  Compressed sensing is a novel technique where one can recover sparse signals
from the undersampled measurements. This paper studies a $K \times N$ partial
Fourier measurement matrix for compressed sensing which is deterministically
constructed via cyclic difference sets (CDS). Precisely, the matrix is
constructed by $K$ rows of the $N\times N$ inverse discrete Fourier transform
(IDFT) matrix, where each row index is from a $(N, K, \lambda)$ cyclic
difference set. The restricted isometry property (RIP) is statistically studied
for the deterministic matrix to guarantee the recovery of sparse signals. A
computationally efficient reconstruction algorithm is then proposed from the
structure of the matrix. Numerical results show that the reconstruction
algorithm presents competitive recovery performance with allowable
computational complexity.
