A Dirty Paper Coding Scheme for the Multiple
Input Multiple Output Broadcast Channel
Balakrishna Saradka, Srikrishna Bhashyam, Andrew Thangaraj
Department of Electrical Engineering
Indian Institute of Technology Madras
Chennai 600036
Email:{ee09s007, skrishna, andrew}@ee.iitm.ac.in
Abstract—Dirty paper coding (DPC) is known to achieve the
capacity region of a Gaussian Multiple Input Multiple Output-
Broadcast channel (MIMO-BC). Practical DPC schemes using
finite length codes are still being actively studied. In this paper, we
design a zero-forcing DPC (ZF-DPC) scheme using trellis shaping
and Low Density Parity Check (LDPC) codes for a MIMO-BC
with two transmit antennas and two users (receivers), each with
one antenna. This is an extension of an earlier design for the
single antenna Gaussian broadcast channel. One of the important
aspects of the DPC code design is the introduction of a one block
delay that enables the channel encoder (and decoder) and the
shaping en coder (and decoder) to operate independently. In the
ZF-DPC method, the MIMO precoder ensures that one user has
no interference. The other user uses DPC to combat i nterference.
The performance of this method is compared using simulations
with the capacity limit and simpler precoder based methods like
Minimum Mean Square Error-Vector Perturbation (MMSE-VP)
and zero-forcing beamforming (ZF-BF).
I. INTRODUCTION
In the cellular downlink, base stations and mobiles use
multiple antennas in order to increase spectral efficiency. This
downlink channel is a non-degraded Multiple Input Multiple
Output Broadcast (MIMO-BC) channel. The sum capacity
for the non-degraded Gaussian-MIMO broadcast channel was
derived in [1] and shown to be achieved using Dirty Paper
Coding (DPC). Later, in [2], it was shown that the DPC
rate region coincides with the capacity region. Practical code
designs for DPC have been studied in [3]–[7]. In the designs
in [3]–[6], joint shaping and coding encoders (or iterative
decoders) are required at the transmitter (or receiver). In [7],
for the single antenna Gaussian broadcast channel, joint en-
coding and iterative decoding between the shaping and channel
codes are avoided by introducing a one-codeword delay at the
transmitter and the shaping of symbols from current message
bits combined with parity bits from the previous codeword. In
this work, we design codes for the MIMO-BC using this idea
from [7].
Other simpler sub-optimal linear and non-linear precoding
schemes have also been proposed for the MIMO-BC. The
simplest form of linear precoding is Zero-Forcing Beam-
Forming (ZF-BF). In ZF-BF, the precoder is simply the
pseudo-inverse of the channel matrix. However, this method
suffers when there is a large spread of the singular values of the
channel matrix. Regularized inversion of the channel matrix
and vector perturbation (a non-linear precoding method) can
help to address this problem [8], [9].
In this paper, we design a zero-forcing DPC (ZF-DPC)
scheme using trellis shaping and Low Density Parity Check
(LDPC) codes for a MIMO-BC with two transmit antennas
and two users each with one antenna. This is an extension of
the design for the single antenna Gaussian broadcast channel
in [7]. BER performance of the proposed DPC scheme is
compared with the capacity limit and Minimum Mean Square
Error-Vector Perturbation (MMSE-VP) and ZF-BF techniques.
Since the shaping and channel coding encoders and decoders
do not have to be joint, the complexity of the proposed scheme
is moderate compared to other DPC designs.
II. MODEL
We consider the downlink scenario with the transmitter
having 2 antennas and 2 receiving users, each with single
antenna. There is no cooperation channel among the users.
A block diagram of such a system is shown in Fig. 1. This
type of channel can be represented by well known equation
User 2
User 1
Tx
h
2,2
h
2,1
h
1,2
x
1
x
2
h
1,1
y
1
y
2
Fig. 1. MIMO-BC model
y = Hx + w, where x = [x
1
x
2
]
T
is the transmitted vector
and w = [w
1
w
2
]
T
is the complex Additive White Gaussian
Noise (AWGN) vector with mean zero and covariance I (2 ×
2) identity matrix. The channel matrix H
2×2
with elements
h
i,j
is fixed and this perfect channel state information (CSI)
is assumed to be known completely to the transmitter as well
as to both users. Each user’s information bits are shaped and
channel-coded, and the symbols are chosen from a M-QAM
constellation to get symbol vector d. The transmit vector x is
obtained from precoding the symbol vector d by a MIMO
precoder matrix P such that x = Pd. The transmitter has
average power constraint of P
t
units so that E{||x||
2
} < P
t
.
The received vector y = [y
1
y
2
]
T
for the 2 users can be
written as:
y
1
=h
11
x
1
+ h
12
x
2
+ w
1
(1)
y
2
=h
21
x
1
+ h
22
x
2
+ w
2
(2)
III. CAPACITY
In [1], Caire and Shamai derived the sum capacity (total
throughput) for the above channel to be:
R =
log(1 + |h
1
|
2
P
t
), P
t
≥ P
1
log
P
t
det(HH
H
+ tr(HH
H
)
2
− 4|h
2
h
H
1
|
2
4 det(HH
H
)
, P
t
≤ P
1
,
where h
1
, h
2
are the rows of channel matrix H, P
1
=
|h
1
|
2
−|h
2
|
2
det(HH
H
)
and it is assumed that ||h
1
|| ≥ ||h
2
||. The sum-rate
can be obtained using above equation for any given channel H.
When rate for each user is fixed the minimum power required
to achieve this can be obtained from the capacity region in [2].
Fig. 2 is a plot of the sum rate as a function of transmitted
power for two cases: (a) sum capacity as in [1], (b) maximum
sum rate when the two users are required to transmit the same
rate. In Fig. 2, we choose
H =
0.7071e
2.37j
0.8660e
2.14j
e
2.37j
0.5e
0.87j
.
0 2 4 6 8 10 12 14 16 18 20
0
2
4
6
8
10
Pt −−−(dB)−−>
SumRate (R1+R2)−−>
Caire Shamai sum capacity bound
Optimal bound for R1=R2
Fig. 2. Capacity bounds 2 × 2 MIMO-BC for the given channel H
IV. ZF-DPC WITH TRELLIS SHAPING AND LDPC CODE
MIMO Precoding
The zero-forcing DPC (ZF-DPC) method proposed in [1] for
MIMO-BC is used. The channel matrix H can be decomposed
using Gram-Schmidt orthogonalization as H = GQ, where G
is a lower triangular matrix and Q is an orthonormal matrix
such that QQ
H
= I (I is a 2 × 2 identity matrix). The
transmitter precodes the vector u to get x = Q
H
Ru (See Fig.
3), where the power allocation matrix, R =
r
11
0
0 r
22
.
Suppose G =
g
11
0
g
21
g
22
, we get the received vector to be
y = GRu + w, which simplifies to:
y
1
=g
11
r
11
u
1
+ w
1
(3)
y
2
=g
22
r
22
u
2
+ g
21
r
11
u
1
+ w
2
(4)
Thus, we get one interference-free channel and another chan-
nel with interference. Here, u
1
carries the message for user
1, and u
2
carries the message for user 2. User 1 sees no
interference. User 2 sees an interference term g
21
r
11
u
1
. In
the case of a MIMO broadcast channel, both the messages are
known at the transmitter non-causally. Therefore, the effect
of interference at user 2 can be removed using DPC for the
message of user 2, i.e., u
2
is the DPC-coded transmission for
user 2. Power allocation matrix entries are computed as in [5].
DPC Scheme
Dirty paper coding [10] is a technique of canceling known
interference at the transmitter. For a channel Y = X + S +
W, where X = [X
1
, X
2
...X
L
] is the sequence of power-
limited transmitted symbols with power P
x
=
E[||X| |
2
]
L
,
S = [S
1
, S
2
...S
L
] is a known interference sequence at the
transmitter and W is colored or white Gaussian noise vector
with P w =
E[||W||
2
]
L
, the capacity is the same as if we have no
interference S, i.e, Y = X + W. A lattice based DPC strategy
was proposed in [4] where the transmitter sends [v − αS]
mod M with elements of v restricted to the symbol set of the
elements of X and α = P
x
/(P
w
+ P
x
) is an MMSE scaling
factor [10]. The receiver decodes using
ˆ
Y = (αY ) mod M .
The interference at user 2 is known at transmitter and can
be canceled using DPC by selecting the value β = −α
g
21
r
11
g
22
r
22
,
where α is the MMSE factor.
The rest of the DPC scheme shown in Fig. 3 is based on
the design in [7]. The lattice based DPC scheme proposed for
degraded broadcast channel [7] is extended to the 2×2 MIMO-
BC case. A lattice-based method that uses a combination of
a convolutional code for sign-bit shaping and an LDPC code
for channel coding is used. The details are explained in the
rest of this section.
Combining Shaping and Channel Coding
Sign-bit shaping [11] is a method of constellation shaping
that modifies the sign of transmitted symbols using redundant
bits to choose cosets of convolutional codes. The redundant
bits are chosen for the whole block of bits using a trellis of a
convolutional decoder so as to optimize a given constraint. The
power of the transmitted symbols is usually taken as the mini-
mization constraint. In the interference-free case, this results in
choosing constellation points with Gaussian-like distribution.
In DPC, the minimization of the power implements the lattice-
quantization after subtraction of interference.
The block diagram of the proposed scheme is shown in Fig.
3. The encoder works with a block of L symbols at a given
time. The i
th
user has J information bits that are split as P
MOD− M
LDPC
Encoder
shaped sign bits
Delay
parity bits
M−
QAM
or
PAM
M−
QAM
or
PAM
Delay
Viterbi
Algorithm
16−state
shaped sign bits
message bits
message bits
K+J−P)
N,(
K+J−P)
N,(
LDPC
parity bits
Encoder
β
x
2
x
1
P sign bits
P sign bits
H
−T
H
−T
π
π
J − P message bits
J − P message bits
P/K
P/K
s
1
m
1
c
1
Q
H
R
u
1
u
2
d
1
d
2
m
2
p
1
a
1
m
2
s
2
p
2
s
2
c
2
s
2
m
1
a
2
Fig. 3. MIMO Trellis Shaping DPC for 2 × 2
sign bits a
T
i
= [a
i
1
a
i
2
· · · a
i
P
] and J − P message bits
m
T
i
= [m
i
1
m
i
2
· · · m
i
J−P
]. A rate 1/2 non-systematic
convolutional code with 4 states is used for sign-bit shaping.
The sign bits are passed through the rate-1/2 inverse-syndrome
former having polynomial H
−T
= [D, 1 + D] to map to a
coset leader. The output of inverse-syndrome former c
T
i
=
[c
i
1
c
i
2
· · · c
i
K
] is the sequence of bits taken as initial sign-
bit sequence. A joint 16-state Viterbi Algorithm (VA) decoder
for the convolutional code C
U
with generator matrix
G
U
=
1 + D
2
, 1 + D + D
2
[0, 0]
[0, 0]
1 + D
2
, 1 + D + D
2
determines the final shaped-bit sequence s
T
i
=
[s
i
1
s
i
2
· · · s
i
K
] for each user jointly. The branch
metric for Viterbi algorithm is chosen to minimize of
the sum of the transmitted power in the two antennas. A
(N, K + J − P ) LDPC encoder with rate (K + J − P )/N
computes parity bits p
T
i
= [p
i
1
p
i
2
· · · p
i
N
] from the
sign-bits and message-bits. In order to combine shaping and
channel coding, the parity bits are been delayed by one block
of l symbols. This means that the parity bits of the previous
block are used in the current block to determine the sign
bits and the modulated symbol. The delayed parity bits and
message bits are interleaved in an interleaver π before being
modulated to M-QAM symbols. The number of symbols in
a given block L =
N
log
2
M
is assumed as an integer. Careful
selection of the bits-to symbol mapping is done such that the
shaped-bits determines the sign of the constellation symbols
d
i
for each user. The bits to symbol mapping is chosen to be
the same for both dimensions of the M-QAM constellation.
The mapping [000, 001, 010, 011, 100, 101, 110, 111] −→
A [1/2, 3/2, 7/2, 5/2, −7/2, −5/2, −1/2, −3/2] in each axis
is used for our 64-QAM simulation. This bits-to-symbol
mapping is chosen because: (i) flipping of the sign-bit
provides a significant change in the transmitted energy of the
symbols and (ii) bits are almost as gray coded so that Bit
Interleaved Coded Modulation (BICM) for LDPC coding is
feasible.
Decoder
The decoder for this scheme is similar for all users except
the scaling factor and is shown in Fig. 4. Bits are decoded
by computing
ˆ
Y
i
= (αY
i
) mod M . The receiver mod
operation can be replaced by searching
ˆ
Y in a M-QAM
constellation repeated multiple times in both dimensions [3].
The repeated constellation in each dimension A
R
= {A −
rM, · · · , A − M, A, A + M, · · · , A + rM} where the A is
the transmitter constellation and r is taken to be a sufficiently
large value. The receiver computes LLR for each bit as
L
i
=
X
sǫA
R
,biti=0
exp
−(
ˆ
Y
j
− s)
2
2β
!
X
sǫA
R
,biti=1
exp
−(
ˆ
Y
j
− s)
2
2β
!
.
The LLRs for the sign bits and message bits of block T are
used with the LLRs of the parity bits in block T + 1 for
LDPC decoding. Thus, the one block delay of parity bits at
the transmitter is compensated before decoding. The inverse
permutation for parity bits and message bits are also applied
before the LDPC decoder decodes the message and shaped
bits. The sign bits are recovered using the syndrome former
H
T
from the shaped bits to get original sign bits.
V. SIMULATION RESULTS
We simulate the BER (averaged over the two users) per-
formance of the proposed ZF-DPC scheme for a sum rate
of 4 bits/s/Hz (each user with equal rate) and plot it in Fig.
Decoder
Channel
DeMap
(Bit−LLR
for
Extended
Constelation)
Delay
shaped bits
message bits
π
−1
LLR sign bits
LLR parity bits
ˆ
Y
sign bits
shaped bits
Delay
H
T
LLR message bits
Fig. 4. User module
5. The channel matrix H =
0.7071e
2.37j
0.8660e
2.14j
e
2.37j
0.5e
0.87j
is taken as in [5] for comparison purposes. A shaping code
with code rate 1/2 (= P/K) is used for shaping. An irregular
LDPC code (30000,15000) with code rate 0.5 having bit-
node degree distribution of 0.4691x + 0.3171x
2
+ 0.0103x
5
+
0.05085x
6
+ 0.1096x
8
+ 0.04305x
2
9 (edge perspective) and
check-node distribution 0.0489x
7
+ 0.9511x
8
is used as the
channel code. These degree distributions are optimized for bi-
nary transmission over AWGN channels. Further optimization
for the M-QAM constellation is a possible extension of this
work. The parameters as shown in the block diagram Fig. 3
are P = 5000, J = 10000, K = 10000 and M = 64 for
our simulations. A simple row-column interleaver of row size
174 is used for interleaving parity and message bits before
QAM modulation. The MMSE factor, α = 0.71 is selected
by simulation. The overall rate for each individual user is 2
bits/channel use.
We also simulated a ZF-DPC design where we employing
our ZF-DPC scheme for the 2 quadrature channels (I and
Q) individually using 8-PAM in each dimension, thereby
achieving the same overall rate. The performance of this
scheme was slightly worse than the 64-QAM design. Note that
in the 64-QAM design the sign bits for both the dimensions
are jointly optimized
We compare our method with the following: (i) TCQ/TTCM
ZF-DPC method proposed in [5], (ii) sub-optimal precoding
methods such as MMSE-VP and Zero Forcing Beam Form-
ing (ZF-BF) [9], and (iii) the capacity limit. The result for
TCQ/TTCM ZF-DPC is taken from the simulation result in
[5]. In case of MMSE-VP, a sphere encoder is used to select
the transmitted vector from an extended constellation. We
choose an LDPC code (30000,10000) of rate-1/3 with bit
node distribution 0.239268x + 0.109947x
2
+ 0.345878x
3
+
0.304904x
1
4 (edge perspective) and check node degree dis-
tribution 0.454772x
4
+ 0.390687x
5
+ 0.154541x
7
to code the
bits for the individual user rate of 2 bit/sec/Hz in 64-QAM
constellation. In case of ZF-BF, the precoding matrix is taken
as H
H
(HH
H
)
−1
R. The ZF-DPC method is observed to be
1.8 dB better than MMSE-VP and 2.4 dB better than ZF-
BF at around 3 × 10
−4
BER. Based on the capacity limit,
the minimum transmitter power P
t
required for a sum rate
of 4 bit/sec/Hz is found to be 8.3 dB. From Fig. 2 and our
simulation performs 4.3 dB away from the capacity limit. This
method performs 1.9 dB poorer than the best known coding
method for MIMO-BC called TTCM/TCQ method with ZF-
DPC in [5]. The major loss components are the loss due to
LDPC coding, shaping loss and modulo loss. Also, note that
our method does not require joint shaping and coding as in
[5] using a joint trellis. Further improvement in our scheme is
possible by optimization of the LDPC code for 64-QAM and
the bit mapping used in our design. At present, the coding loss
of the LDPC code used seems to be the major component of
the overall loss.
9 10 11 12 13 14 15 16
10
−4
10
−3
10
−2
10
−1
10
0
Pt −−− (dB) −−>
BER−−−−−>
ZF−BF,Rate=0.5 LDPC, 64−QAM
MMSE−VP,Rate=1/3, 64QAM
ZF−DPC, Rate =0.5, 8PAM in 2−axis
ZF−DPC, Rate=0.5 LDPC,64QAM
Capacity Limit= 8.3(dB)
TCQ/TTCM ZF−DPC
Fig. 5. Comparison results of the scheme for 2 × 2 MIMO-BC for sum-rate
of 4 bits/sec/Hz
VI. CONCLUSION
We designed a ZF-DPC scheme for a non-degraded 2 × 2
MIMO-BC channel and studied its BER performance using
simulations. The main advantage of our scheme is that we
have integrated a shaping technique as well as an LDPC code
with the introduction of a simple block delay component for
the parity bits at the encoder and a ZF-DPC. This removes the
requirement for joint shaping and channel coding. We also
compare our method with MMSE-VP and ZF-BF methods
and show the gains. The gap from the capacity has also
been calculated. Reducing this gap with further LDPC code
optimization is currently under study.
Other future directions are: (1) Optimal DPC as in [5]
instead of ZF-DPC using our design with LDPC codes and
sign-bit shaping, (2) Optimizing the LDPC degree distribution
specifically for the chosen M-QAM modulation and bit-to-
symbol mapping, (3) Designs for higher rates.
ACKNOWLEDGMENT
The authors would like to thank Shilpa Gadiraju for valuable
discussions.
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