Asymptotic behavior of maximum likelihood estimates of superimposed exponential signals

Strong consistency and asymptotic normality are derived for the maximum-likelihood estimates (MLEs) of the unknown parameters (ω ₁,. . .,ω_p), (α₁,. . ., α_p), and σ² in the superimposed exponential model for signals, Y_t=Σ α exp (itω_k)+e_t, where the summation is from k=1 to p, t=0, 1, . . ., n-1, and σ² is the variance of the complex normal distribution of e_t. As a by-product, it is found that the MLEs of the parameters attain the Cramer-Rao lower bound for the asymptotic covariance matrix