  Compressed sensing with sparse frame representations is seen to have much
greater range of practical applications than that with orthonormal bases. In
such settings, one approach to recover the signal is known as
$\ell_1$-analysis. We expand in this article the performance analysis of this
approach by providing a weaker recovery condition than existing results in the
literature. Our analysis is also broadly based on general frames and
alternative dual frames (as analysis operators). As one application to such a
general-dual-based approach and performance analysis, an optimal-dual-based
technique is proposed to demonstrate the effectiveness of using alternative
dual frames as analysis operators. An iterative algorithm is outlined for
solving the optimal-dual-based $\ell_1$-analysis problem. The effectiveness of
the proposed method and algorithm is demonstrated through several experiments.
