama, M. Imaki, Y. Hirano, S. Ueno, S. Kawakami, D. Sakaizawa, T. Kimura, and M. Nakajima, “Feasibility study on 1.6 μm continuous-wave modulation laser absorption spectrometer system for measurement of global CO2 concentration from a satellite,” Appl. Opt. 50, 2055–2068 (2011).
    [CrossRef]
  13. M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).
  14. M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).
  15. J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).
  16. D. Sakaizawa, S. Kawakami, M. Nakajima, T. Tanaka, I. Morino, and O. Uchino, “An airborne amplitude-modulated 1.57 μm differential laser absorption spectrometer: simultaneous measurement of partial column-averaged dry air mixing ratio of CO2 and target range,” Atmos. Meas. Tech. 6, 387–396 (2013).
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  18. C. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Academic, 1967).
  19. A. Kononov, L. Ulander, and L. Eriksson, Design of Optimum Weighting Functions for LFM Signals Convergence and Hybrid Information Technologies, M. Crisan, ed. (Intech, 2010). http://www.intechopen.com/books/convergence-and-hybrid-informationtechnologies/
  20. Y. Pan, S. Peng, K. Yang, and W. Dong, “Optimization design of NLFM signal and its pulse compression simulation,” in Proceedings, Radar Conference, 2005 IEEE International (IEEE, 2005) pp. 383–386.
  21. A. W. Doerry, “Generating precision nonlinear FM chirp waveforms,” Proc. SPIE 6547, 6547D (2007).
    [CrossRef]
  22. L. R. Varshney and D. Thomas, “Sidelobe reduction for matched filter range processing,” in Proceedings of IEEE Radar Conference (IEEE, 2003).
  23. C. Lesnik, “Nonlinear frequency modulated signal design,” Acta Physica Polonica A 116, 351–354 (2009).
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    [CrossRef]

2013 (1)

D. Sakaizawa, S. Kawakami, M. Nakajima, T. Tanaka, I. Morino, and O. Uchino, “An airborne amplitude-modulated 1.57 μm differential laser absorption spectrometer: simultaneous measurement of partial column-averaged dry air mixing ratio of CO2 and target range,” Atmos. Meas. Tech. 6, 387–396 (2013).

2012 (1)

2011 (3)

S. Kameyama, M. Imaki, Y. Hirano, S. Ueno, S. Kawakami, D. Sakaizawa, T. Kimura, and M. Nakajima, “Feasibility study on 1.6 μm continuous-wave modulation laser absorption spectrometer system for measurement of global CO2 concentration from a satellite,” Appl. Opt. 50, 2055–2068 (2011).
[CrossRef]

J. F. Campbell, N. S. Prasad, and M. A. Flood, “Pseudorandom noise code–based technique for thin-cloud discrimination with CO2 and O2 absorption measurements,” Opt. Eng. 50, 126002 (2011).
[CrossRef]

J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
[CrossRef]

2010 (1)

2009 (2)

C. Lesnik, “Nonlinear frequency modulated signal design,” Acta Physica Polonica A 116, 351–354 (2009).

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

2008 (1)

2007 (1)

A. W. Doerry, “Generating precision nonlinear FM chirp waveforms,” Proc. SPIE 6547, 6547D (2007).
[CrossRef]

2005 (1)

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

2004 (1)

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Abshire, J. B.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Agishev, R.

O. Batet, F. Dios, A. Comeron, and R. Agishev, “Intensity-modulated linear-frequency-modulated continuous-wave lidar for distributed media: fundamentals of technique,” Appl. Opt. 49, 3369–3379 (2010).
[CrossRef]

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

Ahmed, S.

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

Allan, G. R.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Amzajerdian, F.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Bai, Y.

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

Barnes, B. W.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Batet, O.

Bernfeld, M.

C. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Academic, 1967).

Beyon, J. Y.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Blume, N.

M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).

Braun, M.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

Browell, E. V.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

Campbell, J.

Campbell, J. F.

J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
[CrossRef]

J. F. Campbell, N. S. Prasad, and M. A. Flood, “Pseudorandom noise code–based technique for thin-cloud discrimination with CO2 and O2 absorption measurements,” Opt. Eng. 50, 126002 (2011).
[CrossRef]

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

Chen, S.

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

Choi, Y.

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

Comeron, A.

Cook, C.

C. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Academic, 1967).

Davis, R. E.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Dios, F.

Dobbs, M.

M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).

Dobbs, M. E.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

Dobler, J.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

Dobler, J. T.

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

Doerry, A. W.

A. W. Doerry, “Generating precision nonlinear FM chirp waveforms,” Proc. SPIE 6547, 6547D (2007).
[CrossRef]

Dong, W.

Y. Pan, S. Peng, K. Yang, and W. Dong, “Optimization design of NLFM signal and its pulse compression simulation,” in Proceedings, Radar Conference, 2005 IEEE International (IEEE, 2005) pp. 383–386.

Eriksson, L.

A. Kononov, L. Ulander, and L. Eriksson, Design of Optimum Weighting Functions for LFM Signals Convergence and Hybrid Information Technologies, M. Crisan, ed. (Intech, 2010). http://www.intechopen.com/books/convergence-and-hybrid-informationtechnologies/

Fenn, M. A.

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

Flood, M. A.

J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
[CrossRef]

J. F. Campbell, N. S. Prasad, and M. A. Flood, “Pseudorandom noise code–based technique for thin-cloud discrimination with CO2 and O2 absorption measurements,” Opt. Eng. 50, 126002 (2011).
[CrossRef]

Gilerson, A.

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

Gregory, D.

M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).

Gross, B.

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

Harrison, F. W.

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

Hasselbrack, W. E.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Hirano, Y.

Hodson, W. D.

J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
[CrossRef]

Holm, W. A.

M. A. Richards, J. A. Scheer, and W. A. Holm, Principles of Modern Radar: Basic Principles (SciTech Publishing, 2010).

Imaki, M.

Ismail, S.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Kameyama, S.

Kavaya, M. J.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Kawa, S. R.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Kawakami, S.

Kimura, T.

Koch, G. J.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Kononov, A.

A. Kononov, L. Ulander, and L. Eriksson, Design of Optimum Weighting Functions for LFM Signals Convergence and Hybrid Information Technologies, M. Crisan, ed. (Intech, 2010). http://www.intechopen.com/books/convergence-and-hybrid-informationtechnologies/

Kooi, S. A.

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

Lesnik, C.

C. Lesnik, “Nonlinear frequency modulated signal design,” Acta Physica Polonica A 116, 351–354 (2009).

Mao, J.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

McGregor, D.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

Meadows, B. L.

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

Moore, B.

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

Morino, I.

D. Sakaizawa, S. Kawakami, M. Nakajima, T. Tanaka, I. Morino, and O. Uchino, “An airborne amplitude-modulated 1.57 μm differential laser absorption spectrometer: simultaneous measurement of partial column-averaged dry air mixing ratio of CO2 and target range,” Atmos. Meas. Tech. 6, 387–396 (2013).

Moshary, F.

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

Nagel, J.

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

Nakajima, M.

Overbeck, J.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

Pan, Y.

Y. Pan, S. Peng, K. Yang, and W. Dong, “Optimization design of NLFM signal and its pulse compression simulation,” in Proceedings, Radar Conference, 2005 IEEE International (IEEE, 2005) pp. 383–386.

Peng, S.

Y. Pan, S. Peng, K. Yang, and W. Dong, “Optimization design of NLFM signal and its pulse compression simulation,” in Proceedings, Radar Conference, 2005 IEEE International (IEEE, 2005) pp. 383–386.

Petros, M.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Petway, L. B.

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

Prasad, N. S.

J. F. Campbell, N. S. Prasad, and M. A. Flood, “Pseudorandom noise code–based technique for thin-cloud discrimination with CO2 and O2 absorption measurements,” Opt. Eng. 50, 126002 (2011).
[CrossRef]

J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
[CrossRef]

Pruitt, J.

M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).

Richards, M. A.

M. A. Richards, J. A. Scheer, and W. A. Holm, Principles of Modern Radar: Basic Principles (SciTech Publishing, 2010).

Riris, H.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Sakaizawa, D.

Scheer, J. A.

M. A. Richards, J. A. Scheer, and W. A. Holm, Principles of Modern Radar: Basic Principles (SciTech Publishing, 2010).

Sharp, W.

M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).

Singh, U. N.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Sun, X.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Tanaka, T.

D. Sakaizawa, S. Kawakami, M. Nakajima, T. Tanaka, I. Morino, and O. Uchino, “An airborne amplitude-modulated 1.57 μm differential laser absorption spectrometer: simultaneous measurement of partial column-averaged dry air mixing ratio of CO2 and target range,” Atmos. Meas. Tech. 6, 387–396 (2013).

Temyanko, V. L.

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

Thomas, D.

L. R. Varshney and D. Thomas, “Sidelobe reduction for matched filter range processing,” in Proceedings of IEEE Radar Conference (IEEE, 2003).

Uchino, O.

D. Sakaizawa, S. Kawakami, M. Nakajima, T. Tanaka, I. Morino, and O. Uchino, “An airborne amplitude-modulated 1.57 μm differential laser absorption spectrometer: simultaneous measurement of partial column-averaged dry air mixing ratio of CO2 and target range,” Atmos. Meas. Tech. 6, 387–396 (2013).

Ueno, S.

Ulander, L.

A. Kononov, L. Ulander, and L. Eriksson, Design of Optimum Weighting Functions for LFM Signals Convergence and Hybrid Information Technologies, M. Crisan, ed. (Intech, 2010). http://www.intechopen.com/books/convergence-and-hybrid-informationtechnologies/

Varshney, L. R.

L. R. Varshney and D. Thomas, “Sidelobe reduction for matched filter range processing,” in Proceedings of IEEE Radar Conference (IEEE, 2003).

Vay, S.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Vay, S. A.

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

Weaver, C. J.

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Yang, K.

Y. Pan, S. Peng, K. Yang, and W. Dong, “Optimization design of NLFM signal and its pulse compression simulation,” in Proceedings, Radar Conference, 2005 IEEE International (IEEE, 2005) pp. 383–386.

Yu, J.

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Zaccheo, T.

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

Zaccheo, T. S.

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

Acta Physica Polonica A (1)

C. Lesnik, “Nonlinear frequency modulated signal design,” Acta Physica Polonica A 116, 351–354 (2009).

Am. J. Phys. (1)

J. F. Campbell, M. A. Flood, N. S. Prasad, and W. D. Hodson, “A low cost remote sensing system using PC and stereo equipment,” Am. J. Phys. 79, 1240–1245 (2011).
[CrossRef]

App. Opt. (1)

G. J. Koch, B. W. Barnes, M. Petros, J. Y. Beyon, F. Amzajerdian, J. Yu, R. E. Davis, S. Ismail, S. Vay, M. J. Kavaya, and U. N. Singh, “Coherent differential absorption lidar measurements of CO2,” App. Opt. 43, 5092–5099 (2004).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

R. Agishev, B. Gross, F. Moshary, A. Gilerson, and S. Ahmed, “Atmospheric CW-FM-LD-RR ladar for trace-constituent detection: a concept development,” Appl. Phys. B 81, 695–703 (2005).
[CrossRef]

Atmos. Meas. Tech. (1)

D. Sakaizawa, S. Kawakami, M. Nakajima, T. Tanaka, I. Morino, and O. Uchino, “An airborne amplitude-modulated 1.57 μm differential laser absorption spectrometer: simultaneous measurement of partial column-averaged dry air mixing ratio of CO2 and target range,” Atmos. Meas. Tech. 6, 387–396 (2013).

Opt. Eng. (1)

J. F. Campbell, N. S. Prasad, and M. A. Flood, “Pseudorandom noise code–based technique for thin-cloud discrimination with CO2 and O2 absorption measurements,” Opt. Eng. 50, 126002 (2011).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

A. W. Doerry, “Generating precision nonlinear FM chirp waveforms,” Proc. SPIE 6547, 6547D (2007).
[CrossRef]

Tellus B (1)

J. B. Abshire, H. Riris, G. R. Allan, C. J. Weaver, J. Mao, X. Sun, W. E. Hasselbrack, and S. R. Kawa, “Pulsed airborne lidar measurements of atmospheric CO2 column absorption,” Tellus B 62, 770–783 (2009).
[CrossRef]

Other (12)

NRC, Earth Science and Applications from Space: National Imperatives for the Next Decade and Beyond (The National Academies, 2007).

M. Dobbs, J. Pruitt, N. Blume, D. Gregory, and W. Sharp, “Matched filter enhanced fiber-based lidar for earth, weather and exploration,” NASA ESTO Conference, June (2006).

M. E. Dobbs, J. Dobler, M. Braun, D. McGregor, J. Overbeck, B. Moore, E. V. Browell, and T. Zaccheo, “A modulated CW fiber laser-lidar suite for the ASCENDS mission,” in Proceedings, 24th International Laser Radar Conference, Boulder, CO, 24–29 July (2008).

J. T. Dobler, J. Nagel, V. L. Temyanko, T. S. Zaccheo, E. V. Browell, F. W. Harrison, and S. A. Kooi, “Advancements in a multifunctional fiber laser lidar for measuring atmospheric CO2 and O2,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

M. A. Richards, J. A. Scheer, and W. A. Holm, Principles of Modern Radar: Basic Principles (SciTech Publishing, 2010).

C. Cook and M. Bernfeld, Radar Signals: An Introduction to Theory and Application (Academic, 1967).

A. Kononov, L. Ulander, and L. Eriksson, Design of Optimum Weighting Functions for LFM Signals Convergence and Hybrid Information Technologies, M. Crisan, ed. (Intech, 2010). http://www.intechopen.com/books/convergence-and-hybrid-informationtechnologies/

Y. Pan, S. Peng, K. Yang, and W. Dong, “Optimization design of NLFM signal and its pulse compression simulation,” in Proceedings, Radar Conference, 2005 IEEE International (IEEE, 2005) pp. 383–386.

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne laser CO2 column measurements: evaluation of precision and accuracy under wide range of conditions,” presented at Fall AGU Meeting, San Francisco, CA, 5–9 December (2011).

E. V. Browell, J. T. Dobler, S. A. Kooi, M. A. Fenn, Y. Choi, S. A. Vay, F. W. Harrison, and B. Moore, “Airborne validation of laser CO2 and O2 column measurements,” Proceedings, 16th Symposium on Meteorological Observation and Instrumentation, 92nd AMS Annual Meeting, New Orleans, LA, 22–26 January (2012).

S. Chen, Y. Bai, L. B. Petway, B. L. Meadows, J. F. Campbell, F. W. Harrison, and E. V. Browell, “Digital Lock-in detection for multiple-frequency intensity-modulated continuous wave lidar,” Proceedings, 26th International Laser Radar Conference, S1P-38, Porto Heli, Greece, 25–29 June (2012).

L. R. Varshney and D. Thomas, “Sidelobe reduction for matched filter range processing,” in Proceedings of IEEE Radar Conference (IEEE, 2003).

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Figures (11)

Fig. 1.
Fig. 1.

Simplified instrument block diagram.

Fig. 2.
Fig. 2.

Result of cross correlation of the return signal to obtain range and amplitude.

Fig. 3.
Fig. 3.

(a) Frequency and (b) phase as a function of time for linear sweep case.

Fig. 4.
Fig. 4.

Single linear sweep sine wave.

Fig. 5.
Fig. 5.

(a) Channel 1 autocorrelation function and (b) channel 1-2 cross correlation cross talk.

Fig. 6.
Fig. 6.

(a) Single sweep nonlinear sweep and (b) phase for different values of k .

Fig. 7.
Fig. 7.

Comparison between linear and strongly nonlinear case shows that as k approaches 1 the frequency profile approaches a Gaussian shape in a way that the area under the power curve is preserved.

Fig. 8.
Fig. 8.

Comparison between linear sweep and nonlinear sweep for 512 sweep length and k = 0.91 shows highest sidelobe in nonlinear case is about 34 dB compared to 13 dB in the linear case.

Fig. 9.
Fig. 9.

(a) Highest sidelobe is 55 dB for k = 0.999 and 4096 points per sweep. (b) Highest sidelobe is below 80 dB for k = 0.99999 and 2 17 points per sweep.

Fig. 10.
Fig. 10.

(a) Channel 1 nonlinear autocorrelation function for k = 0.91 and a sweep length of 512 shows reduced sidelobes and (b) channel 1-2 cross correlation cross talk is as good as the linear case.

Fig. 11.
Fig. 11.

Quadrature/lock-in demodulation used for signal detection of unswept sine wave modulation.

Equations (36)

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

M on = 1 + m cos ( ϕ 1 ( t ) ) , M off = 1 + m cos ( ϕ 2 ( t ) ) .
ϕ 1 ( t ) = 2 π 0 t f 1 ( t ) d t , ϕ 2 ( t ) = 2 π 0 t f 2 ( t ) d t ,
I on ( t ) = K r 2 P on exp ( 2 ε 0 r β ( r ) d r ) exp ( 2 ξ ) exp ( 2 ξ ) ( 1 + m cos ( ϕ 1 ( t 2 r / c ) ) ) I off ( t ) = K r 2 P off exp ( 2 ε 0 r β ( r ) d r ) exp ( 2 ξ ) ( 1 + m cos ( ϕ 2 ( t 2 r / c ) ) ) ,
S ( t ) = C 1 cos ( ϕ 1 ( t 2 r / c ) ) + C 2 cos ( ϕ 2 ( t 2 r / c ) ) ,
C 1 = K r 2 P on exp ( 2 ε 0 z β ( r ) d r ) exp ( 2 ξ ) exp ( 2 ξ ) C 2 = K r 2 P off exp ( 2 ε 0 z β ( r ) d r ) exp ( 2 ξ ) ,
ξ = 1 2 ln ( C 2 P on C 1 P off ) .
f j ( t ) = f 0 j + f sweep ( t τ int ( t / τ ) ) ,
ϕ sweep ( t ) = 2 π 0 t f sweep ( t ) d t .
ϕ j ( t ) = 2 π f 0 j t + ϕ sweep ( τ ) int ( t / τ ) + ϕ sweep ( t τ int ( t / τ ) ) .
R ( ref , data ) = 1 N m = 0 N 1 ref * ( m ) data ( m + n ) = DFT 1 ( DFT * ( ref ) DFT ( data ) ) ,
R ( exp ( i ϕ j ) , cos ( ϕ k ) ) = 0 , j k ,
f j = f 0 j + f sweep = f 0 j + 1 2 π τ ϕ sweep ( τ ) ,
f 01 = n 1 2 M τ ϕ sweep ( τ ) 2 π τ > 0 , f 02 = f 01 = n 2 2 M τ , , f 0 K = f 01 + n K 2 M τ ,
f sweep ( t ) = Δ f t τ , 0 t < τ ,
δ r = c 2 Δ f ,
r max = c τ 2 .
f j ( t ) = f 0 j + Δ f t τ int ( t / τ ) τ ,
ϕ j ( t ) = 2 π [ f 0 j t + 1 2 Δ f τ int ( t / τ ) + Δ f 2 τ ( t τ int ( t / τ ) ) 2 ] .
| R ( exp ( i ϕ j ) , cos ( ϕ j ) ) | | 1 τ 0 τ t exp ( i ϕ j ( t ) ) exp ( i ϕ j ( t + t ) ) d t + 1 τ τ t τ exp ( i ϕ j ( t ) ) exp ( i ϕ j ( t + t ) ) d t | = | sin [ π Δ f t ( 1 | t | τ ) ] 2 π Δ f t ( 1 | t | τ ) | , τ < t < τ .
f sweep ( t ) = 1 2 Δ f Δ f 1 k 2 2 1 2 t / τ 1 k 2 ( 1 2 t / τ ) 2 ,
f j ( t ) = f 0 j + 1 2 Δ f Δ f 1 k 2 2 1 2 t τ 1 k 2 ( 1 2 t / τ ) 2 , t = t τ int ( t / τ ) .
lim k 0 f j = f 0 j + Δ f t τ , lim k 1 f j = f 0 j + 1 2 Δ f .
ϕ sweep ( t ) = 2 π [ 1 2 Δ f t + Δ f τ 4 ( 1 k 2 ) k 2 ( 1 1 k 2 ( 1 2 t τ ) 2 ( 1 k 2 ) ) ] .
ϕ ( t ) = 2 π [ ( f 0 j + 1 2 Δ f ) t + Δ f τ 4 ( 1 k 2 ) k 2 ( 1 1 k 2 ( 1 2 t τ ) 2 ( 1 k 2 ) ) ] , t = t τ int ( t / τ ) .
S ( t ) = C 1 cos ( 2 π f 1 t + ϕ 1 ) + C 2 cos ( 2 π f 2 t + ϕ 2 ) = A 1 sin ( 2 π f 1 t ) + B 1 cos ( 2 π f 1 t ) + A 2 sin ( 2 π f 2 t ) + B 2 cos ( 2 π f 2 t ) ,
A i = C i cos ( ϕ i ) , B i = C i sin ( ϕ i ) .
x i = 1 N n = 0 N 1 sin ( 2 π f i τ n ) S ( n Δ t ) , y i = 1 N n = 0 N 1 cos ( 2 π f i τ n ) S ( n Δ t ) ,
1 N n = 0 N 1 sin ( 2 π f i n Δ t ) sin ( 2 π f i n Δ t ) = 1 4 N sin ( ( 2 N 1 ) π ( f i f j ) Δ t ) sin ( π ( f i f j ) Δ t ) 1 4 N sin ( ( 2 N 1 ) π ( f i + f j ) Δ t ) sin ( π ( f i + f j ) Δ t ) , i j = 1 4 N ( 2 N 1 ) 1 4 N sin ( 2 ( 2 N 1 ) π f i Δ t ) sin ( 2 π f i Δ t ) , i = j
1 N n = 0 N 1 cos ( 2 π f i n Δ t ) cos ( 2 π f j n Δ t ) = 1 2 N + 1 4 N sin ( ( 2 N 1 ) π ( f i f j ) Δ t ) sin ( π ( f i f j ) Δ t ) + 1 4 N sin ( ( 2 N 1 ) π ( f i + f j ) Δ t ) sin ( π ( f i + f j ) Δ t ) , i j = 1 4 N ( 2 N + 1 ) + 1 4 N sin ( 2 ( 2 N 1 ) π f i Δ t ) sin ( 2 π f i Δ t ) , i = j
1 N n = 0 N 1 cos ( 2 π f i n Δ t ) sin ( 2 π f j n Δ t ) = 1 2 N sin ( N π ( f i f j ) Δ t ) sin ( ( N 1 ) π ( f i f j ) Δ t ) sin ( π ( f 1 f 2 ) Δ t ) + 1 2 N sin ( N π ( f i + f j ) Δ t ) sin ( ( N 1 ) π ( f i + f j ) Δ t ) sin ( π ( f i + f j ) Δ t ) , i j = 1 2 N sin ( 2 N π f i Δ t ) sin ( 2 ( N 1 ) π f i Δ t ) sin ( 2 π f i Δ t ) , i = j
sin ( N π ( f 1 f 2 ) Δ t ) = 0 , sin ( N π ( f 1 + f 2 ) Δ t ) = 0 ,
f 1 = 1 2 N Δ t ( p q ) , f 2 = 1 2 N Δ t ( p + q ) ,
1 N ( p + q ) < 1 .
x i = n = 0 N 1 sin ( 2 π f i n Δ t ) S ( n Δ t ) = A i 2 y i = n = 0 N 1 cos ( 2 π f i n Δ t ) S ( n Δ t ) = B i 2 ,
x 1 2 + y 1 2 = 1 2 A 1 2 + B 1 2 = 1 2 C 1 x 2 2 + y 2 2 = 1 2 A 2 2 + B 2 2 = 1 2 C 2 .
f 01 = n 1 2 N Δ t , f 02 = n 2 2 N Δ t , , f 0 K = n K 2 N Δ t .

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