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2021/08/13
Synthetic-Aperture GNSS Signal Processing
By Thomas Pany, Nico Falk, Bernhard Riedl, Carsten Stöber, Jón O. Winkel, and Franz-Josef Schimpl
INNOVATION INSIGHTS by Richard Langley
A SYNTHETIC APERTURE? WHAT’S THAT? Well, an aperture in optics is just a hole or opening through which light travels. Those of us into photography know that the amount of light reaching the camera’s imaging sensor is controlled by the shutter speed and the size of the lens opening or aperture (called the f-stop). And a correct combination of the aperture setting and shutter speed results in a correct exposure. For an optical telescope, its aperture is the diameter of its main, light-gathering lens or mirror. A larger aperture gives a sharper and brighter view or image.
In the radio part of the electromagnetic spectrum, the term aperture refers to the effective collecting (or transmitting) area of an antenna. The gain of the antenna is proportional to its aperture and its beamwidth or resolution is inversely proportional to it.
Astronomers, whether using optical or radio telescopes, often seek higher and higher resolutions to see more detail in the objects they are investigating. Conventionally, that means larger and larger telescopes. However, there are limits to how large a single telescope can be constructed. But by combining the light or radio signals from two or more individual telescopes, one can synthesize a telescope with a diameter equal to the baseline(s) connecting those telescopes. The approach is known as interferometry. It was first tried in the optical domain by the American physicist Albert Michelson who used the technique to measure the diameter of the star Betelgeuse. Radio astronomers developed cable- and microwave-connected interferometers and subsequently they invented the technique of very long baseline interferometry (VLBI) where atomic-clock-stabilized radio signals are recorded on magnetic tape and played back through specially designed correlators to form an image. (VLBI has also been used by geodesists to precisely determine the baselines between pairs of radio telescopes even if they are on separate continents.)
A similar approach is used in synthetic-aperture radar (SAR). Mounted on an aircraft or satellite, the SAR beam-forming antenna emits pulses of radio waves that are reflected from a target and then coherently combined. The different positions of the SAR, as it moves, synthesize an elongated aperture resulting in finer spatial resolution than would be obtained by a conventional antenna.
But what has all of this got to do with GNSS? In this month’s column, we take a look at a novel GNSS signal-processing technique, which uses the principles of SAR to improve code and carrier-phase observations in degraded environments such as under forest canopy. The technique can simultaneously reject multipath signals while maximizing the direct line-of-sight signal power from a satellite. Along with a specially programmed software receiver, it uses either a single conventional antenna mounted, say, on a pedestrian’s backpack for GIS applications or a special rotating antenna for high-accuracy surveying. Want to learn more? Read on.
“Innovation” is a regular feature that discusses advances in GPS technology andits applications as well as the fundamentals of GPS positioning. The column is coordinated by Richard Langley of the Department of Geodesy and Geomatics Engineering, University of New Brunswick. He welcomes comments and topic ideas.
Over the past few years, we have been developing new GNSS receivers and antennas based on an innovative signal-processing scheme to significantly improve GNSS tracking reliability and accuracy under degraded signal conditions. It is based on the principles of synthetic-aperture radar. Like in a multi-antenna phased-array receiver, GNSS signals from different spatial locations are combined coherently forming an optimized synthetic antenna-gain pattern. Thereby, multipath signals can be rejected and the line-of-sight received signal power is maximized. This is especially beneficial in forests and in other degraded environments.
The method is implemented in a real-time PC-based software receiver and works with GPS, GLONASS, and Galileo signals. Multiple frequencies are generally supported.
The idea of synthetic-aperture processing is realized as a coherent summation of correlation values of each satellite over the so-called beam-forming interval. Each correlation value is multiplied with a phase factor. For example, the phase factor can be chosen to compensate for the relative antenna motion over the beam-forming interval and the resulting sum of the scaled correlation values represents a coherent correlation value maximizing the line-of-sight signal power. Simultaneously, signals arriving from other directions are partly eliminated.
Two main difficulties arise in the synthetic-aperture processing. First, the clock jitter during the beam-forming interval must be precisely known. It can either be estimated based on data from all signals, or a stable oscillator can be used. In one of our setups, a modern oven-controlled crystal oscillator with an Allan variance of 0.5 × 10-13 at an averaging period of 1 second is used. Second, the precise relative motion of the antenna during the beam-forming interval must be known. Again it can be estimated if enough sufficiently clean signals are tracked. The antenna trajectory is estimated directly from the correlator values as shown later in this article. In more severely degraded environments, the antenna may be moved along a known trajectory. We are developing a rotating antenna displacement unit. (see FIGURE 1). The rotational unit targets forestry and indoor surveying applications. The relative motion of the antenna is measured with sub-millimeter accuracy.
FIGURE 1. Artist’s impression of the synthetic-aperture GNSS system for surveying in a forest.
After beam-forming, the code pseudoranges and the carrier phases are extracted and used in a conventional way. That is, they are written into Receiver Independent Exchange (RINEX) format files and standard geodetic software can be used to evaluate them. In the case where the artificial movement antenna is used, the GNSS signal processing removes the known part of the movement from the observations, and the observations are then like those from a static antenna. As a result, common static positioning algorithms, including carrier-phase ambiguity fixing, can be applied. The presented method therefore prepares the path for GNSS surveying applications in new areas. An important point is the mechanical realization of the antenna movement. This has to be done in a cost-efficient and reliable way. Lubrication-free actuators are used together with magnetic displacement sensors. The sensors are synchronized to the software receiver front end with better than 1 millisecond accuracy. The rotating antenna uses slip rings to connect the antenna elements. The rotating antenna can also be used to map the received signal power as a function of elevation and azimuth angles. This is beneficial for researchers. For example, it could be used to estimate the direction of arrival of a spoofing signal or to determine which object causes multipath in an indoor environment. For the latter purpose, the rotating antenna can be equipped with left-hand and right-hand circularly polarized antennas on both ends of the rotating bar. The rotating antenna is mounted on a geodetic tripod. See Further Reading for reports of initial studies of the rotating antenna.
Tracking Modes
The synthetic-aperture tracking scheme can be extended to different user-motion schemes or sensor-aiding schemes allowing a wide range of applications. This is reflected in the algorithm implementation within the modular structure of the software receiver. The base module “µ-trajectory & Clock Estimator” in Figure 2 prepares the synthetic-aperture tracking scheme. Different implementations derive from this base class. Each derived module is used for a different user motion scheme and makes use of a different sensor.
FIGURE 2. Different µ-trajectory motion estimators used by the synthetic-aperture processing.
Basically, the modules differ in the way they estimate the relative antenna motion over the beam-forming interval. This relative motion is called the µ-trajectory. Usually the µ-trajectory covers time spans from a few hundreds of milliseconds to a few seconds.
The µ-trajectories have the following characteristics:
The pedestrian motion estimator does not rely on any sensor measurements and fits a second-order polynomial into the user µ-trajectory of a walking pedestrian. A second-order polynomial is good for representing the motion for up to a quarter of a second.
The sensor input to the rotating antenna estimator is the relative angular displacement of the rotating antenna. The estimator estimates the absolute direction, which is stable in time. Thus the number of µ-trajectory parameters equals one.
The vertical antenna motion estimator retrieves the vertical position of the antenna and does not estimate any µ-trajectory parameters. Only clock parameters are estimated.
Finally, the inertial navigation estimator uses accelerometer and gyro measurements and estimates the 3D user motion. The µ-trajectory parameters consist of accelerometer biases, the gyro biases, attitude errors, and velocity errors. The estimation process is much more complex and exploits the timely correlation of the parameters.
Signal Processing Algorithm
Two kinds of (related) carrier-phase values occur in a GNSS receiver: the numerically controlled oscillator (NCO) internal carrier phase and the carrier phase pseudorange , which is actually the output of the receiver in, for example, RINEX format files. Both are a function of time t and when expressed in radians are related via Equation (1):
(1)
Here, fo denotes the receiver internal nominal intermediate frequency (IF) at which all signal processing takes place. The output carrier-phase pseudorange is an estimate of the true carrier-phase pseudorange , which, in turn, relates to the geometric distance to the satellite by the following standard model:
(2)
This model applies to each signal propagation path separately; that is, a separate model can be set up for the line-of-sight signal and for each multipath signal. In Equation (2), λ denotes the nominal carrier wavelength in meters, ρ(t) is the geometric distance in meters between transmitting and receiving antennas, fRF is the nominal carrier frequency in hertz, dtsat(t) and dtrec(t) are the satellite and receiver clock errors in seconds, N is the carrier-phase ambiguity, and T(t) contains atmospheric delays as well as any hardware delays in meters. Here, no measurement errors are included, because we are considering the relationship between true values.
Defining now a reference epoch t0, we will describe a procedure to obtain an improved carrier-phase estimate for this epoch using data from an interval [t0 – TBF, t0]. The beam-forming interval TBF can be chosen to be, for example, 0.2–2 seconds but should be significantly longer than the employed predetection integration time (the primary one, without beam forming).
Correlator Modeling. In this sub-section, the relationships between phase, correlator values, and geometric distances will be established. These relationships apply for each propagation path individually. In the next section these relationships will be applied to the total received signal, which is the sum of all propagation paths plus thermal noise. To model the correlator output we assume that any effect of code or Doppler-frequency-shift misalignment on carrier-phase tracking can be neglected. This is reasonable if the antenna motion can be reasonably well predicted and this prediction is fed into the tracking loops as aiding information. Then the prompt correlator output is given as
. (3)
Again, any noise contribution is not considered for the moment. Here a(t) denotes the signal amplitude and d(t) a possibly present navigation data bit. The carrier phase difference Δφ is given as
(4)
where φ(t) is the true carrier phase and φNCO(t) is the NCO carrier phase used for correlation.
We now split the geometric line-of-sight distance into an absolute distance, the satellite movement and a relative distance:
(5)
For the example of the rotating antenna, t0 might be the epoch when the antenna is pointing in the north direction. The term ρ0(t0) is the conventional satellite-to-reference-point distance (for example, to the rotation center) and ρsat(t0,t) accounts for the satellite movement during the beam-forming interval.
The term Δρµ(t) is the rotational movement and may depend on the parameter µ. The parameter µ represents, for the rotating antenna, the absolute heading but may represent more complex motion parameters. The absolute term ρ0(t0) is constant but unknown in the beam-forming interval. We assume that approximate coordinates are available and thus Δρµ(t) can be computed for a given set of µ (that is, the line-of-sight projection of the relative motion is assumed to be well predicted even with only approximate absolute coordinates). The same applies also to ρsat(t0,t).
Let’s assume that the NCOs are controlled in a way that the satellite movement is captured as well as the satellite clock drift and the atmospheric delays:
. (6)
Then
(7)
and
.(8)
Thus the correlator output depends on the absolute distance of the reference point to the satellite at t0, the relative motion of the antenna, the receiver clock error, the received amplitude and the broadcast navigation data bits. Satellite movement and satellite clock drift are absent.
Let us now denote m as the index for the different satellites under consideration. The index k denotes correlation values obtained during the beam-forming interval at the epoch tk. Then:
.(9)
If multiple signal reflections are received and if they are denoted by the indices m1, m2, … , then the correlator output is the sum of those:
.(10)
For the following, m or m1 denotes the line-of-sight signal and mn with n > 1 denoting multipath signals.
Estimation Principle. It seems natural to choose receiver clock parameters dtrec and trajectory parameters µ in a way that they optimally represent the receiver correlation values. This approach mimics the maximum likelihood principle. The estimated parameters are:
.(11)
Data bits are also estimated in Equation (11). Once this minimization has been carried out, the parameters µ and dtrec are known as well as the data bits. The real-time implementation of Equation (11) is tricky. It is the optimization of a multi-dimensional function. Our implementation consists of several analytical simplifications as well as a highly efficient implementation in C code. The pedestrian estimator has been ported to a Compute-Unified-Device-Architecture-capable graphics processing unit exploiting its high parallelism.
Equation (11) realizes a carrier-phase-based vector tracking approach and the whole µ-trajectory (not only positions or velocity values) is estimated at once from the correlation values. This optimally combines the signals from all satellites and frequencies. The method focuses on the line-of-sight signals as only line-of-sight signals coherently add up for the true set of µ-trajectory and clock parameters. On the other hand, multipath signals from different satellites are uncorrelated and don’t show a coherent maximum.
Purified Correlator Values. The line-of-sight relative distance change Δρµm(t) due to the antenna motion is basically the projection of the µ-trajectory onto the line-of-sight. Multipath signals may arrive from different directions, and is the antenna motion projected onto the respective direction of arrival.
Let the vector denote the phase signature of the nth multipath signal of satellite m based on the assumed µ-trajectory parameters µ:
.(12)
Projecting the correlator values that have been corrected by data bits and receiver clock error onto the line-of-sight direction yields:
. (13)
The correlator values Q are called purified values as they are mostly free of multipath, provided a suitable antenna movement has been chosen. This is true if we assume a sufficient orthogonality of the line-of-sight signal to the multipath signals, and we can write:
.(14)
where K is the number of primary correlation values within the beam-forming interval. The projection onto the line-of-sight phase signature is then
.(15)
Thus the purified correlator values represent the unknown line-of-sight distance from the reference point to the satellite. Those values are used to compute the carrier pseudorange. The procedure can similarly also be applied for early and late correlators. The purified and projected correlation values represent the correlation function of the line-of-sight signal and are used to compute the code pseudorange.
Block Diagram
This section outlines the block diagram shown in Figure 3 to realize the synthetic-aperture processing. The signal processing is based on the code/Doppler vector-tracking mode of the software receiver.
FIGURE 3. Synthetic-aperture signal processing.
The scheme has not only to include the algorithms of the previous section but it has also to remove the known part of the motion (for the rotating antenna, say) from the output observations. In that case, the output RINEX observation files should refer to a certain static reference point. This is achieved by a two-step process.
First, the known and predictable part of the motion is added to the NCO values. By doing that, the correlation process follows the antenna motion to a good approximation, and the antenna motion does not stress the tracking loop dynamics of the receiver. Furthermore, discriminator values are small and in the linear region of the discriminator. Second, the difference between the current antenna position and the reference point is projected onto the line-of-sight and is removed from the output pseudoranges and Doppler values. For further details on the processing steps of the block diagram, see the conference paper on which this article is based, listed in Further Reading.
Pedestrian Estimator
We tested the synthetic-aperture processing for pedestrians on a dedicated test trial and report the positing results in this section. These results are not final and are expected to improve as more GNSSs are included and general parameter tuning is performed.
Test Area. To test the pedestrian estimator, we collected GPS L1 C/A-code and GLONASS G1 signals while walking through a dense coniferous forest. The trees were up to 30–40 meters high and are being harvested by a strong local lumber industry. The test was carried out in May 2012. We staked out a test course inside the forest and used terrestrial surveying techniques to get precise (centimeter accuracy) coordinates of the reference points. Figure 4 shows a triangular part of the test course.
FIGURE 4. Triangular test course in a forest.
Measurement data was collected with a geodetic-quality GNSS antenna fixed to a backpack. This is a well-known style of surveying. We used a GNSS signal splitter and a commercial application-specific-integrated-circuit- (ASIC-) based high-sensitivity GNSS receiver to track the signals and to have some kind of benchmark. The algorithms of this ASIC-based receiver are not publicly known, but the performance is similar to other ASIC-based GNSS receivers inside forests.
We came from the west, walked the triangular path five times, left to the north, came back from the north, walked the triangular path again five times clockwise, and left to the west. We note that the ASIC-based receiver shows a 3–5 meter-level accuracy with some outliers of more than 10 meters. We further note that the use of the geodetic antenna was critical to achieve this rather high accuracy inside the forest.
µ-trajectory Estimation. As mentioned before, the pedestrian estimator uses a second-order polynomial to model the user motion over an interval of 0.2 seconds. If we stack the estimated µ-trajectories over multiple intervals, we get the relative motion of the user. An example of the estimated user motion outside (but near) the forest is shown in Figure 5.
FIGURE 5. Estimated relative user trajectory over 5 seconds outside the forest; user walking horizontally.
The figure clearly shows that the walking pattern is quite well estimated. An up/down movement of ~10 cm linked to the walking pattern is visible. Inside the forest, the walking pattern is visible but with less accuracy.
Synthetic-Aperture Antenna Pattern. It is possible to estimate the synthetic antenna gain pattern for a given antenna movement (see “Synthetic Phased Array Antenna for Carrier/Code Multipath Mitigation” in Further Reading). The gain pattern is the sensitivity of the receiver/antenna system to signals coming from a certain direction. It depends on the known direction of the line-of-sight signal and is computed for each satellite individually. It adds to the normal pattern of the used antenna element.
We assume that the system simply maximizes the line-of-sight signal power for an assumed satellite elevation of 45° and an azimuth of 135°. We model the pedestrian movement as horizontal with a constant speed of 1 meter per second, and an up/down movement of ± 7.5 centimeters with a period of 0.7 seconds. Employing a beam-forming interval of 2 seconds yields the synthetic antenna gain pattern of Figure 6.The pattern is symmetric to the walking direction. It shows that ground multipath is suppressed.
FIGURE 6. Synthetic antenna aperture diagram for a walking user and beam-forming interval of 2 seconds.
Positioning Results. Our receiver implements a positioning filter based on stacking the estimated µ-trajectory segments. As already mentioned, the stacked µ-trajectory segments represent the relative movement of the user. GNSS code pseudorange observations are then used to get absolute coordinates. Basically, an extended Kalman filter is used to estimate a timely variable position offset to the stacked µ-trajectory segments. The Kalman filter employs a number of data-quality checks to eliminate coarse outliers. They are quite frequent in this hilly forested environment.
The positioning results obtained are shown in Figure 7. They correspond to the same received GPS+GLONASS signal but three different beam-forming intervals (0.2, 1, and 2 seconds) have been used. The position output rate corresponds to the beam-forming interval. Blue markers correspond to the surveyed reference positions, and the yellow markers are estimates when the user is at those reference markers. For each marker, there are ten observations.
FIGURE 7. Estimated user trajectory with 0.2, 1, and 2 seconds beam-forming interval (blue: surveyed reference markers).
The triangular walking path is clearly visible. We observe a bias of around 3 meters and a distance-root-mean-square of 1.2 meters if accounting for this bias (the values refer to the 2-second case). The reason for the bias has not yet been investigated. It could be due to ephemeris or ionospheric errors, but also possibly multipath reflections.
For the short beam-forming interval of 0.2 seconds, we observe noisier walking paths, and we would also expect less accurate code observations. However, the code observation rate is highest in this case (5 Hz), and multipath errors tend to average out inside the Kalman filter. In contrast, the walking paths for the 1-second or 2-second case are straighter. The beam-forming seems to eliminate the multipath, and there are fewer but more precise observations.
Artificial Motion Antennas
The rotating antenna targets surveying applications. It fits standard geodetic equipment. The antenna is controlled by the software receiver, and the rotational information is synchronized to the received GNSS signal.
Synthetic-Aperture Antenna Pattern. With the same methodology as referenced previously, it is possible to estimate the synthetic antenna gain pattern. We assume that the pattern simply maximizes the line-of-sight signal power for an assumed satellite elevation angle of 45° and an azimuth of 135°. We use a rotation radius of 50 cm. The antenna has a really high directivity, eliminating scattered signals from trees. The gain pattern is symmetric with respect to the horizon and ground multipath of perfectly flat ground would not be mitigated by the synthetic aperture. Ground multipath is only mitigated by the antenna element itself (for example, a small ground plane can be used). However, mostly the ground is not flat, and in that case the rotating antenna also mitigates the ground multipath.
Results with a Simulator. The rotating antenna has been tested with simulated GNSS signals using an RF signal generator. The signal generator was configured to start with the antenna at rest, and at some point the antenna starts rotating with a speed of 15 revolutions per minute. Six GPS L1 C/A-code signals have been simulated.
The signal-processing unit has to estimate the antenna state (static or rotating) and the north direction. The quality of the estimation can be visualized by comparing the complex argument of the prompt correlator values to the modeled correlator values. Two examples are shown in FIGURES 8 and 9. In Figure 8, the differences are at the millimeter level corresponding to the carrier-phase thermal noise. This indicates that the absolute heading and receiver clock parameters have been estimated to a high precision.
FIGURE 8. Carrier-phase residuals for all satellites observed with the rotating antenna without multipath. Time is in seconds and all data contributing to the RINEX observation record has been considered.
FIGURE 9. Carrier-phase residuals for all satellites observed with the rotating antenna with multipath. Time is in seconds and all data contributing to the RINEX observation record has been considered.
If multipath from a reflection plane is present (see Figure 9), the phase residuals show the multipath reflection. For example, around t = -0.65 seconds in the figure, the antenna is moving parallel to the reflection plane and the phase residuals are constant over a short time span. As the distance of the antenna to the reflection plane changes, the phase residuals start to oscillate. Generally, the estimation of the absolute heading and of the receiver clock parameters works even with strong multipath signals, but the parameters are not as stable as in the multipath-free case.
In the case when the antenna is rotating, signal processing has to remove the rotation from the code and carrier observations. To check if this elimination of the artificial motion is done correctly, we use carrier-smoothed code observations to compute a single-point-positioning solution. Only if the antenna is rotating can the system estimate the absolute heading and refer the observations to the rotation center. Before that point, the observations refer to the antenna position. The antenna position and the rotation center differ by the radius of 0.5 meters. Since the position is stable for t > 100 seconds, we conclude that the elimination of the artificial motion has been done correctly.
Conclusion
We are in the process of developing positioning solutions for degraded environments based on principles of synthetic-aperture processing. The tools target operational use as an end goal, supporting standard geodetic form factors (tripods) and the software receiver running on standard laptops, and producing data in standardized formats (such as RINEX or the National Marine Electronics Association (NMEA) standards).
Acknowledgments
The research leading to the results reported in this article received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 287226. This support is gratefully acknowledged. It also received funding from the Upper Bavarian Administration Aerospace Support Program under the contract number 20-8-3410.2-14-2012 (FAUSST), which is also thankfully acknowledged. This article is based on the paper “Concept of Synthetic Aperture GNSS Signal Processing Under Canopy” presented at the European Navigation Conference 2013, held in Vienna, Austria, April 23–25, 2013.
Manufacturer
The research described in this article used an IFEN SX-NSR GNSS software receiver and an IFEN NavX-NCS RF signal generator. The rotating antenna displacement unit was designed and manufactured by Blickwinkel Design & Development.
THOMAS PANY works for IFEN GmbH in Munich, Germany, as a senior research engineer in the GNSS receiver department. He also works as a lecturer (Priv.-Doz.) at the University of the Federal Armed Forces (FAF) Munich and for the University of Applied Science in Graz, Austria. His research interests include GNSS receivers, GNSS/INS integration, signal processing and GNSS science.
NICO FALK received his diploma in electrical engineering from the University of Applied Sciences in Offenburg, Germany. Since then, he has worked for IFEN GmbH in the receiver technology department, focusing on signal processing, hardware, and field-programmable-gate-array development.
BERNHARD RIEDL received his diploma in electrical engineering and information technology from the Technical University of Munich. Since 1994, he has been concerned with research in the field of real-time GNSS applications at the University FAF Munich, where he also received his Ph.D. In 2006, he joined IFEN GmbH, where he is working as the SX-NSR product manager.
JON O. WINKEL is head of receiver technology at IFEN GmbH since 2001. He studied physics at the universities in Hamburg and Regensburg, Germany. He received a Ph.D. (Dr.-Ing.) from the University FAF Munich in 2003 on GNSS modeling and simulations.
FRANZ-JOSEF SCHIMPL started his career as a mechanical engineer and designer at Wigl-Design while studying mechanical engineering. In 2002, he founded Blickwinkel Design & Development with a focus on prototyping and graphic design.
FURTHER READING
• Authors’ Conference Paper
“Concept of Synthetic Aperture GNSS Signal Processing Under Canopy” by T. Pany, N. Falk, B. Riedl, C. Stöber, J. Winkel, and F.-J. Schimpl, Proceedings of ENC-GNSS 2013, the European Navigation Conference 2013, Vienna, Austria, April 23–25, 2013.
• Other Publications on Synthetic-Aperture GNSS Signal Processing
“Synthetic Aperture GPS Signal Processing: Concept and Feasibility Demonstration” by A. Soloviev, F. van Graas, S. Gunawardena, and M. Miller in Inside GNSS, Vol. 4, No. 3, May/June 2009, pp. 37–46. An extended version of the article is available online: http://www.insidegnss.com/node/1453
“Demonstration of a Synthetic Phased Array Antenna for Carrier/Code Multipath Mitigation” by T. Pany and B. Eissfeller in Proceedings of ION GNSS 2008, the 21st International Technical Meeting of The Institute of Navigation, Savannah, Georgia, September 16–19, 2008, pp. 663-668.
“Synthetic Phased Array Antenna for Carrier/Code Multipath Mitigation” by T Pany, M. Paonni, and B. Eissfeller in Proceedings of ENC-GNSS 2008, the European Navigation Conference 2013, Toulouse, France, April 23–25, 2008.
• Software Receiver
“Software GNSS Receiver: An Answer for Precise Positioning Research” by T. Pany, N. Falk, B. Riedl, T. Hartmann, G. Stangl, and C. Stöber in GPS World, Vol. 23, No. 9, September 2012, pp. 60–66.
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