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Robust time-hopping pseudolite signal acquisition method based on dynamic Bayesian network

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Abstract

The time-hopping direct sequence spread spectrum (TH-DSSS) signal has been widely used in Pseudolites Positioning Systems to overcome the near-far problem. To capture the TH-DSSS signal, an additional parameter representing the time-hopping (TH) rules should be estimated in addition to the PRN code phase and carrier Doppler. However, the techniques of estimating a TH parameter in existing TH-DSSS signal acquisition methods have significant issues in poor signal quality environments. Here, we propose a robust and general TH-DSSS signal acquisition method to reduce the impact of signal degradation. In this method, we first capture every short pulse to obtain the code phase and carrier Doppler. After sufficient successful pulse acquisitions, we model the process of TH parameter acquisition as a dynamic Bayesian network. The so-called state confidence that describes the probability of each candidate TH parameter is then introduced to infer the real TH parameter. Finally, this method has been seen, both theoretically and experimentally, to be both general and effective to compensate for harsh signal environments. Simulation results show that compared with baseline algorithms, this method provides a significant improvement in detection probability and considerable reduction in acquisition time.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (NSFC), under Grant 61771272.

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Correspondence to Zheng Yao.

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Appendix: derivation of confidence

Appendix: derivation of confidence

To calculate the three probabilities on the right side of (12), knowledge of DBN and TH pattern is needed. Simplifying the first term \(\Pr \left( {z_{n} |x_{i}^{\left( n \right)} = 1,z_{0} ,z_{1} , \cdots ,z_{{n{ - }1}} } \right)\) with the Markov property of DBN that the measurement at time \(n\) is conditionally independent of previous measurements gives

$$\Pr \left( {z_{n} |x_{i}^{\left( n \right)} = 1,z_{0} ,z_{1} , \cdots ,z_{{n{ - }1}} } \right) = \Pr \left( {z_{n} |x_{i}^{\left( n \right)} = 1} \right)$$
(28)

Furthermore, the conditioned Bayesian rule is applied to (28) by inserting a binary variable denoted as \(h_{i}^{\left( n \right)}\), which represents whether the timeslot at time \(n\) is active or not given that \({\text{IFI}} = i\) and is known for a pre-designed TH pattern sequence. Then, (28) becomes

$$\begin{gathered} \Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1} \right) \hfill \\ = \Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 1} \right)\Pr \left( {h_{i}^{\left( n \right)} = 1|x_{i}^{\left( n \right)} = 1} \right) \hfill \\ + \Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 0} \right)\Pr \left( {h_{i}^{\left( n \right)} = 0|x_{i}^{\left( n \right)} = 1} \right) \hfill \\ { = }\Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 1} \right)h_{i}^{\left( n \right)} \hfill \\ + \Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 0} \right)\left( {1 - h_{i}^{\left( n \right)} } \right) \hfill \\ \end{gathered}$$
(29)

and

$$\begin{gathered} \Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1} \right) \hfill \\ = \Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 1} \right)\Pr \left( {h_{i}^{\left( n \right)} = 1|x_{i}^{\left( n \right)} = 1} \right) \hfill \\ + \Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 0} \right)\Pr \left( {h_{i}^{\left( n \right)} = 0|x_{i}^{\left( n \right)} = 1} \right) \hfill \\ { = }\Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 1} \right)h_{i}^{\left( n \right)} \hfill \\ + \Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 0} \right)\left( {1 - h_{i}^{\left( n \right)} } \right) \hfill \\ \end{gathered}$$
(30)

Simplifying (29) and (30) by the following definitions

$$\Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 1} \right) = P_{d}$$
(31)
$$\Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 0} \right) = P_{fa}$$
(32)
$$\Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 1} \right) = 1{ - }P_{d}$$
(33)
$$\Pr \left( {z_{n} = 0|x_{i}^{\left( n \right)} = 1,h_{i}^{\left( n \right)} = 0} \right) = 1{ - }P_{fa}$$
(34)

we write the likelihood probability in (28) with a simpler form as

$$\Pr \left( {z_{n} = 1|x_{i}^{\left( n \right)} = 1} \right) = P_{d} h_{i}^{\left( n \right)} + P_{fa} \left( {1 - h_{i}^{\left( n \right)} } \right)$$
(35)
$$\Pr (z_{n} = 0|x_{i}^{\left( n \right)} = 1) = (1 - P_{d} )h_{i}^{\left( n \right)} + (1 - P_{fa} )(1 - h_{i}^{\left( n \right)} )$$
(36)

Note that the DBN is a cause–effect system which means that given the measurements before the \(n^{{{\text{th}}}}\) epoch, only the state at time \(n - 1\) is available but the forward state keeps unknown. In other words, given \(z_{n}\) is unknown, state at time n keeps the same as that at time \(n - 1\). So, the second term reduces to

$$\begin{gathered} \, \Pr (x_{i}^{\left( n \right)} = 1|z_{0} ,z_{1} , \cdots ,z_{n - 1} ) \hfill \\ = \Pr (x_{i}^{{\left( {n - 1} \right)}} = 1|z_{0} ,z_{1} , \cdots ,z_{n - 1} ) \triangleq p_{i}^{{\left( {n - 1} \right)}} \hfill \\ \end{gathered}$$
(37)

Lastly, the denominator of (12) is a normalized factor calculated by

$$\begin{gathered} \eta^{\left( n \right)} = \Pr \left( {z_{n} |z_{0} ,z_{1} , \cdots ,z_{n - 1} } \right) \\ = \sum\limits_{i} {\Pr \left( {z_{n} |x_{i}^{\left( n \right)} = 1} \right)p_{i}^{{\left( {n - 1} \right)}} } \\ \end{gathered}$$
(38)

Herein, inserting (35)–(38) back into (12), we obtain the final expression of confidence in (13).

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Liu, X., Yao, Z. & Lu, M. Robust time-hopping pseudolite signal acquisition method based on dynamic Bayesian network. GPS Solut 25, 38 (2021). https://doi.org/10.1007/s10291-020-01066-y

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