  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.
