Spectral analysis of the DC kernel for regularized system identification

System identification with regularization methods has attracted increasing attention recently and is a complement to the current standard maximum likelihood/prediction error method. In this paper, we focus on the kernel-based regularization method and give a spectral analysis of the so-called diagonal correlated (DC) kernel, one family of kernel structures that has been proven useful for linear time-invariant system identification. In particular, using the theory of Bessel functions, we derive the eigenvalues and corresponding eigenfunctions of the DC kernel. Accordingly, we derive the Karhunen-Loève expansion of the stochastic process whose covariance function is the DC kernel.