Abstract

We analyze the minimum achievable mean-square error in frequency-modulated continuous-wave range estimation of a single stationary target when photon-counting detectors are employed. Starting from the probability density function for the photon-arrival times in photodetectors with subunity quantum efficiency, dark counts, and dead time, we derive the Cramér–Rao bound and highlight three important asymptotic regimes. We then derive the maximum-likelihood (ML) estimator for arbitrary frequency modulation. Simulation of the ML estimator shows that its performance approaches the standard quantum limit only when the mean received photons are between two thresholds. We provide analytic approximations to these thresholds for linear frequency modulation. We also compare the ML estimator’s performance to conventional Fourier transform (FT) frequency estimation, showing that they are equivalent if the reference arm is much stronger than the target return, but that when the reference field is weak the FT estimator is suboptimal by approximately a factor of 2 in root-mean-square error. Finally, we report on a proof-of-concept experiment in which the ML estimator achieves this theoretically predicted improvement over the FT estimator.

© 2013 Optical Society of America

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References

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  1. M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
    [CrossRef]
  2. H. L. V. Trees, Detection, Estimation and Modulation Theory, Part 1 (Prentice Hall, 2001).
  3. N. Satyan, A. Vasilyev, G. Rakuljic, V. Leyva, and A. Yariv, “Precise control of broadband frequency chirps using optoelectronic feedback,” Opt. Express 17, 15991–15999 (2009).
    [CrossRef]
  4. Z. W. Barber, W. R. Babbitt, B. Kaylor, R. R. Reibel, and P. A. Roos, “Accuracy of active chirp linearization for broadband frequency modulated continuous-wave ladar,” Appl. Opt. 49, 213–219 (2010).
    [CrossRef]
  5. L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).
    [CrossRef]
  6. D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.
  7. K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
    [CrossRef]
  8. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
    [CrossRef]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  10. L. A. Jiang and J. X. Luu, “Heterodyne detection with a weak local oscillator,” Appl. Opt. 47, 1486–1503 (2008).
    [CrossRef]
  11. T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
    [CrossRef]
  12. S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).
    [CrossRef]
  13. T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).
    [CrossRef]
  14. Throughout this paper, an ideal laser field refers to a paraxial and quasi-monochromatic optical field in a single spatial and polarization mode, which is also in a coherent state of the quantized field operator, such that it gives rise to Poisson statistics when an ideal photon-counting measurement is performed on it [9].
  15. This characterization of the photodetector output implies that the detector has infinite electrical bandwidth, allowing the arrival times to be precisely resolvable. In practice, there is little loss in adopting this idealization if the photodetector impulse response is significantly narrower than the mean photoelectron interarrival time.
  16. R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).
  17. D. L. Snyder, Random Point Processes (Wiley, 1975).
  18. Tensor-product coherent states incident on a beam splitter yield tensor-product coherent-state outputs [19]. Because the propagation paths in Fig. 1 can be modeled as a sequence of beam splitters (including the loss elements), the fields incident on the photodetectors will also be tensor-product coherent states. It is well known that in this case the semiclassical theory of photodetection and quantum measurement theory predict exactly the same statistics for the output photocurrent [9].
  19. Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).
    [CrossRef]
  20. The square-bracketed term in Eq. (8) is approximated as εϕ/2 to obtain this result.
  21. Here we continue to assume ψ0=0, consistent with our treatment in the previous section.
  22. P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).
    [CrossRef]
  23. F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).
    [CrossRef]
  24. B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.
  25. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
    [CrossRef]
  26. The mean incident photon number, NI, is estimated from the mean registered photoelectron counts, ndet, by accounting for the incident photons lost to dead time. ndet is equal to the sum and integral of the mean arrival rates given in Eq. (A4) in Appendix A. To a good degree of approximation, ndet=ηNI(1+α(1−β2/2))/(1+2α+α2(1−β2/2)).
  27. The (appropriately normalized) mean of the two estimators at this flux level are separated by approximately 1 Hz, whereas the standard deviation of the estimates is on the order of 10 Hz. Thus, the contribution of the bias term to the MSE is negligible as long as the true range to the target is in the vicinity of the mean estimates generated by the two estimators. We believe that both estimators having significant bias at the highest flux level is unlikely.
  28. G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
    [CrossRef]

2010 (1)

2009 (1)

2008 (1)

2007 (2)

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).
[CrossRef]

T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).
[CrossRef]

2005 (1)

F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).
[CrossRef]

2004 (1)

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).
[CrossRef]

2001 (1)

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

1998 (1)

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

1996 (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

1995 (1)

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).
[CrossRef]

1992 (1)

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

1987 (1)

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).
[CrossRef]

1983 (1)

1978 (1)

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Amann, M.-C.

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Asaka, K.

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

Athley, F.

F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).
[CrossRef]

Babbitt, W. R.

Barber, Z. W.

Bosch, T.

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Boyer, E.

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).
[CrossRef]

Chan, V. W. S.

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Cousineau, M.

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

Duncan, B. D.

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).
[CrossRef]

Dupuy, D.

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

Erkmen, B. I.

B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.

Forster, P.

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).

Gibson, G. J.

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).
[CrossRef]

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Green, T. J.

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Hernandez-Marin, S.

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).
[CrossRef]

Hirano, Y.

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

Hong, C. K.

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).
[CrossRef]

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Jiang, L. A.

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).

Karr, T. J.

T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).
[CrossRef]

Kasahara, K.

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

Kaylor, B.

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Larzabal, P.

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).
[CrossRef]

Lescure, M.

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

Leyva, V.

Luu, J. X.

Mandel, L.

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mark, M. B.

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).
[CrossRef]

Masters, L. T.

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).
[CrossRef]

Moision, B.

B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.

Myllylä, R.

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).
[CrossRef]

Rakuljic, G.

Reibel, R. R.

Rioux, M.

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

Roos, P. A.

Satyan, N.

Shapiro, J. H.

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley, 1975).

Tajime, T.

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

Tatsumi, K.

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

Teich, M. C.

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Trees, H. L. V.

H. L. V. Trees, Detection, Estimation and Modulation Theory, Part 1 (Prentice Hall, 2001).

Vannucci, G.

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Vasilyev, A.

Wallace, A. M.

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Yariv, A.

Yuen, H. P.

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

T. J. Karr, “Atmospheric phase error in coherent laser radar,” IEEE Trans. Antennas Propag. 55, 1122–1133(2007).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Hernandez-Marin, A. M. Wallace, and G. J. Gibson, “Bayesian analysis of lidar signals with multiple returns,” IEEE Trans. Pattern Anal. Mach. Intell. 29, 2170–2180 (2007).
[CrossRef]

IEEE Trans. Signal Process. (2)

P. Forster, P. Larzabal, and E. Boyer, “Threshold performance analysis of maximum likelihood DOA estimation,” IEEE Trans. Signal Process. 52, 3183–3191 (2004).
[CrossRef]

F. Athley, “Threshold region performance of maximum likelihood direction of arrival estimators,” IEEE Trans. Signal Process. 53, 1359–1373 (2005).
[CrossRef]

Opt. Commun. (2)

G. Vannucci and M. C. Teich, “Effects of rate variation on the counting statistics of dead-time-modified Poisson processes,” Opt. Commun. 25, 267–272 (1978).
[CrossRef]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Relation between input and output states for a beam splitter,” Opt. Commun. 63, 118–122 (1987).
[CrossRef]

Opt. Eng. (3)

T. J. Green and J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

M.-C. Amann, T. Bosch, M. Lescure, R. Myllylä, and M. Rioux, “Laser ranging: a critical review of usual techniques for distance measurement,” Opt. Eng. 40, 10–19 (2001).
[CrossRef]

L. T. Masters, M. B. Mark, and B. D. Duncan, “Analysis of ladar range resolution enhancement by sinusoidal phase modulation,” Opt. Eng. 34, 3115–3121 (1995).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Rev. (1)

K. Asaka, Y. Hirano, K. Tatsumi, K. Kasahara, and T. Tajime, “A pseudo-random frequency modulation continuous wave coherent lidar using an optical field correlation detection method,” Opt. Rev. 5, 310–314 (1998).
[CrossRef]

Other (13)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

D. Dupuy, M. Lescure, and M. Cousineau, “A FMCW laser range-finder based on a delay line technique,” in Proceedings of IEEE Instrumentation and Measurement Technology Conference (IEEE, 2001), pp. 1084–1088.

H. L. V. Trees, Detection, Estimation and Modulation Theory, Part 1 (Prentice Hall, 2001).

B. I. Erkmen and B. Moision, “Maximum likelihood time-of-arrival estimation of optical pulses via photon-counting photodetectors,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1909–1913.

The mean incident photon number, NI, is estimated from the mean registered photoelectron counts, ndet, by accounting for the incident photons lost to dead time. ndet is equal to the sum and integral of the mean arrival rates given in Eq. (A4) in Appendix A. To a good degree of approximation, ndet=ηNI(1+α(1−β2/2))/(1+2α+α2(1−β2/2)).

The (appropriately normalized) mean of the two estimators at this flux level are separated by approximately 1 Hz, whereas the standard deviation of the estimates is on the order of 10 Hz. Thus, the contribution of the bias term to the MSE is negligible as long as the true range to the target is in the vicinity of the mean estimates generated by the two estimators. We believe that both estimators having significant bias at the highest flux level is unlikely.

The square-bracketed term in Eq. (8) is approximated as εϕ/2 to obtain this result.

Here we continue to assume ψ0=0, consistent with our treatment in the previous section.

Throughout this paper, an ideal laser field refers to a paraxial and quasi-monochromatic optical field in a single spatial and polarization mode, which is also in a coherent state of the quantized field operator, such that it gives rise to Poisson statistics when an ideal photon-counting measurement is performed on it [9].

This characterization of the photodetector output implies that the detector has infinite electrical bandwidth, allowing the arrival times to be precisely resolvable. In practice, there is little loss in adopting this idealization if the photodetector impulse response is significantly narrower than the mean photoelectron interarrival time.

R. M. Gagliardi and S. Karp, Optical Communications (Wiley, 1976).

D. L. Snyder, Random Point Processes (Wiley, 1975).

Tensor-product coherent states incident on a beam splitter yield tensor-product coherent-state outputs [19]. Because the propagation paths in Fig. 1 can be modeled as a sequence of beam splitters (including the loss elements), the fields incident on the photodetectors will also be tensor-product coherent states. It is well known that in this case the semiclassical theory of photodetection and quantum measurement theory predict exactly the same statistics for the output photocurrent [9].

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Figures (6)

Fig. 1.
Fig. 1.

Block diagram of FMCW ranging. Optical fields are denoted with thick (orange) lines and electrical signals are denoted with thin (black) lines.

Fig. 2.
Fig. 2.

Exact CRB , normalized by T , is shown as the solid (blue) line. The dashed (red) line is the dark noise asymptote, the dash-dotted (green) line indicates the shot-noise asymptote, and the dotted (cyan) line is the dead-time asymptote.

Fig. 3.
Fig. 3.

RMSE normalized to the width of the range uncertainty window. The simulated ML (magenta) and FT (cyan) estimator performances, the CRB (dashed red), the analytic RMSE model (dash-dotted green), and the FT RMSE asymptote (dotted blue) are plotted. δ r is the fractional position of the object in its support.

Fig. 4.
Fig. 4.

RMSE normalized to the width of the range uncertainty window. Plot legend is identical to that of Fig. 3.

Fig. 5.
Fig. 5.

Block diagram of FMCW ranging experiment.

Fig. 6.
Fig. 6.

Performance of the ML estimator and the FT estimator, plotted as a function of η N I . The ML error model, the CRB (dashed red), and the asymptotic RMSE of the FT estimator (dash-dotted blue) are plotted for comparison.

Tables (1)

Tables Icon

Table 1. L 2 Norms of Various Frequency Modulations with Identical Frequency Detuning Range Δ f a

Equations (37)

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

E D , m ( t ) = E S , 2 ( t ) + ( 1 ) m + 1 E R , 2 ( t ) 2 ,
Λ m ( t ; t 0 ) = η N I 2 T [ 1 + ( 1 ) m + 1 β cos ( D ( t ; t 0 ) + ψ 0 ) ] ,
β 2 N S , 2 N R , 2 N S , 2 + N R , 2
p ( n 1 , u 1 , , u n 1 , n 2 , v 1 , , v n 2 ; t 0 ) = [ i = 1 n 1 Λ 1 ( u i ; t 0 ) ] [ j = 1 n 2 Λ 2 ( v j ; t 0 ) ] e T / 2 T / 2 d t [ Λ 1 ( t ; t 0 ) + Λ 2 ( t ; t 0 ) ] .
CRB 2 log p ( n 1 , u 1 , , u n 1 , n 2 , v 1 , , v n 2 ; t 0 ) t 0 2 1 ,
CRB = 1 η N I [ 1 T T / 2 T / 2 d t β 2 sin 2 ( D ( t ; t 0 ) ) 1 β 2 cos 2 ( D ( t ; t 0 ) ) ϕ ˙ 2 ( t t 0 ) ] 1 .
CRB 1 η N I [ 1 T T / 2 T / 2 d t ϕ ˙ 2 ( t t 0 ) ] 1 ,
CRB 1 β 2 η N I [ 1 T T / 2 T / 2 d t sin 2 ( D ( t ; t 0 ) ) ϕ ˙ 2 ( t t 0 ) ] 1 ,
γ 1 + N R , 1 / N S , 1 1 + 4 r S 2
Λ m , d ( t ; t 0 ) = η N I , d 2 T [ 1 + ( 1 ) m + 1 β d cos ( D ( t ; t 0 ) ) ]
p ( n 1 , u 1 , , u n 1 , n 2 , v 1 , , v n 2 ; t 0 ) = p d ( n 1 , u 1 , , u n 1 ; t 0 ) p d ( n 2 , v 1 , , v n 2 ; t 0 ) ,
p d ( n m , u 1 , , u n m ) [ i = 1 n m Λ m , d ( u i ; t 0 ) ] × exp { T / 2 u 1 d t Λ m , d ( t ; t 0 ) i = 2 n m + 1 u i 1 u i d t Λ m , d ( t ; t 0 ) w τ ( t u i 1 ) }
CRB { 2 N d ( 1 + α ) η N I 2 β 2 K 1 1 ( t 0 ) N I 2 N d and α 1 1 η N I K 2 1 ( β , t 0 ) N I 2 N d and α 1 τ 2 T K 3 1 ( β , t 0 ) N I 2 N d and α 1 ,
K 1 ( t 0 ) = 1 T T / 2 T / 2 d t sin 2 ( D ( t ; t 0 ) ) ϕ ˙ 2 ( t t 0 ) ,
K 2 ( β , t 0 ) = 1 T T / 2 T / 2 d t β 2 sin 2 ( D ( t ; t 0 ) ) 1 β 2 cos 2 ( D ( t ; t 0 ) ) ϕ ˙ 2 ( t t 0 ) ,
K 3 ( β , t 0 ) = 1 T T / 2 T / 2 d t β 2 sin 2 ( D ( t ; t 0 ) ) [ 1 + β 2 cos 2 ( D ( t ; t 0 ) ) ] [ 1 β 2 cos 2 ( D ( t ; t 0 ) ) ] 2 ϕ ˙ 2 ( t t 0 ) .
t ^ 0 argmax t 0 T log p ( n 1 , u 1 , , u n 1 , n 2 , v 1 , , v n 2 ; t 0 ) ,
t ^ 0 = argmax t 0 T T / 2 T / 2 d t [ i 1 ( t ) h 1 ( t ; t 0 ) + i 2 ( t ) h 2 ( t ; t 0 ) ] ,
h m ( t ; t 0 ) = log ( 1 + ( 1 ) m + 1 β d cos ( D ( t ; t 0 ) ) ) + ( 1 ) m + 1 α β d cos ( D ( t ; t 0 ) ) ,
t ^ 0 arg max t 0 T T / 2 T / 2 d t ( i 1 ( t ) i 2 ( t ) ) cos ( D ( t ; t 0 ) ) ;
t ^ 0 arg max t 0 T R { e i Δ f t 0 2 / ( 2 T ) T / 2 T / 2 d u ( i 1 ( u ) i 2 ( u ) ) e i Δ f t 0 u / T } ;
( t 0 t 0 ( FT ) ) 2 2 CRB
( t 0 t ^ 0 ) 2 = σ A 2 P A + CRB ( 1 P A ) ,
σ A 2 ( T 2 T 1 ) 2 12 + ( T 2 + T 1 2 t 0 ) 2
P A σ A 2 = ( 1 P A ) CRB .
P A = P ( n < 2 ) + P ( n 2 ) P A | 2 ,
P A | 2 = 1 P ( k * C ( t k * ) > C ( t ) ) ,
P A | 2 1 [ 1 Q ( η N I β / 2 ) ] n bin ,
N = 8 η β 2 W 1 ( 1 4 ε ϕ σ A 2 n bin ) ,
η N u K 3 ( β , t 0 ) T 5 K 2 ( β , t 0 ) τ ,
N u max { T 5 η τ , N } .
log p ( n 1 , u 1 , , u n 1 , n 2 , v 1 , , v n 2 ; t 0 ) = log f 1 + log f 2 + C 1 ,
log f m T / 2 T / 2 d t i m ( t ) ( log Λ m , d ( t ) + τ Λ m , d ( t ) ) .
CRB 1 = m = 1 2 T / 2 T / 2 d t i m ( t ) [ Λ ˙ m , d 2 ( t ) Λ m , d 2 ( t ) Λ ¨ m , d ( t ) Λ m , d ( t ) τ Λ ¨ m , d ( t ) ] ,
i m ( t ) = Λ m , d ( t ) 1 + τ Λ m , d ( t ) .
CRB 1 = T / 2 T / 2 d t Λ ˙ 1 , d 2 ( t ) [ m = 1 2 1 Λ m , d ( t ) ( 1 + τ Λ m , d ( t ) ) ] .
CRB = T η N I , d [ T / 2 T / 2 d t β d 2 sin 2 ( D ( t ; t 0 ) ) 1 β d 2 cos 2 ( D ( t ; t 0 ) ) ϕ ˙ 2 ( t t 0 ) 1 + α [ 1 + β d 2 cos 2 ( D ( t ; t 0 ) ) ] 1 + 2 α + α 2 [ 1 β d 2 cos 2 ( D ( t ; t 0 ) ) ] ] 1 ,

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