In this paper we show that if the restricted isometry constant δk​ of
the compressed sensing matrix satisfies δk​<0.307, then k-sparse
signals are guaranteed to be recovered exactly via ℓ1​ minimization when
no noise is present and k-sparse signals can be estimated stably in the noisy
case. It is also shown that the bound cannot be substantively improved. An
explicitly example is constructed in which δk​=2k−1k−1​<0.5,
but it is impossible to recover certain k-sparse signals