Least-squares Khatri-Rao factorization of a polynomial matrix

The Khatri-Rao product is extensively used in array processing, tensor decomposition, and multi-way data analysis, where many applications rely on a least-squares (LS) Khatri-Rao factorization. In broadband sensor array problems, polynomial matrices effectively model frequency-dependent behaviors, necessitating extensions of conventional linear algebra techniques. This paper generalizes the LS Khatri-Rao factorization from ordinary to polynomial matrices by applying it to the discrete Fourier transform (DFT) sample points of polynomial matrices. Phase coherence across bin-wise Khatri-Rao factors is ensured via a phase-smoothing algorithm. The proposed phase smoothing method is validated through broadband angle-of-arrival (AoA) estimation for uniform planar arrays (UPAs), where the steering matrix is a polynomial matrix and can be represented as a Khatri-Rao product between decoupled steering matrices in azimuth and elevation directions.