Noisy 1-Bit Distributed Compressed Sensing with Partial Gaussian Circulant Matrices

在过去大多数的研究中，带噪1-bit分布式压缩感知局限于使用高斯随机测量矩阵进行信号的重构。但是，这类矩阵内存消耗大，计算速度慢，阻碍了实际应用的发展。因此，作为改进，本文考虑将结构化的部分高斯循环矩阵应用于带噪1-bit分布式压缩感知当中。部分高斯循环矩阵可以通过快速傅里叶变换显著降低计算复杂度，提高恢复效率。为此，我们提出了新的鲁棒1-bit分布式压缩感知恢复算法。该算法通过计算符号不一致的数量，可以自适应地检测到符号翻转的位置，并进行修正。数值实验表明，在存在测量噪声和传输噪声的情况下，部分高斯循环矩阵和高斯随机矩阵的恢复性能相当。在此基础上，部分高斯循环矩阵的恢复时间更短，重构效率更高。In most previous studies, noisy 1-bit distributed compressed sensing has been limited to the use of Gaussian random measurement matrices for signal reconstruction. However, such matrices consume significant memory and have slow computational speeds, which hinder the development of their practical application. Therefore, as an improvement, this paper considers the application of structured partial Gaussian circulant matrices to noisy 1-bit distributed compressed sensing. Partial Gaussian circulant matrices can significantly reduce computational complexity and improve recovery efficiency through fast Fourier transform. To this end, we propose a new robust 1-bit distributed compressed sensing recovery algorithm. This algorithm can adaptively detect and correct sign flip positions by calculating the number of sign inconsistencies. Numerical experiments demonstrate that, in the presence of measurement noise and transmission noise, the recovery performance of partial Gaussian circulant matrices is comparable to that of Gaussian random matrices. Furthermore, the recovery time of partial Gaussian circulant matrices is shorter, and the reconstruction efficiency is higher.