Estimating complex Fourier coefficients of periodic functions from additively error disturbed observations is discussed. This is the first step in most methods for estimating the parameters of linear dynamic systems from noise disturbed responses to periodic perturbation signals. In this type of estimation problem the period is known. Estimating complex Fourier coefficients is also the first step in procedures for reconstructing periodic images from noisy observations. In these problems the period is usually unknown and the Fourier coefficients and the period have to be estimated jointly. Complex valued estimators are proposed for both types of problem. The real and imaginary parts of the unknown complex valued parameters are not treated as separate real parameters. Statistical and numerical properties of the proposed estimators are described.