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摘要:
电路系统测试响应信号具有周期性强、分布较稀疏的特点，针对电路系统测试响应信号的压缩重构问题进行了研究，提出了基于梯度方向追踪的K奇异值分解（GP-KSVD）稀疏重构算法。结合单一响应信号以及混合信号其自身特点进行字典训练，利用更新后字典对含噪信号进行梯度追踪稀疏表征，通过对含噪信号的重构，实现了去噪的目的，算法计算复杂度低，储存量小，具有较好的重构效果。仿真中将GP-KSVD表征与使用随机字典、离散余弦字典（DCT）的表征进行比较，从信噪比（SNR）以及相对均方误差（RMSE）2项指标中得出使用KSVD字典具有更好的重构去噪效果；此外将GP-KSVD稀疏重构算法与正交匹配追踪正交匹配追踪（OMP）-KSVD、预处理共轭梯度追踪（PCGP）算法进行比较，得出GP-KSVD的计算时间最短、重构精度更高的结论，并且进行了实测验证。算法可用来对测试响应信号进行预处理，为电路系统设备性能的评估分析提供了理论依据。
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关键词:
- 压缩感知 /
- GP-KSVD /
- 稀疏表征 /
- 电路测试响应信号重构 /
- 去噪
Abstract:Response signals in circuitry system always have the characteristics of high periodicity and sparse distribution. In order to realize response signals reconstruction in circuitry system, an algorithm combining gradient pursuit and K singular value decomposition (GP-KSVD) was proposed. Dictionary was trained according to the features of single and mixed signal. Making use of the updated dictionary and gradient pursuit to sparse representation on noisy signal, the reconstruction achieves the aim of de-noising. The algorithm has excellent reconstruction results with low computing complexity and storage capacity. In simulation, GP-KSVD dictionary was compared with both random and discrete cosine dictionary (DCT) dictionary, and the results show that the denoising effect of sparse representation with KSVD dictionary is the best depending on the indices of signal to noise ratio (SNR) and root mean square error (RMSE). GP-KSVD sparse representation was compared with orthogonal matching pursuit(OMP)-KSVD and preconditioning conjugate gradient pursuit(PCGP) algorithms. The simulation results prove that GP-KSVD has the minimum computer running time and the highest reconstruction precision, and the measurement verification proves the universality of the algorithm. This algorithm can be applied to response signal preprocessing, which provides theoretical basis for circuitry system equipment performance evaluation analysis.
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图 2 正弦信号1 kHz以及KSVD字典基原子
Figure 2. 1 kHz sine signal and basic atom of KSVD dictionary
图 3 GP-KSVD对1 kHz正弦信号的重构以及系数表征
Figure 3. 1 kHz sine signal reconstruction and coefficient representation by GP-KSVD
图 4 不同字典处理正弦信号的SNR以及RMSE指标的比较
Figure 4. Comparison of index of SNR and RMSE of sine signal processed with different dictionaries
图 5 几种不同算法对正弦信号的重构对比
Figure 5. Comparison of sine signal reconstruction among different algorithms
图 6 15 MHz正切信号以及KSVD字典基原子
Figure 6. 15 MHz tangent signal and basic atom of KSVD dictionary
图 7 GP-KSVD对15 MHz正切信号的重构以及系数表征
Figure 7. 15 MHz tangent signal reconstruction and coefficient representation by GP-KSVD
图 8 不同字典处理正切信号的SNR以及RMSE指标的比较
Figure 8. Comparison of indices of SNR and RMSE of tangent signal processed with different dictionaries
图 9 几种不同算法对正切信号的重构对比
Figure 9. Comparison of tangent signal reconstruction among different algorithms
图 14 GP-KSVD对电磁辐射信号的去噪重构
Figure 14. Denoising reconstruction of electromagnetic radiation signal by GP-KSVD
表 1 算法复杂度和存储量
Table 1. Algorithms complexity and storage
算法 计算复杂度 存储量 OMP-Cholesky 2D+3n2+3M+(D+N) n (n+1) /2+(D+M+2n+N) GP D+n+3M+(D+N) M+(D+M+2n+N) 
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表 2 对正弦信号不同算法的SNR指标以及运行时间
Table 2. Index of SNR and running time of different algorithms on sine signal
算法 SNR/dB 信噪比增值/dB 运行时间/s PCGP 12.7 7.2 0.162 89 OMP-KSVD 11.0 5.5 0.205 86 GP-DCT 7.1 1.6 0.111 33 GP-KSVD 12.9 7.4 0.097 656
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表 3 对正切信号不同算法的SNR指标以及运行时间
Table 3. Index of SNR and running time of different algorithms on tangent signal
算法 SNR/dB 信噪比增值/dB 运行时间/s PCGP 10.6 7.7 0.008 984 4 OMP-KSVD 9.3 6.4 0.014 063 GP-DCT 4.4 1.5 0.069 531 GP-KSVD 10.6 7.7 0.005 468 7
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