The relation between range sampling rate, system bandwidth, and variance reduction in spectral averaging for meteorological radar

Periodograms are frequently used in radar meteorology to get spectral moments of weather echo. Each range trace is sampled and the set of samples at each of the specific ranges is used to obtain a pulse-to-pulse periodogram from which the spectral moments are estimated. Over some range interval, the separate periodograms would be averaged to get smoother estimates to be reported. The averaging will reduce the variance of the spectral estimates but the resulting variance is not quite inversely proportional to the number of periodograms because of the correlation introduced by a sampling rate that preserves range resolution. The authors derive the equivalent number M_eq of uncorrelated range-samples when M range samples are actually averaged. The factors involved are the 3-dB system half-bandwidth W, the number M, and the range sampling frequency f_s . The formula is M_eq=M[1+2Σ(_r=1)^M-1 (1-r/M)exp(-28.48r²W²/f_s² )]^-1. A table is provided of M_eq versus M