Structured Sparse Signal Recovery for mmWave Channel Estimation: Intra-vector Correlation and Modulo Compressed Sensing
This thesis contributes new theoretical results and recovery algorithms for the area of sparse signal recovery motivated by applications to the problem of channel estimation in mmWave communication systems. The presentation is in two parts. The first part focuses on the recovery of sparse vectors with correlated non-zero entries from their noisy low dimensional projections. Such structured sparse signals can be recovered using the technique of covariance matching. Here, we first estimate the covariance of the signal from the compressed measurements, and then use the obtained covariance matrix estimate as a plug-in to the linear minimum mean squared estimator to obtain an estimate of the sparse vector. We present a novel parametric Gaussian prior model, inspired by sparse Bayesian learning (SBL), which captures the underlying correlation in addition to the sparsity. Based on this prior, we develop a novel Bayesian learning algorithm called Corr-SBL, using the expectation-maximization procedure. This algorithm learns the parameters of the prior and updates the posterior estimates in an iterative fashion, thereby yielding a sparse vector estimate upon convergence. We present a closed form solution for the hyperparameter update based on fixed-point iterations. In case of imperfect correlation information, we present a pragmatic approach to learn the parameters of the correlation matrix in a data-driven fashion. Next, we apply Corr-SBL to the channel estimation problem in mmWave multiple-input multiple-output systems employing a hybrid analog-digital architecture. We use noisy low dimensional projections of the channel obtained in the pilot transmission phase to estimate the channel across multiple coherence blocks. We show the efficacy of the Corr-SBL prior by analyzing the error in the channel estimates. Our results show that, compared to a genie-aided estimator and other existing sparse recovery algorithms, exploiting both sparsity and correlation results in significant performance gains, even under imperfect covariance estimates obtained using a limited number of samples. In the second part of the presentation, we consider the sparse signal recovery problem when low-resolution ADCs with finite resolution are used in the measurement acquisition process. To counter the effect of signal clipping in these systems, we use modulo arithmetic to fold the measurements crossing the range back into the dynamic range of the system. For this setup, termed as modulo-CS, we answer the fundamental question of signal identifiability, by deriving conditions on the measurement matrix and the minimal number of measurements required for unique recovery of sparse vectors. We also show that recovery using the minimum required number of measurements is possible when the entries of the measurement matrix are drawn independently from any continuous distribution. Finally, we present an algorithm based on convex relaxation, and formulate a mixed integer linear program (MILP) for recovery of sparse vectors under modulo-CS. Our empirical results show that the minimum number of measurements required for the MILP is close to the theoretical result, for signals with low variance.