In this paper, we present a novel technique for the estimation of the high frequency components (4-8kHz) of speech signals from narrow-band (0-4 kHz) signals using convolutive Non-Negative Matrix Factorisation (NMF). The proposed technique utilizes a brief recording of simultaneous broad band and narrow band signals from a target speaker to learn a set of broad-band non-negative "bases" for the speaker. The low-frequency components of these bases are used to determine how the high-frequency components must be combined in order to reconstruct the high-frequency components of new narrow-band signals from the speaker. Experiments reveal that the technique is able to reconstruct broadband speech that is perceptually virtually indistinguishable from true broadband recordings.