Convergence of Iterative Decoding in Channels with Memory

In [1], the authors give a criterion based on the Bhattacharyya parameter to show the convergence of symbol error rate in an iterative estimation-decoding system containing LDPC codes, known as the weak sense iterative decoding limit. This result has implications not only for iterative decoding, but for the convergence of the density evolution procedure, and our main result extends it to channels with memory. We consider estimation-decoding schemes that are defined on factor graphs, using the sum-product algorithm (SPA). A channel with memory is any channel where there is some dependence in either the noise process or the channel output from channel use to channel use. If the channel inputs are the symbols from an error-correcting code, channel state estimation can be performed jointly with decoding to improve performance. A common way to combine these tasks is to use factor graphs and the SPA [2]. See [3] for an example of an estimation-decoding scheme in the Gilbert-Elliott channel. We now review the result from [1]. Let ψZ,0(z) (resp., ψZ,1(z)) represent the probability density function (PDF) of a particular extrinsic message in sum-product decoding, assuming the underlying symbol is 0 (resp., 1). The Bhattacharyya parameter (BP) corresponding to this pair of densities is given by B = � ∞ � ψZ,0(z)ψZ,1(z)dz. It is easy to show that B is an upper bound on the probability of symbol error for an optimal decision using the densities ψZ,0(z) and ψZ,1(z). A similar expression is used for the channel message, though we use the symbol A. That is, A = � ∞ � ψC,0(c)ψC,1(c)dc. For decoding in memoryless channel, B will change with the number of iterations, so we will write B (ℓ) to represent its value at the ℓth iteration. For a regular LDPC code in a memoryless channel with variable degree J and check degree K, where J ≥ 3, there exists a bound on B (ℓ) given b