A robust MISO training sequence design

In this paper, the problem of robust training sequence design for multiple-input single-output (MISO) channel estimation is investigated. The mean-squared error (MSE) of the channel estimates is considered as a performance criterion to design an optimized training sequence which is a function of channel covariance matrix. In practice, the channel covariance matrix is not perfectly known at the transmitter side. Our goal is to take such imperfection into account and propose a robust design following the worst-case philosophy which results in finding the optimal training sequences for the least favorable channel covariance matrix within a deterministic uncertainty set. In this work, we address the formulated minimax design problem under different assumptions of the uncertainty set, and we show that for a unitarily-invariant uncertainty set, the optimally robust training sequence shares its eigenvectors with the channel covariance matrix. Furthermore, we give analytical closed-form solutions for robust training sequences if the spectral norm or nuclear norm are considered as constraints to bound the existing uncertainty.