Digital estimation of autocovariance functions and power spectra from randomly sampled data using a lag product technique

Abstract

The estimation of autocovariance functions and power spectra from randomly sampled data is a signal processing problem with applications in several areas. A practical autocovariance estimator produces estimates at integer values of lag by using random sample times to determine the nearest integer number of lag units between a given pair of samples. The average sample rate can be sub-Nyquist. A power spectrum estimate is computed as the discrete Fourier transform of the autocovariance estimate. An analysis is conducted assuming Poisson sampling to determine the quality of the estimators and to give insight in the choice of parameter values. The analysis is in terms of the profitability density function of the random lag between samples and the discretized lag variable. The autocovariance estimator is found to be approximately unbiased and consistent, while the power spectrum estimator is biased. An experimental study shows typical estimates and the behavior of the error in the estimates as the estimator parameters change. Together, the analysis and experimental results show the effects of discretizing the lags and how the parameter values should be chosen when implementing the estimators.