Compressed Sensing Using Prior Information

Compressed sensing has recently emerged as a technique allowing a discrete-time signal with a sparse representation in some domain to be reconstructed with theoretically perfect accuracy from a limited number of linear measurements. Current applications range from sensor networks to tomography and general medical imaging. In this paper, we show that the amount of samples which must be taken from a signal with sparse Discrete-time Fourier Transform (DFT) can be reduced compared to the original compressed sensing approach if information on the support of the sparse domain can be employed. More precisely, the required number of samples in time domain is reduced by exactly the amount of known frequencies associated to non-zero coefficients. Our results additionally provide a link between the so-called fractional Fourier transform and compressed sensing framework, when the positions of all the non-zero components are known.