Analysis and design of Raptor codes using a multi-edge framework

TL;DR

An efficient Raptor code design method using the multi-edge framework, where the component codes corresponding to the inner code and the precode are decoded in parallel and provide information to each other, and an exact expression for the stability of Raptor codes is derived.

Abstract

The focus of this paper is on the analysis and design of Raptor codes using a multi-edge framework. In this regard, we first represent the Raptor code as a multi-edge type low-density parity-check (METLDPC) code. This MET representation gives a general framework to analyze and design Raptor codes over a binary input additive white Gaussian noise channel using MET density evolution (MET-DE). We consider a joint decoding scheme based on the belief propagation (BP) decoding for Raptor codes in the multi-edge framework, and analyze the convergence behavior of the BP decoder using MET-DE. In joint decoding of Raptor codes, the component codes correspond to inner code and precode are decoded in parallel and provide information to each other. We also derive an exact expression for the stability of Raptor codes with joint decoding. We then propose an efficient Raptor code design method using the multi-edge framework, where we simultaneously optimize the inner code and the precode. Finally we consider performance-complexity trade-offs of Raptor codes using the multi-edge framework. Through density evolution analysis we show that the designed Raptor codes using the multi-edge framework outperform the existing Raptor codes in literature in terms of the realized rate.

Figures (6)

• Figure 1: Graphical representation of a Raptor code.
• Figure 2: Graphical representation of the LT code component of the Raptor code ensemble at $p$th decoding attempt with incremental decoding strategy using the multi-edge framework. $x$ denotes the fraction of output bits used from previous decoding attempts. $\mathcal{R}_1$ and $\mathcal{R}_2$ respectively denote realized rates at the $(p-1)$th decoding attempt and the $p$th decoding attempt. Not shown are the ($1-x$) fraction of output bit nodes not used in this decoding round.
• Figure 3: Rate efficiency results computed using MET-DE and finite-length simulations for the existing Raptor codes in literature with fixed LDPC code rate of 0.98, evaluated at different SNRs.
• Figure 4: Rate efficiency results computed using MET-DE for the Raptor codes designed using the multi-edge framework, evaluated at the designed SNR.
• Figure 5: Rate efficiency results computed using MET-DE for the Raptor codes designed at -5dB with fixed LDPC code rate of 0.95, designed/evaluated using joint decoding and tandem decoding.

Theorems & Definitions (5)

• Theorem 1: Sufficient condition for the stability of Raptor codes decoded with joint decoding based on the BP decoding
• Theorem 2: Stability of Raptor codes decoded with tandem decoding based on the BP decoding
• Theorem 3: Sufficient condition for the stability of the LDPC code component of the Raptor code decoded with tandem decoding based on the BP decoding
• Lemma 1: Degree-one variable nodes RichardsonW2002multi
• Remark 1: Stability of MET-LDPC codes with degree-one variable nodes richardsonBOOK2008modern