# Direction of Arrival Estimation of Coherent Sources via a Signal Space Deep Convolution Network

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## Abstract

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## 1. Introduction

- We propose a novel signal space deep convolution (SSDC) network that learns angular features of coherent signals to address the coherent DOA estimation problem.
- Since the conventional neural network frames are unable to effectively convey information in the complex domain, we divide the covariance matrix of the input signal space into real and imaginary parts and perform two-dimensional convolution operations separately, which can make the features of the input data fully utilized.
- The proposed SSDC network also considers the spatial sparsity of the array output and performs a spectral peak search on the output to determine the interested DOAs.

## 2. Problem Formulation

#### 2.1. Signal Model

#### 2.2. Signal Sparse Representation

## 3. Proposed SSDC Network

## 4. Simulation Results

#### 4.1. Experiment Setup

#### 4.2. MSE during Training and Validation

#### 4.3. Experiment with the Same Number of Coherent Sources in the Test Set As in the Training Set

#### 4.4. Experiment with a Different Number of Coherent Sources in the Test and Training Sets

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**RMSE for various SNRs for two coherent sources with DOAs $-{5.5}^{\circ}$ and ${8.5}^{\circ}$.

**Figure 6.**MSE versus different MC trials for ${\theta}_{1}$ and ${\theta}_{1}+\Delta \phi $, $T=256$, SNR = 0 dB. (

**a**) $\Delta \phi ={6}^{\circ}$. (

**b**) $\Delta \phi ={21}^{\circ}$.

Method | ASL [29] | DNN [19] | DCN [20] | SSDC | SS-MUSIC [26] | ||
---|---|---|---|---|---|---|---|

Time | |||||||

Item | |||||||

Total params | 585,276 | 32,466 | 1801 | 6774 | ∖ | ||

Train time | 1236.9480 s | 1390.2328 s | 1023.6829 s | 713.2126 s | ∖ | ||

Test time | 0.0688 s | 0.2340 s | 0.0960 s | 0.0722 s | 0.3288 s |

**Table 2.**RMSE for various SNRs for two coherent sources with DOAs $-{5.5}^{\circ}$ and ${8.5}^{\circ}$.

Snapshot | Snapshot = 200 | Snapshot = 400 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE | ||||||||||||

Method | ||||||||||||

SNR | DCN | ASL | DNN | SS-MUSIC | SSDC | DCN | ASL | DNN | SS-MUSIC | SSDC | ||

−4 | 0.5448 | 0.4530 | 1.1791 | 1.5217 | 0.4204 | 0.5462 | 0.4145 | 0.9382 | 0.8892 | 0.3695 | ||

−2 | 0.5317 | 0.4142 | 0.9368 | 0.8740 | 0.3809 | 0.5288 | 0.4074 | 0.8720 | 0.6553 | 0.3428 | ||

0 | 0.5165 | 0.4052 | 0.8745 | 0.6542 | 0.3457 | 0.5099 | 0.4052 | 0.8142 | 0.5423 | 0.3092 | ||

2 | 0.5125 | 0.4026 | 0.8328 | 0.5332 | 0.3188 | 0.4828 | 0.4020 | 0.7721 | 0.5049 | 0.2795 | ||

4 | 0.4924 | 0.4004 | 0.7884 | 0.5280 | 0.2894 | 0.4486 | 0.4019 | 0.7571 | 0.5020 | 0.2621 | ||

6 | 0.4568 | 0.3968 | 0.7605 | 0.5190 | 0.2624 | 0.4104 | 0.4002 | 0.7538 | 0.5000 | 0.2323 | ||

8 | 0.4319 | 0.3967 | 0.7523 | 0.5125 | 0.2446 | 0.3683 | 0.3995 | 0.7522 | 0.5000 | 0.2124 | ||

10 | 0.4136 | 0.3940 | 0.7522 | 0.5000 | 0.2225 | 0.3571 | 0.3969 | 0.7521 | 0.5000 | 0.2065 | ||

12 | 0.4024 | 0.3938 | 0.7519 | 0.5000 | 0.2174 | 0.3414 | 0.3948 | 0.7518 | 0.5000 | 0.2009 |

Snapshot | Snapshot = 200 | Snapshot = 400 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

RMSE | ||||||||||

Method | ||||||||||

SNR | DCN | ASL | DNN | SSDC | DCN | ASL | DNN | SSDC | ||

−14 | 9.8974 | 10.3946 | 10.0370 | 9.6787 | 9.0068 | 8.3523 | 9.3075 | 7.6015 | ||

−12 | 9.1955 | 8.2625 | 9.4923 | 7.6144 | 9.1493 | 6.3520 | 8.5037 | 4.7888 | ||

−10 | 9.2857 | 5.9264 | 8.1795 | 4.3582 | 9.4124 | 4.3264 | 7.9674 | 3.9796 | ||

−8 | 7.6149 | 4.18250 | 7.8453 | 3.3594 | 7.9726 | 3.4661 | 7.1206 | 2.7717 | ||

−6 | 5.4049 | 3.1088 | 6.8769 | 2.3168 | 5.4628 | 2.2495 | 6.5506 | 1.8838 | ||

−4 | 4.2463 | 2.2809 | 6.3749 | 1.8961 | 4.3471 | 1.7000 | 5.7563 | 1.5841 | ||

−2 | 3.3147 | 1.8345 | 5.5767 | 1.1746 | 3.2389 | 1.6672 | 4.7569 | 1.3516 | ||

0 | 2.5506 | 1.6609 | 4.9036 | 1.1926 | 2.2688 | 1.6304 | 3.3722 | 1.2355 | ||

2 | 2.1431 | 1.6710 | 3.7398 | 1.2104 | 2.0703 | 1.6321 | 2.2262 | 1.2239 |

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**MDPI and ACS Style**

Zhao, J.; Gui, R.; Dong, X.; Zhao, Y.
Direction of Arrival Estimation of Coherent Sources via a Signal Space Deep Convolution Network. *Symmetry* **2024**, *16*, 433.
https://doi.org/10.3390/sym16040433

**AMA Style**

Zhao J, Gui R, Dong X, Zhao Y.
Direction of Arrival Estimation of Coherent Sources via a Signal Space Deep Convolution Network. *Symmetry*. 2024; 16(4):433.
https://doi.org/10.3390/sym16040433

**Chicago/Turabian Style**

Zhao, Jun, Renzhou Gui, Xudong Dong, and Yufei Zhao.
2024. "Direction of Arrival Estimation of Coherent Sources via a Signal Space Deep Convolution Network" *Symmetry* 16, no. 4: 433.
https://doi.org/10.3390/sym16040433