  In this paper, we develop a framework to design sensing matrices for
compressive sensing applications that lead to good mean squared error (MSE)
performance subject to sensing cost constraints. By capitalizing on the MSE of
the oracle estimator, whose performance has been shown to act as a benchmark to
the performance of standard sparse recovery algorithms, we use the fact that a
Parseval tight frame is the closest design - in the Frobenius norm sense - to
the solution of a convex relaxation of the optimization problem that relates to
the minimization of the MSE of the oracle estimator with respect to the
equivalent sensing matrix, subject to sensing energy constraints. Based on this
result, we then propose two sensing matrix designs that exhibit two key
properties: i) the designs are closed form rather than iterative; ii) the
designs exhibit superior performance in relation to other designs in the
literature, which is revealed by our numerical investigation in various
scenarios with different sparse recovery algorithms including basis pursuit
de-noise (BPDN), the Dantzig selector and orthogonal matching pursuit (OMP).
