Table Of ContentProbabilistic Shaping and Non-Binary Codes
Joseph J. Boutros∗, Fanny Jardel†, and Cyril Me´asson‡
∗Texas A&M University, 23874 Doha, Qatar
†Nokia Bell Labs, 70435 Stuttgart, Germany
‡Nokia Bell Labs, 91620 Nozay, France
[email protected], fanny.jardel,cyril.measson @nokia.com
{ }
Abstract—Wegeneralizeprobabilisticamplitudeshaping(PAS) implementations of contemporary coding schemes that have
withbinarycodes[1]tothecaseofnon-binarycodesdefinedover been proven to asymptotically achieve capacity with constant
prime finite fields. Firstly, we introduce probabilistic shaping
complexity per information unit [17], [21]–[23], it is then
via time sharing where shaping applies to information symbols
natural to combine them with efficient shaping methods.
only.Then,wedesigncircularquadratureamplitudemodulations
(CQAM) that allow to directly generalize PAS to prime finite Inprobabilisticshaping,forlineardigitalmodulations,thea
fields with full shaping. prioriprobabilitydistributionofmodulationpointsismodified
7
1 to match a discrete Gaussian-like distribution, namely the
0 I. INTRODUCTION Maxwell-Boltzmann distribution [3]. The method aims at
2 Shaping refers to methods that adapt the signal distribu- maximizing the mutual information with respect to the same
n tion to a communication channel for increased transmission modulationwhereallpointsareequallylikely.Forspecial2m-
a efficiency. Shaping is eventually important for optimal infor- ASK and 2m-QAM constellations with linear binary codes,
J
mation transmissions [4] and various solutions starting with this method is equivalent to probabilistic amplitude shaping
7
non-linear mapping over asymmetric channel models towards (PAS) where uniformly-distributed parity bits are assigned to
2
pragmatic proposals involving shaped QAM signaling have the sign of a constellation point [1], [2]. In this paper, we
] been investigated and/or implemented over the years. generalize this method to the non-binary case. The goal is
T More precisely, building upon early works on, e.g., many- to permit the use of efficient non-binary codes in order to
I to-one mapping, research efforts from the 70s towards the enablelow-latencyprocessing(reducingthe needfor‘Turbo’-
.
s 90s derive conceptual frameworks and methods to reduce detection[17]).Also,fromanalgebraicviewpoint,acharacter-
c
[ the shaping gap in communication systems. Exploiting the isticp>2ofthefinitefieldFpm onwhichcodingisbuiltleads
principles of coded modulation, a sequence of works [5]– to new interesting problems such as distribution matching in
1
v [12] present operational methods to reduce the shaping gap. Fpm and assigning a constellation points to elements in Fpm.
76 iCnogmgpaairnedistoaccuhbieivcacbolnesuteslilnagtiownse,llu-apdtaoptπ6eed≈si1g.n5a3lidnBg.oSfismhapple- finIend tohviesrpFap,erw, hceordeesp aisreansuopdpdosperdimtoe.bFeirsltilnye,aerxacnepdt dfoer-
p
9 methods such as trellis shaping or shell mapping permit to codes with a sparse generator matrix, we show in Section II
7 recover a significant fraction of the 1.53dB figure. Exam- that parity symbols are asymptotically uniformly-distributed
0 ples of applications include the ITU V.34 modem standard over F . This fact is used to derive two new methods for
. p
1 recommendations that uses shell mapping to recover 0.8dB. probabilistic shaping. Time sharing is proposed in Section III
0
While several shaping schemes are based on the structural wheresymbolsofap-arycodearemappedintop-ASKpoints.
7
propertiesoflattices[13]–[15],morerandomizedschemesalso Hence,intimesharing,probabilisticshapingisperformedonly
1
: emerge after the re-discovery of probabilistic decoding in the when information symbols are transmitted. Full probabilistic
v
90s. With the adventof efficientbinarycodes,differentcoded shaping is described in Section IV where circular QAM
i
X modulationschemeswere proposedofferingflexible and low- (CQAM) constellations of size p2 points are introduced. This
r complexsolutions[17].Inthe2000s,despitetheimportantde- second method assigns a constellation shell to a Maxwell-
a
velopmentofwirelesscommunications,theneedforadvanced Boltzmann-distributed information symbol and then a parity
shaping methods seems to have remained marginal. From symbol selects a point within that shell. Numerical results
a technological viewpoint, this may have been justified by for p-ASK-based time sharing and p2-CQAM probabilistic
the high variations of the channel in wireless communication shapingareshowninSectionV.Similartothebinarycase[1],
networks. From an academic viewpoint, schemes have been a gap to channel capacity of 0.1 dB or less is observed for
analyzed and match the capacity-achieving distribution of a CQAM constellations.
channelindifferenttheoreticalscenarios[18]–[20].Inthe last
few years, industrialapplicationsofshapingmethodshavere- II. SUM OF RANDOMVARIABLES IN APRIME FIELD
gainedinterest.Thisconcernsareaswherecurrenttechnologies Lemma 4.1 in [16] established the expressions of the
operate close to fundamental limits. For example, different probability of a sum in F . We translate this result to a
2
methods have been experimented in optical communications prime field F = Z/pZ. Principles of this extension to the
p
[26], [27]. Hence, because there are already efficient VLSI non-binary case are implicit from Chapter 5 of [16] with
the use of the z-transform. Nevertheless, we give here the It remains to summarize this observation in a theorem.
exact expression of the probabilityof a sum of prime random Theorem 1: Let p be a prime and F = 0,1, ,p 1
p
{ ··· − }
symbols modulo p. This expression is directly related to be the associated Galois field. Consider a sequence s
ℓ ℓ≥1
Hartmann-Rudolph symbol-by-symbol probabilistic decoding of independent random symbols over F with re{spe}ctive
p
[24] in the special case of a single-parity check code and its probability distributions (q (0),q (1), ,q (p 1)) 1
ℓ ℓ ℓ ℓ
{ ··· − } ≥
generalization to characteristic p [25]. such that limsup max (q (p)) <1. Then
Lemma 1: Let p be a prime and F = 0,1, ,p 1 ℓ→∞{ p ℓ }
mbe itnhdeeapsesnodceianttedsyfimnbitoelsfioelvde.rCFonsinidewrhpaicsheqt{hueenℓc-eth·{·ss·yℓm}mℓb=−o1loi}sf ∀k ∈Fp, ml→im∞Pr{ m sℓ =k}= p1. (6)
p ℓ=1
β F with probability X
∈ p For error-correction over a prime field, this observation is
Pr sℓ =β =qℓ(β). interestingasfollows.ThelimittheoremoverFpindicatesthat
{ }
the non-systematic symbols obtained from a linear encoder
Then, for any k F , the probabilitythat the sum of the s ’s
∈ p ℓ associated with a dense generator matrix will tend to have a
equals k is
uniform distribution independently on the input distribution.
m 1+ p−1 m p−1q (β)ωiβ−k+1
Pr s =k = i=1 ℓ=1 β=0 ℓ , III. PROBABILISTIC SHAPING VIA TIMESHARING OVER
{ ℓ } P Q (cid:16)Pp (cid:17) PRIME FIELDS
ℓ=1
X (1) A common mapping between non-binary codes and non-
binarymodulationsistoselectaconstellationandafinitefield
where ω d=efexp(2π√ 1/p) indicates the p-th root of unity. of identical size. Let p be a prime integer, p > 2. Consider
−
Proof: Consider the enumerator function in t, the set of p points shown in Figure 1, known as p-ASK mod-
m p−1 ulation. This p-ASK set = p−1,..., 1,0,1,...,p−1
Q(t)d=ef qℓ(β)tβ mod (tp 1). (2) is isomorphicto the finiteAfield{F−p (2a ring is−omorphismin2Z)}.
− Symbols from F are one-to-one mapped into p-ASK points.
ℓY=1(cid:16)βX=0 (cid:17) p
We embed F into Z such that a symbol s F and its
Observethatif this isexpandedintoa polynomialin t (where p ∈ p
degreeoperationsaretakenmodp),thecoefficientoftk isthe corresponding point in A satisfy x−s=0 mod p.
probability that the sum of m symbols is k, which we write
m Fp={0,1,...,p−1}
Pr s =k =coef[Q(t),tk]. (3)
ℓ
{ }
one to onemapping
Xℓ=1 − −
Let us also define for any k 0,1,2, ,p 1 the
p-ASK
∈ { ··· − }
function
Q (t)=t−kQ(t) mod tp 1. (4) p−1 1 0 1 +p−1
k − − 2 − 2
In anidenticalmannerasforQ(t), expandingQk(t) would Fig. 1: Real p-ASK constellation isomorphic to F .
p
enumerate the probabilities of the sum of the s ’s. Recall
ℓ
that the p-th root of unity in the complex plane satisfies
There are many advantages for such a simple structure
p−1ωki = 0 for any k 1,2, ,p 1 . Then, for any
k i=F0 , we have ∈ { ··· − } where the source is p-ary, the linear code is over Fp, and
P∈ p p-ary modulation points are transmitted over the channel.
m p−1 Firstly, a probabilistic detector needs no conversion between
1
Pr sℓ =k = Qk(ωi) (5) modulation points and code symbols. A channel likelihood,
{ } p
ℓ=1 i=0 after normalization, is directly fed as a soft value to the
X X
because all but the constant polynomial terms are annihilated input of a probabilistic decoder. Secondly, turbo detection-
from the fact that the roots of unity sum to zero. It remains decoding between the constellation and the code C is not
A
to evaluate the expression observing that ω0 = 1 to get the requiredasforbinarycodeswithnon-binarymodulations[17].
results. (cid:3)
Remark 1: This lemma shows that, if a s is uniformly Consider a systematic linear code C over F where parity
ℓ p
distributedoverF ,thenthesumisalsouniformlydistributed. symbolssatisfyTheorem1,i.e.,checknodesusedforencoding
p
Remark 2: For any ℓ, it is straightforward from convexity have a relatively high degree. Many practical error-correcting
argumentsthatthe weightedsum p−1q (β)ωiβ in the com- codes do satisfy this property, such as LDPC codes over
β=0 ℓ
plex plane lies inside the unit circle in the strict sense if and F . Let R = k/n be the coding rate of C, where n is
p c
P
onlyifoneoftheprobabilitydistributionq isnotdegenerated the code length and k is the code dimension. Assume that
ℓ
inonesingularpoint.Therefore,assumingthatthenormtends information symbols s ,s ,...,s at the encoder input are
1 2 k
to be smaller and bounded away from 1, the distribution of identically distributed according to an a priori probability
the infinite sum tends to be uniform. distribution π p−1. Let P (x,ν) exp( ν x2) be a
{ i}i=0 MB ∝ − | |
discrete Maxwell-Boltzmann distribution [3] with parameter The key idea in our new coded modulation is to assign
ν 0. The a priori distribution π is taken to be the parity symbol to p modulation points with the same
i
≥ { }
amplitude. This is a directgeneralizationof the sign mapping
π =P (0,ν) 1, (7)
0 MB ∝ to p-ary mapping. The linear p-ary code is assumed to be
πi =πp−i =PMB(i,ν) exp( νi2), (8) systematic. Its information symbols become amplitude labels
∝ −
in the modulation. We propose a bi-dimensionalconstellation
fori=1...p−1.Theaverageenergyperpointforthep-ASK
2 withp2 points,referredtoasp2-circularquadratureamplitude
constellation, denoted by E , is given by
s modulation (p2-CQAM). A circle containing CQAM points
(p−1)/2 of the same amplitude will be called a shell. The p2-CQAM
E = P (x,ν) x2 = 2 π i2. (9) includespshellswithppointspershell.Suchabi-dimensional
s MB i
| |
x∈A i=1 constellation is not unique. Indeed, many ways do exist to
X X
build p shells and populate each shell with p points. As a
From Theorem 1 and (7)&(8), a fraction R of transmit-
c
consequence, we introduce a figure of merit for a constella-
ted ASK points corresponding to information symbols are
tion [6] and we build a specific p2-CQAM constellation that
Maxwell-Boltzmann-shaped and a fraction 1 R of ASK
c
− maximizes this figure of merit.
pointscorrespondingtoparitysymbolsisuniformlydistributed
Definition 1: Consider a finite discrete QAM constellation
in the constellation. We refer to this coding scheme as proba-
bilistic shapingviatime sharing.The targetrate shouldbe the A ⊂ C. Assume that x∈Ax = 0. Let Es = Px∈|AA||x|2 be
averageinformationrate(expressedinbitsperrealdimension) the average energy per point, assuming equiprobable points.
P
Rt =Rclog2(p)=RcI(Xs;Y)+(1−Rc)I(Xp;Y), (10) Lsqeutadre2EdmEinu(cAlid)e=anmdiisntaxn,xc′e∈Abe,xtw6=xe′e|nxt−hexp′o|2inbtseotfheA.mAinifimguumre
of merit for is defined by the following expression:
where the two random variablesXs,Xp satisfy Xs πi FM A
∈A ∼
and X 1/p. The random variable Y represents the output d2 ( )
ofareapl∼additivewhiteGaussiannoisechannel,whereadditive FM(A)= EmEin A ·log2(|A|). (13)
s
noise has variance σ2 = N0. For a given target rate R ,
2 t Thelog ( )factorisarbitrary,itisusedintheabovedefini-
the Maxwell-Boltzmann parameter ν is chosen such that the 2 |A|
tiontonormalizethesquaredminimumEuclideandistanceby
signal-to-noise ratio γ = E /N attaining R is minimized.
s 0 t bit energy instead of point energy. This may be useful when
Let γ be that minimum. We also define two signal-to-noise
A comparing two constellations of different sizes.
ratios γcap and γunif such that Now,we builda p2-CQAM constellation thatmaximizes
A
1 ( ) by populating the p shells as follows:
M
R = log(1+2γ ), (11) F A
t 2 cap 1) For the first shell, draw p uniformly-spaced points on
and the unit circle. The points are xℓ = exp(ℓ2pπ√−1),
for ℓ = 0...p 1. Here, we impose the constellation
R =I(X ;Y), for γ =γ . (12)
t p unif −
minimum distance to be the distance between two con-
Then, the gap to capacity and the shaping gain (expressed in secutive points of the first shell,
decibels) are respectively given by γA(dB) γcap(dB) and π
− d ( )=2sin( ). (14)
γ (dB) γ (dB).Inthistimesharingscheme,probabilis- Emin
unif A A p
−
tic shaping is made only duringa fraction R of transmission
c 2) Assume that shells 1 to i 1 are already built. Let
time. This method is attractive due to isomorphism between −
x = ρ exp(φ √ 1) be the first point of the i-th
the fieldFp andthe p-ASKconstellation.From(10), onemay shipell. Thei p 1i r−emaining points on this shell are
quickly conclude that high coding rate is recommended to −
x = ρ exp((φ + ℓ2π)√ 1), ℓ = 1,...p 1.
approachfull-timeprobabilisticshaping.However,atRc close Liept+dℓ2 = mi in i xp −x 2 be the minim−um
to 1,the mutualinformationI(X;Y), forX , approaches i ℓ=0...ip−1| ip − ℓ|
∈A distance between the first point of the current shell and
its asymptote log ( ) = log (p) and the required signal-to-
2 A 2 all previously constructed points. The radius ρ and the
noise ratioγ goesfar awayfromγ . Thisis clearlyshown i
A cap
phase shift φ are determinedby an incrementalsearch:
in the numericalresults presentedin Section V. A methodfor i
full probabilistic shaping is proposed in the next section. • Start with ρi =ρi−1 and increment by a step ∆ρ.
• At each radius increment, vary φi from π/p to
IV. PROBABILISTIC SHAPING VIAp2-CIRCULAR QAM π/p.
−
OVERPRIME FIELDS • Stop incrementing the radius ρi and the phase shift
φ when d2 d2 ( ). Now, x is found.
We proposein this section a codedmodulationscheme that i i ≥ Emin A ip
allows full probabilistic shaping of all transmitted symbols 3) Repeat the second construction step until completing
withanon-binarylinearcodeoverF .Probabilisticamplitude the p-th shell of the p2-CQAM constellation.
p
shaping with binary codes maps uniformly-distributed parity
bits into the sign of an ASK point [1]. This sign mapping is Thep2-CQAMobtainedwiththeconstructiondescribedabove
not valid with a prime finite field F , p>2. has the circular symmetry required by PAS over F .
p p
3 3 3
2 2 2
1 1 1
0 0 0
-1 -1 -1
-2 -2 -2
-3 -3 -3
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
Fig. 2: Bi-dimensional p2-CQAM constellation for p=5,7,11 from left to right.
Examples of circular QAM modulations for probabilistic Given the a priori distribution π p−1 of symbols in the
amplitude shaping are shown in Figure 2, for p = 5,7,11 finite field F , the linear code C{[ni},ik=]0 and the p2-CQAM
p p
respectively. Points are drawn as small circles in red. Blue constellation should be combined together as illustrated in
segments connect points located at minimum Euclidean Figure 3. Suppose that R = 1/2 and say that s F
c 1 p
distance. By the given construction, the inner radius of the is an information symbol (encoder input) and p F∈is a
1 p
∈
p2-CQAM is ρ = 1, p. The outer radius ρ varies paritysymbol.Then,s shouldbebeshapedbythedistribution
in out 1
slightly with p but lim ∀ρ =ρ ( ) 3.6. This limit matcher (DM) according to π p−1 [2]. The symbol s will
p→∞ out out ∞ ≈ { i}i=0 1
exists because the sequence ρ (p) is increasing with p and be used to select a shell in the p2-CQAM and the parity
out
bounded from above by 1 + (p 1)d ( ) 1 + 2π. symbolp willuniformlyselectapointonthatshell.Similarly,
Emin 1
− A ≤
The limitation of the Maxwell-Boltzmann probability mass supposethatR =2/3andconsiderfourinformationsymbols
c
function to amplitudes between ρ = 1 and ρ ( ) s ,s ,s ,s and two parity symbols p ,p . The DM shall
in out 1 2 3 4 1 2
is a major drawback. This short interval [1,ρ ( )[∞is generate s ,s ,s according to the distribution π p−1 and
out ∞ 1 2 3 { i}i=0
shifted away from the origin and is not large enough to select the shell of three p2-CQAM points. The symbol s
4
yield a good Gaussian-like discrete distribution. In the is read directly from a uniform i.i.d p-ary source. Given the
next section, the shells radii are modified to get a wider shellsofthreepoints,uniformly-distributedsymbolss ,p ,p
4 1 2
amplituderange,thep2-CQAMphaseshiftsarekeptinvariant. constitute the pointsindicesinside those shells. In the general
case,n/2symbolsinF withprobabilitydistribution π are
p i
{ }
Lets ,s ,...,s bei.i.d.informationsymbolswithapriori read from the DM and mapped into a shell number for n/2
1 2 k
probability distribution π p−1, as in the previous section. CQAM points. The probabilistic shaping is due to these n/2
{ i}i=0
Then, for points x , i,ℓ = 0...p 1, the prior symbols On the other hand, k n/2 uniformly-distributed
ip+ℓ
∈ A − −
distribution becomes symbols are directly read from the source. Together with
n k parity symbols, i.e., a total of n/2 symbols, uniformly-
π P (x ,ν)
π(xip+ℓ)= pi = MB |pip| . (15) di−stributed symbols in Fp determine the phase of CQAM
points within constellation shells.
In presence of the above distribution, the signal-to-noise
ratio should be defined with an average energy per point
Etpu2sa-lC=iQnPfAoMrpim=−01faatπicoiil|nix.taiTpte|h2se.tFhgueernntehuremarlmeeroxicrpear,leteshvseiaolcnuiracotiufolnIar(oXsfy;amYvme)reawtgrieythmofpu2a- DM s1,...,sn2Encoder select psMh2el-alCppQerAM n2 points
integraltermsreducestoptermsonly.Themutualinformation sn2+1,...,sk overFp p1,...,pn−k Merge sweiltehcitn p sohienltl
I(X;Y) is given by
Fig. 3: Full probabilistic amplitude shaping with p2-CQAM.
p−1
p(y x )
ip
π p(y x )log | dy.
Xi=0 iZy∈C | ip 2 ℓp=2−01π(xℓ)p(y|xℓ)! Ourp-arycodedmodulationsuitedforprobabilisticshaping
The Maxwell-Boltzmann paramPeter ν in (15) is chosen such assumes that Rc 1/2, i.e., k n/2. Coding rates in the
≥ ≥
that 2R = 2R log (p) = I(X;Y) at a minimal value of interval [0,1/2[ are less attractive for probabilistic amplitude
t c 2
signal-to-noiseratio E /N =γ , where R is the targetrate shaping because, for small R , a constellation with equiprob-
s 0 A t c
per real dimension. The gap to capacity is determined by the able points already shows a rate that is too close to channel
differenceγ γ withγ satisfying2R =log (1+γ ). capacity in terms of signal-to-noise ratio.
A− cap cap t 2 cap
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