Wave-Like Solutions of General 1-D Spatially Coupled Systems
We establish the existence of wave-like solutions to spatially coupled graphical models which, in the large size limit, can be characterized by a 1-D real-valued state. This is extended to a proof of the threshold saturation phenomenon for all such models, which includes spatially coupled irregular low-density parity-check codes over the binary erasure channel (BEC), but also addresses hard-decision decoding for transmission over general channels, the code division multiple access problem, compressed sensing, and some statistical physics models. For traditional uncoupled iterative coding systems with two components and transmission over the BEC, the asymptotic convergence behavior is completely characterized by the EXIT curves of the components. In particular, the system converges to the desired fixed point, which is the one corresponding to perfect decoding, if and only if the two EXIT functions describing the components do not cross. For spatially coupled systems whose state is 1-D a closely related graphical criterion applies. Now the curves are allowed to cross, but not by too much. More precisely, we show that the threshold saturation phenomenon is related to the positivity of the (signed) area enclosed by two EXIT-like functions associated to the component systems, a very intuitive, and easy-to-use graphical characterization. In the spirit of EXIT functions and Gaussian approximations, we also show how to apply the technique to higher dimensional and even infinite-dimensional cases. In these scenarios, the method is no longer rigorous, but it typically gives accurate predictions. To demonstrate this application, we discuss transmission over general channels using both the belief-propagation as well as the min-sum decoder.