Time-resolved speckle effects on the estimation of laser-pulse arrival times

Bin-Ming Tsai; Chester S. Gardner

doi:10.1364/JOSAA.2.000649

Journal of the Optical Society of America A
Vol. 2,
Issue 5,
pp. 649-656
(1985)
•https://doi.org/10.1364/JOSAA.2.000649

Time-resolved speckle effects on the estimation of laser-pulse arrival times

Bin-Ming Tsai and Chester S. Gardner

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Bin-Ming Tsai and Chester S. Gardner, "Time-resolved speckle effects on the estimation of laser-pulse arrival times," J. Opt. Soc. Am. A 2, 649-656 (1985)

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Abstract

When laser pulses are reflected by targets that have range spreads larger than the transmitted pulse width, the width of the received pulses will be longer than the correlation length of the speckle-induced fluctuations. As a consequence, speckle will cause random small-scale fluctuations within the received pulse that will distort its shape. This phenomenon is called time-resolved speckle. In laser ranging and altimetry, the random pulse distortion caused by time-resolved speckle can seriously degrade the timing accuracy of the receivers. In this paper, we study the statistical properties of time-resolved speckle and the problem of estimating the arrival times of laser pulses in its presence. The maximum-likelihood (ML) estimator of the pulse arrival time is derived, and its performance is evaluated for pulse reflections from flat diffuse targets. The performance of the ML estimator is compared with the performance of several suboptimal estimators. When the signal level is high, speckle noise places a fundamental limit on the accuracy of the suboptimal estimators. It is shown that the ML estimator performs considerably better than the suboptimal estimators and that its accuracy improves as the width of the receiver observation interval increases.

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Estimator	Timing Variances
${\hat{τ}}_{SPEC} = arg max_{τ} {\sum_{i = 1}^{n} [k_{i} + M_{i} (τ) - 1 / 2] ln [k_{i} + M_{i} (τ)] - \sum_{i = 1}^{n} k_{i} ln [1 + \frac{M_{i} (τ)}{{\bar{k}}_{i} (τ)}]}$	${\sum_{i = 1}^{n} {\dot{\bar{k}}}_{i}^{2} / [{\bar{k}}_{i} (1 + \frac{{\bar{k}}_{i}}{M_{i}})]}^{- 1}$
${\hat{τ}}_{SHOT} = arg max_{τ} \sum_{i = 1}^{n} k_{i} ln {\bar{k}}_{i} (τ)$	${(\sum_{i = 1}^{n} \frac{{\dot{\bar{k}}}_{i}^{2}}{{\bar{k}}_{i}})}^{- 1} + \sum_{i = 1}^{n} \frac{{\dot{\bar{k}}}_{i}^{2}}{M_{i}} / {(\sum_{i = 1}^{n} \frac{{\dot{\bar{k}}}_{i}^{2}}{{\bar{k}}_{i}})}^{2}$
${\hat{τ}}_{CENT} = \sum_{i = 1}^{n} i T k_{i} / \sum_{i = 1}^{n} k_{i}$	$\frac{1}{{〈 N 〉}^{2}} \sum_{i = 1}^{n} {(i - \bar{i})}^{2} T^{2} ({\bar{k}}_{i} + \frac{{\bar{k}}_{i}^{2}}{M_{i}})$

Estimator	Timing Variance	Speckle-Limited Timing Variance
${\hat{τ}}_{SPEC}$	$\frac{{σ_{T}}^{2}}{〈 N 〉} \frac{1}{g (〈 N 〉)}$	$6 \sqrt{π} {(\frac{2 σ_{T}}{n T})}^{3} \frac{{σ_{T}}^{2}}{2 K_{s}}$
${\hat{τ}}_{SHOT}$	$\frac{{σ_{T}}^{2}}{〈 N 〉} + \frac{{σ_{T}}^{2}}{2 K_{s}}$	$\frac{{σ_{T}}^{2}}{2 K_{s}}$
${\hat{τ}}_{CENT}$	$\frac{{σ_{T}}^{2}}{〈 N 〉} + \frac{{s_{T}}^{2}}{2 K_{s}}$	$\frac{{σ_{T}}^{2}}{2 K_{s}}$

Estimator	Timing Variances
${\hat{τ}}_{SPEC} = arg max_{τ} {\sum_{i = 1}^{n} [k_{i} + M_{i} (τ) - 1 / 2] ln [k_{i} + M_{i} (τ)] - \sum_{i = 1}^{n} k_{i} ln [1 + \frac{M_{i} (τ)}{{\bar{k}}_{i} (τ)}]}$	${\sum_{i = 1}^{n} {\dot{\bar{k}}}_{i}^{2} / [{\bar{k}}_{i} (1 + \frac{{\bar{k}}_{i}}{M_{i}})]}^{- 1}$
${\hat{τ}}_{SHOT} = arg max_{τ} \sum_{i = 1}^{n} k_{i} ln {\bar{k}}_{i} (τ)$	${(\sum_{i = 1}^{n} \frac{{\dot{\bar{k}}}_{i}^{2}}{{\bar{k}}_{i}})}^{- 1} + \sum_{i = 1}^{n} \frac{{\dot{\bar{k}}}_{i}^{2}}{M_{i}} / {(\sum_{i = 1}^{n} \frac{{\dot{\bar{k}}}_{i}^{2}}{{\bar{k}}_{i}})}^{2}$
${\hat{τ}}_{CENT} = \sum_{i = 1}^{n} i T k_{i} / \sum_{i = 1}^{n} k_{i}$	$\frac{1}{{〈 N 〉}^{2}} \sum_{i = 1}^{n} {(i - \bar{i})}^{2} T^{2} ({\bar{k}}_{i} + \frac{{\bar{k}}_{i}^{2}}{M_{i}})$

Estimator	Timing Variance	Speckle-Limited Timing Variance
${\hat{τ}}_{SPEC}$	$\frac{{σ_{T}}^{2}}{〈 N 〉} \frac{1}{g (〈 N 〉)}$	$6 \sqrt{π} {(\frac{2 σ_{T}}{n T})}^{3} \frac{{σ_{T}}^{2}}{2 K_{s}}$
${\hat{τ}}_{SHOT}$	$\frac{{σ_{T}}^{2}}{〈 N 〉} + \frac{{σ_{T}}^{2}}{2 K_{s}}$	$\frac{{σ_{T}}^{2}}{2 K_{s}}$
${\hat{τ}}_{CENT}$	$\frac{{σ_{T}}^{2}}{〈 N 〉} + \frac{{s_{T}}^{2}}{2 K_{s}}$	$\frac{{σ_{T}}^{2}}{2 K_{s}}$

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Equations (56)

Journal of the Optical Society of America A