Construction of Convolution Compressed Sensing Measurement Matrices Based on Cyclotomic Classes
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摘要:
卷积压缩感知是近年来兴起的新型压缩感知技术。卷积压缩感知选用循环矩阵作为测量矩阵，其采样可以简化为卷积的过程，因此大大降低算法复杂度。该文基于分圆类构造适用于卷积压缩感知的测量矩阵，测量值通过利用确定性序列循环卷积信号，然后进行随机2次采样获得。该文构造的测量矩阵的相关性小于已有文献构造的测量矩阵的相关性。模拟仿真结果表明，该文构造的测量矩阵与同等条件下的随机高斯矩阵相比，可以更好地恢复稀疏信号；所构造的矩阵还可以应用于信道估计以及2维图像的重构。
Abstract:Convolutional compressed sensing emerging in recent years is a new type of compressed sensing technology. By using cyclic matrix as measurement matrices, the sampling in convolutional compressed sensing can be simplified into convolution process, thus the complexity of the algorithm is greatly reduced. In this paper, a construction of measurement matrices for convolutional compressed sensing based on cyclotomic classes is proposed. The measurements are obtained by using the circulate convolution signal of the deterministic sequence and then by random subsampling. The correlation of the measurement matrix constructed in this paper is smaller than that of the existing constructions in the literature. The simulation results show that the measurement matrix constructed in this paper can recover the sparse signal better than the random Gaussian matrix under the same conditions. The proposed matrix can also be applied to channel estimation and reconstruction of two-dimensional images.
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Key words:
- Signal processing /
- Compressed sensing /
- Convolution /
- Cyclotomic class /
- Random sampling
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表 1 与已有序列构造的矩阵相关性比较
对角向量$\sigma $ 序列长度$N$ 相关性参数$\mu ({{A}})$ 文献[10] 抽样Sidelnikov序列 $N = \dfrac{ { {p^m} - 1} }{c}$, $c$为偶数 $\sqrt {c + {{\rm{1}} / N}} + {1 / {\sqrt N }}$, $N$为偶数 $\sqrt {c + {1 / N}} $, $N$为奇数 文献[11] Extended Frank-Zadoff-Chu(扩展FZC)序列 $N$为偶数 $4 + {4 / {\sqrt N }}$ $N$为奇数 $2.69 + {{8.15} / {\sqrt N }}$ Extended Golay(扩展Golay)序列 $N = {2^{{k_1}}}{10^{{k_2}}}{26^{{k_3}}}$, $N$为偶数，${k_1},{k_2},{k_3}$为整数 $2 + {2 / {\sqrt N }}$ $N = {2^{{k_1}}}{10^{{k_2}}}{26^{{k_3}}} \pm 1$, $N$为奇数，${k_1},{k_2},{k_3}$为整数 $2 + {1 / {\sqrt N }}$ 本文 由2阶分圆类得到的序列 $N = p$为奇素数，$p \equiv 1({\rm{mod}}4)$ $1 + {1 / {\sqrt N }}$ 由$e > 2$阶分圆类得到的序列 $N = p$为奇素数 $\mu ({{A} }) \le 2$ 
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