Ergodicity; Temporal and Spectral Estimations
In many practical cases, only one realization of a random signal is available and treatments by ensemble averaging are not possible. One is led to estimate the statistical properties of the signal using this single realization. The signal should necessarily possess qualities of stationarity. Ergodicity qualifies the equivalence between expectations and time averages. As a record of this signal has necessarily a limited length, ergodicity can only be reached asymptotically when the sample length tends to infinity. Integrals on time-limited intervals can only provide estimates of intrinsic statistical properties. We study the estimation of the mean of a random signal by the sum of consecutive samples and the variance of the estimator. An estimator of the power spectral density is taken as the Fourier transform of one estimator of the correlation function. We show that the poor quality of the raw estimator of the psd results from the poor estimation of the correlation function for large time delays. We present the methods that are used to improve largely the spectral estimation. The chapter ends with the presentation of methods for extracting one or several harmonic components in a noisy spectrum. The Capon maximum likelihood super resolution method and Pisarenko method are discussed.