Fourier series expansions of noisy signals — consistent estimation of the whole spectrum

摘要

AbstractIn this note the problem of estimating the Fourier series coefficients of a deterministic signal measured in a noise is discussed. Firstly, it is shown that if random errors are not taken into account, then the mean square error between the true spectrum and its commonly used estimator is infinite. The method proposed for overcoming this difficulty is based on multiplying the estimates by the geometric sequence. It is shown that if this sequence depends on the number of observations and is appropriately chosen, then consistent estimation of the whole spectrum is possible. The method introduces a bias which is shown to be asymptotically vanishing. For smooth signals an upper bound for the bias is also derived.