Unveiling Bias Compensation in Turbo-Based Algorithms for (Discrete) Compressed Sensing

In Compressed Sensing, a real-valued sparse vector has to be recovered from an underdetermined system of linear equations. In many applications, however, the elements of the sparse vector are drawn from a finite set. Adapted algorithms incorporating this additional knowledge are required for the discrete-valued setup. In this paper, turbo-based algorithms for both cases are elucidated and analyzed from a communications engineering perspective, leading to a deeper understanding of the algorithm. In particular, we gain the intriguing insight that the calculation of extrinsic values is equal to the unbiasing of a biased estimate and present an improved algorithm

Bias Compensation in Iterative Soft-Feedback Algorithms with Application to (Discrete) Compressed Sensing

In all applications in digital communications, it is crucial for an estimator to be unbiased. Although so-called soft feedback is widely employed in many different fields of engineering, typically the biased estimate is used. In this paper, we contrast the fundamental unbiasing principles, which can be directly applied whenever soft feedback is required. To this end, the problem is treated from a signal-based perspective, as well as from the approach of estimating the signal based on an estimate of the noise. Numerical results show that when employed in iterative reconstruction algorithms for Compressed Sensing, a gain of 1.2 dB due to proper unbiasing is possible

Low-Complexity Iterative Algorithms for (Discrete) Compressed Sensing

We consider iterative (`turbo') algorithms for compressed sensing. First, a unified exposition of the different approaches available in the literature is given, thereby enlightening the general principles and main differences. In particular we discuss i) the estimation step (matched filter vs. optimum MMSE estimator), ii) the unbiasing operation (implicitly or explicitly done and equivalent to the calculation of extrinsic information), and iii) thresholding vs. the calculation of soft values. Based on these insights we propose a low-complexity but well-performing variant utilizing a Krylov space approximation of the optimum linear MMSE estimator. The derivations are valid for any probability density of the signal vector. However, numerical results are shown for the discrete case. The novel algorithms shows very good performance and even slightly faster convergence compared to approximative message passing