Methods are presented for locating threshold points by using laser pulse analysis. It was found that there are errors involved in the determination of each of these quantities, and an attempt was made to separate their effects on the overall range correction. Several series of corrected range measurements for fixed reflectors and satellites were obtained. Residuals were computed by fitting the range measurements to either fixed-reflector distances or short arcs of satellite orbits. Root mean square values of these residuals are presented

Eldred, R. J.

Lehr, C. G.

Mendes, G. M.

Tsiang, C. R. H.

English

NASA Technical Reports Server

LASER PULSE ANALYSIS
C. G. Lehr, C. R. H. Tsiang, G. M. Mendes, and R. J. Eldred
-
t------
To be presented at the First International Symposium
on Use of Artificial Satellites
for Geodesy and Geodynamics
Athens, 14-21 May 1973
N75-17657
(NASA-CR-1 3 6 7 5 9 ) LASER PULSE ANALYSIS N5 
(Smithsonian Astrophysical Observatory)
14 p HC $3.25 Unclas
-- _G3/36 11851
May 1973
Smithsonian Institution
Astrophysical Observatory
Cambridge, Massachusetts 02138
305-037
https://ntrs.nasa.gov/search.jsp?R=19750009585 2020-03-22T23:06:53+00:00Z
LASER PULSE ANALYSIS
C. G. Lehr, C. R. H. Tsiang, G. M. Mendes, and R. J. Eldred
1. INTRODUCTION
In a pulsed laser ranging system, the range error for a single-electron return is
equal to the standard deviation of a probability distribution having the shape of the
transmitted pulse. A typical Q-switched ruby laser transmits a pulse whose shape is
closely approximated by a normal-density distribution
(t) = -1(2T)-1/2 exp (-t 2 /2 r2) , (1)
where the center of the pulse is at t = 0 and the pulse's standard deviation is a. For
a typical pulse, the value of T, the time between half-intensity points, is about 20 ns,
and the corresponding a value is T/2(21n 2)1/2 or a 8. 5 ns. When a single photo-
electron is received, the range error is this a value times half the velocity of light,
0. 15 m/ns, or 1. 3 m. If, instead of a single electron, the return contains N electrons,
the returned pulse gives N independent measurements of the range. The average of
these measurements has a range error that is smaller than the single-electron meas-
urement by a factor of N1/2. Hence, it can be seen that the range error of a 20-ns laser
can be less than 0. 5 m when there are 7 or more electrons in the return and less than
0. 1 m when the number of electrons is 163 or more.
It is seldom practical to resolve each electron in the return, determine its respec-
tive range, and compute the average. Instead, the digital range counter stops at a point
on the pulse's leading edge, this point being determined by the threshold-voltage setting
of the input discriminator of the counter's STOP circuit. A series of incremental
ranges referenced from this threshold point are then measured to various points on the
pulse, and these points are referred to the center of the pulse. If there is a significant
305-037
,-.je-to-pulse variation in the shape or size of the transmitted pulse, a similar cor-
rection for the time between the START threshold and the transmitted pulse center can
be obtained by the same procedure.
At the observing stations of the Smithsonian Astrophysical Observatory (SAO),
oscilloscopic presentations of the transmitted and received pulses have been photo-
graphed in a study of the laser system's ranging accuracy. Figure 1 is a block
diagram of the range-measuring equipment. When the counter's START and STOP
thresholds are reached, the sweep circuits of oscilloscope channels 1 and 2, respectively,
are triggered, and the transmitted and received pulses are displayed and photographed.
Figure 2 shows typical transmitted and received pulses on a time axis, with the
amplitude of the transmitted pulse greatly attenuated. The counter measures the time
T c between the threshold points TTT and RT. Since the corrected range is the time
T R between the pulse centers TTC and TRC, a correction A must be added to TC
where
A= (TRC-TRT) - (T -TC TT) (2)
This correction is known once the threshold points and pulse centers have been deter-
mined. Figure 3 shows a typical pair of transmitted and received pulses as displayed
on an oscilloscope photograph, measured, and plotted by a digital computer.
This paper is concerned with methods for locating the points TTT, TTC, TRT,
and TRC on presentations like those of Figure 3. There are errors involved in the
determination of each these quantities, and an attempt is made to separate their effects
on the overall range correction. Several series of corrected range measurements for
fixed reflectors and satellites were obtained. Residuals were computed by fitting the
range measurements to either fixed-reflector distances or short arcs of satellite
orbits. Rms values of these residuals are presented but not enough data have yet been
analyzed to be sure that the results are either typical or optimum.
2
2. THE EXPERIMENTAL DATA
The oscilloscope photographs of the transmitted and received pulses are obtained
with a Tektronix model 454 oscilloscope, whose bandwidth is 150 MHz. The pulses
were originally photographed on Polaroid film; later, 35-mm cameras were used. At
first, the x-axis sweep was 50 ns, but this was later changed to 100 ns to avoid the
truncation of the leading and trailing edges of the pulses. The Polaroid films were
measured by hand at 50 points uniformly spaced along the x axis. The 35-mm films
were measured at 100 time points by projecting the film on a screen and using a meas-
uring machine. The measurements were punched on cards, read into a digital computer,
and stored on magnetic tape. Two sets of the data used in this report were obtained
from only 30 to 40 measured points per pulse. These were converted to 100 data points
by constructing a complete Fourier series of 15 harmonics. Tests showed that data so
obtained are nearly identical to those obtained by the other method. The data for each
frame are stored on magnetic tape as 200 amplitudes for 100 equally spaced time points
on the transmitted and received pulses. For the 50-ns sweep, the points were 0. 5 ns
apart, whereas for the 100-ns sweep, they were 1 ns apart.
3. ANALYSIS OF THE EXPERIMENTAL DATA
In computing the range correction A, the transmitted and received pulses must be
located in time with respect to the starting and the stopping of the range counter.
These locations must be determined to within a small fraction of the pulse durations.
Hence, some point on the received pulse must be identified with a similar point on the
transmitted pulse. The nearly normal shape of the transmitted pulse makes its "center"
an obvious choice for this point. However, as will be shown below, the corresponding
center of the irregular received pulse can be specified in several different ways. It is
also possible to align the 100-point sets of the transmitted and received pulses by using
a correlation technique that does not require the pulse center to be identified.
One way to find the center of the pulse is to compute its centroid. If the pulse
shape defines a density distribution on the time axis, then the centroid is the mean
The fixed-reflector returns at the station in Arizona and the BE-C satellite returns
at the station in Peru.
3
value of this distribution. If RP(t) is a continuous function representing the received
pulse and t is its centroid, then the probability density distribution is
f(t) = RP(t)/f / RP(t) dt
00
and
t, = tf(t) dt (3)
-00
When the integral is evaluated on the computer, summation, rather than integration,
is used.
The centroid determination does not explicitly use the fact that the transmitted
pulse and the probability-density function producing the received pulse are symmetric
about their mean values. To find this symmetric center for an irregularly shaped
received pulse, we set its time origin (t' = 0) at successive points t, t 2, ... , ti,...,t 100
For each of these origins, we write f(t) as a sum of an even and an odd function, viz,
f(t') = fe(t') + f 0 (t') , (4)
where fe(t') and f 0 (t') can be determined as follows:
fe(t') = [f(t') + f(-t')] (5a)
f0 (t') = I [f(t') - f(-t')] (5b)
The symmetric center, ts, is defined as that value of t i that minimizes the odd func-
tion in the sense that
Here we are neglecting the distortion that is due to the nonsymmetric configurations
of retroreflectors on satellites currently in orbit. Since the amount of this distortion
varies as the satellite transits, it cannot readily be taken into account by the methods
we are now using.
4
f If0 (t')I dt' = min (6)
0
In practice, the t. values are taken at 0. 5- or 1-ns intervals (depending on whether an
1
oscilloscope sweep of 50 ns or 100 ns is used), and a linear interpolation near the
minimum is used to obtain t s precisely.
Another method uses the symmetry of the received pulse by finding the point th
that divides the area under RP(t) into two equal halves.
The correlation method does not give a center of the received pulse, but instead
determines the time TCL that the received pulse must be shifted back toward the trans-
mitted pulse to achieve the best correlation between the two pulses. Since
TCL = TRC - TTC and it is this difference in pulse centers, rather than their individual
values, that is used in determining the range correction A, the method is applicable
to the purpose at hand. The value of TCL is determined as that value that maximizes
the cross-correlation function, i. e., the value that gives
00
STP(t) - RP(t + TCL ) = max (7)
-cO
In practice, the integration is replaced by summation, and TCL is interpolated between
two discrete values.
Once TRC and TTC or their difference has been determined, it remains to deter-
mine either the threshold values TTT and TRT, or TTT - TRT. Since the oscilloscope
sweeps are started by trigger pulses from the counter's START and STOP circuits,
TTT - TRT is constant to within the jitter of the electronic circuits involved. If this
constant has been included in the calibration of the system's electronic delay, the
range correction becomes TRC 
- TTC'
If the jitter is appreciable, it can be eliminated from the range correction by
determining TTT and TRT from the corresponding threshold voltages VT and VR .
5
Although it is difficult to measure these voltages directly because they are somewhat
dependent on the slope of the pulse's leading edge, they can be obtained experimentally
by ranging to a reflector in a fixed position. The method, which works best when
there are large pulse-to-pulse variations in the shape of the leading edge, is the
following: a series of threshold voltages are assumed, and the corresponding threshold
times are determined from the photographs for each pulse in the series. Then the
sample mean and the sample variance of the threshold time are computed for each
assumed voltage. The threshold corresponding to the smallest sample variance cor-
responds to the actual threshold. Without jitter, the threshold should occur at a fixed
position on the x axis of the oscilloscope and the variance should be zero. Since there
is jitter, the sample variance will not be zero at the actual threshold voltage, but equal
to the rms value of the jitter between the threshold activation of the counter and the
start of the sweep.
When the values of VT and VR are precisely known, TTT and TRT can be deter-
mined from the oscilloscope photographs, and the jitter between counter and oscilloscope
does not enter into the measurement of the range correction. There is a method that
can be used to determine VT and VR without assuming that TTT and TRT have fixed
positions on the oscilloscope photographs. This method uses range measurements
made to a reflector in a fixed position. A series of threshold voltages are again
assumed for the transmitted and received pulses. For the measurements correspond-
ing to each pair of assumed thresholds, the range correction A is calculated and added
to the counter reading. Then, the sample mean and the sample variance are calculated
for all the fixed-reflector measurements. Because this range is fixed, the variance
is a minimum when the assumed thresholds are equal to the actual thresholds. This
minimum variance is a measure of the range-correction error for the received signal
strength (in photoelectrons) that was used for the experiment.
4. RESULTS
Tables 1 and 2 summarize initial results obtained from range measurements made
in accordance with the methods described in previous sections. These results are
In practice, the transmitter threshold voltage is determined with sufficient accuracy
by the previously described procedure, and only variations in the receiver threshold
are assumed.
6
illustrative of the procedures involved. They show that the corrections made from the
oscilloscope photographs do improve the accuracy of the counter readings. The results
are consistent with those expected from the equipment in current use at the SAO observ-
ing stations, but they are not representative of the ultimate accuracy to be expected
eventually with improved equipment and procedures. It can be seen from Table 1 that
l, the jitter between counter and oscilloscope, is about 1 ns, with most of this error
associated with the received pulse. Although the counter's threshold discrimination
and trigger-generating circuits are identical in both START and STOP channels, they
respond differently to the smooth leading edge of the transmitted pulse than they do to
the irregular leading edge of the received pulse. Hence, it appears reasonable that
the jitter associated with the received pulse is larger than that associated with the
transmitted pulse.
The range error o1 applies to range corrections obtained by setting A = TRC - TTC
These corrections are based on the assumption that TTT - TRT = const. Hence, 01
includes the effect of the synchronization jitter. Table 1 shows that al is about 1. 5 ns.
The quantity a2 applies to measurements made by using the correction
A = (TRC - TRT ) - (TTC - TTT), with VT and VR selected to minimize U2. The U2
values in Table 1 are about 1 ns, a value that appears to be reasonable for a return
pulse of 500 electrons. For a return of this strength, we would expect an error of
8. 5/5001/2 - 0.4 ns in determining the center of the pulse and an additional error of
1/121/2 - 0. 3 ns for a counter with a uniform probability density over its 1-ns resolu-
tion interval.
Table 2 shows that the residuals of three satellite transits were reduced by apply-
ing the corrections obtained from oscilloscope photographs. Range corrections for
several other satellite transits were also computed, but improvements in the short-
arc residuals could not be realized because in some cases the variance of the uncorrected
range residuals was considerably larger than that of the range corrections. In one case,
the average number of electrons per pulse was too small to make an effective determina-
tion of the center of the received pulse.
The satellite BE-C shows the greater decrease in residual (0. 7 to 0.4 m), a result
that might be expected because the retroreflectors on this satellite are arranged in such
7
a way that they produce less of a random variation in range with changes in the satel-
lite's aspect angle than do those on Geos 1. For the measured pulse width of the
BE-C returns, the corresponding a of a normal pulse is about 8. 5 ns; hence, the error
in determining the center of a 38-electron pulse is 8.5/381/2 = 1. 4 ns (0. 2 m). This
error combined with the 1. 7-ns synchronization jitter in Peru gives
(1.42 + 1. 72)1/2 = 2. 2 ns (0. 33 m), a value comparable to the 0. 39 m shown in
Table 2.
In computing the short-are residuals shown in Table 2, a 4- to 5-day orbit was
first obtained by using laser data from all stations. Then, best-fitting short arcs
for the given pass were obtained by varying inclination, eccentricity, mean anomaly,
and mean motion. All range observations used in the orbit computations were corrected
for the nonrandom displacements between pulse center and the center of mass of the
satellite. These corrections, which depend on the angle between the laser beam and
the axis of the satellite's retroreflector, were computed by Arnold (1972).
5. ACKNOWLEDGMENTS
The authors acknowledge with appreciation the computer programing done by
Mrs. Diane M. Hills, the assistance of Mr. Frank Keresztes-Fischer in measuring
films, helpful discussions with other SAO staff members, and the support of NASA
under grant NGR 09-015-002.
6. REFERENCE
Arnold, D. A.
1972. Calculation of retroreflector array transfer functions. Final Tech. Rep.
NASA grant NGR 09-015-196, 57 pp., December.
8
Table 1. Summary of range measurements to fixed reflectors.
Average Measured pulse Synchronization jitter
Oscilloscope number of durations Range error
sweep electrons Transmitted Received
duration Number in received Transmitted Received pulse pulse c 1 2
Station (ns) of pulses pulse (ns) (ns) (ns) (ns) (ns) (ns) (ns)
Brazil 50 17 560 22 26 0.1 1. 3 2.1 2.1 (C T) 0.7 (CL)
Peru 50 18 463 26 32 0.2 1.3 1.7 1.5 (CT) 1.1 (CT)
Arizona 100 20 416 21 26 0.3 0.9 1.7 1.3 (S) 1.3 (CL)
The a's are sample standard deviations of the series of pulses: ac represents the uncorrected counter
readings, and a1 and a2 represent corrected counter readings. For G1, the counter and oscilloscope were
assumed to be in exact synchronism. For r2, the receiver threshold was determined by using the fact that
the range is constant. The o1 and u2 values given above are the lowest for the four methods of pulse-center
determination: centroid (CT), symmetry (S), half-area (A), or correlation (CL).
Table 2. Summary of satellite-range corrections. Method used for pulse-center determination:
centroid (CT), symmetry (S).
Average Measured pulse Short-arc residuals
Oscilloscope number of widths
sweep electrons Uncorrected Corrected
Station duration Number of in received Transmitted Received observations observations
Date location Satellite (ns) observations pulse (ns) (ns) (m) (m)
7 June 1972 Brazil BE-C 50 9 38 24 22 0.73 0.45 (CT)
17 June 1972 Brazil Geos 1 50 35 85 20 21 0.71 0.57 (S)
17 March 1973 Peru BE-C 100 19 276 23 19 0. 74 0. 39 (CT)
305-037
COUNTER
(I-ns resolution)
START STOP
TRIGGERS
[ I.
OSCILLOSCOPE
(150-MHz bandwidth)
SAMPLE OFTRANSMITTED DIODE CHANNEL ITRANSMITTED DETECTOR
DETECTORPULSE CHANNEL 2
RECEIVED PHOTOMULTIPLIER VARIABLE VIDEONAO AM
PULSE (10 gain) (0 to 50 dB) (31 dB)
Figure 1. Block diagram for pulse detection.
305-037
TC
TR
STRANSMITTED PULSE RECEIVED PULSE
z
V VT R
TTT TTC TRT TRC TIME
Figure 2. Diagram for the range correction A, where TR = TC + A and A= (TRC - TRT) (TTC - TTT)
'b "
t L
IMI to
C><
O,
I 
WO
0 x
x x
x x,
<4.4<
,ii )<
Figure 3. Digitized representation of the photographed pulses: * = transmitted pulse; x = received pulse. Sweep
duration= 100 ns.


Laser pulse analysis

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