Table Of ContentDirty 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,[email protected]
Abstract—Dirty paper coding (DPC) refers to methods for The main source of complexity in lattice-based DPC designs
0 pre-subtraction of known interference at the transmitter of a is combining shaping and coding encoders/decoders at the
multiuser communication system. There are numerous applica-
1 transmitter/receiver. Simple concatenation schemes are not
tionsforDPC,includingcodingforbroadcastchannels.Recently,
0 applicable because of the following reasons - outer shaping
lattice-basedcodingtechniqueshaveprovidedseveraldesignsfor
2
DPC.Inlattice-basedDPC,therearetwocodes-aconvolutional followed by inner coding results in unshaped parity symbols
n code that defines a lattice used for shaping and an error that increase transmitted energy, while outer coding followed
a correction code used for channel coding. Several specific designs by inner shaping results in a poor inner code that needs to be
J have been reported in the recent literature using convolutional
iteratively decoded at the receiver with the outer code.
0 and graph-based codes for capacity-approaching shaping and
In [6], encoding is done on a combined trellis of the source
2 codinggains.Inmostofthereporteddesigns,eithertheencoder
works on a joint trellis of shaping and channel codes or the code (Turbo TCQ) and a channel code (Turbo TCM). At the
] decoder requires iterations between the shaping and channel receiver, decoding is done for Turbo TCM followed by syn-
T decoders. This results in high complexity of implementation. In drome computation to recover message bits. The transmitter
I this work, we present a lattice-based DPC scheme that provides
is complex in [6] because of the use of the joint trellis. The
.
s goodshapingandcodinggainswithmoderatecomplexityatboth
DPC method proposed in [7] is similar to that of [6]. In [5],
c the encoder and the decoder. We use a convolutional code for
[ sign-bit shaping, and a low-density parity check (LDPC) code multilevel coding is used, and there are different codes for
forchannelcoding.Thecrucialideaistheintroductionofaone- different bits of the symbols. At the receiver, iterations have
1
codeworddelayandcarefulparsingofthebitsatthetransmitter, to be performed between decoders for some of the channel
v
which enable an LDPC decoder to be run first at the receiver.
6 codesandtheshapingdecoder.In[8]and[9],shapingfollows
This provides gains without the need for iterations between the
7 channel coding and the receiver performs iterations between
shapingandchanneldecoders.Simulationresultsconfirmthatat
4 high rates the proposed DPC method performs close to capacity the shaping and channel decoders.
3
withmoderatecomplexity.AsanapplicationoftheproposedDPC In this work, we propose a lattice-based method that uses a
.
1 method,weshowadesignforsuperpositioncodingthatprovides novelcombinationofaconvolutionalcodeforsign-bitshaping
0 ratesbetterthantime-sharingoveraGaussianbroadcastchannel. and a low density parity check (LDPC) code for channel
0
coding. As shown in specific designs and simulations, the
1
method provides good shaping and coding gains at moderate
v: I. INTRODUCTION complexity. The main idea for reducing complexity at the
i
X Situations where interference is known non-causally at receiver is the introduction of a one-codeword delay at the
the transmitter but not at the receiver model several useful transmitter, and the shaping of symbols from current message
r
a multiuser communication scenarios. In [1], Costa introduced bits combined with parity bits from the previous codeword.
andstudiedcodingforsuchsituationsandcalledit“writingon This enables the LDPC decoder to be run first at the receiver
dirtypaper”.Dirtypapercoding(DPC)isnowrecognizedasa (with a one-codeword delay) without any need for iterations
powerful notion central to approaching capacity on multiuser withashapingdecoder.Asanapplication,weusetheproposed
channels. DPCmethodto designcodesforsuperposition codingintwo-
Lattice-based ideas for DPC were suggested and shown to user Gaussian broadcast channels. By simulations, we show
be capacity-approaching in [2], [3]. Recently, many designs that rate points outside the time-sharing region are achieved.
of lattice-based DPC schemes have been proposed in [4]–[8]. The rest of the paper is organized as follows. After a brief
Lattice-based schemes typically use cosets of a convolutional review of the lattice-based DPC coding method in Section
code for lattice-quantizing or shaping to minimize the energy II, we present the proposed DPC method in Section III.
of the difference of the coded symbols and the interfering This is followed by description and simulation of specific
symbols. A part of the message bits is used to choose the designs of DPC codes in Section IV. In Section V, design
specific coset used in the minimization. In addition to the of a superposition scheme using the proposed DPC method
shaping convolutional code, an error correction code needs is described and simulation results are presented. Concluding
to be used to obtain coding gain and approach capacity. remarks are made in Section VI.
II. LATTICEDIRTYPAPERCODES III. PROPOSEDSCHEME
In a Gaussian dirty paper channel, the received symbol The proposed scheme uses a convolutional code for sign-
vector Y =[Y1 Y2···Yn] is modeled as bit shaping [10] and low density parity check (LDPC) codes
forchannelcoding.WeassumeaM-PAMsignalconstellation
Y =X+S+N,
withacarefullychosenbit-to-symbolmappingthatiscompati-
where X = [X X ···X ] denotes the transmitted vector, blewithsign-bitshapingandbit-interleavedcodedmodulation
1 2 n
S = [S S ···S ] denotes the interfering vector assumed (BICM) [11]. For M =16, the constellation and mapping are
1 2 n
to be known non-causally at the transmitter and N denotes shown in Fig. 1. The mapping in Fig. 1 is suited for sign-
the additive Gaussian noise vector. The transmit power is bit shaping, since a flip of the most significant bit results in a
assumedtobeupper-boundedby 1E[|X|2]≤P persymbol, significantchangeinsymbolvalueforallpossible4-bitinputs.
n X
and the interference power is denoted 1E[|S|2] = P per Also, the mapping is mostly Gray except for a few symbol
n S
symbol. The noise variance per symbol is denoted P . In transitions. Gray mapping is known to be the most effective
N
[1], Costa shows that the capacity of the dirty paper channel mappingforBICMwithLDPCcodes.Thisheuristicchoiceof
is 1log 1+ PX i.e. known interference can be canceled mapping enables the possibility of good shaping and coding
perf2ectly at thePNtransmitter. gainstobeobtainedsimultaneously.Asexpected,largervalues
(cid:16) (cid:17)
TheinterferingvectorSisusedasaninputintheencoding of M will result in larger shaping gains in our design, and
process and plays an important role to determine a suitable we stick to the 16-PAM shown in Fig. 1 for illustration and
transmit vector X. A coding strategy for choosing X needs simulation.
to overcome the imminent addition of S and protect the
A. Encoder Structure
transmittedinformationfromtheadditionofthenoiseN.Such
The encoder structure for the proposed scheme is as
codingstrategiesarecalleddirtypapercoding(DPC)methods.
shown in Fig. 2. We describe the operations in the en-
In [3], a dirty paper coding (DPC) scheme based on
coder at time step T or in the T-th block. A k-bit mes-
lattice strategies was proposed and shown to achieve the
sage m = [m m ···m ] is encoded into a s-symbol
capacity of the dirty paper channel. We follow [4] for a 1 2 k
vector u = [u u ···u ] from the M-PAM constellation
brief review of the transmitter and receiver structure in the 1 2 s
A = {−(M −1)/2,··· ,−1/2,1/2,··· ,(M −1)/2}, where
lattice DPC method [3]. Let Λ denote an n-dimensional
s = n is assumed to be an integer. Let l = log M and
lattice with fundamental Voronoi region ν having averaged log M 2
2
second moment P (Λ)=PX and normalized second moment let fM : {0,1}l → A denote the bit-to-symbol mapping. The
G(Λ). Also let U ∼ Unif(ν) i.e. U is a random variable bits that map to the i-th symbol are denoted zia2ia3i···ali;
(dither) uniformly distributed over ν. The lattice transmission thesign-bitvectorisdenotedz=[z1 z2···zs],andwedefine
approach of [3] [4] is as follows. vectors aj =[aj1 aj2···ajs] for 2≤j ≤l. Finally, we have
v = f (za ···a ), where f operates component-wise on
M 2 l M
a vector input.
• Transmitter: The input alphabet X is restricted to ν. For
Let us assume that the vectors a , 2 ≤ j ≤ l are available
any v∈ν, the encoder sends j
at the encoder. The sign-bit shaping convolutional code is
X =[v−αS−U]mod Λ, (1) used to determine the sign-bit vector z as follows. A part
where α= PX is a MMSE scaling factor [4]. of the message m(cid:48) = [m1 m2···mk(cid:48)] with k(cid:48) < k bits is
PX+PN mapped to a coset leader of the convolutional code using an
• Receiver: The receiver computes
inverse syndrome former [10]. Note that we need the rate of
Y(cid:48) =[αY+U]mod Λ (2) theconvolutionalcodetobe1−k(cid:48)/s.Letthecosetchosenby
The channel from v to Y(cid:48) defined by (1) and (2) is equivalent m(cid:48) be denoted C(m(cid:48)). The sign-bit vector z is chosen from
C(m(cid:48)) so as to minimize the squared sum (energy) of the
in distribution to
vector (v−αS) mod M, where α= PX is the MMSE
Y(cid:48) =[v+N(cid:48)]mod Λ, (3) factor and S is the interference vector. TPXha+tPiNs,
where z=arg min |(f (ua ···a )−αS) mod M|2. (6)
M 2 l
N(cid:48) =[(1−α)U+αN]mod Λ. (4) u∈C(m(cid:48))
The minimization in (6) is implemented using the Viterbi
Lower bounds on achievable rates for the above equivalent
algorithm [10].
channel is shown in [4] to be equal to
Thea ,2≤j ≤laredeterminedasfollows.An(n,k−k(cid:48)+
j
1 1
I(V;Y(cid:48))≥ log (1+SNR)− log (2πeG(Λ)). (5) s)LDPCcodeisusedattheencoderwithasystematicencoder
2 2 2 2 E : {0,1}k−k(cid:48)+s → {0,1}n. Let m(cid:48)(cid:48) = [z m ···m ] be
k(cid:48)+1 k
For optimal shaping, G(Λ)→ 1 and we approach capacity input to the systematic LDPC encoder to obtain the codeword
2πe
of the dirty paper channel. Note that the dither is assumed to E(m(cid:48)(cid:48))=[m(cid:48)(cid:48) p ], where p is the parity-bit vector for the
T T
be known at the transmitter and receiver (say, through the use T-th block. The parity-bit vector is delayed by one time step.
of a common seed in a random number generator). FortheT-thblock,then−s=s(l−1)bitsin[m ···m ]
k(cid:48)+1 k
is as shown in Figure 3.5. We used the mapping shown in figure instead of two’s
compliment mapping because in coded modulation system gray mapping (or almost
gray mapping) gives better coding gain than two’s compliment mapping. This con-
stellation ordin arily supports 8 bits per 2-D. In sign bit shaping we modify only
the sign bit z in each dimension. Modifying the sign bits z results in 4 different
j
constellation points with different energies per 2-dimensions as shown in Figure 3.4.
(1100,0100) (0100,0100)
(1100,1100) (0100,1100)
2
1.8
1.6
Figure 3.4: 256-QAM constellation with equivalent points shown
1.4
1.2
1000 1001 1011 1010 1110 1111 1101 1100 0000 0001 0011 0010 0110 0111 0101 0100
1
-15/2-13/2-11/2 -9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/2 11/2 13/2 15/2
0.8
0F.6igure 3.5: 16-PAM constellation with equivalent points shown
Fig.1. 16-PAMconstellation.
0.4
n−s
0.2 ParityplusMessage
Bits
0
−8 −6 −4 −2 0 2 4 6 8
Less Significant bits abc
Mesks(cid:48)age s=Biltosgn2M 3(cid:63)nBits Shasped -
Bits (cid:45) H−1T z (cid:45) ? Bits (cid:45) ConMstaepllpaetrion nVαalue-s(cid:45)CDNheloaayinsnye(cid:45)l α map−1 zab-c SDheacpoidnegr sab-c
kBsits- H−1T nB0its-ShcaoSpmdhiiAanenpVligrimgniotgierziretCbhsiomtndheeryBits-(cid:18)(cid:19)P6(cid:17)(cid:16)nzBits -EoLnfDcroPadCteer n−(PkBa−riittsyk(cid:48)+s) k−k(cid:48) (cid:45) (cid:81)b abc(cid:45) Mbapbtopibngb v (cid:45)(cid:19)−α(cid:63)bS(cid:16)bbb
energyof (k−k(cid:48)+s)/n Messagebits mMap−(zPaAbMc) (cid:18)(cid:80)(cid:63)(cid:17)
(v−αS) modM
modM
Figure 3.6: Modulation system using Sign bit Shaping
(cid:45)
(k−k(cid:48)) s z
(cid:63)
Message (cid:54) ShapedBits (v−αS) modM
Bits
The sign-bit shaping operation at the encoder is shown in Figure 3.6. In sign-bit
Fig.2. Encoderstructure.
trealnlidspshaparienrgear(rafnogred1b6y aPpAerMmut)ati3onnΠ+tokfoirnmptuhet b[4i]t.sThaerreeforde,ivthiedreecdeiviendtcoonssteellqatuioennAceiss aa,repbl,icactedand
T−1 R
vectors a , 2 ≤ j ≤ l. This permutation is necessary in an version of the M-PAM constellation A used at the transmitter
j
implementation of BICM [11]. (assuming that scaling factors have been corrected at the
Note that both the shaping and coding objectives have receiver). That is,
19
been met at the encoder. The transmitted symbols v − αS
mod M have minimal energy in the lattice defined by sign- AR ={A−rM,··· ,A−M,A,A+M,··· ,A+rM}.
bit shaping using the convolutional code. Selected bits in
successive blocks of symbols form codewords of the LDPC The number of replications r is chosen so that the average
code.Insummary,theencoderstructureachievesDPCshaping powerofAR isapproximatelyequaltothetotalaveragepower
and LDPC coding with bit-interleaved modulation. PX +PS, 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
B. Decoder Structure
i-th bit in the j-th symbol Y is computed according to the
j
The decoder for the proposed scheme is as shown in Fig.3. constellation A using the following formula:
R
The demapper computes log likelihood ratios (LLRs) for the (cid:98)
bitsfromthereceivedsymbolsinY =αY+U.TheLLRsof 2
Y −a
the (k−k(cid:48)) message bits after a delay of one time step, and exp −1 j
theLLRsofthen−(k−k(cid:48)+s)p(cid:98)aritybitsarede-interleaved. 2(cid:16) αPN(cid:17)
Thes= n LLRsofthesignbitsafteradelayononetime L = a∈AR(cid:88):biti=0 (cid:98) .
log2M i 2
step,andthen−soutputLLRsofthede-interleaveraregiven Y −a
1 j
astheinputtotheLDPCdecoder.TheLDPCdecoderoutputs exp −
2(cid:16) αP (cid:17)
k −k(cid:48) message bits and s bits of the sign bit vector of the a∈AR(cid:88):biti=1 (cid:98) N
previous block. Now, the s-bit sign vector is passed through
thesyndromeformertorecovertheremainingk(cid:48) messagebits. Since the constellation mapping is nearly Gray, iterations
ThedemapperfunctionatthereceiverhastocalculateLLRs with the demapper do not provide significant improvements
taking into account the modulo M operation at the encoder in coding gain, particularly for large M.
n−(k−k(cid:48)+s)
LLRsof s
ParityBits Bits k(cid:48)
(cid:45) (cid:45)
−1 (cid:45) HT (cid:45) Message
Y=αY+U (cid:45) Delay (cid:45) (cid:45) LDPC Bits
(cid:45) Demapper (k−k(cid:48)) (cid:81)
(cid:98) LLRsofMessageBits Decoder k−k(cid:48)
(cid:45) Delay (cid:45) (cid:45) Message
n
log2M Bits
LLRsofsignbits
Fig.3. Decoderstructure.
IV. SIMULATIONRESULTS plotforillustration.WeseethataBERof10−5 isachievedat
For simulations, we have taken n = 40000, k = 30000, a SNR of 19.45dB with interference, and at a SNR of 19.33
k(cid:48) = 5000 with M = 16; this results in s = 10000. dB without interference. We have simulated 1000 blocks of
The constellation mapping is as given in Fig. 1. We length 40000 to obtain sufficient statistics for a BER of 10−5.
have chosen a rate-1/2 memory 8 (256 state) non- TheAWGNcapacityatanSNRof10log10(26−1)=17.99
systematic convolutional code with generator polynomials dB for a rate of 3 bits per channel use. This shows that we
D8+D5+D4+D2+D+1, D8+D7+D4+D2+1 are 1.46 dB away from ideal dirty paper channel capacity.
as the sign-bit shaping code. A non-systematic convolutional The granular gain G(Λ) = 2C∗/6Sx is computed from the
(cid:0)code is used to avoid error propagation problems. (cid:1) simulations to be 1.282dB [4], where C∗ = 3.5 is the rate
A randomly constructed irregular LDPC code (40000, beforechannelcoding,andSx isthetransmitpower(obtained
35000) of rate 7/8 with variable node degree distribution: through simulations). From this, the shaping loss is calculated
0.1256x+0.7140x2+0.1604x9 and check node degree dis- as follows:
tribution x31 is used as the channel code. The overall rate of 2πeG(Λ)22C∗ −1
transmission is seen to be 30000 × 4 = 3 bits per channel 10log =0.2548dB. (7)
40000 10 22C∗ −1
use. Fig. 4 shows BER plots over an AWGN channel and a
DPC channel with interference known at the transmitter. The 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.
10−2 V. APPLICATIONTOGAUSSIANBROADCASTCHANNEL
DPC
AWGN We use the proposed scheme for superposition coding in
a two-user Gaussian broadcast channel Y = X + N and
1 1
Y = X +N with P > P . We let P = (1−β)P
2 2 N1 N2 X1
and P = βP, where P is the total transmit power. Here,
10−3 X2
User2iscodedusingDPCconsideringUser1asinterference.
e
at User 1 is shaped using sign-bit shaping and coded using an
or r LDPC code over M-PAM. Fig. 5 shows a block diagram of
Bit err t1heistraasnsinmiFttiegr. 2anwdirthecethiveeirns.teTrfheereenncceovdeecrtostrruSct=ure0.foHreUnsceer,
10−4
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 using the following formula.
1j
1(Y −a)2
10−5 p(a)exp −2βP1j+P
18 18.5 19SNR(dB1)9.5 20 20.5 Li = a∈A(cid:88):biti=0(cid:40) (cid:40) N1(cid:41)(cid:41),
1(Y −a)2
p(a)exp − 1j
2βP +P
a∈A:biti=1(cid:40) (cid:40) N1(cid:41)(cid:41)
Fig.4. BERplotforDPCandAWGN. (cid:88)
wherep(a)fora∈Arepresentstheaprioriprobabilityofthe
interferingvectorwasgeneratedatrandomfordifferentpower M-PAM symbol a. At the receiver, we approximate p using
i
levels. The plot with interference did not change appreciably a Gaussian distribution with variance P assuming that the
S
forallpowerlevelsofinterference,andwehaveprovidedone distribution of M-PAM symbols is approximately Gaussian.
5
Receiver1 Broadcast capacity
(cid:0)(cid:18)
MeUsssaegre1bits(cid:45) LSDhPapCedcoadnedd Y1=X+(cid:0)N1(cid:0)(cid:0)(cid:0) el use) 4.45 Time sharing
X1 (cid:0) nn
MeUsssaegre2bits(cid:45) DPCc(cid:63)odaeUsdXsSer1f2or (cid:18)(cid:19)(cid:80)(cid:63)(cid:54)X2(cid:17)(cid:16)X (cid:45)Y(cid:0)(cid:64)2=(cid:0)(cid:64)X(cid:64)+(cid:64)N2(cid:64)(cid:64)(cid:64) of User 1 in bits/cha 23..355 (2.474, 3()3, 2.6(30,8( 33)), 3.409(3).451, 3)
(cid:64)(cid:82) Receiver2 ate 2
R
R (11.5
1
Fig.5. Blockdiagramforatwo-userGaussianbroadcastchannel.
0.5
0
We simulated a two user degraded broadcast channel with 0 1 2 3 4 5 6 7
R (Rate of User 2 in bits/channel use)
2
P = 0.9 and P = 0.09 using the proposed scheme with
N1 N2
parameters from Section IV. The total transmit power, power
forUser1andpowerforUser2requiredforabiterrorrateof Fig.6. Two-userGaussianbroadcastchannelcapacityregion.
10−5 (atbothreceivers)areestimatedfromthesimulationand
denoted P, P and P , respectively. The SNR for Receiver
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