Methods for Sparse Signal Recovery Using Kalman Filtering With Embedded Pseudo-Measurement Norms and Quasi-Norms

Abstract

We present two simple methods for recovering sparse signals from a series of noisy observations. The theory of compressed sensing (CS) requires solving a convex constrained minimization problem. We propose solving this optimization problem by two algorithms that rely on a Kalman filter (KF) endowed with a pseudo-measurement (PM) equation. Compared to a recently-introduced KF-CS method, which involves the implementation of an auxiliary CS optimization algorithm (e.g., the Dantzig selector), our method can be straightforwardly implemented in a stand-alone manner, as it is exclusively based on the well-known KF formulation. In our first algorithm, the PM equation constrains the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm of the estimated state. In this case, the augmented measurement equation becomes linear, so a regular KF can be used. In our second algorithm, we replace the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm by a quasi-norm <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lp</i> , 0 ¿ p < 1. This modification considerably improves the accuracy of the resulting KF algorithm; however, these improved results require an extended KF (EKF) for properly computing the state statistics. A numerical study demonstrates the viability of the new methods.