Subspace-based spectrum estimation in frequency-domain by regularized nuclear norm minimization

• Subspace-based methods are effective to estimate MIMO systems from noisy spectrum samples.
• A critical step is splitting of causal/noncausal invariant subspaces to determine model order.
• Splitting requires mirror image symmetry of eigenvalues.
• We propose a robust model order selection scheme based on regularized nuclear norm optimization.

Subspace-based methods have been effectively used to estimate multi-input/multi-output, discrete-time, 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 two invariant spaces, required by the subspace algorithms, is lost due to low signal-to-noise ratio, unmodeled dynamics, and insufficient amount of data. Consequently, the choice of model order is not straightforward. In this paper, we propose a new model order selection scheme that is insensitive to noise and undermodeling and based on the regularized nuclear norm optimization in combination with a recently developed subspace-based spectrum estimation algorithm which uses non-uniformly spaced, in frequencies, spectrum measurements. A detailed simulation study shows the effectiveness of the proposed scheme to large amplitude noise over short data records. Examples illustrating application of the proposed scheme to real-life problems are also presented. The proposed scheme can be readily integrated into frequency-domain instrumental variable subspace algorithms to estimate auto-power spectral density or cross-power spectral density function matrices from non-uniformly spaced, in frequencies, spectrum measurements.