PROCEEDINGS OF THE IEEE, VOL. 5 8 , NO. 10, OCTOBER 1970

lo3 ,

0’ IO2 i I a P

c n - E - a W E IO’

I 00 io3 IO’ IO7 lo9

INFORh$ATION RATE, BITSIS

Fig. 2. Optimum value of major system parameters for optimization examples.

SUhmARY The COPTRAN optical communication system optimiza-

tion program has been used by NASA Electronics Research Center for more than two years. It enables rapid design,

1741

evaluation, and comparison of communication systems and acts as a convenient reservoir for system design data.

REFERENCES “Parametric analysis of microwave and laser systems for communi- cation and tracking,” Hughes Aircraft Company, Culver City, Calif., Rept. P67-09, December 1966. L. S. Stokes, K. L. Brinkman, W. K. Pratt, and J. W. Weber, “Study and development of a mathematical analysis for the performance assessment of space communication systems parameters,” Hughes Aircraft Company, Culver City, Calif., Rept. P68-267, 1968. W. K. Pratt, Laser Communication Systems. New York: Wiley, 1968. R. S. Schechter, The Variational Method in Engineering. New York:

M. J. Forray, Variational Calculus in Science and Engineering. New York: McGraw-Hill, 1968. W. K. Pratt and R. J. Norton, “An experimental optical polarization modulation communication system,” Proc. 1966 IEEE Internatl. Conference on Communications, pp. 261269. W. K. Pratt, “Binary detection in an optical polarization modulation communication channel,” IEEE Trans. Commun. Technol., vol. COM-14, pp. 664-665, October 1966. J. I. Marcum, “A statistical theory of target detection by pulsed radar,” IRE Tram. Inform. Theory, vol. IT-6, pp. 159-160, April 1960. M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. W. K. Pratt, “Partial differentials of Marcum’s Q function,” h o c . IEEE (Letters), vol. 56, pp. 122Ck1221, July 1968.

McGnw-Hill, 1967.

Error Analysis of Optical Range Measurement Systems

HOWARD C . SALWEN, MEMBER, IEEE

Abstract-The ranging error of short pulse optical tracking sys- tems is investigated. The relationships between the tracker character- istics, the characteristics of the received signal and background radiation, and the maasurement accuracy are determined. It is found that the effects of characteriatics of the tracker, the propawing medium, and/or the target to be tracked will predominate in the derived accuracy expreasions. This result is baaed on the awumption that the transmitted puke duration is short compared to the delay fluctuations induced by at leaat one of these three effects.

INTRODUCTION DVANCES in laser implementation techniques have

stimulated interest in high-precision optical range measurement systems. These implementation tech-

Manuscript received August 15, 1969; revised June 5, 1970. This work was supported by Electronic Research Center, NASA, Cambridge, Mass., under Contract NAS 12-2058.

The author was with ADCOM, Teledyne Company, Cambridge, Mass. He is now with Signatron, Inc., Lexington, Mass. 02173.

niques provide subnanosecond pulses at moderate effective radiated power levels, thereby satisfying the prerequisite for ultra-high accuracy range measurement. The usefulness of high-accuracy range measurement systems is not limited to radar and navigation applications. Recent develop ments in the area of optical communications [ 1 ] have shown that high-speed digital ‘modems specifically optimized for the optical communication channel require high-accuracy range tracking capabilities to provide adequate bit syn- chronization. This paper presents an analysis of error for range measurement techniques whch are applicable to optical radar, navigation, and communication systems.

The range measurement error analysis employed in the following discussion parallels that typically employed for microwave pulsed radar systems [2]. The goal of the anal- ysis is to determine the ranging accuracy achievable by pulsed optical systems using state-of-the-art tracking im-

1142 PROCEEDINGS OF THE IEEE, OCTOBER 1970

plementations. In particular, the tracking technique as- sumed for the analysis consists of a phase-locked loop with a split-gate timing error detector. The analysis determines the relationships between the split-gate parameters, the tracking loop parameters, the characteristics of the re- ceived signal and background radiation, and the measure- ment accuracy.

The discussion begins with a brief description of a split- gate tracking loop which is suitable for optical range mea- surement systems. A simplikd analysis of error is per- formed in which the effect of observed photoelectrons due to background radiation is ignored. A rectangularly dis- tributed pulse-to-pulse time jitter is assumed for the purpose of this analysis. The results obtained are then expanded to include the effect of background radiation and to consider other reasonable jitter probability distributions.

SPLIT-GATE TRACKER For the purposes of analysis, the received optical signal

is assumed to be a pulse train consisting of subnanosecond pulses repeating with a relatively stable pulse repetition fre- quency r . Such waveforms are conveniently generated us- ing the mode-lock laser implementation and conveniently tracked using a split-gate tracker.

Details of the split-gate implementation can be altered to take advantage of the quantized nature of the receiver’s photodetector output. Consider, for example, the case in which the observed photoelectron rate at the output of the photodetector is much smaller than the transmitted pulse repetition frequency, i.e., most of the received pulses are not observable. Fig. 1 shows a split-gate design which pro- duces a digital phase error output for this case. In this de- vice, the loop error signal is generated by counting and taking the difference of the number of received photoelec- trons which arrive in each half of the gate during the digital loop sampling interval z,, z, >> l/r. The difference is found by sensing the state of the integrator (sample-and-hold) at the end of each gate interval and making a tertiary decision. The contents of the counter are read out and the counter is reset at the sampling rate l / r s . This rate is related to the loop tracking bandwidth which, in turn, is determined by con- straints imposed by noise and target dynamics. The con- tents of the counter are filtered in the compensation net- work and used to advance or retard the phase of the voltage- controlled oscillator (VCO) output so that the split gate is aligned with the center of gravity of the returned pulses, on the average.

SIMPLIFIED ANALYSIS The analysis of the split-gate operation is somewhat

simplified if the effect of background radiation is excluded from the calculations. This effect will be added later.

The average number signal photoelectrons observed in the gate in one sampling interval is defined to be n,z, where n, is the mean signal photoelectron rate. The probability that k photoelectrons are observed in rS seconds is given by

provided that the gate is properly aligned. The number of photoelectrons in the early and late halves of the gate, k , and k - , are also Poisson distributed with means nsz$2. The counter forms the difference k , and k - which clearly has zero mean when the gate is properly aligned. Proper align- ment implies that all received photoelectrons are observable within the duration of the gate. Therefore, the gate should be sufficiently wide such that the instantaneous relative timedelay difference betweewthe received pulse train and the gate trigger pulse train never exceeds half the gate width. The time-delay difference constitutes the system measure- ment error and results from a number of factors. These in- clude jitter induced by fluctuations in signal and back- ground received radiation, instability in the pulse repetition frequency of the transmitted signal, and the trigger pulse train and dynamic lag of the tracking loop. To some extent, time-delay fluctuations may be unavoidably induced in tracking systems by characteristic of the target. Such will be the case in optical altimetry systems in which the range to an extended target is measured.

Since the systems under consideration employ extremely short pulses, it is reasonable to assume that the delay fluctua- tions induced by gate jitter, propagation anomalies, and other effects which are normally of secondary importance in comparable microwave systems will predominate in the optical case. This situation is depicted in Fig. 2 where a rectangular distribution is assumed for the relative delay fluctuations. The gate width shown in Fig. 2 is chosen exactly equal to the width of the delay fluctuation distri- bution.

The joint probability density of the number of photo- electrons in the early and late halves of the gate k + and k - is simply the product of their probabilities because these events are independent. Thus

and the expected value of the difference is clearly zero. With mean equal to zero, the variance of the difference is com- puted to be

E[(k+ -IC-)’] = ( k t - 2 k + k - + k?)p(k+)p(k- ) m o D

k + = O k - = O

= nsTs. (3)

Therefore, the contents of the counter has nns fluctuation, on=&. These random counter fluctuations will cause random fluctuations of the gate position AT^^^ which con- stitute ranging measurement noise. Thus the effect of counter fluctuations on on the gate position must be de- termined. It is simpler, conceptually, to derive this relation- ship by determining the gate sensitivity, i.e., the steady- state error signal (counter contents) which would be pro- duced given that the gate is displaced by a small amount AT. Then substituting the random counter fluctuations due

S A L W N : ERROR ANALYSIS OF OPTICAL RANGE MEASUREMENT 1143

Photodetector

Interval

Fig. 1. Modified split-gate design.

I i Probobtltty

Photom Recelved Denslty of S~gnol Durtng rI (Not Photon Delay Sbrnultoneourly) i

Waveform

Fig. 2. Photons counted during 5,.

to the statistical nature of the signal for the steady-state error component in the sensitivity relationship, the resulting delay fluctuations AT- can be determined.

The sensitivity of the gate to a displacement A7 is found as follows : Assume the gate is moved A7 to the left in Fig. 2. The mean value of k , is still

n, 7 s 1 7

7 9 2 E(k+/A7) = - = n g , - 2.

2 (4)

The mean value of k - is reduced because part of the late gate is outside the region in which photoelectron returns are expected, i.e.,

E(k- /Az ) = ns7s - - A7 .

Thus the expected value of the contents of the counter k , - k - given a displacement AT, is nsT,AT/Tg. Combining this with (3), it is found that the rms fluctuations of the con- tent of the counter on will be interpreted as two-way range error with rms value given by

5 l ( 1 ) ( 5 )

Arm = - cA7rms - --. 2J;I,t,

(6)

There are several assumptions implicit in this result, namely : 1 ) effects of background and dark curreware ignored ; 2) sig- nal photoelectron delay fluctuations are rectangularly dis- tributed with width T , ; 3) the gate width T~ is chosen equal to 7,.

EFFECT OF BACKGROUND RADIATION

The effect of background radiation is easily added to the result, (6), using the same calculation technique. Assume a background photoelectron rate nb and a pulse rate r. The average number in each gate is nbtg and the average total number per sample interval is 17tbTgTs. Again the number of

background photoelectrons is Poisson-distributed and the mean in the early and late halves of the gate is mbTgTJ2 regardless of gate displacement. Denoting the number of background photoelectrons in the early and late halves of the gate by I , and I - , respectively, the variance of the con- tents of the counter is

E[(k+ + 1, - k - - 1 - ) ' ] = (n, + r%bTg)ts. (7)

Thus the rms range fluctuation including the effect of back- ground are

When independent range measurements are desired every 7 i seconds where zi = m~,, m = 1,2, . * , the results of rn mea- surements can be averaged in T~ seconds yielding an error

Typically, the loop integration time, or equivalently, the independent sample rate, is expressed in terms of the loop noise bandwidth B,. In particular, the integration time T~ is approximately inversely proportional to the loop noise bandwidth.

Before further generalizhg of the range error expression, (9), it is of interest to examine the behavior of the e m r as a function of the number of signal and background photons per decision, i.e., N,= nszi and Nb=rnbTgTi. Fig. 3 shows a plot of the function , / m J N , versus Nb with N, as a parameter.

FURTHER GENERALIZATION The analysis leading to (9) assumes that the gate duration

zg and the width of the delay distribution T , are equal. For practical reasons such as acquisition considerations, it is often desirable to make the gate width wider than the ex- pected width of the delay distribution. This case is analyzed next.

The number of signal and background photoelectrons have the same statistics as described previously. However, the gate sensitivity calculation must be altered. In particular, the count k + will not be independent of displacement AT.

1744 PROCEEDR‘IGS OF THE IEEE, OCTOBER 1970

1 I , I l l I I I I , I l l ,

I 10 100 0

Nb

Fig. 3. Laser tracker accuracy factor and gate width criterion.

Thus (4) is replaced by age basis. For example, T~ can be chosen such that the criterion is maintained as long as the gate displacement is

E(k+/A.r) = 5 T,(? + AT) (10) within its la value. That is T W

and (5 ) is replaced by

before, the rms- ranging error is found to be

At first glance the accuracy predicted by (12) appears to be better by a factor of 2 than that predicted by (9). But it should be remembered that T ~ > T , in (12). An exact com- parison can be made under conditions of small error. Namely, the ratio of (9) over (12) constitutes an improve- ment given by

Equation (13) shows that in the absence of background radiation the wider gate tracker does provide double the accuracy. It is important to note that when zg is just slightly larger than T,, (13) does not apply. It only applies when AT <(T,- ~,)/2, i.e., all the signal photoelectron expected arrival times fall within the gate. It is thus concluded that zg should be made as small as possible while still maintain- ing this criterion. Since the gate timing error AT is a random variable, this criterion can only be established on an aver-

A plot of the ratio T J T , which meets the la criterion is simply added to Fig. 3. Since gate widths on the order of 20, are sufficiently wide to satisfy the la criterion men- tioned previously under all conditions shown in Fig. 3, it is clear from (13) that improvement factors greater than

are achievable when Nb> N, and that improvement factors approaching 2 are achievable in some cases.

GAUSSIAN JITTER DISTRIBUTION Up to t.his point, the delay distribution density has been

assumed to be rectangular. Of course, other possibilities exist and predominant among these is the Gaussian dis- tribution. For the purposes of analysis a raised cosine given by

is assumed to approximate the Gaussian probability weight- ing. This approximation is employed to simplify the com- putation without siccant loss of generality. The calcula- tion to determine the ranging accuracy of the tracker, as-

SALWEN: ERROR ANALYSIS OF OPTICAL RANGE MEASUREMENT 1145

suming the approximately Gaussian distribution of delays is carried out in the same manner as that of the rectangular distribution. The main difference, in terms of results, is in the sensitivity calculation. Assuming a gate width 7g < n/a, a displacement A7 produces a net output of the integrator (counter) given by

eint(AT) = - - 2ans7s[ - s’ cos2 at dt n - ( r g / 2 ) + A r

- 2JOAr cos2 at dt + cos2 at dt 1 = - [sin 2 a A ~ ( c o s a z g - l)]. ns 7s

n

ing accuracy is thus

../-. (20) Arms = 4a(l - cos a7g)ns7,

The optimum gate width is a function of the signal-to- background ratio Nb/N,. When the background is small 7g = n/a is best. As this ratio is increased, the optimum gate width is reduced. Digital computation of (20) with various values of the ratio NdN, have been performed and it is found that when N,,/N,I 1, zg= n/a is optimum and when 1 < Nb/N, I 3000, zg =0.9 n/a is roughly optimum. (Calcula- tions with Nb/N, > 3000 were not undertaken.)

For small A7 this is approximated by CONCLUSIONS

eint(A7) zz - 2aAz(cos ~7~ - 1). ns7s The analysis presented in the preceding sections has at- II tempted to determine expressions for range error which

This is, as before, set equal to the rms integrator fluctuations due to the random nature of the signal and background, which is given by a modified form of (7). The modification is required because the signal photoelectron mean rate n, is a function of the gate width in the system presently under consideration. As the gate is narrowed, the number of signal photoelectrons counted in the interval 7s is given by

could realistically be expected of optical tracking systems. It is noted that pulsed radar range accuracy expressions typically show a direct proportionality between range error and transmitted pulse duration, while the results obtained for optical tracking systems do not involve the transmitted pulsewidth as a parameter. These latter results are based on the assumption that the transmitted pulse duration is small relative to the width of instrument and channel induced de- lay fluctuations.

ns(7g)7s = REFERENCES

ns7, = - [ ~ 7 ~ + sin UT,] (19) [l] S. Karp and R. M. Gagliardi, “M-ary Poisson detection and optical

[2] D. K. Barton, Radar Systems Analysis, Englewood Cliffs, N. J . : n communications,” NASA Tech. Note D 4 2 3 .

which approaches nS7, as zg approaches n/a. The rms rang- Prentice-Hall, 1964.