Shaping Bits

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by  Albert Guillen i Fabregas

Shaping Bits
Albert Guille´n i Fa`bregas Alfonso Martinez
Engineering Department Centrum Wiskunde & Informatica
University of Cambridge, UK Amsterdam, The Netherlands
[email protected] [email protected]
Abstract—The performance of bit-interleaved coded modula- II. PRELIMINARIES
tion(BICM)withbitshaping(i.e.,non-equiprobablebitprobabil-
A. Blockwise Coded Transmission
ities in the underlying binary code) is studied. For the Gaussian
channel, the rates achievable with BICM and bit shaping are Consider a memoryless channel with input X and output
0 practically identical to those of coded modulation or multilevel Y, respectively belonging to the sets X and Y. A block
1 coding.Thisidentityholdsforthewholerangeofvaluesofsignal-
code M ⊆ XN is a set of |M| vectors (or codewords)
0 to-noise ratio. Moreover, the random coding error exponent of
2 BICM significantly exceeds that of multilevel coding and is very x of length N (the number of channel uses), i. e. x =
n close to that of coded modulation. (x1,...,xN) ∈ XN. The output y =∆ (y1,...,yN) is a
a random transformation of the input with transition probability
J distribution P (y|x). For memoryless channels the distri-
Y|X
8 I. INTRODUCTION bution P (y|x) admits the decomposition
Y|X
1
N
For non-binary modulations in the Gaussian channel, three (cid:89)
] P (y|x)= P (y |x ) (1)
T main constructions for coding schemes achieve information Y|X Y|X k k
I rates close to the channel capacity are known: coded mod- k=1
s. ulation (CM), bit-interleaved coded modulation (BICM), and With no loss of generality, we limit our attention to contin-
c multilevel coding (MLC). CM dates back to the pioneering uous output and identify PY|X(y|x) as a probability density
[
workofUngerbo¨ck[1],andmergescodingandmodulationin function. We adopt the convention that capital letters repre-

1 a single entity. In contrast, BICM separates them, and is built sent random variables, while the corresponding small letters
v aroundthemappingofasimplebinarycodeontoanon-binary correspond to realizations of the variables.
6
modulation [2], [3], [4]. MLC makes use of a layer of binary At the source, a message m drawn with equal probability
3
0 codes, one for each bit in the binary label of the modulation from a message set M is mapped onto a codeword x. We
3 symbol [5], [6]. denote this encoding function by φ, i. e. φ(m) = x. The
1. CM allows for the highest information rates. It is closely correspondingtransmissionrateRisgivenbyR=∆ N1 log|M|.
0 followed by multilevel coding (for equiprobable modulation Atthereceiver,thedecoderdeterminesthecodeworddecoding
0 symbols the rates coincide) and, with a larger loss, by BICM. metric,denotedbyq(x,y),forallcodewords,andoutputsthe
1 Intermsoferrorexponents,thesituationissomewhatreversed, message m(cid:98) whose metric is largest,
:
v with CM again the best, but now BICM beats multilevel
m= argmax q(φ(m),y). (2)
i codingatlowrates.Whereaspreviousanalysesintheliterature (cid:98)
X m∈{1,...,|M|}
assume that the modulation symbols are used equiprobably,
r The metrics we consider are products of symbol decoding
a in this work we lift this assumption and consider shaping,
metrics q(x,y), namely (with some abuse of notation)
whereby the bit or symbol probabilities are arbitrary. We
will see that BICM with shaping achieves both information N
(cid:89)
rates and error exponents very close to those of CM, thus q(x,y)= q(x ,y ). (3)
k k
closing the gap which made multilevel coding better in terms k=1
of information rates. For maximum likelihood (ML) decoders, the decoding
This paper is organized as follows. In Sect. II we introduce metric is given by q(x,y) = P (y|x). More generally, a
Y|X
the necessary concepts and notation describing the various decoder finds the most likely codeword as long as the metric
schemes. In Sect. III we derive the achievable rates for BICM q(x,y)isastrictlyincreasingbijectivefunctionofthechannel
with shaping by using mismatched decoding theory. These transitionprobabilityP (y|x).Instead,ifthemetricq(x,y)
Y|X
resultsareparticularizedfortheGaussianchannelinSect.IV, isnotabijectivefunctionofthechanneltransitionprobability,
which also includes some numerical results. we have a mismatched decoder [7], [8].
Of special interest is the random ensemble correspond-
ing to CM, for which 1) the channel inputs are selected
1ThisworkhasbeensupportedbytheInternationalJointProject2008/R2
oftheRoyalSociety. independently for each codeword component according to a
probabilitydistributionP (x),and2)thedecoderusestheML as if the m bits in a symbol were independent, i.e.,
X
metric. In this case, and for practical reasons, the modulation m
set X is taken finite. Let M =∆ |X| denote the cardinality of qbicm(x,y)= (cid:89)qj(cid:0)bj(x),y(cid:1). (10)
X and m =∆ log M the number of bits required to index a j=1
2
symbol.Thelargestinformationratethatcanbeachievedwith Hence, the BICM receiver uses the following symbol metric
CM under the constraint x∈X is is m (cid:32) m (cid:33)
(cid:89) (cid:88) (cid:89)
Ccm = sup I(X;Y). (4) q(x,y)= PY|X(y|x(cid:48)) PBj(cid:48)(bj(cid:48)(x(cid:48))) .
j=1 x(cid:48)∈Xj j(cid:48)=1
PX(X) bj(x)
(11)
Moreover, for any input distribution P (X), the block error
X
C. Multilevel Coding
probability P satisfies [9]
e
Multilevelcodes(MLC)combinedwithmultistagedecoding
P¯e ≤e−NEr(R) (5) (MSD) have been proposed [5], [6] as an efficient method
to attain the channel capacity by using binary codes. For
where E (R)=sup E (ρ)−ρR and
r 0≤ρ≤1 0 BICM, a single binary code C is used to generate a binary
E (ρ)=∆ −logE(cid:34)(cid:32)(cid:88)P (x(cid:48))(cid:18)PY|X(Y|x(cid:48))(cid:19)1+1ρ(cid:33)ρ(cid:35). cboindaerwyolradb,ewlinhgicfhunisctuiosnedµt.oInseMlecLtCm, othdeuliantpiount bsiynmarbyolcsodbeyCa
0 x(cid:48) X PY|X(Y|X) is the Cartesian product of m binary codes of length N, one
(6) per modulation level, i. e. C = C ×...×C , and the input
1 m
The expectation is carried out according to the joint distribu- distribution for the symbol x(b ,...,b ) has the form
1 j
tion P (x,y)=P (y|x)P (x).
X,Y Y|X X m
(cid:89)
Pmlc(x)=P (b ,...,b )= P (b ). (12)
B. Bit-Interleaved Coded Modulation X B1,...,BM 1 m Bj j
j=1
In practical CM schemes, since the codewords are selected For a fixed input distribution on the bits, MLC achieve the
elements of XN and the alphabet X has typically more than mutual information [5], [6] both with ML joint decoding and
2 elements, the corresponding codes are in some sense non- withmultistagedecoding.Thelargestinformationratethatcan
binary. BICM is a different construction where the underlying be achieved with MLC under the constraint x∈X is
codeisbinary.Originallyanalyzedin[3]undertheassumption
Cmlc = sup I(X;Y). (13)
of infinite-depth interleaving, this restriction was recently
liftedin[4],[10],whereitwasshownthatBICMhasanatural PB1(B1),...,PBm(Bm)
The error exponents of MLC with multistage decoding were
description in terms of mismatched decoding.
derived in [11], [12], [4], [13], where it was also shown the
The BICM encoder generates a vector of mN bits, b =
error exponent is upper bounded by one.
(b ,...,b ), i. e. φ(m) = b. This vector is mapped onto a
1 mN
vector of N modulation symbols according to a labeling rule III. ACHIEVABLERATESWITHBICM
µ:Fm2 →X, such that FortheBICMschemedescribedabove,itwasshownin[4],
(cid:0) (cid:1) [10] that the rate
x =µ b ,...,b , k =1,...,N. (7)
k (k−1)m+1 (k−1)m+1 (cid:32) (cid:34) (cid:81)m q (cid:0)b (X),Y(cid:1)s (cid:35)(cid:33)
Nbeoetneathbasotrtbheediinntetrhliesavdeerscwrihpitciohngoivfethseitesnncoamdeer.tAonBalIoCgMoushlays, Rgmi =ssu>p0 E log M1 (cid:80)xj(cid:48)=(cid:81)1mj=j1qjj(cid:0)bj(x(cid:48)),Y(cid:1)s ,
(14)
we denote the inverse labeling by b , so that b (x) is the j-
j j also named generalized mutual information (GMI), is achiev-
th bit in the binary label of modulation symbol x, for j = able with equiprobable bits, P (b) = 1. The proof is based
1,...,m.Byconstruction,themodulationsymbolsxareused Bj 2
on a simple extension of Gallager’s analysis of ML decoding
with probabilities
in terms of error exponents to mismatched decoding [7], [8].
m References [4], [10] also show that the above rate may be
PXbicm(x)= (cid:89)PBj(cid:0)bj(x)(cid:1). (8) decomposed as the sum of m bit GMI terms, and that it
j=1 coincides with the BICM capacity defined in [3]. The next
result generalizes this result for arbitrary bit probabilities.
In addition to the different code construction, BICM also
Theorem 1: The generalized mutual information of the
differs from CM at the receiver side. First, let us define the
BICM mismatched decoder is equal to the sum of the gen-
sets Xj as those elements of X having bit b in the j-th label
b eralized mutual informations of m binary-input channels,
position, i.e., Xj =∆ {x∈X :b (x)=b}. The BICM symbol
b j  (cid:34) (cid:35)
metricqco(mbb(ixn)es=thbe,ym)=bit(cid:88)metriPcs qj((byj|,xy(cid:48)))Pgbivicemn(xb(cid:48)y), (9) Rgmi =ssu>p0(cid:88)jm=1E log(cid:80)1b(cid:48)=0qqjj((bB(cid:48)jj,,YY))ssPBj(b(cid:48)j) .
j j Y|X X
(15)
x(cid:48)∈Xj
b
The expectation is carried out according to the joint distribu- where the expectation is done according to the joint dis-
tion P (b )P (y|b ), with tribution Pbicm(x)P (y|x). This gives Eq. (17) since the
Bj j j j X Y|X
generalized mutual information is the supremum over all s
P (y|b)=∆ (cid:88) PY|X(y|x)PXbicm(x). (16) [7], [8]. As for Eq. (17), we derive it by noting that, for each
j x∈Xj (cid:80)x(cid:48)∈XbjPXbicm(x(cid:48)) j, the summation over x in the expectation can be split into
b two parts and rearranged as follows,
An alternative expression is
(cid:88) (cid:88) (cid:88)
Rgmi =ssu>p0(cid:88)jm=1E(cid:34)log(cid:80)1b(cid:48)=q0jq(cid:0)jb(jb((cid:48)jX,Y),)YsP(cid:1)Bsj(b(cid:48)j)(cid:35), x f(x)==bj∈(cid:88){0,1}Px∈Xbj(bf()x(cid:88)) f(x) . ((2256))
(17) Bj j P (b )
bj∈{0,1} x∈Xbj Bj j
where the expectation is done according to the joint distribu-
tion PXbicm(x)PY|X(y|x). As PBj(bj)=(cid:80)x(cid:48)∈XbjPXbicm(x(cid:48)) by construction, recovering
Proof: For fixed s and probabilities Pbicm(x) = the expression of f(x) we obtain Pj(y|bj) in Eq. (16).
(cid:81)m P (cid:0)b (x)(cid:1) the GMI can be written as X The following result applies to BICM with the decoding
j=1 Bj j
metric given in Eq. (9).
R (s)=E(cid:20)log qbicm(X,Y)s (cid:21) (18) Corollary 1: For the classical BICM decoder with metric
gmi (cid:80) qbicm(x(cid:48),Y)sPbicm(x(cid:48)) in Eq. (9) the supremum over s is achieved at s = 1, and
=E(cid:34)log(cid:80)x(cid:48) (cid:81)m(cid:81)mjq=(cid:0)1bqj((cid:0)xb(cid:48)j)X(,XY)(cid:1),sYP(cid:1)s(cid:0)b (x(cid:48))(cid:1)(cid:35), RgmiP=ro(cid:80)of:mj=S1iInc(Bejt;hYe).metric qj(bj,y) is proportional to
x(cid:48) j=1 j j Bj j Pj(y|bj), we can identify the quantity
(19)
 
(cid:0) (cid:1)s
where the expectation is carried out according to Elog qj Bj,Y  (27)
PXbWicme(nxo)wPY|hXav(ye|xa).closer look at the denominator in the (cid:80)1b(cid:48)j=0qj(b(cid:48)j,Y)sPBj(b(cid:48)j)
logarithm of (19). The key observation is that the sum over as the generalized mutual information of a matched binary-
the constellation points (x(cid:48) ∈X) of the product of a function input channel with transitions Pj(y|bj). Then, the supremum
f(cid:0)b (x(cid:48))(cid:1) evaluated at all the binary label positions admits an over s is achieved at s = 1 (that is, the mutual information
j
alternative expression, namely I(B ;Y)) and we get the desired result.
j
(cid:32) (cid:33) (cid:32) (cid:33) In the remainder of the paper, for the sake of simplicity and
m m
(cid:88) (cid:89)f(cid:0)b (x(cid:48))(cid:1) = (cid:89) (cid:88) f(b ) . (20) without loss of generality, we focus on the classical BICM
j j
metric. Clearly, the methods and results we present generalize
x(cid:48)∈X j=1 j=1 bj∈{0,1}
to other metrics, in which case, s should also be optimized.
Indeed, after carrying out the product in the right-hand side,
The above results suggest that we can chose the input
we obtain the desired sum over all 2m binary m-tuples
bit distribution that yields the largest GMI, i.e., effectively
(b ,...,b ) of summands of the form f(b )···f(b ).
1 m 1 m implying shaping the bit probabilities in BICM as
Therefore, for the specific choice f(cid:0)b (x(cid:48))(cid:1) =
j
q (cid:0)b (x(cid:48)),Y(cid:1)sP (cid:0)b (x(cid:48))(cid:1) we have the product over all (cid:88)m
j j Bj j Cbicm = sup I(B ;Y). (28)
label positions of the sum of the probabilities of the bit b j
j PB1(B1),...,PBm(Bm)j=1
being zero and one, i.e.,
  For iid codebooks, Cbicm is also the largest rate that can be
m
(cid:88) (cid:89)qj(cid:0)bj(x(cid:48)),Y(cid:1)sPBj(bj(x)) (21) traTnshmisittceadpawciitthyvsahnoisuhldingbeerrcoormpproabreadbiliwtyith[14th].e equivalent
x(cid:48)∈X j=1
quantities on CM and multi-level coding, given in Eqs. (4)
 
m and (13) respectively,
(cid:89) (cid:88)
=  qj(bj,Y)sPBj(b(cid:48)j). (22) Ccm = sup I(X;Y), (29)
j=1 b(cid:48)∈{0,1}
PX(X)
Next, going back to (19), we obtain
Cmlc = sup I(X;Y). (30)
  
Rgmi(s)=Elog(cid:89)m (cid:80)1 qjq(cid:0)b(jb((cid:48)X,Y),)YsP(cid:1)s (b(cid:48)), Note that BICM difPfeBr1s(Bf1ro),m...,PCBMm(Binm)the transmitter, where
j=1 b(cid:48)=0 j j Bj j the modulation symbol probabilities have the specific form
(23) Pbicm(x) = (cid:81)m P (cid:0)b (x)(cid:1), and the receiver, where the
=(cid:88)m E(cid:34)log(cid:80)1 qjq(cid:0)b(jb((cid:48)X,Y),)YsP(cid:1)s (b(cid:48))(cid:35), (24) syXmInbotelrmmsetorifcthinje=Er1aqn.dB(o1jm1)jciosduinsegdefrororrdeexcpoodninegn.t, the analysis
j=1 b(cid:48)=0 j j Bj j in[4],[10]canbemergedwiththepreviousprooftoshowthat
for any input distribution P (x), the block error probability 5
X
P is upper bounded by
e
P¯e ≤e−NErq(R) (31) 4
where Eq(R)=sup Eq(ρ,s)−ρR, and
r 0≤ρ≤1 0
s>0
Eq(ρ,s)=∆ −logE(cid:34)(cid:32)(cid:88)P (x(cid:48))(cid:18)qbicm(x(cid:48),Y)(cid:19)s(cid:33)ρ(cid:35) 3
0 X qbicm(X,Y) C
x(cid:48)
(32) 2 0.45 dB
is a generalized Gallager function. The expectation is carried
out according to the joint distribution P (y|x)P (x).
Y|X X
1
IV. BITSHAPINGFORTHEGAUSSIANCHANNEL
A. Channel Model
1 dB
Weconsiderthetransmissionovercomplex-planesignalsets 0
−2 0 2 4 6 8 10 12
(X ⊂ C, Y = C) in the AWGN channel. It is a memoryless
Eb (dB)
channel satisfying N0
√
Y = snrX +Z , k =1,...,N (33) Fig.1. CapacitiesforGaussianinputs(thicksolid),CM/MLCwithshaping
k k k (thinsolid),BICMwithshaping(dashed)andBICMwithequiprobableinputs
(dottedline)for16-QAMwithGraymappingandbitshapingasafunction
where Z are zero-mean, unit-variance, circularly symmetric
k of Eb (dB).
complexGaussiansamples,andsnristhesignal-to-noiseratio N0
(SNR).WewishtosolvetheoptimizationproblemsinEqs.(4),
resultinthebestperformance,i.e.,MLCisoptimalandBICM,
(13) and (28) with the additional constraints that x ∈ X,
E[X]=0, and E[|X|2]=1. as we shall see, is very close. However, for m > 4 this is
no longer true and the optimization over symbol probabilities
We consider binary reflected Gray mapping2. For shaping,
withoutrestrictionP (x)tobetheproductofbitprobabilities
2m-QAM signal sets are of special interest; this constellation X
is the Cartesian product of two 2m2 -PAM constellations, one could potentially yield larger rates.
for each of the in-phase and quadrature components of the
B. Numerical Examples
channel. Since the optimum input distribution is known to
be Gaussian, a good input distribution over the set X should Figure 1 shows the improvement in BICM capacity de-
approach in some sense a Gaussian density. Symmetry be- rived from shaping for 16-QAM with binary reflected Gray
tween the in-phase and quadrature components and along the mapping. As we observe Cbicm (dashed) is almost indistin-
zero axis (so that the positive and negative plane have equal guishable from Ccm or Cmlc (thin solid) or channel capacity
probability) dictate that the optimization problems in Eqs. (4) itself (thick solid). This shows that shaping bits for BICM
and (13) respectively have can recover the BICM capacity loss for equiprobable bits and
• 2m2−1−1 free parameters for CM, and effectively close the gap with CM and MLC. Remark that the
BICM demodulator is a one-shot non-iterative demodulator.
• m −1 free parameters for BICM and MLC.
2 In general, the decoding complexity of BICM is larger than
For BICM we used the symmetries of binary reflected Gray
that of MLC, since the codes of MLC are shorter. In practice,
mapping and the fact that the most significant bit selects the
however, if the decoding complexity grows linearly with the
positive or negative half-plane, and always has probability 1.
2 number of bits in a a codeword, e. g. with LDPC or turbo
Note that the CM optimization problem does not restrict
codes, the overall complexity of BICM becomes comparable
theinputdistributiontobeP (x)=(cid:81)m P (b (x)),hence
X j=1 Bj j to that of MLC.
being able to achieve potentially larger rates. As we shall
Figure 2 shows the error exponents for CM and BICM,
see, the resulting difference in information rates is however
with and without shaping, for 16-QAM at snr=8 dB. When
marginal. Moreover, note that there is an exponential relation-
shaping is used, the input distribution is the corresponding
ship between the number of free parameters for BICM and
optimalcapacityachievingdistribution.Weobservethatwhen
CM, which can induce rather large computational savings for
shaping is used, in the region near capacity, the overall BICM
largesignalsets.Forexample,sincefor16-QAMthereisonly
error exponent is very close to that of CM, while when
one free parameter for MLC and CM, the optimization will
equiprobable bits are used, the exponent deviates from that
of CM. Remark that, according to [4], [10] the BICM error
2RecallthatthebinaryreflectedGraymappingformbitsmaybegenerated
recursively from themapping for m−1 bits by prefixinga binary 0 to the exponent cannot be larger than that of CM, as opposed to
mappingform−1bits,thenprefixingabinary1tothereflected(i.e.listedin that of the independent parallel channel model. Furthermore,
reverseorder)mappingform−1bits.ForQAMmodulationsintheGaussian
note that the error exponent of BICM is much larger than
channel,thesymbolmappingistheCartesianproductofGraymappingsover
thein-phaseandquadraturePAMcomponents. that of MLC (always being given by the minimum of the
0.25 Theorem 2: ShapingmakesBICMtransmissionoverQAM
modulations with binary reflected Gray mapping first- and
second-order optimal, i.e., c =1 and c =−1.
1 2 2
0.2 The key fact is that the bit probabilities are such that
a QPSK constellation is effectively selected. To see how,
note that for m = 2 we have QPSK with Gray mapping.
0.15
Limitingourselvestoonedimension,thebinaryreflectedGray
)
(R mapping for m +1 bits is constructed from the mapping for
r 2
E m bits by prefixing a binary 0 to the mapping for m − 1
0.1 2 2
bits, then prefixing a binary 1 to the reflected (i. e. listed in
reverse order) mapping for m−1 bits. With shaping, one has
2
the flexibility to fix each of this additional bits to a given
0.05
value, say, 0, so that one is effectively transmitting over a
Equiprobable Shaping BPSK constellation (QPSK over the two quadratures) when
the resulting constellation is normalized in mean and energy.
0
2 2.2 2.4 2.6 2.8 3 Note that this is property does not necessarily hold for other
R
mapping rules.
Fig. 2. Error exponent zoom at capacity for 16-QAM with and without
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