  Kernel-based methods have been successfully introduced in system
identification to estimate the impulse response of a linear system. Adopting
the Bayesian viewpoint, the impulse response is modeled as a zero mean Gaussian
process whose covariance function (kernel) is estimated from the data. The most
popular kernels used in system identification are the tuned-correlated (TC),
the diagonal-correlated (DC) and the stable spline (SS) kernel. TC and DC
kernels admit a closed form factorization of the inverse. The SS kernel induces
more smoothness than TC and DC on the estimated impulse response, however, the
aforementioned property does not hold in this case. In this paper we propose a
second-order extension of the TC and DC kernel which induces more smoothness
than TC and DC, respectively, on the impulse response and a
generalized-correlated kernel which incorporates the TC and DC kernels and
their second order extensions. Moreover, these generalizations admit a closed
form factorization of the inverse and thus they allow to design efficient
algorithms for the search of the optimal kernel hyperparameters. We also show
how to use this idea to develop higher oder extensions. Interestingly, these
new kernels belong to the family of the so called exponentially convex local
stationary kernels: such a property allows to immediately analyze the frequency
properties induced on the estimated impulse response by these kernels.
