Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, USA.
Sensors (Basel). 2022 Dec 28;23(1):307. doi: 10.3390/s23010307.
In this paper, we derive the Cramér-Rao lower bounds (CRLB) for direction of arrival (DoA) estimation by using sparse Bayesian learning (SBL) and the Laplace prior. CRLB is a lower bound on the variance of the estimator, the change of CRLB can indicate the effect of the specific factor to the DoA estimator, and in this paper a Laplace prior and the three-stage framework are used for the DoA estimation. We derive the CRLBs under different scenarios: (i) if the unknown parameters consist of deterministic and random variables, a hybrid CRLB is derived; (ii) if all the unknown parameters are random, a Bayesian CRLB is derived, and the marginalized Bayesian CRLB is obtained by marginalizing out the nuisance parameter. We also derive the CRLBs of the hyperparameters involved in the three-stage model and explore the effect of multiple snapshots to the CRLBs. We compare the derived CRLBs of SBL, finding that the marginalized Bayesian CRLB is tighter than other CRLBs when SNR is low and the differences between CRLBs become smaller when SNR is high. We also study the relationship between the mean squared error of the source magnitudes and the CRLBs, including numerical simulation results with a variety of antenna configurations such as different numbers of receivers and different noise conditions.
在本文中，我们使用稀疏贝叶斯学习（SBL）和拉普拉斯先验来推导到达方向（DoA）估计的克拉美罗下界（CRLB）。CRLB 是估计量方差的下界，CRLB 的变化可以指示特定因素对 DoA 估计器的影响，在本文中，使用拉普拉斯先验和三阶段框架进行 DoA 估计。我们在不同情况下推导出 CRLB：（i）如果未知参数由确定变量和随机变量组成，则推导出混合 CRLB；（ii）如果所有未知参数都是随机的，则推导出贝叶斯 CRLB，并通过边缘化掉杂项参数来获得边际化的贝叶斯 CRLB。我们还推导出三阶段模型中涉及的超参数的 CRLB，并探讨了多个快照对 CRLB 的影响。我们比较了 SBL 的推导 CRLB，发现当 SNR 较低时，边际化的贝叶斯 CRLB 比其他 CRLB 更紧，而当 SNR 较高时，CRLB 之间的差异会变小。我们还研究了源幅度均方误差与 CRLB 之间的关系，包括在各种天线配置下的数值模拟结果，例如不同数量的接收器和不同的噪声条件。
