In a simplified model for a periodic signal that propagates through a random medium, the received signal is mixed with a background noise, and in addition, each period is randomly time shifted and attenuated. In this correspondence, we introduce two methods for retrieving the magnitude spectrum of the nominally periodic waveform from repeated noisy measurements and estimating some of the parameters that characterize the random medium. The flrst method is based on averaging the biperiodograms of the noisy data. We show that the reconstructed magnitude spectrum is an unbiased and consistent estimator if the background noise is white with a symmetric pdf. The second method is based on averaging the periodograms of the noisy data. In this method , it is possible to reconstruct the magnitude spectrum only if the magnitude of the background noise is either known or can be estimated from an independent measurements. Both methods are analyzed, and their performance is demonstrated via Monte Carlo simulations.