Characterization of multi-band GNSS multipath in urban canyons using the 3D ray-tracing method

Abstract

The multipath effect is the major challenge that deteriorates the high precision and robustness of positioning with global navigation satellite system (GNSS) in urban canyons. It leads to signal attenuation, Doppler shift, time delay, and phase rotation in a coupling manner. In addition, the multipath effect also exhibits the inter-frequency correlation features among different signal bands and types. This work derives theoretical multipath features and quantifies the multi-band multipath correlation. The ray-tracing method with 3D city maps is utilized to construct multipath geometry and conduct multipath characterization. The temporal-frequency multipath spectral analysis on real measurements regarding carrier-to-noise ($C/{N}_{0}$), Doppler residuals, and code-minus-carrier (CMC) are provided by using the Hilbert–Huang transform (HHT). To verify the periodic fluctuations and the multi-band correlation in a more realistic situation, the influence of baseband processing is considered and reconstructed. It is confirmed that multipath measurements oscillate at the relative Doppler shift rate, which is proportional to the carrier frequency. This work facilitates multipath identification and mitigation and eventually improves GNSS positioning precision under multipath influence.

Appendices

Appendix

Derivation of the Doppler measurement error

This appendix presents the detailed calculation of the Doppler measurement error. The discrimination algorithm is:

$$\begin{array}{*{20}c} {\delta f_{d,i} \left[ k \right] = \frac{1}{2\pi T}\arctan \left( {\frac{{P_{{\text{cross,i}}} \left[ k \right]}}{{P_{{\text{dot,i}}} \left[ k \right]}}} \right) } \\ \end{array}$$

(A.1)

where ${P}_{cross,i}\left[k\right]$ is computed as:

$$\begin{array}{*{20}c} {P_{{\text{cross,i}}} \left[ k \right] = P_{Q,i} \left[ k \right]P_{I,i} \left[ {k - 1} \right] - P_{I,i} \left[ k \right]P_{Q,i} \left[ {k - 1} \right]} \\ \end{array}$$

(A.2)

and ${P}_{dot,i}\left[k\right]$ is:

$$\begin{array}{*{20}c} {P_{{\text{dot,i}}} \left[ k \right] = P_{I,i} \left[ k \right]P_{I,i} \left[ {k - 1} \right] + P_{Q,i} \left[ k \right]P_{Q,i} \left[ {k - 1} \right]} \\ \end{array}$$

(A.4)

Thus:

$$\begin{aligned}{P}_{cross,i}\left[k\right]&\approx {A}_{LOS,i}{A}_{NLOS,i}{R}_{i}\left(\Delta {\tau }_{i}[k]\right)\left({\text{sin}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right)-{\text{sin}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k-1\right]T\right)\right)\\& +{A}_{NLOS,i}^{2}{R}_{i}^{2}\left(\Delta {\tau }_{i}[k]\right)\left({\text{sin}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right){\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k-1\right]T\right)\right)\\& -{A}_{NLOS,i}^{2}{R}_{i}^{2}\left(\Delta {\tau }_{i}[k]\right)\left({\text{sin}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k-1\right]T\right){\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right)\right)\end{aligned}$$

(A.5)

$$\begin{aligned}{P}_{dot,i}\left[k\right] & \approx {A}_{LOS,i}^{2}\\& + {A}_{LOS,i}{A}_{NLOS,i}{R}_{i}\left(\Delta {\tau }_{i}[k]\right)\left({\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right)+{\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k-1\right]T\right)\right)\\& + {A}_{NLOS,i}^{2}{R}_{i}^{2}\left(\Delta {\tau }_{i}[k]\right)\left({\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right){\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k-1\right]T\right)\right)\\& + {A}_{NLOS,i}^{2}{R}_{i}^{2}\left(\Delta {\tau }_{i}[k]\right)\left({\text{sin}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right){\text{sin}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k-1\right]T\right)\right)\end{aligned}$$

(A.6)

They are simplified to:

$$\begin{aligned}{P}_{cross,i}\left[k\right]&\approx 2{A}_{LOS,i}{A}_{NLOS,i}{R}_{i}\left(\Delta {\tau }_{i}[k]\right){\text{cos}}\left(\Delta {\phi }_{i}\left[k\right]+2\pi\Delta {f}_{d,i}\left[k\right]T\right){\text{sin}}\left(\pi\Delta {f}_{d,i}[k]T\right)\\& + {A}_{NLOS,i}^{2}{R}_{i}^{2}\left(\Delta {\tau }_{i}[k]\right){\text{sin}}\left(2\mathrm{\pi \Delta }{f}_{d,i}[k]T\right)\end{aligned}$$

(A.6)

and

$$\begin{aligned} P_{{{dot,i}}} \left[ k \right]& \approx A_{{{LOS,i}}}^{2} \\ & + 2A_{{{LOS,i}}} A_{{{NLOS,i}}} R_{i} \left( {\Delta \tau_{i} \left[ k \right]} \right)\cos \left( {\Delta \phi_{i} \left[ k \right] + 2\pi \Delta f_{d,i} \left[ k \right]T} \right)\cos \left( {\pi \Delta f_{d,i} \left[ k \right]T} \right) \\ & + A_{{{NLOS,i}}}^{2} R_{i}^{2} \left( {\Delta \tau_{i} \left[ k \right]} \right)\cos \left( {2\pi \Delta f_{d,i} \left[ k \right]T} \right) \\ \end{aligned}$$

(A.7)

Assuming that $2\mathrm{\pi \Delta }{f}_{d,i}[k]T$ is relatively small, then ${\text{sin}}\left(2\pi \Delta {f}_{d,i}\left[k\right]T\right)\approx 2\mathrm{\pi \Delta }{f}_{d,i}[k]T$, and ${\text{cos}}\left(2\pi \Delta {f}_{d,i}\left[k\right]T\right)\approx 1$. Therefore, the Doppler error expression is simplified to:

$$\begin{array}{*{20}c} {\delta f_{d,i} \approx \frac{1}{2\pi T}\arctan \left( {\frac{{\alpha_{i} R_{i} \left( {{\Delta }\tau_{i} \left[ k \right]} \right)\left( {\cos \left( {{\Delta }\phi_{i} \left[ k \right] + 2\pi {\Delta }f_{d,i} \left[ k \right]T} \right) + \alpha_{i} R_{i} \left( {{\Delta }\tau_{i} \left[ k \right]} \right)} \right)}}{{1 + \alpha_{i} R_{i} \left( {{\Delta }\tau_{i} \left[ k \right]} \right)\left( {2\cos \left( {{\Delta }\phi_{i} \left[ k \right] + 2\pi {\Delta }f_{d,i} \left[ k \right]T} \right) + \alpha_{i} R_{i} \left( {{\Delta }\tau_{i} \left[ k \right]} \right)} \right)}}2\pi {\Delta }f_{d,i} \left[ k \right]T} \right)} \\ \end{array}$$

(A.8)

Finally, by taking simplification of ${\text{sin}}\left(2\pi \Delta {f}_{d,i}\left[k\right]T\right)$ and ${\text{cos}}\left(2\pi \Delta {f}_{d,i}\left[k\right]T\right)$, we can obtain the Doppler error based on (A.8).