Abstract

Estimating time and time-lag in time-of-flight velocimeters is investigated. Statistics of a filtered Poisson point process is given. A Maximum Likelihood Estimator is compared with suboptimum estimators in terms of robustness. For a dominating background combined spatial and temporal processing can improve the robustness compared with purely temporal processing. Schemes for the spatial filters are given.

© 1983 Optical Society of America

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References

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  1. R. Schodl, “The Laser-Dual-Focus Flow Velocimeter,” NATO, Advisory Group on Aerospace Research and Development, AGARD Conf. Proc. 21, paper 21.
  2. L. Lading, “The Time-of-Flight Laser Anemometer,” NATO, Advisory Group on Aerospace Research and Development, AGARD Conf. Proc. 21, paper 23.
  3. Special issue on Time Delay Estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, No. 3, Part 2 (1981).
  4. W. T. Rhodes, Proc. IEEE 69, 65 (1981).
    [CrossRef]
  5. B. V. K. Vijaya Kumar, D. Casasent, A. Goutzoulis, Appl. Opt. 21, 3855 (1982).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 144–146.
  7. A. Papoulis, Systems and Transforms with Application in Optics (McGraw-Hill, New York, 1968), p. 434.
  8. A. S. Jensen, L. Lading, The Optimum Code in Single Particle Detection (Risø R-413, Roskilde, Denmark, 1980).
  9. A. D. Wahlen, Detection of Signals in Noise (Academic, New York, 1971), pp. 337–339.
  10. B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978), pp. 8–10, 69.
  11. H. Meyer, IEEE Trans. Commun. COM-24, 331 (1976).
    [CrossRef]
  12. L. Lading, “Processing of Laser Anemometry Signals,” in Dynamic Flow Measurement, P. Buchhave, Ed. (Copenhagen1978), pp. 801–823.
  13. Ref. 7, pp. 83–89, 435.

1982

1981

Special issue on Time Delay Estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, No. 3, Part 2 (1981).

W. T. Rhodes, Proc. IEEE 69, 65 (1981).
[CrossRef]

1976

H. Meyer, IEEE Trans. Commun. COM-24, 331 (1976).
[CrossRef]

Casasent, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 144–146.

Goutzoulis, A.

Jensen, A. S.

A. S. Jensen, L. Lading, The Optimum Code in Single Particle Detection (Risø R-413, Roskilde, Denmark, 1980).

Lading, L.

A. S. Jensen, L. Lading, The Optimum Code in Single Particle Detection (Risø R-413, Roskilde, Denmark, 1980).

L. Lading, “Processing of Laser Anemometry Signals,” in Dynamic Flow Measurement, P. Buchhave, Ed. (Copenhagen1978), pp. 801–823.

L. Lading, “The Time-of-Flight Laser Anemometer,” NATO, Advisory Group on Aerospace Research and Development, AGARD Conf. Proc. 21, paper 23.

Meyer, H.

H. Meyer, IEEE Trans. Commun. COM-24, 331 (1976).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Application in Optics (McGraw-Hill, New York, 1968), p. 434.

Rhodes, W. T.

W. T. Rhodes, Proc. IEEE 69, 65 (1981).
[CrossRef]

Saleh, B.

B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978), pp. 8–10, 69.

Schodl, R.

R. Schodl, “The Laser-Dual-Focus Flow Velocimeter,” NATO, Advisory Group on Aerospace Research and Development, AGARD Conf. Proc. 21, paper 21.

Vijaya Kumar, B. V. K.

Wahlen, A. D.

A. D. Wahlen, Detection of Signals in Noise (Academic, New York, 1971), pp. 337–339.

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

Special issue on Time Delay Estimation, IEEE Trans. Acoust. Speech Signal Process. ASSP-29, No. 3, Part 2 (1981).

IEEE Trans. Commun.

H. Meyer, IEEE Trans. Commun. COM-24, 331 (1976).
[CrossRef]

Proc. IEEE

W. T. Rhodes, Proc. IEEE 69, 65 (1981).
[CrossRef]

Other

L. Lading, “Processing of Laser Anemometry Signals,” in Dynamic Flow Measurement, P. Buchhave, Ed. (Copenhagen1978), pp. 801–823.

Ref. 7, pp. 83–89, 435.

R. Schodl, “The Laser-Dual-Focus Flow Velocimeter,” NATO, Advisory Group on Aerospace Research and Development, AGARD Conf. Proc. 21, paper 21.

L. Lading, “The Time-of-Flight Laser Anemometer,” NATO, Advisory Group on Aerospace Research and Development, AGARD Conf. Proc. 21, paper 23.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 144–146.

A. Papoulis, Systems and Transforms with Application in Optics (McGraw-Hill, New York, 1968), p. 434.

A. S. Jensen, L. Lading, The Optimum Code in Single Particle Detection (Risø R-413, Roskilde, Denmark, 1980).

A. D. Wahlen, Detection of Signals in Noise (Academic, New York, 1971), pp. 337–339.

B. Saleh, Photoelectron Statistics (Springer, Berlin, 1978), pp. 8–10, 69.

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Figures (5)

Fig. 1
Fig. 1

Estimating the temporal position of a light pulse. h0(t) is an even filter response function. o{·} is a linear operator converting an even function to an uneven; o{·} may operate on either the electronic signal (a) or the optical signal (b).

Fig. 2
Fig. 2

Variance of the estimated temporal position of a pulse as a function of the relative filter bandwidth ωfs. The estimate is given by the zero-crossing of the filtered signal after the even pulse has been converted to an uneven. This is done with a (1) d/dt operator, (2) Hilbert filter, (3) spatial lead-lag filter: (a) dominating background normalized with B/A2; (b) no background normalized with 1/A.

Fig. 3
Fig. 3

System for optically converting an even pulse to an uneven by use of the Hilbert transform.

Fig. 4
Fig. 4

Signal generated with the configuration in Fig. 3 having a thin wire move through in the measuring plane. The dotted line is computed.

Fig. 5
Fig. 5

System for optically generating uneven pulses in conjunction with time-of-flight measurements. The sandwich W1 − λ/4 − W2 is placed in the left focal plane of L1. It causes spots of consecutive orthogonal polarizations to appear in the right focal plane. (Only the beam axes are shown.) Beams 1 and 2, and 3 and 4, respectively, are partly overlapping. Detector 1 only detects light from beam 1 and 3 and 2 only from beams 2 and 4. W3 makes sure that the images from beams 1 and 2, and 3 and 4, respectively, are overlapping. This facilitates an easier spatial filtering of the images.

Equations (51)

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r ( t ) = A s ( t + τ ) + B ,
p i ( n i ) = [ r ( t i ) Δ t ] n i n i ! exp [ r ( t i ) Δ t ] .
i = 1 N n i s ( t i + τ ) τ = 0 ,
F = 2 ln p ( n ) τ ] τ 2 .
F = π 2 A 2 B ω s ,
F = 2 π A ω s
q i = j = 1 N n j a j i .
Q i ( s ) j = 1 N exp { r ( t j ) Δ t [ exp ( s a j i ) 1 ] } ,
Q i c ( s ) = j r ( t j ) Δ t [ exp ( s a i j ) 1 ] ,
Q c ( s t ) = r ( t ) { exp [ s h ( t t ) ] 1 } d t ,
var { q ( t ) } = r ( t ) h 2 ( t t ) d t .
p ( n j a j 1 , n i a j 2 ) = p j i ( n j a j 1 ) δ ( n j n i ) .
cov { q ( t 1 ) , q ( t 2 ) } = r ( t ) h ( t 1 t ) h ( t 2 t ) d t .
var { τ ̂ } = var { q ( τ ) } [ d d t { r ( t ) * h ( t ) } ] t = τ 2 .
h 0 ( t ) * o ( t ) = d d t { h 0 ( t ) } ,
h 0 ( t ) * o ( t ) = h 0 ( t + τ 1 ) h 0 ( t τ 1 ) .
r ( t τ ) = A exp [ ½ ( t ω s ) 2 ] + B ,
h 0 ( t ) = ω f 2 π exp [ ½ ( t ω f ) 2 ] .
m 2 2 { u ( x , y ) + ǔ x ( x , y ) + j [ u ( x , y ) + ǔ x ( x , y ) ] } δ ( x x x 1 , y y y 1 ) ,
m 2 2 { u ( x , y ) ǔ x ( x , y ) + j [ u ( x , y ) ǔ x ( x , y ) ] } δ ( x x x 2 , y y y 2 ) ,
i 1 = C m 2 4 ( u 2 + ǔ x 2 + 2 u ǔ x ) ,
i 2 = C m 2 4 ( u 2 + ǔ x 2 + 2 u ǔ x ) ,
i 1 + i 2 = C m 2 2 ( u 2 + ǔ x 2 )
i 1 i 2 = C m 2 u ǔ x .
L R 1 j j 1 .
½ ( u + j ǔ x ) + j 2 ( u ǔ x ) j 2 ( u + j ǔ x ) + ½ ( u ǔ x ) .
i 1 = C m 2 2 ( u 2 + ǔ x + 2 u ǔ x ) , i 1 = C m 2 2 ( u 2 + u x 2 2 u ǔ x ) ,
p ( n ) = i = 1 N p i ( n i ) .
τ ln p ( n τ ) = 0 .
i = 1 N n i ln [ r ( t i ) Δ t ] τ = 0 .
i = 1 N n i s ( t i + τ ) τ = 0 .
r ( t ) = A exp [ ½ ( t ω s ) 2 ] + B ,
h 0 ( t ) = ω f 2 π exp [ ½ ( t ω f ) 2 ] .
r ( t ) * h 0 ( t ) = A ω p ω s exp [ ½ ( t ω p ) 2 ] + B ω f 2 π exp [ ½ ( t ω f ) 2 ] ,
q ( t ) = r ( t ) * { d d t [ h 0 ( t ) ] } .
var { τ ̂ 2 } = [ A 4 π ω f 6 , ( ω s 2 2 + ω f 2 ) 3 / 2 + B 4 π ω f 3 ] / ( A ω p 3 ω s ) 2 .
d dt { s ( t ) * h ( t ) } t = 0 = 2 2 π 0 ω S ( ω ) d ω .
d dt { s ( t ) * h ( t ) } t = 0 = A 2 π ω p 2 ω s .
h 0 2 ( t ) dt = h ˇ 0 2 ( t ) dt ,
var { τ ̂ } = π 2 B A 2 ω f ω s 2 ω p 4 .
r 1 ( t ) = ½ r ( t + τ ) , r 2 ( t ) = ½ r ( t τ ) .
q ( t ) = [ r 1 ( t ) r 2 ( t ) ] * h 0 ( t ) .
var { q ( o ) } = 2 ½ r ( τ ) h 0 2 ( τ t ) dt .
var { q ( o ) } = A 2 π ω f ω q ω s exp [ ½ ( τ ω q ) 2 ] + B ω f 2 π ,
dq ( t ) dt t = 0 = A ω p 3 ω s τ exp [ ½ ( τ ω p ) 2 ] .
var { τ ̂ } = 1 2 π 1 A ω f ω q ω s 2 ω p 6 τ 2 exp [ ½ τ 2 ( ω q 2 2 ω p 2 ) ] + 1 2 π B A 2 ω f ω s 2 ω p 6 τ 2 exp [ ( τ ω p ) 2 ] .
H ( w x , w y ) = { j , w x 2 + w y 2 < ζ 0 2 and w x < 0 , j , w x 2 + w y 2 < ζ 0 2 and w x > 0 , 1 , otherwise .
{ u ( r ) [ 1 + m δ ( r r ) ] } * h ( r ) ǔ x ( r ) + m δ ( r r ) u ( r ) ,
i P + m ǔ x ( r ) u ( r ) ,
ǔ x ( r ) + m δ ( r r ) + m δ ( r r ) * [ g ˇ x ( r ) g ( r ) ] ,
m δ ( r r ) u ( r ) ǔ x ( r ) dA = m ǔ x ( r ) u ( r ) ; mg ( r r ) u ( r ) ǔ x ( r ) da = m ǔ x ( r ) u ( r ) ; m g ˇ x ( r r ) u ( r ) u ( r ) da = 0 .

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