Compressed sensing has captured considerable attention of researchers in the past decades. In this paper, with the aid of the powerful null space property, some deterministic
recovery conditions are established for the previous ℓ 1 -ℓ 1 and ℓ 1 -ℓ 2 methods to guarantee the exact sparse recovery when the side information of the desired signal is available. These obtained results provide a useful and necessary complement to the previous investigation of the ℓ 1 -ℓ 1 and ℓ 1 -ℓ 2 methods that are based on the statistical analysis. Moreover, one of our theoretical findings also shows that, the sharp conditions previously established for the classical ℓ 1 method remain suitable for the ℓ 1 -ℓ 1 method to guarantee the exact sparse recovery. Numerical experiments are also carried out to further verify the recovery performance of both ℓ 1 -ℓ 1 and ℓ 1 -ℓ 2 methods.