Explicit entropy error bound for compressive DOA estimation in sensor array

Compressive sensing (CS) simplifies software and hardware by sampling below the Nyquist rate, making it widely used in array signal processing. For the assessment of the compressive direction-of-arrival (DOA) estimation, a lower bound on the mean square error is essential. However, the most widely utilized Cramér-Rao bound (CRB) is only asymptotically tight. This paper proposes a globally tight bound with a closed-form expression for compressive DOA estimation in the uniform linear arrays employing Shannon information theory. Based on the a posteriori probability density function, we propose the indicator of the entropy error (EE) with compression to assess the DOA estimation. The theoretical EE bounds the compressive DOA estimation performance. Moreover, the explicit EE is derived by approximating the normalized differential entropy, which is comprehensive and captures the effect of the compression ratio, the SNR, the number of elements, and the mean square bandwidth. In Particular, the compression ratio has almost no influence on the EE in low SNR. Additionally, the asymptotic lower bound of the theoretical EE is identical to the CRB. Simulation results illustrate the superiority of EE over CRB in evaluating and predicting compressive DOA estimation performance in the uniform linear arrays.