Variance of Fluctuating Radar Echoes from Thermal Noise and Randomly Distributed Scatterers

In several cases (e.g., thermal noise, weather echoes, ...), the incoming signal to a radar receiver can be assumed to be Rayleigh distributed. When estimating the mean power from the inherently fluctuating Rayleigh signals, it is necessary to average either the echo power intensities or the echo logarithmic levels. Until now, it has been accepted that averaging the echo intensities provides smaller variance values, for the same number of independent samples. This has been known for decades as the implicit consequence of two works that were presented in the open literature. The present note deals with the deriving of analytical expressions of the variance of the two typical estimators of mean values of echo power, based on echo intensities and echo logarithmic levels. The derived expressions explicitly show that the variance associated to an average of the echo intensities is lower than that associated to an average of logarithmic levels. Consequently, it is better to average echo intensities rather than logarithms. With the availability of digital IF receivers, which facilitate the averaging of echo power, the result has a practical value. As a practical example, the variance obtained from two sets of noise samples, is compared with that predicted with the analytical expression derived in this note (Section 3): the measurements and theory show good agreement.