# Combining DCQGMP-Based Sparse Decomposition and MPDR Beamformer for Multi-Type Interferences Mitigation for GNSS Receivers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Signal Model

**a**, representing the steering vector of the GNSS signal whose dimension is $N\times 1$, can be written as:

## 3. The Proposed Method

#### 3.1. Multi-Channel Signal Interference Suppression Method Based on Improved DCQGAMP

#### 3.1.1. Matching Pursuit Decomposition

- Initialization: $l=0$; ${f}_{{x}_{l}}={f}_{x}$; $B=\u2300$.
- Compute the inner products $Z\left(\right)open="("\; close=")">{g}_{r},{f}_{{x}_{l}}$ for all atoms; find the maximum amongst all inner products: ${r}_{n}=argma{x}_{{g}_{r}\in D}\left(\right)open="("\; close=")">Z\left(\right)open="("\; close=")">{g}_{r},{f}_{{x}_{l}}$; and add ${b}_{n}=\left(\right)open="("\; close=")">{z}_{l},{g}_{{r}_{l}}$ to codebook B.
- Compute the residual signal: ${f}_{{x}_{l+1}}={f}_{{x}_{l}}-Z\left(\right)open="("\; close=")">{g}_{{r}_{l}},{f}_{{x}_{l}}$; if precision is reached, then stop; otherwise $l=l+1$, and iterate to Step 2.

#### 3.1.2. Analysis of Interference Detection Performance

**ℵ**can be regarded as a Gaussian noise with zero mean and $\sigma $ variance.

**ℵ**is the Gaussian noise with zero mean and $\sigma $ variance, then ${\mathbf{\Re}}_{n}$ is the Gaussian noise with zero mean and $M{c}_{r}^{2}\sigma $ variance. Additionally, about 99.997 percent of the values drawn from a normal distribution are within four standard deviations ${c}_{r}\sqrt{M\sigma}$ away from the mean. Therefore, the necessary condition for detecting CWI signals is that the number of samples M should satisfy:

#### 3.1.3. The Mutual Influence of Multiple Signals

- When $fr{e}_{1}=fr{e}_{2}$, the two interfering signals can be treated as one interference, and the reason is:$$\begin{array}{c}\hfill \begin{array}{c}{\mathbf{J}}_{cw\_r1}\left(m\right)+{\mathbf{J}}_{cw\_r2}\left(m\right)=\sqrt{{p}_{1}}\mathrm{cos}\left(\right)open="("\; close=")">fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}+{\phi}_{1}\hfill & +\sqrt{{p}_{2}}\mathrm{cos}\left(\right)open="("\; close=")">fr{e}_{2}\left(\right)open="("\; close=")">m{T}_{s}\\ +{\phi}_{2}\end{array}\phantom{\rule{6.0pt}{0ex}}\end{array}$$
- When $fr{e}_{1}\ne fr{e}_{2}$, the best atom corresponding to Interfering Signal 1 should be ${g}_{{r}_{1}}\left(m\right)={c}_{{r}_{1}}\mathrm{cos}\left(\right)open="("\; close=")">fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$, then the influence of Interfering Signal 2 on Interfering Signal 1 can be expressed as:$$\begin{array}{c}\hfill \begin{array}{c}\kappa =\left(\right)open="\langle "\; close="\rangle ">{g}_{{r}_{1}},{\mathbf{J}}_{cw\_r2}={\displaystyle \sum _{m=0}^{M-1}}{c}_{{r}_{1}}\sqrt{{p}_{2}}\mathrm{cos}\left(\right)open="("\; close=")">fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}\hfill & +{\phi}_{1}\\ \mathrm{cos}\left(\right)open="("\; close=")">fr{e}_{2}\left(\right)open="("\; close=")">m{T}_{s}\end{array}+{\phi}_{2}\end{array}=\frac{1}{2}{c}_{{r}_{1}}\sqrt{{p}_{2}}{\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}+fr{e}_{2}\left(\right)open="("\; close=")">m{T}_{s}\hfill & +{\phi}_{1}+{\phi}_{2}$$According to the types of the interfering signals, $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$ and $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$ can be treated as a quadratic function or a linear function of the sampling time.
- When $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$ and $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$ are the quadratic function of the sampling time, let $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}={\alpha}_{1}{m}^{2}+{\beta}_{1}m$ and $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}={\alpha}_{2}{m}^{2}+{\beta}_{2}m$, where $\left(\right)open="|"\; close="|">{\alpha}_{1}$. Then:$$\begin{array}{c}\hfill \begin{array}{c}{\kappa}_{1}={\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">{\alpha}_{1}{m}^{2}+{\beta}_{1}m+{\phi}_{1}+{\phi}_{2}={\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">{\alpha}_{1}{\left(\right)}^{m}2\hfill & -\frac{{\beta}_{1}^{2}}{4{\alpha}_{1}}+{\phi}_{1}+{\phi}_{2}\\ \phantom{\rule{6.0pt}{0ex}}\end{array}\\ =\mathrm{cos}\left(\right)open="("\; close=")">{\phi}_{1}+{\phi}_{2}-\frac{{\beta}_{1}^{2}}{4{\alpha}_{1}}{\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">{\alpha}_{1}{\left(\right)}^{m}2\hfill \\ \phantom{\rule{6.0pt}{0ex}}\end{array}$$Similarly,$$\begin{array}{c}\hfill \begin{array}{c}{\kappa}_{2}=\mathrm{cos}\left(\right)open="("\; close=")">{\phi}_{1}-{\phi}_{2}-\frac{{\beta}_{2}^{2}}{4{\alpha}_{2}}{\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">{\alpha}_{1}{\left(\right)}^{m}2\hfill \\ \phantom{\rule{6.0pt}{0ex}}\end{array}\\ -\mathrm{sin}\left(\right)open="("\; close=")">{\phi}_{1}-{\phi}_{2}-\frac{{\beta}_{2}^{2}}{4{\alpha}_{2}}{\displaystyle \sum _{m=0}^{M-1}}\mathrm{sin}\left(\right)open="("\; close=")">{\alpha}_{1}{\left(\right)}^{m}2\hfill \end{array}$$
- When $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$ and $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}$ are the linear function of the sampling time, let $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}={\beta}_{3}m$ and $fr{e}_{1}\left(\right)open="("\; close=")">m{T}_{s}={\beta}_{4}m$, where $\left(\right)open="|"\; close="|">{\beta}_{3}$. Then:$$\begin{array}{c}\hfill \begin{array}{c}{\kappa}_{1}={\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">{\beta}_{3}m+{\phi}_{1}+{\phi}_{2}\hfill \end{array}=\mathrm{cos}\left(\right)open="("\; close=")">{\phi}_{1}+{\phi}_{2}{\displaystyle \sum _{m=0}^{M-1}}\mathrm{cos}\left(\right)open="("\; close=")">{\beta}_{3}m\hfill & -sin\left(\right)open="("\; close=")">{\phi}_{1}+{\phi}_{2}\\ {\displaystyle \sum _{m=0}^{M-1}}\mathrm{sin}\left(\right)open="("\; close=")">{\beta}_{3}m\end{array}=\mathrm{cos}\left(\right)open="("\; close=")">{\phi}_{1}+{\phi}_{2}\frac{\mathrm{sin}\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">M+\frac{1}{2}}{{\beta}_{3}}-\mathrm{sin}\left(\right)open="("\; close=")">\frac{{\beta}_{3}}{2}\hfill \\ 2\mathrm{sin}\left(\right)open="("\; close=")">\frac{{\beta}_{3}}{2}$$Similarly,$$\begin{array}{c}\hfill \begin{array}{c}{\kappa}_{2}=\mathrm{cos}\left(\right)open="("\; close=")">{\phi}_{1}-{\phi}_{2}\frac{\mathrm{sin}\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">M+\frac{1}{2}}{{\beta}_{4}}-\mathrm{sin}\left(\right)open="("\; close=")">\frac{{\beta}_{4}}{2}\hfill \\ 2\mathrm{sin}\left(\right)open="("\; close=")">\frac{{\beta}_{4}}{2}\end{array}& -\mathrm{sin}\left(\right)open="("\; close=")">{\phi}_{1}-{\phi}_{2}\\ \frac{-\mathrm{cos}\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">M+\frac{1}{2}}{{\beta}_{4}}+\mathrm{cos}\left(\right)open="("\; close=")">\frac{{\beta}_{4}}{2}\end{array}2\mathrm{sin}\left(\right)open="("\; close=")">\frac{{\beta}_{4}}{2}$$

#### 3.1.4. The Improved MP Algorithm and Design Strategy of the Over-Complete Dictionary

#### 3.1.5. The Terminate Condition of MP

#### 3.1.6. The Improved Double Chain Quantum Genetic Algorithm

#### 3.1.7. The Multi-Channel Signal Interference Suppression Method Based on Improved DCQGMP

#### 3.2. MPDR Beamformer

**w**represents the array weight vector;

**a**is defined by Equation (2); $\mathbf{R}$ denotes the spatial covariance matrix of the residual signals obtained by Stage 1, which can be expressed by:

## 4. Simulation Results and Analysis

#### 4.1. Performance of the Improved DCQGA

#### 4.2. Performance of the Proposed Interference Mitigation Method

#### 4.2.1. Influence of the DCQGMP-Based Interference Suppression on the GNSS Signal

#### 4.2.2. Performance of the Multi-Channel Signal Interference Suppression Method Based on Improved DCQGMP

#### 4.2.3. Performance of the Cascade Method for Multi-Type Interferences Mitigation

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

DCQGMP | Double chain quantum genetic matching pursuit |

MPDR | Minimum power distortionless response |

GNSS | Global navigation satellite system |

MP | Matching pursuit |

CWI | Continuous wave interference |

DCQGA | Double chain quantum genetic algorithm |

DoF | Degree of freedom |

MVDR | Minimum variance distortionless response |

STAP | Spatial-temporal adaptive processing |

DOA | Direction of arrival |

NMSE | Normalized mean square error |

S-MPDR | Space-only minimum power distortionless response |

ST-MPDR | Space-time minimum power distortionless response |

DST-MPDR | Distortionless space-time minimum power distortionless response |

NC | Normalized correlation |

## References

- Gao, G.X.; Sgammini, M.; Lu, M. Protecting GNSS Receivers from Jamming and Interference. Proc. IEEE
**2016**, 104, 1327–1338. [Google Scholar] [CrossRef] - Broumandan, A.; Jafarnia-Jahromi, A.; Daneshmand, S. Overview of Spatial Processing Approaches for GNSS Structural Interference Detection and Mitigation. Proc. IEEE
**2016**, 104, 1246–1257. [Google Scholar] [CrossRef] - Mosavi, M.R.; Pashaian, M.; Rezaei, M.J. Jamming mitigation in global positioning system receivers using wavelet packet coefficients thresholding. IET Signal Process.
**2015**, 9, 457–464. [Google Scholar] [CrossRef] - Chien, Y.R. Design of GPS Anti-Jamming Systems Using Adaptive Notch Filters. IEEE Syst. J.
**2015**, 9, 451–460. [Google Scholar] [CrossRef] - Rezaei, M.J.; Abedi, M.; Mosavi, M.R. New GPS anti-jamming system based on multiple short-time Fourier transform. IET Radar Sonar Navig.
**2016**, 10, 807–815. [Google Scholar] [CrossRef] - Mosavi, M.R.; Shafiee, F. Narrowband interference suppression for GPS navigation using neural networks. GPS Solut.
**2016**, 20, 341–351. [Google Scholar] [CrossRef] - Chien, Y.R.; Chen, P.Y.; Fang, S.H. Novel Anti-Jamming Algorithm for GNSS Receivers Using Wavelet-Packet-Transform-Based Adaptive Predictors. IEICE Trans. Fundam. Electron. Commun. Comput. Sci.
**2017**, E100-A, 602–610. [Google Scholar] [CrossRef] - Van, T.; Harry, L. Optimum Array Processing, Detection, Estimation, and Modulation Theory Part IV; John Wiley & Sons: New York, NY, USA, 2002; pp. 428–699. [Google Scholar]
- Fernandez-Prades, C.; Arribas, J.; Closas, P. Robust GNSS Receivers by Array Signal Processing: Theory and Implementation. Proc. IEEE
**2016**, 104, 1207–1220. [Google Scholar] [CrossRef] - Zoltowski, M.D.; Gecan, A.S. Advanced adaptive null steering concepts for GPS. In Proceedings of the 1995 IEEE Conference Record, Military Communications Conference (MILCOM’95), San Diego, CA, USA, 5–8 November 1995; pp. 1214–1218. [Google Scholar]
- Fante, R.L.; Vaccaro, J.J. Wideband cancellation of interference in a GPS receive array. IEEE Trans. Aerosp. Electron. Syst.
**2000**, 36, 549–564. [Google Scholar] [CrossRef] - Daneshmand, S.; Jafarnia-Jahromi, A.; Broumandan, A.; Lachapelle, G. GNSS space-time interference mitigation: Advantages and challenges. In Proceedings of the International Symposium on GNSS (IS-GNSS’15), Kyoto, Japan, 16–19 November 2015. [Google Scholar]
- O’Brien, A.J.; Gupta, I.J. Mitigation of Adaptive Antenna Induced Bias Errors in GNSS Receivers. IEEE Trans. Aerosp. Electron. Syst.
**2011**, 47, 524–538. [Google Scholar] [CrossRef] - Church, C.; Gupta, A.I. Adaptive Antenna Induced Biases in GNSS Receivers. In Proceedings of the 63rd Annual Meeting of The Institute of Navigation, Cambridge, MA, USA, 23–25 April 2007; pp. 204–212. [Google Scholar]
- Li, S.; Zhu, C.; Kan, H. A Compensating Approch for Signal Distortion Introduced by STAP. In Proceedings of the IEEE International Conference on Communication Technology, Guilin, China, 27–30 November 2006; pp. 1–4. [Google Scholar]
- Daneshmand, S.; Jahromi, A.J.; Broumandan, A. GNSS Space-Time Interference Mitigation and Attitude Determination in the Presence of Interference Signals. Sensors
**2015**, 15, 12180–12204. [Google Scholar] [CrossRef] [PubMed] - Chen, F.; Nie, J.; Li, B. Distortionless space-time adaptive processor for global navigation satellite system receiver. Electron. Lett.
**2015**, 51, 2138–2139. [Google Scholar] [CrossRef] - Zhou, Z.; Lu, S.J.; Zhang, E.Y. Interference Excision of GPS Received Signal in Complex Environment. Adv. Mater. Res.
**2013**, 760–762, 350–354. [Google Scholar] [CrossRef] - Xu, J.; Yao, R.; Chen, Y. Cascaded Frequency and Spatial-time Domain Anti-jamming Technique in Navigation Systems. J. Proj. Rocket. Missiles Guid.
**2015**, 2, 137–140. [Google Scholar] - Elad, M.; Elad, M. Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing; Springer: New York, NY, USA, 2010; pp. 1094–1097. [Google Scholar]
- Shao, X.; Gui, W.; Xu, C. Note onset detection based on sparse decomposition. Multimed. Tools Appl.
**2016**, 75, 2613–2631. [Google Scholar] [CrossRef] - Feng, W.; Zhang, Y.; He, X.; Guo, Y. Cascaded clutter and jamming suppression method using sparse representation. Electron. Lett.
**2015**, 51, 1524–1526. [Google Scholar] [CrossRef] - Li, P.C.; Song, K.P.; Shang, F.H. Double chains quantum genetic algorithm with application to neuro-fuzzy controller design. Adv. Eng. Softw.
**2011**, 42, 875–886. [Google Scholar] [CrossRef] - Sun, K.; Jin, T.; Yang, D. An improved time-frequency analysis method in interference detection for GNSS receivers. Sensors
**2014**, 15, 9404–9426. [Google Scholar] [CrossRef] [PubMed] - Kong, H.; Ni, L.; Shen, Y. Adaptive double chain quantum genetic algorithm for constrained optimization problems. Chin. J. Aeronaut.
**2015**, 28, 214–228. [Google Scholar] [CrossRef] - Chen, P.; Yuan, L.; He, Y. An improved SVM classifier based on double chains quantum genetic algorithm and its application in analogue circuit diagnosis. Neurocomputing
**2016**, 211, 202–211. [Google Scholar] [CrossRef] - Guo, Q.; Sun, Y.X. Improved quantum genetic algorithm with double chains in image denoising. J. Harbin Inst. Technol.
**2016**, 48, 140–147. [Google Scholar]

**Figure 4.**Frame of the multi-channel signals interference mitigation method based on improved DCQGMP.

Algorithm | Best Result | Worst Result | Average Result | Convergence Times | Average Time (s) |
---|---|---|---|---|---|

Improved DCQGA | 0.99998 | 0.99028 | 0.995266 | 14 | 0.07731 |

F-DCQGA | 0.99788 | 0.99013 | 0.991806 | 7 | 0.07753 |

DCQGA | 0.99547 | 0.99016 | 0.990603 | 2 | 0.07821 |

Name | The Number of Inner Product for Single Channel Signal | Total Number of Inner Product |
---|---|---|

Conventional MP | $8.25\times {10}^{10}$ | $4.125\times {10}^{11}$ |

Proposed DCQGMP | $2.64\times {10}^{6}$ | $2.64\times {10}^{6}+44$ |

Name | Type of Interference | Center Frequency (MHz) | Bandwidth (MHz) | DOA (°) | Interference to Noise Ratio (dB) | Others |
---|---|---|---|---|---|---|

1 | Narrowband | 2.046 | 0 | 60 | 25 | / |

2 | Narrowband | 1.962 | 0 | 130 | 25 | / |

3 | Narrowband | 2.08 | 0 | 80 | 25 | / |

4 | Linear chirp | 2.046 | 2 | 120 | 32 | ${T}_{LF}=0.058$ ms ${t}_{0}=0.0012$ ms |

5 | Linear chirp | 2.046 | 2 | 80 | 32 | ${T}_{LF}=0.092$ ms ${t}_{0}=0.0095$ ms |

6 | Wideband Gaussian | 2.046 | 2 | 20 | 32 | / |

7 | Wideband Gaussian | 2.046 | 2 | 35 | 32 | / |

8 | Wideband Gaussian | 2.046 | 2 | 165 | 32 | / |

9 | Wideband Gaussian | 2.046 | 2 | 135 | 32 | / |

Iteration Number | 1 | 2 | 3 | 4 |
---|---|---|---|---|

$\rho $ | 73 | 33.2 | 35.5 | 3.14 |

Whether to terminate | No | No | No | Yes |

Name | Channel 1 | Channel 2 | Channel 3 | Channel 4 | Channel 5 |
---|---|---|---|---|---|

CW interference | 0.0216 | 0.0190 | 0.0157 | 0.0235 | 0.0247 |

Residual signal | 0.0182 | 0.0185 | 0.0132 | 0.0197 | 0.0207 |

Name | S-MPDR | ST-MPDR | DST-MPDR | The Proposed |
---|---|---|---|---|

Acquisition factor | 3.2 | 3.6 | 3.2 | 4.7 |

Peak’s position (code delay (chips), Doppler (Hz)) | (0, 2000) | (0, 2000) | (1.875, 2000) | (0, 2000) |

Name | S-MPDR | ST-MPDR | DST-MPDR | The Proposed |
---|---|---|---|---|

Acquisition factor | 1 | 1.26 | 2.0 | 4.0 |

Peak’s position (code delay (chips), Doppler (Hz)) | / | (0, 2000) | (1.875, 2000) | (0, 2000) |

Name | S-MPDR | ST-MPDR | DST-MPDR | The Proposed |
---|---|---|---|---|

Acquisition factor | 1 | 1 | 1 | 2.4 |

Peak’s position (code delay (chips), Doppler (Hz)) | / | / | / | (0, 2000) |

Name | S-MPDR | ST-MPDR | DST-MPDR | The Proposed |
---|---|---|---|---|

Acquisition factor | 1 | 1 | 1 | 2.5 |

Peak’s position (code delay (chips), Doppler (Hz)) | / | / | / | (0, 2000) |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Guo, Q.; Qi, L.
Combining DCQGMP-Based Sparse Decomposition and MPDR Beamformer for Multi-Type Interferences Mitigation for GNSS Receivers. *Sensors* **2017**, *17*, 813.
https://doi.org/10.3390/s17040813

**AMA Style**

Guo Q, Qi L.
Combining DCQGMP-Based Sparse Decomposition and MPDR Beamformer for Multi-Type Interferences Mitigation for GNSS Receivers. *Sensors*. 2017; 17(4):813.
https://doi.org/10.3390/s17040813

**Chicago/Turabian Style**

Guo, Qiang, and Liangang Qi.
2017. "Combining DCQGMP-Based Sparse Decomposition and MPDR Beamformer for Multi-Type Interferences Mitigation for GNSS Receivers" *Sensors* 17, no. 4: 813.
https://doi.org/10.3390/s17040813