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| Compressed sensing is put forward in recent years as a new type of signal transmission theory framework.~Compressed sensing theorymainly includes three aspects:~the sparse representation of signal,~encoding measuring and reconstruction algorithm.~Sparse representation of signal is a priori condition of compressed sensing.~In the measurement of coding,~In order to keep the original structure of the signal,~projection matrix must satisfy restricted isometry conditions,~and then obtain linear projection measurement of the original signal through the product of original signal and measure matrix.~Finally,~reconstruct the original signal by the measured value and the projection matrix using the reconstruction algorithm.~In this paper, a new bound on the restricted isometry conditions for sparse signals recovery is established. For the recovery of high-dimensional sparse signals, this paper considers constraint $ell_1$ minimization methods in the noiseless. It is shown that if the sensing matrix $A$ satisfies the corresponding $RIP$ condition, then all $k$-$sparse$ signals $eta$ can be recovered exactly via the constrained $ell_{1}$ minimization based on $y=Aeta$, which has improved the bound that was established by T. Cai and A. Zhang (IEEE Trans. Inf. Theory, 2014). |
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| Keywords:Computational mathematics,Compressed sensing,Sparse signal recovery, Restricted isometry, $ell_1$ minimization |
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