# Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss

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## Abstract

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## 1. Introduction

- A double-layer compressive sensing framework is proposed to eliminate the adverse effects of block data loss on DOA estimation in WSAN. Specifically, we model the random packet loss during transmission as a passive CS process and introduce an active CS process at each array sensor to address the block data loss problem.
- We present the mutual coherence of the equivalent measurement matrix and the absolute off-diagonal entries’ distribution of the corresponding Gram matrix under the double-layer CS framework, which account for the satisfactory DOA estimation performance.
- A direct DOA estimation technique (DCS-DDOA) is proposed under the double-layer CS framework to avoid the error propagation problem in DCS-DOA. A joint frequency and spatial domain sparse representation of the sensor array data is constructed and exploited to directly perform DOA estimation at the FC.

## 2. Background

#### 2.1. Compressive Sensing

#### 2.2. DOA Estimation

#### 2.3. Lossy Wireless Links

#### 2.4. Block Data Loss

## 3. Double-Layer CS Framework-Based DOA Estimation

#### 3.1. Double-Layer CS Framework

#### 3.2. DOA Estimation by DCS-DOA

#### 3.2.1. Signal Recovery

#### 3.2.2. DOA Estimation

#### 3.3. Improved DOA Estimation by DCS-DDOA

#### 3.3.1. Joint Sparse Representation

#### 3.3.2. Direct DOA Estimation

## 4. Experimental Results

#### 4.1. Simulation Settings

#### 4.2. Performance Analysis for DCS-DOA

#### 4.3. Performance Analysis for DCS-DDOA

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Packet reception rate (PRR) and normalized transmission efficiency E versus payload length under different BER.

**Figure 4.**Histograms of the occurrence number for different sizes of block data loss under different packet sizes.

**Figure 5.**Schematic diagram of the double-layer compressive sensing (CS) framework and the DOA estimation process of DCS-DOA and DCS-direct DOA (DDOA).

**Figure 6.**Examples of the passive CS measurement matrix under different packet sizes, with blue grids denoting one and blank grids denoting zero.

**Figure 7.**Histograms of the absolute off-diagonal elements of the corresponding Gram matrix to the equivalent measurement matrices under the single-layer CS and double-layer CS framework when N = 512 and M = 64.

**Figure 8.**The active CS process at each array sensor using a projection matrix constructed from a permutation or Gaussian matrix.

**Figure 9.**The signal recovery error, DOA estimation RMSE and detection frequency versus the number of received data samples M under different packet sizes when projection (active CS) is not introduced at each array sensor.

**Figure 13.**The comparison of detection frequency between CS-DOA and DCS-DOA under different packet sizes.

**Figure 16.**The comparison of DOA estimation RMSE among CS-DOA, DCS-DOA, DCS-DDOA under different packet sizes.

**Figure 17.**The comparison of detection frequency among CS-DOA, DCS-DOA, DCS-DDOA under different packet sizes.

**Figure 18.**The comparison of DOA estimation RMSE and detection frequency between DCS-DOA and DCS-DDOA under a varying SNR with the packet size being 64 and M being 64.

Notations | Explanations |
---|---|

${\mathbf{x}}_{h}$ | raw data samples at the h-th array sensor |

$\mathsf{\Psi}$ | Fourier sparsifying dictionary |

${\mathsf{\alpha}}_{h}$ | sparse representation of ${\mathbf{x}}_{h}$ under the basis $\mathsf{\Psi}$ |

${\mathsf{\Phi}}_{s}$ | the active CS projection matrix adopted at each array sensor |

${\widehat{\mathbf{x}}}_{h}$ | newly-generated data samples after a projection on ${\mathbf{x}}_{h}$ |

${\mathsf{\Phi}}_{r}$ | the passive CS measurement matrix modeling the packet loss |

${\mathbf{y}}_{h}$ | received data samples at the FC from the h-th array sensor |

${\tilde{\mathbf{x}}}_{h}$ | recovered data samples for the h-th array sensor at the FC |

**Table 2.**The mutual coherence of the equivalent measurement matrix under the single-layer CS and double-layer CS framework when N = 512 and M = 64.

Single-Layer CS | Double-Layer CS | |||
---|---|---|---|---|

Packet Size | $\mathit{\mu}\left(\mathbf{A}\right)$ | ${\mathit{\mu}}_{\mathit{av}}\left(\mathbf{A}\right)$ | $\mathit{\mu}\left(\mathbf{A}\right)$ | ${\mathit{\mu}}_{\mathit{av}}\left(\mathbf{A}\right)$ |

1 | 0.2859 | 0.2297 | 0.2825 | 0.2306 |

8 | 0.5911 | 0.3328 | 0.2905 | 0.2269 |

16 | 0.7489 | 0.4077 | 0.2920 | 0.2311 |

32 | 0.8866 | 0.5118 | 0.2927 | 0.2296 |

64 | 0.9745 | 0.5616 | 0.2929 | 0.2308 |

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**MDPI and ACS Style**

Sun, P.; Wu, L.; Yu, K.; Shao, H.; Wang, Z.
Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss. *Sensors* **2017**, *17*, 1688.
https://doi.org/10.3390/s17071688

**AMA Style**

Sun P, Wu L, Yu K, Shao H, Wang Z.
Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss. *Sensors*. 2017; 17(7):1688.
https://doi.org/10.3390/s17071688

**Chicago/Turabian Style**

Sun, Peng, Liantao Wu, Kai Yu, Huajie Shao, and Zhi Wang.
2017. "Double-Layer Compressive Sensing Based Efficient DOA Estimation in WSAN with Block Data Loss" *Sensors* 17, no. 7: 1688.
https://doi.org/10.3390/s17071688