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Multipath propagation is one of the major sources of error in GPS measurements. In this research, a ray-tracing technique is proposed to study the frequency domain characteristics of multipath propagation. The Doppler frequency difference, also known as multipath phase rate and fading frequency, between direct (line-of-sight, LOS) and reflected (non-line-of-sight, NLOS) signals is studied as a function of satellite elevation and azimuth, as well as distance between the reflector and the static receiver. The accuracy of the method is verified with measured Doppler differences from real data collected in a downtown environment. The use of ray-tracing derived predicted Doppler differences in a receiver, as a means of alleviating the multipath induced errors in the measurement, is presented and discussed.

Ever increasing Global Navigation Satellite System (GNSS) based applications require reliable and accurate navigation solutions in challenging environments such as cities and indoors. In such environments, receiver accuracy and reliability are limited by signal shadowing, blockage, and multipath. These factors lead to increased position errors. Signal shadowing, where the signal is present but attenuated, leads to poor acquisition and tracking performance, while complete signal blockage leads to increased dilution of precision, and, finally, multipath leads to poor measurement accuracy and fading. These challenges and some solutions are discussed in [

Multipath propagation is examined here in the frequency domain. The separation of direct and reflected signals in that domain is studied in [

The use of precise oscillators is limited due to cost, size, and power consumption at this time. Hence, there is not much study of multipath characterization done in the frequency domain. There is hope that the development of Chip Scale Atomic Clocks (CSAC) and nano-/microclocks [

In the kinematic case, the Doppler spread of the reflected signals in urban canyons is studied in [

Nievinski and Larson [

This research proposes a method using the ray-tracing methodology to study the

The concept of separating, or resolving, the composite signal is described in Section

As shown by Irsigler [

CAF of PRN 14 with 10 and 120 s of coherent integration.

Contour plot of the CAF of PRN 14 with 10 and 120 s of coherent integration.

A single uniform reflector causing specular reflection is assumed to develop and verify the ray-tracing approach and the reflection coefficient is assumed to be a complex constant. As the size of the reflector is small compared to the distance to the satellite, the angle of incidence of the satellite signal on the reflector can be assumed to be nearly the same at all points. Therefore, the rate of change of the phase and amplitude of the signal are negligible. Hence, the magnitude of the reflection coefficient becomes insignificant when restricting the study to the Doppler difference. Naturally, its value becomes critical when there is a need to analyze the effect of the reflected signals on the direct ones and in turn on the measurements. In this study, the receiver antenna related effects and edge diffraction effects are not considered. Hence, this method falls into the category of “Geometrical” simulators of “Plates” type as defined by Nievinski and Larson [

The reflector is assumed to be a triangle to simplify the construction of various other shapes, for example, rectangles and other polygons, and to perform the ray-tracing computation efficiently. In reality, the shape of the majority of the reflectors is rectangular which can easily be constructed using two triangles. Three vertices of the triangular reflector are assumed to be known along with the receiver position.

Due to the large distance of over 20,000 km [

This section describes the Doppler frequency difference,

Satellite (

As direct and reflected signals travel along different paths they are observed at different Doppler frequencies at the receiver antenna. The

The velocity of the satellite and the receiver is known and by using the position of the receiver and the satellite, the

Ray-tracing is used commonly in computer graphics for image synthesis. Specifically, the path of a ray of light from its source is followed (or traced) as it bounces multiple times around the scene [

Consider Figure

Ray-tracing diagram showing the reflector (triangle

The process of finding the POI of a ray and computing the range of the direct and reflected signals is continued for every change in satellite position. Then the Doppler frequencies of the direct and the reflected signals are computed as a rate of change of the range over time leading to the determination of the

A simulation was performed using the equations derived in the previous section to study the characteristics of reflected signals in the frequency domain. Table

Static case simulation parameter.

Parameter | Value |
---|---|

Receiver position | 51.07995373°N, 114.13384821°W, 1118 m |

GPS week number | 1799 |

Time of week (TOW) (s) | 208800 |

Simulation duration (hours) | 12 |

Sample interval (s) | 1 |

PRNs | 2, 5, 7, 16, 19, 21, 26, and 30 |

Reflector type | Isosceles triangle with base of 600 and height of 300 m |

Reflector orientation | Surface normal is defined as |

Type and number of reflections | One specular reflection |

Arbitrarily chosen reflector distances from the receiver (m) | 2, 5, 10, 15, 20, 30, 50, 75, and 100 |

Ray-Triangle intersection when the triangular reflector is kept at a distance of 20 m from the receiver (red dot) to illustrate the movement of point of intersection (POI) as PRN 30 moves in its orbit.

To illustrate the movement of the POI, the ray-tracing simulation results for PRN 30, whose elevation and azimuth angles are available in Figure

Doppler differences (

Figure

For a given set of visible PRNs and receiver location and with a reflector kept at a distance of 20 m south of the receiver, the maximum

In reality, the reflectors will not be as large as 600 × 300 m. However, for illustration purpose and to reduce the processing time during the simulation, one single large triangle is chosen instead of multiple smaller triangles. To understand the characteristics of reflected signals in the frequency domain, only a small portion of this triangle is sufficient for a given period of time. Reflectors of smaller sizes, for instance, 50 × 30 m, are more likely and therefore only a small portion of the triangle where the transmit ray intersects the triangle is considered for further analysis. It is interesting to note that the slopes of the

To study the effect of the distance between reflector and receiver, the simulation is carried out with various reflector distances for the case south of the receiver. The visibility of reflected signals at the receiver changes as the reflector distance varies. The visibility is much influenced by the PRN elevation angle. For this case, the effect of azimuth is not observable. As the distance increases, high elevation PRNs no longer experience reflected propagation. For example, for 50 m or greater, reflections are not present for PRN 7 and PRN 30 due to the change in the angle of incidence and in turn in the angle of reflection.

From a point of view of separating the direct and reflected signals in the frequency domain, it is sufficient to observe the absolute

As show in Figure

In real scenarios, to separate the direct and reflected signals in the frequency domain due to the small

The implementation of this method is validated by comparing the results obtained for a large planar ground reflector with that of the simple geometry model. A large planar and horizontal reflector with a surface normal defined as

Simple geometric model versus triangle based ray-tracing method.

Though the focus is on the static case, this method can be directly used for the moving receiver case in an environment of static or dynamic reflectors, as long as the receiver and reflector velocities are known. The factor that causes the difference is the velocity of the receiver, as shown in (

The data was collected in a partially open-sky environment of downtown Calgary and the corresponding sky plot is shown in Figure ^{−13} over 1 to 30 seconds, namely, the

Sky plot of test site. Only PRN 11 was severely affected by reflected signals.

The data was processed using a modified version of GSNRx [

Based on geometry and the environment, PRN 11 is expected to be affected by multipath. To confirm this, the pseudorange errors of PRN 11 due to multipath are obtained by removing other error sources including ionospheric and tropospheric, orbit, and satellite clock, via differential processing using a base station located 10 km away. The pseudoranges are obtained using the normalized noncoherent early-minus-late envelope discriminator [

Multipath induced errors in PRN 11 pseudoranges.

Aided acquisition with coherent integration of 155 s is performed. Using an initial 35 seconds of data, the signals are acquired and tracked as in a standard receiver to obtain time, ephemeris, code delay, and approximate Doppler frequency offset for each PRN. These parameters are then used to initialize the fine search process to reduce the search space. The known position, computed time, and ephemeris are used to derive the Doppler rate to compensate for the change in the Doppler due to satellite motion during the coherent integration period. The search space size is selected to accommodate the possible frequency and code delay spread due to reflected signals. Since the center frequency and code delay for each PRN are derived from the initial data, any residual frequency offset due to the local oscillator frequency and any bias in the code delay are removed. Figure

Frequency domain view of the CAF envelope and corresponding delay in code domain.

Examining the CAF values in the upper subplot of Figure

From Figure

To compare the measurements from the real data case with that of the ray-tracing based simulation, a building that is 43 m south of the receiver is considered a reflector and a simulation is performed. The Doppler frequency difference and multipath delay for PRN 11 at the time that corresponds to the beginning of the 155 s coherent integration are found. The multipath delays and Doppler frequency differences obtained from the simulation and the measurement of real signals are listed in Table

Simulation versus measured reflected delay and Doppler difference.

Method/parameter | Multipath delay (m) | Doppler difference (mHz) |
---|---|---|

Simulation | 70 | −27.5 |

Measured | 87 | −21.5 |

The reflected signals are always delayed relative to the direct signal. However, these reflected signals cause an advance or delay of pseudorange measurements depending on whether they interfere with direct signals destructively or constructively. This is the reason for seeing valid points with both positive and negative delays in Figure

As an example, three ACFs from various different frequency bins of PRN 11 are shown in Figure

Code-domain view of the CAF with selected three ACFs.

The advantage of separating the direct and reflected signals in the frequency domain is that it is independent of the RF signal bandwidth. Traditionally, multipath mitigation has relied on observing only the few meters around the zero delay of the autocorrelation function, where there is little multipath, and the bias induced by multipath that is present is low. When doing so, achieving high discriminator gain requires high signal bandwidth. In contrast, if multipath is resolved in the frequency domain, the requirement for high signal bandwidth can be relaxed. This might allow one to slightly reduce the sampling frequency, which in turn may reduce the processing load.

After resolving the composite signals in the frequency domain, with sufficient frequency resolution, then what remains to be done to generate correct measurements is to identify which one of detected signals correspond to the line-of-sight. In an ideal case, the detected signal with minimum delay should correspond to the direct signal and all the remaining signal components should correspond to the non-line-of-sight propagation. Selection of the earliest signal component is influenced by the code delay resolution which is a function of the RF front-end bandwidth. Hence, depending on the method used to generate the range measurements, the accuracy may remain as, to an extent, a function of the RF front-end bandwidth.

As the choice of coherent integration period depends on the satellite geometry, the location of the reflectors, and the received signal strength, there is no globally optimum choice. However, from Figure

The theoretical and experimental results show that it would be possible to separate direct and reflected signals in the frequency domain using very long coherent integration periods for a static receiver and reflector. This method is well suited for characterizing the multipath environment at permanent fixed sites such as reference stations as it can separate multiple reflected signals and allows one to easily identify the number of reflected signals, their delay, and their power levels relative to direct signal.

This section describes how the ray-tracing algorithm can be used to predict the Doppler difference between direct and reflected signals, which can be used in positioning applications. Three potential uses of the Doppler difference information are suggested including prediction of the position solution convergence time; coherent integration period selection for improved measurement generation; and selective measurement rejection or weighting based on multipath error period.

The Doppler difference between the direct and reflected rays predicted by the ray-tracing algorithm can be used to determine the approximate averaging time required to obtain a position solution in which the multipath error has averaged to near zero, for example, when employing a batch least-squares position estimation.

A location in the downtown of Calgary, Canada, with tall buildings on its north, south, and east, was selected as a test location to illustrate the application. The test location and surrounding area are shown in Figure

Test location and buildings around it from Google Earth.

PRNs sky plot during the test time [

The Doppler differences computed using the ray-tracing algorithm are listed in the second column of Table

Doppler differences comparison.

PRN | Ray-tracing based (mHz) | CMC meas. based (mHz) | 120 s coherent integration CAF based (mHz) |
---|---|---|---|

14 | 60 | 60 | 60 |

19 | 17 | 20 | 50 |

21 | 42 | 57/63/144 | 40 |

27 | 70 | 47/70 | 50 |

Before going into position domain analysis, it is important to confirm presence of reflected signals and the accuracy of the Doppler differences computed by the ray-tracing algorithm. To accomplish this code-minus-carrier (CMC) measurements were generated by processing the IF-data. Data was collected as complex samples at a rate of 20.25 MHz with precise 10 MHz oscillator as reference using Tele Orbit’s front-end. The IF data was processed with the GSNRx software-defined GNSS receiver, and the code and carrier phase measurements were generated at a rate of 1 Hz for a total duration of 10 minutes. The CMC measurements of each PRN are biased by the presence of many errors, including code multipath error, carrier phase multipath error, two times the ionosphere delay, code multipath error, carrier phase multipath error, clock bias, carrier phase integer ambiguity, and code and carrier phase measurement noise. Since the data was collected using a precise oscillator, the clock bias can be treated as constant over the period of data collection and ionosphere delay over the data collection time can be reasonably assumed to be a constant over such a short period. By design, the carrier phase ambiguity term is constant and hence the bias in the CMC measurements is treated as a constant bias and is removed. What remains is the variation caused by the code and carrier phase multipath errors due to constructive and destructive interferences. The carrier-phase multipath, which will be less than 5 cm for the case when the direct signal power is greater than that of the reflected signal [

Code-Minus-Carrier (CMC) measurements after bias removal and their corresponding FFT output.

The normalized (with respect to PRN 14) magnitude of the FFT output of the bias-removed-CMC measurements is shown in the lower subplot of Figure

Having generated these Doppler difference predictions, the goal is to determine a time after which the position solution converges to a solution without large multipath errors. Firstly, to determine the best averaging time at the measurement level to alleviate the multipath errors, moving averages of the bias-removed-CMC measurements were computed for various averaging-window sizes over the entire duration of the data. Secondly, measurement averaging was done at the position level to observe the improvement in the position accuracy with respect to averaging time. The RMS bias-removed-CMC errors for various moving-average window sizes are shown in Figure

RMS errors of bias-removed-CMC measurements for each PRN and RMS 3D position errors for different averaging times (Window sizes).

It is apparent that the RMS errors decrease rapidly with increasing averaging time, up until the averaging time is equal to the inverse of the Doppler difference reported in Table

Nonetheless, from examination of the errors associated with PRN 14 and PRN 19, it is evident that the ray-tracing simulation output can be used to select appropriate measurement averaging times that will ensure significant reduction of the multipath induced errors. For the PRNs with multiple reflected signals, the smallest of all the Doppler differences can be chosen as a conservative estimate of the required averaging time.

To observe the effect of averaging at the position level, the least-squares method of position estimation is employed [

After averaging time of 16.7 s, the error contribution associated with PRNs 14, 21, and 22 has significantly reduced; however the error from PRN 19 continues to dominate. Beyond an averaging time of 59 seconds the error associated with PRN 19 also reduces. Beyond this time the improvement in the position accuracy is small. This suggests that the majority of the error is due to multipath propagation and that it can be effectively averaged out within a period equal to the inverse of the Doppler difference. Hence, an averaging time can be determined by inverting the smallest Doppler difference derived from the ray-tracing simulation.

The predicted Doppler difference from the ray-tracing simulation can also be used to prescribe the length of coherent integration required to reduce effect of the reflected signals on the ACF of the direct signal. To illustrate this, for the selected PRNs, CAFs were generated with various coherent integration times. Examining the maximum value of the CAF, early (

Showing multipath error reduction as a function of coherent integration period for various PRNs.

There may be scenarios where it is not convenient to apply an averaging time equal to the reciprocal of the minimum predicted Doppler difference of all PRNs in view. In such cases, the measurements with Doppler differences that require excessive averaging might simply be rejected or deweighted while computing the solution. Of course, it should first be ascertained whether or not the DOP of the remaining PRNs is acceptable to the positioning application. Although further detailed analysis is required to test this possibility, the potential use of ray-tracing as a means of identifying measurement quality is interesting.

A mathematical model was developed to characterize direct and reflected signals in the frequency domain by adopting a ray-tracing technique. The simulation results were positively verified using results from a ground reflector geometry model. The Doppler differences computed using this model for a city core location match closely measured Doppler differences using long coherent integration of live GNSS data. This data was used to demonstrate how ray-tracing derived Doppler differences can be used to determine the solution convergence time in a static positioning case and to estimate the appropriate coherent integration time to reduce multipath errors in range measurements.

Consider Figure

Three vertices of the reflector, satellite position, and receiver position are used to find the point

Find the center point

Find the direction of the reflected ray

Assuming the direction of the reflected ray at point

The position vector at center point

Since the angle between

The magnitude of

As stated earlier, the direction of

To find point

The authors declare that there is no conflict of interests regarding the publication of this paper.

Mr. Rakesh Kumar is acknowledged for his ray-tracing discussion session and Mr. Ranjith Kumar is acknowledged for providing an IF data set.