Design of lowdensity paritycheck codes for modulation and detection
ABSTRACT A coding and modulation technique is studied where the coded bits of an irregular lowdensity paritycheck (LDPC) code are passed directly to a modulator. At the receiver, the variable nodes of the LDPC decoder graph are connected to detector nodes, and iterative decoding is accomplished by viewing the variable and detector nodes as one decoder. The code is optimized by performing a curve fitting on extrinsic information transfer charts. Design examples are given for additive white Gaussian noise channels, as well as multipleinput, multipleoutput (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 outperforms a scheme employing a parallel concatenated (turbo) code by wide margins when there are more transmit than receive antennas.

 SourceAvailable from: Benjamin Gadat
Conference Paper: Asymptotic analysis and design of LDPC codes for Laurentbased optimal and suboptimal CPM receivers
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ABSTRACT: In this paper, we derive an asymptotic analysis for a capacity approaching design of serially concatenated turbo schemes with low density parity check (LDPC) codes and continuous phase modulation (CPM) based on Laurent decomposition. The proposed design is based on extrinsic mutual information evolution and Gaussian approximation. By inserting partial interleavers between LDPC and CPM and allowing degree1 variable nodes under a certain constraint we show that designed rates are very close to the maximum achievable rates. Furthermore, we discuss the selection of low complexity receivers that works with the same optimized profiles.Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2014, Florence; 05/2014  SourceAvailable from: Tolga M Duman[Show abstract] [Hide abstract]
ABSTRACT: In this work, we exploit the capacity approaching capability of low density parity check (LDPC) codes over wireless relay channels. We consider the classical fullduplex relay channel model under both ergodic and nonergodic scenarios, and propose two practical relaying schemes. By comparing with the theoretical information rate bounds, we show that the LDPC coded relay systems can approach the ergodic/outage information rates (i.e., constrained channel capacities) very closely. Specifically, we show that they can even outperform the existing turbo coded relay systems under appropriate code design. In addition, based on the measure of average mutual information, we analyze the convergence behavior of the proposed schemes which also demonstrates their great potential to perform near capacity over both ergodic and nonergodic wireless relay channels.
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670IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Design of LowDensity ParityCheck
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 lowdensity paritycheck (LDPC)
codearepasseddirectlytoamodulator.Atthereceiver,thevariable
nodes of the LDPC decoder graph are connected to detector nodes,
and iterativedecoding is accomplishedby viewing thevariable and
detectornodesasonedecoder.Thecodeisoptimizedbyperforming
a curve fitting on extrinsic information transfer charts. Design ex
amples are given for additive white Gaussian noise channels, as
well as multipleinput, multipleoutput (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.
IndexTerms—Fading,iterative
paritycheck (LDPC) codes, multipleinput, multipleoutput
(MIMO) detection, mutual information.
decoding,lowdensity
I. INTRODUCTION
I
noisy channels [1]–[5]. 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 [2]–[4] and connect the LDPC variable nodes to detector
nodes.
We deal with the second problem by using a curvefitting
procedure on extrinsic information transfer (EXIT) charts
[6]. The design methodology is illustrated for two types of
channels: additive white Gaussian noise (AWGN) channels
with binary phaseshift keying (BPSK), and multipleinput,
multipleoutput (MIMO) fading channels with quadrature
phaseshift 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 lowdensity paritycheck (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 (email: stenbrink@realtek
us.com).
G. Kramer and A. Ashikhmin are with Bell Laboratories, Lucent
Technologies, Murray Hill, NJ 07974 USA (email: gkr@belllabs.com;
aea@belllabs.com).
Digital Object Identifier 10.1109/TCOMM.2004.826370
be possible using other chart techniques, see, e.g., [5], [7],
and [8]. We refer to [9] for a comparison of some of these
tools. Another alternative is to use numerical optimization with
density evolution
. 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, trelliscoded modulation (TCM)
[10], multilevel coding [11], bitinterleaved coded modulation
(BICM) [12], and space–time blockcoded (STBC) modulation
[13], [14] (see also [15] and references therein). A growing
body of work uses BICM with turbo and LDPC codes, see, e.g.,
[16]–[29]. The EXIT curvefitting approach described here was
motivated by results for erasure channels [30] and appeared in
[31]. Parallel work using similar ideas was reported in [32] and
[33]. The method was used to design repeataccumulate (RA)
codes in [34] and [35].
This paper is organized as follows. In Section II, we develop
the curvefitting 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
mancewithaschemeemployingauniversalmobiletelecommu
nicationssystem(UMTS)standardturbocode.TheLDPCcodes
are shown to perform substantially better for channels having
more transmit than receive antennas. Such a situation is likely
to occur on the basetomobile station link of a wireless com
munication system. Section IV summarizes our results.
II. CODE DESIGN FOR AWGN CHANNELS
ConsideranLDPCcodeoflength
An iterative decoder for this code can be viewed as a graph that
has
variablenodes,anedgeinterleaver,and
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 paritycheck equation. The th check node checks
bits so that it hasedges. The sets of variable and check
nodes are referred to as the variablenode decoder (VND) and
checknode 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
anddesignrate.
checknodes.
00906778/04$20.00 © 2004 IEEE
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TEN BRINK et al.: DESIGN OF LOWDENSITY PARITYCHECK 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 paritycheck codes. This observation
illustrates the close relation between LDPC decoding and other
iterative schemes, such as turbo decoding [36].
A. EXIT Charts
An a posteriori probability (APP) decoder converts channel
and a priori loglikelihood ratios (LLRs or
posteriori
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 [37] for further details on extrinsic
information processing.
An EXIT chart characterizes the decoder’s operation. We use
the notation of [6] 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 [6] for further details on how to interpret
these quantities.
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
fromtheedgeinterleaverandonefromthechannel.Thevari
able node decodes by computing, for
has incoming messages,
(1)
where
node,
able node, and
Consider the AWGN channel with BPSK (
andnoisevariance
.Wedefinethenormalizedsignaltonoise
ratio (SNR) as
is the th a priori
is the th extrinsic value coming out of the vari
is the channelvalue.
value going into the variable
) modulation
. The channel value is
(2)
where
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
conditioned onis
(3)
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
(4)
where the functions
pendix (see also [6]). Fig. 2 plots several variable node curves
when
and
andare given in the Ap
dB. The quantity
(5)
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 paritycheck(or rate
(6)
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672IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
Fig. 3.
swapped, as compared with Fig. 2.
CND EXIT curves. Observe that the a priori and extrinsic axes are
In the
“boxplus” operation
value notation of [37], this can also be written as a
(7)
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 [38], [46] or by simulation. Alternatively, for the binary
erasure channel, a duality property exists [5], [30] that gives the
EXIT curve
of the length
in terms of the EXIT curve
) repetition code, i.e.,
as the output value of an AWGN
single paritycheck code
of the length (or rate
(8)
This property is not exact for BPSK/AWGN a priori inputs, but
it is very accurate [38], [46]. For convenience, we use (8) and
write
(9)
where the second step follows from (4) with
further useful to express (9) in terms of its inverse function, i.e.,
. It is
(10)
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 checkregular LDPC codes, i.e., all
check nodes have degree
(in [39], these are called rightreg
ular codes). After choosing, the remaining LDPC design in
volves specifying the variable node degrees
,.
Let
note these degrees by
node degree is
be the number of different variable node degrees, and de
,. The average variable
(11)
where
the
VND and CND are the same, we have
is the fraction of nodes having degree
must satisfy
. Note that
. Since the number of edges at the
or
(12)
Let
be the fraction of edges incident to variable nodes of
. There are, in total,
such nodes, so we have
degree
edges involved with
(13)
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 [30] and [40], it is
(14)
i.e.,(14)isaweightedsumofthecurvesofFig.2.Notethatonly
edge fractions can be adjusted because we must enforce
(12) and
. Thus, in order to have any flexibility, we
must choose
. We shall see that
surprisingly good results [4, p. 634].
already gives
E. Design Example
Thepaper[30](seealso[40])showsthattoapproachcapacity
onerasurechannels,onemustmatchtheVNDandCNDtransfer
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.
Recallthat
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
curvethatliesabovetheCNDcurve.Furthermore,thisgivesthe
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:
isthechannelcapacity(see
, and choose
dB
so that
). 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
with
, a random edge interleaver, and 100 iterations
shows a turbo cliff at about 0.55 dB (we measure the turbo cliff
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TEN BRINK et al.: DESIGN OF LOWDENSITY PARITYCHECK CODES FOR MODULATION AND DETECTION 673
Fig. 4.Curve fit for an LDPC code with ? ? ???.
at a biterror rate (BER) of
iterations,theturbocliffisat0.5dB,whichverifiestheaccuracy
of the EXIT chart and our approximations.
Weremarkthatbychoosinglarger
curve more closely to the CND curve. One can also model the
a priori
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
that
can yield a good curve fit. We would now again
choose the
to ensure that the
curve.
). For longer codes and more
,onecanmatchtheVND
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
case.
Consider the setup of Fig. 5, where there are
receiveantennas.Eachtransmittersymbolisan
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
transmit and
1vector
so that
coded bits. For example,
Fig. 5.MIMO model.
symbol is limited to
for
The receiver sees
is the
Suppose the entries of
Gaussianrandomvariableswithindependentrealandimaginary
parts each having variance
ized SNR
as
, and we will use
.
1 vectors, where
channel matrix and
are independent, complex, zeromean,
is a 1 noise vector.
. We define the normal
(15)
The value
to keep the
ferent
consider a Rayleigh fading channel, so that the entries of
independent, complex, zeromean, Gaussian random variables
with independent real and imaginary parts each having variance
1/2 [42, Sec. 4]. For a quasistatic channel, the matrix
mains unchanged over long time intervals, while for an ergodic
channel,
changes for every symbol . We consider only the
ergodic model whose capacity is (see [42] and [43])
appears in (15) by convention; this is done simply
at capacity close to each other for dif
. We assume thatis known to the receiver only, and
are
re
(16)
where
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 [34]).
is the identity matrix and
. One achieves capacity with Gaussian dis
is the complexconju
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
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674IEEE 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
all
possible hypotheses on
detector computes the
values
(see [16]–[20]). That is, the
(17)
for
onto the vector symbol
tector EXIT curve cannot be described in closed form, so we
measure it by Monte Carlo simulation [20]. We denote the de
tector by DET and its EXIT curve by
, where
whose channel output is
is the th coded bit mapped
. The de
(18)
Fig. 6 shows simulated EXIT curves of MIMO detectors for
Graymapped 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
(19)
where
random variable, and where we have used (5). We remark that
any 1
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
for fixed
is a zeromean, unitvariance, complex Gaussian
1
curve at, and that
and.
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
to
variable nodes. We further choose allvariable
nodes connected to a common detector node to have the same
degree
. 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., [26] and [27], 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
approach capacity.
Consider the three boxes in Fig. 5 corresponding to one of
the
individual detector/VNDs. We denote the EXIT
curve of this structure by
priori
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
into
(20)
We next approximate the detector curve (18) by a thirdorder
polynomial. For example, the EXIT curve of the 4
detector with QPSK at
approximated as
1 MIMO
is well dB and
dB
(21)
The polynomial (21) lets us express the combined de
tector/VND curve in closed form, which is convenient for curve
fitting.
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
(22)
Finally, inserting (18), (20), and the detector polynomial (21)
into (22), we obtain the desired transfer curve in the form
(23)
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 checkregular LDPC codes by matching the curve
(23)toaCNDcurve(9).Weagainrestrictourselvestojustthree
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.
Fig.9andTableIgivesimulationresultsforLDPCcodeswith
Graymapped QPSK,
,
leaver, and 100 decoder iterations. All schemes operate within
1.25dBoftheirrespectivecapacitylimits(duetothecomplexity
of the APP processing, the number of simulated blocks for the
) and perform the curve
, a random edge inter
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TEN BRINK et al.: DESIGN OF LOWDENSITY PARITYCHECK CODES FOR MODULATION AND DETECTION675
Fig. 7.
and ? ? ???.
Combined 4?1detector and VND EXIT curves at ? ??
? ? dB
TABLE I
LDPC CODE PARAMETERS FOR ? ? ? MIMO CHANNELS, QPSK, AND
? ? ???. OVERALL RATE: ? BITS/CHANNEL USE
4
pacity can be narrowed further by using
fits, and by using longer codes and more iterations.
Fig.9alsoplotstheBERcurvesofBICMemployingaUMTS
standard turbo code, denoted PCC for “parallel concatenated
code.”ThePCChasmemorythreeconstituentcodes,withfeed
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 ? ? ???.
Fig. 9.
concatenated codes or PCC). All schemes have an overall rate of ? bits per
channel use.
BERs for LDPC codes (LDPCC) and turbo codes (parallel
back and feedforward generator polynomials
,respectively.ThemodulationisagainGraymapped
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 turbocoded
scheme by wide margins for
for 4
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 [20].
and
,, and a random in
(the gain is over 5 dB
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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 timedivi
sionmultipleaccess,orwithmulticarriermodulationemploying
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 quasistatic 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 quasistatic
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 [13]–[15] and references
therein). However, the gains come at the expense of spectral
efficiency. For example,consider the 2
QPSK and a rate1/2 LDPC code. The spectral efficiency is
two bits per channel use. Suppose next that we use Alamouti’s
space–time code [13] as an inner code. This scheme obtains a
diversity gain with what is, in effect, a rate1/2 repetition code.
Thus, to achieve two bits per channel use with a rate1/2 outer
code, one must compensate for the innercode rate loss by
changing the constellation from QPSK to 16quadrature ampli
tude modulation (QAM). But recall that 16QAM 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
arate1/2LDPCcode.AcorrespondingSTBC[14],whichis,in
effect, a rate1/4 repetition code, would have to use 256QAM
to achieve four bits per channel use with a rate1/2 outer code.
Furthermore, no rate1/4 orthogonal STBC exists for the 4
MIMO channel [14]. These considerations suggest that diver
sity is best achieved by designing the coding and modulation
together rather than separately.
1 MIMO channel with
1 MIMO
1
IV. SUMMARY
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 [34], [35].
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 quasistatic channels, EXIT
curve fitting for suboptimal detectors, e.g., the detectors of sev
eral papers in [44], and improving the EXIT approximations.
. We remark
APPENDIX
Consider
is zeromean, Gaussian noise with variance
value of (2) is a function of , and we write this as
Note that
conditioned on
mean
and variance
, where and
. The channel
.
is Gaussian with
. We thus have
Letbe the mutual information. We have
(24)
where
tropy of
the same as
is, therefore,
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 [45]) to obtain
is the entropy of
conditioned on
. The capacity of our
and
. Note that
is the en
is
channel
.
into two parts,
andwhere
where
For the inverse
tervals at
function we split the curve into two in
. Our approximation is
where
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TEN BRINK et al.: DESIGN OF LOWDENSITY PARITYCHECK CODES FOR MODULATION AND DETECTION677
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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 multipleantenna systems.
Since July 2003, he has been with Realtek, Irvine,
CA, where he is involved in the development and
standardization of highthroughput WLAN systems.
His research interests include errorcorrecting coding, iterative decoding,
multipleantenna communications, and watermarking.
Page 9
678IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 4, APRIL 2004
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.
FromAugust1998toMarch2000,hewaswithEn
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 errorcorrecting codes.
Dr. Ashikhmin currently serves as an Associate Editor for Coding Theory for
the IEEE TRANSACTIONS ON INFORMATION THEORY.
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