Extended Low-density Parity-check Codes for Cooperative Diversity
Hussain Ali and Maan Kousa
Dept. of Electrical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
Keywords:
Low-density Parity-check Codes, Cooperative Diversity.
Abstract:
Cooperative diversity or user cooperation achieves the diversity gain without adding physical antennas to the
users. The users work in cooperative fashion using their single antennas to create a virtual multiple-input
multiple-output (MIMO) antenna system. The diversity gain achieved by cooperative diversity can be further
improved using error correction codes. Low-density parity-check (LDPC) codes are linear block codes with
good error correction capabilities. We present a novel approach using extended LDPC codes to increase the
diversity gain in cooperative diversity.
1 INTRODUCTION
Wireless communications face the challenges of
channel impairments and fading that severely degrade
the capacity of wireless channels. Numerous spatial
diversity techniques have been in use to combat chan-
nel impairments and fading. One such technique is
cooperative diversity in which the users or mobile sta-
tions cooperate using their single transmitting antenna
in a particular scenario to exploit the availability of
good channels from users to base stations or destina-
tion. In cooperative diversity, generally, the destina-
tion receives multiple packets for the same data from
independent channels creatinga virtual multiple-input
multiple-output channel. Cooperative diversity can-
not guarantee error free transmission, therefore, error
control coding techniques are applied in cooperative
scenario.
User cooperation diversity has been used to
achieve diversity gain using the partners transmitting
antennas (Sendonaris et al., 2003a), (Sendonaris et al.,
2003b). If the channel with one user to the destination
is bad, then the channels from other users, called part-
ners, can be used to send the packet to the destination.
The destination receives two packets of the same data
from two independent channels that may not be noisy
or in deep fade at the same time. The destination pro-
vides decoding by maximal ratio combining on both
packets received and thus achieving spatial diversity
gain in simple repetition schemes. In coded coopera-
tive diversity or cooperation diversity through coding
(Hunter and Nosratinia, 2006), rate-compatible con-
volutional (RCPC) codes were used jointly with coop-
eration. We extend the coded cooperative diversity
work using extended LDPC codes.
Low-density parity-check (LDPC) codes were in-
vented by Gallager in his Ph.D. work (Gallager, 1962)
in 1960. LDPC codes belong to the class of linear
block codes. These codes were ignored due to lack
of appropriate hardware in 1960s. These codes were
rediscovered by MacKay (MacKay, 1999) and oth-
ers. These codes have become more practical due
to the advancements in transistor technology leading
to high computational power of the hardware. These
codes have gain attention due to their near-capacity
performance. These codes can be modified to achieve
rate-compatibility. Extended LDPC codes were in-
troduced in (Bi and Perez, 2006; Li and Narayanan,
2002; Yazdani and Banihashemi, 2004) to achieve
lower rate codes from high rate codes. A joint and
efficient design for puncturing and extension is dis-
cussed in (Li and Narayanan, 2002) which is preferred
in cooperative scenario for its rate adaptability.
The paper is organized as follows: In section 2, we
introduce the coded cooperative diversity. In section
3, we propose the extended LDPC codes for coopera-
tive diversity. The simulation results for the proposed
extended codes for cooperativediversityare presented
in section 4. The last section concludes the paper.
2 CODED COOPERATIVE
DIVERSITY
We assume a time-division based system with two ter-
minals T
1
and T
2
as users and one terminal T
3
as des-
357
Ali H. and Kousa M..
Extended Low-density Parity-check Codes for Cooperative Diversity.
DOI: 10.5220/0004124303570360
In Proceedings of the International Conference on Signal Processing and Multimedia Applications and Wireless Information Networks and Systems
(WINSYS-2012), pages 357-360
ISBN: 978-989-8565-25-9
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

tination. The channels for T
1
and T
2
transmission are
assumed to be orthogonal in time. The codeword N
is divided into two weaker codewords denoted by N
1
and N
2
. The frame transmission for N is divided into
two time slots. The first time slot is reserved for each
user’s own data. For the user T
1
, N
T
1
1
is transmitted
to the destination T
3
and to the partner T
2
where N
T
1
1
is the first codeword for user T
1
. Similarly, T
2
sends
the codeword N
T
2
1
to T
1
and T
3
. Both T
1
and T
2
check
the integrity of data received by applying cyclic re-
dundancy check (CRC). The transmission in the sec-
ond time slot is determined by the success or failure
of decoding of these packets received in the first time
slot for each user. The four cases that arise after the
first time slot transmission are shown in Fig. 1. In
T
1
T
2
T
3
T
1
T
2
T
3
T
1
T
2
T
3
T
1
T
2
T
3
Case 1 Case 2
Case 3 Case 4
T
1
Information
T
2
Information
Figure 1: Four cooperative diversity cases based on trans-
mission in second time slot.
case 1, both users successfully decodes the packet re-
ceived from their partners. Therefore, T
1
will send
N
T
2
2
for T
2
and T
2
will send N
T
1
2
for T
1
in the second
time slot. In case 2, both users fail to decode their
partners transmission and continue to send their own
second codeword N
2
in the second time slot. In case
3, T
1
fails to decode the transmission from T
2
. In this
case, both users will transmit N
T
1
2
for T
1
. In case 4, T
2
fails to decode the transmission from T
1
. In this case,
both users will transmit the codeword N
T
2
2
for T
2
.
3 EXTENDED LDPC CODES
LDPC codes are defined by their parity-check matrix
H with G.H
T
= O where G is called the generator
matrix and O is an all zero matrix. The regular LDPC
codes have constant row and column weight. In this
work, we will use regular LDPC codes with column
weight 3 and row weight 6, denoted as (3,6) regu-
lar code. The (3,6) regular code has the best error
correction capabilities in the class of regular LDPC
codes. Rate-compatible design is required to generate
codewords of different lengths. The design of (Li and
Narayanan, 2002) is capable of embedding higher rate
codewords in lower rate codewords. We exploit this
design to be used in coded cooperative diversity. We
design the extended parity-check matrix according to
the following definitions of matrices
H
2
=

H
1
O
A B

m
′
×n
′
(1)
where m = n− k, m
′
= n
′
− k and H
1
is the (3,6) reg-
ular parity-check matrix of dimensions m× n for the
mother code of rate R = k/n. To extend the code rate
to R
′
= k/n
′
, the extra parity bits in the extended code-
words will be e
bits
= n
′
− n where n
′
is the size of ex-
tended codeword for rate R
′
. The O matrix is an all
zero matrix of size k×e
bits
or k×(n
′
− n). The A ma-
trix is a very sparse matrix of size (n
′
− n) × n with at
least one 1 in each row. The B matrix has dimensions
(n
′
− n) × (n
′
− n) with column weight 3.
The systematic form of H
1
is
H
1
=

P
T
1
I
n−k

m×n
(2)
where P
T
is the transpose of P. The systematic form
of H
2
is given by
H
2
=
"
P
T
1
P
T
2
I
n
′
−k
#
m
′
×n
′
(3)
The O matrix ensures that the higher rate codewords
are embedded in extended lower rate codewords by
keeping the integrity of P
T
1
. The generator matrix for
H
1
and H
2
becomes
G
1
=

I
k
P
1

k×n
(4)
and
G
2
=

I
k
P
1
P
2

k×n
′
(5)
respectively, where P
1
has dimensions k×(n−k) and
P
2
has dimensions k× (n
′
− n).
We exploit this design of extended LDPC codes to
the cooperative diversity framework and modify the
extended LDPC codes to achieve decoding in three
steps at the receiver. The codeword N
1
is generated by
using the generator matrix obtained from H
1
. Using
Eq. 4, N
1
takes the following form
N
1
= [i p
1
]
1×n
(6)
where i is the information part and p is the parity part
in the codeword N
1
. The second codeword N
′
2
is gen-
erated by the generator matrix G
2
mentioned in Eq.
5, in the following form
N
′
2
= [i p
1
p
2
]
1×n
′
(7)
WINSYS2012-InternationalConferenceonWirelessInformationNetworksandSystems
358

where p
1
is the same parity part as in N
1
and p
2
is
the extended parity part. This N
′
2
is modified to gen-
erate a codeword of length n. The second codeword
is transmitted in the following format
N
2
= [i p
2
]
1×n
. (8)
At the receiver, in the first step N
1
is decoded using
H
1
. If the codeword is not successfully recovered in
the first step, then the codeword is decoded using H
2
matrix with erasures inserted at p
1
of N
2
. If the de-
coding fails in the first two steps, then the codeword
is concatenated and jointly decoded using H
2
to re-
cover the codeword N.
4 PRELIMINARY SIMULATION
RESULTS
We assume BPSK modulation for the simulations.
We also assume very slow fading channel in which
the channel fading coefficient are Rayleigh distributed
and remains constant for complete codeword N. The
information block size k = 512 and the codewords N
1
and N
2
transmitted are of length 1024 bits. The sum-
product algorithm (SPA) is used for the decoding. The
received packets are decoded on H
1
= 512 × 1024
for the first step and H
2
= 1024× 1536 for the next
two steps of decoding. The simulation results have
been plotted as BER versus the channel SNR. The
plots with BER versus information bit SNR will be
identical with a shift of 10logR dB. The bit-error rate
(BER) and frame-error rate (FER) for the proposed
extended LDPC codes design for cooperative diver-
sity with varying inter user channel are shown in Fig.
2 and Fig. 3 respectively. The gain is approximately
11 dB for the perfect inter user channel as compared
to worse inter user channel.
5 CONCLUSIONS AND FUTURE
WORK
The proposed extension to LDPC codes successfully
integrates with cooperative diversity. A very high di-
versity gain is achieved when the channel between the
cooperating users is good. The decoding in the first
step is done on a smaller parity-check matrix, hence
reducing the complexity. However, the decoding in
the next two steps is done on larger parity-check ma-
trix but it gives better performance in terms of BER
and FER. The proposed scheme can be further ana-
lyzed on fast fading channels. The proposed scheme
can also be investigated and compared with punctured
0 2 4 6 8 10 12 14 16 18 20
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Average Received SNR at Base Station (both users equal) (dB)
BER
No cooperation
Perfect interuser channel
20dB interuser channel
10dB interuser channel
0dB interuser channel
Figure 2: BER of cooperative diversity with extended
LDPC codes and varying inter user channel.
0 2 4 6 8 10 12 14 16 18 20
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Average Received SNR at Base Station (both users equal) (dB)
FER
No cooperation
Perfect interuser channel
20dB interuser channel
10dB interuser channel
0dB interuser channel
Figure 3: FER of cooperative diversity with extended LDPC
codes and varying inter user channel.
LDPC codes which will have higher decoding com-
plexity because of larger parity-check matrices.
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