Performance analysis of GNSS multipath mitigation using antenna arrays

Although modern GNSS receivers provide high accuracy positioning and navigation solutions in open sky conditions, multipath remains a major error source in many environments. Multipath results in a distorted correlation function that is used to estimate delays and pseudoranges. This results in erroneous navigation solutions. Multipath also leads to incorrect ambiguity resolution affecting carrier phase positioning. If the multipath pseudorange error becomes large, the initial position solution is biased and the carrier phase ambiguity search space can be enlarged, resulting in longer ambiguity resolution time (Joosten et al. 2002). Long-delay code multipath caused by distant reflectors can be mitigated using currently available advanced correlator techniques such as Narrow Correlator™ (Dierendonck et al. 1992), Multipath Estimating Delay Locked Loop (MEDLL) (Van Nee et al. 1994) and Edge Correlator (Garlin et al. 1996) to name a few. However, multipath due to nearby reflectors is still a major problem for correlator-based techniques.

Antenna array processing, a signal processing scheme that exploits the signal spatial features, is proven to be effective in mitigating different types of interference. Even though antenna array processing is well studied for wireless communication systems, the application of these techniques to GNSS differ from those systems. For instance, in most wireless communication systems, increasing the signal to noise ratio to reduce bit error rate is the main focus; for GNSS the focus is to improve time-delay estimation to improve estimated position accuracy. The effectiveness of different beamforming techniques for GNSS applications was studied in (Fern’andez-Prades et al. 2016; Gupta et al. 2016; Broumandan et al. 2016; Cuntz et al. 2016; Amin et al. 2016; Daneshmand et al. 2014; Arribas et al. 2014; Egea et al. 2014; Kalyanaraman and Braasch 2007; Kalyanaraman and Braasch 2010). Most of the distortionless beamforming techniques are developed with the assumption that there is no correlation between desired and interference signals. However, performance of these beamforming techniques degrades in multipath interference because there is a high degree of correlation between desired and multipath signals (Van Trees 2002). The effectiveness of antenna arrays to mitigate multipath interference has been studied through different robust beamforming techniques in GNSS applications (Brown 2000; Fu et al. 2003; Seco-Granados et al. 2005; Sahmoudi and Amin 2007; Konovaltsev et al. 2007; Vicario et al. 2010; Fern’andez-Prades et al. 2011; Daneshmand et al. 2013a; Manosas-Caballu et al. 2013; Rougerie et al. 2011; Rougerie et al. 2012; Lee and Hsiao 2008). Sahmoudi and Amin (2007) used adaptive beamforming and high resolution direction finding methods to improve robustness against multipath and electronic interference. Vicario et al. (2010) analyzed robust beamforming techniques for Galileo ground stations and shown a reduction of tracking errors by 47 %. Fernández-Prades et al. (2011) studied the inherent capability of different eigen beamforming techniques to mitigate multipath through simulations. Some of these techniques assume either a linear array or a large planar array which is not however feasible for practical applications. Efficient maximum likelihood techniques to mitigate multipath are not practical for many applications due to their high computational burden. Even though the results from the previous research have shown that effective multipath mitigation is possible, the performance of antenna array based GNSS receivers in terms of time-delay estimation and position accuracy has not been analyzed extensively. Such performance is therefore assessed herein in terms of measurement and position accuracy through different beamforming techniques.

The focus is on short-range multipath signals with specular reflections. As GNSS signals are below the noise level before the correlation process, spatial processing to mitigate multipath signals is mostly performed after the de-spreading process (i.e., correlation and Doppler removal) (Arribas et al. 2012; Chen et al. 2012). The inherent capability of DAS and MPDR beamformers to mitigate multipath are studied first without any preprocessing to decorrelate the LOS and multipath signals. A preprocessing technique called spatial smoothing is used to decorrelate the signals. This process consists of two stages. In the first stage, spatial smoothing is used later to decorrelate LOS and multipath signals while in the second stage, spatially smoothed signals are combined using the MPDR beamformer. Measurement and position results from simulated and actual GPS signals are provided.

The system model and the main assumptions are outlined in Section II. Effects of multipath on antenna array processing techniques and the decorrelation effect due to spatial smoothing are discussed in Section III. In Section IV, GPS multi-antenna signal simulation methodology using a ray tracing method and beamforming implementation is discussed. The results of multipath mitigation using simulated signals are presented in Section V and actual GPS signal processing results are provided in Section VI. Finally, Section VII summarizes the findings.

This section describes the two different beamforming solutions considered in this research, namely DAS and MPDR with and without spatial smoothing. The effect of correlation between LOS and multipath signals on beamformers is discussed and different numerical simulations are performed to evaluate the performance of these beamforming techniques to mitigate multipath signals for GNSS applications. The main difference between GNSS and other systems is that the measurement quality is of utmost importance beside signal strength improvement. Any type of filtering that distorts measurement quality affects GNSS receiver performance. Hence, special care is required for beamforming design and implementation.

MPDR beamformer with spatial smoothing (MPDRSS)

Consider an M × N array divided into overlapping subarrays of size {J,L}. Assume P subarrays in the x-direction and Q in the y-direction. Let \( {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{p}}\mathit{\mathsf{q}}} \) be the covariance matrix of the [p, q]^th forward subarray. The forward spatially smoothed covariance matrix is the sample means of all the forward subarray covariance matrices and can be computed as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}}=\frac{\mathsf{1}}{\mathit{\mathsf{P}}\mathit{\mathsf{Q}}}{\displaystyle \sum_{\mathit{\mathsf{p}}=\mathsf{1}}^{\mathit{\mathsf{P}}}{\displaystyle \sum_{\mathit{\mathsf{q}}=\mathsf{1}}^{\mathit{\mathsf{Q}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{p}}\mathit{\mathsf{q}}}}} $$

(13)

Similarly, if \( {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{b}}} \) is the backward spatially smoothed covariance matrix, then the forward-backward spatially averaged covariance matrix is given by

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{b}}}=\frac{{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}}+{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{b}}}}{\mathsf{2}} $$

(14)

The optimum weight vector for the MPDR beamformer with spatial smoothing is (Van Trees 2002)

$$ {\text{\textbf{\textsf{w}}}}_{\mathit{\mathsf{MPDRSS}}}=\frac{{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{b}}}^{-\mathsf{1}}{\text{\textbf{\textsf{a}}}}_{\mathsf{11}}}{{\text{\textbf{\textsf{a}}}}_{\mathsf{11}}^{\mathit{\mathsf{H}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{f}}\mathit{\mathsf{b}}}^{-\mathsf{1}}{\text{\textbf{\textsf{a}}}}_{\mathsf{11}}} $$

(15)

where \( {\text{\textbf{\textsf{a}}}}_{\mathsf{11}} \) is the steering vector of the LOS signal for the first subarray.

Beamformer’s performance depends on a number of factors such as the number of antenna elements, array configuration and incoming signal directions of arrival to name a few. The size and number of antenna elements are some of the limitations for practical applications in terms of cost and system complexity. Hence investigation of the performance of an antenna array based GNSS receiver with a limited number of antenna elements while still being able to perform spatial smoothing is important. In this research a Uniform Rectangular Array (URA) with six antenna elements is considered (M = 3, N = 2). The subarray formation for the spatial smoothing is shown in Fig. 2. Due to the limited number of elements in the array, only two subarrays (P = 2, Q = 1) are constructed with size {J = 2, L = 2}. The decorrelation obtained by spatial smoothing and in turn, the performance of the beamformer, is analyzed in the following sections.

Numerical simulations

This section presents numerical simulation results for the array structure shown in Fig. 2 with inter-element spacing of 9.5 cm. The performance of the beamforming techniques in the presence of multipath signals is evaluated using the Signal-to-Multipath Ratio (SMR) (Egea et al. 2014) metric. SMR refers to the ratio between the LOS power and multipath power at the output of the beamformer and is expressed in dB. The pre-beamformer SMR is assumed to be 0 dB. Here, it is assumed that multipath is coming from (15°, 175°) and the LOS signal azimuth is (50°). Beamformer performance for different correlation coefficients of the LOS and multipath signals for different LOS signal elevations is assessed. For the two signals case, r₁ (LOS) and r₂ (multipath) the covariance matrix can be represented as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{y}}\mathit{\mathsf{y}}}=\text{\textbf{\textsf{A}}}{\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{s}}}{\text{\textbf{\textsf{A}}}}^{\mathit{\mathsf{H}}}+{\sigma}_{\eta}^{\mathsf{2}}\text{\textbf{\textsf{I}}} $$

(16)

where \( {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{s}}} \) is the source covariance matrix and \( {\sigma}_{\eta}^{\mathsf{2}} \) is the noise variance. The source covariance can be defined as

$$ {\text{\textbf{\textsf{R}}}}_{\mathit{\mathsf{s}}}=\left[\begin{array}{l}{\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}^{\mathsf{2}}\\ {}{\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}{\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}\rho \end{array}\right.\kern0.6em \left.\begin{array}{l}{\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}{\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}\rho \\ {}{\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}^{\mathsf{2}}\end{array}\right] $$

(17)

where \( {\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}^{\mathsf{2}} \) is the variance of the source signal, \( {\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}^{\mathsf{2}} \) is the variance of the multipath signal and ρ is the correlation coefficient between the LOS and multipath, defined as

$$ \rho =\frac{E\left[{\mathit{\mathsf{r}}}_{\mathit{\mathsf{1}}}{\mathit{\mathsf{r}}}_{\mathsf{2}}^{\mathit{\mathsf{H}}}\right]}{\sqrt{E\left[{\mathit{\mathsf{r}}}_{\mathit{\mathsf{1}}}{\mathit{\mathsf{r}}}_{\mathsf{1}}^{\mathit{\mathsf{H}}}\right]\sqrt{E\left[{\mathit{\mathsf{r}}}_{\mathsf{2}}{\mathit{\mathsf{r}}}_{\mathsf{2}}^{\mathit{\mathsf{H}}}\right]}}} $$

(18)

The power of both LOS and multipath is set to 10 (\( {\sigma}_{\mathit{\mathsf{r}}\mathsf{1}}^{\mathsf{2}}={\sigma}_{\mathit{\mathsf{r}}\mathsf{2}}^{\mathsf{2}}=\mathsf{10} \)) and the noise variance is assumed to be 1. The elevation of the LOS signal varies from 0° to 90° for different magnitudes of the correlation coefficient between the signals and the SMR performance of both MPDR and MPDRSS is shown in Fig. 3. The MPDR performance is the same for different LOS signal elevations for a given correlation coefficient. For very low correlation coefficients, which is the case when LOS and multipath signals are uncorrelated to each other, the MPDR beamformer yields a SMR up to 40 dB. However, as correlation increases, beamformer performance decreases and results in low SMR. As seen in Fig. 3, when the correlation coefficient magnitude is above 0.6, the SMR is nearly 0 dB. The performance of MPDRSS is better for higher elevation satellites when signals are correlated to each other, as compared to MPDR. This is due to the fact that the angular separation of the LOS from multipath signals is higher and spatial smoothing is able to provide better decorrelation. As can be seen in Fig. 3, SMR up to 10 dB can be achieved using MPDRSS for higher elevation satellites even when signals are highly correlated. Since the decorrelation achieved by the spatial smoothing process is a function of the DOA of the incoming signals and the number of antenna elements, the MPDRSS beamformer performance will be different for different signals impinging on the array from different directions. However, it was observed that for the rectangular array considered, MPDRSS beamformer performance improves with an increase in the elevation angle of the LOS signal, considering the multipath signal is coming from a low elevation.

The beampatterns for the DAS, MPDR and MPDRSS beamformers for different correlation coefficients for a higher elevation satellite with multipath from low elevation are shown in Fig. 4. Here it is assumed that LOS is coming from (75°, 50°) and multipath from (15°, 175°). As the DAS beamformer does not rely on the statistics of the received signal, the performance will be same for any correlation between LOS and multipath signals. However, MPDR performance is improved only when the correlation between LOS and multipath is very low. However, MPDRSS provides better attenuation of the multipath signals. Even when signals are highly correlated, MPDRSS can attenuate multipath by up to 10 dB. Based on the LOS signal directions and correlation between LOS and multipath signals, the DAS beamformer performance could be similar to that of MPDR and MPDRSS. In some cases, it could be better than MPDR as correlation can degrade the performance of the latter.

This section describes the multi-antenna GPS signal simulator and receiver architecture used for the analysis in multipath environments.

Multi-antenna GPS signal simulator

The multi-antenna GPS signal simulator can simulate GPS signals for a given user scenario and antenna array configuration. It has the option to simulate different multipath signals utilizing a ray-tracing approach. The main advantage of a software simulator as compared to the use of actual data is the ability to control error sources such as antenna calibration uncertainties, atmosphere, multipath and clock errors. Therefore, the performance of a beamformer can be analyzed for different multipath signals parameters. The basic blocks of the simulator are shown in Fig. 5. The input is the digitized IF samples collected using a data acquisition system either from a hardware similator or actual signals. These digitized samples are free of multipath. The software simulator is configured through two option files. The main option file defines the parameters such as sampling frequency, channel numbers and satellite list. The second option file is related to multipath signal parameters and defines the number of reflectors, reflector coordinates, user motion scenario and the antenna array configuration.

Using the digital samples, visible satellites are acquired and tracked. During the initial state of tracking, satellites are tracked in Phase Locked Loop (PLL) with higher bandwidth and loop order without assisting Delay Locked Loop (DLL). Later, based on the Phase Lock Indicator (PLI), the tracking state is switched to the PLL-assisted DLL mode. The replica signals from this stage are used to generate multi-antenna signals. The replica signal consists of code replica, carrier replica and the navigation data bits. Using ephemeris information, satellite positions are computed. The satellite DOA is then computed using these satellite positions and the known user position with accuracy of a few decimetres or better. Based on the antenna array configuration defined in the option file and satellite signal DOAs, LOS steering vectors are computed for all satellites. The replica LOS signal is then multiplied by the LOS array steering vector to generate multi-antenna signals.

Based on the reflector and the satellite positions, the point of reflection for the multipath signals is computed. Once the reflection point is found, the extra distance travelled by multipath signals is converted to the number of chips, which is then added to the LOS prompt code to generate multipath signals. Due to the additional path travelled by these signals and the reflection location, the Doppler observed by a multipath signal will be different from that of the LOS signal. The multipath signal SMR for different satellites is defined in the multipath option file. Using multipath Doppler information, replica code and the attenuation factor, multipath signals are generated for each visible satellite. Using the point of reflection and known user position, The DOAs of multipath signals are computed and the corresponding steering vectors are generated using Equation (9) The multi-antenna multipath signals thus generated for a particular satellite are then added to the corresponding LOS multi-antenna replica signals. The combined LOS and multipath signals from all visible satellites are added to generate the composite GPS baseband signal. Later, independent noise is added to each antenna signal to have the desired C/N₀ values for the simulated signals.

In order to evaluate the performance of the proposed multi-antenna software simulator, the IF sample files generated from the software simulator (reference antenna IF file) and the Spirent hardware simulator were processed with the GSNRx software receiver (Petovello et al. 2008). The carrier Doppler values from the software receiver were inter-compared and similar performance was observed in terms of C/N₀, signal tracking and navigation solution. The validation process showed that the performance of the multi-antenna GPS software simulator is comparable with that of a hardware simulator.

Multi-antenna GPS receiver

An open source single antenna MATLAB™ based GPS software receiver (Borre et al. 2007) was modified for multi-antenna receiver functionalities. The acquisition, tracking and navigation strategies of the original software receiver were modified. The basic blocks of the multi-antenna receiver are shown in Fig. 6. One of the antenna elements in the array acts as the reference antenna. Satellite signals are acquired and tracked using the digital samples of the reference antenna. The Doppler and code delays thus obtained are used to despread the signals from other antennas so that relative phase values between the antenna elements are maintained. After Doppler and code removal from the digital samples corresponding to each antenna, the prompt correlator values are used to compute the optimum weights using the MPDR beamformer. In order to capture the statistics of the incoming signals, prompt correlation values collected over one second are used to compute the covariance matrix of the MPDR beamformer. Thus its weights are updated every second. The DAS beamformer does not use the statistics of the prompt correlation values as it relies only on the satellite DOA. Weights for the DAS beamformer are also updated every second to capture the LOS signal DOA variations. The weights computed are used to combine 1 ms, early, prompt and late correlator values of the six antennas. The combined correlator arms, namely early-prompt-late, are used by the tracking loops to generate the code and carrier replica signals.

A narrow correlator approach with 0.1 chip spacing between early and late arms and a normalized non-coherent early minus late envelope code discriminator are used. A first order DLL with bandwidth of 0.1 Hz is used in the PLL-assisted DLL mode. The C/N₀ is computed using narrowband power and wideband power as described in (Dierendonck 1996). The least squares method is used to compute the position solution with pseudorange measurements.

The numerical simulation results described in the paper indicate that performance of MPDR and MPDRSS beamformers improves as the correlation between LOS and multipath signals decreases. It was observed that, for a rectangular array with six antenna elements, the MPDRSS beamformer provides better multipath mitigation for higher elevation satellites. The proposed multi-antenna signal simulator was used to generate multipath affected multi-antenna signals for different user environments and the results were compared with realistic multipath scenarios. Using the simulated GPS signals, it was observed that pseudorange errors can be reduced by tens of metres in high multipath environments, thereby improving position accuracy. It was observed that the MPDRSS beamformer performs better than the MPDR and DAS beamformer. With actual GPS L1 signals collected in a moderate specular multipath scenario, a reduction of 10 m in RMS pseudorange error was observed for satellites affected by multipath signals. Pseudorange error reduction was reflected in the position solutions. Finally, it was shown that a six-antenna rectangular array is effective to mitigate short-range multipath signals and provide an improved navigation solution, based on the data used in the analysis. Extensive testing would be required to confirm these enhancements in different environments.