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Joint angle and delay estimation for GNSS multipath signals based on multiple sparse Bayesian Learning

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Abstract

Multipath signals formed by signal reflection coming from objects in the vicinity of Global Navigation Satellite System (GNSS) receivers result in a degradation of the tracking performance and an increase in the positioning error. By estimating the parameters of both line-of-sight signal and the multipath signals, superior multipath mitigation, spoofing suppression, and localization can be attained. We propose using the multiple sparse Bayesian learning method together with the joint angle and delay estimation technique in GNSS multipath environment to fully exploit the sparsity present in both the spatial and the temporal domains. We also extend the techniques to the estimation of fractional Doppler frequency besides the angle and delay. To counteract the intrinsic drawbacks of sparse representations, two different algorithms based on on-grid and off-grid estimators are proposed to either reduce the complexity or enhance the resolution such that the proposed multipath mitigation approach can be adapted to various GNSS practical situations. Subsequently, a third algorithm with improved resolution is obtained by applying the Space Alternating Generalized Expectation–Maximization algorithm to refine the MSBL-based joint angle and delay estimates. Simulation results indicate that the three proposed algorithms can effectively resolve the GNSS multipath signals and have better performance than existing methods even in severe situations, like the cases of signals with low carrier-to-noise-power-density ratio and spatially and temporally correlated multipath.

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Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgment

The work of G. Seco-Granados was supported in part by the Research and Development Projects of Spanish Ministry of Science, Innovation, and Universities under Grants TEC2017-89925-R and TEC2017-90808-REDT, and by the ICREA Academia Program.

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Authors and Affiliations

  1. Xi’an Jiaotong University, Xi’an, People’s Republic of China

    Ning Chang, Wenjie Wang & Xi Hong

  2. Universitat Autònoma de Barcelona, Barcelona, Spain

    José A. López-Salcedo & Gonzalo Seco-Granados

Authors
  1. Ning Chang
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  2. Wenjie Wang
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  3. Xi Hong
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  4. José A. López-Salcedo
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  5. Gonzalo Seco-Granados
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Correspondence to Ning Chang.

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Appendix

Appendix

We calculate (27) through the following two equalities where the sampling point is omitted for short:

$$\begin{array}{l} \left\| {{\varvec{y}_s} - {\varvec{\Phi} _{{\varvec{\beta} _\tau }}}\varvec{\mu} } \right\|_2^2\\ = \left\| {{\varvec{y}_s} - \left( {\varvec{A} \otimes \varvec{C}} \right)\varvec{\mu} - {\varvec{\Xi} _\tau }{\varvec{\beta} _\tau }} \right\|_2^2\\ = \varvec{\beta} _\tau ^T\varvec{\Xi} _\tau ^H{\varvec{\Xi} _\tau }{\varvec{\beta} _\tau } - 2\Re \left\{ {{{\left( {{\varvec{y}_s} - \left( {\varvec{A} \otimes \varvec{C}} \right)\varvec{\mu} } \right)}^H}{\varvec{\Xi} _\tau }} \right\}{\varvec{\beta} _\tau } + {C_1} \end{array}$$
(41)
$$\begin{array}{l} Tr\left\{ {{\varvec{\Phi} _{{\varvec{\beta} _\tau }}}{{\varvec{\Sigma} }}\varvec{\Phi} _{{\varvec{\beta} _\tau }}^H} \right\}\\ = 2\Re \left\{ {Tr\left\{ {\left( {\varvec{A} \otimes \varvec{C}} \right){{\varvec{\Sigma} }}{{\left( {\varvec{A} \otimes \left( {{\varvec{B}_\tau }{\rm{diag}}\left( {{\varvec{\beta} _\tau }} \right)} \right)} \right)}^H}} \right\}} \right\}\\ + Tr\left\{ {\left( {\varvec{A} \otimes \left( {{\varvec{B}_\tau }{\rm{diag}}\left( {{\varvec{\beta} _\tau }} \right)} \right)} \right){{\varvec{\Sigma} }}{{\left( {\varvec{A} \otimes \left( {{\varvec{B}_\tau }{\rm{diag}}\left( {{\varvec{\beta} _\tau }} \right)} \right)} \right)}^H}} \right\} + {C_2} \end{array}$$
(42)

where \(C_{1}\) and \(C_{2}\) are the parts irrelevant to \(\varvec{\beta }_{\tau }\). Equation (42) can be obtained by the following two parts:

$$\begin{array}{l} \Re \left\{ {Tr\left\{ {\left( {\varvec{A} \otimes \varvec{C}} \right){{\varvec{\Sigma} }}{{\left( {\varvec{A} \otimes \left( {{\varvec{B}_\tau }{\rm{diag}}\left( {{\varvec{\beta} _\tau }} \right)} \right)} \right)}^H}} \right\}} \right\}\\= \Re \left\{ {Tr\left\{ {\left( {\varvec{A} \otimes \varvec{C}} \right){{\varvec{\Sigma} }}\left[ {{\text{diag}}\left( {{\varvec{J}_{{N_\theta }{N_\tau } \times {N_\tau }}}{\varvec{\beta} _\tau }} \right)\left( {{\varvec{A}^H} \otimes \varvec{B}_\tau ^H} \right)} \right]} \right\}} \right\}\\ = \Re {\left\{ {{\text{diag}}\left( {\left( {{\varvec{A}^H}\varvec{A} \otimes \varvec{B}_\tau ^H\varvec{C}} \right){{\varvec{\Sigma} }}} \right)} \right\}^T}\left( {{\varvec{J}_{{N_\theta }{N_\tau } \times {N_\tau }}}{\varvec{\beta} _\tau }} \right) \end{array}$$
(43)
$$\begin{array}{l} Tr\left\{ {\left( {\varvec{A} \otimes \left( {{\varvec{B}_\tau }{\rm{diag}}\left( {{\varvec{\beta} _\tau }} \right)} \right)} \right) {{\varvec{\Sigma} }}{{\left( {\varvec{A} \otimes \left( {{\varvec{B}_\tau }{\rm{diag}}\left( {{\varvec{\beta} _\tau }} \right)} \right)} \right)}^H}} \right\}\\ = Tr\left\{ {\left( {\left( {\varvec{A} \otimes {\varvec{B}_\tau }} \right){\rm{diag}}\left( {{\varvec{J}_{{N_\theta }{N_\tau } \times {N_\tau }}}{\varvec{\beta} _\tau }} \right)} \right){{\varvec{\Sigma} }}\left( {{\rm{diag}}\left( {{\varvec{J}_{{N_\theta }{N_\tau } \times {N_\tau }}}{\varvec{\beta} _\tau }} \right)\left( {{\varvec{A}^H} \otimes \varvec{B}_\tau ^H} \right)} \right)} \right\}\\ {\rm{ = }}\varvec{\beta} _\tau ^T\varvec{J}_{{N_\theta }{N_\tau } \times {N_\tau }}^T\left( {{{\varvec{\Sigma} }} \odot {{\left( {\varvec{A} \otimes {\varvec{B}_\tau }} \right)}^H}\left( {\varvec{A} \otimes {\varvec{B}_\tau }} \right)} \right){\varvec{J}_{{N_\theta }{N_\tau } \times {N_\tau }}}{\varvec{\beta} _\tau } \end{array}$$
(44)

Note that \(\varvec{\beta }_{\tau }^{T} \varvec{Q\beta }_{\tau }\) belongs to real domain under the circumstance of a positive semi-definite matrix \({\varvec{Q}}\) and thus leads to a result \(\varvec{\beta }_{\tau }^{T} \varvec{Q\beta }_{\tau } = \Re \left\{ {\varvec{\beta }_{\tau }^{T} \varvec{Q\beta }_{\tau } } \right\} = \varvec{\beta }_{\tau }^{T} \Re \left\{ \varvec{Q} \right\}\varvec{\beta }_{\tau }\) due to the real-valued \(\varvec{\beta }_{\tau }\). Then we have the positive semi-definite matrix \(\varvec{P}_{\tau }\). As for the solution to (31), the derivations can be referred to that of (27).

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Chang, N., Wang, W., Hong, X. et al. Joint angle and delay estimation for GNSS multipath signals based on multiple sparse Bayesian Learning. GPS Solut 25, 64 (2021). https://doi.org/10.1007/s10291-020-01072-0

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