## Maximum-likelihood estimation for frequency-modulated continuous-wave laser ranging using photon-counting detectors |

Applied Optics, Vol. 52, Issue 10, pp. 2008-2018 (2013)

http://dx.doi.org/10.1364/AO.52.002008

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### 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

© 2013 Optical Society of America

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(010.3640) Atmospheric and oceanic optics : Lidar

(030.5260) Coherence and statistical optics : Photon counting

(270.5290) Quantum optics : Photon statistics

(040.1345) Detectors : Avalanche photodiodes (APDs)

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: October 11, 2012

Revised Manuscript: January 22, 2013

Manuscript Accepted: February 15, 2013

Published: March 21, 2013

**Citation**

Baris I. Erkmen, Zeb W. Barber, and Jason Dahl, "Maximum-likelihood estimation for frequency-modulated continuous-wave laser ranging using photon-counting detectors," Appl. Opt. **52**, 2008-2018 (2013)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-52-10-2008

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### References

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