Detection and estimation of weak sine waves with random offset and additive noise

This paper deals with the effects of unknown offset and additive noise on the detection and estimation of weak sine waves after analog-to-digital conversion and windowing. A weak sine wave, for which the problem of detection and estimation makes sense, is by necessity quantized in a roughly manner, i.e., it ranges over only a small number of quantization levels. Thus, for a weak sine wave a low-resolution quantization must be taken into account even if a high-resolution analog-to-digital converter is employed. The first part of this paper shows the influence of an unknown offset on the uncertainty of the estimated sine-wave amplitude due to low-resolution quantization. It is shown that the conventional noise model of quantization in this case can lead to an underestimation of the amplitude uncertainty. The second part of the paper is focused on windowing of a noisy sine wave, and a quantitative criterion is provided for a proper selection of the time window that optimizes the detection probability after a discrete Fourier transform. The results are provided in analytical form as functions of the sampling parameters and the properties of the time window. Analytical results are validated by means of numerical simulations.