  Spatially coupled low-density parity-check codes show an outstanding
performance under the low-complexity belief propagation (BP) decoding
algorithm. They exhibit a peculiar convergence phenomenon above the BP
threshold of the underlying non-coupled ensemble, with a wave-like convergence
propagating through the spatial dimension of the graph, allowing to approach
the MAP threshold. We focus on this particularly interesting regime in between
the BP and MAP thresholds.
  On the binary erasure channel, it has been proved that the information
propagates with a constant speed toward the successful decoding solution. We
derive an upper bound on the propagation speed, only depending on the basic
parameters of the spatially coupled code ensemble such as degree distribution
and the coupling factor $w$. We illustrate the convergence speed of different
code ensembles by simulation results, and show how optimizing degree profiles
helps to speed up the convergence.
