DESIGN OF GOOD LOW-DENSITY PARITY-CHECK CODES FOR BLOCK FADING CHANNELS

For coherent estimation of block fading channels, where the channel phase is known, the continuous-valued fading levels must be estimated. Most of the existing iterative receiver designs take the approach of quantizing the fading levels into a finite number of values [10], [11], [16]. Jin et al. [13] derived an iterative receiver for coherent block fading channels that estimated the continuous value of the fading levels; density evolution was used to analyze the performance of the resulting receiver.
Recently, LDPC codes have been successfully applied to wireless communication channels and other channels with memory. Worthen et al. first applied LDPC codes to a simplified two-state block fading channel with memory and developed a joint channel estimation and decoding algorithm [10]. Eckford et al. [14], [15] analyzed LDPC performance over a class of Markov-modulated channels and designed good irregular LDPC codes for those channels. Jin et al. [16] considered finite-state block fading channels with an iterative channel estimation and LDPC decoding algorithm, and analyzed the performance of the iterative algorithm on both correlated and independent block fading channels using density evolution. Hou et al. [12] used density evolution and differential evolution to design good irregular LDPC codes for Rayleigh fading channels with perfect channel state information (CSI).
Regular LDPC codes were first introduced by Gallager [2]. These were extended to the construction and analysis of random irregular LDPC codes over erasure
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Si Ti1 Vi1
LDPC edge permutation
Ck
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Fig. 1. A fraction of factor graph for a block fading channel with LDPC code, where channel states are fading amplitudes and independent with each other. In the graph, Tij is the channel characteristic function and Vij is the variable node, both are for the jth symbol of the ith frame, and Ck is the kth parity check node.
Graphical models, such as factor graphs [7], [8], [9], provide a unified approach for designing iterative receivers [11]. Given a factor graph representation of the receiver, the iterative sum-product algorithm can be derived on the graph. Performance analysis of the resulting receiver with LDPC codes is also possible by means of density evolution; this leads to the design of good codes matched to the receiver.
The block fading channel is a widely used model for dispersive wireless channels [1]. In this model, the channel state remains fixed over a block (or frame) of a given size; the channel states may be independent or correlated between successive blocks. This model is a reasonable approximation to a wide variety of practical communication systems, such as orthogonal frequency division multiplexing (OFDM), time-division multiple access (TDMA), frequency-hopped spread-spectrum (FHSS) systems and slow fading channels.
I. INTRODUCTION
Joint channel estimation and decoding is the optimal way to achieve reliable communication over time-varying wireless channels. A good approximation to this optimal scheme with reasonable complexity is the so called iterative receiver, which combines the channel with iteratively decodable codes and carries out channel estimation and decoding iteratively. Many such receivers have been proposed recently. However, little work has been done on designing good codes that are well-matched to the structure of iterative receivers. This paper deals with LDPC code design for coherent block fading channels under joint channel estimation and decoding.
DESIGN OF GOOD LOW-DENSITY PARITY-CHECK CODES FOR BLOCK FADING CHANNELS
Xiaowei Jin, Andrew W. Eckford, and Thomas E. Fuja Department of Electrical Engineering, University of Notre Dame
Notre Dame, Indiana, USA 46556
ABSTRACT
The focus of this paper is the design of good irregular low density parity check (LDPC) codes for Rayleigh block fading channels using a strategy of joint channel estimation and LDPC decoding. It is assumed that the amplitude of the fading coefficient is unknown. A density evolution technique employing a semi-Gaussian approximation is used to find good degree sequences for irregular LDPC codes for channels with different block memory sizes. The capacity of the BPSK-modulated Rayleigh block fading channel with unknown fading amplitude is used as a benchmark. For a block fading channel with a memory size of 20 bits, the best irregular LDPC code found in this paper has a threshold that is 0.47dB away from the channel capacity.