Abstract

The Laser Interferometer Space Antenna (LISA) will use Time Delay Interferometry (TDI) to suppress the otherwise dominant laser frequency noise. The technique uses sub-sample interpolation of the recorded optical phase measurements to form a family of interferometric combinations immune to frequency noise. This paper reports on the development of a Pseudo-Random Noise laser ranging system used to measure the sub-sample interpolation time shifts required for TDI operation. The system also includes an optical communication capability that meets the 20 kbps LISA requirement. An experimental demonstration of an integrated LISA phase measurement and ranging system achieved a ≈ 0.19 m rms absolute range error with a 0.5 Hz signal bandwidth, surpassing the 1 m rms LISA specification. The range measurement is limited by mutual interference between the ranging signals exchanged between spacecraft and the interaction of the ranging code with the phase measurement.

© 2010 OSA

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References

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  1. M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D 59(10), 102003 (1999).
    [Crossref]
  2. M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D,  71(4), 041101 (2005).
    [Crossref]
  3. LISA Frequency control study team, “LISA frequency control white paper,” ESA document LISA-JPL-TN-823 (2009).
  4. D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
    [Crossref]
  5. G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
    [Crossref] [PubMed]
  6. S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. 23, 4201 (2006).
    [Crossref]
  7. E. D. Kaplan, Understanding GPS Principles and Applications, (Prentice Hall PTR, 1996).
  8. W. M. Klipstein, R. E. Spero, and D. A. Shaddock, “Anti-aliasing for LISA photoreceiver signals,” JPL Technical Note, (2006)
  9. J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
    [Crossref]
  10. D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).
  11. D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).
  12. Hans-Reiner Schulte “Presentation at LISA Mission Formulation MTR,” (14/15. 4. 2006)
  13. Vinzenz Wand “Interferometry at low frequencies: Optical phase measurement for LISA and LISA pathfinder”. Ph.D Thesis (4 2007). Leibniz UniversitätHannover
  14. J. G. Proakis and M. Salehi, Communications Systems Engineering, (Prentice Hall, 2002).
  15. T. S. Rappaport, Wireless Communications, (Artech House Publishers, 2002).
  16. W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
    [Crossref]
  17. W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

2010 (2)

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

2006 (1)

S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. 23, 4201 (2006).
[Crossref]

2005 (1)

M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D,  71(4), 041101 (2005).
[Crossref]

2004 (1)

D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
[Crossref]

1999 (1)

M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D 59(10), 102003 (1999).
[Crossref]

1997 (1)

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

Armstrong, J. W.

M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D,  71(4), 041101 (2005).
[Crossref]

M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D 59(10), 102003 (1999).
[Crossref]

Bender, P L

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

Bykov, I.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Chong, Y.

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Cruz, R.

W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

Danzmann, K.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

de Vine, G

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

de Vine, G.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

Eichholz, J.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Esteban, J. J.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Folkner, W M

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

Garca, A. F.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Halverson, P.

D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).

Halverson, P. G.

W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

Hechler, F

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

Heinzel, G.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Kaplan, E. D.

E. D. Kaplan, Understanding GPS Principles and Applications, (Prentice Hall PTR, 1996).

Klipstein, W.

W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

Klipstein, W. M.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

W. M. Klipstein, R. E. Spero, and D. A. Shaddock, “Anti-aliasing for LISA photoreceiver signals,” JPL Technical Note, (2006)

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).

McKenzie, K.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Peinado, A. M.

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Peters, R.

W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

Pollack, S. E.

S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. 23, 4201 (2006).
[Crossref]

Proakis, J. G.

J. G. Proakis and M. Salehi, Communications Systems Engineering, (Prentice Hall, 2002).

Rappaport, T. S.

T. S. Rappaport, Wireless Communications, (Artech House Publishers, 2002).

Robison, D.

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Salehi, M.

J. G. Proakis and M. Salehi, Communications Systems Engineering, (Prentice Hall, 2002).

Schulte, Hans-Reiner

Hans-Reiner Schulte “Presentation at LISA Mission Formulation MTR,” (14/15. 4. 2006)

Shaddock, D. A.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
[Crossref]

W. M. Klipstein, R. E. Spero, and D. A. Shaddock, “Anti-aliasing for LISA photoreceiver signals,” JPL Technical Note, (2006)

D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).

W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

Shaddock, D.A.

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Spero, R. E.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
[Crossref]

W. M. Klipstein, R. E. Spero, and D. A. Shaddock, “Anti-aliasing for LISA photoreceiver signals,” JPL Technical Note, (2006)

D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).

Spero, R.E.

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Stebbins, R. T.

S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. 23, 4201 (2006).
[Crossref]

Stecheson, T.

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Sweetser, T H

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

Tinto, M.

M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D,  71(4), 041101 (2005).
[Crossref]

M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D 59(10), 102003 (1999).
[Crossref]

Vallisneri, M.

M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D,  71(4), 041101 (2005).
[Crossref]

D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
[Crossref]

Vincent, M A

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

Wand, Vinzenz

Vinzenz Wand “Interferometry at low frequencies: Optical phase measurement for LISA and LISA pathfinder”. Ph.D Thesis (4 2007). Leibniz UniversitätHannover

Ware, B.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
[Crossref]

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).

Woodruff, C.

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

Class. Quantum Grav. (2)

S. E. Pollack and R. T. Stebbins, “A demonstration of LISA laser communication.”Class. Quantum Grav. 23, 4201 (2006).
[Crossref]

W M Folkner, F Hechler, T H Sweetser, M A Vincent, and P L Bender, “LISA orbit selection and stability,” Class. Quantum Grav. 14, 1405 (1997).
[Crossref]

J. Phys.: Conf. Ser. (1)

J. J. Esteban, A. F. Garca, J. Eichholz, A. M. Peinado, I. Bykov, G. Heinzel, and K. Danzmann, “Ranging and phase measurement for LISA,” J. Phys.: Conf. Ser. 228, 012045 (2010).
[Crossref]

Phys. Rev. D (3)

M. Tinto and J. W. Armstrong, “Cancellation of laser noise in an unequal-arm interferometer detector of gravitational radiation,” Phys. Rev. D 59(10), 102003 (1999).
[Crossref]

M. Tinto, M. Vallisneri, and J. W. Armstrong, “Time-delay interferometric ranging for space-borne gravitationalwave detectors,” Phys. Rev. D,  71(4), 041101 (2005).
[Crossref]

D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri“Postprocessed time-delay interferometry for LISA,” Phys. Rev. D 70(8), 081101 (2004).
[Crossref]

Phys. Rev. Lett. (1)

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the laser interferometer space antenna,” Phys. Rev. Lett. 104(21), 211103 (2010).
[Crossref] [PubMed]

Other (10)

LISA Frequency control study team, “LISA frequency control white paper,” ESA document LISA-JPL-TN-823 (2009).

D.A. Shaddock, B. Ware, R.E. Spero, K. McKenzie, G de Vine, D. Robison, T. Stecheson, Y. Chong, C. Woodruff, and W. M. Klipstein, “LISA Phasemeter Technology Assessment and Report,” LISA Project Document LIMAS 2009-002 (2009).

D. A. Shaddock, B. Ware, P. Halverson, R. E. Spero, and W. M. Klipstein, “Overview of the LISA phasemeter,” in AIP Conf. Proc.873, pg. 654–660 (2006).

Hans-Reiner Schulte “Presentation at LISA Mission Formulation MTR,” (14/15. 4. 2006)

Vinzenz Wand “Interferometry at low frequencies: Optical phase measurement for LISA and LISA pathfinder”. Ph.D Thesis (4 2007). Leibniz UniversitätHannover

J. G. Proakis and M. Salehi, Communications Systems Engineering, (Prentice Hall, 2002).

T. S. Rappaport, Wireless Communications, (Artech House Publishers, 2002).

W. Klipstein, P. G. Halverson, R. Peters, R. Cruz, and D. A. Shaddock, “Clock noise removal in LISA,” in AIP Conf. Proc.873, pg. 312–318 (2006).

E. D. Kaplan, Understanding GPS Principles and Applications, (Prentice Hall PTR, 1996).

W. M. Klipstein, R. E. Spero, and D. A. Shaddock, “Anti-aliasing for LISA photoreceiver signals,” JPL Technical Note, (2006)

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Figures (10)

Fig. 1
Fig. 1

Showing a pulse of phase noise sent between SC 1 and 2 with propagation time T. Local SC clock times t 1(t) and t 2(t) are relative to global time t.

Fig. 2
Fig. 2

A simplified LISA configuration with spacecraft SC 1 and SC 2. The SC 1 and SC 2 lasers, with phases x 1 and x 2, are interfered on SC 1 at beamsplitter A 1 and their phase difference is measured by a phasemeter through photodetector (D 1). A ranging output is produced by the phasemeter and is passed to the MF-DLL for demodulation.

Fig. 3
Fig. 3

(a) The auto-correlation function of a nominal PRN sequence (solid) measured over 50 chips. The auto-correlation of the Rectangular pulse shape (dashed) is also shown for comparison. (b) describes the overlap between the ranging and demodulation pulses with a small timing misalignment and discounting the effects of shot noise. Region A is the ‘wanted’ correlation, while Region B shows the ‘unwanted’ area introduced by the misalignment.

Fig. 4
Fig. 4

Showing a simplified phasemeter Quadrature (Q) with demodulation including mixer, Local Oscillator (LO), Low Pass Filters (LPF) and Controller CPLL . Signal frequencies greater than the Controller bandwidth fu appear in the Quadrature (Q) output, while frequencies lower than fu are suppressed by the controller and imposed onto the LO phase

Fig. 5
Fig. 5

Plot (a) shows the phasemeter pulse responses for a 1 μs chip with Rectangular and Manchester pulse shapes. Plot (b) shows the pulse correlation functions for Manchester and Rectangular pulses with and without phasemeter filtering.

Fig. 6
Fig. 6

Showing the timing of the early and late demodulation correlations. Comparing the demodulated signals gives the DLL timing error signal ε ϕQ,c 2 τ 2) shown in for the Manchester pulse shape. The error signal zero occurs away from ideal timing point Δτ 2 = 0 due to phasemeter induced dispersion.

Fig. 7
Fig. 7

Block diagram of the Closed Loop Timing Control system including ADC, Phasemeter with impulse response hPM , PRN generator with timing estimate τ ^ i , correlation noise sources νPM and νA and Controller C. The timing error signal is calculated using the difference between Early and Late demodulations.

Fig. 8
Fig. 8

(a) shows the raw inter-node pseudo-range measurements. The clock difference causes the node timing to drift, appearing as a range ramp of ∼ 300 m/s. This effect is anti-symmetric and is removed in the round trip estimate. (b) shows a comparison of the detrended GHz Clock sideband and A-B pseudo-range. Differencing the clock sideband and pseudo-range allows estimation of the residual ranging error. The zoom section demonstrates sub-ns pseudo-range error with a 0.5 Hz signal bandwidth.

Fig. 9
Fig. 9

Showing the spectrum of the closed loop timing estimate (blue) achieved by the test bed compared against the GHz clock signal (dark green). The predicted interference (dashed) and shot noise (solid) levels are displayed for comparison. Measurement of a single code (red), with interfering code deactivated, shows the limit achievable without interference. The difference (light green) trace shows the residual ranging noise after subtraction of the clock drift.

Fig. 10
Fig. 10

Showing the implemented data protocol for the optical communications system. Each bit is represented by 50 code chips, which are formed into a data packet. The data packet boundary is identified by the repetition length of the PRN code (2560 bits or 128,000 chips) meeting the data rate requirement of 20 kbps.

Equations (14)

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

ϕ 1 ( t ) = x 2 ( t τ 2 ) x 1 ( t ) + ν PM = β ( c 2 ( t τ 2 ) c 1 ( t ) ) + ν PM
f ( t ) = { 1 , 0 t T P , 0 , else R f ( τ ) = { T P | τ | , T P τ T P , 0 , else
A ϕ 1 , c ^ 2 [ n ] = β τ Corr [ R f ( Δ τ 2 ) N + R f ( ( Δ τ 2 + T P ) mod T P ) C c ^ 2 , c 2 [ n , ± 1 ] R f ( τ ^ 2 mod T P ) C c ^ 2 , c 1 [ n , m i ] R f ( ( τ ^ 2 + 1 ) mod T P ) C c ^ 2 , c 1 [ n , m i + 1 ] ] + ν P M
H PM ( s ) = G ( s ) 1 + G ( s ) C PLL ( s ) s s + α , α 2 π f u = 2 π × 10 5
ϕ Q ( t ) = ( x 2 ( t τ 2 ) x 1 ( t ) ) h P M + ν P M
f M ( t ) = { 1 0 t < T P / 2 , 1 T P / 2 t < T P , 0 else
A ϕ Q , c ^ 2 [ n ] = β τ Corr [ R PM ( Δ τ 2 ) N + R PM ( ( Δ τ 2 + T P ) mod T P ) C c ^ 2 , c 2 [ n , ± 1 ] R PM ( τ ^ 2 mod T P ) C c ^ 2 , c 1 [ n , m i ] R PM ( ( τ ^ 2 + 1 ) mod T P ) C c ^ 2 , c 1 [ n , m i + 1 ] ] + ν PM
A ϕ Q , c ^ 2 [ n ] = β τ Corr [ R PM ( Δ τ 2 ) N + R PM ( ( Δ τ 2 + T P ) mod T P ) ν c ^ 2 , c 2 [ ± 1 ] R PM ( τ ^ 2 mod T P ) ν c ^ 2 , c 1 [ m i ] R PM ( ( τ ^ 2 + 1 ) mod T P ) ν c ^ 2 , c 1 [ m i ± 1 ] ] + ν PM
A ϕ Q , c ^ 2 ( Δ τ 2 ) β τ Corr NR PM ( Δ τ 2 ) + β τ Corr ( ν c ^ 2 , c 2 ν c ^ 2 , c 1 ) + ν PM β τ Corr NR PM ( Δ τ 2 ) + ν A + ν PM
ε ϕ Q , c 2 ( Δ τ 2 ) = A ϕ Q , p ^ 2 late ( Δ τ 2 ) A ϕ Q , p ^ 2 early ( Δ τ 2 )
ε ϕ Q , c 2 ( Δ τ 2 ) = β τ Corr N ( R PM ( Δ τ 2 + T P / 2 ) R PM ( Δ τ 2 T P / 2 ) ) + ν A error + ν PM error
Δ τ 0 = 3 T P / 2 + [ ln ( 2 ) ln ( 1 4 e α T P 2 + 5 e α T P ) ] / α
ε ϕ Q , c 2 ( Δ τ 2 ) = 2 β N τ Corr Δ τ 2 + ν A error + ν ϕ PM error
τ ^ 2 ( s ) = GC II 2 τ 2 1 + GC + C II 2 ( ν A error + ν PM error ) 1 + GG II 2

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