The Null Space of the Delsarte-Goethals Frame

In compressed sensing (CS) [1–3], we wish to sense a signal f ∈ RC by taking its product with a matrix Φ ∈ RN×C to obtain a measurement vector y ∈ R . We refer to the rows of Φ as the projection vectors, as measurements y correspond to projections, or inner products, of the signal f onto the rows of the matrix Φ. When N C, this acquisition scheme effectively compresses the signal f . Since in this case the signal recover problem is ill posed, one must exploit prior information on the signal such as sparsity or compressibility. For example, we say a signal is K-sparse if only K out of the C entries of f are nonzero. CS relies on the use of sparse approximation algorithms, as well as specially tailored signal recovery algorithms based on sparsity, to recover the signal f from the measurements y and the CS matrix Φ. Most work in CS relies on random constructions on the matrix Φ; that is, the entries of the matrix are drawn independently from a suitable probability distribution such as Gaussian or Rademacher. Such matrices have been shown to provide enough information about a K-sparse signal f through the measurements y when N = O(K log C).