This work considers the design of short non-binary low-density parity-check (LDPC) codes over finite fields of order m, for channels with phase noise. In particular, m-ary differential phase-shift keying (DPSK) modulated code symbols are transmitted over an additive white Gaussian noise (AWGN) channel with Wiener phase noise. At the receiver side, noncoherent detection takes place, with the help of a multi-symbol detection algorithm, followed by a non-binary decoding step. Both the detector and decoder operate on a joint factor graph. As a benchmark, finite length bounds and information rate expressions are computed and compared with the codeword error rate (CER) performance, as well as the iterative threshold of the obtained codes. As a result, performance within 1.2 dB from finite-length bounds is obtained, down to a CER of 10 −3 .

In this paper, we study digital transmission over an additive white Gaussian noise (AWGN) channel with mary differential phase-shift keying (DPSK) modulation in the presence of phase noise. At the receiver side, non-coherent iterative detection and decoding is assumed. We present a nonbinary low-density generator matrix (LDGM) code design which is suitable for both coherent and non-coherent channels. The code construction is strongly related to the one of non-binary irregular repeat-accumulate (IRA) low-density parity-check (LDPC) codes.

We propose a decoding algorithm for tail-biting convolutional codes over phase noise channels. It can be seen as a reduced complexity approximation of maximum-likelihood decoding. We target short blocks and extend the wrap-around Viterbi algorithm to trellises describing the random evolution of the phase impairment, for which we adopt two different models: a blockwise non-coherent and a blockwise Wiener channel model. Numerical results show that the performance of the proposed algorithm is within a few tenths of dB or less from maximum likelihood decoding for the setup studied in this paper.

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