Reconstruction of the sequence of Diracs from noisy samples via maximum likelihood estimation

We propose a reconstruction procedure for periodic sequence of K Diracs from noisy uniform measurements based on the maximum likelihood estimation. We first express the noise vector using the measurement vector and estimation parameters. This expression and the probability density function (PDF) for the noise vector allow us to define the (log-) likelihood function. We show that when the PDF is Gaussian, the maximization of the likelihood function is equivalent to finding the nearest sequence to the noisy sequence in the Fourier domain. This problem can be efficiently solved by combining an analytic solution and the so-called particle swarm optimization (PSO) search. Computer simulations show that the proposed method outperforms the conventional methods with computational cost of approximately O(K).