Sparse FIR estimation of low-order systems

This paper discusses estimation of the finite impulse response (FIR) for a linear time-invariant (LTI) system. Specifically, we focus on the case where the FIR sequence is sparse and the system model is low-order; the latter is equivalent to that the Hankel matrix constructed from the FIR sequence is low-rank. These two properties motivate us to propose a unified system identification framework, which minimizes weighted sum of three norms: the ℓ₂ norm of measurement errors for data fidelity, the ℓ₁ norm of FIR sequence for its sparsity, and the nuclear norm of Hankel matrix for its low-rankness. We further develop an optimal algorithm based on the alternating direction method (ADM) for this convex program. Numerical experiments verify the effectiveness of the proposed identification framework and the developed algorithm.