Exploiting Bilinear Forms: Time-varying Wireless Channel estimation via Non-convex and Convex Optimization

Abstract

Bilinear inverse problems such as non-negative matrix factorization and blind deconvolution are of strong current interest due to the possibility of exploiting modern structured signal processing methods such as compressed sensing and matrix completion. Herein, the estimation of a narrowband time-varying channel from its response to a known input signal under the practical assumptions of finite block length and finite transmission bandwidth is investigated. It is shown that the signal after passing through a time-varying narrowband channel, reveals a particular parametric low-rank structure that can be represented as a bilinear form. To estimate the channel, two structured methods are investigated. The first method exploits the low-rank bilinear structure of the channel via a non-convex strategy based on alternating direction optimization between the delay and Doppler directions. Due to the non-convex nature of this approach, a challenge is the presence of local minima. Thus, a novel convex approach based on minimization of the atomic norm is proposed. The number of required measurements for a given estimation resolution is analyzed and shown to scale linearly with the product of the number of channel paths and the maximum number of resolvable delay or Doppler values. Numerical results show that the performance of proposed algorithm is independent of the leakage effect and the new methods achieve significant gains over previously proposed methods that only exploit sparsity.