基于去网格化变分贝叶斯方法的低精度量化线谱估计问题研究

Line spectral estimation (LSE) aims to extract the parameters of a superposition of complex exponential functions from noisy measurements. As a fundamental problem in signal processing fields, it has many applications such as direction of arrival estimation and channel estimation. Because of the larger bandwidths that accompany both high resolution radar and high data rate mmWave communication system, the cost and power consumption are huge due to high precision (e.g., 10~12 bits) analog-to-digital converters (ADCs). Consequently, low precision (e.g., 1~3 bits) ADCs are employed to relieve this ADC bottleneck. Theoretical analysis reveals that there exists plentiful harmonics of the heavily quantized samples. As a result, conventional methods such as fast Fourier transform (FFT) often overestimate the model order. This research resorts to expectation propagation (EP) and grid-less variational Bayesian method. According to EP, the challenging quantized model of line spectral estimation is iteratively approximated via the minimum mean square error (MMSE) module as a sequence of simple pseudo unquantized heteroscedastic models (different components having different variance), a variant of the variational LSE (VALSE) is re-derived. For the MMSE module, it refines the pseudo observations and variances of the unquantized model. While for the pseudo unquantized model, VALSE iteratively refines the frequency estimates. The two modules iteratively exchange the information and improve the estimation gradually. In addition, this project also extends the above algorithm to solve the direction of arrival estimation and channel estimation problems. To sum up, this project is not only important for solving the fundamental LSE from heavily quantized samples, but also helpful for application in array signal processing and communication fields.

线谱估计是指从时域或者空域估计信号的频域信息，其在雷达和通信领域得到了广泛的应用。然而，随着雷达和通信系统带宽的提升，高精度和高采样率的量化器大大提高了系统的经济成本和硬件能耗。一个有效的解决方案即是在接收机采用低精度量化技术。由于低精度量化数据含有丰富的谐波，传统的离散傅里叶变换方法会过高地估计模型阶数，因此低精度量化这种非线性影响必须加以考虑。利用期望传播，本项目拟将此问题的求解分解为无量化异方差模块和最小均方误差估计模块。一方面，最小均方误差估计模块迭代地修正无量化异方差模型的观测值和噪声方差，使其越来越接近真实的无量化模型；另一方面，无量化异方差模块迭代地修正线谱，得到更为准确的外信息送入最小均方误差估计模块。这两个模块迭代地进行消息传递，并逐渐改善线谱估计的性能。在上述研究成果的基础上，项目将其拓展并应用于到达角和信道估计问题。项目的研究成果，必将具有重要的理论意义和应用价值。

低精度量化技术通过在时域对接收信号进行低精度量化，可实现极高速率采样、大幅降低数据量、显著降低时域采样成本。因此，低精度量化线谱估计是信号处理领域中非常重要的问题。由于低精度量化数据含有丰富的谐波，无量化方法应用于低精度量化数据会估计出谐波分量，导致谐波虚警。课题提出了VALSE-EP算法，其利用期望传播，将问题的求解分解为无量化异方差模块和最小均方误差估计模块。一方面，最小均方误差估计模块迭代地修正无量化异方差模型的观测值和噪声方差，使其越来越接近真实的无量化模型；另一方面，无量化异方差模块迭代地修正线谱，得到更为准确的外信息送入最小均方误差估计模块。这两个模块迭代地进行消息传递，逐渐改善线谱估计的性能。在上述研究成果的基础上，项目将其拓展并应用于到达角估计、信道估计、单比特雷达、联合信道估计和符号检测等问题。通过大量的数值仿真、毫米波雷达和水声实测数据以及与其他算法性能进行对比，验证了算法的有效性。从理论层面，项目分析了低精度量化数据的频谱，指出谐波会导致虚警。而且，推导了相应模型的克拉美—罗界。尤其在多测量单音情况下，假设信号相位均匀分布，给出了频率估计误差的渐近表达式。结果表明均方误差反比于观测数目的立方和拍数，高信噪比下反比于信噪比的平方根，低信噪比下反比于信噪比。综上，本项目开展的研究工作获得了较好的研究效果，达到预期目标，为我国未来低精度量化感知和通信技术提供了理论指导和技术解决方案。

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