  In this paper, with the aid of the powerful Restricted Isometry Constant
(RIC), a deterministic (or say non-stochastic) analysis, which includes a
series of sufficient conditions (related to the RIC order) and their resultant
error estimates, is established for the weighted Basis Pursuit De-Noising
(BPDN) to guarantee the robust signal recovery when Partially Known Support
Information (PKSI) of the signal is available. Specifically, the obtained
conditions extend nontrivially the ones induced recently for the traditional
constrained weighted $\ell_{1}$-minimization model to those for its
unconstrained counterpart, i.e., the weighted BPDN. The obtained error
estimates are also comparable to the analogous ones induced previously for the
robust recovery of the signals with PKSI from some constrained models.
Moreover, these results to some degree may well complement the recent
investigation of the weighted BPDN which is based on the stochastic analysis.
