Design of low-density parity-check codes for modulation and detection

Abstract

A coding and modulation technique is studied where the coded bits of an irregular low-density parity-check (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 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 outperforms a scheme employing a parallel concatenated (turbo) code by wide margins when there are more transmit than receive antennas.

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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)
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 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.
IndexTerms—Fading,iterative
parity-check (LDPC) codes, multiple-input, multiple-output
(MIMO) detection, mutual information.
decoding,low-density
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 curve-fitting
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 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-
us.com).
G. Kramer and A. Ashikhmin are with Bell Laboratories, Lucent
Technologies, Murray Hill, NJ 07974 USA (e-mail: gkr@bell-labs.com;
aea@bell-labs.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, trellis-coded modulation (TCM)
[10], multilevel coding [11], bit-interleaved coded modulation
(BICM) [12], and space–time block-coded (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 curve-fitting 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 repeat-accumulate (RA)
codes in [34] and [35].
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-
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 base-to-mobile 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 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
anddesignrate.
checknodes.
0090-6778/04$20.00 © 2004 IEEE

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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 [36].
A. EXIT Charts
An a posteriori probability (APP) decoder converts channel
and a priori log-likelihood 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
.Wedefinethenormalizedsignal-to-noise
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
) modulation
. The channel -value is
(2)
where
function (pdf) evaluated at the output
andbe random variables representing the respective channel
input and output. The variance of
is the channel conditional probability density
given the input . Let
conditioned on is
(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
and are 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 parity-check(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
“box-plus” 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 parity-check 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 check-regular LDPC codes, i.e., all
check nodes have degree
(in [39], these are called right-reg-
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 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
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, zero-mean,
is a1 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, 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
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 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

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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
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
(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 zero-mean, unit-variance, 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 third-order
polynomial. For example, the EXIT curve of the 4
detector with QPSK at
approximated as
1 MIMO
is welldB 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 check-regular 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
Gray-mapped 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-