  Signals with sparse frame representations comprise a much more realistic
model of nature than that with orthonomal bases. Studies about the signal
recovery associated with such sparsity models have been one of major focuses in
compressed sensing. In such settings, one important and widely used signal
recovery approach is known as $\ell_1$-synthesis (or Basis Pursuit). We present
in this article a more effective performance analysis (than what are available)
of this approach in which the dictionary $\Dbf$ may be highly, and even
perfectly correlated. Under suitable conditions on the sensing matrix $\Phibf$,
an error bound of the recovered signal $\hat{\fbf}$ (by the $\ell_1$-synthesis
method) is established. Such an error bound is governed by the decaying
property of $\tilde{\Dbf}_{\text{o}}^*\fbf$, where $\fbf$ is the true signal
and $\tilde{\Dbf}_{\text{o}}$ denotes the optimal dual frame of $\Dbf$ in the
sense that $\|\tilde{\Dbf}_{\text{o}}^*\hat{\fbf}\|_1$ produces the smallest
$\|\tilde{\Dbf}^*\tilde{\fbf}\|_1$ in value among all dual frames
$\tilde{\Dbf}$ of $\Dbf$ and all feasible signals $\tilde{\fbf}$. This new
performance analysis departs from the usual description of the combo
$\Phibf\Dbf$, and places the description on $\Phibf$. Examples are demonstrated
to show that when the usual analysis fails to explain the working performance
of the synthesis approach, the newly established results do.
