Spectral analysis of velocity signals recorded by acoustic-Doppler velocimetry (ADV) and contaminated with intermittent spikes remains a challenging task. In this paper, we propose a new method for reconstructing contaminated time series, which integrates two previously developed techniques for detecting and replacing spurious spikes. The spikes are first detected using a modified version of the Universal Phase-Space-Thresholding technique and subsequently replaced by the last valid data points. The accuracy of the new approach is evaluated by applying it to identify and remove spikes and reconstruct the spectra of two clean data sets which are artificially contaminated with random spikes: (a) high-quality hot-wire measurement; and (b) numerically simulated velocity time series with bi-modal probability density distribution. The technique is also applied to reconstruct the spectra obtained from intentionally contaminated ADV measurements and compare them with ADV spectra at the same point in the flow obtained using proper ADV settings. Special emphasis is placed on testing the ability of the technique to reproduce realistic power spectra in flows with rich coherent dynamics. The results show that the power spectra of the reconstructed time series contain a filtered white noise caused by the steps in the reconstruction technique using the last valid data point. We show that even for a severely contaminated time series, the proposed method can accurately recover the power spectra up to the frequency corresponding to the half the mean sampling rate of the valid data points.