Dirty Paper Coding using Sign-bit Shaping and
LDPC Codes
Shilpa G, Andrew Thangaraj and Srikrishna Bhashyam
Dept of Electrical Engg
Indian Institute of Technology Madras
Chennai 600036, India
Email: andrew,skrishna@ee.iitm.ac.in
Abstract—Dirty paper coding (DPC) refers to methods for
pre-subtraction of known interference at the transmitter of a
multiuser communication system. There are numerous applica-
tions for DPC, including coding for broadcast channels. Recently,
lattice-based coding techniques have provided several designs for
DPC. In lattice-based DPC, there are two codes - a convolutional
code that defines a lattice used for shaping and an error
correction code used for channel coding. Several specific designs
have been reported in the recent literature using convolutional
and graph-based codes for capacity-approaching shaping and
coding gains. In most of the reported designs, either the encoder
works on a joint trellis of shaping and channel codes or the
decoder requires iterations between the shaping and channel
decoders. This results in high complexity of implementation. In
this work, we present a lattice-based DPC scheme that provides
good shaping and coding gains with moderate complexity at both
the encoder and the decoder. We use a convolutional code for
sign-bit shaping, and a low-density parity check (LDPC) code
for channel coding. The crucial idea is the introduction of a one-
codeword delay and careful parsing of the bits at the transmitter,
which enables an LDPC decoder to be run first at the receiver.
This provides gains without the need for iterations between the
shaping and channel decoders. Simulation results confirm that at
high rates the proposed DPC method performs close to capacity
with moderate complexity. As an application of the proposed DPC
method, we show a design for superposition coding that provides
rates better than time-sharing over a Gaussian broadcast channel.
I. INTRODUCTION
Situations where interference is known non-causally at
the transmitter but not at the receiver model several useful
multiuser communication scenarios. In [1], Costa introduced
and studied coding for such situations and called it “writing on
dirty paper”. Dirty paper coding (DPC) is now recognized as a
powerful notion central to approaching capacity on multiuser
channels.
Lattice-based ideas for DPC were suggested and shown to
be capacity-approaching in [2], [3]. Recently, many designs
of lattice-based DPC schemes have been proposed in [4]–[8].
Lattice-based schemes typically use cosets of a convolutional
code for lattice-quantizing or shaping to minimize the energy
of the difference of the coded symbols and the interfering
symbols. A part of the message bits is used to choose the
specific coset used in the minimization. In addition to the
shaping convolutional code, an error correction code needs
to be used to obtain coding gain and approach capacity.
The main source of complexity in lattice-based DPC designs
is combining shaping and coding encoders/decoders at the
transmitter/receiver. Simple concatenation schemes are not
applicable because of the following reasons - outer shaping
followed by inner coding results in unshaped parity symbols
that increase transmitted energy, while outer coding followed
by inner shaping results in a poor inner code that needs to be
iteratively decoded at the receiver with the outer code.
In [6], encoding is done on a combined trellis of the source
code (Turbo TCQ) and a channel code (Turbo TCM). At the
receiver, decoding is done for Turbo TCM followed by syn-
drome computation to recover message bits. The transmitter
is complex in [6] because of the use of the joint trellis. The
DPC method proposed in [7] is similar to that of [6]. In [5],
multilevel coding is used, and there are different codes for
different bits of the symbols. At the receiver, iterations have
to be performed between decoders for some of the channel
codes and the shaping decoder. In [8] and [9], shaping follows
channel coding and the receiver performs iterations between
the shaping and channel decoders.
In this work, we propose a lattice-based method that uses a
novel combination of a convolutional code for sign-bit shaping
and a low density parity check (LDPC) code for channel
coding. As shown in specific designs and simulations, the
method provides good shaping and coding gains at moderate
complexity. The main idea for reducing complexity at the
receiver is the introduction of a one-codeword delay at the
transmitter, and the shaping of symbols from current message
bits combined with parity bits from the previous codeword.
This enables the LDPC decoder to be run first at the receiver
(with a one-codeword delay) without any need for iterations
with a shaping decoder. As an application, we use the proposed
DPC method to design codes for superposition coding in two-
user Gaussian broadcast channels. By simulations, we show
that rate points outside the time-sharing region are achieved.
The rest of the paper is organized as follows. After a brief
review of the lattice-based DPC coding method in Section
II, we present the proposed DPC method in Section III.
This is followed by description and simulation of specific
designs of DPC codes in Section IV. In Section V, design
of a superposition scheme using the proposed DPC method
is described and simulation results are presented. Concluding
remarks are made in Section VI.
ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010
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II. LATTICE DIRTY PAPER CODES
In a Gaussian dirty paper channel, the received symbol
vector Y =[Y
1
Y
2
···Y
n
] is modeled as
Y = X + S + N,
where X =[X
1
X
2
···X
n
] denotes the transmitted vector,
S =[S
1
S
2
···S
n
] denotes the interfering vector assumed
to be known non-causally at the transmitter and N denotes
the additive Gaussian noise vector. The transmit power is
assumed to be upper-bounded by
1
n
E[|X|
2
] ≤ P
X
per symbol,
and the interference power is denoted
1
n
E[|S|
2
]=P
S
per
symbol. The noise variance per symbol is denoted P
N
.In
[1], Costa shows that the capacity of the dirty paper channel
is
1
2
log
1+
P
X
P
N
i.e. known interference can be canceled
perfectly at the transmitter.
The interfering vector S is used as an input in the encoding
process and plays an important role to determine a suitable
transmit vector X. A coding strategy for choosing X needs
to overcome the imminent addition of S and protect the
transmitted information from the addition of the noise N. Such
coding strategies are called dirty paper coding (DPC) methods.
In [3], a dirty paper coding (DPC) scheme based on
lattice strategies was proposed and shown to achieve the
capacity of the dirty paper channel. We follow [4] for a
brief review of the transmitter and receiver structure in the
lattice DPC method [3]. Let Λ denote an n-dimensional
lattice with fundamental Voronoi region ν having averaged
second moment P (Λ) = P
X
and normalized second moment
G (Λ). Also let U ∼ Unif (ν) i.e. U is a random variable
(dither) uniformly distributed over ν. The lattice transmission
approach of [3] [4] is as follows.
• Transmitter: The input alphabet X is restricted to ν.For
any v ∈ ν, the encoder sends
X =[v − αS − U] mod Λ, (1)
where α =
P
X
P
X
+P
N
is a MMSE scaling factor [4].
• Receiver: The receiver computes
Y
=[αY + U] mod Λ (2)
The channel from v to Y
defined by (1) and (2) is equivalent
in distribution to
Y
=[v + N
] mod Λ, (3)
where
N
= [(1 − α) U + αN] mod Λ. (4)
Lower bounds on achievable rates for the above equivalent
channel is shown in [4] to be equal to
I (V;Y
) ≥
1
2
log
2
(1 + SNR) −
1
2
log
2
(2πeG (Λ)) . (5)
For optimal shaping, G (Λ) →
1
2πe
and we approach capacity
of the dirty paper channel. Note that the dither is assumed to
be known at the transmitter and receiver (say, through the use
of a common seed in a random number generator).
III. P
ROPOSED SCHEME
The proposed scheme uses a convolutional code for sign-
bit shaping [10] and low density parity check (LDPC) codes
for channel coding. We assume a M-PAM signal constellation
with a carefully chosen bit-to-symbol mapping that is compati-
ble with sign-bit shaping and bit-interleaved coded modulation
(BICM) [11]. For M =16, the constellation and mapping are
shown in Fig. 1. The mapping in Fig. 1 is suited for sign-
bit shaping, since a flip of the most significant bit results in a
significant change in symbol value for all possible 4-bit inputs.
Also, the mapping is mostly Gray except for a few symbol
transitions. Gray mapping is known to be the most effective
mapping for BICM with LDPC codes. This heuristic choice of
mapping enables the possibility of good shaping and coding
gains to be obtained simultaneously. As expected, larger values
of M will result in larger shaping gains in our design, and
we stick to the 16-PAM shown in Fig. 1 for illustration and
simulation.
A. Encoder Structure
The encoder structure for the proposed scheme is as
shown in Fig. 2. We describe the operations in the en-
coder at time step T or in the T -th block. A k-bit mes-
sage m =[m
1
m
2
···m
k
] is encoded into a s-symbol
vector u =[u
1
u
2
···u
s
] from the M -PAM constellation
A = {−(M − 1)/2, ··· , −1/2, 1/2, ··· , (M − 1)/2}, where
s =
n
log
2
M
is assumed to be an integer. Let l = log
2
M and
let f
M
: {0, 1}
l
→ A denote the bit-to-symbol mapping. The
bits that map to the i-th symbol are denoted z
i
a
2i
a
3i
···a
li
;
the sign-bit vector is denoted z =[z
1
z
2
···z
s
], and we define
vectors a
j
=[a
j
1
a
j
2
···a
j
s
] for 2 ≤ j ≤ l. Finally, we have
v = f
M
(za
2
···a
l
), where f
M
operates component-wise on
a vector input.
Let us assume that the vectors a
j
, 2 ≤ j ≤ l are available
at the encoder. The sign-bit shaping convolutional code is
used to determine the sign-bit vector z as follows. A part
of the message m
=[m
1
m
2
···m
k
] with k
<kbits is
mapped to a coset leader of the convolutional code using an
inverse syndrome former [10]. Note that we need the rate of
the convolutional code to be 1 − k
/s. Let the coset chosen by
m
be denoted C(m
). The sign-bit vector z is chosen from
C(m
) so as to minimize the squared sum (energy) of the
vector (v − αS)modM, where α =
P
X
P
X
+P
N
is the MMSE
factor and S is the interference vector. That is,
z = arg min
u∈C(m
)
|(f
M
(ua
2
···a
l
) − αS)modM|
2
. (6)
The minimization in (6) is implemented using the Viterbi
algorithm [10].
The a
j
, 2 ≤ j ≤ l are determined as follows. An (n, k−k
+
s) LDPC code is used at the encoder with a systematic encoder
E : {0, 1}
k−k
+s
→{0, 1}
n
. Let m
=[z m
k
+1
···m
k
] be
input to the systematic LDPC encoder to obtain the codeword
E(m
)=[m
p
T
], where p
T
is the parity-bit vector for the
T -th block. The parity-bit vector is delayed by one time step.
For the T -th block, the n −s = s(l − 1) bits in [m
k
+1
···m
k
]
ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010
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1000
-15/2
1001
-13/2
1011
-11/2
1010
-9/2
1110
-7/2
1111
-5/2
1101
-3/2
1100
-1/2
0000
1/2
0001
3/2
0011
5/2
0010
7/2
0110
9/2
0111
11/2
0101
13/2
0100
15/2
Fig. 1. 16-PAM constellation.
- -
k
Message
Bits
H
−1
T
(k − k
)
Message
Bits
Algorithm
Viterbi
Shaping Coder
minimizes the
energy of
(v − αS) mod M
-
s
s =
n
log
2
M
Bits
Shaped
Bits
Encoder
LDPC
of rate
(k − k
+ s) /n
6
-
-
-
-
a
b c
n − (k − k
+ s)
Parity
Bits
Delay
k − k
Message bits
-
s
Shaped Bits
z
Mapping
to
M − PAM
map (zabc)
-
?
?
v
−αS
(v − αS) mod M
mod M
?
?
n − s
Parity plus Message
Bits
Fig. 2. Encoder structure.
and p
T −1
are rearranged by a permutation Π to form the
vectors a
j
, 2 ≤ j ≤ l. This permutation is necessary in an
implementation of BICM [11].
Note that both the shaping and coding objectives have
been met at the encoder. The transmitted symbols v − αS
mod M have minimal energy in the lattice defined by sign-
bit shaping using the convolutional code. Selected bits in
successive blocks of symbols form codewords of the LDPC
code. In summary, the encoder structure achieves DPC shaping
and LDPC coding with bit-interleaved modulation.
B. Decoder Structure
The decoder for the proposed scheme is as shown in Fig.3.
Because of the one-codeword delay, parity bits of the T +
1-th block and message plus shaped bits of the T -th block
form a valid LDPC codeword. Therefore, an LDPC decoder
is first run on these symbols corresponding to a codeword. The
decoded shaped bits are passed through a syndrome former to
get message bits used for shaping. Iterations between shaping
decoder and channel decoder are not needed. The demapper
computes log likelihood ratios (LLRs) for the bits from the
received symbols in
Y = αY + U. The LLRs of the (k − k
)
message bits after a delay of one time step, and the LLRs of the
n− (k − k
+ s) parity bits are de-interleaved. The s =
n
log
2
M
LLRs of the sign bits after a delay on one time step, and the
n−s output LLRs of the de-interleaver are given as the input to
the LDPC decoder. The LDPC decoder outputs k−k
message
bits and s bits of the sign bit vector of the previous block. Now,
the s-bit sign vector is passed through the syndrome former
to recover the remaining k
message bits.
The demapper function at the receiver has to calculate LLRs
taking into account the modulo M operation at the encoder [4].
Therefore, the received constellation A
R
is a replicated version
of the M-PAM constellation A used at the transmitter (assum-
ing that scaling factors have been corrected at the receiver).
That is, A
R
= {A−rM,··· ,A−M, A, A+M, ··· ,A+rM}.
The number of replications r is chosen so that the average
power of A
R
is approximately equal to the total average
power P
X
+ P
S
, and the bit mapping of the symbol a + jM
(a ∈ A, 1 ≤ j ≤ r) is the same as that for a. The LLR for
the i-th bit in the j-th symbol
Y
j
is computed according to
the constellation A
R
using the following formula:
L
i
=
a∈A
R
:bit i=0
exp
⎛
⎜
⎝
−
1
2
Y
j
− a
2
αP
N
⎞
⎟
⎠
a∈A
R
:bit i=1
exp
⎛
⎜
⎝
−
1
2
Y
j
− a
2
αP
N
⎞
⎟
⎠
.
ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010
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-
Y = αY + U
Demapper
- -
−1
-
Delay
-
n − (k − k
+ s)
LLRs of
Parity Bits
-
Delay
- -
(k − k
)
LLRs of Message Bits
n
log
2
M
LLRs of sign bits
LDPC
Decoder
-
- -
H
T
s
Bits
k
Message
Bits
k − k
Message
Bits
Fig. 3. Decoder structure.
Since the constellation mapping is nearly Gray, iterations
with the demapper do not provide significant improvements
in coding gain, particularly for large M.
IV. S
IMULATION RESULTS
For simulations, we have taken n = 40000, k = 30000,
k
= 5000 with M =16; this results in s = 10000.
The constellation mapping is as given in Fig. 1. We
have chosen a rate-1/2 memory 8 (256 state) non-
systematic convolutional code with generator polynomials
D
8
+ D
5
+ D
4
+ D
2
+ D +1,D
8
+ D
7
+ D
4
+ D
2
+1
as the sign-bit shaping code. A non-systematic convolutional
code is used to avoid error propagation problems.
A randomly constructed irregular LDPC code (40000,
35000) of rate 7/8 with variable node degree distribution
0.1256x+0.7140x
2
+0.1604x
9
and check node degree distri-
bution x
31
is used as the channel code. We considered several
candidate distributions originally designed for BPSK over
AWGN, and chose the one that provided the best performance.
The overall rate of transmission is seen to be
30000
40000
× 4=3
bits per channel use. Fig. 4 shows BER plots over an AWGN
channel and a DPC channel with interference known at the
transmitter. The interfering vector was generated at random
10 10.5 11 11.5 12 12.5 13
10
−5
10
−4
10
−3
10
−2
Eb/N0(dB)
Bit error rate
AWGN
DPC
Capacity for 3 bits/sym
Fig. 4. BER plot for DPC and AWGN.
for different power levels. The plot with interference did not
change appreciably for all power levels of interference, and
we have provided one plot for illustration. We see that a BER
of 10
−5
is achieved at an SNR of 19.45dB with interference,
and at an SNR of 19.33 dB without interference. We have
simulated 1000 blocks of length 40000 to obtain sufficient
statistics for a BER of 10
−5
.
The AWGN capacity at an SNR of 10 log
10
(2
6
−1)=17.99
dB for a rate of 3 bits per channel use. This shows that we
are 1.46 dB away from ideal dirty paper channel capacity.
The granular gain G(Λ)=2
2R
/6S
x
is computed from the
simulations to be 1.282dB [4], where R =3.5 is the rate
before channel coding, and S
x
is the transmit power (obtained
through simulations). From this, the shaping loss is calculated
as follows:
10log
10
2πeG (Λ) 2
2C
∗
− 1
2
2C
∗
− 1
=0.2548 dB, (7)
where C
∗
=3. So, of the total gap of 1.46 dB, we have
a shaping gap of 0.2548dB, and a coding gap of 1.2052dB
to capacity. We observed that trellis shaping with larger
number of states results in a decrease in shaping loss in other
simulations.
V. A
PPLICATION TO GAUSSIAN BROADCAST CHANNEL
We use the proposed scheme for superposition coding in
a two-user Gaussian broadcast channel Y
1
= X + N
1
and
Y
2
= X + N
2
with P
N
1
>P
N
2
. We let P
X
1
=(1− β) P
and P
X
2
= βP, where P is the total transmit power. Here,
User 2 is coded using DPC considering User 1 as interference.
User 1 is shaped using sign-bit shaping and coded using an
LDPC code over M-PAM. Fig. 5 shows a block diagram of
the transmitter and receivers. The encoder structure for User
1 is as in Fig. 2 with the interference vector S = 0. Hence,
for User 1, the shaping coder minimizes the energy of v. The
demapper at Receiver 1 calculates LLR for the i-th bit in the
j-th receiver symbol Y
1j
using the following formula.
L
i
=
a∈A:bit i=0
p(a) exp{−
1
2
(Y
1j
−a)
2
βP+P
N
1
}
a∈A:bit i=1
p(a) exp{−
1
2
(Y
1j
−a)
2
βP+P
N
1
}
,
where p(a) for a ∈ A represents the a priori probability of the
M-PAM symbol a. At the receiver, we approximate p
i
using
ISIT 2010, Austin, Texas, U.S.A., June 13 - 18, 2010
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-
User 1
Message bits
Shaped and
LDPC coded
?
X
1
as S for
User 2
?
X
1
-
User 2
Message bits
DPC coded
6
X
2
-
X
@
@
@
@
@
@
@
@R
Y
1
= X + N
1
Y
2
= X + N
2
Receiver 1
Receiver 2
Fig. 5. Block diagram for a two-user Gaussian broadcast channel.
a Gaussian distribution with variance P
S
assuming that the
distribution of M-PAM symbols is approximately Gaussian.
We simulated a two user degraded broadcast channel with
P
N
1
=0.9 and P
N
2
=0.09 using the proposed scheme with
parameters from Section IV. The total transmit power, power
for User 1 and power for User 2 required for a bit error rate of
10
−5
(at both receivers) are estimated from the simulation and
denoted P , P
x
1
and P
X
2
, respectively. The SNR for Receiver
1 is computed as 10 log
10
P
X
1
P
X
2
+P
N
1
=19.1791 dB. Since
DPC is done for User 2, the effective SNR at Receiver 2
is computed as 10 log
10
P
X
2
P
N
2
=19.4574 dB. Comparison
with the SNR needed for a single user capacity of 3 bits per
channel use (which is 17.99 dB) shows that the total loss for
both the users is about 2.4642dB. Fig. 6 shows the (3, 3) rate
pair in the capacity region of the two-user Gaussian broadcast
channel with total transmit power P and noise power P
N
1
,
P
N
2
, which is defined by R
1
≤
1
2
log
1+
(1−β)P
βP+P
N
1
, R
2
≤
1
2
log
1+
βP
P
N
2
for 0 ≤ β ≤ 1. We see that the (3,3) rate
point is clearly outside the time-sharing region.
VI. C
ONCLUSIONS
In this work, we have proposed a method for designing
lattice-based schemes for dirty paper coding using sign-bit
shaping and LDPC codes. Simulation results show that the
proposed design performs 1.46dB away from the dirty paper
capacity for a block length of n = 40000 at the rate of
3 bits/channel use. This performance is comparable to other
results in the literature. However, as discussed in this article,
a novel method for combining shaping and coding results
in good gains at lesser complexity in our design, when
compared to other lattice-based strategies. As an application,
we have designed a superposition coding scheme for Gaussian
broadcast channels that is shown to perform better than time-
sharing through simulations.
Out of the 1.46 dB gap to capacity, about 1.2 dB is gap
attributed to a sub-optimal choice of the LDPC code and a
relatively low blocklength. Optimizing the LDPC code using
genetic algorithms and asymmetric density evolution [12]
0 1 2 3 4 5 6 7
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
R
2
(Rate of User 2 in bits/channel use)
R
1
(Rate of User 1 in bits/channel use)
Broadcast capacity
Time sharing
(3, 3.409)
(3.451, 3)
(3, 3)
(2.474, 3)
(3, 2.608)
Fig. 6. Two-user Gaussian broadcast channel capacity region.
along with joint optimization of shaping code and LDPC code
using EXIT charts [8] are topics for future work.
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