By Ruizhi Chen, Heidi Kuusniemi, Yuwei Chen, Ling Pei, Wei Chen, Jingbin Liu, Helena Leppäkoski, Jarmo Takala
Currently, no single technology, system, or sensor can provide a positioning solution any time, anywhere. The key is to utilize multiple technologies. We are now exploring a multi-sensor multi-network (MSMN) approach for a seamless indoor-outdoor solution. Its hardware platform is described in the previous article. The digital signal processor (DSP) is embedded in the GPS module. All sensors are integrated to the DSP that hosts core software for real-time sensor data acquisition and real-time processing to estimate user location. A smartphone handset provides wireless network measurements.
Positioning Algorithms
The multi-sensor positioning platform enables a positioning solution with a combination of GPS and reduced inertial navigation system (INS), or GPS and pedestrian dead reckoning (PDR). The reduced INS consists of a 3D accelerometer and a 2D digital compass, as a low-cost alternative to augment GNSS positioning. The reduced INS combined with GPS uses a loosely coupled Kalman filter for data integration, while the combination of PDR and GPS uses algorithms for estimating the position change with pedestrian step-length estimation.
PDR. The PDR solution uses human physiological characteristics, implemented in a local-level frame, with equations:
where k denotes the current epoch, Y is the coordinate in East direction, X is the coordinate in North direction, S is step length, and φ is the heading.
The PDR positioning algorithm includes step detection, step length estimation, determination of heading, and positioning.
To achieve an accurate heading, compass measurements are corrected with an empirical online estimated error model, which requires some training data.
WLAN and Bluetooth. Figure 1 describes the basic concept of the WLAN or Bluetooth locating solution using a fingerprint database approach. The circles around the access point (AP) in the figure represent the radio coverage area and the color the signal strength. This radio map is a simplified example representing measurements from just one AP.
FIGURE 1. Sample WLAN or Bluetooth fingerprint map, in meters.
For the fingerprinting approach, the received signal strength indicators (RSSIs) are the basic observables. The whole process consists of a training phase and a positioning phase. During the training phase, a radio map of probability distribution of the received signal strength is constructed for the targeted area. The targeted area is divided into a matrix of grids, and the central point of each grid is referred to as a reference point. The probability distribution of the received signal strength at each reference point is represented by a Weibull function, and the parameters of the Weibull function are estimated with the limited number of training observation samples. Based on the constructed radio map, the positioning phase determines the current location using the measured RSSI observations in real time.
Given the observation vector , the problem is to find the most probable location (l) with the maximized conditional probability , maximized by Bayesian theorem as:
We applied an assumption of Hidden Markov Models (HMM) to represent the pedestrian movement process. The locating problem is then translated into finding such a state sequence (locations) that is most likely to have generated the output sequence (the measured RSSIs) assuming the given HMM model. The Viterbi algorithm typically solves these kinds of problems efficiently. This study also utilizes the Viterbi algorithm to trace the user trajectory.
MSMN. The general integration scheme combining the GPS output, sensor measurements, WLAN, or Bluetooth output, and their variance estimates is depicted in Figure 2. A simplified representation of the central filter combining different input sources can be described with typical Kalman filter equations. The measurement model is zk= Hkxk+vk where the state estimate vector is ,
with X, Y, and φ as previously defined, and S the user horizontal velocity (speed). The measurement vector is given as
where g refers to GPS, W to WLAN/Bluetooth, acc to accelerometer, and dc to digital compass. The matrix Hk is the design matrix of the system and the vector vk is the measurement error vector.
FIGURE 2. Integration scheme for multi-sensor, multi-network positioning approach
The recursive sequence includes prediction and update steps. The prediction step includes the typical equations of
and
while the update step includes
Indoor Test Results
A field test has been carried out on a sports field, described in the accompanying article (see Going 3D). An indoor test was carried out in an office-building corridor, but the test started and ended in an outdoor terrace area. During the test, the indoor corridor was covered with eight WLAN and three BT APs.
Figure 3 shows the positioning results of the GPS-only (red), Bluetooth-only (black), and WLAN-only (magenta) solutions; Figure 4 shows that of the integrated multi-sensor multi-network (MSMN) solution (blue) for an outdoor-indoor-outdoor test. A reference trajectory is in green in both figures and building outlines in grey. The position update rate achievable by the WLAN and Bluetooth fingerprinting approach is only 0.1 Hz whereas the GPS-only and the integrated MSMN solutions are obtained every second and thus have a higher availability.
FIGURE 3. Pedestrian test results with GPS-only, BT-only, and WLAN-only positioning approaches with respect to a reference trajectoryFIGURE 4. Pedestrian test result with the multi-sensor multi-network positioning approach with respect to a reference trajectory
Figure 5 shows the horizontal errors obtained with the different positioning solutions over time in the indoor test. A mean horizontal error of 2.2 meters was achieved with the WLAN solution. The Bluetooth solution is not as accurate as the WLAN solution, due to the smaller amount of BT APs; it achieved a mean horizontal error of 5.1 meters. When moving inside the corridor, the GPS solutions are used for the MSMN integration only with very low weights due to their poor quality. GPS is mainly used as a source of location outdoors where the test starts and ends. The mean horizontal error of the GPS-only solutions during the whole test is 8.4 meters. WLAN- and Bluetooth-derived locations and the self-contained sensors are the main sources used inside the building for the MSMN positioning solution: the mean horizontal accuracy o
btained with MSMN is 2.7 meters with a solution availability of 1 Hz.
FIGURE 5. Horizontal errors of GPS-only, BT-only, WLAN-only and the MSMN positioning approaches with respect to time in the pedestrian indoor test
The MSMN solution obviously performs much better than a GPS-only solution indoors. The track of the pedestrian walking inside the corridor can be identified clearly, which is not the case with typical approaches of GPS-only or GPS/low-cost sensors. WLAN fingerprinting provides good position accuracy indoors, but the MSMN solution provides the best result when taking into account positioning accuracy and the solution availabilities in both time and space domains.
Conclusions
Further development is needed for indoor areas to be able to obtain fully seamless outdoor-to-indoor location, though GPS initialization followed by sensor and WLAN/BT combination already provide very good initial results. Additional sensors and more refined pedestrian-specific algorithms will be added to further improve the positioning accuracy.
Enhancing GNSS Receiver Sensitivity by Combining Signals from Multiple Satellites
By Penina Axelrad, James Donna, Megan Mitchell, and Shan Mohiuddin
A new approach to enhancing signal sensitivity combines the received signal power from multiple satellites in a direct-to-navigation solution.
INNOVATION INSIGHTS by Richard Langley
ALTHOUGH I HAVE MANAGED the Innovation column continuously since GPS World’s first issue, it wasn’t until the second issue that I authored a column article. That article, co-written with Alfred Kleusberg, was titled “The Limitations of GPS.” It discussed some of the then-current problems of GPS, including poor signal reception, loss of signal integrity, and limited positioning accuracy.
In the ensuing 20 years, both signal integrity and positioning accuracy have improved significantly. Advances in the GPS control segment’s capabilities to continuously monitor and assess signal performance, together with receiver-autonomous integrity monitoring and integrity enhancement provided by augmentation systems, have reduced worries about loss of signal integrity. The removal of Selective Availability and use of error corrections provided by augmentation systems, among other approaches, have improved positioning accuracy.
But the problem of poor reception due to weak signals is still with us. In that March/April 1990 article, we wrote “[GPS] signals propagate from the satellites to the receiver antenna along the line of sight and cannot penetrate water, soil, walls, or other obstacles very well. … In surface navigation and positioning applications, the signal can be obstructed by trees, buildings, and bridges. … [In] the inner city streets of urban areas lined with skyscrapers, the ‘visibility’ of the GPS satellites is very limited. In such areas, the signals can be obstructed for extended periods of time or even [be] continuously unavailable.”
Poor signal reception in other than open-sky environments is still a problem with conventional GPS receivers. However, extending signal integration times and using assisted-GPS techniques can give GPS some degree of capability to operate indoors and in other restricted environments, albeit typically with reduced positioning accuracy. An antenna with sufficient gain is needed and capable systems are available on the market. The pilot channels of modernized GNSS signals will also benefit signal acquisition and tracking in challenging environments.
In this month’s column, we look at a completely different approach to enhancing signal sensitivity. Rather than requiring each satellite’s signal to be acquired and tracked before it can be used in the navigation solution, the new approach — dubbed “collective detection” — combines the received signal power from multiple satellites in a direct-to-navigation-solution procedure. Besides providing a quick coarse position solution with weak signals, this approach can be used to monitor the signal environment, aid deeply-coupled GPS/inertial navigation, and assist with terrain and feature recognition.
“Innovation” features discussions about advances in GPS technology, its applications, and the fundamentals of GPS positioning. The column is coordinated by Richard Langley, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
Growing interest in navigating indoors and in challenging urban environments is motivating research on techniques for weak GPS signal acquisition and tracking. The standard approach to increasing acquisition and tracking sensitivity is to lengthen the coherent integration times, which can be accomplished by using the pilot channels in the modernized GPS signals or by using assisted GPS (A-GPS) techniques. These techniques operate in the traditional framework of independent signal detection, which requires a weak signal to be acquired and tracked before it is useful for navigation. This article explores a complementary, but fundamentally different, approach that enhances signal sensitivity by combining the received power from multiple GPS satellites in a direct-to-navigation-solution algorithm. As will be discussed in the following sections, this collective detection approach has the advantage of incorporating into the navigation solution information from signals that are too weak to be acquired and tracked, and it does so with a modest amount of computation and with no required hardware changes. This technology is appropriate for any application that requires a navigation solution in a signal environment that challenges traditional acquisition techniques. Collective detection could be used to monitor the signal environment, aid deeply coupled GPS/INS during long outages, and help initiate landmark recognition in an urban environment. These examples are explained further in a subsequent section. In order to understand how the collective detection algorithm works, it is instructive to first consider the traditional approach to acquisition and tracking.
Acquisition Theory and Methods
In a typical stand-alone receiver, the acquisition algorithm assesses the signal’s correlation power in discrete bins on a grid of code delay and Doppler frequency (shift). The correlation calculations take the sampled signal from the receiver’s RF front end, mix it with a family of receiver-generated replica signals that span the grid, and sum that product to produce in-phase (I) and quadrature (Q) correlation output. The correlation power is the sum of the I and Q components, I2 + Q2. Plotting the power as a function of delay and frequency shift produces a correlogram, as shown in FIGURE 1. It should be noted that both correlation power and its square root, the correlation amplitude, are found in the GPS literature. For clarity, we will always use the correlation power to describe signal and noise values.
If a sufficiently powerful signal is present, a distinct peak appears in the correlogram bin that corresponds to the GPS signal’s code delay and Doppler frequency. If the peak power exceeds a predefined threshold based on the integration times and the expected carrier-to-noise spectral density, the signal is detected. The code delay and Doppler frequency for the peak are then passed to the tracking loops, which produce more precise measurements of delay — pseudoranges — from which the receiver’s navigation solution is calculated. When the satellite signal is attenuated, however, perhaps due to foliage or building materials, the correlation peak cannot be distinguished and the conventional approach to acquisition fails.
The sensitivity of traditional tracking algorithms is similarly limited by the restrictive practice of treating each signal independently. More advanced tracking algorithms, such as vector delay lock loops or deeply integrated filters, couple the receiver’s tracking algorithms and its navigation solution in order to take advantage of the measurement redundancy and to leverage information gained from tracking strong signals to track weak signals. The combined satellite detection approach presented in this article extends the concept of coupling to acquisition by combining the detection and navigation algorithms into one step.
Collective Detection
In the collective detection algorithm, a receiver position and clock offset grid is mapped to the individual GPS signal correlations, and the combined correlation power is evaluated on that grid instead of on the conventional independent code delay and Doppler frequency grids. The assessment of the correlation power on the position and clock offset grid leads directly to the navigation solution. The mapping, which is key to the approach, requires the receiver to have reasonably good a priori knowledge of its position, velocity, and clock offset; the GPS ephemerides; and, if necessary, a simplified ionosphere model. Given this knowledge, the algorithm defines the position and clock offset search grid centered on the assumed receiver state and generates predicted ranges and Doppler frequencies for each GPS signal, as illustrated in FIGURE 2. The mapping then relates each one of the position and clock offset grid points to a specific code delay and Doppler frequency for each GPS satellite, as illustrated in FIGURE 3. Aggregating the multiple delay/Doppler search spaces onto a single position/clock offset search space through the mapping allows the navigation algorithm to consider the total correlation power of all the signals simultaneously. The correlation power is summed over all the GPS satellites at each position/clock-offset grid point to create a position domain correlogram. The best position and clock-offset estimates are taken as the grid point that has the highest combined correlation power. This approach has the advantage of incorporating into the position/clock-offset estimate information contained in weak signals that may be undetectable individually using traditional acquisition/tracking techniques.
It should be noted that a reasonable a priori receiver state estimate restricts the size of the position and clock-offset grid such that a linear mapping, based on the standard measurement sensitivity matrix used in GPS positioning, from the individual signal correlations, is reasonable. Also, rather than attempt to align the satellite correlations precisely enough to perform coherent sums, noncoherent sums of the individual satellite correlations are used. This seems reasonable, given the uncertainties in ranging biases between satellites, differences and variability of the signal paths through the ionosphere and neutral atmosphere, and the large number of phases that would have to be aligned.
Applications
The most obvious application for collective detection is enabling a navigation fix in circumstances where degraded signals cause traditional acquisition to fail. The sweet spot of collective detection is providing a rapid but coarse position solution in a weak signal environment. The solution can be found in less time because information is evaluated cohesively across satellites. This is especially clear when the algorithm is compared to computationally intensive long integration techniques.
There are several ways that collective detection can support urban navigation. This capability benefits long endurance users who desire a moderate accuracy periodic fix for monitoring purposes. In some circumstances, the user may wish to initiate traditional tracking loops for a refined position estimate. However, if the signal environment is unfavorable at the time, this operation will waste valuable power. The collective detection response indicates the nature of the current signal environment, such as indoors or outdoors, and can inform the decision of whether to spend the power to transition to full GPS capabilities.
In urban applications, deeply integrated GPS/INS solutions tolerate GPS outages by design. However, if the outage duration is too long, the estimate uncertainty will eventually become too large to allow conclusive signal detection to be restored. Running collective detection as a background process could keep deeply integrated filters centered even in long periods of signal degradation. Because collective detection approaches the acquisition problem from a position space instead of the individual satellite line-of-sight space, it provides inherent integrity protection. In the traditional approach, acquiring a multipath signal will pollute the overall position fix. In collective detection, such signals are naturally exposed as inconsistent with the position estimate.
Another use would be to initialize landmark correlation algorithms in vision navigation. Landmark correlation associates street-level video with 3D urban models as an alternative to (GPS) absolute position and orientation updates. This technique associates landmarks observed from ground-level imagery with a database of landmarks extracted from overhead-derived 3D urban models. Having a coarse position (about 100 meters accuracy) enhances initialization and restart of the landmark correlation process. Draper Laboratory is planning to demonstrate the utility of using collective detection to enable and enhance landmark correlation techniques for urban navigation.
In all of these applications, collective detection is straightforward to implement because it simply uses the output of correlation functions already performed on GPS receivers.
Simulations and Processing
The new algorithm has been tested using live-sky and simulated data collected by a Draper Laboratory wideband data recorder. A hardware GPS signal simulator was used to simulate a stationary observer receiving 11 equally powered GPS signals that were broadcast from the satellite geometry shown in FIGURE 4. The data recorder and the signal simulator were set up in a locked-clock configuration with all of the simulator’s modeled errors set to zero. No frequency offsets should exist between the satellites and the receiver. A clock bias, however, does exist because of cable and other fixed delays between the two units. The data recorder houses a four-channel, 14-bit A/D module. It can support sample rates up to 100 MHz. For this work, it was configured to downconvert the signal to an IF of 420 kHz and to produce in-phase and quadrature samples at 10 MHz.
Results and Discussion
To combine satellites, a position domain search space is established, centered on the correct location and receiver clock bias. A grid spacing of 30 meters over a range of ± 900 meters in north and east directions, and ± 300 meters in the vertical. In the first simulated example, the correlation power for all the satellites is summed on the position grid using a single 1-millisecond integration period. In this case, the true carrier-to-noise-density ratio for each signal is 40 dB-Hz. The results are shown in FIGURE 5. The plots in the left panel show the individual signal correlations as a function of range error. The four plots in the upper-right panel show several views of the combined correlation as a function of position error. The upper-left plot in the panel shows the correlation value as a function of the magnitude of the position error. The upper-right plot shows the correlation as a function of the north-east error, the lower-left the north-down error, and the lower-right the east-down error. Notice how the shape of the constant power contours resembles the shape of the constant probability contours that would result from a least-squares solution’s covariance matrix. The final plot, the bottom-right panel, shows a 3D image of the correlation power as a function of the north-east error. It is clear in these images that in the 40 dB-Hz case each satellite individually reaches the highest correlation power in the correct bin and that the combined result also peaks in the correct bin. In the combined satellite results, each individual satellite’s correlation power enters the correlogram as the ridge that runs in a direction perpendicular to the receiver-satellite line-of-sight vector and represents a line of constant pseudorange.
FIGURE 6 shows a similar set of graphs for a simulator run at 20 dB-Hz. The plots in the left panel and the four plots in the upper-right panel show the individual and combined correlations, as in Figure 5. In the lower-right panel, the 3D image has been replaced with correlations calculated using 20 noncoherent 1-millisecond accumulations. The indistinct peaks in many of the individual correlations (left panel) suggest that these signals may not be acquired and tracked using traditional methods. Those signals, therefore, would not contribute to the navigation solution. Yet in the combined case, those indistinct peaks tend to add up and contribute to the navigation solution. These results indicate the feasibility of using the information in weak signals that may not be detectable using traditional methods and short acquisition times. The situation is further improved by increasing the number of noncoherent integration periods.
Impact of Reduced Geometry. Of course, it is a bit unrealistic to have 11 satellites available, particularly in restricted environments, so we also considered three subsets of four-satellite acquisitions, under the same signal levels. FIGURE 7 compares the position domain correlograms for the following 20 dB-Hz cases: (1) a good geometry case (PRNs 3, 14, 18, 26), (2) an urban canyon case where only the highest 4 satellites are visible (PRNs 15, 18, 21, 22), and (3) a weak geometry case where just a narrow wedge of visibility is available (PRNs 18, 21, 26, 29). As expected, the correlation power peak becomes less distinct as the satellite geometry deteriorates. The pattern of degradation, morphing from a distinct peak to a ridge, reveals that the position solution remains well constrained in some directions, but becomes poorly constrained in others. Again, this result is expected and is consistent with the behavior of conventional positioning techniques under similar conditions.
Focusing on Clock Errors. In some real-world situations, for example, a situation where a receiver is operating in an urban environment, it is possible for the position to be fairly well known, but the clock offset and frequency to have substantial uncertainty. FIGURE 8 shows how the combined satellites approach can be used to improve sensitivity when viewed from the clock bias and frequency domain. The figure presents example 1-millisecond correlograms of clock bias and clock drift for three 20 dB-Hz cases: (1) a single GPS satellite case; (2) a four-satellite, good geometry case; and (3) an 11-satellite, good geometry case. The assumed position solution has been offset by a random amount (generated with a 1-sigma of 100 meters in the north and east components, and 20 meters in the up component), but no individual satellite errors are introduced. These plots clearly show the improved capability for acquisition of the clock errors through the combining process.
Live Satellite Signals. FIGURE 9 shows combined correlograms derived from real data recorded using an outdoor antenna. The first example includes high-signal-level satellites with 1.5-second noncoherent integration. The second example includes extremely attenuated satellite signals with a long noncoherent integration period of six seconds.
The plots in the upper-left and upper-right panels show combined correlograms as a function of the north-east position error for satellite signals with carrier-to-noise-density ratios of 48 dB-Hz or higher. The plots in the lower-left and lower-right panels show combined correlograms resulting from much weaker satellites with carrier-to-noise-density ratios of roughly 15 to 19 dB-Hz, using a coherent integration interval of 20 milliseconds and a noncoherent interval of six seconds. FIGURE 10 shows one of the individual single-satellite correlograms. In this attenuated case, the individual satellite power levels are just barely high enough to make them individually detectable. This is the situation in which collective detection is most valuable.
Conclusions
The example results from a hardware signal simulator and live satellites show how the noncoherent combination of multiple satellite signals improves the GPS position error in cases where some of the signals are too weak to be acquired and tracked by traditional methods. This capability is particularly useful to a user who benefits from a rapid, but coarse, position solution in a weak signal environment. It may be used to monitor the quality of the signal environment, to aid deeply coupled navigation, and to initiate landmark recognition techniques in urban canyons. The approach does require that the user have some a priori information, such as a reasonable estimate of the receiver’s location and fairly accurate knowledge of the GPS ephemerides. Degradation in performance should be expected if the errors in these models are large enough to produce pseudorange prediction errors that are a significant fraction of a C/A-code chip. Absent that issue, the combined acquisition does not add significant complexity compared to the traditional approach to data processing. It can be used to enhance performance of existing acquisition techniques either by improving sensitivity for the current noncoherent integration times or by reducing the required integration time for a given sensitivity. Further development and testing is planned using multiple signals and frequencies.
Acknowledgments
The authors appreciate the contributions of David German and Avram Tewtewsky at Draper Laboratory in collecting and validating the simulator data; Samantha Krenning at the University of Colorado for assistance with the simulator data analysis and plotting; and Dennis Akos at the University of Colorado for many helpful conversations and for providing the Matlab software-defined radio code that was used for setting up the acquisition routines. This article is based on the paper “Enhancing GNSS Acquisition by Combining Signals from Multiple Channels and Satellites” presented at ION GNSS 2009, the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation, held in Savannah, Georgia, September 22–25, 2009.
The work reported in the article was funded by the Charles Stark Draper Laboratory Internal Research and Development program.
Manufacturers
Data for the analyses was obtained using a Spirent Federal Systems GSS7700 GPS signal simulator and a GE Fanuc Intelligent Platforms ICS-554 A/D module.
PENINA AXELRAD is a professor of aerospace engineering sciences at the University of Colorado at Boulder. She has been involved in GPS-related research since 1986 and is a fellow of The Institute of Navigation and the American Institute of Aeronautics and Astronautics.
JAMES DONNA is a distinguished member of the technical staff at the Charles Stark Draper Laboratory in Cambridge, Massachusetts, where he has worked since 1980. His interests include GNSS navigation in weak signal environments and integrated inertial-GNSS navigation.
MEGAN MITCHELL is a senior member of the technical staff at the Charles Stark Draper Laboratory. She is involved with receiver customization for reentry applications and GPS threat detection.
SHAN MOHIUDDIN is a senior member of the technical staff at the Charles Stark Draper Laboratory. His interests include GNSS technology, estimation theory, and navigation algorithms.
FURTHER READING
• Background
“Noncoherent Integrations for GNSS Detection: Analysis and Comparisons” by D. Borio and D. Akos in IEEE Transactions on Aerospace and Electronic Systems, Vol. 45, No. 1, January 2009, pp. 360–375 (doi: 10.1109/TAES.2009.4805285).
“Impact of GPS Acquisition Strategy on Decision Probabilities” by D. Borio, L. Camoriano, and L. Lo Presti in IEEE Transactions on Aerospace and Electronic Systems, Vol. 44, No. 3, July 2008, pp. 996–1011 (doi:10.1109/TAES.2008.4655359).
“Understanding the Indoor GPS Signal” by T. Haddrell and A.R. Pratt in Proceedings of ION GPS 2001, the 14th International Technical Meeting of the Satellite Division of The Institute of Navigation, Salt Lake City, Utah, September 11–14, 2001, pp. 1487–1499.
“The Calculation of the Probability of Detection and the Generalized Marcum Q-Function” by D.A. Shnidman in IEEE Transactions on Information Theory, Vol. 35, No. 2, March 1989, pp. 389–400 (doi: 10.1109/18.32133).
• Weak Signal Acquisition and Tracking
“Software Receiver Strategies for the Acquisition and Re-Acquisition of Weak GPS Signals” by C. O’Driscoll, M.G. Petovello, and G. Lachapelle in Proceedings of The Institute of Navigation 2008 National Technical Meeting, San Diego, California, January 28-30, 2008, pp. 843–854.
“Deep Integration of Navigation Solution and Signal Processing” by T. Pany, R. Kaniuth, and B. Eissfeller in Proceedings of ION GNSS 2005, the 18th International Technical Meeting of the Satellite Division of The Institute of Navigation, Long Beach, California, September 13–16, 2005, pp. 1095–1102.
“Deeply Integrated Code Tracking: Comparative Performance Analysis” by D. Gustafson and J. Dowdle in Proceedings of ION GPS 2003, the 16th International Technical Meeting of the Satellite Division of The Institute of Navigation, Portland, Oregon, September 9–12, 2003, pp. 2553–2561.
“Block Acquisition of Weak GPS Signals in a Software Receiver” by M.L. Psiaki in Proceedings of ION GPS 2001, the 14th International Technical Meeting of the Satellite Division of The Institute of Navigation, Salt Lake City, Utah, September 11–14, 2001, pp. 2838–2850.
• General Combining Techniques
“Coherent, Non-Coherent, and Differentially Coherent Combining Techniques for the Acquisition of New Composite GNSS Signals” by D. Borio, C. O’Driscoll, and G. Lachapelle, in IEEE Transactions on Aerospace and Electronic Systems, Vol. 45, No. 3, July 2009, pp. 1227–1240.
“Comparison of L1 C/A-L2C Combined Acquisition Techniques” by C. Gernot, K. O’Keefe, and G. Lachapelle in Proceedings of the European Navigation Conference ENC-GNSS 2008, Toulouse, France, April 23–25, 2008, 9 pp.
Performance Analysis of the Parallel Acquisition of Weak GPS Signals by C. O’Driscoll, Ph.D. dissertation, National University of Ireland, Cork, 2007; available on line: <http://plan.geomatics.ucalgary.ca/papers/phd_thesis_cillian_odriscoll_jan07.pdf>.
• Coherent Combining of Signals from Multiple Satellites
“GPS PRN Code Signal Processing and Receiver Design for Simultaneous All-in-View Coherent Signal Acquisition and Navigation Solution Determination” by R. DiEsposti in Proceedings of The Institute of Navigation 2007 National Technical Meeting, San Diego, California, January 22–24, 2007, pp. 91–103.
To meet the challenges inherent in producing a low-cost, highly CPU-efficient software receiver, the multiple offset post-processing method leverages the unique features of software GNSS to greatly improve the coverage and statistical validity of receiver testing compared to traditional, hardware-based testing setups, in some cases by an order of magnitude or more.
By Alexander Mitelman, Jakob Almqvist, Robin Håkanson, David Karlsson, Fredrik Lindström, Thomas Renström, Christian Ståhlberg, and James Tidd, Cambridge Silicon Radio
Real-world GNSS receiver testing forms a crucial step in the product development cycle. Unfortunately, traditional testing methods are time-consuming and labor-intensive, particularly when it is necessary to evaluate both nominal performance and the likelihood of unexpected deviations with a high level of confidence. This article describes a simple, efficient method that exploits the unique features of software GNSS receivers to achieve both goals. The approach improves the scope and statistical validity of test coverage by an order of magnitude or more compared with conventional methods.
While approaches vary, one common aspect of all discussions of GNSS receiver testing is that any proposed testing methodology should be statistically significant. Whether in the laboratory or the real world, meeting this goal requires a large number of independent test results. For traditional hardware GNSS receivers, this implies either a long series of sequential trials, or the testing of a large number of nominally identical devices in parallel. Unfortunately, both options present significant drawbacks.
Owing to their architecture, software GNSS receivers offer a unique solution to this problem. In contrast with a typical hardware receiver application-specific integrated circuit (ASIC), a modern software receiver typically performs most or all baseband signal processing and navigation calculations on a general-purpose processor. As a result, the digitization step typically occurs quite early in the RF chain, generally as close as possible to the signal input and first-stage gain element. The received signal at that point in the chain consists of raw intermediate frequency (IF) samples, which typically encapsulate the characteristics of the signal environment (multipath, fading, and so on), receiving antenna, analog RF stage (downconversion, filtering, and so on), and sampling, but are otherwise unprocessed. In addition to ordinary real-time operation, many software receivers are also capable of saving the digital data stream to disk for subsequent post-processing. Here we consider the potential applications of that post-processing to receiver testing.
FIGURE1. Conventional test drive (two receivers)
Conventional Testing Methods
Traditionally, the simplest way to test the real-world performance of a GNSS receiver is to put it in a vehicle or a portable pack; drive or walk around an area of interest (typically a challenging environment such as an “urban canyon”); record position data; plot the trajectory on a map; and evaluate it visually. An example of this is shown in Figure 1 for two receivers, in this case driven through the difficult radio environment of downtown San Francisco.
While appealing in its simplicity and direct visual representation of the test drive, this approach does not allow for any quantitative assessment of receiver performance; judging which receiver is “better” is inherently subjective here. Different receivers often have different strong and weak points in their tracking and navigation algorithms, so it can be difficult to assess overall performance, especially over the course of a long trial. Also, an accurate evaluation of a trial generally requires some first-hand knowledge of the test area; unless local maps are available in sufficiently high resolution, it may be difficult to tell, for example, how accurate a trajectory along a wooded area might be.
In Figure 2, it appears clear enough that the test vehicle passed down a narrow lane between two sets of buildings during this trial, but it can be difficult to tell how accurate this result actually is. As will be demonstrated below, making sense of a situation like this is essentially beyond the scope of the simple “visual plotting” test method.
FIGURE 2. Test result requiring local knowledge to interpret correctly.
To address these shortcomings, the simple test method can be refined through the introduction of a GNSS/INS truth reference system. This instrument combines the absolute position obtainable from GNSS with accurate relative measurements from a suite of inertial sensors (accelerometers, gyroscopes, and occasionally magnetometers) when GNSS signals are degraded or unavailable. The reference system is carried or driven along with the devices under test (DUTs), and produces a truth trajectory against which the performance of the DUTs is compared.
This refined approach is a significant improvement over the first method in two ways: it provides a set of absolute reference positions against which the output of the DUTs can be compared, and it enables a quantitative measurement of position accuracy. Examples of these two improvements are shown in Figure 3 and Figure 4.
FIGURE 3. Improved test with GPS/INS truth reference: yellow dots denote receiver under test; green dots show the reference trajectory of GPS/INS.FIGURE 4. Time-aligned 2D error.
As shown in Figure 4, interpolating the truth trajectory and using the resulting time-aligned points to calculate instantaneous position errors yields a collection of scalar measurements en. From these values, it is straightforward to compute basic statistics like mean, 95th percentile, and maximum errors over the course of the trial. An example of this is shown in Figure 5, with the data (horizontal 2D error in this case) presented in several different ways. Note that the time interpolation step is not necessarily negligible: not all devices align their outputs to whole second boundaries of GPS time, so assuming a typical 1 Hz update rate, the timing skew between a DUT and the truth reference can be as large as 0.5 seconds. At typical motorway speeds, say 100 km/hr, this results in a 13.9 meter error between two points that ostensibly represent the same position. On the other hand, high-end GPS/INS systems can produce outputs at 100 Hz or higher, in which case this effect may be safely neglected.
FIGURE 5. Quantifying error using a truth reference
Despite their utility, both methods described above suffer from two fundamental limitations: results are inherently obtainable only in real time, and the scope of test coverage is limited to the number of receivers that can be fixed on the test rig simultaneously. Thus a test car outfitted with five receivers (a reasonable number, practically speaking) would be able to generate at most five quasi-independent results per outing.
Software Approach
The architecture of a software GNSS receiver is ideally suited to overcoming the limitations described above, as follows.
The raw IF data stream from the analog-to-digital converter is recorded to a file during the initial data collection. This file captures the essential characteristics of the RF chain (antenna pattern, downconverter, filters, and so on), as well as the signal environment in which the recording was made (fading, multipath, and so on). The IF file is then reprocessed offline multiple times in the lab, applying the results of careful profiling of various hardware platforms (for example, Pentium-class PC, ARM9-based embedded device, and so on) to properly model the constraints of the desired target platform. Each processing pass produces a position trajectory nominally identical to what the DUT would have gathered when running live. The complete multiple offset post-processi
ng (MOPP) setup is illustrated in Figure 6.
FIGURE 6. Multiple Offset Post-Processing (MOPP).
The fundamental improvement relative to a conventional testing approach lies in the multiple reprocessing runs. For each one, the raw data is processed starting from a small, progressively increasing time offset relative to the start of the IF file. A typical case would be 256 runs, with the offsets uniformly distributed between 0 and 100 milliseconds — but the number of runs is limited only by the available computing resources, and the granularity of the offsets is limited only by the sampling rate used for the original recording. The resulting set of trajectories is essentially the physical equivalent of having taken a large number of identical receivers (256 in this example), connecting them via a large signal splitter to a single common antenna, starting them all at approximately the same time (but not with perfect synchronization), and traversing the test route.
This approach produces several tangible benefits.
The large number of runs dramatically increases the statistical significance of the quantitative results (mean accuracy, 95th percentile error, worst-case error, and so on) produced by the test.
The process significantly increases the likelihood of identifying uncommon (but non-negligible) corner cases that could only be reliably found by far more testing using ordinary methods.
The approach is deterministic and completely repeatable, which is simply a consequence of the nature of software post-processing. Thus if a tuning improvement is made to the navigation filter in response to a particular observed artifact, for example, the effects of that change can be verified directly.
The proposed approach allows the evaluation of error models (for example, process noise parameters in a Kalman filter), so estimated measurement error can be compared against actual error when an accurate truth reference trajectory (such as that produced by the aforementioned GPS/INS) is available. Of course, this could be done with conventional testing as well, but the replay allows the same environment to be evaluated multiple times, so filter tuning is based on a large population of data rather than a single-shot test drive.
Start modes and assistance information may be controlled independently from the raw recorded data. So, for example, push-to-fix or A-GNSS performance can be tested with the same granularity as continuous navigation performance.
From an implementation standpoint, the proposed approach is attractive because it requires limited infrastructure and lends itself naturally to automated implementation. Setting up handful of generic PCs is far simpler and less expensive than configuring several hundred identical receivers (indeed, space requirements and RF signal splitting considerations alone make it impractical to set up a test rig with anywhere near the number of receivers mentioned above). As a result, the software replay setup effectively increases the testing coverage by several orders of magnitude in practice. Also, since post-processing can be done significantly faster than real time on modern hardware, these benefits can be obtained in a very time-efficient manner.
As with any testing method, the software approach has a few drawbacks in addition to the benefits described above. These issues must be addressed to ensure that results based on post-processing are valid and meaningful.
Error and Independence
The MOPP approach raises at least two obvious questions that merit further discussion.
How accurately does file replay match live operation?
Are runs from successive offsets truly independent?
The first question is answered quantitatively, as follows. A general-purpose software receiver (running on an x86-class netbook computer) was driven around a moderately challenging urban environment and used to gather live position data (NMEA) and raw digital data (IF samples) simultaneously. The IF file was post-processed with zero offset using the same receiver executable, incorporating the appropriate system profiling to accurately model the constraints of real-time processing as described above, to yield a second NMEA trajectory. Finally, the two NMEA files were compared using the methods shown in Figure 4 and Figure 5, this time substituting the post-processed trajectory for the GPS/INS reference data. A plot of the resulting horizontal error is shown in Figure 7.
FIGURE 7. Quantifying error introduced by post-processing.
The mean horizontal error introduced by the post-processing approach relative to the live trajectory is on the order of 2.5 meters. This value represents the best accuracy achievable by file replay process for this environment.
More challenging environments will likely have larger minimum error bounds, but that aspect has not yet been investigated fully; it will be considered in future work. Also, a single favorable comparison of live recording against a single replay, as shown above, does not prove that the replay procedure will always recreate a live test drive with complete accuracy. Nevertheless, this result increases the confidence that a replayed trajectory is a reasonable representation of a test drive, and that the errors in the procedure are in line with the differences that can be expected between two identical receivers being tested at the same time.
To address the question of run-to-run independence, consider two trajectories generated by post-processing a single IF file with offsets jB and kB, where B is some minimum increment size (one sample, one buffer, and so on), and define FJK to be some quantitative measurement of interest, for example mean or 95th percentile horizontal error. The deterministic nature of the file replay process guarantees FJK = 0 for j = k. Where j and k differ by a sufficient amount to generate independent trajectories, FJK will not be constant, but should be centered about some non-negative underlying value that represents the typical level of error (disagreement) between nominally identical receivers. As mentioned earlier, this is the approximate equivalent of connecting two matched receivers to a common antenna, starting them at approximately the same time, and driving them along the test trajectory.
Given these definitions, independence is indicated by an abrupt transition in FJK between identical runs ( j = k) and immediately adjacent runs (|j – k| = 1) for a given offset spacing B. Conversely, a gradual transition indicates temporal correlation, and could be used to determine the minimum offset size required to ensure run-to-run independence if necessary. As shown in Figure 8, the MOPP parameters used in this study (256 offsets, uniformly spaced on [0, 100 msec] for each IF file) result in independent outputs, as desired.
FIGURE 8. Verifying independence of adjacent offsets (upper: full view; lower: zoomed top view)
One subtlety pertaining to the independence analysis deserves mention here in the context of the MOPP method. Intuitively, it might appear that the offset size B should have a lower usable bound, below which temporal correlation begins to appear between adjacent post-processing runs. Although a detailed explanation is outside the scope of this paper, it can be shown that certain architectural choices in the design of a receiver’s baseband can lead to somewhat counterintuitive results in this regard.
As a simple example, consider a receiver that does not forcibly align its channel measurements to whole-second boundaries of system time. Such a device will produce its measurements at slightly different times with respect to the various timing markers in the incoming signal (epoch, subframe, and frame boundaries) for each different post-processing offset. As a result, the position solution at a given time point will differ slightly between adjacent post-processing runs until the offset size becomes smaller than the receiver’s granularity limit (one packet, one sample, and so on), at which point the outputs from successive offsets will become identical. Conversely, altering the starting point by even a single offset will result in a run sufficiently different from its predecessor to warrant its inclusion in a statistical population.
Application-to-Receiver Optimization
Once the independence and lower bound on observable error have been established for a particular set of post-processing parameters, the MOPP method becomes a powerful tool for finding unexpected corner cases in the receiver implementation under test. An example of this is shown in Figure 9, using the 95th percentile horizontal error as the statistical quantity of interest.
FIGURE 9. Identifying a rare corner case (upper: full view; lower: top view)
For this IF file, the “baseline” level for the 95th percentile horizontal error is approximately 6.7 meters. The trajectory generated by offset 192, however, exhibits a 95th percentile horizontal error with respect to all other trajectories of approximately 12.9 meters, or nearly twice as large as the rest of the data set. Clearly, this is a significant, but evidently rare, corner case — one that would have required a substantial amount of drive testing (and a bit of luck) to discover by conventional methods.
When an artifact of the type shown above is identified, the deterministic nature of software post-processing makes it straightforward to identify the particular conditions in the input signal that trigger the anomalous behavior. The receiver’s diagnostic outputs can be observed at the exact instant when the navigation solution begins to diverge from the truth trajectory, and any affected algorithms can be tuned or corrected as appropriate. The potential benefits of this process are demonstrated in Figure 10.
FIGURE 10. Before (top) and after (bottom) MOPP-guided tuning (blue = 256 trajectories; green = truth)
Limitations
While the foregoing results demonstrate the utility of the MOPP approach, this method naturally has several limitations as well. First, the IF replay process is not perfect, so a small amount of error is introduced with respect to the true underlying trajectory as a result of the post-processing itself. Provided this error is small compared to those caused by any corner cases of interest, it does not significantly affect the usefulness of the analysis — but it must be kept in mind.
Second, the accuracy of the replay (and therefore the detection threshold for anomalous artifacts) may depend on the RF environment and on the hardware profiling used during post-processing; ideally, this threshold would be constant regardless of the environment and post-processing settings.
Third, the replay process operates on a single IF file, so it effectively presents the same clock and front-end noise profile to all replay trajectories. In a real-world test including a large number of nominally identical receivers, these two noise sources would be independent, though with similar statistical characteristics. As with the imperfections in the replay process, this limitation should be negligible provided the errors due to any corner cases of interest are relatively large.
Conclusions and Future Work
The multiple offset post-processing method leverages the unique features of software GNSS receivers to greatly improve the coverage and statistical validity of receiver testing compared to traditional, hardware-based testing setups, in some cases by an order of magnitude or more. The MOPP approach introduces minimal additional error into the testing process and produces results whose statistical independence is easily verifiable. When corner cases are found, the results can be used as a targeted tuning and debugging guide, making it possible to optimize receiver performance quickly and efficiently.
Although these results primarily concern continuous navigation, the MOPP method is equally well-suited to tuning and testing a receiver’s baseband, as well its tracking and acquisition performance. In particular, reliably short time-to-first-fix is often a key figure of merit in receiver designs, and several specifications require acquisition performance to be demonstrated within a prescribed confidence bound. Achieving the desired confidence level in difficult environments may require a very large number of starts — the statistical method described in the 3GPP 34.171 specification, for example, can require as many as 2765 start attempts before a pass or fail can be issued — so being able to evaluate a receiver’s acquisition performance quickly during development and testing, while still maintaining sufficient confidence in the results, is extremely valuable.
Future improvements to the MOPP method may include a careful study of the baseline detection threshold as a function of the testing environment (open sky, deep urban canyon, and so on). Another potentially fruitful line of investigation may be to simulate the effects of physically distinct front ends by adding independent, identically distributed swaths of noise to copies of the raw IF file prior to executing the multiple offset runs.
Alexander Mitelman is GNSS research manager at Cambridge Silicon Radio. He earned his M.S. and Ph.D. degrees in electrical engineering from Stanford University. His research interests include signal quality monitoring and the development of algorithms and testing methodologies for GNSS.
Jakob Almqvist is an M.Sc. student at Luleå University of Technology in Sweden, majoring in space engineering, and currently working as a software engineer at Cambridge Silicon Radio.
Robin Håkanson is a software engineer at Cambridge Silicon Radio. His interests include the design of optimized GNSS software algorithms, particularly targeting low-end systems.
David Karlsson leads GNSS test activities for Cambridge Silicon Radio. He earned his M.S. in computer science and engineering from Linköping University, Sweden. His current focus is on test automation development for embedded software and hardware GNSS receivers.
Fredrik Lindström is a software engineer at Cambridge Silicon Radio. His primary interest is general GNSS software development.
Thomas Renström is a software engineer at Cambridge Silicon Radio. His primary interests include developing acquisition and tracking algorithms for GNSS software receivers.
Christian Ståhlberg is a senior software engineer at Cambridge Silicon Radio. He holds an M.Sc. in computer science from Luleå University of Technology. His research interests include the development of advanced algorithms for GNSS signal processing and their mapping to computer architecture.
James Tidd is a senior navigation engineer at Cambridge Silicon Radio. He earned his M.Eng. from Loughborough University in systems engineering. His research interests
include integrated navigation, encompassing GNSS, low-cost sensors, and signals of opportunity.
Seven technologies that put GPS in mobile phones around the world — the how and why of location’s entry into modern consumer mobile communications.
By Frank van Diggelen, Broadcom Corporation
Exactly a decade has passed since the first major milestone of the GPS-mobile phone success story, the E-911 legislation enacted in 1999. Ensuing developments in that history include:
Snaptrack bought by Qualcomm in 2000 for $1 billion, and many other A-GPS startups are spawned.
Commercial GPS receiver sensitivity increases roughly 30 times, to 2150 dBm (1998), then another 10 times, to 2160 dBm in 2006, and perhaps another three times to date, for a total of almost 1,000 times extra sensitivity. We thought the main benefit of this would be indoor GPS, but perhaps even more importantly it has meant very, very cheap antennas in mobile phones. Meanwhile:
Host-based GPS became the norm, radically simplifying the GPS chip, so that, with the cheap antenna, the total bill of materials (BOM) cost for adding GPS to a phone is now just a few dollars!
Thus we see GPS penetration increasing in all mobile phones and, in particular, going towards 100 percent in smartphones.
This article covers the technology revolution behind GPS in mobile phones; but first, let’s take a brief look at the market growth. This montage gives a snapshot of 28 of the 228 distinct Global System for Mobile Communications (GSM) smartphone models (as of this writing) that carry GPS.
Back in 1999, there were no smartphones with GPS; five years later still fewer than 10 different models; and in the last few years that number has grown above 200. This is that rare thing, often predicted and promised, seldom seen: the hockey stick!
The catalyst was E-911 — abetted by seven different technology enablers, as well as the dominant spin-off technology (long-term orbits) that has taken this revolution beyond the cell phone.
In 1999, the Federal Communications Commission (FCC) adopted the E-911 rules that were also legislated by the U.S. Congress. Remember, however, that E-911 wasn’t all about GPS at first.
It was initially assumed that most of the location function would be network-based. Then, in September 1999, the FCC modified the rules for handset technologies. Even then, assisted GPS (A-GPS) was only adopted in the mobile networks synchronized to GPS time, namely code-division multiple access (CDMA) and integrated digital enhanced network (iDEN, a variant of time-division multiple access).
The largest networks in the world, GSM and now 3G, are not synchronized to GPS time, and, at first, this meant that other technologies (such as enhanced observed time difference, now extinct) would be the E-911 winners. As we all now know, GPS and GNSS are the big winners for handset location. E-911 became the major driver for GPS in the United States, and indirectly throughout the world, but only after GPS technology evolved far enough, thanks to the seven technologies I will now discuss.
Technology #1. Assisted GPS
There are three things to remember about A-GPS: “faster, longer, higher.” The Olympic motto is “faster, stronger, higher,” so just think of that, but remember “faster, longer, higher.”
The most obvious feature of A-GPS is that it replaces the orbit data transmitted by the satellite. A cell tower can transmit the same (or equivalent) data, and so the A-GPS receiver operates — faster.
The receiver has to search over a two-dimensional code/frequency space to find each GPS satellite signal in the first place. Assistance data reduces this search space, allowing the receiver to spend longer doing signal integration, and this in turn means higher sensitivity (Figure 1). Longer, higher.
FIGURE 1. A-GPS: reduced search space allows longer integration for higher sensitivity.
Now let’s look at this code/frequency search in more detail, and introduce the concepts of fine time, coarse time, and massive parallel correlation. Any assistance data helps reduce the frequency search. The frequency search is just as you might scan the dial on a car radio looking for a radio station — but the different GPS frequencies are affected by the satellite motion, their Doppler effect. If you know in advance whether the satellite is rising or setting, then you can narrow the frequency-search window.
The code-delay is more subtle. The entire C/A code repeats every millisecond. So narrowing the code-delay search space requires knowledge of GPS time to better than one millisecond, before you have acquired the signal. We call this “fine-time.”
Only two phone systems had this time accuracy: CDMA and iDEN, both synchronized to GPS time. The largest networks (GSM, and now 3G) are not synchronized to GPS time. They are within 62 seconds of GPS time; we call this “coarse-time.” Initially, only the two fine-time systems adopted A-GPS. Then came massive parallel correlation, technology number two, and high sensitivity, technology number three.
#2, #3. MPC, High Sensitivity
A simplified block diagram of a GPS receiver appears in Figure 2. Traditional GPS (prior to 1999) had just two or three correlators per channel. They would search the code-delay space until they found the signal, and then track the signal by keeping one correlator slightly ahead (early) and one slightly behind (late) the correlation peak. These are the so-called “early-late”correlators.
FIGURE 2. Massive parallel correllation.
Massive parallel correlation is defined as enough correlators to search all C/A code delays simultaneously on multiple channels. In hardware, this means tens of thousands of correlators. The effect of massive parallel correlation is that all code-delays are searched in parallel, so the receiver can spend longer integrating the signal whether or not fine-time is available.
So now we can be faster, longer, higher, regardless of the phone system on which we implement A-GPS.
Major milestones of massive parallel correlation (MPC):
In 1999, MPC was done in software, the most prominent example being by Snaptrack, who did this with a fast Fourier transform (FFT) running on a digital signal processor (DSP). The first chip with MPC in hardware was the GL16000, produced by Global Locate, then a small startup (now owned by Broadcom).
In 2005, the first smartphone implementation of MPC: the HP iPaq used the GL20000 GPS chip. Today MPC is standard on GPS chips found in mobile phones.
#4. Coarse-Time Navigation
We have seen that A-GPS assistance relieves the receiver from decoding orbit data (making it faster), and MPC means it can operate with coarse-time (longer, higher).
But the time-of-week (TOW) still needed to be decoded for the position computation and navigation: for unambiguous pseudoranges, and to know the time of transmission. Coarse-time navigation is a technique for solving for TOW, instead of decoding it. A key part of the technique involves adding an extra state to the standard navigation equation, and a corresponding extra column to the well known line-of-sight matrix.
The technical consequence of this technique is that you can get a position faster than it is possible to decode TOW (for example, in one, two, or three seconds), or you can get a position when the signals are too weak to decode TOW. And a practical consequence is longer battery life: since you can get fast time-to-first-fix (TTFF) always, without frequently waking and running the receiver to maintain it in a hot-start state.
#5. Low Time-of-Week
A parallel effort to coarse-time navigation is low TOW decode, that is, lowering the threshold at which
it is possible to decode the TOW data. In 1999, it was widely accepted that -142 dBm was the lower limit of signal strength at which you could decode TOW. This is because -142 dBm is where the energy in a single data bit is just observable if all you do is integrate for 20 ms.
However, there have evolved better and better ways of decoding the TOW message, so that now it can be done down to -152 dBm. Today, different manufacturers will quote you different levels for achievable TOW decode, anywhere from -142 to -152 dBm, depending on who you talk to. But they will all tell you that they are at the theoretical minimum!
#6, #7. Host-Based GPS, RF-CMOS
Host-based GPS and RF-CMOS are technologies six and seven, if you’re still counting with me. We can understand the host-based architecture best by starting with traditional system-on-chip (SOC) architecture. An SOC GPS may come in a single package, but inside that package you would find three separate die, three separate silicon chips packaged together: A baseband die, including the central processing unit (CPU); a separate radio frequency tuner; and flash memory. The only cost-effective way of avoiding the flash memory is to have read-only memory (ROM), which could be part of the baseband die — but that means you cannot update the receiver software and keep up with the technological developments we’ve been talking about. Hence state-of-the-art SOCs throughout the last decade, and to date, looked like Figure 3.
FIGURE 3. Host-based architecture, compared to SOC.
The host-based architecture, by contrast, needs no CPU in the GPS. Instead, GPS software runs on the CPU and flash memory already present on the host device (for example, the smartphone). Meanwhile, radio-frequency complementary metal-oxide semi-conductor (RF-CMOS) technology allowed the RF tuner to be implemented on the same die as the baseband. Host-based GPS and RF- CMOS together allowed us to make single die GPS chips.
The effect of this was that the cost of the chip went down dramatically without any loss in performance.
Figure 4 shows the relative scales of some of largest-selling SOC and host- based chips, to give a comparative idea of silicon size (and cost). The SOC chip (on the left) is typically found in devices that need a CPU, while the host-based chip is found in devices that already have a CPU.
FIGURE 4. Relative sizes of host-based, compared to SOC.
In 2005, the world’s first single-die GPS receiver appeared. Thanks to the single die, it had a very low bill of materials (BOM) cost, and has sold more than 50 million into major-brand smartphones and feature phones on the market.
Review
We have seen that E-911 was the big catalyst for getting GPS into phones, although initially only in CDMA and iDEN phones. E-911 became the driver for all phones once GPS evolved far enough, thanks to the seven technology enablers:
A-GPS >> faster, longer, higher
Massive parallel correlation >> longer, higher with coarse-time
High-sensitivity >> cheap antennas
Coarse time navigation >> fast TTFF without periodic wakeup
Low TOW >> decode from weak signals
Host-based GPS, together with
RF-CMOS g single die.
Meanwhile, as all this developed, several important spin-off technologies evolved to take this technology beyond the mobile phone. The most significant of all of these was long-term orbits (LTO), conceived on May 2, 2000, and now an industry standard.
Long-Term Orbits
Why May 2, 2000? Remember what happened on May 1, 2000: the U.S. government turned off selective availability (SA) on all GPS satellites. Suddenly it became much easier to predict future satellite orbits (and clocks) from the observations made by a civilian GPS network. At Global Locate, we had just such a network for doing A-GPS, as illustrated in Figure 5. On May 2 we said, “SA is off — wow! What does that mean for us?”And that’s where LTO for A-GPS came from.
FIGURE 5. Broadcast ephemeris and long-term orbits.
Figure 5 shows the A-GPS environment with and without LTO. The left half shows the situation with broadcast ephemeris only. An A-GPS reference station observes the broadcast ephemeris and provides it (or derived data) to the mobile A-GPS receiver in your mobile phone. The satellite has the orbits for many hours into the future; the problem is that you can’t get them.
The blue and yellow blocks in the diagram represent how the ephemeris is stored and transmitted by the GPS satellite. The current ephemeris (yellow) is transmitted; the future ephemeris (blue) is stored in the satellite memory until it becomes current. So, frustratingly, even though the future ephemeris exists, you cannot ordinarily get it from the GPS system itself.
The right half of the figure shows the situation with LTO. If a network of reference stations observes all the satellites all the time, then a server can compute the future orbits, and provide future ephemeris to any A-GPS receiver. Using the same color scheme as before, we show here that there are no unavailable future orbits; as soon as they are computed, they can be provided. And if the mobile device has a fast-enough CPU, it can compute future orbits itself, at least for the subset of satellites it has tracked.
Beyond Phones. This idea of LTO has moved A-GPS from the mobile phone into almost any GPS device. Two of most interesting examples are personal navigation devices (PNDs) in cars, and smartphones themselves that continue to be useful gadgets once they roam away from the network. Now, of course, people were predicting orbits before 2000 — all the way back to Newton and Kepler, in fact. It’s just that in the year 2000, accurate future GPS orbits weren’t available to mobile receivers. At that time, the International GNSS Service (IGS) had, as it does now, a global network of reference stations, and provided precise GPS orbits organized into groups called Final, Rapid and Ultra-Rapid. The Ultra-Rapid orbit had the least latency of the three, but, in 2000, Ultra-Rapid meant the recent past, not the future.
So for LTO we see that the last 10 years have taken us from a situation of nothing available to the mobile device, to today where these long-term orbits have become codified in the 3rd Generation Partnership Project (3GPP) and Secure User Plane Location (SUPL) wireless standards, where they are known as “ephemeris extension.”
Imagine
GPS is now reaching 100 percent penetration in smartphones, and has a strong and growing presence in feature phones as well. GPS is now in more than 300 million mobile phones, at the very least; credible estimates range above 500 million.
Now, imagine every receiver ever made since GPS was created 30 years ago: military and civilian, smart-bomb, boat, plane, hiking, survey, precision farming, GIS, Bluetooth-puck, personal digital assistant, and PND. In the last three years, we have put more GPS chips into mobile phones than the cumulative number of all other GPS receivers that have been built, ever!
Frank van Diggelen has worked on GPS, GLONASS, and A-GPS for Navsys, Ashtech, Magellan, Global Locate, and now as a senior technical director and chief navigation officer of Broadcom Corporation. He has a Ph.D. in electrical engineering from Cambridge University, holds more than 45 issued U.S. patents on A-GPS, and is the author of the textbook A-GPS: Assisted GPS, GNSS, and SBAS.
The September “Innovation” column in this magazine, “It’s Not All Bad: Understanding and Using GNSS Multipath,” by Andria Bilich and Kristine Larson, mentions the use of multipath in studying soil moisture, ocean altimetry and winds, and snow sensing. An experiment the authors conducted, designed to study soil moisture, yielded a surprise bonus: a new methodology for measuring snow depth via GPS multipath. It has important implications for weather and flood forecasting, and could also bring new insight to bear on GPS antenna design.
In the “Innovation” column, the authors wrote, “Motivated by our studies showing that multipath effects could clearly be seen in geodetic-quality data collected with multipath-suppressing antennas, we proposed that these same GPS data could be used to extract a multipath parameter that would correlate with changes in the reflectance of the ground surface. . . .
“We carried out an experiment designed to more rigorously demonstrate the link between GPS signal-to-noise ratio (SNR) and soil moisture. Specifically, we were interested in using GPS reflection parameters to determine the soil’s volumetric water content — the fraction of the total volume of soil occupied by water, an important input to climate and meteorological models. Traditional soil moisture sensors (water content reflectometers) were buried in the ground at multiple depths (2.5 and 7.5 centimeters) at a site just south of the University of Colorado.”
Here Comes the Storm. During the experiment, two late-season snowstorms swept over Boulder. Larson and colleagues discovered that changes in multipath clearly correlated with changes in the snow’s depth, as measured by hand and with ultrasonic sensors at the test site. While it has been long recognized that snow can affect a GPS signal, this demonstrates for the first time that a standard GPS receiver, antenna, and installation — deliberately designed to suppress multipath — can be used to measure snow depth.
On September 11, Geophysical Research Letters, published by the American Geophysical Union, featured an article titled “Can We Measure Snow Depth with GPS Receivers?” by Larson and Felipe Nievinski of the Department of Aerospace Engineering Sciences, University of Colorado; Ethan Gutmann and John Brown of the National Center for Atmospheric Research; Valery Zavorotny of the National Oceanic and Atmospheric Administration; and Mark W. Williams, from UC’s Department of Geography, all based in Boulder.
The authors adapted an algorithm used for modeling GPS multipath from bare soil to predict GPS SNR for snow, introducing a uniform planar layer of the snow on the top of soil. The algorithm treats both direct and surface-reflected waves at two opposite circular polarizations as plane waves that sum up coherently at the antenna. They write:
“The amplitude and the phase of the reflected wave is driven by a polarization-dependent, complex-value reflection coefficient at the upper interface of such a combined medium with a known vertical profile of the dielectric permittivity e. The reflection coefficient is calculated numerically using an iterative algorithm in which the medium is split into sub-layers with a constant e. For the soil part, we use a known soil profile model that depends on the soil type and moisture. For frozen soil, soil moisture (liquid water) is low, as for very dry soil. For the snow part, we take a constant profile with e, considering relatively dry and wet snow layer thicknesses.
“After calculating the complex amplitude of the reflected wave at each polarization, we multiply it by a corresponding complex antenna gain. The same procedure is applied to the complex amplitude of the direct wave. After that, the modulation pattern of the received power, or the SNR, as a function of the GPS satellite elevation angle is obtained by summing up coherently all the signals coming from the antenna output and taking the absolute value square of the sum.”
Figure 1(a) shows GPS SNR measurements for one satellite on the day immediately before and the day immediately after an overnight snowfall of 35 centimeters (roughly 10 inches). Figure 1(b) shows the corresponding model predictions for multipath. The two figure portions amply demonstrate that the multipath has a significantly lower frequency if snow is present as compared with bare soil. The authors further noted that the model amplitudes do not show as pronounced a dependence on satellite elevation angle as the observations, and state the necessity of further work on antenna gains in order to use model amplitude predictions.
Figure 1. (a) GPS SNR measurements for PRN 7 observed at Marshall GPS site on days 107 (red) and 108 (black) after direct signal component has been removed. Approximately 35 centimeters of snow had fallen by day 108. (b) Model predictions for GPS multipath from day 107 with no snow on the ground (red), and day 108 after 35 centimeters of new snow fall had accumulated (black) using an assumed density of 240 kg m-3 (figures reproduced by permission of American Geophysical Union).
How Deep the Snow. The authors propose that the hundreds of geodetic GPS receivers operating in snowy regions of the United States, originally installed for plate deformation studies, surveying, and weather monitoring, could also provide a cost-effective means to estimate snow depth.
Currently, a few conventional monitor points measure snow depth, but only at that point, and the data does not extrapolate well. Snow forms an important component of the climate system and a critical storage component in the hydrologic cycle. Accurate data of the amount of water stored in the snowpack is critical for water supply management and flood control systems. As more snow falls at higher elevations, varying greatly even within one valley or watershed, current remote-sensing snow monitors do not supply adequate data. Further, snow may be redistributed by wind, avalanches, and non-uniform melting, so that continuous data would be very helpful.
Using GPS multipath to map snow depth could improve watershed analyses and flood prediction — and, carried steps further, produce data to help better understand multipath, bringing innovation to future antenna designs.
FIGURE 2. Snow depth derived from GPS (red squares), the three ultrasonic snow depth sensors (blue lines), and field measurements (black diamonds). Bars on field observations are one standard deviation. GPS snow-depth estimates during the first storm (doy 85.5–86.5) are not shown (gray region) because the SNR data indicate that snow was on top of the antenna.
Kristine Larson was featured as one of the “50 GNSS Leaders to Watch” in the May 2009 issue of GPS World.
Manufacturer
For the experiment a Trimble NetRS receiver was used with a TRM29659.00 choke-ring antenna with SCIT radome, pointed at zenith.
AURORA BOREALIS seen from Churchill, Manitoba, Canada. Ionospheric scintillation research can benefit from this new method. (Photo: Aiden Morrison)Photo: Canadian Armed Forces
By Aiden Morrison, University of Calgary
Two broad user groups will find important consequences in this article:
Time synchronization and test equipment manufacturers, whose GPS-disciplined oscillators have excellent long-term performance but short- to medium-term behavior limited by the quality, and therefore cost, of the integrated quartz device. This article portends a family of devices delivering oven-controlled crystal oscillator (OCXO) performance down to the 10-millisecond level, with an oscillator costing pennies, rather than tens or hundreds of dollars. Applications include ionospheric scintillation research (above).
High-performance receiver manufacturers who design products for high-dynamic or high-vibration environments (see cover) where the contribution of phase noise from the local oscillator to velocity error cannot be ignored. In these areas, the strategy outlined here would produce equipment that can perform to higher specifications with the same or a lower-cost oscillator.
The trade-off requires two tracking channels per satellite signal, but this should not pose a problem. At ION GNSS 2009, manufacturers showed receivers with 226 tracking channels. There are currently only 75 live signals in the sky, including all of GPSL1/L2/L5 and GLONASS L1/L2. — Gérard Lachapelle
If the channel data within a GNSS receiver is handled in an effective manner, it is possible to form meaningful estimates of the local-oscillator phase deviations on timescales of 10 milliseconds (ms) or less. Moreover, if certain criteria are met, these estimates will be available with related uncertainties similar to the deviations produced by a typical oven-controlled crystal oscillator (OCXO). The processing delay required to form this estimate is limited to between 10 and 20 ms. In short, it becomes possible in near-real-time to remove the majority of the phase noise of a local oscillator that possesses short-term instability worse than an OCXO, using standalone GNSS. This represents both a new method to accurately determine the Allan deviation of a local oscillator at time scales previously impractical to assess using a conventional GNSS receiver, and the potential for the reduction in observable Doppler uncertainty at the output of the receiver, as well as ionospheric scintillation detection not reliant on an expensive local OCXO.
Concept. Inside a typical GNSS receiver, the estimate of the error in the local oscillator is formed as a component of the navigation solution, which is in turn based on the output of each satellite-tracking channel propagating its estimate of carrier and code measurements to a common future point. While this method of ensuring simultaneous measurements is necessary, it regrettably limits the resolution with which the noise of the local oscillator can be quantified, due to the scaling of non-simultaneous samples of local oscillator noise through the measurement propagation process. To bypass these shortcomings requires a method of coherently gathering information about the phase change in the local oscillator across all available satellite signals: to use the same samples simultaneously for all satellites in view to estimate the center-point phase error common across the visible constellation.
To explain how this is feasible, we must first understand the limitations imposed by the conventional receiver architecture, with respect to accurately estimating short-term oscillator behavior, and subsequently to determine the potential pitfalls of the proposed modifications, including processing delays needed for bit wipe-off, expected observation noise, and user dynamics effects.
Typical Receiver Shortfalls
In a typical receiver, while information about local time offset and local oscillator frequency bias may be recovered, information about phase noise in the local oscillator is distorted and discarded, as a consequence of scaling non-simultaneous observations to a common epoch.
As shown in FIGURE 1, coherent summation intervals in a receiver are used to approximate values of the phase error, including oscillator phase, measured at the non-simultaneous interval centrers in each channel, which are then propagated to a common navigation solution epoch. Each channel will intrinsically contain a partially overlapping midpoint estimate of oscillator noise over the coherent summation interval that will then be scaled by the process of extrapolation. As these estimates are scaled and partially overlapping, they do not make optimal use of the information known about the effects of the local oscillator, and form a poor basis for estimating the contributions of this device to the uncertainty in the channel measurements. As shown in Figure 1, the phase error measured in each channel will be distorted by an over unity scaling factor.
FIGURE 1. Propagation and scaling of phase estimates within a typical receiver.
Depending on implementation decisions made by the designers of a given GNSS system, the average value of the propagation interval relative to the bit period will have different expected values. Assuming the destination epoch is the immediate end of the furthest advanced (most delayed) satellite bitstream, and that integration is carried out over full bit periods, the minimum propagation interval for this satellite would be ½-bit period.
For the average satellite however, the propagation delay would be this ½-bit period plus the mean skew between the furthest satellite and the bitstreams of other space vehicles. Ignoring further skew effects due to the clock errors within the satellites, which are typically limited well below the ms level, the skew between highest and lowest elevation GPS satellites for a user on the surface of earth would be approximately 10 ms. The average value of this skew due to ranging change over orbit, assuming an even distribution of satellites in the sky at different elevation angles, would therefore be 5 ms.
Combining the minimum value of the skew interval with the minimum propagation interval of the most delayed satellite yields a total average propagation interval of 15 ms. In turn, this gives a typical scaling factor of 1.75, used from this point forward when referring to the effects of scaling this quantity.
Proposed Implementation
Overcoming limitations of a typical receiver requires recording the approximate bit-timing and history of each tracked satellite as well as a short segment of past samples. This retained data guarantees that the bit-period boundaries of the satellites will not pose an obstacle to forming common N-ms coherent periods between all visible satellites, over which simultaneous integration may proceed by wiping off bit transitions. Using this approach as shown in FIGURE 2, all available constellation signal power is used to estimate a single parameter, namely the epoch-to-epoch phase change in the local oscillator.
FIGURE 2. Common intervals over which to accurately estimate local oscillator phase changes.
Having viewed the existence of these common periods, it becomes evident that it is conceptually possible to form time-synchronized estimates of the phase contribution of the common system oscillator alternately across one N-ms time slice, then the next, in turn forming an unb
roken time series of estimates of the phase change of the system oscillator. Forming the difference between the adjacent discriminator outputs will provide the following information:
The ΔEps (change in the noise term in the local loop)
The ΔOsc (change in the phase of the local oscillator, the parameter of interest)
The ΔDyn (change in the untracked/residual of real and apparent dynamics of the local loop/estimator)
Noticing that term 1 may be considered entirely independent across independent PRNs (GPS, Galileo, Compass) or frequency channels (GLONASS), and that the value of term 3 over a 10-ms period is expected to be small over these short intervals, it becomes obvious that term 2 can be recovered from the available information. To determine the weighting for each satellite channel, the variance of the output of the discriminator is needed.
Performance Determination
To allow the realistic weighting of discriminator output deltas, it becomes desirable to estimate at very short time intervals the variance of the output of the phase discriminator. In the case of a 2-quadrant arctangent discriminator, this means one wishes to quantify the variance
Letting Q/I 5 Z, recall that if Y 5 aX then
Applying this to the variance of the input to the arctangent discriminator in terms of the in phase and quadrature accumulators, this would give
Rather than proceed with a direct evaluation from this point onward to determine the expression for the variance at the output of the discriminator, it is convenient to recognize that simpler alternatives exist since
The implication is that since the slope of the arctangent transfer function is very nearly equal to 1 in the central, typical operating region, and universally less than 1 outside of this region, it is easy to recognize that the variance at the output of the arctangent discriminator is universally less than that at the input, and can be pessimistically quantified as the variance of the input, or σ2(Z). This assumption has been verified by simulation, its result shown in FIGURE 3, where the response has been shown after taking into account the effect of operating at a point anywhere in the range ±45 degrees. While the consequence of the simplification of the variance expression is an exaggeration of discriminator output variance, FIGURE 4 shows output variance is well bounded by the estimate, and within a small margin of error for strong signals.
FIGURE 3. Predicted variances at the output of the ATAN2 discriminator versus C/N0.FIGURE 4. Difference between actual and predicted variance at output of discriminator.
The gap between real and predicted output variance may also be narrowed in cases where Q>I by using a type of discriminator which interchanges Q and I in this case and adds an appropriate angular offset to the output as
Proceeding in this vein, the next required parameter is the normalized variance of the in-phase and quadrature arms.
The carrier amplitude A can be roughly approximated as
Resulting in a carrier power C
Further, the noise power is given as
Expressing bandwidth B as the inverse of the coherent integration time, and rearranging now gives noise density N0 as
Combining this expression, and the one previously given for the carrier power C results in the following expression for the carrier to noise density ratio:
This latest expression can be rearranged to find the desired variance term. Assuming the 10-ms coherent integration time discussed earlier is used, this yields
Normalizing for the carrier amplitude gives the normalized variance in terms of radians squared:
In any situation where the carrier is sufficiently strong to be tracked, it is likely that the carrier power term employed above can be gathered from the immediate I and Q values, ignoring the contribution of the noise term to its magnitude.
Oscillator Phase Effect. Determining the expected magnitude of the local oscillator phase deviation requires only three steps, assuming that certain criteria can be met. The first requirement is that the averaging times in question must be short relative to the duration, at which processes other than white phase and flicker phase modulation begin to dominate the noise characteristics of the oscillator. Typically the crossover point between the dominance of these processes and others is above 1 s in averaging interval length, when quartz oscillators are concerned. Since this article discusses a specific implementation interval of 10 ms within systems expected to be using quartz oscillators, it is reasonable to assume that this constraint will be met.
The second requirement is that the Allan deviation of the given system oscillator must be known for at least one averaging interval within the region of interest. Since the Allan deviation follows a linear slope of -1 with respect to averaging interval on a log-log scale within the white-phase noise region, this single value will allow an accurate prediction of the Allan deviation at any other point on the interval and, in turn, of the phase uncertainty at the 10 ms averaging interval level.
Letting σΔ(τ) represent the Allan deviation at a specific averaging interval, recall that this quantity is the midpoint average of the standard deviation of fractional frequency error over the averaging interval τ. Scaling this quantity by a frequency of interest results in the standard deviation of the absolute frequency error on the averaging interval:
By integrating this average difference in frequency deviations over the coherent period of interest, one obtains a measure of the standard deviation in degrees, of a signal generated by this reference:
Note that the averaging interval τ must be identical to the coherent integration time.
Turning to a practical example, if the oscillator in question has a 1 s Allan Deviation of 1 part per hundred billion (1 in 1011), a stability value between that of an OCXO and microcomputer compensated crystal oscillator (MCXO) standard, and shown to be somewhat pessimistic, this would scale linearly to be 1e-9 at a 10-ms averaging interval, under the previous assumption that the oscillator uncertainty is dominated by the white phase-noise term at these intervals. Also, for illustration purposes, if one assumes the carrier of interest to be the nominal GPS L1 carrier, the uncertainty in the local carrier replica due to the local oscillator over a 10-ms coherent integration time becomes
When stated in a more readily digested format, this represents roughly 15 centimeter/second in the line-of-sight velocity uncertainty. In an operating receiver, two additional factors modify this effect. The first is the previously discussed scaling effect that will tend to exaggerate this effect by a typical factor of 1.75, as previously discussed. The second factor is that this noise contribution is filtered by the bandwidth-limiting effects of the local loop filter, producing a modification to the noise affecting velocity estimates, as well as reduced information about the behaviour of the local oscillator.
Impact of Apparent Dynamics. When considering the error sources within the system, it is important to realize which individual sources of error will contribute to estimation errors, and which will not. One area of potential concern would appear to be the errors in the satellite ephemerides, encompassing both the satellite-orbit trajectory misrepresentation and the satellite clock error. While the errors in the satellite ephemerides are of concern for point positioning, they are not of consequence to this application, as the apparent error introduced by a deviation of the true orbit from that expressed in the broadcast orbital parameters does not affect the tracking of that satellite at the loop level.
Additionally, while the satellite clock will add uncertainty to the epoch-to-epoch phase change within each channel independently, the magnitude of this change is minimal relative to the contribution of uncertainty due to the variance at the output of the discriminator guaranteed by the low carrier-to-noise density ratio of a received GNSS signal. Since this contribution is uncorrelated between satellites and relatively small compared to other noise contributions affecting these measurements, even when compared to the soon-to-be-discontinued Uragan GLONASS satellites that had generally less stable onboard clocks, it is likely safe to ignore. When compared to the more stable oscillators aboard GPS or GLONASS-M satellites, it is a reasonable assumption that this will be a dismissible contribution to received signal-phase uncertainty change.
While atmospheric effects present an obstacle which will directly affect the epoch-to-epoch output of the discriminators, it is believed that under conditions that do not include the effects of ionospheric scintillation the majority of the contribution of apparent dynamics due to atmospheric changes will have a power spectral density (PSD) heavily concentrated below a fraction of 1 Hz. The consequence of this concentration is that the tracking loops will remove the vast majority of this contribution, and that the difference operator that will be applied between adjacent phase measurements, as in the case of dynamics, will nullify the majority of the remaining influence.
Impact of Real Dynamics. Real dynamics present constraints on performance, as do any tracking loop transients. For example, a low-bandwidth loop-tracking dynamics will have long-lasting transients of a magnitude significant relative to levels of local oscillator noise. For this reason it is necessary to adopt a strategy of using the epoch-to-epoch change in the discriminator as the figure of interest, as opposed to the absolute error-value output at each epoch. This can reasonably be expected to remove the vast majority of the effects of dynamics of the user on the solution.
To validate this assumption under typical conditions calls for a short verification example. Assuming the use of a second-order phase-locked loop (PLL) for carrier tracking, with a 10-Hz loop bandwidth the effects of dynamics on the loop are given by these equations:
Letting Bn be 10 Hz, one can write
Recall that the dynamic tracking error in a second-order tracking loop is given by
Given the choices above, this would result in a constant offset of 0.00281 cycles, or 1.011 degrees of constant tracking error due to dynamics, following from the relation between line-of-sight acceleration and loop bandwidth to tracking error. Since this constant bias will be eliminated by the difference operator discussed earlier, it is necessary to examine higher order dynamics.
Further, if one used a coherent integration interval of 10 ms as assumed earlier, and let the dynamics of interest be a jerk of 1 g/s, this results in a midpoint average of 0.005 g on this interval:
Substituting this result into equation 16 produces the associated change in dynamic error over the integration interval, which is in this case:
This value will be kept in mind when evaluating capabilities of the estimation approach to determine when it will be of consequence. As the estimation process will be run after a short delay, an existing estimate of platform dynamics could form the basis of a smoothing strategy to reduce this dynamic contribution further.
Estimated Capabilities
In the absence of the influence of any unmodeled effects, the expected performance of this method is dependent on only the number of satellite observables and their respective C/N0 ratios. Across each of these scenarios we assume for simplicity’s sake that each satellite in view is received at a common C/N0 ratio and over a common integration period of 10 ms.
If the assumption of minimal dynamic influences is met, the situation at hand becomes one in which multiple measures of a single quantity are present, each containing independent (due to CDMA or FDMA channel separation) noise influences with a nearly zero mean. When one can express the available data form:
x[n] = R + w[n]
where x[n] is the nth channel discriminator delta which includes the desired measure of the local oscillator delta (R), as well as w[n], a strong, nearly white-noise component, there are multiple approaches for the estimation of R.
The straightforward solution to estimate R in this case is to use the predicted variances of each measure to serve as an inverse weighting to the contribution of each individual term, followed by normalization by the total variance, as expressed by
Now, since it is desired to bound the uncertainty of the estimate of R, the variance of this quantity should also be noted. This uncertainty can be determined as
To determine the performance of the estimation method for a given constellation configuration, with specific power levels and available carrier signals, it is necessary to utilize the predicted variances plotted in Figure 3 as inputs to equations 20 and 21. To provide numerical examples of the performance of this method, three scenarios span the expected range of performance.
Scenario 1 is intended to be char-acteristic of that visible to a single-freq-uency GPS user under slight attenuation. It is assumed that 12 single-frequency satellites are visible at a common C/N0 of 36 dB-Hz, yielding from the simulation curves a value for each channel of 0.0265 rad2. When substituted into equation 24, this predicts an estimation uncertainty of
This is a level of estimation uncertainty similar to that assumed to be intrinsic to the local oscillator in the previous section. The result implies that with this minimally powerful set of satellites, it becomes possible to quantify the behavior of the local oscillator with a level of uncertainty commensurate with the actual uncertainty in the oscillator over the 10 ms averaging interval.
Consequentially, this indicates that the Allan deviation of this system oscillator could be wholly evaluated under these conditions at any interval of 10 ms or longer. Further, if the system oscillator were in fact the less stable MCXO from the resource above, this estimate uncertainty would be significantly lower than the actual uncertainty intrinsic to the oscillator, providing an opportunity to “clean” the velocity measurements.
Scenario 2 is intended to be characteristic of a near future multi-constellation single-frequency receiver. It is assumed that eight satellites from three constellations are visible on a single frequency each, with a common C/N0 of 42 dB-Hz, yielding a value for each channel of 6.4e-3 rad2, leading to an estimation uncertainty of
Scenario 3 is intended to serve as an optimistic scenario involving a future multi-frequency, multi-constellation receiver. It is assumed that nine future satellites are available from each of three constellations, each with four independent carriers, all received at 45 dB-Hz, yielding a value for each channel of 3.2e-3 rad2, leading to an estimation uncertainty of
Application to Observations
The theoretical benefit of subtracting these phase changes from the measurements of an individual loop prior to propagating that measurement to the common position solution epoch ranges from moderate to very high depending on the satellite timing skew relative to the solution point. The most beneficial scenario is total elimination of oscillator noise effects (within the uncertainty of the estimate), which is experienced in the special case (Case A, FIGURE 5), where the bit period of a given satellite falls entirely over two of the 10-ms subsections. The uncertainty would increase to 2x the level of uncertainty in the estimate in the special case (Case B) where the satellite bit period straddles one full 10-ms period and two 5-ms halves of adjacent periods, and would lie somewhere between 1 and 2 times the level of uncertainty for the general case where three subintervals are covered, yet the bit period is not centered (Case C).
FIGURE 5. Special cases of oscillator estimate versus bit-period alignment.
While the application to observations of the predicted oscillator phase changes between integration intervals does not appear immediately useful for high-end receiver users with the exception of those in high-vibration or scintillation-detection applications, it could be applied to consumer-grade receivers to facilitate the use of inexpensive system clocks while providing observables with error levels as low as those provided by much more expensive receivers incorporating ovenized frequency references.
Further Points
While the chosen coherent integration period may be lengthened to increase the certainty of the measurement from a noise averaging perspective, this modification risks degrading the usefulness of said measurement due to dynamics sensitivities. Additionally, as the coherent integration time is increased, the granularity with which the pre-propagation oscillator contribution may be removed from an individual loop will be reduced. While this may be useful in cases of very low dynamics where the system is intended to estimate phase errors in a local oscillator with high certainty, it would be of little use if one desires to provide low-noise observables at the output.
For this reason, it is recommended that increases in coherent integration time be approached with caution, and extra thought be given to use of dynamics estimation techniques such as smoothing, via use of the subsequent n-ms segment in the formation of the estimate of dynamics for the “current” segment. This carries the penalty of increased processing latency, but could greatly reduce dynamics effects by enabling their more reliable excision from the desired phase-delta measurements.
Acknowledgments
The author thanks his supervisors, Gerard Lachapelle and Elizabeth Cannon, and the Natural Sciences and Engineering Research Council of Canada, the Alberta Informatics Circle of Research Excellence, the Canadian Northern Studies Trust, the Association of Canadian Universities for Northern Studies, and Environment Canada for financial and logistical support.
AIDEN MORRISON is a Ph.D. candidate in the Position, Location, and Navigation (PLAN) Group, Department of Geomatics Engineering, Schulich School of Engineering at the University of Calgary, where he has developed a software-defined GPS/GLONASS receiver for his research.
GPS receivers must deal with measurements and models that have some degree of error, which gets propagated into the position solution. If the errors are systematically different for the different simultaneous pseudoranges, as is typically the case when trying to correct for ionospheric and tropospheric effects, these errors propagate into the receiver solution in a way that is fundamentally different from the way that random errors propagate. So in addition to dilution of precision, we need a companion measure of systematic effects. In this month’s column, we introduce just such a measure.
INNOVATION INSIGHTS by Richard Langley
WE LIVE IN AN IMPERFECT WORLD. We know this all too well from life’s everyday trials and tribulations. But this statement extends to the world of GPS and other global navigation satellite systems, too. A GPS receiver computes its three-dimensional position coordinates and its clock offset from four or more simultaneous pseudoranges. These are measurements of the biased range (hence the term pseudorange) between the receiver’s antenna and the antenna of each of the satellites being tracked. The receiver processes these measurements together with a model describing the satellite orbits and clocks and other effects, such as those of the atmosphere, to determine its position. The precision and accuracy of the measured pseudoranges and the fidelity of the model determine, in part, the overall precision and accuracy of the receiver-derived coordinates.
If we lived in an ideal world, a receiver could make perfect measurements and model them exactly. Then, we would only need measurements to any four satellites to determine our position perfectly. Unfortunately, the receiver must deal with measurements and models that have some degree of error, which gets propagated into the position solution. Furthermore, the geometrical arrangement of the satellites observed by the receiver — their elevation angles and azimuths — can significantly affect the precision and accuracy of the receiver’s solution, typically degrading them. It is common to express the degradation or dilution by dilution of precision (DOP) factors. Multiplying the measurement and model uncertainty by an appropriate DOP value gives an estimate of the position error.
These estimates are reasonable if the measurement and model errors are truly random. However, it turns out that this simple geometrical relationship breaks down if some model errors are systematic. If that systematic error is a constant bias and if it is common to all pseudoranges measured simultaneously, then the receiver can easily estimate it along with its clock offset, leaving the position solution unaffected. But if the errors are systematically different for the different simultaneous pseudoranges, as is typically the case when trying to correct for ionospheric and tropospheric effects, these errors propagate into the receiver solution in a way that is fundamentally different from the way that random errors propagate. This means that in addition to DOP, we need a companion measure of systematic effects.
In this month’s column, Dennis Milbert introduces just such a measure — the error scale factor or ESF. ESF, combined with DOP, forms a hybrid error model that appears to more realistically portray the real-world GPS precisions and accuracies we actually experience.
“Innovation” features discussions about advances in GPS technology, its applications, and the fundamentals of GPS positioning. The column is coordinated by Richard Langley, Department of Geodesy and Geomatics Engineering, University of New Brunswick.
The recent edition of the Standard Positioning Service (SPS) Performance Standard (PS) and the corresponding document for the Precise Positioning Service (PPS) both emphasize a key element. They only specify the GPS signal-in-space (SIS) performance. Since these standards do not define performance for any application of a GPS signal, it becomes even more important to understand the relationship of signal statistics to positioning accuracy. Historically, as well as in Appendix B of the SPS-PS and PPS-PS, this relationship is modeled by covariance elements called dilution of precision (DOP).
Many references are available which describe DOP. The core of DOP is the equation of random error propagation:
Qx= ( AtQ-1A ) -1
where, for n observations, A is the n x 4 matrix of observation equation partial differentials, Q is the n x n covariance matrix of observations, and Qx is the 4 x 4 covariance matrix of position and time parameters (X, Y, Z, T) used to compute DOPs. This equation describes the propagation of random error (noise) in measurements into the noise of the unknown (solved for) parameters. Elements of the Qx matrix are then used to form the DOP.
The equation above is linear for any measurement scale factor of Q. For example, halving the dispersion of the measurements will halve the dispersion of the positional error. This scaling behavior is exploited when forming DOP where, by convention, Q is taken as the identity matrix, I. DOPs then become unitless, and are treated as multipliers that convert range error into various forms of positional error. Thus, we see relationships in the SPS-PS Appendix B such as:
UHNE = UERE x HDOP
where UERE is user equivalent range error, HDOP is horizontal dilution of precision, and UHNE is the resulting user horizontal navigation error.
DOP is a model relationship between signal statistics and position statistics based on random error propagation. But, since the cessation of Selective Availability (SA), the GPS signal in space now displays less random dispersion than the average systematic effects of ionosphere and troposphere propagation delay error. It’s useful to test if a random error model can capture the current behavior of GPS positioning on the ground.
The Federal Aviation Administration collects GPS data at the Wide Area Augmentation System (WAAS) reference stations and analyzes GPS SPS performance. These analyses are documented in a quarterly series called the Performance Analysis (PAN) Reports. To test horizontal and vertical accuracy, the 95th percentile of positional error, taken comprehensively over space and time, without any subsetting whatsoever, is chosen. This measure is always found in Figures 5-1 and 5-2 of the PAN reports. Note that the Appendix A 95% “predictable accuracy” in the reports through PAN report number 51 refers to a worst-site condition and cannot be considered comprehensive. The PAN report 95th percentiles of positional error measured since the cessation of SA are reproduced in FIGURE 1.
Figure 1. Accuracy (95th percentile) of horizontal and vertical L1-only point positioning. GPS data are gathered at WAAS reference stations, analyzed quarterly, and published in the PAN reports. The red line is vertical accuracy and the blue line is horizontal accuracy.
By the DOP error model, the positional error should be the product of the underlying pseudorange error times HDOP
or vertical DOP (VDOP). It is convenient to form the vertical to horizontal positional error ratio, V/H, shown in FIGURE 2. This error ratio should, formally, be independent of the magnitude of the range error. The error ratio should reflect the GPS constellation geometry. One expects the positional error ratio, V/H, to be relatively uniform, and it should also equal the VDOP/HDOP ratio. However, Figure 2 shows a number of spikes (from PAN Reports 37, 40, 44, 64) in the error ratio, and a general increase over the past nine years. The positional error ratios in Figure 2 do not portray the uniform behavior expected for a DOP error model based on random error propagation.
Figure 2. Ratio of the vertical/horizontal accuracy (95th percentile). The spikes indicate effects that are not caused by constellation geometry or signal-in-space error.
The PAN reports form a challenge to our ability to understand and describe the measured performance of the GPS system. In the past, when SA was imposed on the GPS signal, the measured pseudorange displayed random, albeit time-correlated, statistics. DOP was effective then in relating SA-laden range error to positional error. Now, with SA set to zero, the role of DOP should be revisited.
In this article, I will introduce a hybrid error model that takes into account not only the effects of random error but also that of systematic error due to incomplete or inaccurate modeling of observations. But first, let’s examine predicted GPS performance based on DOP calculations alone.
Random Error Propagation
FIGURE 3 displays detail of a 24-hour HDOP time series. Considerable short wavelength structure is evident. Spikes as thin as 55 seconds duration can be found at higher resolutions. Given the abrupt, second-to-second transitions in DOP, and given that the GPS satellites orbit relative to the Earth at about 4 kilometers per second, one may suspect that short spatial scales as well as short time scales are needed to describe DOP behavior.
Figure 3. All-in-view HDOP, July 20, 2007, near the Washington Monument, 5° elevation angle cutoff. Note the abrupt transitions, and that HDOP is around 1.0. VDOP (not pictured) is about 1.5.
To investigate DOP transitions, the conterminous United States (CONUS) was selected as a study area. HDOP and VDOP, with a 5° elevation-angle cutoff, were computed using an almanac on a regular 3 minute by 3 minute grid over the region 24°-53° N, 230°-294° E. These DOP grids were computed at 2,880 30-second epochs for July 20, 2007, yielding more than two trillion DOP evaluations. This fine time/space granularity was selected to capture most of the complex DOP structure seen in Figure 3.
FIGURE 4 plots the HDOP distribution over CONUS and parts of Canada and Mexico at 02:40:30 GPS Time. This epoch was selected to show an HDOP excursion (HDOP 4 2.58) seen in the red zone just north of Lake Ontario. DOPs are rather uniform within zones, and these zones have curved boundaries. The boundaries are sharply delineated and move geographically in time, which explains the jumps seen in high-rate DOP time series (as in Figure 3). The broad, curved boundaries seen in Figure 4 are the edges of the footprints of the various GPS satellites. The gradual variation in hue within a zone shows the gradual variation of DOP as the spatial mappings of the local elevation angles change for a given set of GPS satellites in a region.
Figure 4. HDOP, July 20, 2007, 02:40:30 GPS Time, 5° cutoff. The curved boundaries, which show abrupt transitions in DOP, are the edges of the footprints of various GPS satellites.
The 2,880 color images of HDOP (and VDOP) were converted into an animation that runs 4 minutes and 48 seconds at 10 frames per second. The effect is kaleidoscopic, as the various footprints cycle across one another, and as the zones change color. The footprint boundaries transit across the map in various directions and create a changing set of triangular and quadrilateral zones of fairly uniform DOP. There is no lower limit to temporal or spatial scale of a given DOP zone delimited by three transiting boundaries. The size of a zone can increase or shrink in time. Zones can take a local maximum, a local minimum, or just some intermediate DOP value. And the DOP magnitude in a given zone often changes in time. The animation shows that the DOP maximums are quite infrequent, and the DOPs generally cluster around the low end of the color scale. The animations are available.
To get a quantitative measure of distribution, the HDOPs (and VDOPs) are histogrammed with a bin width of 0.01 in FIGURE 5. Tabulations of various percentiles, computed from the bin counts, are displayed in TABLE 1. HDOP ranges from 0.600 to 2.685 and VDOP ranges from 0.806 to 3.810.
Figure 5. HDOP, July 20, 2007, 5° cutoff. DOP has a strong central tendency and a tail showing rare instances of large DOP. Here HDOP ranges from 0.600 to 2.685.
Chart: GPS World
Since the DOP zone boundaries are related to satellites rising and setting, it is natural to expect a relation to a selected cutoff limit of the elevation angle. As a test, DOP was recomputed with a 15° cutoff limit, and histogrammed with a bin width of 0.01 in FIGURE 6. Tabulations of various percentiles, computed from the bin counts, are displayed in TABLE 2. HDOP ranges from 0.735 to 26.335, and VDOP ranges from 1.045 to 72.648.
Figure 6. HDOP, July 20, 2007, 15° cutoff. DOP is sensitive to cutoff angle. Here HDOP ranges from 0.735 to 26.335. This is a large increase over the HDOP with a 5° cutoff.
The Figures 5 and 6 and Tables 1 and 2 show that DOPs are markedly sensitive to cutoff angles. The histogram tails increase and the maximum DOPs dramatically increase as the cutoff angle is increased. The 95th percentile HDOP increases by about 50 percent when the cutoff angle increases from 5° to 15°. The solutions weaken to some degree and the poorer solutions get much worse. The effect is somewhat greater for VDOP.
One normally considers DOP as a property of the satellite constellation that has a space-time mapping. DOP is seen to strongly depend upon horizon visibility. This is a completely local property that is highly variable throughout the region. Clearly, DOP depends on the antenna site as well as the constellation.
Systematic Error Propagation
It is known that certain error sources in GPS are systematic. Such errors will display different behaviors from random error. For example, the impact of ionosphere and troposphere error on GPS performance has been recognized in the literature (see “Further Reading”). DOP is not successful in modeling systematic effects. A new metric for systematic positional error is needed.
Consider a systematic bias, b, in measured pseudorange, R. One may propagate the bias through the weighted least-squares adjustment:
(AtQ-1A) x =AtQ-1y
by setting the n x 1 vector, y = b. Vector x will then contain the differential change (error) in coordinates (δx, δy, δz, δt) induced by the bias. The coordinate rror can then be transformed into the north, east, and up local horizon system (δN, δE, δU). Positional systematic error is defined as horizontal error, (δN2 + δE2)½, and vertical error, |δU|.
As with DOP, the equations above are linear for any measurement bias scale factor, k, which applies to all satellite pseudoranges at an epoch. For example, if one halves a bias that applies to all pseudoranges (for example, ky), then one will halve the associated coordinate error, kx. Analogous to DOP, we take bias with a base error b = 1, to create a unitless measure that can be treated as a multiplier. We now designate the horizontal error as horizontal error scale factor (HESF) and vertical error as vertical error scale factor (VESF). This adds a capability of developing error budgets for systematic effects that parallels DOP.
Systematic errors in GPS position solutions have a distinctly different behavior than random errors. This is illustrated by a trivial example. If one repeats any of the tests above with a constant value, c, for the bias, one will find that, aside from computer round-off error, no systematic error propagates into the position. The coordinates are recovered perfectly, and the constant bias is absorbed into the receiver time bias parameter, δ t. This is no surprise, since the GPS point position model is constructed to solve for a constant receiver clock bias.
The ionosphere and troposphere, on the other hand, cause unequal systematic errors in pseudoranges. These systematic errors are greater for lower elevation angle satellites than for higher elevation angle satellites. So, unlike the trivial example above, these errors cannot be perfectly absorbed into δ t. The systematic errors never vanish, even for satellites at zenith. One may expect some nonzero positional error that does not behave randomly.
The systematic effect of the ionosphere and troposphere differ through their mapping functions. These are functions of elevation angle, E, and are scale factors to the systematic effect at zenith (E = 90°). Because of the different altitudes of the atmospheric layers, the mapping functions take different forms. For this reason, systematic error scale factors (ESFs) for the ionosphere and troposphere must be considered separately.
Ionosphere Error Scale Factor. Following Figure 20-4 of the Navstar GPS Space Segment/Navigation User Interfaces document, IS-GPS-200D, the ionospheric mapping function associated with the broadcast navigation message, F, is
F = 1.0 + 16.0 (0.53 – E)3
where E is in semicircles and where semicircles are angular units of 180 degrees and of π radians. Since the base error is considered to be b = 1 for ESFs, y is simply populated with the various values of F appropriate to the elevation angles, E, of the various satellites visible at a given epoch. The resulting HESF and VESF values will portray how systematic ionosphere error will be magnified into positional error, just as DOPs portray how random pseudorange error is magnified into positional error.
As was done with the DOPs, more than two trillion ionosphere HESFs (and VESFs) were computed for CONUS and histogrammed in FIGURE 7. Tabulations of various percentiles, computed from the bin counts, are displayed in TABLE 3. Ionosphere HESF ranges from 0.0 to 0.440 and VESF ranges from 1.507 to 2.765.
Figure 7. HESF, ionosphere, July 20, 2007, 5° cutoff. The HESF-I are much smaller than the HDOP. The VESF-I (not depicted) have an average larger magnitude than the VDOP.
The distribution of the HESF-I in Figure 7 differs profoundly from HDOP. Ionosphere error is seen to have a weak mapping into horizontal positional error, with HESF-I values approaching zero, and having a long tail. The VESF-I is roughly comparable to the magnitude of the ionosphere mapping function at a low elevation angle. The VESFs also fall into a fixed range, without long tails, and are skewed to the right. The percentiles in Table 3 show ionosphere error has a greater influence on the height than that predicted by DOP.
Systematic Range Error and Height. Both troposphere and ionosphere propagation error leads to error in height. The mechanism underlying the behavior in Table 3 is not obvious. Consider the simplified positioning problem in FIGURE 8, where we solve for two unknowns: the up-component of position and receiver bias, dt (which includes effects common to all pseudoranges measured at the same time, such as the receiver clock offset). The atmosphere will cause the pseudoranges AO, BO, and CO to measure systematically longer. However, the ionosphere error will be about three times larger at low elevation angles than at the zenith. (Troposphere error will be about 10 times larger at low elevation angles than at the zenith.)
Figure 8. Schematic of pseudorange positioning. Computing up and receiver clock bias through 3 pseudoranges (AO, BO, CO), BO is biased by +5 meters ionosphere; AO and CO are biased by +15 meters ionosphere. Clock bias will absorb the +15 meters from the conflicting horizontal pseudoranges, and overcorrect the BO pseudorange by 10 meters.
In this simplified example, assume the zenith pseudorange, BO, measures 5 meters too long because of unmodeled ionosphere delay. Then the near-horizon pseudoranges, AO and CO, will measure 15 meters too long. AO and CO can’t both be 15 meters too long at the same time, so that bias is absorbed by the receiver bias term, dt. That dt term is also a component of the up solution from BO. While the AO and CO pseudoranges have superb geometry in establishing receiver clock bias, they also have terrible geometry in establishing height. The height is solved from the BO pseudorange that is overcorrected by 10 meters. Point O rises by 10 meters. The presence of the receiver bias term causes atmospheric systematic error to be transferred to the height. It also shows that the horizontal error will largely be canceled in mid-latitude and equatorial scenarios.
Troposphere Error Scale Factor. A variety of troposphere models and mapping functions are available in the literature. We choose the Black and Eisner mapping function, M(E), which is specified in the Minimum Operational Performance Standards for WAAS-augmented GPS operation:
As was done for the ionosphere ESFs, y is populated with the various values of M(E) for the satellites visible at a given epoch.
The troposphere HESFs (and VESFs) are computed for CONUS and histogrammed in FIGURE 9. Tabulations of various percentiles, computed from the bin counts, are displayed in TABLE 4. Troposphere HESF ranges from 0.0 to 5.203, and VESF ranges from 1.882 to 13.689.
Figure 9. HESF, troposphere, July 20, 2007, 5° cutoff. The HESF-Ts are significantly larger than the HESF-Is, showing that unmodeled troposphere propagation error can more readily influence horizontal position. The VESF-Ts are substantially larger than the VDOPs and VESF-Is.
The troposphere HESFs in Table 4 have similarities with, and differences from, the ionosphere HESFs of Table 3. Troposphere error maps more strongly into the horizontal coordinates than ionosphere error. The VESFs are much larger than the HESFs. And the VESFs still fall into a fixed range, without long tails.
Unlike DOP, which is derived from random error propagation, ESF is constructed for systematic error propagation. A good “vest pocket” number for the tropospheric delay of pseudorange at zenith is 2.4 meters at mean sea level. Thus, without a troposphere model, one can expect horizontal error of 1.80 x 2.4 meters = 4.32 meters or less 95 percent of the time according to Table 4.
Cutoff Angle. We now briefly consider the behavior of ESF under an increased elevation angle cutoff. The ionosphere ESFs with a 10° cutoff show minor improvements. This is a distinct difference from DOP (see Table 2), which showed degraded precision with a larger cutoff angle. The troposphere ESFs with a 10° cutoff angle are computed from histogram bin counts (TABLE 5). 10° cutoff troposphere HESF ranges from 0.0 to 3.228 and VESF ranges from 1.161 to 9.192.
Comparing Table 5 to Table 4 demonstrates a substantial improvement in troposphere ESF with a 10° cutoff. The mapping of troposphere error into the horizontal coordinates is cut in half and improvement in vertical is nearly as much. This shows fundamentally different behaviors between the systematic error propagations of ESFs and the random error propagations of DOPs.
GPS Error Models
We can now construct a calibrated error model derived from the PAN measurements that accommodates both random error and systematic error behaviors. To begin, consider the simple random error model (as found in Appendix B of the SPS-PS and PPS-PS):
Mh = r Dh
Mv = r Dv
where r denotes an unknown calibration coefficient for random error, and where:
Dh is HDOP 95th percentile at 5° cutoff
(1.24 by Table 1)
Dv is VDOP 95th percentile at 5° cutoff (1.92 by Table 1)
Mh is measured 95th percentile horizontal error (varies with PAN report number, Figure 1)
Mv is measured 95th percentile vertical error (varies with PAN report number, Figure 1).
One immediately sees by inspection that we have not one, but two estimates of r for each PAN report. And these estimates are inconsistent.
Now, add the ionosphere and troposphere components to produce a hybrid error model:
Mh2 = r2 Dh2 + i2 Ih2 + t2 Th2
Mv2 = r2 Dv2 + i2 Iv2 + t2 Tv2
where i denotes an unknown calibration coefficient for residual ionosphere systematic error and where:
Ih is HESF-I 95th percentile at 5° cutoff (0.162 by Table 3)
Iv is VESF-I 95th percentile at 5° cutoff (2.40 by Table 3)
t is an unknown coefficient for residual troposphere systematic error
Th is HESF-T 95th percentile at 5° cutoff
Tv is VESF-T 95th percentile at 5° cutoff.
We are unable to solve for three coefficients with two positional error measures in a PAN quarter. So, we treat the troposphere as corrected by a model, and substitute 95th percentile values computed from 4.9 centimeters of residual troposphere error:
Mh2 = r2 Dh2 + i2 Ih2 + (0.01)2
Mv2 = r2 Dv2 + i2 Iv2 + (0.60)2
This leads to a 2 x 2 linear system for each PAN quarter. The r and i coefficients are solved for and displayed in FIGURE 10.
Figure 10. Hybrid model of random and ionosphere error by PAN report number. Red line is random error; blue line is ionosphere. Gaps in the plot indicate inconsistent coefficient solutions.
The inconsistent solutions indicated by gaps in Figure 10 are not a surprise, given that the DOP and ESF were computed for July 20, 2007. Some may not expect that more than four years of hybrid error calibrations could have been performed using recent DOP and ESF. Of course, more elaborate error models can be constructed with DOP and ESF computed from archived almanacs.
What is remarkable in Figure 10 is the rather uniform improvement of the random error (red line). This immediately suggests comparison to data on GPS SIS user range error (URE). Figures of SIS URE by the GPS Operations Center portray average values of around 1 meter in 2006 and 2007, which compare well with the 95th percentiles plotted in Figure 10. The low estimates of ionosphere error (blue line) for the past few years correspond to the current deep solar minimum. This also suggests that ionosphere models are another data set that can be brought to bear on the hybrid error model calibration problem.
This hybrid error model is just a first attempt at simultaneously reconciling random and systematic effects. It shows some capability to distinguish ionosphere error from other truly random noise sources. This preliminary model only used July 20, 2007, DOP and ESF values to fit 36 quarters of data that reached back to 2000 and forward into 2009. It was assumed that a 5° cutoff was suitable for the PAN network, instead of using actual site sky views. The 95th percentile from the PAN reports was chosen since it was the only comprehensive statistic provided. A 50th percentile, if it had been available, is a more robust statistic.
Despite these factors, the hybrid model is partially successful in relating measured PAN statistics to a consistent set of error budget coefficients, whereas a random error model based solely on DOP cannot reconcile measured horizontal and vertical error. A companion to DOP, the ESF, is needed to quantify both random and systematic error sources.
Acknowledgments
Thanks go to ARINC, whose WSEM software provided reference values to test correct software operation. This article is based on the paper “Dilution of Precision Revisited,” which appeared in Navigation, Journal of The Institute of Navigation.
DENNIS MILBERT is a former chief geodesist of the National Geodetic Survey, National Oceanic and Atmospheric Administration, from where he retired in 2004. He has a Ph.D. from The Ohio State University. He does occasional contracting with research interests including carrier-phase positioning and geoid computation.
FURTHER READING
• Dilution Of Precision
“Dilution of Precision Revisited” by D. Milbert in Navigation, Journal of The Institute of Navigation, Vol. 55, No. 1, 2008, pp. 67–81.
“Dilution of Precision” by R.B. Langley in GPS World, Vol. 10, No. 5, May 1999, pp. 52–59.
“Satellite Constellation and Geometric Dilution of Precision” by J.J. Spilker Jr. and “GPS Error Analysis” by B.W. Parkinson in Global Positioning System: Theory and Applications, Vol. 1, edited by B.W. Parkinson and J.J. Spilker Jr., Progress in Astronautics and Aeronautics, Vol. 163, American Institute of Aeronautics and Astronautics, Washington, D.C., 1996, pp. 177–208 and 469–483.
“Post-Modernization GPS Performance Capabilities” by K.D. McDonald and C.J. Hegarty in Proceedings of the IAIN World Congress and the 56th Annual Meeting of The Institute of Navigation, San Diego, California, June 26–28, 2000, pp. 242–249.
“The Residual Tropospheric Propagation Delay: How Bad Can It Get?” by J.P. Collins and R.B. Langley in Proceedings of ION GPS-98, 11th International Technical Meeting of the Satellite Division of The Institute of Navigation, Nashville, Tennessee, September 15–18, 1998, pp. 729–738.
“The Role of the Clock in a GPS Receiver” by P.N. Misra in GPS World, Vol. 7, No. 4, April 1996, pp. 60–66.
“The Effects of Ionospheric Errors on Single-Frequency GPS Users” by R.L. Greenspan, A.K. Tet[e]wsky, J. I. Donna, and J.A. Klobuchar in ION GPS 1991, Proceedings of the 4th International Technical Meeting of the Satellite Division of the Institute of Navigation, Albuquerque, New Mexico, September 11–13, 1991, pp. 291–298.
• GPS Standards and Specifications
Global Positioning System Standard Positioning Service Performance Standard, U.S. Department of Defense, Washington, D.C., September 2008.
Global Positioning System Precise Positioning Service Performance Standard, U.S. Department of Defense, Washington, D.C., February 2007.
Navstar Global Positioning System Interface Specification, IS-GPS-200D, Revision D, IRN-200D-001, by ARINC Engineering Services, LLC for GPS Joint Program Office, El Segundo, California, March 2006.
Telltale signs of multipath are the fluctuations in the signal-to-noise ratios (SNRs) reported by some GNSS receivers. In this month’s column, the authors look at how an analysis of SNR values can be used to map the multipath environment surrounding an antenna so that models of multipath can be constructed to further minimize its effect. Also, although an annoyance for most GNSS users, it turns out that multipath has its positive points.
INNOVATION INSIGHTS by Richard Langley
CAST YOUR MIND BACK 30 OR 40 YEARS. (Sorry, students, this exercise is for the older folks.) What was one of the most striking features of the suburban landscape? Virtually every house was topped with a Yagi TV antenna. The only way to receive TV signals before cable and satellite TV was directly from the transmitter tower. And, unless you had one of those fancy antenna rotors, reception wasn’t always that great. Not only did we have to put up with weak signals, there was the problem of multipath. Besides a direct signal from the transmitter, the antenna could pick up a signal reflected off a nearby building, say, resulting in a delayed ghost image to the right of the main image on the TV screen. Even those out in the country weren’t immune from multipath as a fluttery image might be seen caused by reflections from passing aircraft.
These days, with TV signals primarily delivered by cable and satellite, we don’t see multipath much anymore. But we do hear it in our cars, from time to time, while listening to FM radio. (Students can tune back in now.) Although the FM “capture effect” provides some margin against multipath, it is not uncommon to lose stereo reception or to experience fading out of the signal while driving in built-up areas as a result of reflections.
This same multipath phenomenon also affects GNSS signals. Unlike satellite TV antennas, the antennas feeding our GNSS receivers are omnidirectional. So we have the possibility of not only receiving a direct, line-of-sight signal from a GNSS satellite but also any indirect signal from the satellite that gets reflected off nearby buildings or other objects or even the ground. GNSS antenna and receiver manufacturers have developed techniques to minimize the impact of multipath on the GNSS observables. Nevertheless, there is typically some residual multipath afflicting the pseudorange and carrier-phase observables that limits the precision and accuracy of position determinations.
Telltale signs of multipath are the quasi-periodic fluctuations in the signal-to-noise ratios (SNRs) reported by some GNSS receivers, and in this month’s column, we learn how an analysis of SNR values can be used to map and better understand the multipath environment surrounding an antenna.
And, although an annoyance for most GNSS users, it turns out that multipath is not all bad. By analyzing the SNR fluctuations due to multipath, characteristics of the reflector can be deduced. If the reflector is the ground, then the amount of moisture in the soil can be measured. GNSS for measuring soil moisture? Who would have thought?
“Innovation” is a regular column that features discussions about recent advances in GPS technology and its applications as well as the fundamentals of GPS positioning. The column is coordinated by Richard Langley of the Department of Geodesy and Geomatics Engineering at the University of New Brunswick.
We often hear “multipath” blamed as the last great source of unmodeled errors in GNSS observations, and therefore positions. But what is multipath? And what can we do about it? Can we remove multipath, or understand its temporal and spatial nature, or use it in new and novel ways? In this article, we address some of these outstanding multipath questions through the lens of the signal-to-noise ratio, or SNR. This article begins with background on the multipath phenomenon and discusses how carrier-phase multipath is related to SNR, an observable that is routinely collected by GNSS receivers but rarely used. The remainder of the article details a few new applications of SNR observations for multipath analysis. With this single observable type and a few assumptions about its relation to tracking loops and the environment surrounding the antenna, we can understand the multipath environment, remove multipath errors from carrier-phase measurements, and in some cases even transform this error into a new source of environmental information.
Multipath is exactly what it sounds like — a signal that travels along more than one path. When GNSS radio waves propagate from the GNSS satellite toward the receiving antenna, it is possible for the incoming signal to travel more than one path via reflection, diffraction, scattering, or a combination of these. Although all these phenomena contribute to multipath, in this article we limit multipath to reflections of a specular nature. Specular reflections occur when an electromagnetic wave hits an object (such as the surface of the Earth, a building, or a car) that is smooth relative to the signal wavelength. Upon reflection from the smooth surface, the outgoing energy is coherent, discrete, and sent in a single direction. From this point forward, multipath is taken to mean specular reflections from a large object.
When received by a GNSS antenna, this coherent reflected signal will disturb the tracking loops and distort the measured code and phase. The code and phase distortions occur because the GNSS receiver tracks a composite signal, which is the sum of the direct or line-of-sight signal and one or more multipath signals. The composite signal is biased from the direct signal simply because the multipath signal travels a longer path length than the desired direct signal. GNSS tracking and positioning rely upon the assumption of direct line-of-sight between satellite and receiver, thus tracking a composite signal will result in mismeasurement of the carrier and code ranges.
Why is multipath still an unsolved problem with GNSS positioning? As discussed below, multipath is a site-specific phenomenon — each GNSS site or satellite or vehicle will have a unique multipath-generating environment. Multipath is also dynamic — errors evolve with motion of the GNSS satellites and change as the reflecting surfaces (such as growing vegetation, moving cars, dry or damp ground) around the receiving antenna also change. Multipath errors cannot be simply differenced away — multipath at one station will not cancel out upon differencing with observables from another station. Nor can multipath always be “averaged out” — with real-time or rapid static GNSS positioning, the spatial and temporal complexity of site-specific multipath environments can adversely affect position determination.
Simplified Multipath Model
On the most basic level, multipath errors are driven by the geometric relationships between the receiving point (the GNSS receiver antenna), the sending point (the GNSS satellite antenna), and the reflecting object. We illustrate these geometric relationships using simple ray tracing; for a more involved ray-tracing technique, see the paper “Development and Testing of a New Ray-Tracing Approach to GNSS Carrier-Phase Multipath Modelling” listed in Further Reading. The geometric relationships between the satellite, receiving antenna, and reflecting objects dictate the additional path length traveled by the multipath signal, and how this path length changes as the satellite moves.
In an ideal, multipath-free world, this geometry is described only by the line-of-sight betwxeen satellite and receiver, which we describe via the azimuth and elevation angle of the satellite relative to the receiver. The geometry becomes more complicated when a reflecting/multipath object is introduced. TABLE 1 introduces some multipath terms and FIGURE 1 shows how these factors combine to create a forward-scatter multipath environment where a single reflected signal is received by the GNSS antenna. This illustration shows an antenna receiving two signals from one GNSS satellite, the desired direct ray and a second ray that reflects off a tilted, planar object before reception. For this example, we assume all angles are coplanar and disregard the third dimension.
FIGURE 1. (a) Forward-scatter multipath geometry, where the red arrows indicate the longer path traveled by the multipath signal relative to the direct signal. See Table 1 for definition of terms. (b) Signal amplitudes after including antenna gain pattern (green line) effects and attenuation upon reflection at a surface; see Table 2 for definition of terms.
Using the multipath terms listed in Table 1 and the geometric relationships depicted in Figure 1a, the additional distance traveled by the reflected/multipath signal relative to the direct one is the path delay. The phase of the multipath signal (again, relative to the direct signal) is the angular equivalent of path delay:
[1]
Already, we see that the path delay and multipath relative phase are a function of the antenna-reflector distance (h) and the angle of reflection from the surface (β), and that the same multipath object will result in different multipath phases for different GNSS signals due to the dependence on λ.
As discussed below, the time-varying nature of multipath is key to understanding and mitigating its effects. Thus we examine the multipath frequency, that is, the rate of change of the multipath phase:
[2]
If we assume a single stationary reflecting object, the only time-varying factor in Figure 1 is the satellite — as the satellite moves relative to the receiving antenna, the reflection point also moves, changing the path delay and multipath relative phase. Substituting the angular relationships (see Figure 1a) between the satellite, receiver, and reflecting object into the previous equation makes this more obvious:
[3]
But how is “multipath frequency” related to quantities measured by our GNSS receivers: the code range, carrier phase, and signal-to-noise ratio (SNR)? To answer that question, we must introduce another set of multipath quantities, which describe the dominant signal strength factors (TABLE 2) for the direct and multipath signals; we ignore thermal noise, cable losses, etc.
The amplitude of the direct signal (Ad) is equivalent to the GNSS signal strength as it is received and is affected by the antenna gain pattern (Figure 1b). The multipath signal comes through the antenna gain pattern at a different angle; by design, most GNSS antennas will apply less gain at angles consistent with common multipath geometries, such as below the antenna horizon. The multipath signal will also experience some amount of attenuation upon reflection; the combination of attenuation and antenna gain yields the amplitude of the multipath signal (Am). Note that the broadcast GNSS signals are right-hand circularly polarized (RHCP), which are largely converted to left-hand polarization upon reflection. Thus the simplified “gain pattern” introduced here must incorporate both RHCP and LHCP patterns.
Under the simplified model of GNSS receiver response to tracking direct plus short-delay (smaller than 1.5 code chips) reflected signals, the multipath relative phase and signal amplitudes describe both the code and carrier-phase multipath errors, respectively denoted ρMP and δφ:
[4]
. [5]
These equations are derived from code and carrier tracking behavior in the presence of multipath. Look in Further Reading for precise derivations and additional background material.
In addition to carrier phase and code observables, GNSS receivers routinely record SNR (or the related carrier-to-noise-density ratio — C/N0) for each satellite. As the term indicates, SNR is a ratio of signal power to the noise floor of the GNSS observation, and has conventionally been used only for comparison of signal strengths between channels and satellites or to assess interference. Like code and carrier-phase multipath errors, SNR is a function of multipath phase and signal strengths:
. [6]
If we remove the effects of the direct signal, the remaining SNR is due only to multipath and is reduced to a simple function of multipath signal amplitude, relative phase, and a time-invariant phase offset:
. [7]
Note that the equations for code multipath, carrier-phase multipath, and SNR contain the cosine or sine of the multipath relative phase, ψ. Therefore all three GNSS observables will have quasi-sinusoidal behavior driven by ω. To illustrate this, FIGURE 2 gives an example for a rising satellite reflecting off horizontal ground 1.0 meters below the antenna. All three GNSS observables oscillate at the same frequency; however, pseudorange error and SNR are in phase while carrier-phase error is 90 degrees out of phase.
FIGURE 2. Simulated carrier-phase error, code error, and SNR (recorded direct-plus-multipath SNR in green; SNR due to multipath alone in blue in linear amplitude units for a horizontal surface 1.0 meters below the antenna, assuming Rs 5 0.2 reflection coefficient and a choke ring antenna gain pattern.
In this article, we use SNR observations to understand and quantify multipath effects. We choose SNR over the other observable types because multipath effects on SNR have the most unambiguous relationship to multipath. Typical levels of pseudorange noise will swamp all but the most extreme of multipath errors; carrier-phase data are more precise, but extracting multipath from these data requires first modeling clocks, orbits, and atmospheric delays. SNR data are directly related to carrier-phase multipath, are largely independent of the above effects, and are determined independently for individual satellites. Unfortunately, not all GNSS receivers provide SNR data with the requisite precision and accuracy to clearly observe the multipath relationships; see “Scientific Utility of the Signal-to-Noise Ratio (SNR) Reported by Geodetic GPS Receivers” in Further Reading for information on high-utility SNR. When SNR data are of sufficient quality, they can provide a unique and direct window on the multipath errors affecting the code and carrier observations.
SNR Multipath Applications
A number of new scientific applications of SNR data are evolving to exploit the above multipath relationships. In the following sections, we describe three different SNR-multipath applications and provide relevant (although not exhaustive) references. All of these applications draw upon the above relationships and require precise and accurate SNR data that conform to the simplified multipath model described above.
Multipath Corrections. Recall that the multipath errors in GNSS observables are simply a function of signal amplitudes and the relative phase between direct and multipath signals. It stands to reason that if these amplitudes and phases can be estimated, we can model and remove multipath errors from our code and carrier observations. SNR data allow us to do just that. After extracting the direct signal (Ad) to reveal the SNR due only to multipath (SNRMP), this remaining time series depends only on Am and ψ. As shown in Figure 2 and Equation 7, SNR due to multipath oscillates with a constituent frequency ω, which is the time derivative of ψ, and has an amplitude envelope equivalent to Am. Therefore, from SNR due to multipath we can estimate multipath relative phase and multipath amplitude as a function of time.
This idea of modeling SNR data to estimate multipath parameters as time-varying quantities was first explored in a multi-antenna differential environment. This concept was extended to undifferenced SNR data so that carrier-phase errors at single-antenna GPS stations could be modeled and removed. In our implementation, we used wavelet analysis to first separate the direct amplitude from the multipath signal, then estimated the frequency content ω(t) of SNRMP as a function of time. Using as the primary input to an adaptive least-squares algorithm, we then estimated multipath amplitude and relative phase as a function of time. Substituting these Ad, Am, and ψ estimates into Equation 5 for carrier-phase multipath yielded a multipath-error correction profile.
A simple example from the Salar de Uyuni, a large salt flat in Bolivia, illustrates the process. For PRN8 observed during September 2002 with an antenna about 1.4 meters above the salt surface, the SNR due to multipath has very clear oscillations with a constituent frequency of approximately 0.0021 Hz (470 second period) (see FIGURE 3). Using frequency estimates as an input, the adaptive estimation algorithm estimates direct and multipath signal amplitudes as well as the multipath relative phase, which is approximately linear with time due to the relatively constant frequency estimate. Figure 3 shows that the modeled SNRMP closely matches the SNR data, and the carrier phase correction profile closely matches the phase errors.
FIGURE 3. SNR modeling example from the Salar de Uyuni data set, PRN8, ascending arc, in seconds since the beginning of the satellite pass. Real data are given in black, while estimated quantities are colored lines; estimation uses SNR due only to multipath, i.e., after the direct signal has been removed, in linear amplitude units. The goal of SNR modeling is to generate a phase-multipath correction profile, shown in the bottom panel as a red line overlaying phase residuals.
SNR-based phase-error estimation techniques show great promise for removing multipath errors from phase data. For the Salar de Uyuni test session, we derived SNR-based carrier-phase corrections for all satellites in view. By applying these corrections, we achieved a reduction in carrier-phase postfit residual root-mean-square error of up to 20 percent for static positioning, and 1–7 dB reduction in spectral power at multipath periods for kinematic positions.
Power Spectral Maps. Sadly, the complex and time-varying nature of multipath error cannot always be removed. In those cases, a better understanding of the multipath environment (the direction of and distance to reflecting objects) may aid the GNSS analyst. With this information, an analyst could discern the effect of multipath on position solutions, or de-weight multipath-corrupted observations, or simply choose one solution strategy (static, real-time kinematic or RTK, long vs. short occupation, etc.) over another to minimize or avoid multipath effects. For example, short duration but high-frequency multipath errors would be unimportant to someone solving for a single position using 24 hours of data, but that same multipath source could wreak havoc in an RTK survey. A method to evaluate the multipath environment at different frequencies and with a sense or orientation is therefore of great value.
As with the phase-error modeling example above, we accomplish multipath characterization via the frequency content of SNR oscillations, but this time backing out the distance, h (see Equation 3). This distance is directly related to the multipath frequency — nearby objects yield low-frequency errors, distant objects lead to high-frequency errors. By relating the distance, h, to angles (θ,γ) describing the direction and orientation of reflecting objects (Figure 1a), we can fully describe the multipath environment.
In this application, dubbed power spectral mapping, a wavelet transform is applied to each satellite’s SNR time series to extract multipath power estimates over a range of frequencies or height values. The 3-D power vs. frequency vs. spatial coordinate data cube is then sliced into frequency bands of interest (i.e., height ranges), and all data contributing to a frequency band are combined. The signal power is assigned to the satellite’s location and projected onto a “sky plot.” This type of plot has four quadrants for north, south, east and west; concentric rings indicate satellite elevation angle; the center of the plot is the zenith while the outer ring is the horizon. This combination and projection process forms a map depicting the multipath characteristics of a GPS site.
These maps can help the analyst determine the source of multipath errors. For example, at first glance the permanent International GNSS Service (IGS) GPS station MKEA (see PHOTO) on Mauna Kea volcano in Hawaii seems to be multipath-free as it is surrounded by nothing but jagged rocky ground — uneven ground (relative to the GNSS wavelength) should create a diffuse multipath signature.
Mauna Kea GPS station MKEA, facing northwest
The SNR data tell a different story, with strong coherent oscillations (see FIGURE 4) over a range of frequencies. By conducting wavelet analysis for all satellites in view, the combined power spectral maps (see FIGURE 5) show very strong reflections coming from the south-southeast and northwest, the location of volcanic cinder cones. Although rocky, these cinder cones generate strong multipath reflections. The sloped hillsides can be broken into a set of discrete reflectors at different distances, creating multipath oscillations at different frequencies over each satellite pass. For a more in-depth discussion of MKEA multipath and other power spectral map examples, see “Mapping the GPS Multipath Environment Using the Signal-to-Noise Ratio (SNR),” listed in Further Reading.
FIGURE 4. Example SNR profile from MKEA (top panel) as a function of time, in linear amplitude units after direct signal contributions have been removed. The bottom panels show wavelet power at different periods (colored lines), which are averaged together to form the wavelet power over 30–60 and 60–90 seconds-period bands of interest (heavy black lines).FIGURE 5. GPS L1 power spectral maps for MKEA SNR data for four different frequency bands (given as periods in upper right-hand corner of each plot). Figure is reproduced from “Mapping the GPS Multipath Environment Using the Signal-to-Noise Ratio (SNR).”
Soil Moisture. Manuel Martin-Neira is credited with introducing the idea, in 1993, that reflected GPS signals could be used for scientific studies. Since then, GPS reflection studies for ocean altimetry and winds, soil moisture, and snow sensing have all been discussed in the literature. These studies typically use an antenna pointed to optimize Earth reflections and specifically designed to track reflected (LHCP) signals. This means that antennas designed to suppress ground reflections, such as those used by the geophysical, geodetic, and surveying communities, are not used.
Motivated by our studies showing that multipath effects could clearly be seen in geodetic-quality data collected with multipath-suppressing antennas, we proposed that these same GPS data could be used to extract a multipath parameter that would correlate with changes in the reflectance of the ground surface. In our initial study, we used data from an existing IGS GPS site in Tashkent, Uzbekistan, and concentrated on SNR reflectance changes caused by rain and subsequent drying of the soil. While the correlation between the SNR data and precipitation models was strong, we lacked proper ground instrumentation to demonstrate that we were measuring true soil moisture changes.
Subsequently, together with other colleagues, we carried out an experiment designed to more rigorously demonstrate the link between GPS SNR and soil moisture. Specifically, we were interested in using GPS reflection parameters to determine the soil’s volumetric water content — the fraction of the total volume of soil that is occupied by water, an important input to climate and meteorological models. Traditional soil moisture sensors (water content reflectometers) were buried in the ground at multiple depths (2.5 and 7.5 centimeters) at a site just south of the University of Colorado in Boulder. Precipitation data were also collected. Using a fixed frequency, Equation 7 was used to model the SNR data and estimate an amplitude and phase offset on each day. FIGURE 6 shows phase estimates converted to water content for six satellites that pass over the same ground south of the GPS antenna. We specifically concentrated on these six satellites because they transmit the new L2C signal, which yields superior SNR data compared to the L1 C/A-code signal.
FIGURE 6. Variation in volumetric water content (VWC) from multiple GPS satellites (colored dots) and water content reflectometers buried at 2.5-centimeter depth (data range given by grey shaded region). Daily precipitation totals in blue. Figure is reproduced from “Use of GPS Receivers as a Soil Moisture Network for Water Cycle Studies.”
Figure 6 shows excellent agreement between in situ sensors and the GPS multipath parameters. Soil moisture values rise within hours of a precipitation event, and then drop over approximately one week as the soil dries. It is important to note that the GPS SNR data are sensing much larger spatial regions (hundreds of square meters) whereas the soil probes measure values over a very small soil region (100 centimeters square). Climate scientists desire soil moisture measurements that have large footprints, and SNR data from some existing GPS stations are uniquely poised to provide this scale of soil moisture measurements.
Conclusions
Under the simplified multipath model discussed here, SNR data have a defined relationship to both carrier-phase and pseudorange multipath errors. Although SNR is traditionally used only as a measure of signal tracking, we have demonstrated some applications that use this common but underutilized observable to identify potential multipath sources, model and remove phase multipath errors, or retrieve soil moisture content from ground reflections. All of these applications are predicated upon accurate and precise SNR measurements, which conform to the simplified multipath model. Not all receivers are created equal in this respect, thus care must be taken in selecting reliable SNR data for analysis.
Acknowledgments
We acknowledge technical support from UNAVCO and funding from the National Science Foundation. We thank our colleagues Eric Small, John Braun, Ethan Gutmann, Valery Zavorotny, and Penina Axelrad.
Manufacturers
The Salar de Uyuni and Mauna Kea data sets were obtained from Ashtech (now Magellan Professional) Z-12 receivers using Allen Osborne Associates (acquired by ITT Communications Systems) AOAD/M_T element antennas while the soil moisture experiment data set was from a Trimble NetRS receiver fed by a model TRM29659.00 choke ring antenna with SCIT radome.
ANDRIA BILICH is a geodesist with the National Geodetic Survey’s Geosciences Research Division in Boulder, Colorado. Her research interests include GPS multipath characterization, antenna calibration, and precision improvements to high-rate positioning for geoscience applications. She received her B.S. in geophysics in 1999 from the University of Texas and a Ph.D. in aerospace engineering in 2006 from the University of Colorado. Dr. Bilich was the recipient of the 2007 Parkinson Award from The Institute of Navigation for her dissertation titled Improving the Precision and Accuracy of Geodetic GPS: Applications to Multipath and Seismology.
KRISTINE M. LARSON received a B.A. in engineering sciences from Harvard University in 1985 and a Ph.D. in geophysics from the Scripps Institution of Oceanography, University of California at San Diego, in 1990. Since 1990, she has been a faculty member in the Department of Aerospace Engineering Sciences at the University of Colorado at Boulder. The primary focus of her work is developing and improving GPS applications for measuring plate tectonics, episodic slip, volcanic deformation, ice-sheet motion, timing, seismic waves, soil moisture, and snow depth.
“GPS Receiver Architectures and Measurements” by M.S. Braasch and A.J. van Dierendonck in Proceedings of the IEEE, Vol. 87, No. 1, January 1999, pp. 48–64.
“Conquering Multipath: The GPS Accuracy Battle” by L.R. Weill in GPS World, Vol. 8, No. 4, April 1997, pp. 59–66.
“Multipath Effects” by M.S. Braasch in Global Positioning System: Theory and Applications, edited by B.W. Parkinson, J.J. Spilker Jr., P. Axelrad, and P. Enge, Vol. 1, Chp. 14, American Institute of Aeronautics and Astronautics, Washington, D.C., 1996.
• Multipath Ray Tracing
“Development and Testing of a New Ray-Tracing Approach to GNSS Carrier-Phase Multipath Modelling” by L. Lau and P.A. Cross in Journal of Geodesy, Vol. 81, No. 11, pp. 713–732, 2007 (d
oi: 10.1007/s00190-007-0139-z).
• Assessing and Modeling Multipath Using Signal-to-Noise Ratios
“Mapping the GPS Multipath Environment Using the Signal-to-Noise Ratio (SNR)” by A. Bilich and K. M. Larson in Radio Science, Vol. 42, RS6003, 2007 (doi:10.1029/2007RS003652).
“Scientific Utility of the Signal-to-Noise Ratio (SNR) Reported by Geodetic GPS Receivers” by A. Bilich, P. Axelrad, and K. M. Larson in Proceedings of ION GNSS 2007, the 20th International Technical Meeting of the Satellite Division of The Institute of Navigation, Fort Worth, Texas, September 26–28, 2007, pp 1999-2010.
“Modeling GPS Phase Multipath with SNR: Case Study from Salar de Uyuni, Bolivia” by A. Bilich, K. M. Larson, and P. Axelrad in Journal of Geophysical Research, Vol. 113, B04401, 2008 (doi:10.1029/2007JB005194).
• Using GPS to Estimate Soil Moisture
“Using GPS Receivers to Measure Soil Moisture Fluctuations: Initial Results” by K.M. Larson, E. E. Small, E. Gutmann, A. Bilich, P. Axelrad, and J. Braun in GPS Solutions, Vol. 12, No. 3, pp. 173–177, 2008 (doi: 10.1007/s10291-007-0076-6).
“Use of GPS Receivers as a Soil Moisture Network for Water Cycle Studies by K.M. Larson, E. E. Small, E. D. Gutmann, A. L. Bilich, J. J. Braun, and V. U. Zavorotny in Geophysical Research Letters, Vol. 35, L24405, 2008 (doi:10.1029/2008GL036013).
• Measuring Reflected GPS Signals from Space
“Reflecting on GPS: Sensing Land and Ice from Low Earth Orbit” by S.T. Gleason in GPS World, Vol. 18, No. 10, October 2007, pp. 44–49.
“A Passive Reflectometry and Interferometry System (PARIS): Application to Ocean Altimetry” by M. Martin-Neira in ESA Journal, Vol. 17, No. 4, 1993, pp. 331–355.
The AsteRxi is the company’s first multi-sensor GNSS receiver. AsteRxi processes high-quality GNSS measurements with IMU-measurements to generate an enhanced integrated position.
“Traditionally, professional receivers have been integrated with expensive fiber-optic gyroscopes or similar inertial sensors, making solutions prohibitively expensive for many applications,” said Peter Grognard, managing director of Septentrio. “With the integration of the high-quality MEMS Inertial Measurement Units (IMUs) such as the MTi from Xsens with the high-precision AsteRx receivers, the benefits of integrated inertial/GNSS systems become available for a host of new industrial applications.”
Besides tracking GPS and GLONASS satellites, resulting in improved availability, the integration with IMU measurements allows AsteRxi to deliver precise position data in places where conventional GNSS receivers can’t. Additionally, the integrated solution provides position data at up to 50 Hz as well as attitude measurements, making it the ideal product for high-dynamic applications, delivering robust performance under obstructions, continuous operation under tree foliage, superior accuracy in urban canyons and much higher multipath rejection, the company said.
To optimally address technical and economical requirements of a variety of applications, AsteRxi is designed with an interface that facilitates integration by Septentrio of different IMU-sensors, depending on the application requirements. AsteRxi is delivered standard with Xsens IMUs.
AsteRx2eH Introduced. Septentrio also released the AsteRx2eH, a single-board dual-frequency GPS/GLONASS dual-antenna heading and position receiver designed for machine control, marine survey, photogrammetry, antenna pointing, and other demanding multi-antenna applications.
Asterx2eH provides reliable heading measurements without being susceptible to magnetic interference, or requiring constant recalibration to maintain its accuracy, the company said, adding that its single-board multi-antenna architecture provides unequalled performance and robustness for GNSS-based heading applications.
As member of the AsteRx-family of compact OEM and packaged GNSS receivers, AsteRx2eH is built around the same GNSS chipset and GPS and GLONASS tracking, and advanced signal processing and positioning algorithms for robust tracking and high-precision positioning, especially in challenging environments. Moreover, the electrical and communication interfaces are identical to those of other AsteRx series products, making integration of AsteRx2eH receivers easy, the company said.
“With the evolution of GPS and GLONASS systems and the proliferation of our heading technology into increasingly demanding applications, the added navigation signals offered by the different GNSS systems increase our product’s reliability and accuracy in any application,” Grognard said. “AsteRx2eH is the successor to our popular PolaRx2eH receiver. The more compact form-factor, which requires less power to operate, incorporates the superior performance capabilities of our latest ASIC technology, and all the other innovations that our AsteRx2e product family brings.”
Chefs from U.S. Air Force 746th Test Squadron GPS Test Center of Expertise cook up a winning formula for a booth display, a RECIPE of Resources, Experience, Capability, Innovation, Process, and Efficiency. Visitors were asked to solve an intriguing word puzzle to receive a delicious prize.GPS World’s booth offered a Wii bowling challenge for fun and exciting prizes.A young bowler from Bavaria (above) and ION President Mikel Miller’s family (below) enjoy an evening at the GPS World bowling alley.A visitor to NovAtel’s booth tries his piloting hand in a 3-D attitude solution with NovAtel’s GPS/INS SPAN-CPT product. Used as a controller in the flight simulator, it mimics an actual SPAN application as profiled in the September GPS World.
Miniature autonomous vehicles navigating the mean streets of a LEGO town take center stage on the show floor to highlight the Institute of Navigation’s first Mini-Urban Challenge and upcoming second challenge, now seeking sponsors.Northrop Grumman personnel Jill Rigler, Ed Chou, George Vannick, and Velvet Hoover explain the company’s programs to booth visitors on the show floor. Northrop Grumman was a Gold Sponsor of GPS World’s Leadership Dinner.Larry Skorski of the U.S. Coast Guard took first place in GPS World’s Wii Bowling Challenge (score, 225). Editor Alan Cameron presents him with a copy of Brad Parkinson and James Spilker’s book, The Global Positioning System, signed by the first author.
By Marcel Baracchi-Frei,
Grégoire Waelchli, Cyril Botteron,
and Pierre-André Farine
The idea of a software receiver is to replace the data processing implemented in hardware with software and to sample the analog input signal as close as possible to the antenna. Thus, the hardware is reduced to the minimum — antenna and analog-to-digital converters (ADCs) — while all the signal processing is done in software. As current mobile devices (such as personal digital assistants and smartphones) include more and more computing power and system features, it becomes possible to integrate a complete GNSS receiver with very few external components.
One advantage of a software receiver clearly lies in the low-cost opportunity, as the system resources such as the calculation power and system memory can be shared. Another advantage resides in the flexibility for adapting to new signals and frequencies. Indeed, an update can easily be performed by changing some parameters and algorithms in software, while it would require a new redevelopment for a standard hardware receiver.
Updating capabilities may become even more important in the future, as the world of satellite navigation is in complete effervescence: Europe is developing its own solution, Galileo, foreseen to be operational in 2013; China has undertaken a fundamental redevelopment of its current Compass navigation system; Russia is investing huge sums of money in GLONASS to bring it back to full operation; and the U.S. GPS system will see some fundamental improvements during the next few years, with new frequencies and new modulation techniques. At the same time, augmentation systems (either space-based or land-based) will develop all over the world.
These future developments will increase the number of accessible satellites available to every user — with the advantage of better coverage and higher accuracy. However, to take full advantage of the new satellite constellations and signals, new GNSS receivers and algorithms must be developed.
Definition and Types
The definition of a software receiver (SR) always brings some confusion among researchers and engineers in the field of communications and GNSS. For example, a receiver containing multiple hardware parts which can be reconfigured by setting a software flag or hardware pins of a chipset are regarded by some communication engineers to be a SR. In this article, however, we will consider the widely accepted SR definition in the field of GNSS; that is, a receiver in which all the baseband signal processing is performed in software by a programmable microprocessor.
Nowadays, software receivers can be grouped in three main categories:
field programmable gate arrays (FPGAs), which are sometimes also referred to the domain of SR. These receivers can be reconfigured in the field by software.
post-processing receivers include, among others, countless software tools or lines of code for testing new algorithms and for analyzing the GNSS signal, for example, to investigate GPS satellite failure or to decrypt unpublished codes.
real-time-capable software receivers group that will be further considered here.
A modern GNSS receiver normally contains a RF front-end, a signal acquisition, a tracking, and a navigation block. A hardware-based receiver accomplishes the residual carrier removal, PRN code-despreading, and integration at the system sampling rate. Until the late 1990s, due to the limited processing power of microprocessors, these signal functions could only be practically implemented in hardware.
The GNSS SR boom really started with the development of real-time processing capability. This was first accomplished on a digital signal processor (DSP) and later on a commercial conventional personal computer (PC). Today, DSPs are increasingly replaced by specialized processors for embedded applications.
Challenges
Data rate. The ideal software receiver would place the ADC as close as possible to the antenna to reduce hardware parts to a minimum. In that sense, the most straightforward approach consists of digitizing the data directly at the antenna, without pre-filtering or pre-processing. But as the Nyquist theorem must be fulfilled (that is, sampling with at least twice the highest signal frequency), this translates into a data rate that is, for the time being, too high to be processed by a microcontroller.
Considering the GPS L1 signal and assuming 1 quantization bit per sample, this leads to the following values:
In order to reduce the data throughput, a solution such as a low intermediate frequency (IF) or a sub-sampling analog front-end must be chosen. In a low IF front-end, the incoming signal is down-converted to a lower intermediate frequency of several megahertz. This allows working with a sampling (and data) rate that can be more easily handled by a microcontroller. With the new BOC signal modulations (used for the Galileo E1 and the modernized GPS L1 signals) that have no energy at and near DC, a zero-IF or homodyne architecture is also possible without SNR degredation due to DC offset, flicker noise, or even-order distortions.
The sub-sampling technique exploits the fact that the effective signal bandwidth in a GNSS signal is much lower than the carrier frequency. Therefore, not the carrier frequency but the signal bandwidth must be respected by the Nyquist theorem (assuming appropriate band-pass filtering). In this case, the modulated signal is under-sampled to achieve frequency translation via intentional aliasing. Again, if the GPS L1 signal is taken as an example with assuming 1 quantization bit per sample, this leads to the following values:
However, as the sub-sampling approach is still difficult to implement due to current hardware and resources limitations, a more classical solution based on an analog IF down-conversion is often chosen. That means that the signal is first down-converted to an intermediate frequency and afterwards digitized.
Baseband Processing. Considering an IF-based architecture, the ADC provides a data stream (real or complex), which is first shifted into baseband by at least one complex mixer. The signal is then multiplied with several code replicas (generally early, prompt, and late) and finally accumulated. Figure 1 shows an example of a real data IF architecture.
FIGURE 1. Real IF architecture
In hardware receivers, the local code and carrier are generally generated in real-time by means of a numerically controlled oscillator (NCO) that performs the role of a digital waveform generator by incrementing an accumulator by a per-sample phase increment. The resulting value is then converted to the corresponding amplitude value to recreate the waveform at any desired phase offset. The frequency resolution is typically in the range of a few millihertz with a 32-bit accumulator, and a sampling frequency in the range of a few megahertz.
Assuming that a look-up table (LUT) address can be obtained with two logical operations (one shift and one mask), and the corresponding LUT value reads with 1 memory access — which is quite optimistic — the amount of operations needed to generate the complex waveforms per channel is given in Table 1.
Source: Marcel Baracchi-Frei, Grégoire Waelchli, Cyril Botteron, and Pierre-André Farine
The real-time carrier generation is computationally expensive and is consequently not suitable for a one-to-one software implementation. Earlier studies [Heckler, 2004] demonstrated that, assuming that an integer operation and a multiplication take one and 14 CPU cycles, respectively (for an Intel Pentium 4 processor), the baseband operations (without carrier and code generation or navigation solution) would require at least a 3 GHz Intel Pentium 4 processor with 100 percent CPU load. Therefore, under these conditions, real-time operations are not suitable for embedded processors. Therefore standard hardware receiver architectures cannot be translated directly into software, and consequently new strategies must be developed to lower the processing load.
Status
A major problem with the software architecture is the important computing resources required for baseband processing, especially for the accumulation process. As a straightforward transposition of traditional hardware-based architectures into software would lead to an amount of operations which is not suitable for today’s fastest computers, two main alternate strategies have been proposed in the literature: the first relies on single-instruction multiple-data (SIMD) operations, which provide the capability of processing vectors of data. Since they operate on multiple integer values at the same time, SIMD can produce significant gains in execution speed for repetitive tasks such as baseband processing. However, SIMD operations are tied to specific processors and therefore severely limit the portability of the code.
The second alternative consists in the bitwise parallel operations (sometimes also referred to as vector processing in the literature), which exploit the native bitwise representation of the signal. The data bits are stored in separate vectors, one sign and one or several magnitude vectors, on which bitwise parallel operations can be performed. The objective is to take advantage of the universality, high parallelism, and speed of the bitwise operations for which a single integer operation is translated into a few simple parallel logical relations. While SIMD operations use advanced and specific optimization schemes, the latter methodology exploits universal CPU instructions set. The drawback of the bitwise operations is the different representation of the values. To be able to perform integer operations, a time consuming conversion is needed.
Single-Instruction Multiple-Data
In 1995, Intel introduced the first instance of SIMD under the name of Multi Media Extension (MMX). The SIMD are mathematical instructions that operate on vectors of data and perform integer arithmetic on eight 8-bit, four 16-bit, or two 32-bit integers packed into a MMX register (see Figure 2).
FIGURE 2. Single-instruction single-data versus single-instruction multiple-data.
On average, the SIMD operations take more clock cycles to execute than a traditional x86 operation. Anyhow, since they operate on multiple integers at the same time, MMX code can produce significant gains in execution speed for appropriately structured algorithms. Later SIMD extensions (SSE, SSE2, and SSE3) added eight 128-bit registers to the x86 instruction set. Additionally, SSE operations include SIMD floating point operations, and expand the type of integer operations available to the programmer.
SIMD operations are well suited to parallelize the operations of the baseband processing (BBP) stage. In particular, they can be used to allow the PRN code mixing and the accumulation to be performed concurrently for all the code replicas. With the help of further optimizations such as instruction pipelining, more than 600 percent performance improvement with the SIMD operations compared to the standard integer operations can be observed [Heckler, 2006].For this reason, most of the software receivers with real-time processing capabilities use SIMD operations [Heckler; Pany 2003; Charkhandeh, 2006 ].
Bitwise Operations. Bitwise operation (or vector processing) was first introduced into the SR domain in 2002 [Ledvina]. The method exploits the bit representation of the incoming signal, where the data bits are stored in separate vectors on which bitwise parallel operations can be performed. Figure 3 shows a typical data storage scheme for vector processing.
Source: Marcel Baracchi-Frei, Grégoire Waelchli, Cyril Botteron, and Pierre-André Farine
The sign information is stored in the sign word while the remaining bit(s) representing the magnitude is (are) stored in the magn word(s). The objective is to take advantage of the high parallelism and speed of the bitwise operations for which a single integer addition or multiplication is translated into simple parallel logical operations. The carrier mixing stage is reduced to one or a few simple logical operations which can be performed concurrently on several bits. In the same way, the PRN code removal only affects the sign word.
In a U.S. patent by Ledvina and colleagues, the complete code and carrier removal process requires two operations for each code replica (early, prompt, and late). The complexity can be even further reduced by more than 30 percent by considering one single combination of early and late code replicas (typically early-minus-late). This way, the authors claim an improvement of a factor of 2 for the bitwise method compared to the standard integer operations.
The inherent drawback of this approach is the lack of flexibility: the complexity of the process becomes bit-depth dependent and the signal quantification cannot be easily changed (while performing BBP with integers allows the signal structure to change significantly without code modification).
To overcome this limitation, a combination of bitwise processing and distributed arithmetic can be used [described in Waelchli, 2009]. The power-consuming operations are performed with bitwise operations, and to be able to keep the flexibility of the calculations, standard integer operations are used after the code and carrier removal. The conversion between the two methods is performed with distributed arithmetic that offers an extremely efficient way to switch between the two representations.
Another important aspect in a software receiver is the code and carrier generation. As these tasks represent a huge processing load, new solutions must be developed in this domain.
Code Generation
The pseudorandom noise (PRN) codes transmitted by the satellites are deterministic sequences with noise-like properties that are typically generated with tapped linear feedback shift registers (for GPS L1 C/A) or saved in memory (for Galileo E1). But in order to save processing power, it is preferable for software applications to compute off-line the 32 codes and store them in memory.
One method stores the different PRN codes in their oversampled representation (the code are pre-generated) [Ledvina, 2002]. As the incoming signal code phase is random, the beginning of the first code chip is in general not aligned with the beginning of a word and may occur anywhere within it. To overcome this issue, either all the possible phases can be stored in memory, or the code can be shifted appropriately during the tracking. While the first approach increases the memory requirements, the second requires further data processing in function of the phase mismatch. Regarding the Doppler compensation, all the PRN codes in the table are assumed to have a zero Doppler shift. The code phase errors due to this hypothesis are eliminated by choosing a replica code from the table whose midpoint occurs at the desired midpoint time. The only other effect of the zero Doppler shift assumption is a small correlation power loss which is not more than 0.014 dB if the magnitude of the true Doppler shift is less than 10 kHz [Ledvina patent]. This approach is very popular in the SR domain and can be found in several solutions.
Carrier Generation
The generation of a local carrier frequency is necessary to perform the Doppler removal. The standard trigonometric functions or the Taylor decompositions for the sines and cosines computation are too heavy for a software implementation and are seldom considered.
However, several other techniques exist to reduce the computational load for the carrier generation: the values for the carrier can be pre-generated and then stored in lookup tables. As this would require several gigabytes of memory to store all the possible frequencies, the values are recorded on a coarse frequency grid with zero phases and at the RF front-end sampling frequency. The carrier will thus be available in a sampled version. The limited number of available carrier frequencies introduces a supplementary mismatch in the Doppler removal process. This error can be compensated with a simple phase rotation of the accumulation results. This method is very popular in the SR domain, and many solutions take advantage of it to avoid the power-hungry real-time carrier generation.
Based on the same principle as above, Normark (2004) proposed a method that pre-computes a set of carrier frequency candidates to be stored in memory. The grid spacing is selected so as to minimize the loss due to Doppler frequency offset. Furthermore, to provide phase alignement capabilities of the carriers, a set of initial phases is also provided for each possible Doppler frequency, as illustrated in Figure 4.
FIGURE 4. Set of carrier frequency candidates.
Contrarily to the Ledvina approach and thanks to the phase alignement capabilities, the number of sampling points must not obligatorily correspond to an entire acquisition period. Therefore, the length of the frequency candidate vectors can be chosen with respect to the available memory space and becomes quasi independent of the sampling frequency.
Another approach consists in removing concurrently the Doppler from all received satellite signals [Petovello, 2006]. The algorithm is implemented as a look-up table containing one single frequency, and the carrier removal is performed for all channels with the same frequency, but the frequency error results normally in an unacceptable loss. To overcome this problem, the integration interval is split into sub-intervals for which a partial accumulation is computed.
The result is rotated proportionally to the frequency mismatch in the same way as in the method described above. The algorithm can be applied recursively and with an appropriate selection of the sub-intervals, and the total attenuation factor can be limited to a reasonable value. The author claims an improvement of up to 30 percent compared to the standard look-up table method with respect to the total complexity for both Doppler removal and correlation stages. Regarding the computational complexity, the Doppler removal stage remains unchanged, with the difference that it is only performed once for all satellites. But the rotation needs to be done for each of the sub-intervals. However, this algorithm remains difficult to implement (number of samples varies in one or more full C/A code chip, and the data alignment is different than the sub-interval boundaries).
Available Receivers
Today, software receivers can be found at university and commercial levels. The development not only includes programming solution but also the realization of dedicated RF front-ends. As these RF front-ends are able to capture more and more frequencies with increasing bit-rates and band-widths, the PC-based software receivers require a comparably complex interface to transfer the digitized IF samples into the computer’s memory.
Two classes of PC-based GNSS SR front-end solutions can be found. The first one uses commercially available ADCs that are either connected directly to the PC (for example, via the PCI bus) or that are working as stand-alone devices. The ADC directly digitizes the received IF signal, which is taken from a pure analog front-end. This solution is often found at the university and research institute level, where a high amount of flexibility is required; for example, at the Department of Geomatics Engineering of the University of Calgary, Cornell University, and the University FAF Munich’s Institute of Geodesy and Navigation.
The second solution is based on front-ends that integrate an ADC plus a USB 2.0 interface. Currently, an impressive number of commercial and R&D front-ends are available for the GNSS market. NordNav (acquired by CSR) and Accord were among the first to provide USB-based solutions. Another interesting development comes from the University of Colorado, which in an OpenGPS forum published all details on the RF and USB sections. More companies announced and continue to announce front-ends that are not only capable of capturing a single frequency, but several different bands. To be able to deal with this increasing bandwidth, the USB port is very well suited for SR development, and its maximum theoretical transfer rate of 480 MBit/s allows realizing GPS/Galileo multi-frequency high bandwidth front-ends.
Embedded Market. As mentioned in the introduction, the embedded market will gain increasing importance during the next few years. A growing number of receivers are developed for this market, supporting different embedded platforms (for example, Intel XScale, ARM-based, and DSP-based). Several companies offer commercial software receivers for the embedded market, among others NordNav and SiRF (acquired by CSR), ALK Technologies Inc., and CellGuide.
Commercial PC-Based Receivers. The first commercial GPS/Galileo receiver for a PC platform was presented in 2001 by NordNav. This SR can be compared to a normal GPS receiver, although the CPU load of this solution is still quite impressive. Several other solutions have been presented more recently. One of the first (car) navigation solutions was presented by ALK Technologies under the name CoPilot. The CPU load was drastically reduced, and this solution works on a standard commercial personal computer. The client does not really see a difference compared to a solution that is based on a hardware receiver.
Research Activities. Use in teaching and training is one of the most valuable and obvious application for software GNSS receivers. Receivers, for which the source code is available, allow the observation and inspection of almost every signal data by the researcher.
Several textbooks have been published related to software GNSS receivers. The pioneer in this area is James Bao-yen Tsui, who in 2000 wrote the first book on software receivers, Fundamentals of Global Positioning System Receivers: A Software Approach (Wiley-Interscience, updated in 2004). Kai Borre and co-authors published in 2006 a book that comes with a complete (post-processing) software receiver written in Matlab: A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach (Birkhäuser Boston, 1st edition).
The European Union is financing development of receivers for Galileo. One project was the Galileo Receiver Analysis and Design Application (GRANADA) simulation tool. Running under Matlab, GRANADA is realized as a modular and configurable tool with a dual role: test-bench for integration and evaluation of receiver technologies, and SR as asset for GNSS application developers.
Other companies provide toolboxes (in Matlab or C) that allow testing of new algorithms in a working environment and inspecting almost all data signals; for example, Data Fusion Corporation and NavSys.
Outlook
Software receivers have found their place in the field of algorithm prototyping and testing. They also play a key role for certain special applications. What remains unclear today is if they will enter and drastically change the embedded market, or succeed as generic high-end receivers.
A software GNSS receiver offers advantages including design flexibility, faster adaptability, faster time-to-market, higher portability, and easy optimization at any algorithm stage. However, a major drawback persists in the slow throughput and the high CPU load.
Many different companies and universities have projects running that seek to optimize and develop new algorithms and methods for a software implementation. The developments not only consider the software levels, but also extend in the direction of using additional hardware that is already available on a standard PC; for example, using the high performance graphic processing unit (GPU) for calculating the local carrier [Petovello, 2008].
On the opposite end of the spectrum from the mass market, the following factors seem to ensure that, sooner or later, high-end software receivers will be available:
High bandwidth signals (GPS and Galileo) can already be transferred into the PC in real time and processed.
The processing power is increasing, allowing real-time processing with a limited amount of multi-correlators. The introduction of new multi-core processors will be advantageous for software receivers.
Post-processing is one of the most important benefits of a software receiver, as it enables a re-analysis of the signal several times with all possible processing options. Increasing hard disk capacity facilitates storage of several long data sequences.
Some signal-processing algorithms such as frequency-domain tracking or maximum-likelihood tracking are much easier to implement in software than in hardware, as they require complex operations at the signal level.
History
During the 1990s, a U.S. Department of Defense (DoD) project named Speakeasy was undertaken with the objective of showing and proving the concept of a programmable waveform, multiband, multimode radio [Lackey, 1995]. The Speakeasy project demonstrated the approach that underlies most software receivers: the analog to digital converter (ADC) is placed as near as possible to the antenna front-end, and all baseband functions that receive digitized intermediate frequency (IF) data input are processed in a programmable microprocessor using software techniques rather than hardware elements, such as correlators. The programmable implementation of all baseband functions offers a great flexibility that allows rapid changes and modifications. This property is an advantage in the fast-changing environment of GNSS receivers as new radio frequency (RF) bands, modulation types, bandwidths, and spreading/dispreading and baseband algorithms are regularly introduced.
In 1990, researchers at the NASA/Caltech Jet Propulsion Laboratory introduced a signal acquisition technique for code division multiple access (CDMA) systems that was based on the Fast Fourier Transform (FFT) [van Nee, 1991]. Since then, this method has been widely adopted in GNSS SR because of its simplicity and efficiency of processing load.
In 1996, researchers at Ohio University provided a direct digitization technique — called the bandpass sampling technique — that allowed the placing of ADCs closer to the RF portions of GNSS SRs. Until this time, the implemented SRs in university laboratories post-processed the data due to the lack of processing power mentioned earlier.
Finally, in 2001, researchers at Stanford University implemented a real-time processing-capable SR for the GPS L1 C/A signal [Akos, 2001].
However, the GNSS SR boom really started with the development of real-time processing capability. This was first accomplished on a digital signal processor (DSP) and later on a commercial conventional personal computer (PC). Today, the DSPs are increasingly replaced by specialized processors for embedded applications.
Marcel Baracchi-Frei received a physics-electronics degree from the University of Neuchâtel, Switzerland, and is working as a project leader and Ph.D. candidate in the Electronics and Signal Processing Laboratory at the Swiss Federal Institute of Technology (EPFL).
GRÉGOIRE WAELCHLI received his degree of physics-electronics from the University of Neuchâtel and is now at EPFL for a Ph.D. thesis in the field of GNSS software receivers.
CYRIL BOTTERON received a Ph.D. with specialization in wireless communications from the University of Calgary, Canada, and now leads the EPFL GNSS and UWB research subgroups.
PIERRE-ANDRÉ FARINE is professor and head of the Electronics and Signal Processing Laboratory at EPFL, and associate professor at the University of Neuchâtel.
Using Microwaves and Laser Ranging for Precise Orbit Determination
By Erik Schönemann, Tim A. Springer, Michiel Otten, and Matthias Becker
Though Galileo’s GIOVE-A is a test satellite not necessarily ready for scientific use, orbit analyses with a reduced accuracy can help to identify weaknesses and suggest improvements. This month, the authors share work being carried out to precisely determine the orbit of GIOVE-A using SLR and microwave observations. This preliminary investigation will benefit the procedures to be implemented for the future Galileo constellation.
INNOVATION INSIGHTS by Richard Langley
WE USE THEM FOR LISTENING TO MUSIC, for routine surgeries, for making a point in a presentation, and even for hanging pictures straight. Of course, I’m talking about lasers. Invented in 1960, the laser (an acronym for light amplification by the stimulated emission of radiation) has become ubiquitous in modern society. Every CD and DVD player has one. Many printers use them. But lasers are also used in a wide range of industrial and scientific applications including determining the orbits of satellites through satellite laser ranging (SLR).
In the SLR technique, pulses of laser light from a ground reference station are directed at satellites equipped with an array of corner-cube retroreflectors, which direct the pulses back towards a collocated receiving telescope. By accurately measuring the two-way travel times of the pulses and knowing the location of the station and other operating parameters, the positions of the satellites can be determined. A network of SLR reference stations around the globe is used to monitor the orbits of satellites over time and their variations have been used by scientists to improve our knowledge of the Earth’s gravity field; to study the long term dynamics of the solid Earth, oceans, and atmosphere; and even to verify predictions of the General Theory of Relativity.
The first SLR measurements were obtained from the Beacon Explorer-B satellite, which was launched in October 1964. Since then, dozens of satellites equipped with corner-cube retroreflectors have been launched including a number of radio-navigation satellites. Every GLONASS satellite is equipped with retroreflectors and two GPS satellites have been equipped—SVN35/PRN05 and SVN36/PRN06. The COMPASS-M1 satellite in medium Earth orbit carries retroreflectors, as do both GIOVE-A and –B, the Galileo test satellites.
Precise orbit determination of radio-navigation satellites using SLR has the advantage of being unaffected by any onboard satellite electronics and associated signal biases. Radiometric observations of a satellite’s microwave signals, on the other hand, are influenced by the satellite’s clock, for example, and its effect must be estimated to obtain precise (and accurate) satellite orbits for navigation and positioning. Therefore, a comparison of SLR- and microwave-derived orbits can be very useful for studying the performance of the data measurement and orbit-determination processes of both techniques.
In this month’s column, we take a look at some work being carried out to precisely determine the orbit of the GIOVE-A test satellite using SLR and microwave observations. This preliminary investigation will benefit the procedures to be implemented for the future Galileo constellation.
“Innovation” is a regular column that features discussions about recent advances in GPS technology and its applications as well as the fundamentals of GPS positioning. The column is coordinated by Richard Langley of the Department of Geodesy and Geomatics Engineering at the University of New Brunswick, who welcomes your comments and topic i deas. To contact him, see the “Contributing Editors” section on page 6.
The navigation office of the European Space Operations Centre (ESOC) is engaged in various activities using observations of the Galileo test satellite, GIOVE-A (Galileo In-Orbit Validation Element-A), recorded at the Galileo Experimental Sensor Stations (GESS). The work includes the assessment of the quality and performance of GIOVE satellite observables and the testing and improvement of orbit-determination software. These activities support the long-term goal of advancing the scientific applications of the future Galileo constellation.
Since the launch of GIOVE-A on December 28, 2005, various tests have been carried out to analyze the quality of the new code (pseudorange) and carrier-phase observables derived from tracking the satellite’s microwave signals. All of these tests demonstrate the advantages of the new signal structure compared to that of legacy GPS signals. In general, the reduction of the noise by factor of 4-5 as well as a reduction of the code multipath by approximately a factor of 1.2 (GPS C1C versus GIOVE-A C1B/C1C) could be seen.
As the comparison of observations is done indirectly (GPS and GIOVE-A have different orbits) and the databases used for most analyses published up to now is sparse, a deeper analysis of the signal quality parameters seems appropriate, especially as data quality has a direct impact on the precision of orbit determination. Our analyses, presented in the first half of this article, are based on a broad base of data from most of the stations in the GESS network. Because of the difficulty in accessing the phase multipath directly, we first evaluated the signal strength and the code multipath, which gave the first hint of the multipath behavior. In order to compare GPS and GIOVE-A data directly, only data received from the same elevation angles and azimuths were used. Subsequently, we present an analysis of the phase residuals derived by precise point positioning.
The second part of this article focuses on the precise orbit determination or POD of the GIOVE-A spacecraft. The Navigation Package for Earth Observation Satellites (NAPEOS) software used at the ESOC Navigation Support Office allows microwave (radiometric) and satellite laser ranging (SLR) observations to be used either separately or together. The two methods are different due to different tracking networks and the different sensitivity of the observables to atmospheric effects and in their noise levels. We will present the orbit results focusing on internal orbit consistency checks and SLR validation of the microwave-based orbits.
Data Analysis
We first describe the procedures used for analyzing the microwave data followed by those used for the SLR data.
Microwave Analysis. For the GIOVE-A signal analysis and precise orbit determination we used the RINEX data from all of the GESS stations available from the GIOVE archiving facility (see TABLE 1). All stations are equipped with GPS/Galileo antennas, built by Space Engineering S.p.A. and Galileo Experimental Test Receivers (GETRs), built by Septentrio. The data, containing tracking data of all GPS satellites and the GIOVE-A satellite, is given in the RINEX 3.00 data format with a sampling interval of 1 second. To save on storage space for the long-term analyses, such as orbit determination, the RINEX data is decimated to 30-second samples and Hatanaka-compressed, using a test version of the Hatanaka software for the RINEX 3.00 format.
The signal analyses shown here were carried out using GNU Octave, an open-source program for performing numerical computations similar to Matlab, and different scripts developed by the Institut für Physikalische Geodäsie at the Technische Universität Darmstadt. These analyses cover a selection of the designated Galileo signals recorded by the GESS within the time span from December 16 to 27, 2006. Within this time period, the current GPS signals, as well as the GIOVE-A signals E1 and E5, shown in TABLE 2, were recorded. The table also shows the signal components as well as the RINEX observation-type identifiers, which we use in this article.
The stations used for the analyses show a quite similar level of performance in general. There are stations with different behaviors for single signals, as for example GIEN with a stronger code multipath behavior on C1B and C1A, but no station with a considerably different performance level could be identified. The averaging over the data from all sites reduces the station-dependent effects such as multipath and the atmosphere to a large extent, and gives a good indication of the mean signal performance.
The analyzed phase residuals were taken from the processing carried out for the second part of this article. Hence, they include observation data over an extended period of 149 days and were limited to the GIOVE-A C1C/L1C and C7Q/L7Q signals.
This extended data period is from December 12, 2006 (day of year 346), until May 26, 2007 (day of year 146). During this interval, there is a period where no GIOVE-A data was available due to maintenance of the spacecraft. This gap occurred from February 12 to 28, 2007. So in total we have analyzed 149 days of microwave data. Because there are some differences between the results before and after this gap in February, many of the statistics are given for the first and second part separately. The first part covers December 12, 2006, until February 11, 2007; the second part covers March 1, 2007, until May 26, 2007.
We performed the precise orbit determination using the NAPEOS software, a general-purpose software package for orbit determination, prediction, and control, supporting all phases of an Earth-observation mission in terms of mission preparation and operations.
For the GIOVE-A analysis, the three main NAPEOS programs we used are GnssObs, Bahn, and Multiarc. GnssObs reads, cleans, and decimates the RINEX data and converts the data into the NAPEOS internal tracking-data format. The NAPEOS tracking-data format contains the ionosphere-free linear combination, for both code and phase, of the RINEX observations. For GPS, the ionosphere-free linear combination is based on the combination of C1P and C2P code and L1P and L2P phase measurements. GIOVE-A offers several different observables allowing for many different ionosphere-free observations. For most of the work presented in this article, we have used the ionosphere-free linear combination of the C1C and C7Q and L1C and L7Q observations for code and phase respectively.
The next module, Bahn, performs the parameter estimation. In this step, we use the ionosphere-free code and phase observations at a sampling interval of 5 minutes, and we have applied an elevation angle cut-off of 5 degrees. The data is processed in batches of 24 hours, thus resulting in 1-day-arc solutions. The estimated parameters in these daily solutions are the GIOVE-A state vector (position and velocity), five dynamical orbit parameters from the extended Center for Orbit Determination in Europe (CODE) orbit model, a GIOVE-A clock offset for each epoch, all receiver clock offsets for each epoch, one GPS-GIOVE-A “intersystem bias” parameter per day for each station except for a selected reference station, and the carrier-phase ambiguities (integers not resolved). The station coordinates are estimated but tightly constrained (1 millimeter) to their a priori value. We obtained the a priori station coordinates by combining the full set of daily solutions.
Despite the fact that the 13 GESS stations provide very good global coverage, it is expected that 24-hour solutions will not give the most precise GIOVE-A orbit estimates. To generate longer arc solutions, we have used the Multiarc program. This is a tool that has recently been added to the NAPEOS software package. It allows for a rigorous combination of normal equations, also referred to as normal equation stacking, which are generated by Bahn. During the normal equation combination, the satellite orbit parameters may also be rigorously combined, thus effectively leading to multi-day orbital arcs. For the work presented in this article, we have used Multiarc to generate solutions with arc lengths of 1, 2, 3, 4, and 5 days. We also used Multiarc to compute accurate a priori station coordinates by stacking all available 1-day normal equations.
Satellite Laser Ranging
Besides the 13 GESS stations, GIOVE-A is also tracked by more than 17 different SLR stations around the world. For most periods of the mission, the tracking has been consistent enough to allow for GIOVE-A POD using only the SLR data. As the SLR data is completely independent of the microwave data, the resulting orbit solutions will be to a large extent independent as well and thus can be used to give an indication of the achieved precision of the different microwave solutions.
The orbit determination strategy used for the SLR solutions is very similar to the one used for the microwave orbits with the main difference being the increased arc-length of 7 days. The same satellite parameters are estimated as with the microwave solutions: the GIOVE-A state vector and five dynamical orbit parameters from the extended CODE orbit model. No further parameters need to be estimated and all corrections applied to the SLR data are according to the International Earth Rotation and Reference Systems Service 2003 standards and, for station coordinates, we used those from the rescaled International Terrestrial Reference Frame 2005 solution. As the noise level of the SLR data is very low, the measurements can also be directly used to give an indication of the precision of the radial position components of the different microwave solutions by computing the SLR residuals without using them in the estimation process itself.
Combined Microwave and SLR Analysis. In this step, the SLR data was added to the microwave data in the 24-hour solutions. For the data weighting, we used 100 millimeters for SLR and 1000 millimeters and 10 millimeters for GIOVE-A and GPS code and phase observables respectively. The only change in the analysis strategy in this case was that we now processed the SLR data in 24-hour solutions and not in 7-day batches. All the processing options remained as described in the two previous sections. The resulting 1-day solutions, or rather the associated normal equations, were used in Multiarc to generate combined solutions of different arc lengths.
Microwave Data Quality
We now take a detailed look at the quality of the microwave data in terms of signal-to-noise ratio (SNR), code-tracking noise and multipath, carrier-phase-tracking noise, and carrier-phase residuals.
Signal-to-Noise Ratio. The SNR (or equivalently carrier-to-noise-density ratio, C/N0) is strongly dependent on the satellite transmitter, the signal path through the atmosphere, and the receiver configuration (ground station, antenna, receiver, cable, etc.). Hence the SNR cannot be seen as an absolute value. The SNR is specific to the position, the equipment, and the time. Furthermore, the determination of the SNR values depends on the receiver and the firmware used. As a result, SNR values from different receivers cannot be readily compared. Nevertheless, using only one type of receiver, assuming similar effects on all the different signals at the same epoch, and taking averages over a long time span, we expect the relationships among the signals to be constant. Based on this assumption, we can use the SNR values given in the GESS RINEX files without adjustment.
To compare the GPS with the GIOVE-A SNR values, we ordered the corresponding SNR values of all stations on all days by satellite position into a grid with widths of 5 degrees in azimuth and 5 degrees in elevation angle. For the evaluation, we took the grid cells occupied by both GPS and GIOVE-A values and computed the median over all the cells of equal elevation angle. The median per elevation-angle bin for each signal is shown in FIGURE 1.
FIGURE 1. Signal-to-noise ratio, GPS versus GIOVE-A
As can be seen from the figure, the signal strength of the GIOVE-A C8Q observable ranks best, followed by the GPS C1C, GIOVE-A C7Q, C5I/C5Q, C1A, and C1B/C1C. The weakest signal is found for the GPS C1P/C2P observable, with a maximum signal strength of 40 (receiver-dependent unit, approximately dB-Hz) at the zenith. Comparing the GPS open signals versus GIOVE-A, GPS C1C is considerably stronger than the GIOVE C1B/C1C. According to the GPS and Galileo interface control documents, GIOVE-A C1B/C1A should show up with a stronger signal strength than GPS C1C. The power levels guaranteed on the Earth’s surface are -160 dBW for GPS and -158 dBW for the future Galileo satellite signals except for the BOC(10,5) and BOC(n,m) modeled signals, for which a power level of even -155dBW is guaranteed. But looking at the SNR values shown in Figure 1, we see that the GIOVE-A C1B/C1C is worse by approximately 4 dB than the GPS C1C. But keeping in mind that GIOVE-A is an experimental satellite, an increase of the signal power for the future operational Galileo satellites should improve the signal performance above that shown in this article.
Code-Tracking Noise. For signals containing data and pilot components, as in the case of those from GIOVE-A, the code-tracking noise can easily be computed as the difference between the data and the pilot signal. The advantage of this computation scheme is that both signals are influenced by identical error sources (atmospheric errors, multipath errors, receiver errors, etc.). Based on the assumption of equal uncertainties in the two components, we divided the resulting noise values by the square root of two to specify the noise level of each part according to the laws of error propagation. TABLE 3 shows the code-tracking noise for the two analyzed GIOVE-A codes sorted by elevation angle. The median code-tracking noise is 0.62 meters for C1B/C1C and 0.35 meters for C5I/C5Q, for observations below an elevation angle of 5 degrees. For the C1B and C1C code measurements, the noise median stays below 0.2 meters for an elevation angle above 25 degrees, whereas the median for the C5I and C5Q code measurements for elevation angles above 35 degrees even comes down below 0.1 meters. The results discussed above are consistent with the code-tracking noise values published previously.
Code Multipath. We computed the relative code multipath effects as code minus phase differences assuming the amplitude of phase multipath to be insignificant compared to the amplitude of the code multipath. Ionospheric effects were taken into account by using the phase measurements on two frequencies in the usual way:
In this equation, CMPx is the estimate of the multipath error on the code, Px and Lx are the code and phase measurements of the same frequency, while Ly is the phase measurement used to correct the frequency-dependent ionospheric effect. The constant, , describes the relationship of the ionospheric behavior for the two frequencies.
In order to compare the code multipath level of GPS versus GIOVE-A, we sorted the multipath values using a grid covering the sky with widths of 5 degrees for both elevation angle and azimuth as before. FIGURE 2 shows the median standard deviation of the code multipath values, derived in each grid cell per day and station, versus the elevation angle. No significant difference between GPS C1C and GIOVE-A C1B and C1C, the open code signals on G1/E1, could be found. The code multipath behavior of the GPS precise codes are comparable with the GIOVE-A C5I, C5Q, and C7Q, whereas the C8Q shows the least code multipath effects closely followed by the GIOVE-A C1A, the public regulated service signal.
FIGURE 2. Code multipath, GPS versus GIOVE-A
Carrier-Phase-Tracking Noise Analyses. In the same manner as that carried out with the code, we computed the GIOVE-A carrier-phase-tracking noise as the difference of the two components (pilot minus data). To accommodate the effect of error propagation, the resulting errors were divided by the square root of two. The resulting phase-tracking noise values were sorted by elevation angle and can be found in TABLE 4.
In conformity with the theory that the phase-tracking noise is independent of the modulation scheme, both signals (L1B/L1C and L5I/L5Q) show the same results in units of cycles. Looking at the results in units of distance, GIOVE-A L1B/L1C shows up with a mean phase noise of 0.7 millimeters and L5I/L5Q with 0.9 millimeters. These values confirm those of previous studies.
Carrier-Phase Residuals. Phase residuals contain the phase tracking noise, multipath, as well as all unmodeled remaining errors such as antenna calibration inaccuracy and tropospheric effects. The magnitude of the residuals can be seen as an indicator for the observation and model accuracy as well as for measurement quality.
The following analyses are based on the ionosphere-free linear combination (GPS L1C/L2P, GIOVE-A L1C/L7Q), computed with NAPEOS. The analyses include data of the 13 GESS over a period of 149 days.
To compare the GPS and GIOVE-A residuals, we sorted them into a grid with a width of one degree in both satellite azimuth and elevation angle. Only data in overlapping grid locations were compared to make sure the data was affected in a similar way by multipath or other disturbances.
To properly interpret the results, we should mention that for GIOVE-A, 0.06 percent of the ambiguities (2501) were not fixed correctly whereas for GPS all ambiguities were fixed correctly. Looking at the GIOVE-A observations that were correctly fixed, we find a significantly larger number of rejected observations. The number of rejected observations is less by one third for GPS (6 percent) as for the GIOVE-A (9 percent) data.
Due to the small number of GIOVE-A observations for elevation angles above 86 degrees, the outlier-cleaned mean as well as the standard deviation at this elevation-angle range are not meaningful. For all elevation angles, GIOVE-A residuals show a lower standard deviation than GPS, indicating a superior performance of GIOVE-A signals.
Phase and Code Validation in Processing. Looking at the quality of the code and phase measurements on the different signals, it is conspicuous that GIOVE-A C1A/L1A and C8Q/L8Q rank best, whereas for the current processing of GIOVE-A data, usually the C1C and C7Q signals are used. This leads to the question of which is the best signal combination for GIOVE-A. Hence, we processed 10 days of GIOVE-A data, using different signal combinations. Presently the processing of the C8Q/L8Q signals is not yet implemented in NAPEOS. However, we were able to process the GIOVE-A C1A/L1A – C7Q/L7Q combination. The root-mean-square (RMS) of the code results were reduced by a factor of approximately 1.4 using L1A/C1A compared to L1C/C1C, whereas the RMS of the phase observations showed only a minor improvement. Furthermore, there is a higher number of rejected observations with L1A/C1A. Further analyses have to be carried out to evaluate the potential benefits of the different signal combinations.
Orbit Quality
In this section, we assess the quality of our precise orbit determination solutions. We have three sets of different orbit solutions. Set 1 is made up of the 7-day solutions based solely on SLR observations. Set 2 consists of the solutions based on the microwave observations using 1- to 5-day arcs. Set 3 consists of the solutions based on a joint analysis of the microwave and SLR observations also using 1- to 5-day arcs.
First, we assess the orbit quality by looking at the internal consistency of the solutions. For the two sets using microwave observations, the internal orbit consistency is done using an orbit fit. This will not tell us much about the absolute quality of the solutions but it will indicate the optimal arc length and whether adding the SLR observations to the microwave data improves the orbit estimates.
Secondly, we validate the orbits by determining the SLR residuals. Of course, the solutions that used SLR observations should perform better than the microwave-only solutions. However, the validation of the microwave orbits against the SLR observations will give us a good impression of the absolute accuracy of our orbits.
As a third test, we compare the best orbit (best arc length) of each of the three sets (set 1 only has one arc length) against each other. This should give us another indication of the quality of the orbits.
Internal Orbit Consistency. To determine the internal orbit consistency of the different solutions we make an orbit fit. For this orbit fit test, we used the middle 24 hours of two consecutive solutions and fit one 48-hour arc through these two parts. The satellite orbit was modeled by estimating the satellite state vector and all nine parameters of the extended CODE orbit model. The RMS of this fit gives us an indication of the internal consistency of the orbit estimates. For longer arcs, the RMS of fit should go down because the solutions are not fully independent of each other. So a lower RMS for the longer arc solutions is expected. On the other hand, this means that if the RMS does not go down with increasing arc length that we have reached the limit of our modeling capabilities. Furthermore, comparing the internal orbit consistencies of equal length solutions will tell us which solution has a better internal consistency. The results of this internal orbit consistency check are given in TABLE 5. The table gives the mean of the 2-day RMS over all processed days. The mean is given separately for the first and second part of the observation interval (see above) and also for the total observation interval.
Table 5 shows several interesting results. First of all, it shows that the results of part 2 of the observation interval are significantly better than the results from part 1. The reason for this is unclear since the statistics from the 1-day solutions, such as the residual RMS and number of observations, did not change significantly after the observation gap. The improvement, however, is very significant. The second observation is that the results including the SLR data are significantly better compared to those using only the microwave data. This is true for all arc lengths! As expected, we see a significant improvement of the internal consistency when going from 1-day arcs to 3-day arcs. The 4-day arcs show only a slight improvement compared to the 3-day arcs. The 5-day arcs do not show a significant improvement. This indicates that with the current observations and modeling techniques, the optimal arc length for precise orbit determination seems to be around 3 to 4 days.
SLR Validation. In this section, we look at the SLR residuals obtained from the different orbit solutions. We generated a clean SLR dataset by using the SLR-only orbit to remove any outliers in the SLR observations. The total number of valid SLR normal points for the entire period is 3520 observations from 17 different SLR stations. (A normal point is an average of a number of individual laser returns.) The number of observations for the first part of the observation period is 796 points from 12 stations and for the second part, there were 2724 normal points from 17 stations. For two of the three solutions, the SLR data has been used in the orbit determination process so the residuals will give a too-optimistic indication of the orbit quality.
As can be seen from TABLE 6, the 3-day solution based on the microwave-only data has the lowest SLR residuals and indicates a radial precision of around 100 millimeters. A similar behavior can be seen in the microwave plus SLR solution with the exception of the 1-day solution (and to a smaller extent also the 2-day solution) where the orbit solution is mainly driven by the SLR data, but the quality as can be seen from the internal consistency of the orbit is poor. Interestingly, there is a large improvement in SLR residuals for the microwave plus SLR solution, although the number of SLR data points is only 2 percent of the total tracking data in the combined solution. The values for the SLR-only solution are included in the table to give an indication of the lowest possible SLR residuals one could expect by combining the microwave and SLR data.
Orbit Comparison. To get an indication of the overall orbit quality, the best solutions were compared against each other for the second period of observation. TABLE 7 gives the RMS differences between the SLR only (SLR), 3-day microwave only (micro), and the 3-day microwave and SLR solution (micro+SLR) for the radial, along-track, and cross-track position components as well as the norm (3D).
As expected, the largest difference is between the SLR-only and microwave-only solutions giving a total (norm) orbit difference of 652 millimeters. As a major part of the SLR tracking from GIOVE-A comes from European stations, the quality of the SLR solutions is directly correlated with the ability of the European stations to track GIOVE-A. Bad weather over Europe can lead to data gaps for more than 24 hours, affecting the orbit quality. It is interesting to see the large impact the SLR data has on the combined solution. As mentioned before, the SLR data is only around 2 percent of the total tracking data but has a significant impact on the orbit solution as can be seen from the difference between the microwave-only and microwave-plus-SLR solution.
Based on the analysis presented above, we conclude that the 3-day solution using both microwave and SLR observations has provided the best orbit estimates.
Conclusion
The analyses of the observation data quality (signal quality) confirmed the good results from prior analyses for code multipath behavior and code noise. GPS C1C and the GIOVE-A C1B/C1C show a comparable multipath behavior, whereas the GPS precise codes C1P/C2P are comparable to the GIOVE-A C5I, C5Q, and C7Q. The least code multipath behavior could be found for GIOVE-A C8Q observable, closely followed by the GIOVE-A C1A. Based on this, the combination C1A/L1A – C8Q/L8Q should show the best noise behavior within the data processing scheme.
The results given in this article demonstrate that the 13-station GESS network allows us to determine the orbit of the GIOVE-A satellite quite accurately (~200 millimeters) using only microwave observations. The SLR validation of the microwave orbits gives an RMS of 100 millimeters (one-way range RMS). This result gives an absolute value for the orbital error. Of course, the SLR observations mainly tell us something about the radial orbit errors; the along- and cross-track errors could be much higher. To obtain accurate GIOVE-A orbit estimates, we need to keep the orbits and clocks of the GPS satellites, tracked simultaneously with the GIOVE-A satellite, fixed using the International GNSS Service (IGS) final orbit and clock products. Furthermore, an arc length of 3 days should be used. The microwave-based orbit estimates may be improved by adding the available SLR observations into the orbit-estimation process. Although there are relatively few SLR observations, they do have a significant positive effect on the orbit estimates, improving the internal consistency from 52 to 41 millimeters. Also, the validation of the orbits using the SLR observations shows a significant improvement. However, this is not an independent validation because the same SLR observations were used in the orbit determination.
The results presented in this article, even though based on observations from the GIOVE-A test satellite, can be considered as a first attempt towards establishing an optimal data processing approach for the future Galileo satellite constellation.
Acknowledgments
This article is based on the paper “GIOVE-A Precise Orbit Determination from Microwave and Satellite Laser Ranging Data – First Perspectives for the Galileo Constellation and Its Scientific Use” presented at the 1st Colloquium on the Scientific and Fundamental Aspects of the Galileo Program, held in Toulouse, France, October 1-7, 2007.
ERIK SCHÖNEMANN studied geodesy at the Technische Universität Darmstadt (TUD), Germany, writing his diploma thesis at the University of New South Wales, Sydney, Australia. Since receiving his diploma from TUD in April 2005, he has been working for the Institute of Physical Geodesy at TUD on GNSS station calibration and validation and analyses of GIOVE-A and GIOVE-B data.
TIM SPRINGER received his Ph.D. in physics from the Astronomical Institute of the University of Berne (AIUB) in 1999. He has been a key person in the development of the Center for Orbit Determination in Europe, one of the IGS analysis centers, located at AIUB. Since 2004, he has been working for the Navigation Support Office (NSO) at the European Space Operations Centre (ESOC) of the European Space Agency (ESA) in Darmstadt. In this group, he has led the development of the new ESOC GNSS software, which is used for most GNSS activities at NSO including GIOVE-A and -B analyses.
MICHIEL OTTEN obtained a degree in aerospace engineering from Delft University of Technology in 2001. He has been working for ESOC’s NSO since 2002. His main role within NSO is the precise orbit determination of low Earth-orbiting satellites equipped for SLR, DORIS, and GPS tracking. He is also responsible for ESA’s International DORIS Service Analysis Centre activities.
MATTHIAS BECKER is a full professor of geodesy and director of the Institute of Physical Geodesy, TUD. He received his diploma and Ph.D. in geodesy from TUD in 1979 and 1984, respectively. He is responsible for research and teaching in the fields of physical geodesy and satellite geodesy.
FURTHER READING
• GIOVE-A
“Meet GIOVE-A: Galileo’s First Test Satellite” by E. Rooney, M. Unwin, A. Bradford, P. Davies, G. Gatti, V. Alpe, G. Mandorlo, and M. Malik in GPS World, Vol. 18, No. 5, May 2007, pp. 36–42.
“Galileo Signal Experimentation” by M. Hollreiser, M. Crisci, J.-M. Sleewaegen, J. Giraud, A. Simsky, D. Mertens, T. Burger, and M. Falcone in GPS World, Vol. 18, No. 5, May 2007, pp. 44-50.
• GIOVE Tracking Network
“GIOVE Mission Sensor Station Receiver Performance Characterization: Preliminary Results” by M. Crisci, M. Hollreiser, M. Falcone, M. Spelat, J. Giraud, and S. La Barbera in Proceedings of Navitec 2006, the 3rd ESA Workshop on Satellite Navigation User Equipment Technologies, Noordwijk, The Netherlands, December 11-13, 2006.
• GIOVE Tracking Performance
“Performance Assessment of Galileo Ranging Signals Transmitted by GSTB-V2 Satellites” by A. Simsky, J.-M. Sleewaegen, M. Hollreiser, and M. Crisci in Proceedings of ION GNSS 2006, the 19th International Technical Meeting of the Satellite Division of The Institute of Navigation, Fort Worth, Texas, September 26-29, 2006, pp. 1547–1559.
“Code and Carrier Phase Tracking Performance of a Future Galileo RTK Receiver” by T. Pany, M. Irsigler, B. Eissfeller, and J. Winkel in Proceedings of ENC-GNSS 2002, the European Navigation Conference, Copenhagen, Denmark, May 27-30, 2002.
• Multipath Mitigation in Modernized GNSS
“Comparison of Multipath Mitigation Techniques with Consideration of Future Signal Structures” by M. Irsigler and B. Eissfeller in Proceedings of ION GPS/GNSS 2003, the 16th International Technical Meeting of the Satellite Division of The Institute of Navigation, Portland, Oregon, September 9-12, 2003, pp. 2584–2592.
• GIOVE Orbit Determination
“Estimation and Prediction of the GIOVE Clocks” by I. Hidalgo, R. Píriz, A. Mozo, G. Tobias, P. Tavella, I. Sesia, G. Cerretto, P. Waller, F. González, and J. Hahn in Proceedings of the 40th Annual Precise Time and Time Interval (PTTI) Meeting, Reston, Virginia, December 1-4, 2008.
• Satellite Laser Ranging
“GIOVE’s Track: Satellite Laser-Ranging Campaigns” by M. Falcone, D. Navarro-Reyes, J. Hahn, M. Otten, R. Piriz, and M. Pearlman in GPS World, Vol. 17, No. 11, November 2006, pp. 34–37.
“The International Laser Ranging Service: Current Status and Future Developments” by W. Gurtner, R. Noomen, and M.R. Pearlman in Advances in Space Research, Vol. 36, No. 3, 2005, pp. 327–332 (doi:10.1016/j.asr.2004.12.012).
“Laser Ranging to GPS Satellites with Centimeter Accuracy” by J.J. Degnan and E.C. Pavlis in GPS World, Vol. 5, No. 9, September 1994, pp. 62–7.
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