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The paper discusses the recovery of signals in the case that signals arenearly sparse with respect to a tight frame $D$ by means of the $l_1$-analysisapproach. We establish several new sufficient conditions regarding the$D$-restricted isometry property to ensure stable reconstruction of signalsthat are approximately sparse with respect to $D$. It is shown that if themeasurement matrix $\Phi$ fulfils the condition $\delta_{ts}<t/(4-t)$ for$0<t<4/3$, then signals which are approximately sparse with respect to $D$ canbe stably recovered by the $l_1$-analysis method. In the case of $D=I$, thebound is sharp, see Cai and Zhang's work \cite{Cai and Zhang 2014}. When $t=1$,the present bound improves the condition $\delta_s<0.307$ from Lin et al.'sreuslt to $\delta_s<1/3$.

New sufficient conditions of signal recovery with tight frames via $l_1$-analysis