ZHU Yaowei, XU Zhaobin,*, JIN Xiaojun, GUO Xiaoxu, and JIN Zhonghe

1. Micro-Satellite Research Center, Zhejiang University, Hangzhou 310027, China;

2. School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

Abstract: The advancement of small satellites is promoting the development of distributed satellite systems, and for the latter, it is essential to coordinate the spatial and temporal relations between mutually visible satellites. By now, dual one-way ranging (DOWR) and two-way time transfer (TWTT) are generally integrated in the same software and hardware system to meet the limitations of small satellites in terms of size, weight and power(SWaP) consumption. However, studies show that pseudo-noise regenerative ranging (PNRR) performs better than DOWR if some advanced implementation technologies are employed. Besides,PNRR has no requirement on time synchronization. To apply PNRR to small satellites, and meanwhile, meet the demand for time difference measurement, we propose the round-way time difference measurement, which can be combined with PNRR to form a new integrated system without exceeding the limits of SWaP. The new integrated system can provide distributed small satellite systems with on-orbit high-accuracy and high-precision distance measurement and time difference measurement in real time. Experimental results show that the precision of ranging is about 1.94 cm, and that of time difference measurement is about 78.4 ps, at the signal to noise ratio of 80 dBHz.

Keywords: time difference measurement, time synchronization,inter-satellite ranging, satellite formation autonomous flying.

1. Introduction

In recent years, the use of microelectronics and microsystems technologies, especially the application of compact commercial-off-the-shelf (COTS) components, have dismissed the trade-off between the functionality and the limitations of small satellites in terms of size, weight and power (SWaP) consumption [1–3]. As a result, the manufacturing cost and launch price per small satellite are much cheaper and the development time is much shorter than deploying a traditional monolithic satellite. These characteristics make small satellites very suitable for new scientific and technological research in the satellite field[1,4]. For instance, adopting small satellites in a distributed satellite system (DSS) [5,6] such as satellite formations, swarms, and constellations can dramatically reduce the overall cost and deployment time [2,7].

Compared with a single satellite system, a DSS is more reliable and redundant, with wider surveillance areas, and the mission design can be more flexible, but one of the challenges is high-precision relative navigation [8,9]. It is essential to coordinate the spatial and temporal relations between each two mutually visible satellites in a DSS. On the one hand, an accurate inter-satellite ranging system is required for avoiding satellite collisions [10], maintaining a distributed configuration or fulfilling specific tasks[11]. On the other hand, an accurate inter-satellite time difference measurement system is required because the states of the satellites should be coordinated to obey the event schedule correctly, and the measurement data of different satellites should be marked with a unified timestamp, otherwise, the measurement result cannot be accurate enough [5]. Satellites TerraSAR-X and TanDEM-X formed the first configurable synthetic aperture radar(SAR) interferometer employing autonomous formation flying in 2010 [12]. The satellite formation is equipped with a TanDEM-X autonomous formation flying (TAFF)system, which can take over the in-plane formation maintenance activities from the ground segment [13]. TAFF utilizes the data from on-board global positioning system(GPS) receivers to control the inter-satellite distance, and the typical formation control accuracy is 30 m. In the gravity recovery and climate experiment (GRACE [14])mission, the inter-satellite distance is measured by dual one-way ranging [15] (DOWR) method, and two GPS receivers are used to provide unified timestamps for each of those one-way measurements. To ensure high ranging accuracy, the timestamp synchronization accuracy is further improved by post-interpolation processing. If the time-tag error is 200 ps, the overall accuracy of the range rate is well below 1 μm/s [15]. In the gravity recovery and interior laboratory (GRAIL) mission [16] which is like GRACE, but where GPS service is not available, a pair of dedicated S-band transponders [17] is used for inter-satellite two-way time transfer (TWTT) [18].

For a DSS composed of small satellites, combining inter-satellite ranging and time synchronization and implementing on the same device can reduce the payloads.The combination can be achieved by using the modern digital transceiver design based on field programmable gate array (FPGA). In [19], a system that combined TWTT and the dual-frequency carrier ranging method was illustrated, and the ambiguity of carrier ranging was resolved by using pseudo-noise (PN) code ranging. The precision of ranging and time synchronization are irrelevant, which are 1 cm and (8.3/8) ns respectively. In [20], a precise ranging and time synchronization system was implemented based on DOWR. The time difference measurement method is equivalent to TWTT. And the precision of ranging and time synchronization are 1.038 m and 3.46 ns respectively when the two terminals using different frequency references, 2.788 mm/9.3 ps when using a common frequency reference. It seems that the frequency deviation has a great influence on the measurement precision. Inspired by [20] and [21], both DOWR and TWTT can be implemented into an inter-satellite direct-sequence spectrum spread (DSSS) communication system,thereby forming a comprehensive inter-satellite measurement and communication system. However, the measurement precision and accuracy of DOWR rely on the synchronization accuracy of the measurement time of the two one-way measurements [22,23]. It is the same for TWTT unless the two signal transmission paths remain reciprocal and stationary over time [18,24]. It has been analyzed in the GRAIL mission that the time difference measurement error caused by unsynchronized measurement time is bound to several nanoseconds, and the error caused by the non-reciprocal transmission paths is about hundreds of nanoseconds [25].

PN regenerative ranging [26] (PNRR) is a coherent ranging method which is widely used in missions of deep space exploration. The ranging precision and accuracy of PNRR are irrelevant to the accuracy of time synchronization and the clock source deviation. Additionally, the composite PN sequence used by PNRR has a higher sequence transition density than that of the PN sequence used by a DSSS system, such as Gold code. Thus, the measurement system based on composite PN code can achieve a better measuring precision and accuracy than the measurement system based on the DSSS system by adopting the non-commensurate sampling technique and the double-loop tracking structure [27].

In this paper, we propose the round-way time difference measurement (RWTDM) method based on PNRR to form a new integrated inter-satellite distance and time difference measuring system for micro-satellite formations. The new integrated system enables micro-satellites to measure inter-satellite distance and time difference in real time with high-precision and high-accuracy, without breaking the SWaP constraints. Experimental results show that when the signal power-over-noise power spectral density ratio (SNR) is 80 dBHz and the chip rate is 1 Mbps, the ranging precision is 5.76 cm and the frequency offset measurement precision of 1 s interval is about 1 Hz under the condition that the two test boards use separate clocks, while the ranging precision is 1.94 cm and the time difference measuring precision is 78.4 ps under the condition of using a common clock. The common clock experiment is an equivalent to the situation that the time difference compensation system (TDCS) [21] is applied.And the measurement precision can be further improved if the pseudo-range is smoothed with carrier phase [28].

The rest of this paper is organized as follows. Section 2 is the introduction of PNRR and RWTDM integrated system, as well as the explanation of working principles.Section 3 is the implementation and performance analysis of the measurements. Section 4 presents the experimental results, and Section 5 presents the conclusions.

2. System introduction

2.1 System overview

We have successfully implemented the PNRR and RWTDM integrated system in the inter-satellites communication transceivers (shown in Fig. 1) of two microsatellites. Fig. 2 shows the software framework, where transceiver M-sat serves as the position and time reference of transceiver S-sat, and the latter is responsible for initiating the measurements. The baseband processing(BBP) module is for digital modulation and demodulation, besides, both M-sat and S-sat maintain a local composite PN sequence generator driven by local oscillator(OSC), track the incoming PN sequence through the code tracking loop (CTL) [26], and measure the phase difference between the local PN sequence and the incoming PN sequence. When ready to start measurement, S-sat transmits its local PN sequence to M-sat. However, M-sat does not do so, instead, it echoes the S-sat PN sequence back to S-sat as soon as the sequence is regenerated. The measurement of S-satisρs,whichindicatesthe round-way delayofthe S-satPNsequence.Andthemeasurementof M-sat is ρm, indicating the one-way phase difference (in seconds) between PN sequences of M-sat and S-sat.Then, one-way measurement ρmis transmitted to S-sat through the inter-satellite communication channel for calculating the inter-satellite time difference in real time,while the inter-satellite distance is calculated by using ρs.

Fig. 1 Micro-satellite communication transceiver

Fig. 2 System software framework

2.2 Principles of PNRR and RWTDM

Generally, a one-way phase difference measurement contains information about the inter-satellite distance and time difference, but only the distance information is included in the round-way phase difference measurement of PNRR. Thus, the time difference can be decoupled from the one-way measurement by using the round-way measurement.

To illustrate the principles of PNRR and RWTDM in detail, assume that M-sat and S-sat generate the local PN sequences at equal chip ratefPN, and the time of M-sat is ahead that of S-sat by ΔT. Then, the PN signal phase of M-sat and S-sat at timetcan be expressed as

In Fig. 3, the signal phases of M-sat and S-sat are equal to φ at timet1andt2, respectively, which are

Fig. 3 Measurement process of PNRR and RWTDM

Fig. 3 shows the measurement process of PNRR and RWTDM by tracking the transfer process of phase φ that is sent from S-sat. Simply put, phase φ leaves S-sat for Msat at timet2, arrives at M-sat at timet3, and also leaves M-sat for S-sat at timet3. Finally, phase φ returns to Ssat at timet4.

Suppose the measurement of phase difference can be done instantly no matter at M-sat or S-sat and ignore the thermal noise. Then the one-way measurement of M-sat attimet3aswell as the round-way measurement of S-sat attimet4are whereR,DandIrepresent the signal transmission path delay, device delay, and ionospheric delay respectively,with the subscript indicating the path direction.

Observed at S-sat, the forward and backward transmission paths of phase φ are reciprocal, since phase φ leaves as soon as it reaches M-sat. It is like throwing a weightless ping-pong ball at the speed of light against a wall from S-sat, then it bounces back as soon as it hits the wall and returns to S-sat in the velocity of light, too. In the view of S-sat, the forward path and the backward path of the ping-pong ball are reciprocal with their length equal to the distance between S-sat and the wall at the moment the ping-pong ball hits the wall, no matter the wall is moving or not. Thus, for phase φ,

The delay of phase φ at M-sat due to hardware and software processing is considered in the device delay.Thus, according to (6), the measurement equation of intersatellite distanceris

where c is the velocity of light.

Substitute (8) into (5) to decouple the inter-satellite time difference from the one-way measurements:

Equation (9) verifies the feasibility of RWTDM in measuring the inter-satellite time difference. It also shows that RWTDM does not require ρmand ρsto be measured at the same time, which means there is no need to synchronize the measurement time of the two measurements, making RWTDM suitable for working onboard in real time.

3. Implementation and performance analysis

3.1 Implementation of RWTDM

ThoughRWTDMdoesnotrequire ρmand ρstobe measuredsimultaneously, itis necessarytofindtherightpair ofρmandρsto calculate timedifference.Weproposea simple and effective way tomatch ρmandρswithout recordingthe measurement time of ρmand ρswhich is uselessifthetime of S-satandM-sat has notbeensynchronized yet. The solution is to control the measurement of ρmand ρswith a trigger signal delivered by S-sat through the inter-satellite communication channel when the measurement is about to start. The trigger signal is always transferred along with the measuring signal, serving as a label attached to phase φ which could be arbitrary. Then M-sat measures ρmand echoes the trigger signal back to S-sat as soon as it is received and S-sat measures ρswhen the trigger signal returns. Eventually, one-way measurement ρmand round-way measurement ρsthat are triggered by the same trigger signal can be used for calculating ΔT.

3.2 Implementation and performance of phase difference measurement

The actual implementation of measuring ρsand ρmdetermines how precisely the inter-satellite distance and time difference can be obtained, which involves the selection of measuring signals and the measurement method.In this paper, the Tausworthe,v=4 (T4B) ranging code is used as the measuring signal of the integrated system.The T4B code is composed of a strong clock-code component and five sub-code components, so it is preferable for high measurement accuracy applications [26].

Taking ρsas an example, it can be expressed as follows in general:

whereTcis the chip interval;Nandnare the integer ambiguity and the decimal part of phase difference measurement respectively.

The integer ambiguityNcan be determined by using Chinese remainder theorem [29,30]. During one measurement process, it is unlikely thatNis affected by random noise. On the contrary, the decimal partnis more susceptible to random noise. There are two architectures in[26] for determiningn, which are closed loop architecture and open loop architecture.

Fig. 4 shows the open loop architecture, wherex[i] is the digital sample of the received measuring signal that is output from the Q branch of the carrier recovery loop.Thenx[i] correlates with the local in-phase and midphase clock codes respectively. As shown in (11), when in the sine-square mismatched case [26], the open loop architecture has no dependence on CTL, which means the measurement jitter mainly depends on the integration intervalTiand the SNR of the clock code. If a smaller measurement jitter is wanted,Tican be increased,without modifying the loop bandwidth of CTL and affecting its tracking performance. Consequently, the open loop architecture is preferable.

Fig. 4 Open loop architecture

wherefclkis the frequency of the clock-component code,fclk=1/2Tc, andPclk/N0is the SNR of the incoming signal.

Thus, the jitters of one-way measurement ρmand roundway measurement ρsare

Then, according to (8) and (9), the measurement jitters of inter-satellite distance and inter-satellite time difference are

3.3 Error analysis and calibration

As shown in (8) and (9), to accurately measure the distance and time difference, the device delay and ionospheric delay have to be calibrated. Besides, the derivation of (8) and (9) is based on the assumptions that M-sat and S-sat generate their own PN sequences at the same chip ratefPN, and the phase difference is measured instantly. These assumptions do not accord with the actual situation. In fact, the frequency drift of oscillators and the frequency offset between them, as well as the time taken by the phase difference measurement, will cause deviations between the measured values and their true values.3.3.1 Device delay and ionospheric delay calibration Comparing (8) and (9) with the distance measurement equation in [15] and the time difference measurement equation in [18], respectively, it can be found that the effects of device delay and ionospheric delay on PNRR and RWTDM are the same as on DOWR and TWTT. The device delays and ionospheric delays of the two signal transfer paths are added together for ranging but differentiated for time difference measuring.

Dsmin (8) and (9) is composed of the transmission delay of S-satDtx,sand reception delay of M-satDrx,m,andDmsis composed of the transfer delay of M-satDtx,mand the receive delay of S-satDrx,s. In [31], a delay calibration device that contained a frequency mixer was used to transform the signal from the transmitting port of S-sat to its receiving port, then the self-ranging result was equal to the delay of calibration board plus the transmission delay and reception delay of the ranging board.Therefore, this method is useful for determining the sum ofDsmandDms, but useless for determining the difference. In [32], a calibration device called satellite simulator was used for measuring the transmission delay and the reception delay of the earth station separately. Using this method,Dtx,s,Drx,s,Dtx,mandDrx,mcan be measured one by one, thus the sum ofDsmandDmsor difference betweenDsmandDmscan be easily got, too.

The ionospheric effects cannot be avoided in the nearearth space. For ranging, the ionospheric delays on the two paths always add up. However, for time difference measuring, the ionospheric delays can be offset if the paths are symmetric. In [15] and [33], the dual-frequency method was used for eliminating the ionosphere effect. In GPS, the L1 and L2 signals can be combined to model the ionospheric delay [34]. In [35], the Klobuchar model was used for ionospheric delay calibration in a single frequency global navigation satellite system (GNSS). According to [36], the correction accuracy of the dual-frequency method is the highest, followed by the grid method and the Klobuchar model. Thus, the dual-frequency method should be utilized for high-accuracy ranging and time synchronization.

Letf1andf2denote the dual frequencies that are used,ρ1and ρ2denote the corresponding measurements, andρ denote the ionosphere-free measurement. Then the ionosphere effect can be eliminated by using the combination[15] as follows:

3.3.2 Oscillator frequency offset and drift error

Driven by non-ideal oscillators, the instantaneous chip rate of the PN sequence generated by M-sat and S-sat can be modeled as follows:

whereFis the rate offset of the PN sequence, and α(t) is the rate drift function.

To analyze the influence of oscillator frequency offset and drift on the measurement error, assume that M-sat and S-sat are relatively static, and ignore the phase difference measurement error, device delay, and ionospheric delay. Then, according to (8) and (9),

where

The ranging error is mainly caused by the deviation between the actual frequency of the S-sat oscillator and the nominal frequency, which is

For time difference measurement, the error is determined by comparingwith the time difference when the measurement process ends:

where

Suppose M-sat and S-sat both use the oscillator with a nominal frequency of 40 MHz, stability of 0.5 ppm, and a drift rate of no more than 1 Hz/s. And the chip rate is 1 Mbps. Thus,

On the other hand, the time of a complete measurement process is related to the inter-satellite distance, usually no more than hundreds of milliseconds in the nearearth space. Take the inter-satellite distance of 900 km as an example. Besides, the oscillator frequency drift is a slow process, therefore, it can be regarded as a constant during the short measurement interval. In this case, according to (23) and (24), the maximum ranging error due to the frequency offset of the S-sat oscillator is about 45 cm, and for time difference measurement, the maximum error is about 3 ns.

An effective method for reducing the measurement error of PNRR and RWTDM is to calibrate the measurement results with the exact value of frequency offset and drift. However, it is impossible to measure frequency offset and drift of the oscillators onboard. A lookup table about frequency offset and drift of each oscillator under vacuum circumstances and different temperatures must be established when the measurement system is tested on the ground.

If RWTDM is successfully applied to the time sync system [21] as a time difference measurement module, it is possible to keep

Then, the maximum error of measuring time difference is about 0.1 ps, which is negligible. It seems that if the clock frequencies of the two satellites are synchronized, then most of the time difference measurement error can be canceled out. However, the ranging error is not reduced.

3.3.3 Phase difference measurement error

As shown in Fig. 4, the phase difference measurement involves integrating the correlation results of the incoming signal and the local signal in an intervalTi. This design has some advantages, such as reducing the impact of random jitters on the measurement results and improving the measurement precision. However, there are some disadvantages, such as introducing a deviation between the measurement result and the instantaneous value in dynamic scenes.

To assess the phase difference measurement error in dynamic scenes, the impacts due to quantization, oscillator frequency offset and drift are ignored, and the digital measurement system is regarded as an analog system.The clock-code component of T4B is equivalent to a sine or cosine wave after being sinewave shaped. Thus, the incoming signalx(t) can be expressed as

wherefdis the Doppler frequency offset, and θ0is the phase difference when the measurement starts.

In the sine-square mismatched case [26],

Approximate the square wave with its first harmonic,then,

Thus, the measured phase differenceis

where θendrepresents the phase difference when the measurement ends, and θmeas,eis the measurement error. It turns out that the measurement of the phase difference will always deviate from the instantaneous value in dynamic scenes, and the amount is relevant to the relative velocity between the two satellites and the integration intervalTi.

Suppose there is a dynamic scene with linear uniform motion as shown in Fig. 5. This supposition is reasonable because the relative motion between satellites in a DSS will not change drastically during a short integration interval which is usually hundreds of milliseconds.

Fig. 5 Dynamic scene

According to (35), the one-way phase difference measurement error and round-way phase difference measurement error in seconds are

where

Then, the measurement error of inter-satellite distance and time difference are

Equation (40) shows that the error of inter-satellite distance measurement can be calibrated if the relative speed is known, and (41) shows that the measurement error of inter-satellite time difference is nearly zero because any error that appear in one-way measurement will double in round-way measurement and could be canceled out by then.

4. System demonstration

To verify the feasibility of PNRR and RWTDM integrated system, we carry out some experiments in the testbench as shown in Fig. 6. In these experiments, two micro-satellite transceivers used for inter-satellite communication and measurement are adopted, one is labeled as S-sat and the other as M-sat. It should be noted that the examination of the measurement accuracy is absent in this paper. We mainly focus on the measurement precision of the integrated system.

Fig. 6 Testbench

Fig. 7 shows the schematic diagram for separate clock experiments, where S-sat and M-sat are driven by independent clock sources. This testbench simulates the actual application scenario. The nominal frequency of the two OSCs is 40 MHz. Generally, offset exists between the frequencies of the two OSCs. And the offset makes the time difference measurement a linear function of time,while the distance measurement is a constant with additive noise.

Fig. 7 Testbench of separate clock experiments

Fig. 8 is an example of the measurement results when SNR is 70 dBHz, and the mean values are removed for better display. The measurement results of PNRR are the evidence to its immunization to the frequency and time asynchronization between two transceivers.

For PNRR, the standard deviation of measured values is used to indicate the measurement precision. Fig. 9 shows the measurement precision of PNRR at different SNRs, and the theoretical values are calculated according to (14). The relative deviations between the experimentalvalues and theoretical values are small in the case of a low SNR and large in the case of a high SNR, and the measurement precision is not significantly improved when the SNR changes from 70 dBHz to 80 dBHz. This is because in these experiments SNR refers only to the ratio between signal power and thermal noise power spectral density (PSD) and does not involve noise of the frequency source. When the SNR is high, the PSD of the frequency source noise overwhelms that of the thermal noise. Thus, the actual SNR within the system bandwidth is lower than its preset value. As a result, the measurement precision in a high SNR is worse than expected. The results of the experiments show that the measurement precision of PNRR is better than 10 cm if the SNR exceeds 65 dBHz.

Fig. 8 Measurements of separate clock experiments when SNR is 70 dBHz

Fig. 9 Measurement precision of PNRR of separate clock experiments

For RWTDM, the standard deviation of the measured data is meaningless, but the slope of the measured curve equals the normalized frequency offset between the two independent OSCs, so Allan standard deviation can be used for estimating how well the RWTDM measurements reflect the frequency offset. Fig. 10 shows the Allan standard deviation curve of RWTDM measurements at different SNRs. Allan standard deviation reflects the normalized frequency offset measurement precision, but it seems that there is no clear relationship between the measurement precision and the SNR. Sometimes the measurement precision increases as the SNR increases,sometimes the opposite is true, depending on the value of periments,Tiequals0.1049s. Table1 lists themeasurementprecision offrequency offsetat different SNRswhen τ. The value ofτisan integer multiple ofTi. In theexτis about 1 s.To evaluate the measurement precision of RWTDM,the clock frequencies of S-sat and M-sat must be synchronized. We have not finished replacing the time difference measurement module of the time sync system [21]yet, instead, we try to use a common clock source to drive S-sat and M-sat as shown in Fig. 11. Of course, this approximation is a little over ideal, but it is appropriate to examine the upper bound of measurement precision that RWTDM can achieve.

Fig. 10 Allan standard deviation of RWTDM measurements

Table 1 Frequency offset measurement precision

Fig. 11 Testbench of common clock experiments

Fig. 12 is an example of the measurement results when the SNR is preset to 70 dBHz, and the mean values are removed for better display. Fig. 13 and Fig. 14 show the measurement precision of PNRR and RWTDM at different SNRs, respectively. The best case is at the SNR of 80 dBHz, in which the measurement precision of PNRR is 1.94 cm, and that of RWTDM is 78.4 ps.

Fig. 12 Measurements of common clock experiments when SNR is 70 dBHz

Fig. 13 Measurement precision of PNRR of common clock experiments

Fig. 14 Measurement precision of RWTDM

Comparing Fig. 13 and Fig. 9, it can be found that the measurement precision of PNRR is improved in the case of a high SNR because the frequency synchronization between S-sat and M-sat allows most of the frequency source noise to be eliminated. For example, the ranging precision improves to 5.54 cm at an SNR of 70 dBHz. As shown in Fig. 14 and Table 2, the results of the common clock experiments fit well with the theoretical estimates.The experimental measurement precision is a little better than theoretical prediction at SNRs of 75 dBHz and 80 dBHz. It seems that the actual SNR exceeds the preset value by no more than 1 dBHz, which is reasonable because the minimum adjustment resolution of SNR in the experiments is 1 dBHz. The data about TWTT listed in Table 2 is got from [21]. The comparison between RWTDM and TWTT shows that the measurement precision of RWTDM is comparable to the theoretical measurement precision of TWTT at the same SNR. However,as illustrated in Section 1, the integration of PNRR and RWTDM is better than the integration of DOWR and TWTT.

Table 2 Comparison between RWTDM and TWTT

5. Conclusions

In this paper, we propose a new inter-satellite time difference measurement method, RWTDM, based on PNRR.The measurement equation proves the feasibility of RWTDM theoretically. We integrate RWTDM and PNRR together and implement it in the micro-satellite transceivers. The new integrated system can provide micro-satellites onboard distance measurement and time difference measurement simultaneously with a high precision and a high accuracy in real time, without the need of complex post data processing algorithms or assistance from other satellite systems. The system can be applied to missions such as earth observation or planetary science that uses small satellites and meets the needs of on-orbit inter-satellite distance maintenance and scientific data synchronization. Besides, according to the fact that PNRR can be used to measure the long distance between a deep space probe and its supporting ground station,RWTDM can be used to measure the time difference between the deep space probe and the ground station.Some of the significant implementation details of the system have been described, and the performance is also analyzed. It turns out that errors caused by frequency drift and offset of the oscillators, and inter-satellite relative motion on the two signal transmission paths are added together for PNRR but differentiated for RWTDM. Finally,the results of the separate clock experiments show that

PNRR as a coherent ranging method does not require the time between two satellites to be synchronized to achieve high-accuracy ranging, even though there is a deviation between the clock frequencies. The best result of ranging precision is 5.76 cm at the SNR of 80 dBHz. RWTDM reliably shows that the time error of the two satellites increases with time when driven by clock sources of different frequencies. Further, to assess the measurement precision of RWTDM, we carry out the common clock experiments. The experimental results show that if clock frequencies of the two involved satellites are synchronized,the measurement precision of time difference is 78.4 ps,while the ranging precision is 1.94 cm, at the SNR of 80 dBHz.