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This paper describes a method for utilizing the quasi-periodicity of speech in a minimum mean-square error (MMSE) estimation of the discrete Fourier transform (DFT) log-amplitude, either for speech enhancement or for noise-robust speech recognition. The estimator takes into account the periodicity by conditioning the estimate of voiced speech on the distance between the frequency of any given DFT coefficient and the nearest harmonic: if the DFT coefficient lies in the vicinity of a harmonic, the a priori probability distribution (PD) of its amplitude centers around higher values than if it lies halfway between two harmonics. Thus, knowing the pitch narrows down the a priori PD, improving the estimate. The DFT estimator is combined with a mixture model for the broadband spectral PD, so that correlations between distant frequencies are partially taken into account. The algorithm has been tested with computer-room noise using an MSE criterion for the spectral envelope, defined by Mel-scale filter-bank log energies, and in recognition experiments. The incorporation of correlations in the broadband spectrum improves recognition accuracy significantly; the periodicity conditioning reduces the MSE for voiced speech, but recognition accuracy is not improved because the overwhelming majority of errors occur in unvoiced speech.