The DUET blind source separation algorithm can demix an arbitrary number of speech signals using M = 2 anechoic mixtures of the signals. DUET however is limited in that it relies upon source signals which are mixed in an anechoic environment and which are sufficiently sparse such that it is assumed that only one source is active at a given time frequency point. The DUET-ESPRIT (DESPRIT) blind source separation algorithm extends DUET to situations where M ≥ 2 sparsely echoic mixtures of an arbitrary number of sources overlap in time frequency. This paper outlines the development of the DESPRIT method and demonstrates its properties through various experiments conducted on synthetic and real world mixtures.