Joint Decoding of LDPC Codes and Finite-State Channels Via Linear-Programming

Abstract

This paper considers the joint-decoding problem for finite-state channels (FSCs) and low-density parity-check (LDPC) codes. In the first part, the linear-programming (LP) decoder for binary linear codes is extended to joint-decoding of binary-input FSCs. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs in AWGN. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the joint LP decoder also has the maximum-likelihood (ML) certificate property and all integer-valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional-valued JD-PCWs. After deriving these results, we discovered some elements were equivalent to earlier work by Flanagan on linearprogramming receivers.In the second part, we develop an efficient iterative solver for the joint LP decoder discussed in the first part. In particular, we extend the approach of iterative approximate LP decoding, proposed by Vontobel and Koetter and analyzed by Burshtein, to this problem. By taking advantage of the dual-domain structure of the joint-decoding LP, we obtain a convergent iterative algorithm for joint LP decoding whose structure is similar to BCJR-based turbo equalization (TE). The result is a joint iterative decoder whose per-iteration complexity is similar to that of TE but whose performance is similar to that of joint LP decoding. The main advantage of this decoder is that it appears to provide the predictability of joint LP decoding and superior performance with the computational complexity of TE. One expected application is coding for magnetic storage where the required block-error rate is extremely low and system performance is difficult to verify by simulation.

Linear Programming Decoding of Binary Linear Codes for Symbol-Pair Read Channel

“…However, this assumption may not hold for some cases. The LP decoding for finite-state channels is proposed in [13] and we can consider the symbol-pair read channel with finite state. To develop the LP decoding algorithm for such channel is a future work.…”

An Iterative Joint Linear-Programming Decoding of LDPC Codes and Finite-State Channels

“…In this paper, we consider the problem of low-complexity joint linear-programming (LP) decoding of low-density The dashed curves are computed using the union bound based on joint-decoding pseudo-codewords observed at 2.67 dB as described in [11] and the dash-dot curves are obtained using the state-based JIMPD described in [23]. The circle-solid curves are computed using Algorithm 1.…”

“…IV, the decoder performance results and conclude in Sec. V. Due to space limitations we omit many of the proofs, but they can be found in [20]. …”

Joint Detection and Decoding in LDPC-Based Space-Time Coded MIMO-OFDM Systems via Linear Programming

“…It is observed that the ESD channel matrix is column-orthogonal under OSTBC, satisfying . Thus, the ZF detection is simply (11) Note that the symbol is unquantized and thus can be further processed by soft-input soft-output decoder. Under Q-OSTBC, nonetheless, ZF detection does not admit such a simple form, due to the reason that the columns of channel matrix are not fully orthogonal.…”

“…For transmissions with higher-order modulations, the authors suggest transforming to an equivalent system with QPSK. On the other hand, Flanagan [10] proposed a unified framework for LP-based receivers, while Kim and Pfister considered the joint-decoding problem for finite-state channels [11]. Furthermore, without introducing a large number of extra variables, the authors in [12] extended the treatment of non-affine QAM mapping proposed in [10] for MIMO-OFDM systems.…”