基于鲁棒中国余数定理的欠采样下信号的频率估计方法研究

Frequency estimation is a basic problem of signal processing, which is common existing in radar system, image processing, wireless based distance estimation, etc. Usually, the frequency to be detected is high, which may result in an inconvenience of the system designing if we use the traditional sampling. One of the practical methods to estimate the high frequency is from the detected wrapped frequencies with the under-sampling rates. In this project, we attempt to estimate the frequency from its wrapped frequencies, where the conditions of the frequency or the sampling rates are relaxed. The contradiction of the high estimation precision and the heavy computation load is attempted to be solved also. By the aspect of detecting and estimating, we attempt to obtain the optimal estimation of the single frequency or multiple frequencies by using the Chinese remainder theorem and its generalization. To be specific, we attempt to solve the following questions. First, the optimal frequency is determined by using the robust Chinese remainder theorem when the detected remainders have errors. Second, since the sampling rates of the existing robust Chinese remainder theorem are strict, we attempt to relax these conditions and obtain the more general case. Some open problems are attempted to be solved, such as, the influence of grouping different sampling rates, the rule of grouping, and the robustness of the proposed method. Third, the unknown multiple frequencies can be determined from their residue sets by using the generalized Chinese remainder theorem, where the frequencies have no other restriction.

对信号中的频率进行估计是信号处理中的一个基本问题，其广泛存在于雷达系统、图像处理、无线电测距等问题中。通常要估计的信号频率较高，若直接去估计频率则需更高的采样率以致于给后端的处理及系统的设计带来诸多的不便。为此，在欠采样率下进行估计就变得很有必要。本项目从信号检测与估计的角度出发，拟解决现有的频率估计方法存在的诸多缺陷，如对频率及采样率的条件限制、高的估计精度与低的运算复杂度不可兼得的问题。具体地，解决如下问题：当“折叠”后频率中的“大误差”与“小误差”同时存在时，给出单个频率估计最优的方法；针对现有的基于鲁棒的中国余数定理方法对欠采样率要求苛刻的问题，研究更一般欠采样率下的分组方式对频率估计的影响、最优分组准则以及单个频率最优估计的鲁棒性；针对现有方法对多个频率估计时的条件限制，利用鲁棒的广义中国余数定理来解决无条件限制时的多个频率估计问题。

信号中的频率进行估计是信号处理中的一个基本问题，其广泛存在于雷达系统、图像处理、无线电测距等问题中。通常要估计的信号频率较高，若直接去估计频率则需更高.的采样率以致于给后端的处理及系统的设计带来诸多的不便。为此，在欠采样率下进行估计就变得很有必要。本项目从信号检测与估计的角度出发，一定程度上解决了现有的频率估计方法存在的诸多缺陷，如对频率及采样率的条件限制、高的估计精度与低的运算复杂度不可兼得的问题。在理论方面，给出了从余数组中最优的分组理论及最优的公共余数估计方法；给出了鲁棒估计的条件。在应用方面，提高了欠采样下多个频率的估计精度；将鲁棒的中国余数定理应用于互质阵列的多源角估计问题中，提高了估计精度的同时，降低了运算复杂度；将广义中国余数定理应用于图像的保密传输问题中，在不引入冗余的同时，达到了安全传输的目的。另外，就基于波束形成等技术的随机网络编码以及物理层的保密传输等问题进行了相关研究。