36 th Annual Precise Time and Time Interval (PTTI Meeting DEVELOPMENT OF CARRIER-PHASE-BASED TWO-WAY SATELLITE TIME AND FREQUENCY TRANSFER (TWSTFT Blair Fonville, Demetrios Matsakis Time Service Department U.S. Naval Observatory Washington, DC 39, USA Alexander Pawlitzki and Wolgang Schaeer TimeTech GmbH D-7569 Stuttgart, Germany Abstract For the dissemination o precise time and requency, the use o the Two-Way Satellite Time and Frequency Transer (TWSTFT method has, in recent years, become increasingly valuable. By the application o spread-spectrum technology, a client station, located anywhere within a common satellite s ootprint, can link to a reerence station and compare or synchronize its clock to the reerence clock within nanoseconds or better with high levels o conidence. But as the requency stabilities o today s sophisticated atomic-level clocks improve, so must the stability o a Two-Way tem s method o time and requency transer. Carrier-phase inormation holds the promise o improving the stabilities o TWSTFT measurements, because o the great precision at which requency transers can be achieved. The technique requires that each site observe both its own satellite-translated signal and that o the cooperating site. As was reported in earlier papers, the translation requency o the satellite itsel is an additional unknown actor in the measurement, and it must be taken into consideration. However, it can be shown that simple estimates o the satellite s local oscillator (LO requency will suice. Recent work has been conducted with implementation o the signal carrier-phase in operational TWSTFT links. The purpose o this paper is to discuss this recent work, with emphasis on the tem s development. I. INTRODUCTION At the present state o the art, one o the primary tools or the dissemination o precise time and requency is a Two-Way Satellite Time and Frequency Transer (TWSTFT tem. With such a tem, a client station and a reerence station each employ a code division multiple access (CDMA scheme to lock onto each other and establish communication. The spreading unction used or the CDMA, while providing the spread-spectrum capability and all o the associated beneits, also provides time markers rom each station s respective timescale. This allows the client station to synchronize its time to that o the linked reerence station. Such a tem is illustrated below in Figure []. 49
36 th Annual Precise Time and Time Interval (PTTI Meeting TX Figure. A basic Two-Way tem where τ i is the signal time delay through the transmit equipment RX U at station i; τ i is the signal time delay through the receive equipment at station i; τ i is the signal time D delay or transit rom station i to the satellite; τ i is the signal time delay or transit rom the satellite to station i; T i is the phase o Clock i; and Ti is the dierence, measured at station i, between the phase o Clock i and the phase o the received signal. A CDMA-based TWSTFT tem measures the time dierence between two clocks, and can provide requency by dierentiating the time-dierence measurements. However, the CDMA-based TWSTFT method utilizes the carrier wave o the signal and not the superimposed lower-requency pseudorandom noise (PRN code. Thereore, CDMA-based TWSTFT suers more rom multipath eects, which is a unction o eective wavelength, than does the carrier signal. Additionally, it is only the change o the relative phase that matters; an initial phase oset induced by multipath and the ionosphere has no relevance. For the same reason, other time delays introduced into the tem, such as Sagnac eects and uncalibrated static time delays within user equipment, will have less eect on a carrier-phase tem than in the code-based TWSTFT coniguration. Previous eorts to use the carrier-phase in TWSTFT have developed the tem s undamental equations, and the results have shown some promise []. In this paper, we discuss the derivation o the tem s undamental equations, ollowed by the results o two test runs: a collocated test at the U.S. Naval Observatory (USNO, and an intercontinental test between the Physikalisch-Technische Bundesanstalt (PTB and USNO. II. CREATION OF THE SOLUTION EQUATION SET In general, the carrier requency o a received signal will be oset rom the requency generated by the transmitter. The magnitude o the requency oset can be mostly attributed to our tem-variables (illustrated below in Figure :, k, k, and d [], which are deined in the discussion below. For the remainder o this paper, the notation, or TX, will be used to indicate a transmitted signal, and rx, or RX, will be used to indicate a received signal. 5
36 th Annual Precise Time and Time Interval (PTTI Meeting V S = = d clk clk + Figure. Carrier-Phase Two-Way. As shown in the above igure, the reerence station s (Station time is based on a requency and the requency o Clock is oset rom by a value d. Realization o the oset between the requencies o Clock and Clock is the basis o this paper. A simple comparison o the transerred signals will not yield a true value o d due to three requency perturbations (or requency shits to which the carriers are subjected: two caused by the motion o the satellite and one caused by the satellite s unknown local oscillator requency, or. As an example, consider a one-sided carrier-requency transer depicted in Figure, where Station is transmitting a signal to Station. The signal, which is relayed to Station by a satellite, is subject to the Doppler eects resulting rom the slight motion o the satellite. The irst-order Doppler coeicients are described by the ollowing two equations. = V c ( k = V c ( k where V n is a projection o the velocity vector o the satellite in the direction o Station n and c is the speed o light through the transmission medium. Note that in Figure, the velocity o the satellite has been represented as a vector V S. Suppose that Station transmits a signal with a requency, where is deined with respect to (w.r.t. the tem requency. At the satellite, the requency o the received signal is centered at the Doppler-shited requency + k, which is then mixed with the satellite s local oscillator (LO and retransmitted at the carrier requency given by the ollowing equation: SV = + k. (3 The carrier requency is then observed Doppler-shited at the downlink site as: d = + k + k ( + k. (4 ( 5
36 th Annual Precise Time and Time Interval (PTTI Meeting All o the requencies given in Equation (4 are reerenced to the tem requency. However, the value o d will be measured with reerence to the receiver s local clock. Equation (4 must, thereore, be rewritten such that its terms are expressed with reerence to the clock local to Station. Figure 3. System rames o reerence. For the purpose o converting a given requency, as deined by a rame o reerence m, to its equivalent requency, in a rame o reerence n, a transormation equation has been derived: n re re m m =. (5 n m where is the value o the requency (w.r.t. the transmitter s clock, which is to be transormed; n is m the value to be determined; re is the instantaneous requency o the reerence clock at the transmitter (given w.r.t. tem time; and re n is the instantaneous requency o the reerence clock at the receiver (given w.r.t. tem time. Applying this transormation equation to the requency transer given by Equation (4 produces F = ( + k( + k ( + (6 d d + + k where the notation Fmn is used to denote a requency that has been transmitted by Station m and measured by Station n. Similarly, we may derive the requency transer rom Station to Station as ollows: F + d = ( + k( + k ( +. (7 k In addition to the two transers given by Equations (6 and (7, each station may receive its own signal. These are modeled by the ollowing two loopback equations: F ( + k = + k ( (8 F = ( + k ( + (9 d + k 5
36 th Annual Precise Time and Time Interval (PTTI Meeting With theoretically ideal measurements, the values F, F, F, and F are independent; however, under the limitation o the tem equipment s quantization noise, the ollowing equality is satisied: F + + ( F = F F This equality allows or an immediate data integrity check. Alternatively, it will be shown that it also allows the user to solve the tem using only three o the our equations, thus sparing a receiver channel. With the two carrier transers and the two loopbacks deined above, the carrier-phase tem is ully described and we now have our equations rom which we may resolve our our tem variables. III. UNDERSTANDING THE SYSTEM EQUATION SET Generally speaking, the requencies F, F, F, and F are provided by the two spread-spectrum modems employed by the tem, and the requencies are reported in the IF-band. Because the carrierphase tem is based on these measurements, it may be insightul to examine the construction o the values. Consider the ollowing tem, where the satellite is stationary with respect to both TWSTT stations: Figure 4. RF transceiver signal propagations. Relating the above igure to the carrier-phase tem, the illustrated signals have the ollowing associated requencies: CLK CLK + TX TX + y LO A LO LO d B LOC + y LOD LO + y LO, where the instantaneous ractional requency y is deined as d y =. ( 53
36 th Annual Precise Time and Time Interval (PTTI Meeting As shown, the RF/IF mixers are supplied tunable synthesized requencies rom their respective LO s. It can be shown that the output requencies o the synthesizers will have the same ractional requencies as their inputs, with regard to. Now i the signal is traced rom let to right (i.e. rom the signal in the orm o TX to orm RX, the ollowing equation is obtained: F = + LO y (, ( where F is the requency o the signal denoted in the igure as RX and is given with respect to the tem timescale. In general, the values LO and are tuned such that their dierence models the nominal value o. I we make the assumptions that the satellite s LO is ideal and that the tuning is perect, Equation ( becomes. (3 F = y Using the transormation Equation (5, the transmitted requency would be directly measured at the receiving end (with no up/down requency translation or satellite relay as. (4 F = = y + d + y Since y > y, the observed requency shit (3 is greater than the shit expected rom the transormation (4 by a actor m, which can be derived as ollows: nr m m m m nr nr nr nr y = y = + y = = ( y + y y + + y + + y + * + d + d +. ( a ( b ( c ( d (5 Note that the right side o Equation (5a is simply Equation (3 transormed through the use o the transormation Equation (5. Similar to the derivation o Equation (, proceeding rom right to let produces = + y + LO + y F (, (6 LO which, under the same assumptions as beore, becomes F = + y + (7 ( LO 54
36 th Annual Precise Time and Time Interval (PTTI Meeting and it is seen that when the non-reerence station is the transmitting station, the ractional requency bias is multiplied by a actor o where LO m r = + (8 mr is a coeicient o the clock s ractional requency, as seen by the receiving reerence station. From the above discussion we see that, due to the ractional requency multiplication eect o the Carrier- Phase Two-Way tem, there is a reduction in the eective measurement quantization and, thereore, a reduction in the tem s sensitivity to noise. As a concrete example, consider a receiving station B with a clock that is oscillating mhz slower than the clock at a transmitting Station A, where both clocks have nominal requencies o 5 MHz. This equates to a ractional requency deviation o -. Suppose urther that the receiving Station B is capable o making requency measurements rounded to the nearest tenth o a hertz. Now, i both stations are collocated, no satellite communication is necessary and Station B can take a direct measurement o the Station A output. With an unlimited resolution and in the absence o measurement noise, Station B would record a value o 7,,.4 Hz (assuming an IF nominal transmit requency o 7 MHz. However, the actual value recorded is rounded to 7,,. Hz and the clocks are, thereore, erroneously determined to be in syntonization. Now consider a ull K u -band satellite transer (as opposed to the collocated scenario where the transmitter LO and the receiver LO are tuned to model a satellite turnaround requency o GHz, with the nominal values o 3.93 GHz and.93 GHz, respectively, to produce a 4/ GHz transmit/receive pair when mixed with the 7 MHz IF. Using the coeicient derived in Equation (5, the ractional requency o the received signal is multiplied rom - to approximately -3.43-8 and the rounded measurement F B is then equal to 7,,.4 Hz. From this value, the two Stations will deduce that their ractional requency deviation is FB (7,, 7,,.4 Hz y = = -.99998x -, (9 m 7.43* 7,, Hz nr and the result is very close to the actual clock requency deviation (i.e. -. IV. SOLVING THE SYSTEM The our nonlinear tem equations (Equations (6-(9 may be solved by several dierent numerical techniques. One method begins by linearizing the our equations about some initial estimation vector x = [, k, k, d ] as ollows [3]: δf δf δf δf ( + k ( + k + d = ( + k + d ( + k + + ( + k ( + k + d ( + k + d + ( + k + d ( + k + d + d ( + k ( + k δ ( + d δk + k ( ( + k δk ( + d δd + k + k ( ( 55
36 th Annual Precise Time and Time Interval (PTTI Meeting where For simplicity, Equation ( is re-written as δ F = F, k, k, d F (, k, k, d. ( i i ( i δ F = Gδx ( where G is generally reerred to as the geometry matrix. The state estimate error-vector δ x, o the above equation, contains updates to the estimation vector x and summing the two improves the accuracy o the values contained in x. This method, commonly known as the Newton-Raphson method, may then be iterated to converge δ x toward zero. However, the determinant o matrix G is equal to and the linearized tem is, thereore, singular. As such, no unique solution exists and the matrix tem given by Equation ( is not directly solvable, but necessitates some additional numerical method, such as Singular Value Decomposition (SVD. However, it is known that the values k, k, and d will be very close to, and it can be shown that the tem is resistant to imperect approximations o. Thereore, i the estimation k = k = d = is made, and is assumed to be known and constant at its nominal value, a single iteration will suice and Equation ( is orced to the static equation set + F + + F + = F + + F + + k k. (3 d I Equation (3 is urther reduced by treating k as an independent variable, the tem may be generalized by the geometric igure illustrated below. Figure 5. The three solution points and their average. As shown in Figure 5, Equation (3 produces three lines with three dierent points o intersection: 56
36 th Annual Precise Time and Time Interval (PTTI Meeting d d d P P P3 = = = ( ( ( F ( F ( F F F + ( + ( ( F ( F ( ( F F F ( F F (4 where, i the equality o Equation ( holds, each o these three solution points is expected to be equal. This allows the user the option to use just one equation, involving any three o the F mn, thus sparing a receiver channel or unrelated business. Alternatively, to minimize the eect o noise, the three points may be averaged as ( F F + ( F F 3 ( F F d AVG =. (5 6 ( I all our channels are used, an integrity check can be made based on the consistency o the three solutions in Equations (4. V. RESIDUAL ERRORS Our previous assumption that is static and nominal will cause our resultant approximate solution to deviate rom its ideal position. Likewise, inaccuracies o the linearizations (i.e. truncation o the highorder terms rom the Taylor Series expansions will cause errors in the solution. Other sources o error include (but are not limited to troposphere dynamics, ionosphere dynamics, tem noise, measurement noise, and multipath. An attempt to approximate the eects o some o these error sources is provided below in Table [4,5]. These error sources are non-gaussian. The magnitudes used in the table depict worst-case, but observable, variations. For tau < seconds, the dominating error is the mhz quantization o the TWSTFT modems, which currently provides mhz resolution at IF. Future modem revisions are expected to improve this igure. VI. RESULTS Two preliminary tests have been prepared or inclusion in this report; the results o these test runs make it clear that, while the theoretical aspects o the tem are sound, more work is required beore carrierphase TWSTFT can become ully operational. 57
36 th Annual Precise Time and Time Interval (PTTI Meeting Table. Sensitivity o derived requency transer to modeling errors. The irst test case was conducted using two collocated stations at USNO. The 86,4 ractional requency data points, shown in Figure 6 (below, was calculated with the d P3 result o Equation (4. The data show a modulating requency with a rate o approximately.8 mhz, or about minutes (see Figures 7 and 8, and this is relected in the Allan deviation plot shown in Figure 9. Subsequent tests have suggested that the problem may be traced to low-level requency generations within the spreadspectrum modems. Speciically, i a requency which is to be generated within a modem s direct digital synthesis (DDS block is not an exact multiple o the DDS s minimum incremental resolution, it is compensated or by the phase-lock loop and an artiicial requency modulation can occur at the PLL output. However, it is also possible that the witnessed data variations are simply due to temperature luctuations or other environmental eects. These possibilities are currently under investigation. Figure shows the Allan deviation rom a day o carrier-phase TWFTFT sessions involving stations at both USNO and PTB. While we were unable to produce enough uninterrupted data to provide stability calculations beyond 4 seconds, it is evident that there exist oscillations within these data as well. It is interesting to note, that the oscillations in this data set are at a higher rate than the.8 mhz requency seen in the collocated data. 58
36 th Annual Precise Time and Time Interval (PTTI Meeting VII. CONCLUSION In this paper, we have reined previously published techniques or calculating the carrier-phase two-way solution. The paper suggests a method o solving the tem and then presents a simple error analysis. Worst-case error dependencies related to quantization, linearization, and atmospheric eects o ionosphere and troposphere are shown. We conclude that, with an ideal carrier-phase modem, a -5 requency transer at second is possible. Two-Way modems used at many o today s TAI laboratories have limitations due to the modems current measurement resolution. Our bench tests and live experiments demonstrate these limitations. While it is unclear what is causing the additional noise, several possibilities, including requency synthesis within the modems, are being explored. These unknown eects must be well understood beore the carrier-phase TWSTFT tem becomes ully operational, in order to maintain consistency and accuracy within the tem s error budget. VIII. ACKNOWLEDGMENTS The authors would like to thank Ed Powers, Angela McKinley, Alan Smith, and Paul Wheeler o USNO or their ongoing assistance with this project, and PTB or their contribution o the Two-Way data. IX. REFERENCES [] D. Kirchner, 99, Two-Way Time Transer Via Communication Satellites, Proceedings o the IEEE, 79, 983-99. [] W. Schäer, A. Pawlitzki, and T. Kuhn,, New Trends in Two-Way Time and Frequency Transer via Satellite, in Proceedings o the 3 st Annual Precise Time and Time Interval (PTTI Systems and Applications Meeting, 7-9 December 999, Dana Point, Caliornia, USA (U.S. Naval Observatory, Washington, D.C., pp. 55-54. [3] E. Kaplan, 996, Understanding GPS Principles and Applications (Artech House Publishers, Norwood, Massachusetts, pp. 45-47. [4] P. Misra and P. Enge,, Global Positioning System: Signals, Measurements, and Perormance (Ganga-Jamuna Press, Lincoln, Massachusetts. [5] IGS Central Bureau, IGS Products, International GPS Service, January 5, http://igscb.jpl.nasa.gov/components/prods.html 59
36 th Annual Precise Time and Time Interval (PTTI Meeting Figure 6. Fractional requency collocated data (-second data points collected at USNO. Outliers that exceed a sigma actor o. were removed. Figure 7. A 4-point running average o the collocated ractional requency data accentuates the modulating.8 mhz requency. 6
36 th Annual Precise Time and Time Interval (PTTI Meeting Figure 8. The power spectrum o the collocated ractional requency data shows that the requency o the modulating signal is approximately.8 mhz. Figure 9. Alan deviation plot o the collocated results (-second data. 6
36 th Annual Precise Time and Time Interval (PTTI Meeting Carrier-Phase TWSTFT USNO - PTB Aug, 4 E- E-3 Allan Deviation E-4 E-5 Averaging Time (Seconds Figure. Carrier-phase TWSTFT session between USNO and PTB. 6
36 th Annual Precise Time and Time Interval (PTTI Meeting QUESTIONS AND ANSWERS TOM BAHDER (U.S. Army Research Laboratory: When you set up the whole model, what reerence rame are you in a rotating, or are you in an ECI rame? BLAIR FONVILLE: Well, the satellite is geostationary, so we set up each equation and derive all requencies with respect to a tem reerence station. BAHDER: But then you are implementing everything in a rotating rame, right? And my other question is then there must be some kind o a Sagnac eect that you should be seeing. FONVILLE: Yes, we may see some slight drits due to uncorrected Sagnac in the long term, but we haven t been able to take data over 5-minute spans. Like I say, diurnal eects are also something we need to investigate. BAHDER: You could avoid the Sagnac term i you did the whole implementation in the ECI rame. But then you would have relative velocities o the ground stations and the satellites, and that would have to come into the problem. FONVILLE: The relative velocities o the ground stations to the satellite is something we do account or. Thank you or that comment. CHRISTINE HACKMAN (University o Colorado: You put up a viewgraph where you were talking about all the errors, and you showed two parts in 5 or the troposphere. FONVILLE: Yes, I knew you were going to ask something about that ater your presentation. HACKMAN: So are you assuming it is not going to cancel? Could you elaborate on that a little? FONVILLE: The troposphere will not cancel because we are operating at two dierent sites. HACKMAN: I guess what I normally understood in regular Two-Way, you are kind o going or this. FONVILLE: Oh, I see. Yes, well, we have a transer rom one site to the other and vice versa, but we also have two loopbacks. And so those two loopbacks are where it does not cancel. We have a loopback on one side, we have a loopback on the other; they have dierent tropospheric paths, so it is not going to cancel there. WLODZIMIERZ LEWANDOWSKI (Bureau International des Poids et Mesures: I have a short question. What is your eeling about the uture o this approach? When can we expect the operation technique? FONVILLE: That is a good question. This is all very preliminary, but the results are encouraging. So it does give us some motivation to continue the eort. It can be in operation relatively quickly. But it is contingent on how the longer periods look. For instance, we have had some data that well, in the 5 minutes that I showed, it seems to be dropping without any problem. But we have concatenated contiguous sets o data, and tried to do some numerical calculations on it. And it seems to level o. 63
36 th Annual Precise Time and Time Interval (PTTI Meeting So we may still have some processes that we have not yet ully investigated. So that is a diicult question to answer. 64