# Small PN-Code Lidar for Asteroid and Comet Missions—Receiver Processing and Performance Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. PN Code Ranging

_{b}is the number of bits in the PN code, T

_{p}is the laser pulse duration, E

_{t}is the transmitter pulse energy, T

_{s}is the code period, y(τ) is the received signal and k(τ + t) is the PN code kernel. The peak amplitude of the correlation is proportional to the return signal strength. The unambiguous range is calculated from the location of the peak in the cross-correlation of the PN code and kernel corresponds to the time of the flight of the code. Without additional information the lidar is unable to determine the integer number of code periods that have elapsed between transmission and reception. The absolute range is determined using the lidar’s unambiguous range from the receiver’s correlation process combined with the coarse range (i.e., the estimated range to within ±½ the unambiguous range) that is provided by the spacecraft ephemeris.

#### 2.2. Surface Reflectance Measurement

_{r}is the received pulse energy, E

_{t}is the transmitted pulse energy, ρ is the surface reflectance at the laser wavelength, A

_{r}is the receiver telescope area, R is the range from the lidar to the surface, η

_{r}is the receiver optics transmission, and η

_{det}is the detector quantum efficiency. This offers insight into the geology of the target, its space weathering history, the presence of re-surfacing processes, and may indicate the presence of surface volatiles [27]. For laser altimeters such as the lunar orbiter laser altimeter (LOLA), surface reflectance is calculated by measuring the pulse energy from individual laser shots via the return pulse waveform [28].

#### 2.3. Sampling Rate Considerations

#### 2.3.1. Sampling Error

#### 2.3.2. Correlation Length and Code Length

^{N}− 1 bits, where N is an integer. The cross-correlation and SNR for the 2

^{N}− 1 code is shown in Figure 2A. To maximize the lidar measurement rate in onboard processing, we implemented the receiver cross-correlation using the fast Fourier transform (FFT) on an FPGA. However, the FFT correlation method requires a record length of 2

^{N}. Figure 2 shows the SNR and cross-correlation result for several methods of code extension to a length of 2

^{16}. We can either append one bit (512 length) of zeros to the end of the PN code and kernel (Figure 2B), append four zeros to every bit and eight zeros to the end (Figure 2C), or adjust the sampling clock rate such that each digitized code is 2

^{16}samples long (Figure 2D).

#### 2.4. SALi Performance Model and Simulations

#### 2.4.1. Signal and Kernel Generation and Mode Selection

#### 2.4.2. Simulating Doppler Shift

^{16}length to match the FPGA implementation. We used the selected instrument mode, range, integration time, and instrument and target characteristics given in Table 2 to determine the signal and noise photon detection rates.

#### 2.4.3. Simulation of the Detected Signal and Noise Events

#### 2.5. Bennu Orbit Simulation

#### 2.6. Reflectance Measurement

^{6}photons/s and eventually saturates, at which point the technique is no longer sensitive to surface reflectance. Our plan for the SALi lidar includes generating these calibration curves during hardware testing and implementing them as a look-up table in FPGA memory. The laser power is then adjusted via feedback from the cross-correlation results to keep the signal photon rate within the linear regime.

## 3. Results

#### 3.1. Simulated PN Code Lidar Performance

#### 3.2. Simulated SALi Performance

#### 3.3. Doppler Effect

#### 3.4. Simulation Results for a Bennu Orbit

#### 3.5. Reflectance Measurement

#### 3.6. Data Processing Time with a XCKU060 FPGA

^{16}and the code was circularly shifted simulating a time-of-flight. The code was then passed through a comparator to simulate the 1-bit digitizer present in the SALi design. The delayed code was passed to a NI digital IO module and transmitted and received in a loopback fashion. The received digital code was accumulated into onboard FPGA memory (10-bit depth, 65,536 length) for the requested number of code lengths before being transmitted to the correlation stage. The cross-correlation processes were compiled at the highest clock rate possible (200 MHz) to determine the fastest single-channel measurement rate (latency and throughput) possible under the hardware constraints.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Results of sampling rate test using the PN code performance simulation for the code in Table 1. The range error standard deviation was computed from a population of 25 PN ranging trials, each with a 10 msec integration time. The exact range for each trial was random but all were 50 km ± 1 m.

**Figure 2.**Cross-correlation results comparing various methods of extending PN code and kernel to 2

^{N}. All codes were given a synthetic range of 10,000 prior to cross-correlation. (

**A**) PN code correlation with 2

^{N}− 1 bits and other parameters from Table 1 (length: 65,024). (

**B**) Correlation result after zero-padding of length 512 at the end of the code and kernel. (

**C**) Correlation result after lengthening bit duration to 516 bins and zero-padding of length four at the end of the code and kernel. (

**D**) Correlation result after resampling the code and kernel used in (

**A**) to length 2

^{N}. The SNR is the peak value of the cross-correlation peak divided by the standard deviation of the cross-correlation at all other indices.

**Figure 3.**Flowchart of the performance simulation for the PN code lidar. The steps in this chart are repeated for each distinct measurement in the simulation. The bold text indicates inputs for the simulation. The colored boxes correspond to simulation stages shown in Figure 4.

**Figure 4.**Selected samples from the performance simulation. All horizontal axes correspond to time in microseconds. (

**A**) Transmitted PN code as a function of time. (

**B**) Detector analog signal for a single code length prior to digitization is shown in red. The shorter blue signal corresponds to the range-shifted code. (

**C**) Subset of (

**B**) showing the individual photon detection events. The black, dotted line shows the threshold setting. (

**D**) Digitized signal after the comparator stage. (

**E**) Histogram of 154 accumulated digitized codes corresponding to a 10-ms integration time. (

**F**) Cross-correlation of the histogram in (

**E**) with the kernel, showing the correlation peak at 3.4 µs.

**Figure 5.**Track location for SALi 101955 orbital simulation on the OLA v20 PTM shape model with a 9 pixel per degree basemap from Bennet et al. (2021) [37]. The ground track location is shown in blue and is ~475 m in length. The track runs from 80.5793°S and 222.81°E to 80.99088°N and 207.642°E.

**Figure 6.**One and two-photon simulated analog to digital conversion at 1.00787 GHz exhibiting the origin of the comparator bias and detected pulse width bias. (

**A**) In the single photon case the comparator records six 1’s above the threshold for a single photon. Each 1 contributes to the peak amplitude of the cross-correlation, which leads to a positive bias if uncorrected. (

**B**) Analog detector signal from two-photon detection, with the photons arriving 3 ns apart. The comparator records 10 1’s above the threshold, which is less than twice the single-photon value, leading to a negative bias.

**Figure 7.**Cross-correlation amplitude as a function of incident signal photon rate for 100 Hz measurement rate. For these simulations, the detector dark noise and solar background noise levels were set to zero.

**Figure 8.**Plot of SNR as a function of accumulated signal photons under various noise photon accumulation levels. The SNR is calculated as described in Section 2.1. An SNR < 6 corresponds to a probability of detection of <~80%. Each point corresponds to the average of five independent simulations. The lines denote best fit error functions for each accumulated noise level.

**Figure 9.**Simulated SALi performance for the Reconnaissance mode conditions. (

**A**) Full cross-correlation results. (

**B**) Cross-correlation region near the cross-correlation peak. The range denotes the ambiguous range as determined from the PN code parameters. The SNR was determined as described in Section 2.1. The error is the absolute difference between the measured range via the threshold crossing method and the input synthetic range. The peak width is the difference between the rising and falling edge threshold crossings.

**Figure 10.**Simulation results of the root mean squared ranging error and mean SNR as a function of target range for the SALi Mapping mode under (

**A**) daytime and (

**B**) nighttime conditions. Each point corresponds to five independent simulations. The exact range for each simulation was generated via a base range (ex. 100 km) and an additional randomly generated range between −1 and 1 m. The signal photon rates were calculated from the mean-removed cross-correlation peak, the laser pulse rate, and the integration time. The simulation assumed a flat surface (no range spreading). The vertical dotted line corresponds to the maximum range.

**Figure 11.**Simulated SALi performance under nominal Mapping mode conditions following the conventions of Figure 9. (

**A**) Full cross-correlation results. (

**B**) Cross-correlation region near the cross-correlation peak. The simulation assumed a flat surface (no range spreading).

**Figure 12.**Simulation results of SALi ranging performance as a function of target range for the SALi Mapping mode under (

**A**) daytime and (

**B**) nighttime conditions following the conventions of Figure 10. The simulation assumed a flat surface (no range spreading).

**Figure 13.**PN ranging performance under varying levels of accumulated Doppler shift. The accumulated Doppler shift is normalized here as the ratio of the accumulated shift to the pulse length. (

**A**) RMS ranging error and average SNR of five independent measurements at each accumulated Doppler shift following the conventions of Figure 10 and Figure 12. (

**B**) Plots of a subset of cross-correlation peaks from the simulation in (

**A**) showing the peak-spreading due to the Doppler the shift.

**Figure 14.**Range from the spacecraft altitude to the Bennu surface and the retrieved range from the SALi performance simulation as a function of distance along track. Each blue marker represents one 10-ms measurement. The red line was generated from the interpolated ground track elevation. (inset) A portion of the track from 170 m to 300 m showing the range measurements tracking the elevation over sharp features in the surface terrain.

**Figure 15.**Range residuals (measured minus actual) from the data shown in Figure 14. The ground-truth elevation value was taken as the mean of all ranges within the footprint.

**Figure 16.**A select region of the track exhibiting high range residuals. The blue markers and red line are the same as in Figure 14 and denote the SALi retrieved range and Bennu ground track from the shape model, respectively. The gold line denotes the size of the SALi footprint on the surface for the measurement at 183 m along the track. As the results are plotted as a function of range as measured by the lidar the Bennu surface normal points down in this plot.

**Figure 17.**Reflectance measurement results using the pre-computed calibration curves and laser power control. Each point denotes the average retrieved reflectance for five trials at each reflectance value. Error bars correspond to 1σ. The dashed black line has a slope of 1.

**Figure 18.**Simplified block diagram of real-time processing breadboard. All modules were housed in an NI PXIe chassis and data passed between modules occurred over the backplane.

**Figure 19.**FPGA cross-correlation results of PN code and kernel. (

**A**) Continuous output from the IFFT core of the SALi processing pipeline normalized to the peak value. The cycles correspond to clock cycles of the pipelined IFFT code. (

**B**) Cross-correlation peak from (

**A**) translated into lag between code and kernel (i.e., ambiguous range). The circles represent the discrete IFFT values from which the centroid, width, and SNR were calculated.

Parameter | Value |
---|---|

Code Length | 127 bits |

Bit Duration | 512 ns |

Code Period | 65,024 ns |

Pulse Duration | 8 ns |

RZ Ratio | 64 |

Laser Duty Cycle | 0.78125% |

Bit rate (MHz) | 1.953125 MHz |

Unambiguous Range | 9746.9 m |

Signal Sampling Rate | 1.00787 GHz |

**Table 2.**Instrument and target parameters used in the performance simulations for the SALi instrument.

Parameter | Value |
---|---|

Laser Wavelength | 1550 nm |

Average Laser Output Power | 2 W |

Solar Distance | 1 A.U. |

Target Reflectance | 5% |

Telescope Diameter | 6.4 cm |

Bandpass Filter FWHM | 1 nm |

Detector | 16-pixel HgCdTe APD |

Pixel Layout | 2 × 8 |

Pixel IFOV | 60 µrad |

Detector Quantum Efficiency | 60% |

Detector Dark Counts per Pixel | 250 kHz |

Detector Response Time | 3 ns |

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**MDPI and ACS Style**

Cremons, D.R.; Sun, X.; Abshire, J.B.; Mazarico, E.
Small PN-Code Lidar for Asteroid and Comet Missions—Receiver Processing and Performance Simulations. *Remote Sens.* **2021**, *13*, 2282.
https://doi.org/10.3390/rs13122282

**AMA Style**

Cremons DR, Sun X, Abshire JB, Mazarico E.
Small PN-Code Lidar for Asteroid and Comet Missions—Receiver Processing and Performance Simulations. *Remote Sensing*. 2021; 13(12):2282.
https://doi.org/10.3390/rs13122282

**Chicago/Turabian Style**

Cremons, Daniel R., Xiaoli Sun, James B. Abshire, and Erwan Mazarico.
2021. "Small PN-Code Lidar for Asteroid and Comet Missions—Receiver Processing and Performance Simulations" *Remote Sensing* 13, no. 12: 2282.
https://doi.org/10.3390/rs13122282