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Communications Letters
1
Raptor Codes with Descending Order Degrees for
AWGN Channels
Wei Yu, Krishna Narayanan, Fellow, IEEE, Jun Cheng, Member, IEEE, Jun Wu, Senior Member, IEEE
Abstract—We propose a novel construction of rateless codes
for the additive white Gaussian noise (AWGN) channel with high
order quadrature amplitude modulation (QAM). The proposed
construction is structurally similar to a Raptor code consisting of
an inner Luby transform (LT)-style code and an outer low density
parity check (LDPC) code. The main novelty is that the output
degree distribution for the LT part is time-varying and the degree
of the parity bits decreases with time, parity bits with the same
degree are clustered together rather than random, which is totally
different from traditional fixed degree distributions. Simulation
results with a joint iterative decoding algorithm demonstrate that
both the goodput and the bit error rate (BER) performance of
our proposed scheme are better than those of newly reported
Raptor codes for the AWGN channel in most cases.
Index Terms—Raptor code, LT code, AWGN, Joint iterative
decoding.
I. INTRODUCTION
Fountain codes [1] or rateless codes are a class of error
correction codes that can generate an unlimited number of
coded bits given a finite number of information bits. Luby
Transform (LT) [2] codes and Raptor codes [3] [4] are two
commonly used classes of rateless codes. A Raptor code is the
concatenation of a high rate low density parity check (LDPC)
outer code and a rateless LT inner code, so the performance
is better than pure LT codes. Rateless codes were originally
proposed for erasure channels, but quickly extended to other
channels including binary-input additive white Gaussian noise
(BIAWGN) channels [5]–[9]. Optimization of Raptor codes for
a given signal-to-noise ratio (SNR) was studied in [6] and it
has been shown that an universal optimal degree distribution
does not exist for the class of AWGN channels. Kuo et al.
[8] proposed that Raptor codes can be designed with different
degree distributions for different channel SNRs. Jayasooriya
et al. [7] proposed a reconfigurable Raptor codes for a wider
SNR ranges based on a multi-edge framework, where the
The work of K. Narayanan was supported in part by the National Science
Foundation under grant ECCS-1611285. The work of J. Cheng was supported
in part by JSPS through the Grant-in-Aid for Scientific Research (C) under
Grant 18K04132, and in part by the Ministry of Internal Affairs and Commu-
nications Japan through SCOPE under Grant 195007002. The work of W. Yu
and J. Wu was supported in part by the National Natural Science Foundation
China under Grant 61571329, Grant 61831018, and Guangdong Province Key
Research and Development Program Major Science and Technology Projects
under Grant 2018B010115002.
W. Yu and J. Wu (Corresponding author) are with the College of Electronics
and Information Engineering, Tongji University, China (e-mail: {2014yuwei,
wujun}@tongji.edu.cn).
K. Narayanan is with the Department of Electrical and Computer Engineer-
ing, Texas A&M University, US (e-mail: krn@tamu.edu).
J. Cheng is with the Department of Intelligent Information Engineering and
Science, Doshisha University, Japan (e-mail: jcheng@mail.doshisha.ac.jp).
degree distribution changes in real time as more codeword
bits are transmitted [7]. Kharel et al. [9] proposed to use
discretized density evolution method to optimize different
degree distributions for different coding rates.
In order to improve the spectral efficiency of the whole
communication system, Raptor codes for high order mod-
ulation and AWGN channels are proposed in [10]–[12]. In
[10], degree distributions are separately optimized for the
2 sub-channels of 4-PAM; however, its performance is not
significantly improved compared to the original Raptor code.
Shirvanimoghaddam et al. [11] proposed a type of multi-
layer Raptor codes based on superposition modulation, which
divides an AWGN channel into several identical sub-channels
with very low SNR. A well-optimized Raptor code for low
SNRs is used in each sub-channel. Then the codeword of all
sub-channels are superimposed and transmitted. Jayasooriya
et al. [12] proposed a design of Raptor codes for higher-
order modulation using a multi-edge framework at a given
SNR. Where, distinct Raptor code degree distributions are
simultaneously optimized for distinct bit levels. Both [11] and
[12] require an idealized assumption that the channel SNRs
are perfectly obtained by the transmitter.
The randomized construction of existing LT and Raptor
codes makes them perform well for all erasure patterns which
provide enough mutual information (MI) to decode regardless
of the distribution of erasures in the channel (we will refer to
this as the general erasure channel), i.e., the channel does not
have to be memoryless. In this letter, we consider QAM and
a memoryless AWGN channel whose SNR is unknown at the
transmitter. In a memoryless AWGN channel, there is no need
to protect the codeword from bursty error patterns and this
implies that rateless codes for AWGN channels do no have to
be generated randomly.
Based on our insight, we propose a novel construction of
Raptor codes for AWGN channels, whose salient features are
- the encoder is systematic, the output degree distribution for
the LT part is time-varying and the (average) degree of the
parity bits decreases with time. Although our scheme can
be considered as a reconfigurable coding, its code design
method is essentially different from the existing reconfigurable
Raptor codes such as [7] and [9]. The differences are mainly
embodied in the following aspects. (1) The schemes in [7] and
[9] only use a limited number of degree distributions. In our
scheme, it can be considered that there is an infinite number
of degree distributions. When the decoder is unable to decode
the information bits, the schemes in [7] and [9] generate new
codeword bits by changing the degree distribution from one
to another, while in our scheme, the approach is to reduce the
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