Sinusoidal signals with random amplitude: least-squares estimators and their statistical analysis

The asymptotic properties of constrained and unconstrained least-squares estimates of the parameters of a random-amplitude sinusoid are analyzed. An explicit formula for the asymptotic covariance matrix of the estimation errors is derived for both the constrained and unconstrained estimators. Accuracy aspects are investigated with the following main results. For a certain weighting matrix, which is shown to be the same for the constrained and unconstrained methods, the estimation errors achieve their lower bounds. It is proven that in the optimal case, the constrained method always outperforms the unconstrained method. It is also proven that the accuracy of the optimal estimators improves as the number of least-squares equations increases. A formula for the sample length needed for the asymptotic theory to hold is derived, and its dependence on the lowpass modulating sequence is stressed. Simulations provide illustrations of the difference between the constrained and unconstrained estimators as well as the difference between the optimal and basic estimates. The influence of the number of least-squares equations and the characteristics of the lowpass envelope on the estimation accuracy is also investigated.