Application of Protographbased LDPCCodes for UWB Short Range Communication
 [Show abstract] [Hide abstract]
ABSTRACT: We investigate the performance of the protograph lowdensity paritycheck (LDPC) codes, which have been shown to possess simple structures and outstanding error performance over additive white Gaussian noise (AWGN) channels, over partial response (PR) channels using the finitelength extrinsic information transfer (EXIT) algorithm. Due to the intersymbol interference (ISI) caused by the PR channels, we observe that the conventional protograph LDPC codes do not perform well in terms of error rates. We further propose a new design scheme and construct three new types of protograph LDPC codes. Unlike conventional protograph LDPC codes in which the highestdegree variable nodes are punctured, the new protograph LDPC codes have their lowestdegree variable nodes punctured. Moreover, some edge reconnections are made in one of the proposed codes. The EXITchart analysis, the convergence analysis and the biterrorrate simulation have shown that all three new codes outperform the conventional protograph LDPC codes. Moreover, two of the proposed codes are superior to the regular columnweight3 LDPC code and thus they are good alternatives compared to other errorcorrection codes for use in data storage systems.IEEE Transactions on Communications 01/2012; 60(10):28092819. · 1.75 Impact Factor
Page 1
Application of Protographbased LDPC Codes for
UWB Short Range Communication
Patrick Grosa, Andre Fonseca dos Santos, Michael Lentmaier, Wolfgang Rave and Gerhard Fettweis
Vodafone Chair Mobile Communications
Dresden University of Technology (TU Dresden), 01062 Dresden, Germany
Emails: {grosa, andre.santos, michael.lentmaier, rave, fettweis}@ifn.et.tudresden.de
Abstract—We studied the behavior of an iterative receiver
using protograph based LDPC codes (PGLDPCC) in an UWB
short range communication system. An EXITChartAnalysis is
applied by means of density evolution of protographs. We show
that AWGNoptimized protograph ensembles perform also well
on frequencyflat UWB channels, outperform ratecompatible
punctured convolutional codes (RCPC), but degrade with in
creasing intersymbol interference (ISI). Slight modifications of
the underlying protograph can improve this performance, but
lead to a tradeoff between the minimum required SNR and the
minimum achievable BER. It is shown that both thresholds can
be predicted by the applied tools.
IndexTerms—TurboEqualizer,
tographs, EXITCharts, UWB
I. INTRODUCTION
The demand for wireless high data rate services has been
rapidly growing for years. Ultrawideband (UWB) communi
cation has been considered to satisfy this need in the field
of shortrange communication. However, achieving data rates
of about 1 Gbit/s typically requires a large bandwidth. Such
ultrawide band systems experience a long channel impulse
response with a high number of resolvable paths. One way to
mitigate the effects of ISI is the usage of turbo equalization,
i.e. a continuous improvement of the equalization due to an it
erative exchange of information between equalizer and channel
decoder. However, such systems require the adjustment of the
behavior of both components in order to optimize the overall
performance. Several different approaches have been discussed
in this area, e.g. in [1] or [2].
Throughout this work, we study the interaction of so
called protograph based lowdensity paritycheck codes (PG
LDPCC) and a soft equalizer (consisting of a soft mapper and
demapper and equalizer) as described in [3]. In our case, the
analysis of LDPC codes is done in a very accurate way by den
sity evolution, that neither reduces the inner decoder to a single
iteration as in [2] nor has to rely on the assumption of Gaussian
distributed messages within the channel decoder. However, the
original version of density evolution does not consider the
structure of the code. It is possible to overcome this issue
by using protographs. Hence, we can view PGLPDCCs as
structured LDPC codes. Another benefit of protographs is the
possibility to construct quasicyclic codes with lower hardware
complexity .
In the following section the considered communication system
is described. Subsequently, a brief introduction into protgraphs
and their analysis is given. The results of this analysis are used
Density Evolution,Pro
to characterize the overall system. The predicted performance
is compared with simulations and discussed.
II. SYSTEM DESCRIPTION
We consider a single carrier transmission system with one
antenna at transmitter and receiver side. A data word d ∈
{0,1}K×1is encoded and the resulting vector of coded bits
v ∈ {0,1}N×1is randomly interleaved, yielding the vector
x. Subsequently, the coded interleaved bit vector is linearly
mapped onto the elements of the complex symbol vector sd=
?sd1... sdp... sdP
?Tof length N/Q = P, where 2Qis the
sd
sx
size of the modulation alphabet.
EncoderInterleaver
QAM
Cyclic
Prefix
CIR
w
+
UWB
Channel
d
vr
Fig. 1.Transmission chain
A cyclic prefix is added to the symbol vector in order to
enable equalization in the frequency domain. The transmitter
is illustrated in Figure 1 including the channel. We assume
a multipath channel, with channel impulse response (CIR)
given by h = [h(0) ... h(l) ... h(L − 1)]. We assume block
fading, i.e. the CIR does not change during the transmission of
one block. In addition, white Gaussian noise samples w ∈ CN
with variance σ2
received samples r(i) at time i are observed as:
?L−1
The distorted signal is received and equalized at the receiver
side (Figure 2). Then, the symbol vector is demapped and the
extrinsic loglikelihoodratios (LLRs) LE
for the coded interleaved bits. The deinterleaved vector of
LLRs, denoted by LD
decoder to perform a first decoding attempt. The extrinsic
LLRs of the decoder, i.e. the difference between aposteriori
LLRs, denoted by LD
by LD
iteration, these LLRs are used to improve the equalization of
the signal and the result is forwarded to the decoder, again.
This process is repeated for a fixed number of iterations.
III. PROTOGRAPH ENSEMBLES
A. Protograph LDPC Codes
In Gallager’s original work on LowDensity ParityCheck
codes (LDPCCs) [4], he did not only introduce regular LD
PCCs (same number of ones in each row and each column,
ware added to the symbol vector and the
r(i) =
l=0h(l)s(i − l) + w(i).
(1)
e(x) are computed
a(v), is then forwarded to the channel
p(v) and apriori LLRs, is designated
e(v) and is fed back to the soft equalizer. In the next
Page 2
Equalizer
Soft
Demapper
Soft
Mapper
Soft
Equalizer
De
interleaver
Interleaver
Decoder
r
LE
e(x)
LE
a(x)
LD
a(v)
LD
e(v)
LD
p(d)
LD
p(v)

ˆ d
ˆ s
+
Fig. 2.Receiver chain
respectively) and the belief propagation decoding scheme, but
he also proposed an algorithm to construct structured LDPCCs
composed of individual permutation matrices. In [5], it was
shown that these codes become QuasiCyclicCodes (QC
Codes) if permutation matrices are replaced by circulants.
Since irregular LDPCCs have a very good performance, the in
terest in irregular and structured LDPCCs has led to an LDPC
ensemble description called protographs [6]. A protograph is
a bipartite graph G(V,C,E) that consists of a set of variable
nodes vi ∈ V of degrees dvi, i = 1,..,NP, a set of check
nodes cr ∈ C of degrees dcr, r = 1,..,MP, and a set of
edges et∈ E, t = 1..tmaxthat connect the nodes. We further
distinguish between the set of edges evi
and ecr
vi and check node cr, respectively. The variable node vi is
connected by its jth edge to the check node cr with its k
th edge if evi
j
= ecr
k
= et. As an example, the socalled
AccumulateRepeatJaggedAccumulate (ARJA) protographs
(proposed in [7]) are illustrated in Figure 3, where nodes with
a plus represent check nodes and all other nodes are variable
nodes. Empty variable nodes will be punctured.
j∈ E(vi), j = 1,..,dvi
k∈ E(cr), k = 1,..,dcremanating from variable node
+
+
+
+
+
+
+
+
(a) ARJA (R=1/2)
+
+
+
+
+
+
+
+
+
+
+
+
(b) ARJA (R=1/3)
+
+
+
+
+
+
+
+
+
+
(c) ARJA (R=2/3)
Fig. 3.ARJA protographs for different rates
A corresponding representation of a protograph is an MP×
NP biadjacency matrix B, which is called the base matrix
of a protograph. Each row and each column corresponds to a
check node and a variable node, i.e. the entry in column i and
row r is equal to the number of edges between variable node
viand check node cr.
The construction of derived LDPCCs is done by replacing
each 0 in B by an allzeromatrix and each a > 0 of B by
the addition of a permutation matrices, whereas the position
of the one in each row must be different for every permutation
matrix of size T. This procedure is equivalent to the copyand
permute method proposed in [6]. Both procedures preserve
the structure of the protograph, but multiple edges are spread
between the copies. Both algorithms are used to lift a given
protograph. The set of matrices H that can be derived from
a given protograph by all possible combinations of size T
permutation matrices defines an ensemble of protograph based
LDPCC (PGLDPCC) codes of length N = TNP. The main
point is that both procedures conserve the structure of the
protograph. However, these procedures do not specify the
permutation matrices, which can than be arbitrarily chosen.
For that purpose two similar approaches, the Progressive
EdgeGrowth (PEG) [8] algorithm and the ACE algorithm [9],
were introduced and adapted to the protograph case. Moreover,
both algorithms can be combined by using the result of the
first algorithm as a protograph for the second algorithm.
B. Density Evolution for Protograph Ensembles
Introduced in [10] the density evolution is used for the
construction of irregular LDPC codes. Since all protograph
based construction methods of LDPC codes conserve the
structure of the protograph, it is reasonable to apply the density
evolution within the protograph. The basic idea of density
evolution is to track the probability density function (pdf) of
the LLRs during the decoding process. Therefore, equivalent
pdf calculations for the update functions must be found. Since
specifying a transformation of the check node update in closed
form is not feasible, it is necessary to use the quantized version
of the density evolution, which is explained in detail, e.g. for
protograph based LDPC codes in [6].
IV. EXITCHART ANALYSIS
For the study of an iterative (also called Turbo) system, the
amount of information that is exchanged between the different
components plays an important role. An appropriate way to
analyze the system and to visualize the amount of information
is the socalled Extrinsic Information Transfer (EXIT) Chart.
Introduced in [11] for parallel concatenated codes it became
also popular for LDPC codes and other iterative systems as it
is shown, e.g. in [12].
Starting point is the extrinsic information transfer function, i.e.
the function that maps the apriori mutual information (MI)
Ia to the extrinsic MI Ie of a component. An EXITChart
illustrates the transfer function of two components in a single
graph, whereas the axes of the second component are swapped
by using the inverse mapping from Ie to Ia. The straight
forward way to estimate the EXIT function of a component
is to generate random variables with a specific pdf, yielding
to different values of apriori MI, at the input of a component
and to measure the extrinsic MI at its output. This approach is
used to determine the transfer function of the equalizer (based
on [12], detailed explanations for the used equalizer can be
found in [3]). A common way to analyze LDPC decoder is
to assume Gaussian distributed messages that circulate inside
the decoder. Then, the transfer function can be calculated by
means of the Jfunction. This approach is explained in more
detail in e.g. [13].
A. EXIT Function by means of Density Evolution
However, we use a more precise way to obtain the EXIT
function by means of density evolution, where we use the
general relationship of the MI between the equally likely bit
V ∈ {±1} and the respective LLRs L(V ) for symmetric and
consistent Lvalues, that is given by
?
I(L(V );V ) = 1 − Elog2
?
1 + exp−L(V )??
.
(2)
Page 3
Due to the fact that the density evolution tracks the pdfs of
the LLRs during the decoding process, we even know the pdf
of the apriori LLRs and the pdf of the extrinsic LLRs after
decoding. Hence, we can use this knowledge to compute the
apriori MI and the extrinsic MI by Eq. (2).
V. ANALYSIS AND SIMULATION RESULTS
For the study and simulation we use two different channel
types. The first one is the NLOS desktop channel model from
[14], which is a relatively frequencyflat channel. The second
one has an exponential power delay profile with 10 taps and
a decay rate of 10/tap (pdp(l) = e−l
As a starting point for our studies, we used the earlier
mentioned protographs known to have a good performance
for an AWGN channel. A useful property of this protograph
is the possibility to derive different code rates with only slight
changes. These protographs are illustrated in Figure 3. At first
we lift the protographs by the PEG algorithm with T = 7(14)
and afterwards by the ACE algorithm with T = 128, where
we define the permutation matrices of the second step to be
circulants, to achieve two different block lengths of N =
4480(8960). For comparison, we use ratecompatible punc
tured convolutional (RCPC) codes as introduced in [15]. All
codes in this family are obtained from a mother code of rate
1/4 using the generator polynomials Goct = {23,35,27,33}.
We use a block length of 6000(12000) bits, for which it is
known that the performance of a convolutional code does not
depend on the block length.
A. Low ISI Channels
The first performance estimate is obtained by the EXIT
Chart analysis. We determine the EXIT function of the NLOS
desktop channel model for different signaltonoise ratios
(SNR) as depicted in Figure 4. As one can see, the transfer
functions of the equalizer and the ARJAbased code fit well
together, i.e. match for such boundary conditions of low ISI.
Hence, both curves intersect only at very small SNR values.
A different behavior can be found for the convolutional codes,
which still intersect for high SNR values.
10; l = 0,..,9).
0 0.20.40.60.81
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ia
E,Ie
D
Ie
E,Ia
D
4dB
Soft Equalizer
AR11Z
AR11Z2
ARJA
RCPC
6dB
2dB
Fig. 4. UWB desktop channel  EXITChart
In the biterrorrate (BER) simulation we use QPSK modu
lation. The result is illustrated in Figure 5. As already expected
from the EXIT chart analysis the LDPC codes outperform the
convolutional codes for all rates. At a BER of 10−4the gain
amounts to approximately 2 − 3 dB. A small additional gain
is possible with longer LDPC code words.
100
1234567
10−6
10−4
10−2
Eb / N0 [dB]
BER
RCPC codes
AR4JA codes
R = 1/3
R = 1/2
R = 2/3
R = 1/3
R = 2/3
R = 1/2
Fig. 5.UWB desktop channel  BitErrorRate Simulation
B. Strong ISI Channels
For the strong ISI channel we again determine the EXIT
function for different SNRs. The corresponding EXITChart
is illustrated in Figure 6, where we only consider the rate1/2
code.
00.20.40.60.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Ia
E,Ie
D
Ie
E,Ia
D
Eq SNR=7dB
Eq SNR=6.6dB
Eq SNR=6.65dB
AR11Z2
ARJA
RCPC
AR11Z
Fig. 6.10 tap channel  EXITChart
Because the matching of EXIT functions is an indication
for the performance of a system, we also expect a smaller gain
of the codes for the channel that is impaired by significant
ISI. One can see that there is still an intersection for relatively
high values of Ia, but in case of the ARJA code also in
the intermediate region. Therefore, we slightly changed the
protograph (cmp. Figure 7a7b) by reducing the number of
parallel edges to influence the shape of the transfer functions
and to obtain a better curve fit, which can be observed in the
figure.
+++
+++++
(a) AR11Z
Fig. 7.
+++
+++++
(b) AR11Z2
Proposed protographs
Page 4
In this context, two regions are of special interest for the
performance of such an iterative system. The first one is the
threshold region at Ia = 0.5 and the other one is the error
floor region at Ia≈ 1. The first region determines the starting
point of a strong decay (waterfall) in the BER curve, i.e. the
amount of exchanged information is sufficient to eliminate a
large fraction of error events due to the good curve fit (no
intersection). The match of the EXIT curves in the second
region strongly influence the achievable BER, because the
values for Iamust be very close to one to obtain very good
BERs.
10
−4
10
−3
10
Eq
−2
10
−1
10
0
10
−2
10
−1
10
0
1−Ie
D,1−Ia
1−Ia
D,1−Ie
Eq
Eq SNR=6.65dB
Eq SNR=7dB
Eq SNR=8.25dB
ARJA
AR11Z2
AR11Z
RCPC
Fig. 8.10 tap channel  gaptoone EXIT Chart (loglogscale)
In Figure 8, one can see the different behavior in more detail.
While the transfer functions of the RCPC and the AR11Z
code intersect the equalizer EXIT functions, the curves for
the ARJA and the AR11Z2 code do not intersect them in that
region. Hence, we expect a small decaying BER curve or even
an error floor for the first ones. The required SNR and the BER
can be estimated by means of the EXITChart analysis.
6 6.57 7.58 8.59
10
−8
10
−6
10
−4
10
−2
10
0
Eb/N0 [dB]
BER
RCPC
AR11Z2
AR11Z
ARJA
Fig. 9.10 tap channel  BitErrorRate Simulation
Corresponding simulation results of the BER are illustrated
in Figure 9. One can see that, a gain of about 1.5dB can be
obtained but not for both protograph codes. The slope of the
AR11Z based code starts to decrease at lower Eb/N0than the
AR11Z2 based code, but is getting flatter at 8dB.
The RCPC codes also show nearly an error floor, due to the
intersection of the EXIT function. By appropriate design of
the protographs this intersection should be avoidable. It is
likely that we can construct codes with better performance,
but the proposed ones illustrate the major problems of the
code construction for iterative systems.
VI. SUMMARY AND CONCLUSIONS
We studied the performance of existing AWGNoptimized
PGLDPCCs in terms of their applicability for turbo equaliza
tion in UWBchannels. Due to the fact that the performance of
the equalizer is strongly influenced by the considered channel,
we have investigated a frequencyflat channel at first and
extended the study to a channel that leads to strong ISI.
In order to analyze and predict the performance we used
EXIT charts as analysis and visualization tool of the iterative
information exchange and the density evolution as analysis
tool for the PGLDPCCs. By these means, we generalized
the analysis so that we did not have to assume Gaussian
distributed messages in the decoding process. We have shown
that the performance of such optimized codes degrade with
increasing ISI. In order to avoid degradation, we proposed two
exemplary protographs that have a better curve fit and a better
performance. The analysis and simulation results indicate that
an interesting problem for future investigations is the trade
off between the minimum achievable signaltonoiseratio and
minimum achievable biterrorrate.
REFERENCES
[1] M. T¨ uchler, R. Koetter, and A. Singer, “Turbo equalization: principles
and new results,” Communications, IEEE Transactions on, vol. 50, no. 5,
pp. 754 –767, may 2002.
[2] M. Franceschini, G. Ferrari, and R. Raheli, “EXIT Chartbased Design
of LDPC Codes for InterSymbol Interference Channels,” in Proc. IST
Mobile Summit Dresden, 2005.
[3] A. F. dos Santos, “Efficient receiver methods for coded systems under
channel uncertainty,” Ph.D. dissertation, Dresden University of Technol
ogy, 2010.
[4] R. Gallager, LowDensity ParityCheck Codes.
[5] R. Tanner, D. Sridhara, A. Sridharan, T. Fuja, and D. Costello Jr., “LDPC
block and convolutional codes based on circulant matrices,” Information
Theory, IEEE Transactions on, vol. 50, no. 12, pp. 2966 – 2984, dec.
2004.
[6] J. Thorpe, “Lowdensity paritycheck (LDPC) codes constructed from
protographs,” JPL, IPN Progress Report 42154, Aug 2003.
[7] D. Divsalar, C. Jones, S. Dolinar, and J. Thorpe, “Protograph based
LDPC codes with minimum distance linearly growing with block size,”
vol. 3, nov.2 dec. 2005, p. 5 pp.
[8] X.Y. Hu, E. Eleftheriou, and D. Arnold, “Regular and irregular progres
sive edgegrowth tanner graphs,” Information Theory, IEEE Transactions
on, vol. 51, no. 1, pp. 386 –398, jan. 2005.
[9] T. Tian, C. Jones, J. Villasenor, and R. Wesel, “Construction of irregular
LDPC codes with low error floors,” vol. 5, may 2003, pp. 3125 – 3129
vol.5.
[10] S.Y. Chung, “On the construction of some capacityapproaching coding
schemes,” Ph.D. dissertation, Massachusetts Institute Of Technology,
September 2000.
[11] S. ten Brink, “Convergence behavior of iteratively decoded parallel
concatenated codes,” Communications, IEEE Transactions on, vol. 49,
no. 10, pp. 1727 –1737, oct 2001.
[12] J. Hagenauer, “The EXIT Chart  Introduction to Extrinsic Information
Transfer,” in in Iterative Processing, In Proc. 12th Europ. Signal Proc.
Conf (EUSIPCO), 2004, pp. 1541–1548.
[13] S. ten Brink, G. Kramer, and A. Ashikhmin, “Design of lowdensity
paritycheck codes for modulation and detection,” Communications,
IEEE Transactions on, vol. 52, no. 4, pp. 670 – 678, april 2004.
[14] IEEE 802.15 Task Group 3c, “Channel modeling subcommitee
final report,” IEEE, Tech. Rep., March 2007. [Online]. Available:
http://www.ieee802.org/15/pub/TG3c.html
[15] J. Hagenauer, “Ratecompatible punctured convolutional codes (RCPC
codes) and their applications,” Communications, IEEE Transactions on,
vol. 36, no. 4, pp. 389 –400, apr 1988.
MIT Press, 1963.