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We present an algorithm to derive 7 kHz wideband speech from narrowband "telephone speech". A statistical approach is used that is based on a hidden Markov model (HMM) of the speech production process. A new method for the estimation of the wideband spectral envelope is proposed, using nonlinear state-specific techniques to minimize a mean square error criterion. In contrast to common memoryless estimation methods, additional information from adjacent signal frames can be exploited by utilizing the HMM. A consistent advantage of the new estimation rule is obtained compared to previously published HMM-based hard or soft classification methods.