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Compressed sensing is a novel technique where one can recover sparse signalsfrom the undersampled measurements. In this paper, a $K \times N$ measurementmatrix for compressed sensing is deterministically constructed viamultiplicative character sequences. Precisely, a constant multiple of a cyclicshift of an $M$-ary power residue or Sidelnikov sequence is arranged as acolumn vector of the matrix, through modulating a primitive $M$-th root ofunity. The Weil bound is then used to show that the matrix has asymptoticallyoptimal coherence for large $K$ and $M$, and to present a sufficient conditionon the sparsity level for unique sparse solution. Also, the restricted isometryproperty (RIP) is statistically studied for the deterministic matrix. Numericalresults show that the deterministic compressed sensing matrix guaranteesreliable matching pursuit recovery performance for both noiseless and noisymeasurements.

Deterministic Compressed Sensing Matrices from Multiplicative Character Sequences