Bilinear Matrix Factorization Methods for Time-Varying Narrowband Channel Estimation: Exploiting Sparsity and Rank

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In this paper, the estimation of a narrowband time-varying channel 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 useful parametric low-rank structure that can be represented as a bilinear form. To estimate the channel, two strategies are developed. The first method exploits the low-rank bilinear structure of the channel via a non-convex strategy based on an alternating direction optimization between the delay and Doppler directions. While prior Wirtinger flow methods have exhibited good performance with proper initialization, this is not true in the current scenario. Due to the non-convex nature of this approach, this first approach is sensitive to local minima. Furthermore, the convergence rate of the Wirtinger flow method is shown to be provably modest. Thus, a novel convex approach based on the minimization of the atomic norm using measurements of the signal at the time domain is proposed based on a second bilinear parametrization of the channel. For the convex approach, optimality and uniqueness conditions, and a theoretical guarantee for noiseless channel estimation with small number of measurements are characterized. Numerical results show that the performance of the proposed algorithm is independent of the leakage effect and the new methods can achieve a 5–12 dB improvement, on average, compared to a classical $l_1$ -based sparse approximation method that does not consider the leakage effect, and 2 dB improvement over a basis expansion method that considers the leakage effect.