IEEE-STD-1057 Three Parameter Sine Wave Fit for SNR Estimation: Performance Analysis and Alternative Estimators

Abstract

This paper considers the three-parameter-fit sine wave model. Under a Gaussian noise assumption, it is known that the three-parameter fit given in IEEE Standards 1057 and 1241 coincides with the method of maximum likelihood (ML), which is known for its favorable properties in large samples. Under coherent sampling assumption, the Cramer–Rao Bound of an unbiased estimator (UE) of the signal-to-noise ratio (SNR) is derived followed by an exact finite-sample analysis of the ML estimator of SNR derived from the three-parameter fit, revealing its nonsymmetric F-distribution. Exact expressions for the bias, variance, and the mean squared error (MSE) of the ML estimator are then derived, revealing that the ML estimator in finite samples is far from optimal in terms of precision and accuracy. With the ML estimator as a starting point, several alternative estimators are derived, which outperform the method of ML. In particular, a UE is derived, with lower variance compared with the ML for small sample size. In addition, estimators are derived based on constrained minimization of the MSE. The theoretical findings are illustrated by simulations, showing an excellent agreement between theory and practice. Simulations using quantized data are also used to show the performance of the derived estimators mimicking an analog-to-digital converter (ADC) testing scenario. Furthermore, the derived estimators are applied to coherently and noncoherently sampled measurement data from a 12-bit ADC and, for small number of samples, all are shown to outperform the original estimate, showing the practical relevance of the theoretical findings.