High-precision sinusoidal frequency estimator based on weighted least square method

The estimation of the frequency of a sine wave is a fundamental task in many test and measurement systems. In this report, a frequency estimator based on the weighted least-square method (WLSM) is derived. The estimator is optimal in the sense that it is unbiased, and it has the smallest standard deviation among the whole class of linear estimators. A well-approximated normalized standard deviation of the estimator is obtained in closed form. It is shown that the normalized standard deviation of the optical estimator is of the form 2.2σ/[m×n³]^1/2, where m is the average number of samples per period, n is the number of half periods covered by the observation interval, and σ is the standard deviation of the additive noise. There are three potential applications of this result: precision frequency measurement in white-noise environment; confidence interval display in frequency meter; and waveform digitizer effective-bits characterization.