Subspace-based spectrum estimation by nuclear norm minimization

Subspace-based methods have been effectively used to estimate multi-input/multi-output, linear-time-invariant systems from noisy spectrum samples. In these methods, a critical step is splitting of two invariant subspaces associated with causal and non-causal eigenvalues of some structured matrices built from spectrum measurements via singular-value decomposition in order to determine model order. Mirror image symmetry with respect to the unit circle between the eigenvalue sets of the invariant spaces, required by these algorithms, is lost due to low signal-to-noise ratio, unmodelled dynamics, and insufficient amount of data. Consequently, the choice of model order is not straightforward. In this paper, we propose a robust model order selection scheme based on regularized nuclear norm optimization in combination with a recent subspace algorithm, which uses non-uniformly spaced, in frequencies, spectrum measurements. A simulation example shows the effectiveness of the proposed scheme to large amplitude noise over short data records. Then, the proposed scheme is used to design a linear-shape filter for random road excitations.