Deterministic Matrices with a Restricted Isometry Property for Partially Structured Sparse Signals

Compressive sampling has become an important tool in diverse applications. One of its main challenges, the construction of deterministic sensing matrices with restricted isometry property (RIP) in the optimal sparsity regime, is still an open problem, despite being of crucial importance for practical system designs. The only known work constructing deterministic RIP matrices beyond the square root bottleneck is due to Bourgain et al. The aim of this paper is to construct sensing matrices consisting of two orthogonal bases and to analyse their RIP properties based on the flat-RIP. Using a known estimation on exponential sums due to Karatsuba, we deduce an RIP result for signals which are restricted to a certain sparse structure. Without any assumption on the sparsity structure, we end up facing a known open problem from number theory regarding exponential sums.