BEP Estimation by Weighted Least Squares From the Ratios of Frame Syncword Error Rates

Abstract

For telemetry data processing systems, estimating the bit error probability of received data is important in terms of data quality. For systems in which the number of allowable errors in an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -bit frame syncword (FS) of a telemetry data frame is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> and the numbers of FSs with no error, one error, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\dots$</tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> errors, and more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> errors are given, respectively, it is known from the literature that conventional methods, including the maximum-likelihood estimator (MLE), are not generally presented as a closed-form expression excluding specific cases. This article proposes a weighted least squares estimator (WLSE) by taking the ratios of the observed FS error rates to minimize the squared discrepancies between the observed and the predicted values, and the WLSE is obtained by straightforward calculation. We analyze the characteristics of the bias and the variance and derive optimal weights that minimize the variance considering that the mean squared error (MSE) of the proposed estimator depends on the variance rather than the bias. Based on the derived optimal weights, a method is proposed to sequentially obtain the weights close to optimum. The analytical and simulation results verify that the MSE of the proposed estimator is only slightly larger or even less than those of the existing methods, while the proposed estimator has a significantly lower computational complexity than those of the conventional schemes.