Interior Point Decoding for Linear Vector Channels Based on Convex Optimization

In the present paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, which is referred to hereinafter as interior point decoding, is designed for linear vector channels. The linear vector channels include several practically important channels, such as inter-symbol interference channels and partial response (PR) channels. It is shown that the maximum likelihood decoding (MLD) rule for a linear vector channel can be relaxed to a convex optimization problem, which is called a relaxed MLD problem. The proposed decoding algorithm is based on a numerical optimization technique known as the interior point method with barrier functions. Approximate variations of an interior point method based on the gradient descent and Newton methods are used to solve the relaxed MLD problem. Compared with a conventional joint message-passing decoder, from computer simulations, it is observed that the proposed decoding algorithm achieves better BER performance on PR channels with less decoding complexity in several cases. Furthermore, an extension of the proposed algorithm for high-order modulation formats, such as PAM and QAM, is presented.