Data recovery with sub-Nyquist sampling: fundamental limit and a detection algorithm

While the Nyquist rate serves as a lower bound to sample a general bandlimited signal with no information loss, the sub-Nyquist rate may also be sufficient for sampling and recovering signals under certain circumstances. Previous works on sub-Nyquist sampling achieved dimensionality reduction mainly by transforming the signal in certain ways. However, the underlying structure of the sub-Nyquist sampled signal has not yet been fully exploited. In this paper, we study the fundamental limit and the method for recovering data from the sub-Nyquist sample sequence of a linearly modulated baseband signal. In this context, the signal is not eligible for dimension reduction, which makes the information loss in sub-Nyquist sampling inevitable and turns the recovery into an under-determined linear problem. The performance limits and data recovery algorithms of two different sub-Nyquist sampling schemes are studied. First, the minimum normalized Euclidean distances for the two sampling schemes are calculated which indicate the performance upper bounds of each sampling scheme. Then, with the constraint of a finite alphabet set of the transmitted symbols, a modified time-variant Viterbi algorithm is presented for efficient data recovery from the sub-Nyquist samples. The simulated bit error rates (BERs) with different sub-Nyquist sampling schemes are compared with both their theoretical limits and their Nyquist sampling counterparts, which validates the excellent performance of the proposed data recovery algorithm. 奈奎斯特频率是一般带限信号进行无损采样的采样率下界, 然而在某些情景中, 亚奈奎斯特频率也足以进行无损采样和信号恢复. 以往对亚奈奎斯特采样的研究主要集中在利用信号变换来降低信号维度, 但是亚奈奎斯特采样信号的结构并没有得到充分研究. 本文针对线性调制基带信号的亚奈奎斯特采样, 研究其信号恢复性能极限与算法. 该问题中, 原信号维度无法降低, 因此亚奈奎斯特采样不可避免会带来信息损失, 信号恢复也变成一个欠定线性问题. 本文采用两种亚奈奎斯特采样方法对线性调制基带信号进行采样, 分别研究了两种采样方法下的性能极限和信号恢复算法. 首先, 针对两种亚奈奎斯特采样方法, 分别计算了采样序列之间的最小归一化欧氏距离, 以此作为最优性能的指标. 然后, 在基带信号有限符号集的限制条件下, 采用改进的时变维特比算法从亚奈奎斯特采样序列中恢复原信号. 将仿真得到的亚奈奎斯特采样的误比特率与其性能的理论极限比较, 并与奈奎斯特采样性能对比, 验证了时变维特比算法在信号恢复问题中的优良性能.