A Practical Dirty Paper Coding Applicable for Broadcast Channel

1.2 MB·English

by  Srikanth Pai Bantwal

A Practical Dirty Paper Coding Applicable for
Broadcast Channel
Srikanth B. Pai B. Sundar Rajan
Coding and Modulation Lab, Dept of ECE, Coding and Modulation Lab, Dept of ECE,
Indian Institute of Science, Indian Institute of Science,
Bangalore 560012, India Bangalore 560012, India
Email:[email protected] Email:[email protected]
0 Abstract—In this paper, we present a practical dirty paper Dirty Paper X
1 codingschemeusingtrelliscodedmodulationforthedirtypaper M + + Y
0 channelY =X+S+W,E{X2}≤P,whereW iswhiteGaussian Coding
2 noise with power σw2, P is the average transmit power and S
is the Gaussian interference with power σs2 that is non-causally
n knownatthetransmitter.Weensurethatthedirtinourscheme
a remains distinguishable to the receiver and thus, our designed
S W
J
scheme is applicable to broadcast channel. Following Costa’s

8 idea, we recognize the criteria that the transmit signal must be
1 as orthogonal to the dirt as possible. Finite constellation codes
are constructed using trellis coded modulation and by using a Fig.1. AGaussianDirtyPaperChannel

] Viterbi algorithm at the encoder so that the code satisfies the
T
design criteria and simulation results are presented with codes
I constructed via trellis coded modulation using QAM signal sets fraction of allowed transmit power for the weaker user. The
.
s to illustrate our results. codeword of the weaker user X(m ) is non-causally known
c 2
[ to the transmitter. Since it was decided that we will add the
I. INTRODUCTION
codewordsand thebroadcastingtransmitter knowstheweaker
1
The dirty paper channel model is receiving increased at- user’s codewords, the channel for the stronger user will look
v
7 tention due to its utility in communication over Gaussian like a dirty paper channel. That is, the channel looks like
0 broadcast channels and design of good methods for digital Y = X(m ) + X(m ) + W and while coding for the
1 1 2 1
1 watermarking and information embedding. In this communi- strong user, the additive interference X(m ) is known to the
2
3 cation channel model, characterised by transmitter. The authors in [6] show that there is no loss in
.
1 Y =X+S+W, E(X2)≤P (1) rate when the stronger user performs dirty paper coding, by
0
achieving the well known capacity of the 2 user Gaussian
0
aGaussianinterferenceS,whoserealizationsarenon-causally broadcast region. In the dirty paper coding problem, our
1
: known to the transmitter (but not to the receiver), is added to intentions are only to transmit the message across the channel
v the transmitted signal X along with white Gaussian noise W without worrying about the status of the dirt at the receiver. It
i
X asshownbyFig.1.Itisassumedthatthenoiseisindependent is important to observe, at this point, that the dirt in the case
r oftheinterferenceandtransmitsignal.Costa,in1983,proved ofthebroadcastchannelistheweakeruser’sinformationand
a that the capacity of this channel is the same as the capacity protecting it’s integrity is crucial to communicating over the
of the dirty paper channel without the interference [1]. This broadcast channel. Thus a good solution to the dirty paper
result is, perhaps, surprising because it suggests that there is coding problem that reduces the distinguishability of the dirt
no loss in power, as for as the capacity is concerned, when may not be a good solution to the broadcast coding problem.
the transmitter adapts to the interfering signal. A similar argument holds for digital watermarking also [8].
It was shown that the two user Gaussian broadcast capacity A large amount of work has been done in designing
for the point to point case can be achieved by the use of structured codebooks which achieve Costa’s promised rate
dirty paper coding in [6]. The two user broadcast problem asymptotically (in block length). Zamir et al quantize the
is that a transmitter wants to convey a separate signal X interference into bins and then use a capacity achieving
m1
and X reliably to each of it’s two receivers simultaneously. additivewhiteGaussiannoise(AWGN)codeforeachbin[10].
m2
We will term the user with less noise variance (say user 1) They use lattices as the framework for this idea and postulate
the “strong user” and the remaining user the “weak user” the existence of a good lattice. This approach is pursued from
and choose to add the codewords of these two users (i.e. a practical view point in [11]. For the case of finite alphabets,
the transmit signal X = X + X ) to explain their Garibyetalhaveexploredboundsonthecapacityofthedirty
m1 m2
idea. The transmitter chooses a Gaussian codebook with a paperchannel,assumingaPulseAmplitudeModulated(PAM)
constellation [14]. All these approaches attempt to find the showntoperformclosetoatrelliscodeonanAWGNchannel
maximum, reliable rate of transmission with a constrained for high signal to noise ratio (SNR) empirically. As we have
probability of error. Recently, Skoglund and Larsson have been pointing out, in most applications of dirty paper coding,
investigated optimal modulation schemes for the dirty paper the integrity of dirt cannot be compromised. Let us focus on
channel [13], from the view point of error performance for a the behavior of dirt in the THP example. Observe that the
fixed rate. They optimize the modulator and the demodulator receiver can only know (s − s mod L) and thus the THP
jointly for the case of a one-dimensional modulation scheme. method compromises the distinguishability of s to achieve
The design involves solving an optimization problem for performance independent of dirt power. The main drawback
every value of the interference. Since it is not feasible for ofTHPbasedmethodsisthatitdoesirrecoverabledamageto
a modulator to solve it, they quantize the interference and the dirt and this can prove costly when it is used in Gaussian
construct a lookup table for different values of interfering broadcast channels and information embedding problems. We
signals. will, henceforth refer to such schemes as dirt lossy.
Thechanneldescribedby(1)willbereferredto,ingeneral,
The most popular finite constellation coding approach asthedirtypaperchannelandtheparticularcaseofthechannel
to dirty paper coding uses Tomlinson-Harashima precoding described by (1) with zero dirt power will be called the corre-
(THP) introduced in [2],[3]. Originally, THP was a single sponding interference free channel. While the corresponding
dimensional pre-equalization technique designed to combat interference free channel is just an AWGN channel, the term
intersymbol interference (ISI). The basic idea of THP is corresponding is used to denote that both the dirty paper
to subtract interference, quantize the subtracted signal so channel and the corresponding interference free channel have
that the power is constrained within the required value. The the same Gaussian noise variance. In this paper, we consider
received signal is then decoded with respect to the quantized the problem of finite constellation coding for the dirty paper
framework. The main motivation to use THP in dirty paper channel with focus on reducing the probability of error for a
coding is to design a coding scheme for the dirty paper given rate without affecting the dirt significantly. We mainly
channel that performs independent of the dirt power in an use Costa’s work [1] to draw insights into designing codes
attempttoreinforceCosta’scapacityresult.In[9],theauthors for the dirty paper channel in this paper. Our comparison of
pointed out the connections between THP and the dirty paper the methods used for achievability of rates for the dirty paper
result. They also introduce the idea of partial interference channelwiththemethodsusedforachievabilityofratesforthe
presubtraction (PIP) and show that it outperforms completely AWGN channel motivates us to use a codeword, from the set
subtracting interference. Our following explanation is taken of good partial interference presubtracted AWGN codewords,
from [7] which is a thorough approach to the dirty paper that is orthogonal to the given realization of the dirt. From
coding problem that uses the THP framework. The concept the perspective of minimizing the pairwise probability of
can be easily explained for the case of one dimension as error, the goodness of an AWGN code is determined by the
follows: Suppose we wish to transmit u∈(cid:2)−L,L(cid:1) for some pairwise Euclidean distances between the codewords. Thus, if
2 2
L∈Roverthedirtypaperchannelgivenby(1).Letusforget the pairwise Euclidean distances of a code are large, we may
about the additive noise W, for the moment, to establish the termthecodeagoodAWGNcode.Itiswellknownthatcodes
distinguishabilityofthesignalu.Lettheinterferenceknownat obtained by the trellis coded modulation (TCM) techniques
the transmitter be s∈R. Instead of transmitting u, we intend introducedbyUngerboeckin[5]havelargepairwiseEuclidean
to transmit u−s so that the effect of the dirt is minimized. distances. Thus, we design a scheme that chooses a TCM
However,thepower ofu−s mightbevery largeandmaynot codeword(fromalargebinofTCMcodewords)thatisalmost
satisfy the transmit power constraint. The real trick in THP is perpendicular to the dirt. We find the dirt-orthogonal TCM
toconstraintheu−swithinthefiniteinterval(cid:2)−L,L(cid:1).Thus codeword using the fast Viterbi algorithm. We empirically
2 2
the transmitter transmits u−s modulo-L. In other words, all show that increasing the bin size (by increasing the number
theu−sthatdifferbyanintegermultipleofLareregardedas of states in the trellis and the spectral efficiency for which the
the same symbol. Now, in the absence of noise, the received trellis is designed) results in an increased coding gain.
signal is u−s mod L+s=u mod L+(s−s mod L)= The main contributions of this paper are:
u+(s−s mod L).Clearly[s−(s mod L)] mod Lvanishes • A design of a practical dirty paper coding scheme, for
and even though the receiver does not know s, the receiver spectral efficiencies greater than 1 bit per channel use,
recovers u by performing a modulo-L operation. The utility whose construction mimics Costa’s information theoretic
of this approach is that the transmit signal is approximately proof and the performance of which is close to that of
uniformly distributed between (cid:2)−L,L(cid:1), if the interference the AWGN channel.
2 2
power is large, in which case the transmit power is L2. Thus • The observation that in most applications of dirty paper
12
THP is a simple scheme that performs independent of the dirt coding, like Gaussian broadcast channels, the dirt is
power and does not compromise the distinguishability of the actually valuable information and distorting the dirt to
message. This approach is generalized to higher dimensions rendergoodperformanceisnotagoodidea.Ourdesigned
in [9] using a lattice vector quantizer and using a practical scheme maintains the distinguishability of both the dirt
trellis precoding scheme in [7]. In [7], the designed scheme is and the transmit signal with high probability while the
existingschemes(likelatticebasedmethods[9],[10],[11] II, he showed that for any R such that R < C∗ = 1log(1+
2
and THP based practical methods [7]) do not. P ) there exists length n codes with Q = 2nR that achieve
σ2
The rest of the manuscript is organized as follows: In arwbitrarily small probability of error for large n. And for all
Section II, we introduce the channel model. We explain other R, such a code does not exist.
Costa’s information theoretic proof in Section III and derive Weurgethereadertoreferto[1]toappreciatethefollowing
finiteconstellationcodingdesigninsights.SectionIVexplains construction better. Observe that the above process involves
constructions of explicit finite constellation codes that satisfy constructingtwocodebooksC˜andC.C istheactualcodebook
derived criteria and illustrate the idea by an example. Section made up of points on the transmit sphere shown in fig. 2. The
V addresses the applicability of our scheme to Gaussian elements of the transmit sphere need to satisfy the transmit
broadcast channels. We discuss simulation results in Section powerconstraint.AnelementofC˜ischosenandthenmodified
VI.FinallySectionVIIconcludesourpaperbyexaminingthe accordingtothedirtafterwhichitistransmitted(themodified
deficiencies of this approach and noting the scope for future vector, thus, belongs to C). We will call this modified vector
work. as a “transmitted codeword”. Nevertheless we will call the
Notations: Bold, uppercase letters are used to denote ran- elements of C˜ also as codewords. It is important, for the
dom vectors and bold, lowercase letters are used to denote reader, to bear in mind that a codeword and a transmitted
vector realizations of the corresponding random variables. codeword do not necessarily come from the same set. The
The set of all real numbers are denoted by R. E(.) denotes codegenerationinvolvesconstructingacodebookC˜withlarge
expectation, Pr(.) denotes probability and ||.|| denotes the number of codewords whose co-ordinates are drawn from a
Euclideannorm.ForarealrandomvariableX,X ∼N(0,σ2) zeromeanGaussianrandomvariablewithvarianceP+α2σs2.
denotes that X has a Gaussian distribution with mean 0 and Each codeword, denoted by U∈C˜, is then assigned an index
variance σ2. Q(.) denotes the tail probability of the standard randomly from I and this process is termed binning. Note
(cid:16) (cid:17)
normal distribution, Q(x)= √1 (cid:82)∞exp −u2 du. that this means we can have many codewords that have been
2π x 2 assigned the same index and it may also happen that a certain
II. MODEL index may not get assigned to any codeword. We will refer
to a codeword that has been assigned an index m ∈ I as
Considertheproblemofreliablycommunicatingamessage
a “codeword from the bin m” and the set of codewords that
chosen uniformly from I = {1,2,3,...,Q} over the discrete
havebeenassignedanindexmas“binm”denotedbyB .To
time channel shown in Fig. 1. Realizations of interference m
random variable S ∈ C are known non-causally at the trans- transmit a message m∈I, choose the bin m and then select
a codeword U from B ⊂C˜that is jointly typical with S. In
mitter and the dirty paper coding operation is performed to m
essence,acodewordU∈B ischosensothat|(U−αS)TS|
adaptthetransmittedsignaltotheknownchannelinterference. m
is small enough. In other words, Costa’s idea was to choose a
TherandomvariablesW andS arecomplexnormalGaussian
with zero mean and variances σ2 and σ2 respectively. The transmittedcodewordX=U−αSthatis,ideally,orthogonal
w s tothedirts.Apartialinterferencepresubtraction(PIP)scheme
randomvariablesW,S andM areassumedtobeindependent.
The message M chooses X(M,S) ∈ C based on the known refers to a scheme that transmits a signal that subtracts a
fraction of the dirt vector from a codeword. PIP is performed
realizationS oftheinterferencewhichisthentransmittedover
in an attempt to reduce the effective Gaussian noise seen by
the channel. This constitutes using the dirty paper channel
the codeword. X is clearly a PIP codeword. The decoder just
in one real dimension. We propose block transmission where
attempts to find a codeword U that is jointly typical with
transmission (and hence decoding) is in n complex dimen-
sions. The transmitted signal vector x(M,s) ∈ Cn is subject the received y. It was proved in [1], that the rates achievable
by this method are the same as that of the corresponding
to an average power constraint (averaged over the messages),
for a given vector realization s∈Cn, given by interference free channel by averaging over codebooks.
E(||X(M,s)||2)≤P (2) B. Insights to Finite Constellation Coding Design
For the case of AWGN channel, the strategy of decoding
where,now,M isthemessagechosenfromthemessagesetI.
using joint typicality of the transmit codewords and receive
The received signal vector Y ∈Cn, according to the channel,
signal when transmit codewords were chosen from Gaussian
Y =X(M,S)+S+W (3) generatedcodebooksachievedcapacityoftheAWGNchannel.
Fromapracticalcodingtheoreticperspective,codebooksgen-
is used by the receiver to estimate M. The estimated message
erated using TCM and decoding using minimum Euclidean
index is denoted by M(cid:99). distance decoder (MEDD) (which is actually a maximum
likelihood decoder (MLD) for AWGN channel) yielded good
III. COSTA’SPROOFANDINSIGHTSTOFINITE
performance.
CONSTELLATIONCODINGDESIGN
Suppose we wish to communicate over the dirty paper
A. Costa’s Proof
channel at a spectral efficiency of R and a transmit power
Costa computed the capacity of the channel shown in Fig.1 P using codewords of length n and we know the dirt vector
forthecaseofGaussiandirtin[1].Withnotationsfromsection that will add is s. As we know [1], the strategy to achieve
capacity of the dirty paper channel involves two steps. The
first step involves generating a Gaussian codebook C˜ with a PIP code word from bin 
higher spectral efficiency (in this article, we will refer to it as k most orthogonal to s
Axis orthogonal to s
designspectralefficiency)R˜ =R+R andalargerpower(in
0
thisarticle,wewillrefertoitasdesignpower)P˜ =P+α2σ2.
s
The codewords are then randomly binned into 2nR partitions.
Thesecondstepistopickabincorrespondingtothemessage
and search for a codeword sequence u∗ in the bin that is
Transmit sphere with 
jointly typical with s. The PIP codeword x = u∗ − αs is
radius P
thentransmittedoverthedirtypaperchannel.Asexplainedby
ddiirrtt vveeccttoorr ss
Costa, this operation corresponds to finding a u∗∈C˜so that
|(u∗−αs)Ts| is small. In other words, we want the transmit
PIPcodewordtobealmostperpendiculartherealizationofthe
dirt vector. The decoder decodes a codeword from C˜ that is
A PIP code word from bin k on the surface of 
jointlytypicalwiththereceivedsignal.Thisstrategyisproved the transmit sphere
to be rate optimal.
In what follows we design a practical dirty paper coding
scheme by using the above idea. The MEDD turned out
Fig. 2. An example outlining the method to choose PIP-TCM codewords
to be the probability of error minimizing decoder in the whentransmittingatunitspectralefficiency
case of AWGN channel. Following Costa’s idea and the idea
that a codebook constructed by the TCM method has good
distance properties, we will construct a TCM codebook C˜ is called the transmit sphere. The codewords on the transmit
with elements drawn from a larger average power P˜ and the sphere are shaded differently based on the bin they belong.
spectral efficiency of TCM codebook will be larger than R to The dirt vector denoted by s is marked in the figure. Our
facilitate “binning”. We will then partition these codewords at aim is to find a codeword from a particular bin that is almost
every node of TCM into 2R bins (denoted by Bm for m∈I) orthogonal to the given dirt.
and then choose a codeword u ∈ B so that the transmitted
m
codeword u−αs (clearly a PIP codeword) has the smallest IV. DIRTYPAPERTRELLISCODING
Euclidean inner product with the dirt s for some α ∈ R. A Inthissectionwewilldescribetheconstructionanddecod-
MEDD will be used at the receiver and the decoder will find ing of a dirty paper coding scheme using finite constellation.
the codeword from C˜that is the closest to the received vector
From the previous section we have inferred that we must
y. The receiver knows the partition of the code C˜ and thus
design codes that satisfy the following design criteria:
it finds the bin to which the decoded codeword belongs and
1) Large pairwise Euclidean distance between the code-
decodes the bin index as the transmitted message.
words for every realization of the dirt vector s
Thedistinctionbetweenthetransmittedcodewordandcode- 2) Reduce the Euclidean inner product between the trans-
words from C˜is now clear. The code C˜is the code used by mit PIP codeword and the known dirt vector s
thedecodertodecodemessagesalthoughwhatwastransmitted
To satisfy the first criteria, we will use TCM to design our
wasavectorthathasmodifiedthecodewordfromC˜topartially
codes in the reminder of this section. Suppose we want to
cancel the dirt and yet satisfy the transmit power constraint. transmit messages at a spectral efficiency of r ∈ R+ bits
It should be observed that for the receiver the channel looks
per channel use and the average power constraint on the
like
transmitterisP.ThegeneralideaistoconstructaTCMbased
y=u+(1−α)s+w (4) coding scheme for a design spectral efficiency r0 ∈R+ such
that r > r and the signal points (denoted by u) are drawn
0
whichmeanswhilelookingforuusingaMEDDwearetreat-
from a finite constellation. A TCM scheme designed for a
ing interfering fractional dirt as independent noise (Clearly spectral efficiency of r bits per channel use transmits one
0
this decoder is not a maximum likelihood decoder because of 2r0 different signal points for every channel use and this
the fractional dirt vector is not independent of the transmitted is represented by 2r0 outgoing edges from each state in the
signal).Thisincreasestheeffectivenoiseseenbythecodeword trellis diagram. We partition the set of 2r0 signal points from
u. Since the codeword u is drawn from a constellation with a state in the trellis into 2r disjoint subsets (call it B for
i
larger average power, the hope is that the receive signal i=1,2,3··· ,2r)sothateachsubsethas2r0−r signalpoints.
to noise ratio (SNR) will be large enough yielding good
Thedecoderisassumedtoknowthepartitions.Inourscheme,
performance. amessagemselectsoneofthe2r subsetsateachstepandthen
Fig. 2 illustrates a scenario where we want to communicate thedirtvectorsknowntothetransmitterhelpschooseasignal
at a spectral efficiency of 1 bit per channel use. The PIP u(m,s) among the 2r0−r possible choices. The knowledge of
codewordsarewithinthespherewhoseradiusisP.Thesphere theentirenon-causalrealizationofthedirtvectorsisrequired
to decide the codeword to be transmitted. In order to satisfy 8 TRELLIS STATES
the second criteria with the method explained above for any
0 4 2 6
realization of the dirt vector, we must decide a fast practical
way to find a PIP codeword that is orthogonal to the dirt.
1 5 3 7 The edge denoting the
The decoding is done by running a Viterbi algorithm and possible path when the
reconstructing a codeword. Then we identify the bin to which 4 0 6 2 msg bit is 1
the codeword belongs by deciding the partition at every node.
The problem of finding a PIP-TCM codeword that is or- 5 1 7 3
The edge denoting the
thogonal to the given dirt is a hard problem to solve. We put
possible path when the
down the problem mathematically: 2 6 0 4 msg bit is 0
Problem: Find a TCM codeword u from the chosen bin
(obtained by a walk through the trellis) so that |(u−αs)Hs| 3 7 1 5
is minimized.
Basic Idea: By triangle inequality, 6 2 4 0
i=n
|(u−αs)Hs|<(cid:88)|(u −αs )∗s | 7 3 5 1
i i i
i=1
i=n
(cid:88)
min|(u−αs)Hs|<min |(u −αs )∗s |
i i i
Fig. 3. TCM using 8 State Trellis and 8 PAM constellation for spectral
i=1
efficiencyof2bitsperchanneluse
Thus we try to search for a u∗ that minimizes (cid:80)i=n|(u −
i=1 i
αs )∗s | rather than the original Euclidean inner product. The
i i
problem of minimizing (cid:80)i=n|(u −αs )∗s | can be solved 0 1 2 3 4 5 6 7
i=1 i i i
by running the Viterbi algorithm on the trellis. We use |(u −
i
αs )∗s | as the branch metric and “decode” for a u at each ‐7A ‐5A ‐3A ‐A A 3A 5A 7A
i i i
nodeamongtheTCMcodewordscorrespondingtoaparticular
bin. (cid:113)
Fig.4. Labelled8PAMconstellationwithA= P
Notethatacodewordobtainedbyanarbitrarywalkthrough 21
thetrellisisnotnecessarilyavalidcodewordinoursetting.We
require the message at each instant to choose one of the bins
andonlythesignalpointsfromthechosenbinateachnodeare (corresponding to message 0). When a message sequence and
possible contenders for presubtraction and then transmission. thedirtvectorsaregiven,weusetheViterbialgorithmtofind
This motivates the following definition: apaththroughthetrelliswhosePIPoutputisavalidcodeword
definition For a given message sequence m = almostorthogonaltothedirts.Whilewearewalkingthrough
(m ,m ,··· ,m ), a TCM codeword u = (u ,u ,··· ,u ) the trellis, based on the message, we only consider the edges
0 1 n 0 1 n
for which u ∈ B is called a valid TCM codeword. ofaparticularshadetobetransmitted.ThusournlengthTCM
definition In iwhat fomlloiws, we illustrate our idea clearly by code has 22n codewords which is partitioned into two sets at
the means of an example. each node in the trellis giving 2n bins having 2n codewords
Suppose that we want to transmit at a spectral efficiency each.
of 1 bit per channel use, i.e. r = 1. Consider the 8 PAM The idea behind the working of this scheme is the smart,
constellation shown in fig. 4 and the corresponding trellis for effective use of PIP codeword. The PIP codeword helps in
theTCMschemeinfig3.Thenumberswrittenontheleftside decreasing probability of error by reducing the net Gaussian
of each node denote the labelling of edges so that they satisfy noise power seen by the receiver. However the condition of
heuristic rules given in [5]. We will assume that the average orthogonality between the PIP codeword and the dirt helps
power of the 8 PAM constellation, when all constellation in correlating the codeword u with the dirt so that the
points are equally likely, is the design power P˜. The TCM average power of the codebook containing u is increased (i.e.
scheme,showninfig.3,generatescodeswithgoodEuclidean E(||u||2) = P˜ > P ). Additionally, since we are traversing
distancefortheshownparticularchoiceofsetpartitioningand a subset of the edges at every node, the minimum Euclidean
labeling.Wedirectlyusethe8statetrellis,setpartitioningand distance of our dirty paper trellis code will be higher than
labelingusedintheprimarywork[5].Thetrellisshowninfig usual. Now we can see why our dirty paper code has the
3constructsaTCMschemeforadesignspectralefficiencyof potential to match the performance of a TCM code on the
2 bits per channel use i.e. r =2. We partition the edges into correspondinginterferencefreeAWGNchannel.Anincreased
0
2r =2 sets of 2r0−r =2 signal points each. In the figure, the minimumEuclideandistanceandanincreasedaveragereceive
signalpointsfromthesamesetaredepictedbyeitheradashed power(P˜)trytocompensatefortheadditionalGaussiannoise,
light line (corresponding to message 1) or a solid dark line of power (1−α)2σ2, seen by the signal u.
s
V. APPLICATIONSTOGAUSSIANBROADCAST
Themainaimofourdesignofdirtypapercodingschemeis
to make it applicable for Gaussian broadcast channels. As we MEDD
have seen in the introduction, the dirt in the case of Gaussian
broadcast channel is signal of the weaker user. We will now
Receiver1 
showwhyourscheme doesnotdamagedirtinaninformation (Strong)
lossyway.Armedwiththeknowledgeofthedesignexplained
Dirty Paper   
intheprevioussection,weareinapositiontoappreciatewhy Encoder
we can recover both the required signal and the dirt in our
scheme.
Transmitter MEDD
Theorem 1: Our proposed scheme allows recovery of the
Receiver 2 
dirt at the receiver with high probability.
(Weak)
Proof: We will ignore the noise to check whether the
dirt s is recoverable. Observe that without noise the channel
looks like
y=u+(1−α)s (5) Fig.5. A2userGaussianbroadcastchannelemployingdirtypapercoding
Ourreceiverisdesignedtodecodeuwithverylowprobability
of error even when the additive noise is present. Clearly
component of the codewords look Gaussian. Thus each user’s
from (5), the channel, now, is an AWGN channel with lesser
signal undergoes a noise that is effectively Gaussian, when
noise variance, from the perspective of the decoder. Since
a trellis shaping is used. Each user can then use a MEDD to
our receiver is employing MEDD and u is drawn from
decodetheirmessagereliably.Theadvantageofthismethodis
TCM codebooks, the receiver decodes u with a even lower
that we need not reveal the code of one user to another. Such
probability of error than the case with noise. It should be
an arrangement can be useful when one needs to maintain
clear that the probability of recovering u is the same as the
secrecy of the message while broadcasting.
probability of recovering s. Our proposed scheme is designed
torecoveruwithveryhighprobability.Thusoncethereceiver A naive way to successfully communicate over a Gaussian
recoversu(whichhappenswithhighprobability),thereceiver broadcast channel is to choose zero mean constellations, X1
can subtract it from the received signal to recover s. with average power P1 and X2 with average power P2, for
eachofuserinsuchawaythattheirsumconstellationX +X
The TCM codebooks with Ungerboeck’s method of set 1 2
admits unique decodability. We say that the sum constellation
partitioningandlabelingwithsufficientexpansionofthesignal
has the property of unique decodability when each element
set gives high coding gain on the AWGN channel. In our
of the sum constellation can be obtained by the addition of
case, if the dirt power is large then we will increase the
overall spectral efficiency r sufficiently high so that we will a unique combination of signal points, with one signal point
0
chosenfromeachconstellation(anecessaryandsufficientcon-
have more codewords in each bin. For a given transmit power
constraintP,thiswillletusincreasetheaveragepowerofthe dition for this is |X1+X2|=|X1|.|X2|). This idea is actually
constellation P˜. Thus the receiver will see codewords coming a crude coding theoretic version of superposition coding. For
thecaseofGaussianMACchannel,thisideaisnecessarysince
fromalargeraveragepowerdependingonthedirtpower.Thus
the users transmit independently and there is no sum power
for any dirt power, the receive signal power (or the average
constraint. One can use this type of “superposition coding”
power of the constellation from which the components of the
designed for Gaussian MAC for the case of broadcast (see
codewords are drawn) increases accordingly, allowing us to
recover both u and s with high probability. [15], for instance). However, the performance is likely to be
poor since we are not using the knowledge of the weak user’s
Weuseourschemetocodeforthebroadcastchannel,shown
signal to gain any type of advantage (power or bandwidth).
in fig. 5, in the following way: Suppose we want to transmit
Clearlywecanusejointdecodingofboththesymbolsatboth
at a spectral efficiency of b and b bits per channel use to
s w
thereceiverstodecodeoursymbols.IfX ∈X andX ∈X
the strong user and weak user respectively. We first design 1 1 2 2
are chosen independently, then the average power of the sum
a TCM codebook with a spectral efficiency of b bits per
w
constellation is
channel use for the weak user. Then we use our dirty paper
coding scheme to construct a PIP-TCM codebook, with a E(|X +X |2)=E(|X |2)+E(|X |2)+2E(X∗X ) (6)
1 2 1 2 1 2
spectral efficiency of b bits per channel use and sufficient
s =P +P (7)
1 2
signal set expansion, for the strong user so that the PIP-TCM
codewords to be transmitted are as orthogonal as possible to In such an approach, the average power of the sum constel-
the given codeword of the weak user. By using trellis shaping lation is the sum of the average power of each constellation
([12] is a classic reference on trellis shaping) on the TCM assuming the signal points are independently chosen. On the
codebooks of both the users, we can make the statistics of the otherhand,inourdirtypapercodingapproach,ouraimindirty
8 Trellis States
D0 D4 D2 D6 10−2
The edge denoting the
D1 D5 D3 D7 10−3
possible path when the
msg bit is 1
D4 D0 D6 D2 10−4
D5 D1 D3 D7
The edge denoting the ER10−5
possible path when the B
msg bit is 0
D2 D6 D0 D4 10−6
AWGN
dirt power = 0 dB
D3 D7 D1 D5 10−7 dirt power = 7 dB
dirt power = 10 dB
D6 D2 D0 D4
10−8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010.51111.51212.5
SNR (dB)
D7 D3 D5 D1
Fig.8. BERversusSNRat1bit/channeluseusinga8statetrellisand16
QAMconstellationwithalpha=0.9
Fig. 6. Dirty paper TCM using 8 State Trellis for 16 QAM constellation
indicatingtheusedsetpartition
0 2
AA
3A
0 4 8 12
‐A A
A
1 5 9 13 ‐AA
‐3A ‐A A 3A 1 3
‐A
2 6 10 14
(cid:113)
‐3A Fig.9. Labelled4QAMconstellationA= P
2
33 77 1111 1155
(cid:113)
Fig.7. Labelled16QAMconstellationA= P accrued. The spectral efficiency was fixed at 1 bit per channel
10
use. Thus for each iteration, a block of equally likely bits
of length N were generated as the message sequence. The
paper coding is to reduce |(U−αS)HS| as much as possible. dirt and the message sequence selected a PIP-TCM codeword
In our design of dirty paper coding, we have explained the using a Viterbi algorithm at the encoder. SNR was measured
idea of increasing spectral efficiency to increase the size of as ratio of power of the simulated complex transmit signal
the bin to increase P˜. From Costa’s proof, we know that to power of the simulated complex noise power. We remark
P˜ =P +α2σ2 for the limiting case of infinite alphabet with that we have not used trellis shaping [12] which could have
s
Gaussian distribution. For the sake of analysis, let us assume resulted in an additional gain.
(u−αs)Hs≈0. Thus we have Fig. 8 depicts the simulation performance for case of an
8 state trellis, shown in fig. 6 (the numbers written on the
uHs≈α||s||2 (8)
left side of each node denote the set partition so that they
The codewords in our scheme are aligned to make them satisfyheuristicrulesgivenin[5])with16QAMconstellation,
statistically correlated. This allows the strong user to have showninfig.7.Thesetpartitioningandlabellingfollowedare
an increased design power P˜ for the transmit average power directly used from [5] where D0 = {0,10}, D1 = {6,12},
of P. Thus using dirty paper coding boosts the strong user’s D2 = {5,15}, D3 = {3,9}, D4 = {8,2}, D5 = {4,14},
effective power using the correlation with the weak user’s D6 = {7,13}, D7 = {1,11}. The noise power was fixed as
signal. Superposition coding loses to dirty paper coding in σw2 =1andthedesignpowerP˜ wasvaried.Whileperforming
harvesting this power advantage. PIP, α was chosen to be 0.9 in this case. For a BER =10−5,
with σ2 = 10 the SNR required is 11.6dB, with σ2 = 5 the
s s
VI. SIMULATIONRESULTS SNR required is 10.6dB, with σ2 = 1 the SNR required is
s
The bit error rate (BER) was obtained by simulating our 9.2dB. This shows that the BER performance improves when
scheme in blocks of length N =105 until at least 100 errors the dirt power decreases when the value of α is fixed. The
4 State Trellis 4 State Trellis
0   3 0   5   1   4 The edge denoting the
possible path when the
message bit is 0
1   2 2   7   3   6
3   0 1   4   0   5 The edge denoting the
possiibblle pathh whhen thhe
message bit is 1
2   1 3   6   7   2
Fig.10. TCMusing4StateTrellisfor4QAMconstellationindicatingthe Fig. 12. Dirty paper TCM using 4 State Trellis for 8 QAM constellation
usedsetpartition indicatingtheusedsetpartition
8 TRELLIS STATES
0 5 1 4
100
2 7 3 6 The edge denoting the
dirt power = 0 dB possible path when the
10−1 dirt power = 7 dB 1 4 0 5 msg bit is 1
dirt power = 10 dB
10−2 3 6 7 2 The edge denoting the
possible path when the
5 0 4 1 msg bit is 0
ER 10−3
B
7 2 6 3
10−4
4 1 5 0
10−5 6 3 2 7
10−6
6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12
SNR (dB)
Fig. 13. Dirty paper TCM using 8 State Trellis for 8 QAM constellation
indicatingtheusedsetpartition
Fig.11. BERversusSNRat1bit/channeluseusinga8stateand16QAM
constellationwithP˜ =16.3dB
power varies. The change in transmit power is reflected in
AWGN performance shown in fig. 8 has a BER = 10−5 at the measured SNR. Clearly both the design power and α are
a SNR of 5.9dB and thus our scheme with σ2 = 1 is within designparameters.Asclearlyseenfromfig.11,foraparticular
s
3 dB of AWGN performance. The equivalent AWGN code is BER (say 10−5), with σ2 = 10 the SNR required is 11.2dB,
s
a TCM code designed for a spectral efficiency of 1 bit per withσ2 =5theSNRrequiredis10.7dB,withσ2 =1theSNR
s s
channel use using a 4 QAM constellation, shown in fig. 9, required is 9.7dB. Thus the BER performance improves when
using a 4 state trellis shown in fig. 10. We remark that our the dirt power decreases when the value of design power P˜ is
scheme performs well since even the “dirt lossy” THP based fixed.Also,therelationbetweendesignpowerP˜ andtransmit
trellis precoding scheme constructed in [7] is around 2.5dB power P is unknown analytically and depends on the number
away from AWGN performance with an equivalent code for of states and the design spectral efficiency r .
0
a spectral efficiency of 1 bit per channel use. The overall performance improves as we increase the num-
Fig. 11 depicts the simulation performance for case of an berofstatesandincreasethedesignspectralefficiencyr .This
0
8 state trellis, shown in fig. 6 with 16 QAM constellation, effect is clearly seen in fig. 15. We observe, from fig. 15, that
shown in fig. 7 with a fixed design power P˜ = 16.3dB. The a 4 state trellis, shown in fig. 12 with 8 QAM constellation,
noise power was fixed as σ2 = 1 and α was varied from 0 shown in fig. 14 achieves a BER = 10−5 with σ2 = 1 for a
w s
to 1 in steps of 0.1. This plot explores the effect of varying SNR of 10.8dB. An 8 state trellis, shown in fig. 13 with 8
α. For a fixed design power P˜, as α is varied, the transmit QAM constellation, shown in fig. 14 achieves a BER = 10−5
0 2 4 6 symbols. However if the survivors disagree on the initial
A
path then a good strategy for determining the initial path
is not very clear.
‐3A ‐A A 3A
‐A • Our scheme has implemented random binning and
1 3 5 7 searched for a codeword in a bin using the Viterbi
algorithm.Wehavenotdesignedacriteriaforpartitioning
(cid:113) the signal sets. Since it was seen that certain partitions
Fig.14. Labelled8QAMconstellationA= P
6 behave differently than the others, it might be instructive
to investigate the problem of partitioning signal points at
the nodes to enhance error performance.
• For Gaussian codebooks, the relation between the design
10−2 power P˜ and transmit power P is concrete, i.e. P˜ =
P +α2σ2 and so is the optimal α = P . However
s P+σ2
10−3 for finite constellation, all we can say is P˜w > P and
the optimal α depends on the dirt power. The relation
10−4 between design power P˜ and transmit power P is not
clear analytically. We only know that it depends on the
R
BE10−5 chosenconstellationandthechosentrellis.Itwillbevery
usefulindesigningdirtypaperTCMcodesiftherelation
10−6 between them was characterised analytically. An exact
8 state trellis, 16 QAM
4 state trellis, 8 QAM characterisation or even a relation in terms of bounds on
10−7 8 state trellis , 8 QAM the design power will prove useful.
Information theoretic designs are impractical because de-
5 6 7 8 9 10 11 12 13 14
SNR (dB) coding techniques like joint typical decoding require a brute
force search. Our work interpreted the meaning of typicality
to obtain a nice criteria and then we managed to construct
Fig.15. Thecodinggainobtainedasweincreasethespectralefficiencyof
theTCMschemeandthenumberofstates a fast algorithm to satisfy the criteria, thereby designing a
goodschemeforthedirtypapercodingproblem.Themessage
of this paper is that, while information theoretic approaches
with σ2 = 1 for a SNR of 10.2dB. An 8 state trellis, shown
s that search for codewords in a random bin might seem like
in fig. 6 with 16 QAM constellation, shown in fig. 7 achieves
a hard problem, slight engineering approximations might lead
a BER = 10−5 with σ2 = 1 for a SNR of 9.5dB. Thus it is
s to an easier problem for which a fast algorithm exists. This
clear that increasing the number of states and increasing the
paper introduces a novel approach to coding theory design by
designspectralefficiencyimprovesperformancebyincreasing
additionallyusingthecriteriafrominformationtheoreticview-
the obtained coding gain.
points rather than focus on just traditional coding theoretic
VII. DISCUSSION concepts like deriving design criteria by minimizing pairwise
error probability.
We have presented a novel dirty paper coding scheme
which followed Costa’s paper analytically and allowed the ACKNOWLEDGEMENT
recoverability of the dirt. Our scheme implemented random
ThisworkwaspartlysupportedbytheDRDO-IIScprogram
binninginasystematicmannerusingatrellisandusedViterbi
on Advanced Research in Mathematical Engineering, through
algorithm to quickly obtain a PIP-TCM codeword that is
a research grant to B.S. Rajan.
almostorthogonaltotherealizationofthedirt.Weempirically
demonstratedthatthisschemeperformsclosetoanequivalent
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