  Low-density parity-check (LDPC) convolutional codes (or spatially-coupled
codes) have been shown to approach capacity on the binary erasure channel (BEC)
and binary-input memoryless symmetric channels. The mechanism behind this
spectacular performance is the threshold saturation phenomenon, which is
characterized by the belief-propagation threshold of the spatially-coupled
ensemble increasing to an intrinsic noise threshold defined by the uncoupled
system.
  In this paper, we present a simple proof of threshold saturation that applies
to a broad class of coupled scalar recursions. The conditions of the theorem
are verified for the density-evolution (DE) equations of irregular LDPC codes
on the BEC, a class of generalized LDPC codes, and the joint iterative decoding
of LDPC codes on intersymbol-interference channels with erasure noise. Our
approach is based on potential functions and was motivated mainly by the ideas
of Takeuchi et al. The resulting proof is surprisingly simple when compared to
previous methods.
