Zero Baseline Testing

Navigation Labs (Oliver Montenbruck) Exercise 2: Zero Baseline Test By Hunaiz Ali Hao Ye Jingwei Xie Lina Han November 20, 2014 Earth Oriented Space Science and Technology Technische Universität München 1 Contents 1 Introduction 1.1 The flow of the report . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 2 Basic Mathmatical Analysis 2.1 Mathmatical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mean Value of carrier phase double difference . . . . . . . . . . . . 4 4 5 5 3 Measurement noise of double difference; Analysis 3.1 Satellite Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Comparison of Noise level among Different Observations . . . . . . 6 6 7 4 Estimate DLL and PLL bandwidth 10 4.1 Introduction to DLL and PLL . . . . . . . . . . . . . . . . . . . . . 10 4.2 DLL and PLL Bandwidth calculation . . . . . . . . . . . . . . . . . 11 5 Receiver clock offset 13 5.1 The maximum clock offset difference . . . . . . . . . . . . . . . . . 13 5.2 Asynchronous receivers . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Chapter 1 Introduction The report is based on a pragmatic research task. The task is to analyze the zero baseline tests, where there is a common antenna for two receivers. Mathematical models of double difference measurements are computed and applied on real time data. Using double difference model along with zero baselines lets the study and modeling of measurement noise. We can than explore on what factors does the measurement noise depends on. 1.1 The flow of the report Chapter 1 shows the detail computation of mathematical model for forming double difference. Which effects and errors are eliminated and then the noise of double difference and individual measurement is discussed. Chapter 2 is an implementation of models derive in chapter 1 on real GPS data and noise level comparison along with their trend of double difference of C1, P2, L1 and L2 measurements are made. Chapter 3 explains and computes an estimate of delay lock look (DLL) and phase lock loop (PLL) using data from high and low elevation. Finally the chapter 4 leads the readers to how to assess the maximum clock offset difference between receivers that could be tolerated in a zero baseline test and how to perform a zero-baseline test with asynchronous receivers. The receivers used for the experiment NovAtel OEM-4. 3 Chapter 2 Basic Mathmatical Analysis 2.1 Mathmatical Model We know the phase measurement model for satellite j and receiver A as: LjA = ρjA + cδtA − cδtj + TAj − IAj + bjA + ǫjA (2.1) While ρjA represents the distance between satellite j and receiver A , T represents the tropospheric delay, I means ionospheric delay and ǫ means noise term, b represents the phase ambiguity. For satellite k and receiver B as: LkA = ρkA + cδtA − cδtk + TAk − IAk + bkA + ǫkA (2.2) if receivers A and B individually measure the satellites j and k at the same epoch, then by subtracting equations 2.1 and 2.2 we get the single satellite difference as: jk jk jk jk jk jk ∇Ljk A = ∇ρA + c∇δt + ∇TA − ∇IA + ∇bA + ∇ǫA (2.3) We can observe that the receiver’s clock offset has been eliminated. Similar to equation 2.3 we can get single satellite difference for receiver B and satellites j and k. jk jk jk jk jk jk (2.4) ∇Ljk B = ∇ρB + c∇δt + ∇TB − ∇IB + ∇bB + ∇ǫB Now by subtracting equations 2.3 and 2.4 we get the double difference in Equation 2.5, jk jk jk jk jk ∆∇Ljk AB = ∆∇ρAB + ∆∇TAB − ∆∇IAB + ∆∇bAB + ∆∇ǫAB (2.5) and in this way we get rid of the satellite clock offset. In this lab, we build a zero baseline measurement, which means we have one common antenna with a splitter that provides signals to both receivers A and B at the same time. So we consider receiver A and B to have the same position, troposphere delay and ionosphere delay. Applying the zero baseline condition on double difference model of equation 2.5, we get: jk jk ∆∇Ljk AB = ∆∇bAB + ∆∇ǫAB ∆∇bjk AB is the difference of initial phase ambiguities. ;where If we decompose, we get: 4 (2.6) CHAPTER 2. BASIC MATHMATICAL ANALYSIS j j k k ∆∇bjk AB = bA − bB − bA + bB =λ((NAj − αj + αA ) − (NBj − αj + αB ) − (NAk − αk + αA ) + (NBk − αk + αB ) = λ(NAj − NBj − NAk + NBk ) jk = λ∆∇NAB (2.7) Putting equation 2.7 in equation 2.6, we get the final double difference phase observation model for zero baseline. i.e.: jk jk ∆∇Ljk AB = λ∆∇NAB + ∆∇ǫAB (2.8) In compared to phase observation model we can compute the double difference based on pseudocode range observation. Following the same steps as above, the pseudorange double difference model will be: jk = ∆∇ǫjk ∆∇PAB AB (2.9) By forming double differences in the zero baseline test, satellite and receiver position error, clock error, tropospheric delay, ionospheric delay, multipath errors and some other errors caused by instrumental biases can be eliminated. 2.2 Noise Analysis Here for each measurement between different receiver and satellite, it can be regarded as an independent case, we assume that the noise of each measurement is uncorrected, then the measurement noise model of double difference can be written as: q σ∆∇ε = σ 2 (ǫik ) + σ 2 (ǫil ) + σ 2 (ǫjk ) + σ 2 (ǫjl ) (2.10) Furthermore, here for the zero baseline double difference measurement, we use the same type of receivers and two receivers use the common antenna, so we can also assume that the standard derviation of each measurement noise is the same, and then the formula can be simplified as: σ∆∇ε = p 4σ 2 (ǫ) = 2σ(ǫ) (2.11) where σ(ǫ) is the standard deviation of single measurement noise. 2.3 Mean Value of carrier phase double difference It can be seen from the equation 2.7 that the remained terms after carrier phase double difference are only integer phase ambiguities and the noise. So the relevance of the mean value of the carrier phase double difference is integer phase ambiguities which will stay constant in case of continuous measurement. 5 Chapter 3 Measurement noise of double difference; Analysis In this chapter, the double difference of satellites are computed for C1, P2, L1 and L2 measurements. Later the measurement noise for each individual measurement are shown and discussed. At the end of chapter, the comparison of noise among measurements is made. 3.1 Satellite Selection A pair of satellites with same elevation and Carrier to Noise ratio (C/N0 ) are selected to make a double difference. Having same elevation will let us assume equal signal strength. The argument is supported by choosing equal C/N0 , as can be seen in figure 3.1. Two pairs of satellites are selected for observations; PRN-17 and -32, and PRN-6 and -9. Besides the above mentioned selection criteria, we have choose these pairs to observe how the change in elevation effects on the measurement noise. At the start all four satellites have same elevation. At the end of observation, first pair’s (PRN-17 and -32) elevation decreased while the elevation of pair PRN-6 and -9 increased. We can observe the relation between elevation and measurement noise from the following figures. Noise level is affected by two main factors. First and the most dominant factor is elevation. We can see in figure 3.1 that we started with the same SNR for both pairs of satellites but with the passage of time, elevation of PRN 17 and 32 reduced while the elevation of PRN 6 and 9 is increased. So does the corresponding noise of each pair’s double difference. The results can be interpreted in figures 3.2 and 3.3. So the noise is reacting directly proportional to the elevation. The antenna orientation is vertically up. At 90o elevation of the satellite the antenna gain is maximum while as the elevation decreases the gain is reducing so does the SNR. The antenna gain for the vetically up receiver antenna is also irectly proportional to the elevation angle. Second factor is the transmitter power. L1 signal is trasmitted at higher power compared to the L2 signal. So the SNR of signals at L1 is higher than SNR at L2. Figure 3.4 supports this argument. X-axis represents the satellite number and on y-axis we have the time. The color bars in plot represent each satellite’s SNR 6 CHAPTER 3. MEASUREMENT NOISE OF DOUBLE DIFFERENCE; ANALYSIS Figure 3.1: skyplot and satellite selection corresponding to each epoch. Magnitude ranges of colorbars are self explainatory. 3.2 Comparison of Noise level among Different Observations To explain the noise level of different observations we will first describe them with the help of standard deviation of double difference of each observation type. From the table 3.2 it can be seen that results are quite expected. For both receivers, all observation types react differently but from the magnitude of results it can be concluded that receiver A has more noise compared to receiver A. Table 3.1: Standard deviation of observations PRN-Pair σddo bservation [m] 17 & 32 6 & 9 σddC1 0.5841 0.3164 σddP 2 1.1127 0.2406 σddL1 0.00082 0.00074 σddL2 0.0014 0.00101 We can see that P2 code is more noise than C1 code although both are pseudorange measurement. Similarly L2 carrier phase is more noisy compared to L1. Here it follows the same desciption as discussed in section 3.1 that L1 signals are being transmitte at higher power than L2. For P2 code compared to C1, another possible reason of accuracy is, P2 is encrypted. 7 CHAPTER 3. MEASUREMENT NOISE OF DOUBLE DIFFERENCE; ANALYSIS prn 17 and 32 prn 17 and 32 10 double difference [m] double difference [m] 4 CA 2 0 -2 -4 P2 5 0 -5 -10 0 1000 2000 3000 4000 5000 6000 0 1000 2000 time [s] prn 17 and 32 4000 5000 6000 prn 17 and 32 4 double difference [m] double difference [m] 3 3000 time [s] 2 1 0 L1 -1 2 0 L2 -2 0 1000 2000 3000 4000 5000 6000 0 1000 2000 time [s] 3000 4000 5000 6000 time [s] Figure 3.2: Zero Baseline, PRN 17 and 32 Double Difference prn 6 and 9 prn 6 and 9 2 double difference [m] double difference [m] 2 CA 1 0 -1 -2 P2 1 0 -1 -2 -3 0 1000 2000 3000 4000 5000 6000 0 1000 2000 time [s] prn 6 and 9 4000 5000 6000 prn 6 and 9 0.74 double difference [m] double difference [m] 3.43 3000 time [s] L1 3.428 3.426 3.424 3.422 3.42 L2 0.735 0.73 0.725 0 1000 2000 3000 4000 5000 6000 0 time [s] 1000 2000 3000 4000 5000 6000 time [s] Figure 3.3: Zero Baseline, PRN 6 and 9 Double Difference For the first pair of satellites, several cycleslips can be observed in phase measurement as shown in figure 3.2. So the values shown in table for σddL1 and σddL2 for the PRN pair 17 & 32 are averaged values of 3 parts. 8 CHAPTER 3. MEASUREMENT NOISE OF DOUBLE DIFFERENCE; ANALYSIS Rcvr A:SNRf1 [dB] 0 Rcvr A: SNRf2 [dB] 0 50 50 1000 1000 48 45 46 2000 2000 40 time [s] time [s] 44 42 3000 3000 35 40 38 4000 4000 30 5000 25 36 5000 34 32 5 10 15 20 25 30 5 10 PRN 15 PRN Figure 3.4: SNR at f1 and f2 9 20 25 30 Chapter 4 Estimate DLL and PLL bandwidth In this chapter, an estimate of DLL and PLL bandwidth for C\A code and L1 carrier phase for high and low elevation satellites will be performed. 4.1 Introduction to DLL and PLL A phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is related to the phase of an input signal. The DelayLock Loop (DLL) is similar with PLL, aiming at tracking the phase delay of the incoming GNSS signal. In DLL or PLL, the input signal will be correlated with the replica signal and we look for the phase which makes maximum correlation. 1 chip early-late correlator spacing used to be used in DLL, while nowadays we are able to achieve 0.05-0.1 chip correlator spacing, reducing tracking errors in the presence of both noise and multipath. In order to calculate the bandwidth, we introduce Equation 4.1 and 4.2, which we can usually used in practice to get the expected value of error. From them we can see that for a fixed bandwidth of lock loop operation, error decrease with the growing of signal to noise ratio. s BDLL [Hz] · d[chips]/2 (4.1) σC/A = 293m SA[dB−Hz] 10 10 s BP LL [Hz] λ σP LL = (4.2) 2π 10 SA[dB−Hz] 10 From Equation 4.1 and 4.2 we can get Equation 4.3 and 4.4. BDLL [Hz] = ( SA[dB−Hz] σC/A 2 2 ) · · 10 10 293m d[chips] (4.3) SA[dB−Hz] 2π · σP LL 2 ) · 10 10 λ (4.4) BP LL [Hz] = ( 10 CHAPTER 4. ESTIMATE DLL AND PLL BANDWIDTH Figure 4.1: Satellite position at the end of observation 4.2 DLL and PLL Bandwidth calculation To form a double difference, we need to choose 2 satellites. From Figure 4.1 we can see satellite position at the end of our observation. We choose a pair of high elevation satellites with PRN 9 and 23, and a pair of low elevation satellites with PRN 2 and 13. Since the position of satellites changes with time, observation records measured within 5 minutes before the end of measurement are used, during which we assume elevation of satellites do not change much. Since receiver chip employs a 0.1 chip early-late correlator spacing, d = 0.1[chip] and the wavelength of L1: λ= 299, 792, 458m/s = 0.1903m 1, 575.42M Hz (4.5) [1] Parameter σC/A and σP LL is the thermal noise standard deviation which contribute most for the noise of measurement. We form the double difference using Euqation 2.8 and 2.9. Since the measurement error is doubled from Equation 2.11, we consider half of double difference standard deviation as the standard deviation of C/A and PLL. For SA we just use the signal to noise ratio(SNR) at frequency f1. These parameters are computed in MATLAB and listed in Table 4.1. The signal to noise ratio of high elevation satellite is apearently higher than the lower ones, and the standard deviation of noise is also smaller for higher elevation satellites. Table 4.1: Parameters calculated from measurements Parameter Low el. SAT (SNR 2, 13) High el. SAT (SNR 9, 23) C/Nmean on L1 42.07 50.64 σC/A [m] 0.32 0.08 σL2 [m] 6.27e-4 2.41e-4 11 CHAPTER 4. ESTIMATE DLL AND PLL BANDWIDTH With all above known parameters and equation 4.3 and 4.4, we can get bandwidth of DLL and PLL, listed in Table 4.2. Table 4.2: Range of DLL and PLL bandwidth Low el. SAT (SNR 2, 13) High el. SAT (SNR 9, 23) BDLL [Hz] 0.38 0.18 6.93 7.34 BP LL [Hz] 12 Chapter 5 Receiver clock offset The NovAtel OEM-4 receiver continuously steers its clock to GPS time and residual clock offsets are generally less than 20m. The measurements from two receivers in a zero baseline test are therefore well synchronized and can be differenced without further precautions. Many other receivers (Javad, Ashtech, Septentrio) exhibit clock offsets of up to 0.5 ms and measurements from different receivers are thus collected at slightly different epochs. 5.1 The maximum clock offset difference The goal of this section is to assess the maximum clock offset difference between receivers that could be tolerated in a zero baseline test. Before looking at the effect of the receiver clock offset on performance, it helps to remind ourselves what the clock offset actually is. In brief, the offset represents the difference between what time the receiver thinks it is, and the true time, with the latter determined by the [3] underlying GNSS atomic time scale. In this zero baseline test, the purpose is to determine the noise level of the pseudorange, carrier phase and Doppler measurements as a function of the signal-to-noise [4] ratio. The test is done under the assumption that the receiver clock offset is small enough to be neglected. So only when the effect caused by a receiver clock offset is smaller than the noise of the receiver, the zero baseline test can tolerate the difference. The affect caused by receiver clock offset can be described by the following formula. af f ect = δreceiver ∗ vrelatvive (5.1) Theoretically, the noise level of pseudorange measurement is about 1m while that of phase measurement is about 1mm. And the velocity of a GNSS satellite with [2] respect to the receiver is of the order of 3 km/s. According to formula(5.1), the maximum time interval for both measurement methods can be computed. δpseudo = 1[m]/4[km/s] = 2.5 ∗ 10−4 [s] = 0.25[ms] (5.2) δphase = 1[mm]/4km/s] = 2.5 ∗ 10−7 [s] = 250[ns] (5.3) In our case, the noise analysis is done by the double difference of both methods. This means the clock offset should satisfy both standard, so the maximum offset is 13 CHAPTER 5. RECEIVER CLOCK OFFSET 250[ns]. In other words, if the receiver clock offset is larger than 333[ns], this will tolerate the zero base line test. The receiver used in this project is NovAtel OEM4 receiver. It continuously steers its clock to GPS time and residual clock offsets are generally less than 20m. The the offset can be computed by the following formular. δreceiver = 20[m]/3 ∗ 108 [m/s] = 66.7[ns] (5.4) 66.7[ns] is smaller than the threshold value 333 [ns], so the receiver clock effect is negligible and the test assumption is fulfilled. However, other receiver whose clock offset is 0.5 ms cannot be used directly in the zero baseline test. 5.2 Asynchronous receivers As mentioned previously, the assumption of the test is that the receivers’ closks are synchronous. If they are asynchronous, the receiver clock offset will disturb the result. So we should compute the receiver clock offset. The compution method is Least Square Adjustment. In the following we will take the simple model of [5] peudorange measurement as example. Phase measurement can be done similarly. The pseudorange ρ̃(i) from the i-th satelliteis given by Written on 20/11/2014 14 Bibliography [1] Phillip W. Ward, John W. Betz and Christopher J. Hegarty. Satellite Signal Acquisition and Tracking; chap. 5 in: Kaplan E. D. ed.; Understanding GPS Principles and Applications(1996). Artech House Publishers, 1996. [2] Hugentobler, U. and Gunther, C. ESPACE Lecture on Satellite Navigation and Advance Orbit Mechanics: Observation Equations. TU Munich, 2014. [3] Petovello, M. GNSS solutions: Clock offsets in GNSS receivers. InsideGNSS, 2011. [4] Montenbruck. Performance assessment of the novatel OEM4-G2 receiver for LEO satellite tracking. Technical Note TN, 2003. [5] Christoph, G. Differential Navigation. TU Munich, 2014. 15