Sparse representation and reproduction of speech signals in complex Fourier basis

Sparse representation concerns the task of determining the most compact representation of a signal via a linear combination of bases of an overcomplete dictionary. As the problem is non-convex, it is common to consider approximate suboptimal solutions, and one such method is the orthogonal matching pursuit (OMP) algorithm. OMP is an iterative greedy algorithm, where at each step, the basis vector which is most correlated with the current residual is selected. For the most part, attention in the past has been directed towards using real-valued dictionaries as the considered signal of interest is also real-valued. From the perspective of speech representation, the use of complex dictionaries in sparse representation is intuitively appealing as audio signals are generally assumed to be a mixture of exponentials, with time-varying amplitudes and phases. However, sparse representation of speech signal based on complex dictionary is less investigated mainly because the measurements are normally real-valued. In this paper, we pursue this intuition by modelling the complex dictionary on the popular discrete Fourier transform, and then proceed to introduce a new orthogonalization mechanism in the OMP for such cases. The customization of the conventional OMP algorithm to the complex setting enables high-quality compact representation of the speech signals with low computational complexity. Results from experiments demonstrate that the proposed approach is able to retain high perceptual similarity of the reconstructed speech signals to the original ones.