Bias and variance of averaged and smoothed periodogram-based log-amplitude spectra

摘要

The averaged and smoothed periodogram is the most convenient and intuitive spectral estimate. Its statistical properties are well-known for linear amplitudes, but not for log-amplitude (dB) representations. However, spectral estimates are very often represented by log-amplitude, e.g. in parametrizing ultrasound backscatter spectra, without taking into account that the log-transform may introduce a noticeable additional bias. Thus, I developed expressions for bias and variance of log-amplitude periodogram-based estimates that results from: (1) averaging few independent estimates, (2) smoothed estimates (windowing the autocorrelation sequence or the cepstrum). The resulting expressions give clear advice on how to get unbiased estimates with the lowest possible variance for the first case, and on how to choose shape and length of autocorrelation or cepstrum windows so that we get spectral estimates with low variance and yet low bias for the second case.