  Input design is an important issue for classical system identification
methods but has not been investigated for the kernel-based regularization
method (KRM) until very recently. In this paper, we consider in the time domain
the input design problem of KRMs for LTI system identification. Different from
the recent result, we adopt a Bayesian perspective and in particular make use
of scalar measures (e.g., the $A$-optimality, $D$-optimality, and
$E$-optimality) of the Bayesian mean square error matrix as the design criteria
subject to power-constraint on the input. Instead to solve the optimization
problem directly, we propose a two-step procedure. In the first step, by making
suitable assumptions on the unknown input, we construct a quadratic map
(transformation) of the input such that the transformed input design problems
are convex, the number of optimization variables is independent of the number
of input data, and their global minima can be found efficiently by applying
well-developed convex optimization software packages. In the second step, we
derive the expression of the optimal input based on the global minima found in
the first step by solving the inverse image of the quadratic map. In addition,
we derive analytic results for some special types of fixed kernels, which
provide insights on the input design and also its dependence on the kernel
structure.
