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This paper presents a class of minimum mean-square error (MMSE) estimators for enhancing short-time spectral coefficients of a noisy speech signal. In contrast to most of the presently used methods, we do not assume that the spectral coefficients of the noise or of the clean speech signal obey a (complex) Gaussian probability density. We derive analytical solutions to the problem of estimating discrete Fourier transform (DFT) coefficients in the MMSE sense when the prior probability density function of the clean speech DFT coefficients can be modeled by a complex Laplace or by a complex bilateral Gamma density. The probability density function of the noise DFT coefficients may be modeled either by a complex Gaussian or by a complex Laplacian density. Compared to algorithms based on the Gaussian assumption, such as the Wiener filter or the Ephraim and Malah (1984) MMSE short-time spectral amplitude estimator, the estimators based on these supergaussian densities deliver an improved signal-to-noise ratio.