The Velocity of the Propagating Wave for Spatially Coupled Systems with Applications to LDPC Codes

Abstract: We consider the dynamics of message passing for spatially coupled codes and, in particular, the set of density evolution equations that tracks the profile of decoding errors along the spatial direction of coupling. It is known that, for suitable boundary conditions and after a transient phase, the error profile exhibits a "solitonic behavior". Namely, a uniquely-shaped wavelike solution develops, that propagates with constant velocity. Under this assumption we derive an analytical formula for the velocity in the framework of a continuum limit of the spatially coupled system. The general formalism is developed for spatially coupled low-density parity-check codes on general binary memoryless symmetric channels which form the main system of interest in this work. We apply the formula for special channels and illustrate that it matches the direct numerical evaluation of the velocity for a wide range of noise values. A possible application of the velocity formula to the evaluation of finite size scaling law parameters is also discussed. We conduct a similar analysis for general scalar systems and illustrate the findings with applications to compressive sensing and generalized low-density parity-check codes on the binary erasure or binary symmetric channels.