Ziv-Zakai Bound for DOAs Estimation Under Arbitrary-Bit Quantization

The distribution of quantized observations is inherently non-Gaussian and often implicit, rendering conventional Ziv-Zakai bounds (ZZBs) and other global mean-squared error (MSE) lower bounds inapplicable to direction-of-arrival (DOA) estimation under quantized measurements. To overcome this limitation, we derive a closed-form ZZB for DOA estimation by leveraging the additive quantization noise model (AQNM). The proposed bound yields a unified expression that encompasses fully quantized, mixed-resolution, and unquantized observation scenarios. Furthermore, we introduce a generalized approximation framework based on a beamforming perspective, which enables the proposed ZZB to capture both linear quantization gain and quantization noise effects. Unlike existing bounds, the derived ZZB is valid for arbitrary quantization resolutions and reveals the distinct influence of both the quantization bit depth and the number of low-resolution analog-to-digital converters (ADCs) on the overall MSE. Numerical results demonstrate the accuracy and practical relevance of the proposed bound, highlighting its superiority over existing lower bounds.