[论文解读] Multirate Synchronous Sampling of Sparse Multiband Signals

Michael Fleyer, Amir Rosenthal+2 more|ArXiv.org|Jun 3, 2008

Sparse and Compressive Sensing Techniques参考文献 11被引用 6

一句话总结

本文提出同步多速率采样（SMRS），一种新颖方法，通过少量在不同但同步速率下运行的采样通道，实现对稀疏多带信号的重建。通过利用整数倍采样频率并求解线性方程组，SMRS在总采样率显著低于奈奎斯特率的情况下，实现了高精度、抗噪声的重建，其在通道效率和混叠鲁棒性方面优于多梳齿和非同步多速率方案。

ABSTRACT

Recent advances in optical systems make them ideal for undersampling multiband signals that have high bandwidths. In this paper we propose a new scheme for reconstructing multiband sparse signals using a small number of sampling channels. The scheme, which we call synchronous multirate sampling (SMRS), entails gathering samples synchronously at few different rates whose sum is significantly lower than the Nyquist sampling rate. The signals are reconstructed by solving a system of linear equations. We have demonstrated an accurate and robust reconstruction of signals using a small number of sampling channels that operate at relatively high rates. Sampling at higher rates increases the signal to noise ratio in samples. The SMRS scheme enables a significant reduction in the number of channels required when the sampling rate increases. We have demonstrated, using only three sampling channels, an accurate sampling and reconstruction of 4 real signals (8 bands). The matrices that are used to reconstruct the signals in the SMRS scheme also have low condition numbers. This indicates that the SMRS scheme is robust to noise in signals. The success of the SMRS scheme relies on the assumption that the sampled signals are sparse. As a result most of the sampled spectrum may be unaliased in at least one of the sampling channels. This is in contrast to multicoset sampling schemes in which an alias in one channel is equivalent to an alias in all channels. We have demonstrated that the SMRS scheme obtains similar performance using 3 sampling channels and a total sampling rate 8 times the Landau rate to an implementation of a multicoset sampling scheme that uses 6 sampling channels with a total sampling rate of 13 times the Landau rate.

研究动机与目标

• 解决在低于奈奎斯特率条件下高效采样高带宽、稀疏多带信号的挑战。
• 克服非同步多速率采样（MRS）中各通道同时混叠导致无法重建的局限性。
• 在保持高重建精度和抗噪声能力的同时，减少所需采样通道的数量。
• 通过使用共同时钟的光学下变频和A/D转换，实现实际可行的系统实现。
• 通过最小化各通道间同时混叠，确保对噪声和信号稀疏性的鲁棒性。

提出的方法

• 使用多个采样通道，以共同基频的整数倍频率运行。
• 同步各通道采样，确保一致的时间分辨率，并通过求解线性系统实现混叠消除。
• 在A/D转换前，使用光脉冲序列将全带信号下变频至基带。
• 以不同速率采样下变频后的信号，最高采样率决定时间分辨率。
• 将信号重建表述为线性系统：$\widehat{\mathbf{x}} = \widehat{\mathbf{A}} \mathbf{x}$，其中$\widehat{\mathbf{A}}$将采样数据映射到频谱分量。
• 对于实信号，将问题分解为实部与虚部，利用共轭对称性降低维度，并分别求解两个独立的线性系统。

实验结果

研究问题

• RQ1 具有同步通道的多速率采样方案是否能在总采样率远低于奈奎斯特率的情况下，实现对稀疏多带信号的精确重建？
• RQ2 在通道数量和混叠鲁棒性方面，同步多速率采样与多梳齿及非同步多速率采样相比表现如何？
• RQ3 提高单个通道采样率对多速率采样方案中所需通道数量有何影响？
• RQ4 SMRS方案是否因通道间同时混叠减少而保持对噪声的鲁棒性？
• RQ5 SMRS形成的线性系统在何种条件下能为信号重建提供唯一且稳定的解？

主要发现

• 仅使用三个采样通道，SMRS即实现了对四个实信号（共八个频带）的精确重建。
• SMRS方案的总采样率仅为兰道率的1/8，性能与使用六通道、采样率是兰道率13倍的多梳齿方案相当。
• 重建矩阵的条件数较低，表明具有高数值稳定性及抗噪声能力。
• 随着单个通道采样率提高，所有通道同时发生混叠的概率显著降低，从而提升了重建成功率。
• 在通道数量较少时，SMRS在重建成功率方面优于文献[5]中的多梳齿方案。
• 通过在各通道间共用高速A/D转换器，实现了稳定的时间分辨率，并提升了采样信号的信噪比。