A 1.5-bit Quantization Scheme and Its Application to Direction Estimation

In massive multiple-input multiple-output (MIMO) systems, the balance between cost and performance has made low-bit, especially 1-bit, analog-to-digital converters (ADCs) an indispensable part of the solution. In this paper, a three-level 1.5-bit ADC quantization scheme is proposed, which requires only one additional comparator beyond the 1-bit quantizer. Leveraging the Price theorem and Mehler's formula, we derive the 1.5-bit correlation estimator and analyze the approximation error using a first-order Taylor expansion. Our findings reveal that, at low signal-to-noise ratios (SNRs), the eigenvalues of the 1.5-bit covariance matrix are nearly identical to those of the unquantized covariance matrix. This allows direct parameter estimation without the need to reconstruct the unquantized covariance. Moreover, we show that the approximation error for 1.5-bit measurements is much smaller than that of 1-bit quantization in high SNR conditions. Based on the derived correlation estimator, an algorithm is proposed for recovering the unquantized covariance matrix using a gradient descent method. Simulation results obtained by applying our proposed algorithm to DOA estimation show that, the 1.5-bit scheme is robust to the choice of the threshold value, and significantly outperforms 1-bit quantization without much increase in cost.