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The problem of estimating the parameters in stochastic continuous-time signals, represented as continuous-time autoregressive moving average (ARMA) processes, from discrete-time data is considered. The proposed solution is to fit the covariance function of the process, parameterized by the unknown parameters, to sample covariances. It is shown that the method is consistent, and an expression for the approximate covariance matrix of the estimated parameter vector is derived. The derived variances are compared with empirical variances from a Monte Carlo simulation, and with the Cramer-Rao bound. It turns out that the variances are close to the Cramer-Rao bound for certain choices of the sampling interval and the number of covariance elements used in the criterion function.