Projection Matrix Optimization for Sparse Signals in Structured Noise

We consider the problem of estimating a signal which has been corrupted with structured noise. When the signal of interest accepts a sparse representation, only a small number of measurements are required to retain all the information. The measurements are mapped to a lower dimensional space through a projection matrix. We propose a method to optimize the design of this matrix where the objective is not only to reduce the amount of data to be processed but also to reject the undesired signal components. As a result, we reduce the computation time and the error on the estimation of the unknown parameters of the sparse model, with respect to the uncompressed data. The proposed method has tunable parameters that can affect its performance. Optimal tuning would require a comprehensive study of parameter variations and options. To avoid this learning burden, we also introduce a variant of the algorithm that is free from tuning, without significant loss of performance. Using synthetic data, we analyze the performance of the proposed algorithms and their robustness against errors in the model parameters. Additionally, we illustrate the performance of the method through a radar application using real clutter data with a still target and with a synthetic moving target.