# Low Complexity Robust Data Demodulation for GNSS

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## Abstract

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## 1. Introduction

- We derive a closed-form LLR expression under AWGN channels, which could be directly applied in the following cases: (i) The codeword data are demodulated over an open sky scenario with variations of the signal-to-noise ratio (SNR). (ii) The codeword data are demodulated over an interference scenario where an additive Gaussian noise is added to the entire codeword. To this end, we reformulate the problem of obtaining the LLR values by first computing the joint pdf of symbols and estimated variance, which is then marginalized in order to compute the desired LLRs used at the decoder. To implement such marginalization in practice, we propose to impose a conjugate prior distribution that allows for an analytic closed-form approximation that enables a reduced complexity implementation when compared to the ML solution. Then, assuming statistical knowledge of the estimation error of the noise power per symbol, a closed-from LLR approximation is derived. In our approach, we consider that the noise variance ${\sigma}^{2}$ is not perfectly known, but instead, symbol-wise independent estimations of this noise variance per symbol are available. We further assume a statistical distribution of such estimated variance. In this work, we model the variance of the n-th symbol ${\sigma}_{n}^{2}$ as a random variable, which is characterized by an inverse log-normal distribution (with the aim of taking advantage of the conjugate prior distribution) whose mean and variance are estimated at the decoder.
- We derive a closed-form LLR expression that can be directly applied over a pulsed jamming scenario characterized by a small percentage of codeword symbols disrupted with extra Gaussian noise. Since the Gaussian distribution is known to not fit well the heavy tails caused by pulsed jamming [33], we propose to represent the received symbol distribution with a Laplacian distribution. Then, we compute the marginalized joint pdf of symbols and estimated variance in order to compute the desired LLRs used at the decoder. Again, we propose to compute the marginalization by imposing a conjugate prior distribution that allows for an analytic closed-form expression.

## 2. System Models and Assumptions

#### 2.1. Open Sky Communications

#### LLR Expression

#### 2.2. Communications under Gaussian Jamming

#### LLR Expression

#### 2.3. Communications under Pulsed Jamming

#### LLR Expression

## 3. Closed-Form LLR Expression with Uncertain Noise Variance

#### 3.1. Closed-Form LLR Expression with Statistical CSI on the Noise Variance

#### 3.2. Closed-Form LLR Approximation with First and Second Order Moments of the SNR

## 4. Closed-Form LLR Expression for Pulsed Jamming Scenarios

#### 4.1. Closed-Form LLR Expression under the Likelihood Laplacian Assumption with Statistical CSI on the Noise Variance

#### 4.2. Closed-Form LLR Approximation with First and Second Order Moments of the SNR

## 5. Summary of the State-of-the-Art and Proposed LLR Estimates

## 6. Results

## 7. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Derivation of the LLR Expression with Variance Uncertainty under Conjugacy

## Appendix B. Log-Normal Modeling of the Detail Estimate Considering a Gaussian Distribution

## Appendix C. Calculation of the LLR Values under Gamma Conjugacy and the Laplacian Likelihood

## Appendix D. Normal Modeling of the Detail Estimate Considering a Laplacian Distribution

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**Figure 1.**GPS L1C Subframe 2, FER considering a constant ${\sigma}_{n}^{2}$ for the entire codeword.

**Figure 2.**GPS L1C Subframe 2, FER considering a smooth variation of the variance ${\sigma}_{n}^{2}$ within the codeword.

**Table 1.**Summary of the evaluated LLR methods (the contributions in this paper appear in blue). NWPR, narrowband-wideband power ratio.

Scenarios | |||
---|---|---|---|

Type of CSI Used | Open Sky Scenario | Gaussian Jamming Scenario | Pulsed Jamming Scenario |

Perfect CSI | Equation (4) | Equation (9) | Equation (9) Approx (14) Approx. (16) |

Mismatched CSI using NWPR estimates | Equation (5) with ${\widehat{\sigma}}_{n}^{2}$ estimated with NWPR | Equation (10) with ${\widehat{\sigma}}_{{({N}_{0}+I)}_{n}}^{2}$ estimated with NWPR | Equation (10) with ${\widehat{\sigma}}_{G,n}^{2}$ estimated with NWPR |

Mismatched CSI using ML estimates | Equation (5) with ${\widehat{\sigma}}_{n}^{2}$ estimated with ML | Equation (10) with ${\widehat{\sigma}}_{{({N}_{0}+I)}_{n}}^{2}$ estimated with ML | Not evaluated |

Statistical CSI using proposed approx. and related parameter estimates | Closed-Form (41) | Closed-Form (31) | Closed-Form (31) Closed-Form (31) |

**Table 2.**GPS L1C Subframe 2. ${E}_{s}/{N}_{0}$ to obtain a frame error rate (FER) of ${10}^{-2}$ considering a constant ${\sigma}_{n}^{2}$ for the entire codeword.

CSI (4) | Closed-Form LLR (31) | LLR (5) with NWPR Estimation | LLR (5) ML and L = 10,230 | LLR (5) ML and L = 15 | LLR (5) ML and L = 7 | |
---|---|---|---|---|---|---|

${E}_{s}/{N}_{0}$ | −1.45 dB | −1.44 dB | −1.30 dB | −1.44 dB | −0.96 dB | 0 dB |

**Table 3.**GPS L1C Subframe 2. ${E}_{s}/{N}_{0}$ to obtain an FER of ${10}^{-2}$ considering a smooth variation of the variance ${\sigma}_{n}^{2}$ within the codeword.

CSI (4) | Closed-Form LLR (31) | LLR (5) with NWPR Estimation | LLR (5) ML and L = 10,230 | LLR (5) ML and L = 15 | LLR (5) ML and L = 7 | |
---|---|---|---|---|---|---|

${E}_{s}/{N}_{0}$ | −1.45 dB | −1.44 dB | −1.26 dB | −1.44 dB | −0.94 dB | 0.2 dB |

**Table 4.**GPS L1C Subframe 2. ${E}_{s}/{N}_{0}$ to obtain an FER of ${10}^{-2}$ considering a Gaussian jamming, which harms the entire codeword.

CSI (4) | Closed-Form LLR (31) | LLR (5) with NWPR Estimation | |
---|---|---|---|

${E}_{s}/{N}_{0}$ with $I=1$ dB | −0.44 dB | −0.43 dB | −0.20 dB |

${E}_{s}/{N}_{0}$ with $I=2$ dB | 0.56 dB | 0.57 dB | 0.80 dB |

${E}_{s}/{N}_{0}$ with $I=3$ dB | 1.56 dB | 1.57 dB | 1.80 dB |

${E}_{s}/{N}_{0}$ with $I=5$ dB | 3.56 dB | 3.57 dB | 3.80 dB |

**Table 5.**GPS L1C Subframe 2. ${E}_{s}/{N}_{0}$ to obtain an FER of ${10}^{-2}$ considering a pulsed jamming with $I=5$ dB.

CSI (4) | CSI (14) Gaussian Approx. | CSI (16) Laplacian Approx. | Closed-Form LLR (31) | Closed-Form LLR (41) | LLR (5) NWPR Estimation | |
---|---|---|---|---|---|---|

$P=0.02$ | −1.36 dB | −1.28 dB | −1.17 dB | −1.28 dB | −1.15 dB | 1.15 dB |

$P=0.1$ | −1 dB | −0.6 dB | −0.65 dB | −0.68 dB | −0.62 dB | −0.4 dB |

**Table 6.**GPS L1C Subframe 2. ${E}_{s}/{N}_{0}$ to obtain an FER of ${10}^{-2}$ considering a pulsed jamming with $I=10$ dB.

CSI (4) | CSI (14) Gaussian Approx. | CSI (16) Laplacian Approx. | Closed-Form LLR (31) | Closed-Form LLR (41) | LLR (5) NWPR Estimation | |
---|---|---|---|---|---|---|

$P=0.02$ | −1.32 dB | −0.5 dB | −1 dB | −1 dB | −1 dB | −0.3 dB |

$P=0.1$ | −0.8 dB | 1.95 dB | 0.23 dB | 0.63 dB | −0.21 dB | 2.5 dB |

$P=0.9$ | 8 dB | 8.17 dB | 8.3 dB | 8.17 dB | 8.33 dB | 8.45 dB |

$P=1$ | 8.56 dB | 8.56 dB | 8.72 dB | 8.57 dB | 8.73 dB | 8.77 dB |

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**MDPI and ACS Style**

Ortega, L.; Poulliat, C.; Boucheret, M.L.; Aubault Roudier, M.; Al-Bitar, H.; Closas, P.
Low Complexity Robust Data Demodulation for GNSS. *Sensors* **2021**, *21*, 1341.
https://doi.org/10.3390/s21041341

**AMA Style**

Ortega L, Poulliat C, Boucheret ML, Aubault Roudier M, Al-Bitar H, Closas P.
Low Complexity Robust Data Demodulation for GNSS. *Sensors*. 2021; 21(4):1341.
https://doi.org/10.3390/s21041341

**Chicago/Turabian Style**

Ortega, Lorenzo, Charly Poulliat, Marie Laure Boucheret, Marion Aubault Roudier, Hanaa Al-Bitar, and Pau Closas.
2021. "Low Complexity Robust Data Demodulation for GNSS" *Sensors* 21, no. 4: 1341.
https://doi.org/10.3390/s21041341