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RECENT DEVELOPMENT OF ERROR CONTROL CODES
FUTURE COMMUNICATION AND STORAGE SYSTEMS
FOR
Polar Codes for Soft Decode-and-Forward in HalfDuplex Relay Channels
Fangliao Yang, Kai Niu, Chao Dong, Baoyu Tian
Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications Beijing
100876, China
Abstract: Soft decode-and-forward (DF)
can combine the advantages of both amplify-and-forward and hard DF in relay channels.
In this paper, we propose a low-complexity
soft DF scheme based on polar codes, which
features two key techniques: a low-complexity cyclic redundancy check (CRC) aided list
successive cancellation (CALSC) decoder and
a soft information calculation method. At the
relay node, a low-complexity CALSC decoder
is designed to reduce the computational complexity by adjusting the list size according to
the reliabilities of decoded bits. Based on the
path probability metric of the CALSC decoder,
we propose a method to compute the soft information of the decoded bits in CALSC. Simulation results show that our proposed scheme
outperforms the soft DF based on low-density
parity-check codes and the soft DF with belief propagation or soft cancellation decoder,
especially in the case when the source-relay
channel is at the high signal-to-ratio region.
Keywords: Decode and forward; soft information relaying; Polar codes; relay channels
I. INTRODUCTION
The relay channel, introduced by Van der
Meulen [1], has attracted significant attentions
for over two decades. Various coding techChina Communications • August 2017
niques have been studied [2-6], [26-27] and
applied to relay channels or communication
network. Besides, various relaying protocols
have been proposed to get a trade-off between
power consumption, processing delay and
spectral efficiency. Two of the most widely
used relaying protocols are amplify-and-forward (AF) and decode-and-forward (DF) [7].
Under AF, the relay retains the soft information of the received signal and does not make
any hard decisions. However, it will amplify
the channel noise. Under DF, the relay first
decodes the received signal and re-encodes the
decoded bit to generate an additional part of
parity bits. As DF can eliminate the effect of
noise amplification, it can provide a substantial
improvement in performance. However, when
the source-relay channel condition is poor,
decoding errors will happen at the relay and
error propagation will deteriorate the system
performance if the error bits are transmitted to
the destination. In conclusion, neither AF nor
DF outperforms the other in all scenarios.
In order to combine the advantages of both
AF and DF, an advanced relaying protocol
called the soft DF or soft information relaying
(SIR) was proposed [8], in which the relay
performs the decoding/re-encoding process in
a soft manner and transmits the soft information to the destination (Similarly, we refer to
Received: Feb. 28, 2017
Revised: May 21, 2017
Editor: Ming Xiao
22
the DF with hard decision at the relay as the
hard DF). In [8], [9], a soft-input soft-output
(SISO) encoder with modified BCJR algorithm was proposed. The rate-compatible
low-density parity-check (LDPC) codes were
introduced in the soft DF scheme in [10]. In
[11], a soft decode-compress forward scheme
was proposed. It has been demonstrated that
the soft DF can outperform AF and hard DF
substantially.
Polar codes [12], which have been proven
to be a family of capacity-achieving codes for
the arbitrary binary-input discrete memoryless
channel (B-DMC), have shown better performance than turbo codes and LDPC codes
under cyclic redundancy check (CRC) aided
decoding algorithms with short-to-medium
block length [13], [14], [25]. Polar codes have
been good choices in many communication
scenarios, including relay channels. Polar
codes were first studied in the relay channels in [15]. In the remarkable work [16],
polar codes have been proven to achieve the
capacity of the relay channels with orthogonal receiver components by using a nested
structure. In [17], polar codes were used for
the Gaussian degraded channel with DF and
show substantial performance gain compared
with LDPC codes. In [18], a smart DF scheme
was proposed, in which the relay employs a
threshold of log likelihood ratio (LLR) to determine whether to transmit the signal to the
destination or to remain silent. In [19], polar
codes based on Plotkin’s construction are employed to achieve coded cooperation between
two users. In [20], a class of new DF strategies
named generalized partial information relaying protocol is developed for the degraded
multiple-relay networks that have orthogonal
receiver components based on the nested
structure of polar codes. In [21], a cooperative
polar coding scheme based on partially perfect
bit channels retransmission is proposed. However, the current works for polar codes in relay
channels are all based on hard DF, which are
easily suffered from error propagation. That
is to say, if the condition of the source-relay
channel is poor, the codes will be erroneous-
23
ly decoded and affect the distributed code
construction at the relay, no matter Plotkin’s
construction [19], [21] or nested construction
[15], [16], [17], [18], [20]. The soft DF may
be a good solution.
The main purpose of this paper is to investigate the use of soft DF with polar codes in the
half-duplex relay channels. Here, we address
the main challenge that arises when adopting
the polar codes under the soft DF scheme. As
the soft DF requires that the processing at the
relay should be in a soft manner, the intuitive
idea is to utilize the SISO decoding algorithms, like the belief propagation (BP) and
soft cancellation (SCAN) algorithms [22] to
generate the soft information directly. However, they have some disadvantages. On the one
hand, the complexity of BP is considerable
due to the iterative decoding process. On the
other hand, as the message transmission procedure of BP and SCAN are all based on the
factor graph representation, their performance
are limited by the short-length loops and inferior to other improved SC decoders, like the
list successive cancellation (LSC) decoders
and CRC-aided LSC decoder (CALSC) [14].
The main challenge is that we cannot employ
LSC or CALSC at the relay directly since it
cannot generate soft information.
The contributions of this paper are listed in
the following:
First, we propose a soft DF scheme by utilizing a CALSC decoder at the relay instead of
the BP and SCAN. The relay first decodes the
received signal with a CALSC decoder, then
computes the soft information of decoded bits
of CALSC and finally encodes the soft information with an SISO polar encoder.
Secondly, considering that the relay is usually energy-constrained in practical applications, we propose a low-complexity CALSC
decoder to reduce the computational complexity. In the traditional CALSC decoder, the
number of decoding paths (list size) in each
decoding level (for each bit) is equal. Motivated by the conjecture that if the reliabilities of
the decoded bits are high enough, the number
of decoding paths can be reduced with minor
China Communications • August 2017
performance loss, we propose a low-complexity CALSC decoder with dynamic list size.
Simulation results show that the complexity of
the proposed scheme reduces significantly as
the channel condition becomes better.
Finally, considering that the CALSC decoder employs a hard decision for each decoding
path and cannot generate soft information
of the decoded bits, we propose a method to
compute the soft information based on the
path probability metric of the CALSC decoder. Aided by the check results of CRC, we
modify the soft information by flipping the
sign of the LLRs according to the decoded bits
which pass the CRC. Simulation results show
that the soft DF scheme based on the proposed
method can outperform BP and SCAN. Especially in the case when the condition of source
to relay channel is good, the performance gain
is remarkable.
The rest of the paper is organized as follows. In section II, preliminaries and the
system model are introduced. The proposed
soft DF scheme is developed in section III.
In section IV simulation results are provided
to demonstrate the improved performance of
our proposed scheme. Finally, conclusions are
drawn in section V.
II. PRELIMINARIES AND SYSTEM MODEL
2.1 Notation conventions
In this paper, we use bold letters, such as , to
denote matrix. We use calligraphic characters,
such as , to denote sets, and
to denote the
cardinality of sets. Let
denote the complementary set of .
We use notation
to denote an N-dimen-
sion vector
and
subvector
of
We write
to denote a
,
.
to denote a subvector
.
length of polar codes be
,
.
, where
Encoding:
the uncoded bits,
denotes
denotes the coded bits,
, where
is an N×N permutation
matrix,
is the kernel matrix,
and
denotes the n-th Kronecker power of
.
By recursively applying the techniques of
channel combining and splitting, N independent uses of the B-DMC W can be transformed
into uses of synthesized channels, denoted
. The transition probabili-
as
ty of
is expressed as
(1)
where
It is proven that the synthesized channels
are polarized to reliable ones and unreliable
ones with the increasing code length [12]. The
sets of reliable and unreliable channels are
referred to as the information set
and the
frozen set , respectively. We can utilize the
reliable channels to transmit the information
bits and the unreliable ones to transmit fixed
bits which are known to both encoder and decoder in advance.
Decoding: the decoding can be performed
with successive cancellation (SC) algorithm,
which is described as follows: let
the estimation of
ly from
to
,
denote
is decoded successive-
and is given by
(2)
where
2.2 Polar codes
(3)
Let
denote a B-DMC with input
alphabet , ouput alphabet , and transition
probability
,
,
. Let the
2.3 System model
China Communications • August 2017
In this paper, we consider a typical three-node
relay channel. It consists of a source node, a
24
relay node, and a destination node, which are
denoted as s, r and d, respectively. The transmission on the channel is divided into twotime slots. In the first time slot, the source encodes the information and broadcasts it to the
relay and destination simultaneously. In the
second time slot, the relay processes the signal
received from the first time slot and transmits
the signal to the destination while the source
node remains silent. For simplicity, we assume
that each node has just one antenna and the
modulation scheme is binary phase shift keying modulation (BPSK). Besides, the s-r channel, the r-d channel, and the s-d channel are
all assumed to be the additive white Gaussian
noise (AWGN) channels.
At the source node, the modulated signal
is denoted as
, where
is the length of the
signal.
At the relay node, the received signal and
the re-transmitted signal are denoted as
and
respectively.
is the transmission power of the
source and
is the noise of the s-r channel.
Relay
BP SCAN
BP/SCAN
SCA
Decoder
Decod
w1N
AWG
GN
N
Channel
AWGN
Channel
Destination
Source
Polar
encoder
c1N
BPSK
x1N
AWGN
AWG
N
Channel
Pola
olar
Decooder
LLR
Calculation
Fig. 1a The intuitive soft DF scheme with BP/SCAN decoder
Relay
CALS
SC
Decoder
Decod
Sof
Soft
Info
f rmation
fo
Calculation
Polar
encod
ncoder
c1N
Destination
BPSK
BPS
K
x1N
AWG
AWGN
Channel
Cha
LL
LR
Calculation
Calculatio
Fig. 1b The proposed soft DF scheme with CALSC decoder
25
w1N
AWGN
AWG
Channeel
AWGN
AWG
Chann
annel
Source
SISO
O
Encoder
Encode
Polar
Decoder
and
respectively, which are given by
(5)
(6)
where
is the transmission power of the re-
lay,
and
are the noise of s-d and r-d
channel respectively.
and
are all ze-
ro-mean real Gaussian random variables with
variance
and
, respectively. Throughout
this paper we will assume
.
III. THE PROPOSED SOFT DF SCHEME
BASED ON POLAR CODES
In this section, we will first briefly introduce the framework of the proposed soft DF
scheme based on polar codes. Then, we will
describe the key techniques in detail.
3.1 System framework
is given by
(4)
where
At the destination node, the received signals from the source and relay are denoted as
Unlike the hard DF in which hard decision is
made at the relay node, the soft DF scheme requires that the signals are decoded and re-encoded in a soft manner. Therefore, the polar
decoder and encoder should be a SISO version
or be able to generate the soft information.
An intuitive idea is to utilize a BP or SCAN
decoder at the relay, in which the soft information is directly obtained after the iterative
processing back and forth on the factor graph
and can be transmitted to the destination, as
illustrated in Fig. 1a. However, the message
transmission procedure of both BP and SCAN
are based on the factor graph, the performance
are limited by the short-length loops of the
factor graph. In other words, there is performance gap between the BP/SCAN and other
improved SC decoders, like LSC and CALSC.
Furthermore, the complexity of BP is considerable due to the iterative decoding process.
Considering that the CALSC decoder is of
superior performance, we propose the soft DF
scheme based on CALSC decoder, as illustrated in Fig. 1b.
At the relay node, the signal is first fed into
China Communications • August 2017
SC
Decoding
g
LSC decod
ecoder
with L1 lists
Threshold
old
Decision
n
LLRs with SC
LLRs with GA
10 2
10 1
10 0
10 -1
0
20
40
60
80
100
information bit
120
140
Fig. 3a The LLRs of information bits with erroneous decoding (SNRsr=2.0 dB,
N=256, K=128)
10 3
LLRs with SC
LLRs with GA
10 2
LLR
In practical applications, the relay is usually
energy-constrained. Therefore, we propose a
low-complexity CALSC decoder at the relay
to reduce the complexity of the relay. The
CALSC decoding is the combination of LSC
and CRC decoding. As LSC decoding takes up
most of the processing delay, we focus on how
to reduce the complexity of LSC.
The LSC decoding can be regarded as a
path searching process on a code tree. For polar codes with length , there are levels on
the code tree. In each level, there are nodes,
. Each node has two succeeding
branches each of which stands for a decoded
bit 0 or 1. A decoding path consists of the
continuous branches from the root node to a
leaf node. The path probability metric (PPM)
10 1
can be measured using a
posteriori probability [14] given by
10 0
(7)
where
)
10 3
3.2 The low-complexity CALSC
decoder
of a decoding path
LSC decoder
with L2 lists
wi
Fig. 2 The proposed low-complexity SCL decoder (
LLR
a CALSC decoder. As the output of CALSC
is all hard information, we then propose a
soft information calculation method based
on Bayes rule to calculate the probabilities
of decoded bits. Finally, the calculated soft
information is fed into a SISO polar encoder.
In order to reduce the complexity, a low-complexity CALSC decoder is further proposed. It
is noted that the SISO encoder is the soft version of the encoder at the source node, which
yields an overall repetition coding structure.
At the destination node, the LLRs of the
signals received from the source and relay are
first calculated. After combination, the LLRs
are then passed into the polar decoder for signal reconstruction.
The detailed descriptions are given in the
following subsections.
denotes the received symbols and
can be calculated using the re-
cursive formula in [12]. In the traditional LSC
decoder, the number of surviving decoding
paths, namely the list size, is set to a fixed valChina Communications • August 2017
0
20
40
60
80
100
120
140
information bit
Fig. 3b The LLRs of information bits with correct decoding (SNRsr=2.0 dB,
N=256, K=128)
ue, say . In each level, decoding paths with
the largest PPMs are kept while the others are
26
deleted. LSC decoding can mitigate the effect
of error propagation in SC decoding which has
just one decoding path and cannot correct the
past decoded bits once they were erroneously
decoded.
We argue that if the reliabilities of decoding bits are high enough, it is unnecessary to
employ a large number of decoding paths to
reserve the decoding information. Thus, the
complexity can be reduced. Motivated by this
conjecture, we propose an LSC decoding algorithm with dynamic list size, as illustrated
in Fig. 2. Two LSC decoders with different
list sizes are employed in our algorithm. If the
reliabilities of the decoded bits are higher than
the predesigned threshold, which indicates a
high probability of correct decoding, an LSC
will be applied.
decoder with a small list
Conversely, if the reliabilities of decoded bits
are smaller than the threshold, an LSC decoder
with a large list will be applied.
In the following, we will elaborate how to
measure the reliabilities and how to design the
threshold.
First, we utilize an SC decoding to obtain
the real-time LLRs of decoded bits. As depicted in Fig. 3, the red square points stand for the
absolute LLRs of the decoded bits calculated
by Gaussian approximation (GA) [23] in increasing order, while the blue dots stand for
the absolute LLRs calculated by the SC decoding. The LLRs of GA are calculated point by
point offline in the given SNR range. We can
infer that if the LLR curve of SC is far away
from GA, as illustrated in Fig. 3a, erroneous
decoding tends to occur. Conversely, if the
LLR curve of SC approaches GA, the probability of correct decoding will be high and we
can employ an LSC decoder with a small list
to do the decoding. It is noted that the SC decoding is an error propagation decoding process, which means that if the previous bits are
correctly decoded the LLR of current decoded
bit is reliable, otherwise it is not reliable and
cannot be used for soft information relaying.
Here, we use the deviation degree of LLR
curve of SC and GA to measure the average
degree of reliability and to help to select the
27
LSC decoder with list size or .
We define a deviation metric (DM) to quantize the deviation degree between the LLRs of
SC and GA, which is given by
(8)
and
represent the LLR of SC
where
and GA, respectively. Here,
is used to
take the high reliabilities and low reliabilities
into the same measure scale. A small DM
means a high probability of correct decoding
and high average reliabilities of decoded bits.
We use the DM as the measurement of the reliabilities of decoded bits when performing the
real-time decoding process and use the threshold of DM to determine whether to apply a
small-list LSC decoder or a large-list one.
Here, we give a straightforward method to design the threshold. We perform the
SC decoding off-line and record the DM
each time when correct decoding occurs.
Then we calculate the mean and variance
of DM, denoted as
and
respec-
tively. Assuming the distribution of DM is
Gaussian, we utilize the -law, which holds
, to set the threshold to
.
The performance and the complexity of the
proposed LSC decoder will be given in section
IV.
3.3 Soft information calculation of
the LSC decoder
The output of the LSC decoder is a list of decoding paths with PPMs. However, there is no
soft information about the probabilities of the
decoded bits. In this subsection, we propose a
method to calculate the soft information based
on to the PPMs of the decoding paths.
Let
denote the set of decoded bit sequences. Let
denote the j-th bit sequence,
where
and
. We use
to denote the i-th bit in the j-th sequence, and
use
to denote the PPM of the j-th bit
sequence, abbreviated .
As the PPMs of the decoding paths are normalized to 1 when performing the LSC decod-
China Communications • August 2017
ing, we have
(9)
By using the Bayes rule, we deduce the
LLRs of the decoded bits given by
It is noted that
,
. The
signal transmitted to the destination can be expressed as
(15)
where is the power normalization factor
which can be calculated as
(16)
(10)
where
is the indicative function. The calculated soft information
is then
fed to the SISO encoder which will be introduced in the next subsection.
With CRC aiding, we modify the sign of
LLRs of decoded bits by the decoded sequence which passes the CRC decoding, given
by
(11)
where denotes the i-th bit in the decoded bit
sequence which pass the CRC.
3.4 The SISO encoder based on
polar codes
The processing unit of the SISO polar encoder consists of four nodes, two input
nodes
and
, two output nodes
and
, as depicted in Fig. 4.
The soft information is given by
(12)
and
(13)
By applying (12) and (13) recursively, we
obtain the LLRs of the encoded bits. It is noted that the unit has lots of similarities with the
BP, however, the SISO processing unit merely
has the forward information propagation without the backward one.
As the value of LLR is with high-dynamic,
it is difficult for the relay to forward these signals directly. We use another representation of
the LLR, which is given by
3.5 Calculation of LLRs at the
destination
In our proposed DF scheme, the destination
receives two kinds of signals, i.e., the noised
BPSK modulation symbols from the source
and the soft information from the relay. As the
encoder at the relay is a soft version of the one
at the source and no interleavers are employed,
the overall code structure at the destination is
a simple repetition coding. In the traditional
hard DF scheme with repetition coding, the
signals at the destination are usually combined
with maximum ratio combing (MRC) method.
However, in the soft DF scheme, the two kinds
of signals transmitted by the source and the
relay are not the same type. We have to adopt
different ways to calculate the LLRs for the
two kinds of signals.
For the signal from the source, the LLR of
each bit is computed as
(17)
For the signals from the relay, the calculation of LLRs is not so straightforward. We
have to model the channel between the correct
first.
bit and
Employing the method in [8], we first model the error between the output of the SISO
encoder and the correct bits to be an
equivalent Gaussian channel as
(18)
where
is the equivalent noise introvI (2i  1)
vI (2i )
r2i 1, I
r2i , I
+
=
r2i 1,O
r2i ,O
vO (2i  1)
vO (2i )
(14)
Fig. 4 Processing unit of SISO polar encoder
China Communications • August 2017
28
duced by soft decoding and encoding. The
mean and variance of can be calculate d o ff l i n e a s
and
, respectively.
is the output of the
The received signal
cascaded channel of the equivalent Gaussian
channel and the r-d channel.
For simplicity, we give the expression of
the LLR directly [9] as
(19)
Finally,
is added to
passed to the polar decoder.
and
IV. SIMULATION RESULTS AND
ANALYSIS
In this section, we study the performance
and the complexity of the proposed soft DF
scheme with CALSC decoder.
To justify the good performance, we take
four systems for comparison, which are listed
as follows:
a. AF scheme where MRC is employed to
combine the received signal from the relay
and the source node.
b. Hard DF scheme with nested structure
[16], abbreviated the nested hard DF. In the
nested structure, the information set of polar
codes designed for s-r channel is a subset of
the one designed for r-d channel. Therefore,
the relay extracts and forwards the information which cannot be correctly decoded by the
destination with .
c. The competitive soft DF scheme utilizing
a BP/SCAN decoder at the relay, abbreviated
the soft DF with BP/SCAN. As illustrated in
Fig. 1a, at the relay node, the received signal
is decoded with a BP or SCAN decoder. After
iterative decoding, the LLRs of the bits are
obtained directly without passing into an SISO
encoder. The soft bits are computed by equation (14) and (15).
d. A soft decode-compress-forward relaying scheme based on LDPC codes [11]. This
scheme employs distributed LDPC coding and
29
generates soft parity symbols at the relay node.
The received signal from the source and relay
are defined by a large parity check matrix.
The polar decoder applied at the relay in
nested hard DF is traditional CALSC decoder
with list size
. In soft DF with BP, the
maximum iteration number of BP decoder is
set to 60. As the performance of SCAN decoder has very a limited gain when the iteration
number is larger than 16, we set the maximum
iteration number to 16. In our proposed soft
DF scheme with CALSC, the list size of the
low-complexity LSC is set to
and
respectively, and the threshold of DM
is calculated off-line.
To provide a fair comparison, the polar
decoder employed at the destination in all
schemes is traditional CALSC decoder with
list size
.
In [11], the overall LDPC code has a block
length of N=1224, the overall code rate at the
destination is 1/3. To provide a fair comparison, the length of polar codes in source node
is set to N1=512, the length of information bit
is set to K=341, a CRC-16 code with generator
polynomial g(D)=D16+D12+D5+1 is attached.
As the overall code structure at the destination
is a simple repetition coding, the overall polar
code has a block length of N=N1+N2=1024, the
overall code rate is R=K/(N1+N2)=1/3, where
N1=512 and N2=512 are the code length at the
source and relay, respectively.
In our simulations, the s-r, r-d and s-d
channels are all assumed to be AWGN channels. We fix the signal-to-ratio (SNR) of the
s-r link at different values to represent both
poor and good channel conditions, and fix SNRrd=SNRsd, just as the configurations in [11].
Before the real-time transmission takes
place, the mean and variance of the equivalent
noise are computed numerically and stored
at the destination, so the relay doesn’t need to
transmit extra information.
Fig. 5 shows the bit-error-rate (BER) performance of the proposed scheme when the
SNR of s-r channel is fixed to 2.8 dB (poor
channel condition). The x-axis represents the
SNR of s-d channel. It can be seen that our
China Communications • August 2017
China Communications • August 2017
10 0
AF
Nested hard DF
Soft DF with BP
Soft DF with SCAN
Proposed scheme (L1 =2,L 2 =16)
10 -1
10 -2
Proposed scheme (L1 =16,L2 =16)
Bit Error Rate
LDPC based scheme [11]
10 -3
10 -4
10 -5
10 -6
10 -7
1
1.5
2
2.5
3
3.5
SNRsd /dB
4
4.5
Fig. 5 Bit-error-rate performance of the proposed soft DF scheme under
SNRsr=2.8 dB. The simulations used BPSK over AWGN links, with SNRsd=SNRrd,
N=1024, R=1/3
10 0
10 -1
10 -2
Bit Error Rate
proposed scheme outperforms AF, the nested
hard DF and the soft DF with BP/SCAN. The
performance of soft DF with SCAN is slightly
better than BP because SCAN decoding performs better than BP decoding. Besides, the
proposed scheme has a performance gain of
about 0.35 dB compared with the LDPC-based
soft DF scheme when the BER is at 10-5.
Fig. 6 shows the BER performance of the
proposed scheme when the SNR of s-r channel
is fixed to 5.0 dB (good channel condition).
The proposed schemes with low-complexity
CALSC, BP, and SCAN all outperform the
LDPC-based soft DF because CRC-aided decoding is employed and can greatly improve
the performance of polar codes. Unfortunately,
for soft DF with BP and SCAN, there are error
floors at the high SNR region, because the two
kinds of decoding algorithms are all iterative
decoding based on the factor graph of polar
codes and there are lots of short-loops on the
factor graph. In other word, when the s-r channel is at good condition, error floors will occur
and the decoding error at the relay node will
propagate to the destination. Conversely, when
the SNR of s-r channel is at the low region, as
illustrated in Fig. 5, BP/SCAN decoding does
not reach the point where error floor occurs. At
BER=10-6, our proposed scheme with CALSC
has no error floor and has a performance gain
of more than 1.0 dB compared with the ones
with BP and SCAN.
In addition, Fig. 5 and Fig. 6 both demonstrate that the proposed soft DF with low-complexity CALSC (
and
) has neglected performance loss compared to the soft
DF with traditional CALSC decoder (
and
) applied at the relay. Because
when the s-r channel condition is poor, the
LSC decoder with a large list size
will be more often used than the LSC decoder
with a small list
, the performance of
the proposed low-complexity CALSC decoder
approaches the traditional one with list size
and
. When the s-r channel
condition is good, the performance of LSC
decoder with different list size (
) are almost the same, as illustrated in Fig. 1 in [24].
10 -3
10 -4
10 -5
10 -6
AF
Nested hard DF
Soft DF with SCAN
Soft DF with BP
Proposed scheme (L1 =2,L 2 =16)
10 -7
Proposed scheme (L1 =16,L 2 =16)
10 -8
-1
-0.5
LDPC based scheme [11]
0
0.5
1
SNRsd /dB
1.5
2
2.5
3
Fig. 6 Bit-error-rate performance of the proposed soft DF scheme under
SNRsr=5.0 dB. The simulations used BPSK over AWGN links, with SNRsd=SNRrd,
N=1024, R=1/3
Then we evaluate the complexity of our
proposed scheme. To simplify the complexity
evaluation of polar decoding and encoding,
we measure the average time complexity in
terms of the number of calculation unit which
is employed in CALSC, BP, and SCAN. For
example, for code length N=512, the time
complexity of CALSC decoder with L=16 is
LNlog2N=72728. Fig. 7 shows that the proposed soft DF can achieve a significant reduc-
30
3.5
# 10
5
With traditional CALSC (L1 =32,L 2 =32)
3
With traditional CALSC(L1 =64,L 2 =64)
With proposed CALSC (L 1 =2,L 2 =16)
2.5
Computation units
[2]
With traditional CALSC (L1 =16,L 2 =16)
[3]
With proposed CALSC (L 1 =2,L 2 =32)
With proposed CALSC (L 1 =2,L 2 =64)
2
With BP(60 iterations)
With SCAN(16 iterations)
1.5
[4]
1
0.5
0
-4
[5]
-3
-2
-1
0
SNR sr /dB
1
2
3
4
Fig. 7 The complexity of our proposed soft DF under different SNRs of the
source-relay channel
[6]
[7]
tion in complexity with the increasing SNR
of s-r channel, compared to the soft DF with
traditional CALSC, BP, and SCAN.
[8]
V. CONCLUSIONS
In this paper, we develop a low-complexity soft DF scheme using polar codes in the
half-duplex relay channels. This scheme
utilizes a low-complexity CALSC decoder
at the relay. As the output of CALSC is all
hard information, we then propose a soft
information calculation method to calculate
the probabilities of decoded bits in CALSC.
Simulation results show that the proposed soft
DF with CALSC decoder outperforms the soft
DF with BP/SCAN and the LDPC-based soft
DF in terms of BER and achieves a significant
reduction in complexity when the source-relay
channel condition becomes better.
[9]
[10]
[11]
[12]
ACKNOWLEDGEMENTS
This work is supported by the National Natural
Science Foundation of China (No. 61171099,
No.61671080), Nokia Beijing Bell lab.
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Biographies
Fangliao Yang, received the
B.S. degree in Electronic Engineering from Beijing University
of Posts and Telecommunications (BUPT), Beijing, China, in
2010. He is currently working
towards the Ph.D. degree at
the School of Information and
Communication Engineering of BUPT. His research
interests include information theory, coding theory
and digital signal processing. Email: yangfangliao@
bupt.edu.cn
Kai Niu, received the B.S. degree in information engineering
and Ph.D. degree in signal and
information processing from
Beijing University of Posts and
Telecommunications (BUPT),
Beijing, China, in 1998 and in
2003, respectively. Currently,
he is a professor at the School of Information and
Communication Engineering of BUPT. His research
interests include coding theory and its applications,
space-time codes and broadband wireless communication. Email: niukai@bupt.edu.cn
Chao Dong, received the B.S
and Ph.D. degree in Signal and
Information Processing from
Beijing University of Posts and
Telecommunications (BUPT),
Beijing, China in 2007 and
2012, respectively. Since August 2014, he was a lecturer
of School of Information and Communication Engineering of BUPT. His research interest is mainly on
the MIMO signal processing, multiuser precoding,
decision feedback equalizer and the relay signal processing. Email: dongchao@bupt.edu.cn
Baoyu Tian, received the B.S.
degree in Academy of Military
Engineering, Harbin, in 1969,
and the M.S. degree in information science from Beijing
Institute of Posts and Telecommunications, in 1982. Currently,
he is a professor at the School
of Information and Communication Engineering of
BUPT. His research interests include signal and information processing, digital mobile communication
and source coding theory. Email: tianbaoyu@bupt.
edu.cn
32
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