Approximation and Analysis of the One-Bit Hermite Law

One-bit data with non-zero quantization thresholds suffice to recover the covariance of the original Gaussian data. State-of-the-art recovery methods require computing the one-bit correlation from the estimated covariance. However, this step exhibits trade-offs between the computational complexity and approximation error. This paper proposes the harmonic approximation of the one-bit Hermite law for the one-bit correlation. The proposed approximation features closed forms with finite terms and low computational complexity. Furthermore, its approximation errors are analyzed through the approximation error bounds (AEBs). The AEBs also lead to theoretical insights into the signal statistics, quantization thresholds, and the parameters in the proposed approximation. Numerical examples demonstrate the applicability of the proposed approximation.