Deterministic Compressed Sensing Matrices from Multiplicative Character Sequences

Abstract

Compressed sensing is a novel technique where one can recover sparse signals from the undersampled measurements. In this paper, a $K\times N$ measurement matrix for compressed sensing is deterministically constructed via multiplicative character sequences. Precisely, a constant multiple of a cyclic shift of an $M$-ary power residue or Sidelnikov sequence is arranged as a column vector of the matrix, through modulating a primitive $M$-th root of unity. The Weil bound is then used to show that the matrix has asymptotically optimal coherence for large $K$ and $M$, and to present a sufficient condition on the sparsity level for unique sparse solution. Also, the restricted isometry property (RIP) is statistically studied for the deterministic matrix. Numerical results show that the deterministic compressed sensing matrix guarantees reliable matching pursuit recovery performance for both noiseless and noisy measurements.

References in corpus (6)

• Construction of a Large Class of Deterministic Sensing Matrices that Satisfy a Statistical Isometry Property
• The Sparsity Gap: Uncertainty Principles Proportional to Dimension
• Deterministic Construction of Partial Fourier Compressed Sensing Matrices Via Cyclic Difference Sets
• Deterministic Sampling of Sparse Trigonometric Polynomials
• Reed Muller Sensing Matrices and the LASSO
• Deterministic Compressed Sensing Matrices from Additive Character Sequences