低密度奇偶校验码设计：调制与检测优化

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本文档深入探讨了低密度奇偶校验（Low-Density Parity-Check, LDPC）码在调制与检测中的设计方法。论文发表在2004年4月的《IEEE Transactions on Communications》上，由Stephanten Brink、Gerhard Kramer（IEEE成员）和Alexei Ashikhmin（IEEE成员）共同完成。他们研究了一种编码和调制技术，其中LDPC码的编码比特直接传递给调制器，而非传统的先解码再调制的流程。在接收端，该设计将LDPC解码器图的变量节点连接到检测器节点，通过将变量节点和检测器节点视为一个单一的解码器，实现了迭代解码。这种方法利用了外部信息转移（Extrinsic Information Transfer,EXIT）图表进行优化，通过曲线拟合来调整码的性能。作者关注的是两种主要的通信环境：加性白高斯噪声（Additive White Gaussian Noise, AWGN）信道以及多输入多输出（Multiple Input Multiple Output, MIMO）衰落信道，其中接收端知道但发送端不知道信道状态。在MIMO信道场景下，该LDPC编码与检测技术表现出色，即使在传输天线数量多于接收天线的情况下，也能在接近信道容量的1.25分贝范围内工作。这明显优于采用并行连接（turbo）编码方案，当发射端天线数超过接收端时，性能优势更为显著。论文的关键知识点包括： 1. **直接LDPC码调制**：将编码过程与调制过程整合，减少了信号处理步骤，提高了系统效率。 2. **异步解码与检测**：通过合并解码器和检测器节点，实现了一体化的解码流程。 3. **EXIT图表优化**：利用EXIT图表分析解码性能，通过曲线拟合来优化编码结构。 4. **应用范围**：适用于AWGN信道和MIMO信道，尤其在多径传输环境中表现出优越的性能。 5. **MIMO性能对比**：在多天线设置下，与turbo编码相比，展现出明显的性能提升。总体来说，这篇论文为低密度奇偶校验码在现代通信系统的高效调制与检测设计提供了创新性的解决方案，对无线通信领域的编码理论和技术发展具有重要意义。

670 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004

Design of Low-Density Parity-Check

Codes for Modulation and Detection

Stephan ten Brink, Gerhard Kramer, Member, IEEE, and Alexei Ashikhmin, Member, IEEE

Abstract—A coding and modulation technique is studied where

the coded bits of an irregular low-density parity-check (LDPC)

code are passed directly to a modulator. At the receiver, the variable

nodes of the LDPC decoder graph are connected to detector nodes,

and iterative decoding is accomplished by viewing the variable and

detector nodes as one decoder. The code is optimized by performing

a curve fitting on extrinsic information transfer charts. Design ex-

amples are given for additive white Gaussian noise channels, as

well as multiple-input, multiple-output (MIMO) fading channels

where the receiver, but not the transmitter, knows the channel.

For the MIMO channels, the technique operates within 1.25 dB of

capacity for various antenna configurations, and thereby outper-

forms a scheme employing a parallel concatenated (turbo) code by

wide margins when there are more transmit than receive antennas.

Index Terms—Fading, iterative decoding, low-density

parity-check (LDPC) codes, multiple-input, multiple-output

(MIMO) detection, mutual information.

I. INTRODUCTION

TERATIVE decoding of low-density parity-check (LDPC)

codes is a powerful method for approaching capacity on

noisy channels [1]–[5]. We consider two problems associated

with LDPC codes. The first is how to combine a code with a

modulator and detector. The second is how to design the code

for iterative decoding, i.e., how to choose good degree distribu-

tions for the modulator, channel, and detector.

We approach the first problem by mapping the coded bits of

an irregular LDPC code directly onto a modulation signal set.

The mapping is arranged to facilitate code design. At the re-

ceiver, we consider the graphical representation of an LDPC de-

coder [2]–[4] and connect the LDPC variable nodes to detector

nodes.

We deal with the second problem by using a

curve-fitting

procedure on extrinsic information transfer (EXIT) charts

[6]. The design methodology is illustrated for two types of

channels: additive white Gaussian noise (AWGN) channels

with binary phase-shift keying (BPSK), and multiple-input,

multiple-output (MIMO) fading channels with quadrature

phase-shift keying (QPSK). The MIMO code design can be

be extended in a straightforward way to other modulators,

channels, and detectors. We remark that the curve fitting might

Paper approved by H. El Gamal, the Editor for Space–Time Coding and

Spread Spectrum of the IEEE Communications Society. Manuscript received

July 17, 2002; revised May 30, 2003.

S. ten Brink was with Bell Laboratories, Lucent Technologies, Crawford, NJ.

He is now with Realtek, Irvine, CA 92618 USA (e-mail: stenbrink@realtek-

us.com).

G. Kramer and A. Ashikhmin are with Bell Laboratories, Lucent

Technologies, Murray Hill, NJ 07974 USA (e-mail: gkr@bell-labs.com;

aea@bell-labs.com).

Digital Object Identifier 10.1109/TCOMM.2004.826370

be possible using other chart techniques, see, e.g., [5], [7],

and [8]. We refer to [9] for a comparison of some of these

tools. Another alternative is to use numerical optimization with

density evolution

. Transfer charts and density evolution

complement each other in that the former are easier to visualize

and program, giving insight and good initial code designs,

while the latter can be used to verify the graphical analysis and

to refine the designs.

There are several existing approaches to combining coding

and modulation, for example, trellis-coded modulation (TCM)

[10], multilevel coding [11], bit-interleaved coded modulation

(BICM) [12], and space–time block-coded (STBC) modulation

[13], [14] (see also [15] and references therein). A growing

body of work uses BICM with turbo and LDPC codes, see, e.g.,

[16]–[29]. The EXIT curve-fitting approach described here was

motivated by results for erasure channels [30] and appeared in

[31]. Parallel work using similar ideas was reported in [32] and

[33]. The method was used to design repeat-accumulate (RA)

codes in [34] and [35].

This paper is organized as follows. In Section II, we develop

the curve-fitting procedure for BPSK on the AWGN channel.

In Section III, we extend the technique to other communication

problems, and, in particular, to MIMO fading channels where

the receiver, but not the transmitter, knows the channel. We de-

sign LDPC codes for ergodic fading, and compare their perfor-

mance with a scheme employing a universal mobile telecommu-

nications system (UMTS) standard turbo code. The LDPC codes

are shown to perform substantially better for channels having

more transmit than receive antennas. Such a situation is likely

to occur on the base-to-mobile station link of a wireless com-

munication system. Section IV summarizes our results.

II. C

ODE DESIGN FOR

AWGN CHANNELS

Consider an LDPC code of length

and design rate .

An iterative decoder for this code can be viewed as a graph that

has

variable nodes, an edge interleaver, and check nodes.

The

th variable node represents the th bit of the codeword.

This bit is involved in

parity checks, so that its node has

edges going into the edge interleaver. The edge interleaver

connects the variable nodes to the check nodes, each of which

represents a parity-check equation. The

th check node checks

bits so that it has edges. The sets of variable and check

nodes are referred to as the variable-node decoder (VND) and

check-node decoder (CND), respectively. Iterative decoding is

performed by passing messages between the VND and CND.

The decoder structure is shown in Fig. 1, and its operation is

explained in more detail below. We remark that this structure is

similar to that of an iterative decoder for a serially concatenated

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