The lunar laser communication demonstration time-offlight measurement system: overview, on-orbit performance, and ranging analysis

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1 The lunar laser communication demonstration time-olight measurement system: overview, on-orbit perormance, and ranging analysis The MIT Faculty has made this article openly available. Please share how this access beneits you. Your story matters. Citation As Published Publisher Stevens, M. L. et al. The Lunar Laser Communication Demonstration Time-o-Flight Measurement System: Overview, on-orbit Perormance, and Ranging Analysis. Free-Space Laser Communication and Atmospheric Propagation XXVIII, February 216, San Francisco, Caliornia, USA, edited by Hamid Hemmati and Don M. Boroson, SPIE, March Society o Photo-Optical Instrumentation Engineers SPIE Version Final published version Accessed Wed Jan 16 4:43:18 EST 219 Citable Link Terms o Use Detailed Terms Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please reer to the publisher's site or terms o use.

2 PROCEEDINGS OF SPIE SPIEDigitalLibrary.org/conerence-proceedings-o-spie The lunar laser communication demonstration time-o-light measurement system: overview, onorbit perormance, and ranging analysis M. L. Stevens, R. R. Parenti, M. M. Willis, J. A. Greco, F. I. Khatri, et al. M. L. Stevens, R. R. Parenti, M. M. Willis, J. A. Greco, F. I. Khatri, B. S. Robinson, D. M. Boroson, "The lunar laser communication demonstration time-o-light measurement system: overview, on-orbit perormance, and ranging analysis," Proc. SPIE 9739, Free-Space Laser Communication and Atmospheric Propagation XXVIII, (15 March 216); doi: / Event: SPIE LASE, 216, San Francisco, Caliornia, United States Downloaded From: on 3/16/218 Terms o Use:

3 The Lunar Laser Communication Demonstration time-o-light measurement system: overview, on-orbit perormance and ranging analysis * M. L. Stevens*, R. R. Parenti, M. M. Willis, J. A. Greco, F. I. Khatri, B. S. Robinson, D. M. Boroson Massachusetts Institute o Technology, Lincoln Laboratory 244 Wood Street, Lexington, MA, USA 2124 ABSTRACT The Lunar Laser Communication Demonstration (LLCD) lown on the Lunar Atmosphere and Dust Environment Explorer (LADEE) satellite achieved record uplink and downlink communication data rates between a satellite orbiting the Moon and an Earth-based ground terminal. In addition, the high-speed signals o the communication system were used to accurately measure the round-trip time-o-light () o signals sent to the Moon and back to the Earth. The measured data, sampled at a 2-kS/s rate, and converted to distance, was processed to show a Gaussian white noise loor typically less than 1 cm RMS. This resulted in a precision or relative distance measurements more than two orders-o-magnitude iner than the RF-based navigation and ranging systems used during the LADEE mission. This paper presents an overview o the LLCD system, a summary o the on-orbit measurements, and an analysis o the accuracy o the measured data or the mission. Keywords: time-o-light, deep-space navigation, ree-space laser communication 1. INTRODUCTION Deep-space navigation currently is based on microwave radio measurements utilizing tracking stations located throughout the world 1. One o the key measurements is the round-trip time-o-light () o signals sent rom the ground station up to the spacecrat and back. These navigation systems are generally designed to use specialized waveorms which require the exclusive use o the satellite RF communication system, supplanting the normal command and telemetry data streams or observation times that oten run continuously or hours. In addition to navigation, time-o-light measurements are important or planetary science. Gravitational anomalies o planets and moons can be measured by the small perturbations they cause in the position o orbiting spacecrat. The gravitational anomalies can yield important inormation about the interior o planets and moons 2. There is a desire to reduce the workload on limited ground terminal resources or the navigation measurements 3. NASA is currently working on new concepts, including one which can take advantage o the growing number o interconnected satellites in space. This concept allows or autonomous measurements o the spacecrat state vectors utilizing navigation packets sent between satellites linked in a communication network architecture 4. This paper proposes an alternate novel concept or time-o-light measurements that was demonstrated on the Lunar Laser Communication Demonstration (LLCD) 5 carried out over 15 days in October and November o 213. The LLCD time-o-light used no special waveorms but only the high-speed signals in the duplex communication system. Thus, the time-o-light data was captured whenever duplex laser communications were in operation. The Lunar Laser Communication Demonstration Space Terminal was carried on the Lunar Atmosphere and Dust Environment Explorer (LADEE) 6 satellite launched in September 213. The mission s primary goal was to study the ragile lunar atmosphere and look or the presence o electro-statically transported dust grains. The LADEE satellite was intended to be 1 days in orbit around the Moon. The spacecrat also had suicient capacity to carry the small experimental laser communication terminal o LLCD. *Corresponding author: stevens@ll.mit.edu This work is sponsored by National Aeronautics and Space Administration under Air Force Contract #FA C-2. Opinions, interpretations, recommendations and conclusions are those o the authors and are not necessarily endorsed by the United States Government. Free-Space Laser Communication and Atmospheric Propagation XXVIII, edited by Hamid Hemmati, Don M. Boroson, Proc. o SPIE Vol. 9739, SPIE CCC code: X/16/$18 doi: / Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

4 During the commissioning phase (the irst month in orbit) o the LADEE mission, the spacecrat was placed in a circular orbit about 25 km above the surace o the Moon, and LLCD was allocated about hal o the days in the month to exercise the lasercom links. During this period, LLCD demonstrated the longest ever two-way high-rate laser communications, and the irst between an Earth-based ground terminal and a satellite orbiting the Moon. A record 622 Mbps communication rate was achieved on the downlink rom the Moon to the Earth with a.5-w transmitter and 1- cm aperture at the Moon. The ground terminal receiver, located at White Sands, NM used our 4-cm receive telescopes iber-coupled into our arrays o our super-conducting nanowire single-photon detectors 7. The ground terminal uplink transmitter used our 15-cm telescopes each ed by a 1-W optical ampliier carrying signals at selectable rates o 1 or 2 Mbps 8. These uplink data rates were 5 times higher than the RF-based radios used on all previous missions to the Moon. The increase in data rates made possible by the two-way laser communications enabled real-time high-precision. During the demonstration, data was streamed to the Moon and back. Data types included multiple simultaneous highdeinition video streams, ile transers, commands to the optical terminal at the Moon, and downloads o the entire spacecrat buer in minutes that would have taken days with the RF telemetry link on the spacecrat. In addition to the communication demonstration, LLCD also measured the time-o-light o signals sent to the Moon and back utilizing only the high-speed data clocks implicit in the communication system. No special waveorms were used, so the data was recorded continuously whenever duplex laser communications were in operation. For simplicity, the same raming structure was designed into both the downlink and uplink signals. operation required only that the uplink and downlink clocks be phase-locked to each other, and the raming on the downlink signal be synchronized with the arrival o the rame markers on the received uplink signal at the satellite. 2. LLCD ARCHITECTURE A block diagram o the laser communication system clocks on board the satellite is shown in Figure 1 9, 1. Uplink and downlink data clocks were integer multiples and phase synchronous. A single master clock (VCO) at 5 GHz was phase locked to the received data on the uplink. This same clock was also used to ormat the data and the modulated waveorms on the downlink. Multiple data rates were selectable on both the uplink and downlink. A ixed duration raming structure was used to transmit the data composed o 1, 2, 4, 8, or 16 time-division-multiplexed sub-channels depending on the selected data rate or the downlink and 8 or 16 sub-channels or the uplink. Each sub-channel was composed o a rame alignment sequence and coded data. The rame duration or the downlink was ixed at 6736 cycles o MHz, approximately 195 µs. The uplink used the same rame structure with a ixed duration 32-times the length o the downlink rame, approximately 6.25 ms. The uplink and downlink raming at the satellite terminal were synchronized by a command sent rom the ground terminal. These synchronized clocks and raming provided the timing signals that were the basis o the LLCD measurements. PLL 5 GHz VCO Phase Detector Uplink Symbol Clock Gen From Uplink Receiver Up ink Demodulation Downlink Frame Gen 311 MHz Clock Y 16:1 Serializer FPGA Figure 1. Block diagram o the satellite communication system with measurement capability. To ---- Downlink Transmitter The uplink modulation ormat was 4-PPM with a ixed slot rate o MHz. Each uplink symbol consisted o our slots or the data and an additional 12- or 28-slot dead time where no signal was sent, depending on the selected inormation rates o either Mbps or 9.72 Mbps. (This allowed or a simple, single optical receiver ilter.) Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

5 Degrees The downlink modulation ormat was 16-PPM with a variable slot rate. At the highest downlink data rate o 622 Mbps, a GHz (16 x MHz) slot rate was used, the symbol clock being nominally equal to the slot clock on the uplink, but Doppler shited by the motion o the spacecrat with respect to the ground terminal. Inormation rates on the downlink were selectable rom 622, 311, 156, 78, and 39 Mbps. A block diagram o the ground terminal communication system clocks is shown in Figure A low-noise master clock oscillator at MHz was used to generate the modulation and raming or the uplink. On the downlink receiver side, a 5-GHz clock was phase locked to the received signal. This recovered downlink clock was divided by 16 to produce the symbol clock at nominally MHz, but shited rom the uplink master clock by the two-way Doppler shit resulting rom the relative motion o the spacecrat. Two sensors were built into the downlink receiver yielding so-called Fine and Coarse data. The Fine data sensor was a phase-requency detector (PFD) which measured the instantaneous phase dierence between the two MHz clocks. The PFD had a range o approximately ±3 degrees modulo 36 degrees (one cycle). The analog output o the phase detector was digitized with a 16-bit analog-to-digital converter (ADC) at 2 ks/s, yielding an ADClimited resolution o approximately.23 degrees-per-bit, corresponding to 63µm o round-trip distance. The Coarse data sensor consisted o a measurement o the time delay between the uplink rame header departures and the downlink rame header arrivals. Downlink rame headers were synchronized at the space terminal to the arrival o the uplink rame headers. This plus the phase-locked uplink and downlink clocks at the satellite resulted in a continuous transer o timing inormation over the two-way link at both Fine and Coarse scales. The Coarse delay was measured by clock cycles o the uplink MHz master clock. This measurement, providing coarse two-way data, was sampled at 16 S/s with a resolution o 1 cycle at MHz, equivalent to approximately.96 m. The maximum time delay measurable by the system was the duration o a downlink rame, 6736 cycles o MHz, or about 195 µs, resulting in a range ambiguity o approximately 58.5 km. This ambiguity was resolved during LLCD using ephemerides provided by the LADEE navigation team or LLCD terminal pointing and acquisition. (Future systems could incorporate numbered uplink rame headers or unique uplink data payload markers that could be looped back rom the uplink to the downlink at the satellite terminal, i eliminating the ambiguity is required.) Transmit FPGA Uplink Frame Gen 311 MHz Osc Source clock Frequency stability Expected < 8e-12 at 2.5 seconds Coarse Data Range (58.5 km, 16 S/s) Receive FPGA Frame Clock Compare Downlink Frame Sync 311 MHz Phase Compare 311 MHz Clock 1:16 De- Serializer Fine Data Resolution (63 um, 2 ks/s) From Receiver Detector Read-Out and Aggregation 35 Raw Phase Fine Data Wrapped Raw Data PLL 5-1 Hz BW 5 GHz VCO CM a Mob Samples 58.5 km Figure 2. Block diagram o the ground-based clock system with measurement capability. Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

6 Raw Sample The Coarse and Fine data provided LLCD with a ruler to measure round-trip with an equivalent length o 58.5 km and a smallest tick mark o 63µm. We note that the Fine data alone could be used to generate precision cumulative phase measurements which give rise to relative two-way or distance measurements. Absolute measurements are obtained by adding the oset bias obtained rom the Coarse data (ater resolving the ambiguity using ephemeris predictions and ater careully accounting or latencies in the ground and satellite terminals.) 3. DATA PROCESSING The Fine data sensor output was irst processed to deduce the relative (equivalent to dierential distance) by unwrapping the modulo-36-degree data, a sample o which is shown in Figure 3. Each sawtooth corresponds to one cycle o MHz. At the Doppler null (at closest approach o the satellite to the ground terminal) the phase change can be seen to slow and reverse direction. The range o the phase requency detector was about 6 degrees. Thus, when the phase reached +3 degrees it rolled over to -6 degrees, and when the phase reached -3 degrees it rolled over to +6 degrees. This resulted in a simple linear relationship between the detector output and phase. A cumulativephase time series was thus generated rom the raw ADC samples ater conversion to degrees. The cumulative phase was arbitrarily deined to start at and end with the total cumulative phase over the measurement interval, which we deined as the period o duplex communications with continuous phase locking o both the uplink and downlink receiver clocks (and which was typically a quarter up to the ull 2-25 minute pass deined by the LADEE orbit and depending on the number o communication system parameters chosen or test during the pass.) Artiacts caused by samples that ell in the rollover regions were smoothed by applying a simple straight line interpolation at every rollover in the unwrapped cumulative-phase time series. It was also known that the PFD exhibited a slight non-linearity in its measurements near the extremes o its range. Residual errors rom the smoothing and non-linearity thus led to some eed-through o the beat-requency dierence between the two clocks, but which we removed by iltering ater processing. (An example o the residuals with beat-requency noise beore and ater iltering is shown in Figure 7 in section 4.1). 2.5 x Time [s] Figure 3. The PFD measures the instantaneous phase dierence o the two MHz clocks. Each sawtooth is one cycle o MHz. The phase change slows and then reverses direction at the Doppler null where the satellite makes its closest approach. Figure 4 (a) shows this cumulative phase converted to one-way relative distance (red) and overlaid on the relative distance predictions rom the operational satellite ephemeris (blue). On this scale the total change in distance during this 1-minute measurement interval is seen to be about 9 km. An excellent it with the ephemeris (on such a distance scale) can be seen. We are next interested in the residual noise and variations in our measurements. Spacecrat navigation teams typically perorm this by creating multi-parameter orbital, gravity, and trajectory models and then perorm best its o these to Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

7 Residual Noise (ps, rms) 9 m the data. As this complex calculation was not part o the LLCD mission, we decided to approximate such results by subtracting a polynomial it (i.e. an approximation to an orbit) rom the measured data. Sample residuals are shown in Figure 4 (b). The noise deduced by such measurements (beore any iltering) were ound in this particular data set to be equivalent to 127 ps or 3.8 cm RMS in round-trip distance. We then perormed this approximate analysis on the data rom the entire mission (approximately 1 total passes). We calculated the standard deviation in each 1-second block (to minimize the eects o polynomial itting errors) and averaged over all such intervals to produce the values shown in Figure 5. Averaging over all mission data resulted in 44.3 ps or 1.3 cm (shown by the solid line) or the two-way. x o 14-2 E -3 l -Y Change in Distance m (a) O Data taken 2 Nov :1:52 UTC Ephemeris Measured - (b) Residual Noise Ater Removing Polynomial Fit o s -5 rn -6 i cm rms Elapsed Time [s] Figure 4. Overlay o measured cumulative phase converted to one-way distance and ephemeris prediction Measurement Interval Figure 5. Standard deviation o noise in two-way measurements or the mission. Average or the mission (Red) was 44.3 ps, equivalent to a two-way distance error o 1.3 cm. Absolute distance measurements were extracted by adding the absolute bias oset rom the Coarse data to the Fine data time series. We calculated the absolute oset by irst down sampling and interpolating the Fine data to the same sampling times as the Coarse data. We then computed the mean dierence, over a measurement interval, between the Coarse data (including corrections or the terminal latencies and resolution o the Coarse data ambiguity) and the unwrapped interpolated Fine data. Taking the mean o the dierence o many Coarse-Fine data pairs allowed us to Additional processing was perormed to remove artiacts caused by a GPS receiver that was used as the reerence clock or the ground terminal. Details are provided in Appendix A. Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

8 achieve high precision absolute osets. The quantization noise in the Coarse data has the variance o a uniorm distribution over one cycle o MHz (σ 2 =.768 m 2 ). This variance is reduced, though, by the reciprocal o the number o Coarse samples in the measurement (at 16 samples per second.) The expected error in the absolute oset or a typical measurement interval o 1 1 seconds was thus shown to be mm RMS. We then added this absolute oset to the original Fine data time series to produce an absolute two-way. (We note that only a single absolute oset value is required or any period where the Fine data is continuous.) 4. COMPARISONS WITH LADEE EPHEMERIS Finding a truth against which to compare an absolute range measurement is diicult. We used the LADEE ephemeris data in this comparison. Because the ephemeris samples provided by LADEE came once every ive seconds (or greater), much interpolation was required to compare that data with the LLCD data. We used polynomial, spline, and other itting methods or this exercise, knowing ull well that a real orbit/trajectory modeling approach would be required to give the best results. 4.1 Characterization o the Time-O-Flight range noise.1 Dierence between ephemera and meas ved m E ö Elapsed lime [s] Figure 6. Dierence between the one-way change in ephemeris distance and the one-way change in link distance derived rom the time-o-light measurements. Our analysis o the range data generated rom the Fine data, shown earlier, has shown a Gaussian noise component that is consistent with the 1.3 cm accuracy limit o the round-trip. There are, however, other error sources as well, which we will discuss by examining a representative data set, (recorded on 2 Oct. 213 at 2:13:29 UTC). The unction plotted in Figure 6 represents the dierence between the receiver dierential range estimates or this data sample and the high-rate interpolation to NASA's ephemeris calculations. To properly assess the characteristics o the noise in the time series plotted in Figure 6, several s were taken to isolate random measurement errors rom non-random data artiacts. As described earlier and illustrated in Figure 7, the raw data include an additive component due to the beat-requency output o the PFD. Since these high-requency luctuations lie outside the bandwidth o the measurement noise, their impact can be minimized through the use o a relatively low-pass ilter. We deduced that a Our GPS receiver, used as a clock reerence in the ground terminal, did not meet its assumed speciication and produced artiacts described in Appendix A, resulting in additional uncertainty in the absolute oset. The GPS receiver requency ping produced an equivalent two-way distance variance o σ 2 =.21 m 2. Averaging reduced the variance by the reciprocal o the number o independent 1-second requency s in a measurement interval. Including the GPS receiver artiacts, the expected error or our absolute oset was cm RMS or a typical measurement interval o 1 1 seconds. But we believe the mm value stated here will be valid or uture systems. Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

9 Dierential Distance (cm) 2-Hz ilter was a good trade between removing these artiacts and not destroying useul data. The output o a 2-Hz Butterworth ilter is illustrated by the red overlay in Figure 7. 2 Beat Frequency Artiact Removed by 2 Hz Filter 1 Phase Sensor Noise -1-2 Figure 7. One-way time-o-light measurement segment showing the high-requency noise component (black trace) and the result o low-pass (2 Hz) iltering (red curve). In order to urther isolate the noise terms, we removed slow, linear components o the signal to produce a time series having a zero mean and slope. Figure 8 shows the resulting histogram (black curve) and its best-it Gaussian unction, which was ound to have a standard deviation o.93 cm (red curve) or the one-way or this example. This is a precision or relative distance measurements more than two orders-o-magnitude iner than the RF-based navigation and ranging systems used during the LADEE mission. In uture systems, low-pass iltering could be employed to urther reduce these estimate uncertainties, suggesting that millimeter-class range accuracies should be achievable in similar systems with time-average sampling rates below 1 Hz. Figure 8. One-way residual noise probability density unction. The red overlay indicates a best it Gaussian unction or a standard deviation o.93 cm. Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

10 From this residual noise data, we also calculated the noise power spectrum, which is shown in Figure 9, and which is approximately white between DC and 2 Hz. Figure 9. One-way residual noise power spectrum showing a white noise behavior out to 2 Hz. 4.2 Further discussion o the comparison with LADEE ephemeris Using our method o range analysis, we compared our data with ephemeris-based calculations. When we did this, we ound (see Figure 6) dierences o up to a ew meters in some passes lasting tens o minutes. We can think o three possible reasons or such a non-zero slope o the distance signal. First, the slope component, which in Figure 6 corresponds to 1 mm/sec drit rom the predicted ephemeris, could be attributed to errors in the ephemeris data table or the interpolation o that data. Second, there could be an accumulated bias error in our time-olight measurement. Third, there could be a temporal oset between the ephemeris table and the phase sampling system. I we remove this slow linear component rom each pass, we are let with a new type o variation in some o the data iles, which is possible evidence o real (small) satellite motions relative to the predicted orbital model. We show one example o this in Figure 1, whose data was recorded on 21 Oct. 213 at 1:44:59 UTC. We have removed the linear term rom this data, and have applied a urther.2-hz Butterworth ilter. We see in Figure 1 that several centimeterscale displacements remain. These eatures have time scales o the order o 1 seconds, which corresponds to orbital distances o about 1 km. We have looked at our possible causes or these variations. First, drit in the MHz master oscillator could account or the variations but we have ound that only a ew measurement intervals exhibit this eature. Oscillator drit would be expected to be a eature o all the measurement intervals i it was a signiicant actor. A second possibility could be platorm movement in terms o roll, pitch, and yaw (i.e., motion o the LLCD terminal with respect to the center o mass o LADEE.) But the variations we see here are orders o magnitude greater than would be expected rom known platorm movements. A third possibility could be a temperature variation somewhere in our system. But our pre-launch data showed a very linear correlation between temperature and phase. Looking at the archived temperature telemetry, we ound that the measured temperature change during this period was monotonic, and so likely did not cause the variations ound. Finally, we ound that signal power variations can cause phase changes under certain conditions. But, again, archived telemetry o received signal power at the space terminal showed no correlation with the measured phase variations. The requency ping o the GPS reerence receiver described in Appendix A was a eature o all o the measurement intervals, but this disturbance was periodic in nature and was removed rom the Fine data. Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

11 Thus, we have reason to believe that the measured several centimeter oset rom the ephemeris calculation is real, perhaps having something to do with Lunar gravity anomalies, orbital relativistic eects, Earth tidal orces, or actual spacecrat motions. Figure 1. Low-requency variations observed in a one-way dierential measurement ollowing the removal o the linear slope components and the application o a.2-hz temporal ilter. 5. SUMMARY LLCD demonstrated record uplink and downlink data rates between a satellite orbiting the Moon and an Earth-based ground terminal. These high data rates, implemented on LLCD with optical links, enabled precision two-way measurements using only the communication signals. The LLCD system made real-time phase comparisons o the uplink and recovered downlink signals, and yielded a precision in relative one-way distance measurements typically better than 1 cm RMS with real-time samples at 2 ks/s and 2 Hz iltering. This resulted in a precision or relative distance measurements more than two orders-o-magnitude iner than the RF-based navigation and ranging systems used during the LADEE mission. We have also ound that additional iltering could be applied to yield millimeter precision. LLCD measurements were generated and archived whenever duplex communications were established during the month o LLCD operations, and required no special navigation waveorms or data in the communication stream. We believe that such high-rate communication-signal-based time-o-light systems could be highly useul in uture navigation and science missions. APPENDIX A: GPS RECEIVER ARTIFACTS We used a commercial GPS receiver (not specially designed or the LLCD system) to provide the 1-MHz reerence clock signal at the ground terminal that generated the master clock at MHz as well as the time-o-day clock used to time stamp the data. Internal to the GPS receiver, the local 1-MHz clock oscillator was periodically trimmed by comparison to the Global Positioning Satellite system atomic clocks, and gave perormance known to approach the long-term accuracy o the GPS atomic clocks. But the irmware in the particular GPS receiver unit that was used or LLCD operations actually updated its local clock every 1 seconds resulting in random part-per-billion shits in the reerence clock requency at 1. MHz. (Because o this, the unit did not meet its assumed speciication during the mission, but this was not discovered until ater the mission was over.) The result o each such requency was an apparent ramp in phase, which corresponded to distance on the order o a meter or the approximate 2.6-second round-trip time o the MHz master clock signal rom the Earth to the satellite orbiting the Moon and back. The standard deviation o the requency ping at the 1. MHz reerence was ound to be typically.588 Hz RMS as measured in a typical segment, a portion o which is shown in Figure 11. At the MHz master clock this resulted in.183 Hz RMS deviations. Over the approximate 2.6-second round-trip this resulted in apparent distance changes o: Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

12 Phase Degrees Phase Error [deg] Frequency Oset rom 1 MHz (Hz) c.458 meters RMS, (1) 311.4e6 where c is the speed o light. Again, we discovered that this error had happened regularly, every 1 seconds. Such requency-ping resulted in large variations in the residual two-way phase, shown in the example in Figure 12. The residual phase was generated by subtracting twice the predicted ephemeris distance, converted to phase, rom the measured two-way cumulative phase. (Such a subtraction rom the data was used in order to keep the residual signal to be processed to be only a ew meters. The same data used in this example was shown in Figure 4 with a total range change o approximately 9 km.) A simple model o the measured data converted to two-way distance has the ollowing components: R m N R a c t t dt. (2) Here, R m is the measured two-way range, N is the noise loor described earlier, R a is the actual two-way range, and the phase change due to requency-ping is given by the integral where the requency is deined by: 1 t t t t, (3) 2.E -2 1.E -2.E+O 1 ppb -2 Time Figure 11. Frequency plot o 1. MHz GPS reerence oscillator. 8 8 Corrected Phase Error [deg] 6 Corrected Phase Error [deg] 6 I III Time [s] r _2-4 4o 2 II 3 4 Time [s] Seconds Figure 12. Residual two-way phase error at MHz ater subtracting twice the predicted ephemeris distance (converted to phase) rom the measured data. Frequency ping o the GPS reerence clock is revealed as apparent ramps in phase. with 1 deined to be the (random) inserted by the clock in this particular 1-second interval. The PFD measures the phase dierence between the outgoing clock at time t and the returned clock which carries this requency- inormation at time t-τ. The quantity inside the integral has the values: 7 Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

13 Phase Degrees t t t t t t t t t t 1 t t t. (4) Another requency occurs ten seconds later at time t +1 with a new value or 1. We subtracted the predicted ephemeris distance, R e, rom R m to remove as much as possible the actual range, R a, rom the measurement leaving only noise and the requency ping as shown in the residual phase plot in Figure 12. R m R e N R a R e c t t dt. (5) Our correction o the requency ping was aided by knowledge o the exact 1-second periodicity o the. We irst used a 1-point (~.5-second) running average to reduce the noise. The requency 1 was then estimated by dierentiating to determine the linear slope component between t and t +τ. t t c d tt R R m e 1. (6) We then integrated this estimate o the requency change over the approximate 2.6 second round-trip to derive the distance correction R corr R m c t t dt dt. (7) Good accuracy was achieved with this correction process where agreement between predicted ephemeris distance and our measurements was observed such that between t and t +τ d Ra R dt e c 1 1. (8) We observed this to be the case or most o our measurements where the 1-point running average was expected to suppress the noise (σ) in the correction 2-dB below the residual noise loor o the measurement. We applied our requency correction only over the 2.6-second round-trip time during which the PFD produced the phase ramp. 8 P hase Error [Deg] σ(uncorrected) = 1.6 ns σ(corrected) = 127 ps loo Time [sl Seconds 7 7 Figure 13. Residual two-way phase error at MHz beore (black, blue on-line) and ater (gray, red on-line) requency- correction. Figure 13 shows the residuals beore (black, blue on-line) and ater (gray, red on-line) we applied the requency- correction to the measured data which was used to generate the residuals shown in Figure 12. We subtracted a best-it Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use:

14 polynomial rom the corrected data to generate the residuals ater correction (red). The corrected residuals over this >6 second measurement have a standard deviation o 127 ps, equivalent to 3.8 cm or the two-way or this example **. Thus, although this unplanned error in the hardware complicated our calculation, we believe that our correction method was valid, that the LLCD system still demonstrated the great accuracy possible with such a system, and that careul long-term pre-operations calibration can simpliy the data processing in uture systems. We grateully acknowledge the assistance o David O. Caplan, David J. Geisler, and Stephan P. Bedrosian or assistance in this analysis. REFERENCES [1] J. B. Berner, S. H. Bryant, and P. W. Kinman, Range Measurements as Practiced in the Deep Space Network, Proceedings o the IEEE, 95(11), (27). [2] F. G. Lemoine, S. Goossens, T. J. Sabake et al., High-degree gravity models rom GRAIL primary mission data, Journal o Geophysical Research: Planets, 118, (213). [3] J. Hamkins, P. W. Kinman, H. Xie et al., Telemetry Ranging: Concepts, IPN Progress Report, 42-23, (215). [4] E. Anzalone, C. Becker, D. Crump et al., Multi-spacecrat Autonomous Positioning System: LEO Demo Development, Small Satellite Conerence, (215). [5] D. M. Boroson, B. S. Robinson, D. V. Murphy et al., Overview and Results o the Lunar Laser Communication Demonstration, Proc. SPIE, 8971, 1-11 (214). [6] R. C. Elphic, and C. Russell, [The Lunar Atmosphere and Dust Environment Explorer Mission (LADEE)] Springer International Publishing, Switzerland (215). [7] M. E. Grein, A. J. Kerman, E. A. Dauler et al., An optical receiver or the Lunar Laser Communication Demonstration based on photon-counting superconducting nanowires, Proc. SPIE, (215). [8] D. O. Caplan, J. J. Carney, R. Laon et al., Design o a 4 Watt 1.55 um Uplink Transmitter or Lunar Laser Communications, Proc. SPIE, 8246, (212). [9] M. L. Stevens, and D. M. Boroson, A simple delay-line 4-PPM demodulator with near-optimum perormance, Optics Express, 2(5), (212). [1] S. Constantine, L. E. Elgin, M. L. Stevens et al., Design o a high-speed space modem or the Lunar Laser Communications Demonstration, Proc. SPIE, Free-Space Laser Communication Technologies XXIII, 7923, (211). [11] M. M. Willis, B. S. Robinson, M. L. Stevens et al., Downlink synchronization or the lunar laser communications demonstration, International Conerence on Space Optical Systems and Applications (ICSOS), (211). ** The corrected residuals are also shown in Figure 4 (b). Proc. o SPIE Vol Downloaded From: on 3/16/218 Terms o Use: