Unsupervised spectral unmixing is one of the most important processing tasks performed on hyperspectral imaging data. It is a blind source separation problem where every spectral pixel in the hyperspectral data cube is separated into a set of pure spectra, i.e., endmembers, without any prior knowledge about them. This problem could be solved sparsely using the basis pursuit optimization problem which is a well-known technique used for sparse source recovery. The basis pursuit considers the endmembers sparse in a basis known as dictionary. The contribution of this paper is the fast implementation of the basis pursuit unmixing algorithm due to using the Coiflet orthogonal dictionary. Using Coiflet orthogonal dictionary results in sparser spectral pixels which lead to fast computation by reducing the iterations of the basis pursuit unmixing algorithm. The basis pursuit unmixing algorithm using Coiflet was applied to a synthetic data cube acquired from few materials selected from the given ASTER spectral library.