670IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Design of Low-Density Parity-Check
Codes for Modulation and Detection
Stephan ten Brink, Gerhard Kramer, Member, IEEE, and Alexei Ashikhmin, Member, IEEE
Abstract—A coding and modulation technique is studied where
the coded bits of an irregular low-density parity-check (LDPC)
nodes of the LDPC decoder graph are connected to detector nodes,
and iterativedecoding is accomplishedby viewing thevariable and
a curve fitting on extrinsic information transfer charts. Design ex-
amples are given for additive white Gaussian noise channels, as
well as multiple-input, multiple-output (MIMO) fading channels
where the receiver, but not the transmitter, knows the channel.
For the MIMO channels, the technique operates within 1.25 dB of
capacity for various antenna configurations, and thereby outper-
forms a scheme employing a parallel concatenated (turbo) code by
wide marginswhen there are more transmitthan receiveantennas.
parity-check (LDPC) codes, multiple-input, multiple-output
(MIMO) detection, mutual information.
noisy channels –. We consider two problems associated
with LDPC codes. The first is how to combine a code with a
modulator and detector. The second is how to design the code
for iterative decoding, i.e., how to choose good degree distribu-
tions for the modulator, channel, and detector.
We approach the first problem by mapping the coded bits of
an irregular LDPC code directly onto a modulation signal set.
The mapping is arranged to facilitate code design. At the re-
ceiver, we consider thegraphicalrepresentation ofan LDPCde-
coder – and connect the LDPC variable nodes to detector
We deal with the second problem by using a curve-fitting
procedure on extrinsic information transfer (EXIT) charts
. The design methodology is illustrated for two types of
channels: additive white Gaussian noise (AWGN) channels
with binary phase-shift keying (BPSK), and multiple-input,
multiple-output (MIMO) fading channels with quadrature
phase-shift keying (QPSK). The MIMO code design can be
be extended in a straightforward way to other modulators,
channels, and detectors. We remark that the curve fitting might
TERATIVE decoding of low-density parity-check (LDPC)
codes is a powerful method for approaching capacity on
Paper approved by H. El Gamal, the Editor for Space–Time Coding and
Spread Spectrum of the IEEE Communications Society. Manuscript received
July 17, 2002; revised May 30, 2003.
S. ten Brink was with Bell Laboratories, Lucent Technologies, Crawford, NJ.
He is now with Realtek, Irvine, CA 92618 USA (e-mail: stenbrink@realtek-
G. Kramer and A. Ashikhmin are with Bell Laboratories, Lucent
Technologies, Murray Hill, NJ 07974 USA (e-mail: email@example.com;
Digital Object Identifier 10.1109/TCOMM.2004.826370
be possible using other chart techniques, see, e.g., , ,
and . We refer to  for a comparison of some of these
tools. Another alternative is to use numerical optimization with
. Transfer charts and density evolution
complement each other in that the former are easier to visualize
and program, giving insight and good initial code designs,
while the latter can be used to verify the graphical analysis and
to refine the designs.
There are several existing approaches to combining coding
and modulation, for example, trellis-coded modulation (TCM)
, multilevel coding , bit-interleaved coded modulation
(BICM) , and space–time block-coded (STBC) modulation
,  (see also  and references therein). A growing
body of work uses BICM with turbo and LDPC codes, see, e.g.,
–. The EXIT curve-fitting approach described here was
motivated by results for erasure channels  and appeared in
. Parallel work using similar ideas was reported in  and
. The method was used to design repeat-accumulate (RA)
codes in  and .
This paper is organized as follows. In Section II, we develop
the curve-fitting procedure for BPSK on the AWGN channel.
In Section III, we extend the technique to other communication
problems, and, in particular, to MIMO fading channels where
the receiver, but not the transmitter, knows the channel. We de-
sign LDPC codes for ergodic fading, and compare their perfor-
are shown to perform substantially better for channels having
more transmit than receive antennas. Such a situation is likely
to occur on the base-to-mobile station link of a wireless com-
munication system. Section IV summarizes our results.
II. CODE DESIGN FOR AWGN CHANNELS
An iterative decoder for this code can be viewed as a graph that
The th variable node represents the th bit of the codeword.
This bit is involved in
parity checks, so that its node has
edges going into the edge interleaver. The edge interleaver
connects the variable nodes to the check nodes, each of which
represents a parity-check equation. The th check node checks
bits so that it hasedges. The sets of variable and check
nodes are referred to as the variable-node decoder (VND) and
check-node decoder (CND), respectively. Iterative decoding is
performed by passing messages between the VND and CND.
The decoder structure is shown in Fig. 1, and its operation is
explained in more detail below. We remark that this structure is
similar to that of an iterative decoder for a serially concatenated
0090-6778/04$20.00 © 2004 IEEE
TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION671
Fig. 1. Iterative decoder for an LDPC code.
code that is based on a mixture of inner repetition codes and
a mixture of outer single parity-check codes. This observation
illustrates the close relation between LDPC decoding and other
iterative schemes, such as turbo decoding .
A. EXIT Charts
An a posteriori probability (APP) decoder converts channel
and a priori log-likelihood ratios (LLRs or
-values. This is shown in Fig. 1, where a poste-
riori -values come out of the VND and CND. The a posteriori
-values minus the a priori -values are the extrinsic -values,
which are passed on and interpreted as a priori information by a
second decoder. We refer to  for further details on extrinsic
An EXIT chart characterizes the decoder’s operation. We use
the notation of  and write
tion between the bits on the decoder graph edges (which are the
bits about which extrinsic -values are passed) and the a priori
-values. Similarly, we write
mation between the bits on the graph edges and the extrinsic
-values. We refer to  for further details on how to interpret
-values) into a
for the average mutual informa-
for the average mutual infor-
B. EXIT Curve of the Inner VND
A variable node of degree
able node decodes by computing, for
has incoming messages,
able node, and
Consider the AWGN channel with BPSK (
ratio (SNR) as
is the th a priori
is the th extrinsic -value coming out of the vari-
is the channel-value.
-value going into the variable
. The channel -value is
function (pdf) evaluated at the output
and be random variables representing the respective channel
input and output. The variance of
is the channel conditional probability density
given the input . Let
Fig. 2. VND EXIT curves for ? ?? ? ? dB and ? ? ???.
To compute an EXIT function, we model
-value of an AWGN channel whose input is the th interleaver
bit transmitted using BPSK. The EXIT function of a degree-
variable node is then
as the output
where the functions
pendix (see also ). Fig. 2 plots several variable node curves
andare given in the Ap-
dB. The quantity
is the capacity of the channel at the
which in our case is the capacity of the AWGN channel with
BPSK modulation (see the Appendix).
that is being considered,
C. EXIT Curve of the Outer CND
The decoding of a degree
decoding of a length
code. The output -values are thus (see [37, Sec. II.A])
check node corresponds to the
) single parity-check (or rate
672IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
swapped, as compared with Fig. 2.
CND EXIT curves. Observe that the a priori and extrinsic axes are
-value notation of , this can also be written as a
We again model
channel whose input is the th interleaver bit transmitted using
BPSK. The check node EXIT curves can be computed in closed
form ,  or by simulation. Alternatively, for the binary
erasure channel, a duality property exists ,  that gives the
of the length
in terms of the EXIT curve
) repetition code, i.e.,
as the output-value of an AWGN
single parity-check code
of the length (or rate
This property is not exact for BPSK/AWGN a priori inputs, but
it is very accurate , . For convenience, we use (8) and
where the second step follows from (4) with
further useful to express (9) in terms of its inverse function, i.e.,
. It is
Fig. 3 plots several check node curves. Observe that the curves
are similar to the VND curves of Fig. 2, except that they all start
from the origin.
D. EXIT Curves for Code Mixtures
We will consider only check-regular LDPC codes, i.e., all
check nodes have degree
(in , these are called right-reg-
ular codes). After choosing , the remaining LDPC design in-
volves specifying the variable node degrees
note these degrees by
node degree is
be the number of different variable node degrees, and de-
, . The average variable
VND and CND are the same, we have
is the fraction of nodes having degree
. Note that
. Since the number of edges at the
be the fraction of edges incident to variable nodes of
. There are, in total,
such nodes, so we have
edges involved with
Note that the
shown that the EXIT curve of a mixture of codes is an average
of the component EXIT curves. We must here average using the
(and not the ) because it is the edges that carry the extrinsic
messages. The effective VND transfer curve is thus
must satisfy . In  and , it is
edge fractions can be adjusted because we must enforce
. Thus, in order to have any flexibility, we
. We shall see that
surprisingly good results [4, p. 634].
E. Design Example
curves. Empirically, the same is true for other channels. We il-
lustrate this for BPSK modulation on an AWGN channel, and
in the next section for MIMO channels.
the Appendix). We design a code with
dB, so that
is slightly larger than . We further choose
the CND transfer curve has a reasonable distance from the
axis at . This approach simplifies finding a VND
decoder a good “head start” at the first iteration.
For simplicity, we restrict the VND to have only three dif-
ferent variable node degrees (
can be chosen freely, and Fig. 4 shows a manual curve fit
whose variable node parameters are as follows:
, and choose
). This means that only one
For this simple example, we obtain a convergence threshold of
about 0.5 dB, while the capacity is at 0.19 dB. A simulation
, a random edge interleaver, and 100 iterations
shows a turbo cliff at about 0.55 dB (we measure the turbo cliff
TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION 673
Fig. 4. Curve fit for an LDPC code with ? ? ???.
at a bit-error rate (BER) of
of the EXIT chart and our approximations.
curve more closely to the CND curve. One can also model the
-values more carefully than as outputs of an AWGN
channel. Finally, code design is just as easy for other rates. For
instance, suppose that
start at about
can yield a good curve fit. We would now again
to ensure that the
). For longer codes and more
so that the VND curve should
. Fig. 3 suggests
curve lies above the
III. CODE DESIGN FOR MIMO CHANNELS
We next turn to multiantenna modulation and detection. The
techniques described here can be applied to many other modu-
lators, channels, and detectors.
A. MIMO Channel Model
For MIMO fading, one often distinguishes between three
practical cases based on channel knowledge. The first case is
that both the transmitter and receiver know the channel, the
second that only the receiver knows the channel, and the third
that neither terminal knows the channel [41, pp. 2627–2629].
The first case can be dealt with in a manner similar to the
AWGN channel, with the addition of “water pouring” over the
antennas and time. The third case seems to be the most difficult
and is not dealt with here. We will consider only the second
Consider the setup of Fig. 5, where there are
whose entries take on complex values in a
constellation set. We consider constellations of size
each vector symbol carries
for QPSK, we have
. The average energy per transmit
coded bits. For example,
Fig. 5. MIMO model.
symbol is limited to
The receiver sees
Suppose the entries of
parts each having variance
, and we will use
1 vectors, where
channel matrix and
are independent, complex, zero-mean,
is a 1 noise vector.
. We define the normal-
to keep the
consider a Rayleigh fading channel, so that the entries of
independent, complex, zero-mean, Gaussian random variables
with independent real and imaginary parts each having variance
1/2 [42, Sec. 4]. For a quasi-static channel, the matrix
mains unchanged over long time intervals, while for an ergodic
changes for every symbol . We consider only the
ergodic model whose capacity is (see  and )
appears in (15) by convention; this is done simply
at capacity close to each other for dif-
. We assume that is known to the receiver only, and
gate transpose of
tributed . We will consider only QPSK for simplicity, but the
design procedure described below is the same for other modu-
lation sets and mappings (see ).
is the identity matrix and
. One achieves capacity with Gaussian dis-
is the complex-conju-
B. EXIT Curve of the MIMO Detector
We proceed by showing how to compute EXIT curves for
the MIMO detector. We then describe a combined demodula-
tion/decoding structure that is flexible enough to closely ap-
proach MIMO capacity. This structure automatically specifies
674 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Fig. 6. MIMO detector EXIT curves at ? ?? ? ? dB and ? ? ???.
how the encoding/modulation is done, i.e., one maps the coded
bits directly onto the modulation signal set. Finally, we show
how to match EXIT curves.
The MIMO detector performs APP detection by considering
possible hypotheses on
detector computes the
(see –). That is, the
onto the vector symbol
tector EXIT curve cannot be described in closed form, so we
measure it by Monte Carlo simulation . We denote the de-
tector by DET and its EXIT curve by
whose channel output is
is the th coded bit mapped
. The de-
Fig. 6 shows simulated EXIT curves of MIMO detectors for
Gray-mapped QPSK, and different numbers of transmit and
receive antennas. Observe that most of the curves resemble
straight lines. The 1
1 curve is, in fact, the horizontal line
random variable, and where we have used (5). We remark that
curve is a horizontal line. Furthermore, the
curves meet at
, and the curves decrease with
when . Similarly, one can show analytically that the
curves meet the 1
they decrease with
is a zero-mean, unit-variance, complex Gaussian
curve at, and that
C. EXIT Curve of the Combined MIMO Detector and VND
We combinetheMIMO detectorand theLDPC variablenode
decoder as shown in Fig. 5. The detector consists of
individual detectors (or detector nodes) that are each connected
variable nodes. We further choose allvariable
nodes connected to a common detector node to have the same
. This restriction is not necessary, but it simplifies the
design by focusing on a smaller set of EXIT curves.
The decoder structure specifies that the LDPC coded bits be
mapped directly onto the modulation set. A similar technique
has been considered by others, e.g.,  and , but without
the structured combination of variable and detector nodes, and
without the EXIT curve matching approach to code design. We
show that curve matching gives coded modulations that closely
Consider the three boxes in Fig. 5 corresponding to one of
individual detector/VNDs. We denote the EXIT
curve of this structure by
-values as the output
whose inputs are the interleaver bits transmitted using BPSK.
Using (4), the lower VND in Fig. 5 maps
. We again model the a
-values of an AWGN channel
We next approximate the detector curve (18) by a third-order
polynomial. For example, the EXIT curve of the 4
detector with QPSK at
is welldB and
The polynomial (21) lets us express the combined de-
tector/VND curve in closed form, which is convenient for curve
The third step is to consider the upper VND in Fig. 5. We
use (4) with
detector/VND EXIT curve as
and write the combined
Finally, inserting (18), (20), and the detector polynomial (21)
into (22), we obtain the desired transfer curve in the form
Fig. 7 shows some combined detector and VND transfer curves.
Observe that by setting
detector transfer curves of Fig. 6.
in (23) we recover the “pure”
D. Design Examples
We design check-regular LDPC codes by matching the curve
different variable node degrees (
fit manually. Table I shows the parameters of our curve fitting
for several MIMO channels, and Fig. 8 plots the resulting EXIT
chart for a 4
1 MIMO channel.
leaver, and 100 decoder iterations. All schemes operate within
of the APP processing, the number of simulated blocks for the
) and perform the curve
, a random edge inter-
TEN BRINK et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR MODULATION AND DETECTION 675
and ? ? ???.
Combined 4?1-detector and VND EXIT curves at ? ??
? ? dB
LDPC CODE PARAMETERS FOR ? ? ? MIMO CHANNELS, QPSK, AND
? ? ???. OVERALL RATE: ? BITS/CHANNEL USE
pacity can be narrowed further by using
fits, and by using longer codes and more iterations.
standard turbo code, denoted PCC for “parallel concatenated
cases was limited to 40). We expect that the gaps to ca-
and better curve
Fig. 8.Curve fit for the 4?1 scheme at ? ?? ? ? dB and ? ? ???.
concatenated codes or PCC). All schemes have an overall rate of ? bits per
BERs for LDPC codes (LDPCC) and turbo codes (parallel
back and feedforward generator polynomials
QPSK and the code has
terleaver. The iterative processing is done by performing 20 in-
ternaldecoder iterationsper detector/decoderiteration, and four
detector/decoder iterations. More internal or detector/decoder
iterations hardly improve the BER.
Observe that the LDPC scheme outperforms the turbo-coded
scheme by wide margins for
1 MIMO). This happens because the turbo code EXIT
curve is almost a horizontal line, as will be the case for any
strong code, for the AWGN channel. The code EXIT curve is,
therefore, poorly matched to a steep detector EXIT curve .
, , and a random in-
(the gain is over 5 dB
676IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
E. Fading Model and Finite Modulation Sets
We discuss two practical issues. First, the ergodic model is
suitable if there appear many independent realizations of
and the receiver can accurately estimate these. This occurs in
practice when there is block fading (such as with time-divi-
time/frequency interleaving), and one can afford to code over
many blocks. However, in some cases, the delay constraints are
too severe to code over many blocks, and the quasi-static model
is appropriate [42, Sec. 5]. For such cases, one can still apply
curve fitting, but one now shapes the EXIT curves to minimize
outage probability. The question of how to do this precisely for
given fading statistics is an interesting open problem.
Second, the capacities of both the ergodic and quasi-static
models are achieved with Gaussian input distributions. How-
ever, in practice, one does not use such distributions. Some rea-
sons for this are the transmitter and receiver amplifiers have
peak energy constraints, the amplifiers are linear over limited
ranges of inputs, the receiver must acquire and maintain phase
synchronization, and the APP detector complexity grows expo-
nentially with the number of bits per input symbol. These rea-
sons make small modulation sets such as QPSK attractive for
wirelesscommunication. Ofcourse,thecapacities of suchmod-
ulation sets are smaller than for Gaussian inputs. We have com-
pared the performance of our codes with the finite modulation
set capacities, which is the fairest benchmark.
F. Comparison With STBCs
STBCs are attractive because they improve diversity and
are easy to encode and decode (see – and references
therein). However, the gains come at the expense of spectral
efficiency. For example,consider the 2
QPSK and a rate-1/2 LDPC code. The spectral efficiency is
two bits per channel use. Suppose next that we use Alamouti’s
space–time code  as an inner code. This scheme obtains a
diversity gain with what is, in effect, a rate-1/2 repetition code.
Thus, to achieve two bits per channel use with a rate-1/2 outer
code, one must compensate for the inner-code rate loss by
changing the constellation from QPSK to 16-quadrature ampli-
tude modulation (QAM). But recall that 16-QAM complicates
amplification, synchronization, and detection.
These issues will be even more pronounced for 4
channels. For example, the scheme proposed here achieves a
spectral efficiency of four bits per channel use with QPSK and
effect, a rate-1/4 repetition code, would have to use 256-QAM
to achieve four bits per channel use with a rate-1/2 outer code.
Furthermore, no rate-1/4 orthogonal STBC exists for the 4
MIMO channel . These considerations suggest that diver-
sity is best achieved by designing the coding and modulation
together rather than separately.
1 MIMO channel with
We described a method for combining an irregular LDPC
code with a modulator and detector. We further introduced a
pragmatic technique for designing LDPC degree distributions
that are well matched to detectors. The design was based on
curve fitting on EXIT charts. For MIMO communication, sim-
ulations verified that all our codes operate within 1.25 dB of
their respective capacity limits at a BER of
that EXIT curve fitting can be used to construct equally good
coded modulations with RA codes , .
Finally, the goal of this paper was to present a coding and
modulation technique, as well as a design methodology, that
gives good performance for many communication systems. We
were less ambitious with our code designs. The LDPC degree
distributions listed in Table I can almost certainly be improved
upon, especially for the 4
1 MIMO channel. Other open prob-
lems are EXIT curve fitting for quasi-static channels, EXIT
curve fitting for suboptimal detectors, e.g., the detectors of sev-
eral papers in , and improving the EXIT approximations.
. We remark
is zero-mean, Gaussian noise with variance
-value of (2) is a function of , and we write this as
. The channel
is Gaussian with
. We thus have
Letbe the mutual information. We have
the same as
For computer implementation, we split
corresponding to the intervals
. We used a polynomial fit for the left interval and
an exponential fit for the right interval. We applied the Mar-
quardt–Levenberg algorithm (see ) to obtain
is the entropy of
. The capacity of our
. Note that
is the en-
into two parts,
For the inverse
-function we split the curve into two in-
. Our approximation is
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Stephan ten Brink received the Dipl.-Ing. and the
Dr.-Ing. degrees in electrical engineering from the
University of Stuttgart, Stuttgart, Germany, in 1997
and 2000, respectively.
From 2000 to June 2003, he was with the Wireless
Research Laboratory, Bell Laboratories, Lucent
Technologies, Holmdel, NJ, conducting research
on channel coding for multiple-antenna systems.
Since July 2003, he has been with Realtek, Irvine,
CA, where he is involved in the development and
standardization of high-throughput WLAN systems.
His research interests include error-correcting coding, iterative decoding,
multiple-antenna communications, and watermarking.
678IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004 Download full-text
Gerhard Kramer (S’91–M’94) received the B.Sc.
and M.Sc. degrees in electrical engineering from
the University of Manitoba, Winnipeg, MB, Canada,
in 1991 and 1992, respectively, and the Dr.Sc.
Techn. degree from the Swiss Federal Institute of
Technology (ETH), Zürich, Switzerland, in 1998.
dora Tech AG, Basel, Switzerland, as a communica-
tions engineeringconsultant. SinceMay2000, hehas
been with the Mathematics of Communications Re-
search Department, Bell Laboratories, Lucent Tech-
nologies, Murray Hill, NJ.
Alexei Ashikhmin(M’00) received the Ph.D. degree
in electrical engineering from the Institute for Infor-
mationTransmission Problems, Russian Academyof
Science, Moscow, Russia, in 1994.
From September 1995 to September 1996, he was
with the Mathematics Department, Delft University
of Technology, Delft, The Netherlands. From Jan-
uary 1997 to July 1999, he was a Postdoctoral Fellow
at the Computer, Information, and Communication
Division of Los Alamos National Laboratory, Los
Alamos, NM. Since 1999, he has been with the
Mathematics of Communications Research Department, Bell Laboratories, Lu-
cent Technologies, Murray Hill, NJ. His research interests include information
and communication theory, with emphasis on error-correcting codes.
Dr. Ashikhmin currently serves as an Associate Editor for Coding Theory for
the IEEE TRANSACTIONS ON INFORMATION THEORY.