Check Splitting of Root-Check LDPC Codes over ARQ Block-Fading Channels Sri Krishna Kambhampati, Gottfried Lechner, Terence Chan Institute for Telecommunications Research University of South Australia Adelaide, Australia Sri.Kambhampati@postgrads.unisa.edu.au, Gottfried.Lechner@unisa.edu.au, Terence.Chan@unisa.edu.au Lars K. Rasmussen School of Electrical Engineering KTH - Royal Institute of Technology Stockholm, Sweden lkra@kth.se Abstract—In this paper we extend the concept of check split- ting to root-check LDPC codes, thus providing an incremental redundancy code construction specifically for the ARQ block- fading channel. By construction, the proposed coding scheme achieves a high level of diversity and effectively adapts the transmission rate to the instantaneous channel conditions. I. I NTRODUCTION Due to multi-path propagation and mobility, wireless sys- tems are characterized by time-varying channels with fluctuat- ing signal strength. In applications with no delay constraints, the channel can be considered ergodic, where each transmit- ted symbol experiences independent fading realizations. Con- versely, for applications subject to delay constraints and slowly varying channels, only limited independent fading realizations are experienced. In such non-ergodic scenarios, the channel capacity is zero since there is an irreducible probability, termed outage probability [1], [2], that the transmitted data rate is not supported by the channel. The block-fading channel is a simple and useful model that captures the essential characteristics of non-ergodic channels [1], [2]. The fundamental limit and optimal transmission strategy for a non-ergodic channel depend on the availability of channel state information (CSI) [2]. When perfect CSI is available at the transmitter (CSIT), the optimal coding strategy adapts to the channel conditions by either adjusting the transmission rate or the power allocation. Transmitter CSI may achieve very large gains as shown in [3]–[6] for various constraints on signal constellations, and subject to both short-term and long-term power constraints. Automatic-repeat-request (ARQ) strategies provide partial CSIT through a one-bit feedback, which significantly improves reliability at the expense of increasing the transmission delay [7], [8]. By allowing for multi-bit feedback in the ARQ scheme, it is possible to adapt both rate and power optimally, leading to further diversity gains [9], [10]. The design of fixed-rate outage-approaching codes for non- ergodic channels is complex and involves multi-edge type density evolution [11] and outage regions [12]. It has been shown that maximal distance separable (MDS) codes are necessary to achieve the optimal SNR exponent but not sufficient to approach the outage probability [13], [14]. Using the optimal design criteria turbo-like code constructions have been proposed [13]. Other specific constructions have been proposed for turbo codes [15] and LDPC codes. In particular a class of LDPC codes, termed root-check LDPC codes, has been recently proposed in [14]. These codes are specifically designed to achieve full diversity and a significant coding gain. So far, no specific code design for the ARQ block-fading channel has been proposed. In [8] constructions based on convolutional codes and iterative decoding were investigated for incremental redundance coding over block-fading channels, while check splitting was proposed in [16] as an effective way of generating incremental redundancy coding schemes on the Gaussian channel. A check node with high degree is divided into two checks with almost equal degrees. This division generates a new parity bit so that the resulting parity check equations are satisfied. The contribution in this paper is to extend the concept of check splitting to root-check LDPC codes, thus providing an incremental redundancy code construction specifically for the ARQ block-fading channel. A maximum-rate (for full diver- sity) root-check LDPC code is used for transmission in the first ARQ round, while check splitting is applied to generate redundancy in subsequent rounds. Under the constraint of maximum diversity, we find that the resulting code subject to constrained check splitting corresponds to simple repetition coding. The proposed coding scheme achieves maximum diversity by construction, and effectively adapts the rate to the instantaneous channel conditions. The paper is organized as follows. In Section II, the system model is described and in Section III a brief overview of root- check LDPC codes and check splitting algorithm is given. The extension of check splitting to root-check LDPC codes is proposed in Section IV, and simple examples for block-erasure and block-fading channels are included. II. SYSTEM MODEL We consider an ARQ system where the transmission medium is modelled as a block-fading channel with a coher- AusCTW 2010 978-1-4244-5434-1/10/$26.00 ©2010 IEEE 123 Authorized licensed use limited to: KTH THE ROYAL INSTITUTE OF TECHNOLOGY. Downloaded on August 05,2010 at 08:01:40 UTC from IEEE Xplore. Restrictions apply.