Statistical mechanics of LDPC codes on channels with memory

Introduction. – A common problem in modern mobile telecommunication systems is that the strength of the signal varies over time as a result of e.g. the motion of the receiver with respect to the source and the varying number of obstacles that shadow the signal over time. Channels describing communication of attenuated signals are termed ‘fading channels’. Fading channels are modeled by finite-state Markov channels (FSMC) [1]. These channels have fueled significant research activity (for a recent review on the subject see [2]). In FSMCs there exist a number of different channel states that correspond to the various possible attenuation factors. Each of the states describes a memoryless channel characterized by an error probability, while, the transition from one state to another occurs according to a stationary Markov process. Since there are different states in the FMSC the errorprobabilities between subsequent uses of the channel are correlated, i.e. there is memory in the channel. One of the central problems in the domain of errorcorrecting codes is the design of codes that reach Shannon’s limit. The gap between the Shannon limit and the computational limit was closed by turbo codes [3] and by low-density parity-check codes (LDPC) [4, 5]. For erasure channels it was shown that LDPC can reach the Shannon capacity [6] while for general symmetric channels one can approach the Shannon limit [7]. To design capacity approaching LDPC-codes one uses the density evolution (DE) equations to determine the decoding thresholds [8]. Since channels with memory have a higher capacity [9,10] one would like to introduce memory in the decoding pro-