Abstract

Properties of pulsed optical range finders which are affected by the transmitter pulse waveshape have been analyzed. This was done by deriving the probability density functions of the random variable representing range delay time with respect to the start of the transmitter pulse. These functions were used to predict the calibration or bias time, random range errors, the probability that a range measurement is made, and its entropy, all vs signal energy. Noise and other important system properties are treated as parameters, and it is assumed that photoemission is Poisson-distributed. A number of experiments were performed, including a simulation of geodetic satellite conditions. Although more experimental work remains to be done, the agreement between these measurements and theoretical predictions tends to verify the theory, including the basic assumption that photoemission is accurately described by Poisson statistics under almost all likely conditions.

© 1971 Optical Society of America

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References

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  1. L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
    [CrossRef]
  2. All sets describing first detection at the Ith interval of δ [Eqs. (3) and (4)].
  3. E. Parzen, Modern Probability Theory and Its Applications, (Wiley, New York, 1960), Chap. 5.
  4. G. Raisbeck, Information Theory (MIT Press, Cambridge, 1964), Chap. 1.
  5. The sampling theorem may be invoked to see how the larger number of measurements can be converted into more statistically independent range tracks and these, in turn, averaged to reduce random error.

1958 (1)

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Mandel, L.

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Parzen, E.

E. Parzen, Modern Probability Theory and Its Applications, (Wiley, New York, 1960), Chap. 5.

Raisbeck, G.

G. Raisbeck, Information Theory (MIT Press, Cambridge, 1964), Chap. 1.

Proc. Phys. Soc. (1)

L. Mandel, Proc. Phys. Soc. 72, 1037 (1958).
[CrossRef]

Other (4)

All sets describing first detection at the Ith interval of δ [Eqs. (3) and (4)].

E. Parzen, Modern Probability Theory and Its Applications, (Wiley, New York, 1960), Chap. 5.

G. Raisbeck, Information Theory (MIT Press, Cambridge, 1964), Chap. 1.

The sampling theorem may be invoked to see how the larger number of measurements can be converted into more statistically independent range tracks and these, in turn, averaged to reduce random error.

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Figures (6)

Fig. 1
Fig. 1

Subintervals of the transmitter pulse period.

Fig. 2
Fig. 2

Assumed waveshape of transmitter output pulse.

Fig. 3
Fig. 3

PD vs signal energy for T = 2.

Fig. 4
Fig. 4

EXP[X], SD[X], and PD for R ≈ 10, S ≈ 40.

Fig. 5
Fig. 5

Efficiency vs signal energy for R ≈ 10, S ≈ 40.

Fig. 6
Fig. 6

A comparison of experiment and theory; bias function, EXP[X], vs signal energy.

Tables (2)

Tables Icon

Table I Comparison of Theory and Experiment (Expt. 1)

Tables Icon

Table II Comparison of Theory and Experiment (Expt. 2)

Equations (28)

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P F D [ I ] = P { N [ I ] } ,
{ N [ I ] } = { N [ 1 ] , N [ 2 ] , , N [ A ] , N [ I ] } .
A = 1 I - 1 N [ A ] < T A = 1 I N [ A ] T }             when 1 < I τ / δ .
A = L - τ / δ + 1 L N [ A ] < T ,             for τ / δ L < I A = I - τ / δ + 1 I N [ A ] T }             when τ / δ < I S .
P { N [ I ] } = A = 1 I P ( N [ A ] ) ,
P ( N [ A ] ) = exp ( - J ) J N [ A ] / N [ A ] ! , N [ A ] = 0 , 1 , 2 , ,
J = N ¯ [ A ] = N N ( K [ A ] ) ,
A = 1 S K [ A ] = 1.
I = 1 S P F D [ I ] = P D ,
P F D N [ I ] = P F D [ I ] / P D
P F D [ I , N N , R , R , T ] = P [ I , N N , R , R , T ] × ( 1 - A = 1 I - 1 P F D [ A , N N , R , R , T ] ) ,
P [ I , N N , R , R , T ] = 1 - exp ( - J ) D = 0 T - 1 J D / D ! ,
P F D [ I , N N , R , S , 1 ] = P F D [ I , N N , S , S , 1 ]
P F D [ I , N N , 1 , S , T ] = P [ I , N N , 1 , S , T ] - A = 1 I - 1 P F D [ A , N N , 1 , S , T ] ,
P [ I , N N , 1 , S , T ] = 1 - exp ( - I J ) D = 0 T - 1 ( I J ) D / D ! ,
I J = N N A = 1 I K [ A ] .
EXP [ X , N N , R , S , T , K ] = I - 1 S ( I - 0.5 ) P F D N [ I , N N , R , S , T , K ] ,
S D [ X , N N , R , S , T , K ] = ( I = 1 S ( I - 0.5 ) 2 P F D N [ I , N N , R , S , T , K ] - EXP [ X , N N , R , S , T , K ] 2 ) 1 2
P D [ N N , R , S , T , K ] = I = 1 S P F D [ I , N N , R , S , T , K ] .
H H = I = 1 S P F D N [ I , N N , R , S , T , K ] × log 2 ( 1 / P F D N [ I , N N , R , S , T , K ] ) bits .
t = ( I - 0.5 ) δ = ( I - 0.5 ) T t / S sec ,
t = ( I - 0.5 ) / S , decimal part of T t .
{ T t = 20 nsec , τ = 2 nsec , δ = 1 nsec ;             and             { T t = 100 nsec , τ = 10 nsec , δ = 5 nsec .
H H L = H H M - H H bits ,
H H L = log 2 ( S ) - I = 1 S P F D N [ I ] log 2 ( 1 / P F D N [ I ] ) bits .
= ( H H L ) ( P D ) / N N bits / unit of signal energy / trial
{ ( P D ) ( number of trials ) 1 or ( P D ) > 0.9.
N N E 0 / 0.25 E 1 - 2 photoelectrons , E 0 E 1 ,

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