An Intuitive Derivation of the Coherence Index Relation in Compressive Sensing

Abstract: The existence and uniqueness conditions are a prerequisite for reliable reconstruction of sparse signals from reduced sets of measurements within the Compressive Sensing (CS) paradigm. However, despite their underpinning role for practical applications, existing uniqueness relations are either computationally prohibitive to implement (Restricted Isometry Property), or involve mathematical tools that are beyond the standard background of engineering graduates (Coherence Index). This can introduce conceptual and computational obstacles in the development of engineering intuition, the design of suboptimal practical solutions, or understanding of limitations. To this end, we introduce a simple but rigorous derivation of the coherence index condition, based on standard linear algebra, with the aim to empower signal processing practitioners with intuition in the design and ease in implementation of CS systems. Given that the coherence index is one of very few CS metrics that admits mathematically tractable and computationally feasible calculation, it is our hope that this work will help bridge the gap between the theory and applications of compressive sensing.