Sparse Separation of Under-Determined Speech Mixtures

We are all familiar with the shape of sound from our secondary school science classes; the typical oscillatory form of a string under tension that decays over time is widely know. At first sight, this representation of sound imparts to the observer nothing more than its duration and amplitude. So how does the brain separate different sounds given such a representation? Over millions of years the Mammalian auditory cortex has evolved to effectively understand the sounds of its natural environment. By learning the distinguishing features of a sound, the brain can recognise and classify many different sounds. It has long been desired to replicate this ability using a machine, which has been the genesis for a topic of study known as source separation. In this thesis we utilise two contrasting strategies for the Separation of under-determined speech mixtures, i.e., the case where there are more sources than mixtures. Furthermore, we impose a sparseness requirement on the sources. First, we introduce a blind source separation method called the LOST algorithm, which is based on a Expectation-Maximisation procedure. The LOST algorithm assumes an instantaneous mixing model, and estimates the columns of the mixing matrix by identifying corresponding linear subspaces in a scatter plot. This method combined with a transformation into a sparse domain and an L1-norm minimisation, constitutes a blind source separation algorithm for the under-determined case, where there are at least two mixtures. Second, we investigate Convolutive Non-negative Matrix Factorisation,