Zero-Delay Joint Source-Channel Coding for a Multivariate Gaussian on a Gaussian MAC

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by  Pål Anders Floor

1
Zero-Delay Joint Source-Channel Coding for a
Multivariate Gaussian on a Gaussian MAC
Pa˚l Anders Floor, Anna N. Kim, Tor A. Ramstad, Ilangko Balasingham, Niklas Wernersson, Mikael Skoglund,
Abstract—In this paper, communication of a Multivariate optimalityofuncodedtransmissionisrestricted(tolowSNRin
Gaussian over a Gaussian Multiple Access Channel is studied. general),and infinite complexityanddelayJSCC are required
Distributed zero-delay joint source-channel coding (JSCC) so-
to get close to the distortion lower bound in general. The
4 lutions to the problem are given. Both nonlinear and linear
bivariatecase wasalso treatedin [3], wherea zero-delay(sin-
1 approachesarediscussed.Theperformanceupperbound(signal-
0 to-distortion ratio) for arbitrary code length is also derived and gle letter) constraint was imposed in order to provide simple
2 Zero-delay cooperative JSCC is briefly addressed in order to low complexity and possibly implementable solutions to the
provide an approximate bound on the performance of zero- problem. Two zero-delay nonlinear schemes were proposed,
n delay schemes. The main contribution is a nonlinear hybrid
one discrete scheme and one hybrid discrete-analog scheme.
a discrete-analog JSSC scheme based on distributed quantization
J Both schemes perform well and can improve significantly
and a linear continuous mapping named Distributed Quantizer

1 Linear Coder (DQLC). The DQLC has promising performance on uncoded transmission when the channel signal-to-noise
1 whichimproveswithincreasingcorrelation,andisrobustagainst ratio (SNR) is sufficiently large. Nevertheless, there remains
variations in noise level. The DQLC exhibits a constant gap a constant gap to the distortion lower bound for all SNR due
] to the performance upper bound as the signal-to-noise ratio
T to the zero-delay constraint.
(SNR) becomes large for any number of sources and values of
I correlation. Therefore it outperforms a linear solution (uncoded Themaincontributionofthispaperistoprovideanonlinear
.
s transmission) in any case when the SNR gets sufficiently large. zero-delay JSCC solution for the multivariate version of case
c 2), since there to our knowledge exists few, if any, results
[ Index Terms—Zero-delay joint source-channel coding, multi-
variate Gaussian, Gaussian multiple access channel on this in the literature. We generalize the hybrid discrete-
1 analog scheme from [3] (named SQLC) to the multivariate
v case. The proposed scheme is named DQLC since it consists
8 I. INTRODUCTION
of distributed quantizers and a linear continuous mapping.
6 In this paper we investigate joint source-channel coding
The main advantage of DQLC is that it provides a simple
5
(JSCC) for a multipoint-to-point problem, where multiple
2 JSCCsolutiontotheproblemthatcansignificantlyoutperform
memoryless and inter-correlated Gaussian sources are trans-
. uncoded transmission under sufficiently large channel SNR.
1
mitted distributedly over a memoryless Gaussian multiple ac-
0 To assess the performance of DQLC, the distortion lower
cesschannel(GMAC)withoutmultiplexing.Therearemainly
4 bound for the problem under consideration is derived. Since
1 two cases to consider for this network: 1) Recovery of the this lower bound assumes arbitrary codelength, one must ex-
: common information shared by all sources. 2) Recovery of
v pecta significantbackofffromitwhen assessing performance
each individual source. In Case 1), when the transmit power
i of a zero-delay JSCC scheme. To provide indications on the
X of all sources are equal, the distortion lower bound can be
boundforzero-delaydistributedschemes,andtherebyabetter
r achievedbyasimplezero-delaylinearmapping,oftenreferred
a measure on how well the DQLC perform, known zero-delay
to as uncoded transmission [1].
schemes with cooperative encoders1 are addressed. It is also
In case 2) both common information as well as the in-
shown that DQLC exhibits a constant gap to the distortion
dividual variations of each source are reconstructed at the
lower bound (or performance upper bound, the bound on
receiver.Thebivariatecase(twosources)wastreatedin[2]for
signal-to-distortion ratio) as SNR . This gap increases
arbitrary codeword length. The authors proposed a nonlinear → ∞
somewhatwiththenumberofofsources,butremainsbounded
hybridJSCCschemethatsuperimposesarateoptimal(infinite
as the number of sources becomes large. Since uncoded
dimensional)vectorquantizerwithuncodedtransmission.This
transmissionhasalevelingoff effectathighSNR,DQLCwill
hybrid scheme was shown to achieve the distortion lower
thereforeoutperformuncodedtransmission foranynumberof
bound at high and low channel signal-to-noise ratio (SNR),
sources when the SNR is sufficiently large.
whileasmallgapremainsforotherSNR.Contrarytocase1),
The paper is organizedas follows: In Section II, a problem
P. A. Floor and I. Balasingham are with the Interventional Center, Oslo formulationisgivenandthedistortionlowerboundisderived.
University Hospital and Institute of Clinical Medicine, University of Oslo, Zero-delaycooperativeencodinganduncodedtransmissionare
Oslo, Norway (e-mail: andfl[email protected]). A. N. Kim is with SINTEF
alsointroduced.InSectionIIItheDQLCisintroducedandits
ICT,Oslo,Norway(e-mail:[email protected]).P.A.Floor,T.A.RamstadandI.
BalasinghamarewiththeDepartmentofElectronicsandTelecommunication, distortion is derived mathematically. It is further shown that
NTNU,Trondheim,Norway. DQLC exhibits a constant gap to distortion lower bound as
NiklasWernerssoniswithEricssonResearch,Stockholm,Sweden.Mikael
Skoglund is with the School of Electrical Engineering and the ACCESS
Linnaeus Center, at the Royal Institute of Technology (KTH), Stockholm, 1By cooperation we mean that all source symbols are available at all
Sweden encoders withoutanyadditional useofresources.
2
SNR .InSection IVwe concentrateonthe3 sourcecase an ideal Nyquist channel, where the sampling rate of each
→∞
and optimize the DQLC for general SNR. Its performance is sourceisthesameasthesignallingrateofthechannel.Wealso
comparedto the derivedboundsand uncodedtransmission. A assume ideal synchronization and timing between all nodes.
summary and brief discussion are given in Section V. Our design objective is to find the f and g that minimizes
m m
Note that some of the results in this paper have previously D. The DQLC encoders are asymmetric, where P P
1 2
≥ ≥
been published in [4]. P . Therefore, an average transmit power constraint
M
··· ≥
P =(P +P + P )/M is considered.
1 2 M
···
II. PROBLEM STATEMENTAND BOUNDS
Fig. 1 depicts the communication system under considera-
tion.
B. Distortion lower bound
When no collaboration is allowed among the encoders, the
achievable distortion bound is unknown. One can, however,
derive a lower bound by considering the ideal scenario with
full collaboration among all encoders at no additional cost.
Thisscenarioisthenapoint-to-pointcommunicationproblem,
wherethedistortionlowerboundcanbedeterminedbyequat-
ingtherate-distortionfunctionforanM dimensionalGaussian
source to the rate of the GMAC. The following proposition
quantifies this bound.
Proposition 1: The distortion lower bound for the network
Fig.1. Networkunderconsideration. Amultivariate Gaussianiscommuni-
in Fig. 1, is in the symmetric case D = D = =D =
catedonaGaussianMACwithseparate (distributed) encoders. 1 2 ··· M
D with transmit power P = P = = P = P and
1 2 M
···
correlation ρ =ρ , i,j, given by
ij x
∀
A. Problem statement
σ2 1 P(1+(M−1)ρx)2 P 0, ρx ,
The sources are memoryless discrete time, continuous am- x − P(M2ρx+M(1−ρx))+σn2 σn2 ∈ 1−ρ2x
plitude, zero mean Gaussian random variables x1,...,xM, D  (cid:16) (cid:17) (cid:0) (cid:3)
swidheerreed.thTehmatodise,l txhme s=oursce+s swhmar,ema ∈com{1m,.o..n,Min}forims actoionn- ≥σx2 Ms(cid:0)1P+(MM−+1M)ρ(xM(cid:1)(−11−)ρρxx)M+−σ1n2σn2, σPn2 > 1−ρxρ2x.
csom∼poNne(n0t,wσs2), whil(e0,eσa2ch)s.oAurcseimapllsiofiehdassceannariinodσividu=al  (cid:0) (cid:1) (3)
m ∼ N wm w1
σw2 =···=σwM is assumed here, which implies that σx1 = Proof:LetR∗,D∗ andP∗ denoteoptimalrate,distortion
σ = =σ =σ , making it easier to derive theoretical
x2 ··· xM x and power respectively. Assuming full collaboration, the M
expressions that can be analyzed further. The correlation be-
sources can be considered as a Gaussian vector source of
tween anytwo sourcesis thenρ =E x x /σ2 =ρ , i,j,
ij { i j} x x ∀ dimension M. From [5]
resulting in a simple covariance matrix K = E xxT with
x
{ }
σ2 on the diagonal and σ2ρ in every off-diagonal element,
x x x M
ain=de2i,genv,aMlue.sNλo1te=thσax2t((thMe−sc1h)eρmxe+s1p)reasnednλteid=inσtx2h(i1s−paρpxe)r, D∗(θ,M)= M1 min[θ,λi], (4)
··· Xi=1
can be applied for any case of unequal correlations ρ . The
ij
mathematical analysis becomes more complicated, however.
Each encoder consists of a memoryless encoding function 1 M 1 λ
ym = fm(xi),m ∈ {1,...,M}, and its output is transmitted R∗(θ,M)= M max 0,2log2 θi , (5)
over GMAC with additive noise n∼N(0,σn2). The received Xi=1 (cid:20) (cid:21)
signal is
M where λi is the i-th eigenvalue of the covariance matrix Kx.
z = fm(xm)+n R. (1) Assuming that all M encoders transmit at the same power
∈
m=1 and that the correlation between the encoder outputs, y , are
X m
At the receiver the functions gm(z),m∈{1,...,M}, produce equal to ρx ≥0, the following Lemma results
theestimatesxˆ ,...,xˆ fromthechanneloutputz.We define Lemma 1: The channel capacity per source symbol is
1 M
the end-to-enddistortion D as the mean-squared-error(MSE)
averaged over all source symbols 1 MP∗(1+(M 1)ρ )
C = log 1+ − x . (6)
1 2M 2 σ2
D = E x xˆ 2 + +E x xˆ 2 . (2) (cid:18) n (cid:19)
1 1 M M
M {| − | } ··· {| − | }
The average(cid:0)transmit power for node m is defined as(cid:1) P = Proof: Let Y be the i’th encoder outputat time instant
m i,k
E f (x )2 . We furtherassume ideal Nyquistsamplingand k. Using Lemma C.2 and Theorem C.1 from [2], it can be
m m
{ }
3
shown that: the number of centroids is large and is therefore referred
2 to as S-K mappings in the following. S-K mappings have
n M M n
1 E Y = E[Y2 ] + previously been optimized for memoryless Gaussian sources
n  i,k!  i,k ! and channels when M source symbols are transmitted on N
k=1 i=1 i=1 k=1
X X X X channel uses [6]–[12].
 M  n
1 For the problem at hand, by treating the collaborative en-
2 E[Y Y ]
i,k j,k
n ! codersasone,andtheM sourcesascomponentsofaGaussian
i,j=1,i6=j k=1
X X vector source, S-K mappings with N = 1 can be applied
M n
(a) 1 directly. S-K mappings can not be applied to the distributed
MP +2ρ E[Y2 ] E[Y2 ]
≤ x n i,k j,k ! case, however, as their operation would require knowledge of
i,j=X1,i6=j kX=1q q all source symbols simultaneously at each encoder [6]–[8],
M n n
(b) 1 [11], [12].
MP +2ρ E[Y2 ] E[Y2 ]
≤ x nv i,k v j,k 
i,j=1,i6=j uk=1 uk=1
X uX uX D. Distributed linear JSCC: uncoded transmission
M  t t 
MP +2ρx P (7) A simple way to construct zero-delay distributed encoders
≤ 2
(cid:18) (cid:19) is to let each sensor node scale its observations to satisfy
where (a) comes from [2, Lemma C.2] given ρx ≥ 0, and the power constraint, i.e. fm(xm) = xm P/σx2. With equal
(b) is the Cauchy-Schwartz inequality. The resulting channel transmit power, the received signal becomes
p
capacity is then (6).
M
Now equate R∗ from (5) with C in (6) and calculate the P
z = Ms+ w +n. (9)
corresponding power P∗. We get D ≥ D∗(θ,M) with D∗ sσx2 m=1 m!
given in (4) and X
At the receiver, MMSE decoding given by xˆ =
m
Mmax[λ /θ,1] 1 (E[x z]/E[z2])z, is applied.One can show that the resulting
P =P∗(θ,M)=σ2 1 i − (8) m
n M +M(M 1)ρ end-to-end distortion becomes
Q − x
The max and min in (4) and (8) depends on ρx and the D =σ2 E[xiz]2
SNR. Since the special case ρ = ρ , i,j is treated, there x− E[z2]
ij x ∀ (10)
are two cases to consider: Only the first eigenvalue λ is to P(1+(M 1)ρ )2
1 =σ2 1 − x .
be represented(the commoninformation)and all eigenvalues, x − P(M2ρ +M(1 ρ ))+σ2
λ ,i [1, M], are to be represented. The validity of these (cid:18) x − x n(cid:19)
i ∈ ··· Considering (3), one can see that a linear mapping is optimal
twocasesisthesameasin[2],i.e.SNR=P/σ2 ρ /(1 ρ2)
and SNR=P/σn2 >ρx/(1−ρ2x) respectively.nIf≤onexsolv−e (x8) iwnhfoernmPat/ioσnn2s≤canρxb/e(r1ec−onρs2xtr)u,ctie.ed.. when only the common
withrespecttoθforthesetwocasesandinserttheresultin(4),
the bound (3) results.
III. DISTRIBUTED NONLINEARJSCC: DQLC
Note that (3) becomes a bound for an average transmit
power constraint by setting P =(P +P + +P )/M. Inordertogetclosertotheboundin(3)thanuncodedtrans-
1 2 M
··· mission when P/σ2 > ρ /(1 ρ2), nonlinear mappings are
n x − x
needed. A zero-delay hybrid discrete-analog scheme, DQLC,
C. Zero-delay JSCC with collaborating encoders
where encoders 1 to M 1 are amplitude limited scalar
Since collaboration makes it possible to construct a larger quantizers and encoder M−consist of a limiter followed by
setof encodingoperations,includingalldistributedstrategies, scaling, is presented here.
the performance of distributed coding schemes is upper-
boundedbythosethatallowcollaboration.Optimalzero-delay
A. Formulation of Encoders and Decoders
collaborative schemes therefore serve as a tighter bound for
Encoder m, m [1,...,M 1], first quantizes x , then
singleletterschemescomparedtotheoneswithoutrestrictions ∈ − m
limits the quantizer output to a certain range κ , where
on codeword length. κ R+, and further attenuates the result b±y αm . That
In the case of zero delay, the corresponding optimal col- m ∈ m
laborative encoding operation is the mapping RM R from is fm(xm) = αmℓ±κm[q∆m(xm)]. q∆m(xm) is a uniform
→ midriseormidthreadquantizer,whereq denotescentroidno.
sourcetochannelspace,whichminimizesD atagivenpower im
iofquantizerm.EncoderM onlylimitsx to κ ,thenat-
constraint.Ithasnotyetbeendeterminedhowsuchamapping M ± M
tenuatesthe resultby α . Thatis f (x )=α ℓ [x ].
should be constructed in order to perform optimally. One can M M M M ±κM M
The received signal becomes
anywayget an indicationon how a scheme with collaborative
encoders performs from schemes that are known to operate M−1
close to the distortion lower bound. Examples on known z = αm(ℓ±κm[q∆m(xm)])+αM(ℓ±κM[xM])+n.
schemes with excellent performance are Shannon-Kotel’nikov m=1
X
(11)
mappings (S-K mappings) [6]–[10] and Power Constrained
Channel Optimized Vector Quantizers (PCCOVQ) [11], [12]. In the following we choose α =1 and κ = .
1 1
∞
PCCOVQ can be considered similar to S-K mappings when An example for M = 3 is shown in Fig. 2. The encoders
4
becomes
4 g (z)=arg min
x303 3 3 3 κ3 x30 1−23 1 m m−1 qim,im∈[1,Nqm] M 2
0 0 z α g (z) q α +ρ α .
−∞ −3 −3 −3 −1 −2−1 (cid:13)(cid:18) − l=1 l l (cid:19)− im ·(cid:18) m xk=m+1 k(cid:19)(cid:13)
−3 (cid:13) X X (cid:13)
x20 −κ3 x20 −4−3 (cid:13)(cid:13) (cid:13)(cid:13) (12)
0x x01 The estimate of the M’th source is then found from
1
M−1
(a) (b)
g (z)=β z α g (z) , (13)
M m m
ρx=0: Nq2=6 Channel segment −α3κ3 α3κ3 d2=∆2 (cid:18) −mX=1 (cid:19)
-3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 whereβ is a scaling factor.The MSE is furtherminimizedby
Edge centroid Inner centroid Origin d1=∆1 d2=∆2(α2+ρxα3) computing
ρx=0.95: Nq2=9 α3l3 d1=∆1(1+ρx(α2+α3)) xˆm =[xm|g(x1),··· ,g(xM)], m∈[1,··· ,M] (14)
-4 -3 -2 -1 -3 -2 -1 0 -2 -1 0 1 -1 0 1 2 0 1 2 3 1 2 3 4 Theoptimalparametersα , ∆ andκ needtobefound.
m m m
To ensure that each source is uniquelydecodable,the relation
Decision, encoder 1 Encoder 1 Encoder 2 Encoder 3 betweentheα’smustbe1=α α α α >0.
1 2 3 M
≥ ≥ ≥···≥
(c) When optimizing DQLC, we look at the number of centroids
of encoder m, N , instead of the clipping κ (except
qm m
for encoder M). The parameters −→∆ = [∆ , ,∆ ]T,
1 M−1
···
CFihga.n2n.elDspQaLceC.HfoerrMeκ=2=3.4Sowuhrecnesρpxac=ef0ora:n(da)κρ2x==50w.(hbe)nρρxx==00.9.955..(c) −Na→req o=pt[iNmqiz2e,d··f·o,rNaqnMa−v1e]rTa,ge−→αpo=we[αr2c,o·n·s·tr,aαinMt]PT., WβiathndDκMas
in (2)
min D. (15)
→−∆,−N→q,→−α,β,κM:PMm=1Pm≤MP
firstcreate thesegmentsin sourcespaceasshownin Fig.2(a)
To calculate D and P , the channel output pdf is needed.
m
(ρ = 0) and 2(b) (ρ = 0.95) through quantization and
x x
limitation. These segments are attenuated by α in such a
m B. Calculation of channel output pdf
waythatthechannelspacestructureshowninFig.2(c)results
when GMAC sums over all encoder outputs. To obtain this To determine the relevant pdf, κm, ∆m and αm must
structure, α > α > > α . From Fig. 2(b) one can be chosen so that channel segments do not overlap (the
1 2 M
see how DQLC is affec·t·e·d by correlation. As ρ increases, configuration in Fig. 2(c)). Since the outputs of encoders 1
x
the joint pdf p (x , ,x ) narrows along all its minor to M 1 are discrete, their distributions can be expressed by
axes, effectivelyxlim1iti·n·g· eacMh source segment. The operation pointp−robabilities,whicharestraightforwardtocalculate.For
ℓ then becomes obsolete. This effect results in reduced the output of encoder M, two cases must be considered: 1)
±κm
distortion. As ρx 1 one can let ∆m 0, αm 1 and ρx closeenoughto0forℓ±κM[xM]tobesignificant.2)ρx so
tκrmans→mi∞ssi,o∀nm. ,and→theDQLCbecomeseq→uivalentto→uncoded scolostheattoy1Mth=atfpMx((xx1M,·)·· ,αxMM)xMeff.ectivelylimitsthesegments
≈
Case 1): The whole range of y = f (x ) is now
M M M
To cancel interference at the receiver, sequential decoding
represented on each channel segment, as can be seen by
is used: First an estimate of source 1 is made. This estimate
studying the blue lines in Fig. 2(a) and 2(c). The pdf of
is then subtracted from the channel output to estimate source
encoder M at the channel output is determined by assuming
2, and so on. In order to make the correct decision on the
thatsources1toM 1aresubtracted.Thenthesameanalysis
output from quantizer m, one must take into account that the −
as for the M = 2 case in [3] can be applied. With the mean
midpoint of each channel segment changes with ρ (like d
x 1 given by
and d shown in Fig. 2(c)). Consider source 1: the first order
2
M−2
moment of p(α x ,...,α x x = q ) must be deter-
mined. Consider2a 2sub-divisMionMo|f y1 =[xi1 α x ] into µ=qiM−1(αM−1+ρxαM)+ qim, (16)
1 M M
y = [α x α x ]T and y = x .·B··y sub-dividing mX=1
a 2 2 M M b 1
the covariance··m·atrix K = E yyT according to (45), the same arguments as in [3] lead to
y
{ }
Tρsuxhmxeo1mreeαmd2t1ogi·ne·t·thheerα,AMcpepneTtnr.odiiWdxshgeoivnfesethnEec{oαdtre2arxns21f,o.ar.rm.e,eαsdhMisfotxeuMdrc|bexys1}athr=ee pzM(→−im)(zM)κM = Σ1 Z−ααMMκκMM e−α2Mσx2(z2Mα2M−µσx−2σyn2)2+σn2y2dy
(cid:2) (cid:3) +p p (z µ α κ )+p (z µ+α κ ) ,
sumofallfirstordermoments.Whensource1isdetected,one o n M − − M M n M − M M
(17)
cansubtractitfromz thenusethesameargumentasabovefor (cid:0) (cid:1)
source 2, and so on. Let N denote the numberof centroids where p = Pr x κ , Σ = 2πα σ σ , p is the
qm o { M ≥ M} M x n n
for quantizer m. The estimate of sources m [1,M 1] noise pdf, and z denotes the received signal when sources
M
∈ −
5
1...M 1 are subtracted. Fig. 4(b) shows this pdf when is, ϑ [1,2],dependingon ρ . (17) is valid when l >2κ
x M M
− ∈
µ=0. while (19) is valid when l 2κ .
M M
≤
Case 2): When ρ is close to 1, each source segment, and The total channel output pdf is given by
x
thereforealso each channelsegment, will no longer be equiv-
alent but contain somewhat different(but intersecting) ranges pz(z)= Pr{−→i m}pzM|→−im. (20)
→−
ofyM (thisalsoappliestosources2toM 1giventheothers). Xim
−
This can be seen by studying the blue lines in Fig. 2(b) and p →− is given by (19) or (17) depending on whether
Fig.2(c).Onecannowassumethaty =f (x ) α x . zM|im
M M M ≈ M M l1 > 2κ or not. −→i m = [i1,i2, ,iM−1], and Pr −→i m =
To determine the relevant pdf, p(y q ,...,α q ), ··· { }
after summation over GMAC mustMb|ei1found.MC−o1nsiiMde−r1a Pr{qi1}Pr{qi2|qi1}···Pr{qiM−1|qi1,··· ,qiM−2}. Fig. 4
shows an example of the channel output pdf when M =3 at
sub-division of y = [x α x ] into y = α x
1 ··· M M a M M 30 dB channel SNR.
and y = [x α x α x ]T. By sub-dividing
b 1 2 2 M−1 M−1
···
tThheeocroevmar1ianincethmeaAtrpipxenKdyix g=iveEs{thyeyTse}coancdcoorrddienrgmtoom(e4n5t), 0H.8istoHgirsatomgr afomr ocf hchaannnneell oouutptputut at 30d B Calculated pdf for z3
0.7 Calculated pdf
(M 1)ρ2 0.6
Σ =σ2α2 1 − x . (18) 0.5
aa·b x M(cid:18) − 1+(M −2)ρx(cid:19) 0.4
0.3
By inserting (18) into (46) (in the Appendix) and (16) 0.2
for the mean, the wanted pdf results. After addition 0.1
of noise, the resulting pdf is found by the convolution 0 −3 −2 −1 z0 1 2 3 −00.4 −0.2 z0 0.2 0.4
3
p(y q ,...,α q ) p (n) [13, 181-182], and thus
M| i1 M−1 iM−1 ∗ n (a) (b)
pzM(→−im)(zM)γM = 2π σn21+Σaa·b e−12σ(µn2−+zΣMaa)·2b. (19) Fepnoigcs.oit4div.eerCc1ehinastnrionniedulso(eiuntadpneudxtpN1d)qf2awre=hetrn4a.nMTsmh=eittge3rde.e(fano)rcHbuiorsvttheogesrhnaocmwodswe(rh1e17r)eanw4dhcee2nn.t(rthboe)id(fis1r7osf)t
q showncentered at theorigin (µ=0), that iswhen centroids forencoders 1
The validity of (17) and(cid:0)(19) must b(cid:1)e determined. Since and2aregiven.
the correlation between any two sources is assumed to be the
same, one can focus on the x ,x plane. Fig. 3 provides
M M−1
a geometrical picture for the following discussion. Let l
M C. Distortion and Power calculation
To calculate the distortion, we use an approach similar to
ψ
x M that in [3], where the total distortion is divided into several
M l contributions.
M
1) Distortion and power for source M: The distortion for
e γ
x M M source M can be divided into three contributions: clipping
M−1
∆ distortion, anomalous distortion and channel distortion.
∆M M Clipping distortion: At encoder M we only have distortion
from limitation, ε¯2 , an event with probability Pr x >
κM {| M|
κ and error (x κ )2:
M M M
} −
∞
ε¯2 =2 (x κ )2p (x )dx . (21)
κM M − M x M M
ZκM
Fig.3. DQLCseeninthexM−1,xM-planewhenρx=0.95.Theexpanded The distortion from channel noise can be split into two
rectangleontherightcanbeappliedtocalculateanomalouserrorsforsource
contributions: Channel distortion and anomalous distortion.
M.
Anomalous distortion: results from a threshold effect (see
e.g.[14]or[15])andoccurseverytimethecentroidforoneor
denote the length of the portion of the x axis that contains
M
the significant probability mass2 given x (or q ). severalof quantizers1 to M 1 is erroneouslyselected. This
M−1 iM−1 error leads to a ”jump” from−one channel segment to another
l = 2√ϑ e = 2b √ϑλ , where e = b √λ
M M M M M M M
k k k k (seeFig.2(c))forsourceM,resultinginlargedecodingerrors.
denotes the length of the minor axis of the ellipse depicted
Intheworstcasescenario(ρ =0)largepositiveandnegative
(the source space). b ( 4) is a parameter determining x
M
≈ values are interchanged. The anomalous distortion is difficult
the width of the ellipse shown, and should be chosen so
to calculate exactly. We therefore look at an approximation
that the significant probability mass is within this ellipse.
validaroundtheoptimaloperationofDQLC.Note thatjumps
ϑ = (l /(2 e ))2 depends on ρ : If ρ = 0, then ϑ = 1
M M x x
k k among centroids of in encoder 1 are most fatal since this
since the source space is rotationally invariant (a sphere). If
leads to anomalous distortion for all other sources. Jumps
ρ > 0.7 then ϑ 1/cos2(ψ )= 1/cos2(π/4)= 2. That
x M
≈ ≈ among centroids of encoder 2 lead to anomalous errors for
2“Significantprobabilitymass”meansalleventsexceptthosewithverylow sources 3 to M, and so on. Therefore, the probability for
probability. jumps among centroids of quantizer 1 to M 2 should be
−
6
at least as small as the probabilityfor jumpsamong centroids 2) Distortion and power for source 1 to M 1: Here we
−
of quantizer M 1. By assuming that encoders 1 to M 1 havequantization-andlimitationdistortionfromtheencoding
− −
are constructed correctly, one can calculate an upper bound process and channel distortion from channel noise.
on anomalous distortion for source M by considering jumps Quantizationandlimitation:Thesecontributionsareequiva-
among centroids of quantizer M 1 only. We are then in the lentto granular-andoverloaddistortionfromthequantization
−
samesituationastheM =2casein[3],andcancalculatethe process, that is
anomalous errors from the x ,x plane shown in Fig. 3.
Two cases must be considereMd:−l1 >M2κ and l 2κ . Nq2m−1 i∆ ∆ 2
M M M ≤ M ε¯2 =2 x (i 1)∆ p (x )dx
eveArssyum+e finrst lMd > 2/κ2M=: a∆nomalo(αus erro+rsαhapρpe)n/2wh(seene- q,m Xi=1 Z(i−1)∆(cid:18) m− − − 2(cid:19) x m m
M ≥ M−1 M−1 M−1 M x ∞ Nq ∆ 2
M =3 case in Fig. 2), i.e. the probability for anomalies are +2 x m 1 ∆ p (x )dx .
m x m m
∞ Z Nq2m−1 ∆(cid:18) −(cid:18) 2 − (cid:19) − 2(cid:19)
pthM =2Z∆M2−1(αM−1+αMρx)pzM(→−0m)(zM)κdzM, (22) Cha(cid:0)nnel dis(cid:1)tortion: The distortion from channel nois(e27is)
where p (z ) is given in (17). Since different values given by
zM M κ
of −→i basically shifts p (z ) , the relevant probability
m zM M κ M−fold
can be calculated by setting µ = 0 in (17). p must be
thM
found numerically since the integral in (22) has no closed ε¯cht,m = ··· px(x1,...,xM)p z|y1,...,yM (28)
formsolution.Theanomalouserror’smagnitudesarethesame Zz }|Z{Z
regardless of which segment we are at, and are bounded by y˜ xˆ (g (z),...,g (z)) 2dzdx (cid:0) dx , (cid:1)
m m 1 M 1 M
− ···
(2κ )2,sinceκ getsinterchangedwith κ whenchannel
segmMents start toMoverlap. − M wh(cid:2)ere ym = fm(xm), and y˜m(cid:3) denotes the quantized and
limited x . The relevant pdf can be derived from (20).
Now assume l 2κ : the probability for this event is m
M ≤ M As for the M’th source, the distortion can be divided into
given by
channel distortion and anomalous distortion, where channel
∞ distortion refers to jumps among neighboring centroids and
p˜thM =2Z∆M2−1(αM−1+αMρx)pzM(→−0m)(zM)γMdzM, (23) oanccoumradlouuestodijsutomrtpisonfrroemferosnteoqtuhaenstiitzueadti“osnegwmheenret”latorgaeneortrhoerrs.
awshsuermeeptzhMa(t→−0µ)(=zM0.)γWMheisngρivegnetbsycl(o1s9e),towohneere,tohneeanagoaminalcoauns TnoakiseeMtak=es3:usanaocmroaslisesthoeccduercfiosiroxn2baonrddxer3fwohreenntchoedcehra1nnienl
x
errors,γ ,becomesmaller.ThiscanbeseeninFigs.2and3. Fig. 2 (“segment” now refers to the collection of purple dots
M
Since γ is approximately the same in magnitude regardless between each decision border of encoder 1). Anomalies do
M
ofwhichchannelsegmentwejumpfrom(seeFig.3),itcanbe not occur for source 1, i.e. when a centroid for encoder one
calculatedbyconsideringjumpsbetweenthesegmentsclosest is erroneously detected. For general M, anomalies occur for
to the origin in the source space. The parallelogram shown sourcesxm+1,...,xM whenthereisachannelerrorforsource
on the right hand side of Fig. 3 applies to approximate γM. xm. The anomalous errors for source 2,...,M − 1 can be
Since ψ = π/4, the parallelogram consists of a square and derived in a similar way as for source M, as illustrated in
M
tworighttriangleswithbothcathetiequalto∆ .Thisfurther SectionIV. Channeldistortionforsourcem isproportionalto
M
impliesthatγ l ∆=2b σ ϑ(1 ρ ) ∆ ,where ∆2 anditsprobabilitycanbedeterminedfromtheprobability
M M M x x M m
≈ − − −
ϑ 2 since ρ is large (see Section III-B). foranomalouserrorsforsourcem+1(e.g.theprobabilitiesfor
≈ x p
The anomalous distortion becomes channeldistortionforxM−1canbecalculatedusing(22),(23),
as illustrated in Section IV)
4pthMκ2M, lM >2κM, Power: The power from encoder 1 to M 1 is given by3
ε¯2 = (24) −
anM 
p˜thMγM2 , lM ≤2κM. P =2α2 Nqm/2p (i 1)∆ + ∆m 2 (29)
m m i − m 2
Channel distortion:Let x˜M =ℓ±κM[xM]. With no thresh- Xi=1 (cid:18) (cid:19)
old effect occurring the noise is additive and given by
where p =Pr (i 1)∆ <x i∆ .
i m m m
{ − ≤ }
ε¯2 =E (x˜ (α x˜ +n)β)2
CM { M − M M } (25)
σ2(1 α β)2+β2σ2. D. High SNR analysis
≈ x − M n
From the previous section it is clear that a closed form
The last approximation is based on the assumption that
expressions describing the DQLC in general is hard, if at
E x˜ σ2.
{ M}≈ x all possible to find. One can, however, find closed form
Power: With p =Pr x κ , the output power from
o M M expressionsthat approximatethe distortion well at high SNR.
{ ≥ }
encoder M becomes
These expressions can further be used to determine how well
αMκM
PM = yM2 pyM(yM)dyM +2poα2Mκ2M. (26) 3Note that (27) and (29) is derived for midrise quantizers. A similar
Z−αMκM expressioncanbederived formidthread quantizers.
7
DQLC performs at high SNR as a function of both M and The distortion at high SNR can be approximated by
ρ .
x
α2C (1 ρ )
The performance of DQLC is compared to Performance D 2 2 − x
1
upper bound, i.e. the signal-to-distortion ratio SDR=σx2/D. ≈ 3 MP M α2
Assuming that SNR , the bound (3), for the case − i=2 i
(cid:18) (cid:19)
→ ∞
SNR=P/σ2 >ρ /(1 ρ2), becomes α2 C P(1 ρ ) (31)
n x − x D m+1 m+1 − x , m [2,M 2]
m ≈ 3α2 ∈ −
m
α2 C (1 ρ )+(b σ )2 σ2
SDR≈ Ms 1 MρSxNRM−1 =̺M. (30) DM−1 ≈ M M 3−α2Mx−1 n n , DM ≈ α2Mn ,
−
where C = b2 ϑ2 . For D , the high SNR approximation
(cid:0) (cid:1) m m m M
of (25) was used.
The symbol, ̺M, is introducedto have a compactrepresenta- To determine the optimal performance, the optimal
tion for later derivations. Due to the fact that the code word α2 needs to be found. The obvious way is to solve
m
lleennggtthhiosfsahnoyrt,sttohcehreasitsicavseigctnoirfic[1an6t, vpa.r3ia2n4c]e(faororuanndotrhmeamliezaend ∇sin→−αceDth1e+seDa2re+e·q·u·a+tioDnsMof =ord0erwith4,reasnpaelycttictaol αso2mlu.tiBonust
i.i.d.Gaussianvectorx¯ofdimensionN,Var{kx¯k}=2σx4/N), can(cid:2)not be found, and a di(cid:3)fferent a≥pproach must be chosen.
makingexact analysisdifficult. To obtain closed form expres- From the M = 2 case in [3] is is known that the distortion
sions,onlythedistortiontermsthataredominantathighSNR of DQLC is a constant times the bound ̺ . We therefore
2
aretakenintoaccount.Itisfurtherassumedthatσx =1.Take hypothesize that this is the case for general M as well: By
source M 1: channel errors for this source and anomalous choosing α2 = σ2 ̺ , then D = ̺ . Since we want
errors for −source M can (nearly) be avoided by assuming a D = D =M =nDK M= 1/( ̺M), thKe Mthat satisfies this
1 2 M M
distance ∆M−1 > 2 (αMlM/2)2+(bnσn)2 between each relation must··b·e determined. BKy solvingK1/DM−1 = ̺M,
centroid (the purple dpots in Fig. 2(c)). bn = bM (≈ 4) is a then K
constantthatmustbechosensothatthesignificantprobability
mass of the noise is within 2bnσn. A similar argument can α2 = σn2C (1 ρ ) ̺ 2
be used for the other encoders. To quantify the magnitude M−1 3 M − x K M
(32)
of ∆m, Fig. 5, depicting the x1,x2 plane for the M = 3 = α2MC (1 ρ (cid:0)) ̺ (cid:1) ,
case after quantization, applies. From Fig. 2 and 5 one can 3 M − x K M
(cid:0) (cid:1)
assuming b σ 0. By continuing to solve 1/D = ̺
n n m M
≈ K
withrespecttoα2 formfromM 2to2usingthepreviously
x2 ψ2 Edge centroid derived α2m+1, omne can show that−
∆ l
2 2 Inner centroid
x1 e2 γ2 α2 = α2M M C (1 ρ )M−i ̺ M−i. (33)
∆1 ∆1 i 3M−i(cid:18)j=i+1 j(cid:19) − x K M
∆ Y (cid:0) (cid:1)
2
Source segment Finally, the that makes 1/D1 = ̺M, must be found. Ex-
pandingtheKsum M α2 using (33K), and setting C =1,
i=2 i M+1
one can show that
Fig.5. x1,x2planeofDQLCwithM =3whenρx=0.95.Theexpanded P
rectangleontherightcanbeappliedtocalculateanomalouserrorsforsource M M−1 k−1 M
2. α2 =α2 (1−ρx)K̺M C . (34)
i M 3 j
i=2 k=1 (cid:18) (cid:19) j=M+2−k
X X Y
convince oneself that if ∆ > 2α l /2, the integrals
m m+1 m+1 Letting C = C = C = C (which is the case when
in (28) can be avoided, since no distortion results from 2 3 ··· M
ρ = ρ , i,j at high SNR), the product in (33) becomes
channel noise. For example, if ∆ > 2α l /2 in Fig. 2(c), ij x ∀
1 2 2 C−1Ck−1, and (34) turns into a Geometric series. From the
the green stars will not be confused. One can derive from
sum of a Geometric series
Fig. 5 that l = 2b ϑ (1 ρ ) (as was done for
m+1 m+1 m+1 x
−
tlhMationnesemctiaoynnIeIgIl-eBc)t.tFheorℓplarge[xSN]Ro,peκrmatiobne.coTmheehsioghl-arragtee n qk−1 = 1−qn, (35)
±κm m 1 q
approximation to a scalar quantizer, ∆2 /12, then applies to k=1 −
m X
quantify distortion. To avoid constrained optimization, D is
1 the following high SNR approximation can be derived
further scaled by P =MP M P . For convenience,we
1 − i=2 i
also scale D by P prior to quantization (instead of after).
m m P M α2 (1 ρ )C ̺ M−2
Note that when SNR gets large enough, κ becomes so large α2 m − x K M . (36)
and ∆ becomes so small that P α2 σ2. i ≈ C 3
m m ≈ m x Xi=2 (cid:18) (cid:19)
8
Inserting (36) and (33) with i=2 into the expression for D Loss from bound for DQLC and S−K mapping
1
in (31) 10
α2 (1 ρ )C ̺ M−2
MSNR m − x K M 8
− C 3 ···
(cid:18) (cid:18) (cid:19) (cid:19) (37) B)
(1 ρ )C ̺ 1−M d
− x3 K M =K̺M, R ( 6
(cid:18) (cid:19) D
S DQLC: ρ =0
n x
where SNR= P/σn2. By solving (37) and removing constant s i 4 DQLC: ρ =0.95
terms, then s x
Lo S−K mapping: ρ =0
x
M−1 1− 1 2
3 3 M
= M = , ρ =1. (38)
x
K s C C 6
(cid:18) (cid:19) (cid:18) (cid:19)
0
2 4 6 8 10
(38) quantifies the loss to the performance upper bound. By M
inserting C = b2ϑ and M = 2 in (38), the loss calculated
(a)
in[3]results,andso(38)isa generalization.Onecanobserve
that for any M and ρx, DQLC exhibits a constant gap to Loss from bound for Uncoded transmission
the bound as SNR . The gap grows somewhat with 50
M, however, but is→for∞tunately bounded: Taking the limit ρ =0
x
45
M → ∞, the loss is ≈ 7.2 dB when ρx is close to 0 and ρx=0.95
10.2 dB when ρx is close to 1. From (38) one can see 40
≈
that there are mainly two loss factors. One is due to short B)
code length: When the code length is infinite, b 1 (see (d35
e.g. [17]), whereas when the code length is one, b→≈ 4. The DR 30
nested structureofDQLC resultsin an increasedlosswith M S
whenever b > 1. This accumulation is avoided with the S-K n 25
mappings described in Section II-C, as seen from Fig. 6(a). ss i20
o
The distance to the upper bound is around 0.8-0.95 dB when L
15
ρ = 0, actually decreasing slightly when M increases (the
x
same effect is observed when ρx is close to 1, but the loss 10
is now around 2 dB). This clearly indicates that it is not the
zerodelayrequirementalonethatmakesthelossincreasewith 52 4 6 8 10
M. The reason for the increasing loss is most likely that the M
DQLCis sub-optimal.Alternatively,it maybe thatzerodelay
(b)
distributed JSCC schemes will suffer from an increasing loss
like(38).Thismustbedismissedorconfirmedthroughfurther
Fig. 6. Loss in SDR from upper bound at SNR=100 dB. (a) DQLC.(b)
research,however.Thereasonwhyϑ=2 whenρx is close to Uncodedtransmission.
one is that the source space can not be rotated (decorrelation)
with the choice of distributed encoders, and this implies that
each channel segment gets somewhat longer than necessary. IV. OPTIMIZATIONAND SIMULATION OF DQLCAT
By inserting b 4 and ϑ 1 when ρ is close to zero, ARBITRARY SNR WHENM =3
x
≈ ≈
and ϑ 2 when ρx is close to 1, the performance of DQLC We give an example on how to calculate and optimize
≈
as a functionof M can be plotted.Fig. 6 shows the loss from distortion for all SNR when M =3 by applying the analysis
upper bound for DQLC (high SNR in general) and uncoded in Section III-C. The optimized DQLC is further simulated
transmission at 100 dB SNR. S-K mappings for ρx = 0 case and compared to distortion lower bound, S-K mapping and
isalsoshown(resultsaretakenfrom[11,chapter3]and[18]). uncoded transmission.
Fromtheseplotsonecanseethattheperformanceofuncoded
transmissionwill close in onthe performanceof DQLCwhen
A. Calculation of distortion
around 10 sources are considered. This number will decrease
somewhat for smaller SNR and increase as the SNR gets The distortion for source 3 is found from Section III-C by
higher. Note that for uncodedtransmission the distance to the setting M =3 in (21), (24) and (25), and the channel power
bound will in any case increase as the SNR grows (except isgivenby(26).Furthermore,thedistortionfromquantization
whenρ =1),contrarytoDQLC,andsoDQLCwillimprove andlimitationofsource1and2isgivenby(27)andthepower
x
over uncoded transmission when the SNR gets sufficiently is given by (29). What remains to calculate is the distortion
large, thus fulfilling our objective. due to channel noise for source 1 and 2 given by (28). When
9
analyzing DQLC around its optimal distortion, only jumps to where 2κ =∆ (N 1).
2 2 q2 −
thenearestneighboringcentroidsneedstobeconsidered.This Channeldistortion,source2:Sinceanomalouserrorsresults
will simplify the calculations and speed up the optimization for source 3 whenever channel errors occur for source 2, the
process. As for source 3, (28) can be divided into channel- validity forthese eventsare the same. A similar expressionto
and anomalous distortion. As mentioned in Section III-C2, that in (42) can therefore be derived:
both channel distortion and anomalous distortion may occur Consider first l >2κ : For a given source segment in the
3 3
for source 2, while only channel distortion may occur for x ,x plane an inner centroid has two neighbors, while an
1 2
source 1. edgecentroidhasonlyone(seeFig.5).Theprobabilitythatan
Anomalousdistortionforsource2:Asimilaranalysistothat innercentroidis confusedwith its neighborsis givenby p ,
th3
in Section III-C1 applies here. As for anomalous distortion found by substituting M = 3 in (22). For edge centroids we
for source 3, there are two cases to consider: l2 > 2κ2 and have pth3/2 since channel errors happens if an edge centroid
l2 2κ2. Now l2 = b2σx 2(1 ρx) and calculated in the isexchangedwithaninnercentroid,whereasanomalouserrors
≤ −
samewayaslM inSectionpIII-B,nowusingtheparallelogram happenotherwise(see Fig. 2(c)). When neighboringcentroids
otoftthheereilglhiptsien(FinigF.5ig..b52)iscoanctaoinnsitnagntththeastidgentiefircmaninte(sqtuhaenwtizidedth) aarneeedxgcehacnegnetrdo,idth,etheerrnoris∆22. WithPNq2, the probabilityfor
probability mass.
Assume first that l > 2κ (see ρ = 0 case in Fig. 2). Nq2
2 2 x 2 p
The error we get when anomalies start to happen is the ε¯2 =2∆2p P +∆22P th3
C2 2 th3 i 2 Nq2 2
difference in magnitude between centroid no. 1 and centroid i=2 (43)
X
no.N ,i.e. q q =∆ (N 1).Theprobabilityfor =2∆2p 1/2 P +∆2P p
q2 | (Nq2)2− 12| 2 q2− 2 th3 − Nq2 2 Nq2 th3
anomalies(theprobabilityforcrossingofthedecisionborders =∆2p 1 P .
for encoder 1 in Fig. 2(c)) is given by 2 th3 (cid:0)− Nq2 (cid:1)
Now assume l 2(cid:0)κ : As fo(cid:1)r anomalous distortion, this
pth2 =Pr{|f2(x2)+f3(x3)+n|≥d1/2}=PNq2pdth, (39) case is difficult t3o≤calcul3ate accurately, due to the difficulty
where P = Pr (N 1)∆ /2 < x < is the of determining which centroids are edge centroids. One can,
Nq2 { q2 − 2 2 ∞}
probability for being at an edge centroid (see Fig. 2(c)) and however, upper bound channel distortion by assuming that
p =Pr y +n >d , where jumps from any centroid leads to channel distortion (i.e.
dth {| 3 | th}
d α d none of the possible events leads to anomalous distortion).
dth = 21 − 22 2(Nq2 −1), (40) The probability for channel distortion is then given by p˜th3,
found by substituting M = 3 in (23). We therefore have
is the distance from an edge centroid of encoder 2 to the
ε¯2 ∆2p˜ . This bound is accurate enough to find the
decision border of encoder 1 (see Fig. 2(c)). The distribution C2 ≤ 2 th3
optimal parameters of DQLC.
needed to calculate p is given in (17).
Now consider the dctahse l2 ≤ 2κ2 (ρx = 0.95 scenario in thrCeshhaonlndelefdfeiscttosrthioanp,pesnoufrocer s1o:urScienc1e,Nthqe1 p=rob∞abilaintyd fnoor
Fig. 2). This case is difficult to calculate as accurately as
the l > 2κ case above, since its hard to determine exactly channel distortion for source 1 will be the same as the
2 2
probability for threshold effects for source 2. The magnitude
which centroids that are edge centroids (this can be seen by
comparing the ρx = 0.95 case with the ρx = 0 case in of the error is ∆21. Therefore ε¯2C1 = ∆21pth2 when l2 > 2κ2,
Fig. 2). One can get around this problem by calculating both and ε¯2C1 =∆21p˜th2 when l2 ≤2κ2.
the probability p˜ = Pr f (x )+f (x )+n d /2
th2 2 2 3 3 1
{| | ≥ }
and the resulting anomalous error assuming that y =f (x )
2 2 2 B. Optimization and Simulation
is continuous. By using pzM(→−im)(zM)γM in (19) and the Instead of solving the constrained problem
expression for p(y x ) (see [13, p.223]) one can show that
2 1
| in (15), we choose to scale x by ξ =
∞ 1
p˜th2 =∞Zd1/2(cid:2)pzMe(−→−i12mσ)n2(z+MΣa)a(µγ·b−M+zσM∗x2α)p222((y1−2ρ|x2x)1)(cid:3)(z)dz (41) ptqqoim3((Pg11et−++(aρPρnx2(((αuα∆n22c+,o+αnsα2αt)r3a+))i/n)ξPe)d3in(iαnp(3r1o,o2κrb)d3lee)mrm)/u.tσosxtNdptoehrtceeioondrtehbtoaectoqcrurthheaacennttlgiyzfeaad(ctittohotoner
= dz, im x 2 3
Zd1/2 2π σn2 +Σaa·b+σxα2 2π(1−ρ2x) idnitsetogrrtaitoionnfolirmDitQiLnC(4i1s)Dm=ust(Dalso+bDe ch+anDge)d/)3.,Twhehearveerage
q 1 2 3
where d =∆ (cid:0)1+ρ (α +α ) . Aps in Section (cid:1)III-C1, the
1 1 x 2 3
ε¯2 (∆ )+ε¯2 (∆ ,α ,N )
probability in (4(cid:0)1) is calculated a(cid:1)ssuming µ = 0. With the D1 = q,1 1 C1 1 2 q2 ,
same reasoning as in Section III-C1, one can show that the ξ(∆ ,α ,α ,κ )
2 2 3 3
errorwe getwhen anomaliesstart to happenis approximately D2 =ε¯2q,2(∆2,Nq2)+ε¯2C2(∆2,Nq2,α2,α3,κ3) (44)
γ2T=hel2a−no∆m1al=ousb2dσixstor2ti(o1n−beρcxo)m−es∆1 (see Fig. 5). +ε¯2an2(∆2,Nq2,α2,α3,κ3),
p D =ε¯2 (κ )+ε¯2 (α ,β)+ε¯2 (∆ ,α ,α ,κ ).
p ∆ (N 1) 2, l >2κ , 3 κ3 3 C3 3 an3 2 2 3 3
ε¯2 = th2 2 q2 − 2 2 (42) All parameters must be greater than zero and α α .
an2 p˜ γ(cid:0)2, (cid:1) l 2κ , Some of the distortion terms do not have analytical2sol≥ution3s,
 th2 2 2 ≤ 2

10
oanpdtimthaulspnaurammeeritcearsl.optimizationis necessary to determinethe G50ain fDroQmL Ccorrelation compared to ρx= 0 14 Loss from SDR upper bound
traDnsQmLisCsiiosncofmropmarSedecttoiothneIbI-oDunadnfdrotmheSSec-KtionmIaIp-Bpi,nugnscfordoemd DR (dB)3400 Uncoded transmission DR (dB)11802 UDnQcLoCded transmission
S S
Sraetciotio(SnDIIR-)Cσ.x2A/gDainaswaefucnhcotoiosneotofclohoanknaetlsSiNgnRal=-toP-d/iσstn2or(tainodn Gain in 1200 Loss in 246
ρ ) instead of distortion. The results are shown in Fig. 7 for
ρxx = 0 and ρx = 0.95. The SDR upper bound is given by 00 0.2 0.4 ρx0.6 0.8 1 00 0.2 0.4 ρx0.6 0.8 1
(a) (b)
M=3 sources
30 Fig.8. HowcorrelationaffectsperformancewhenM =3at40dBchannel
SDR Upper bound
SNR.(a)GainfromcorrelationforDQLCanduncodedtransmission.(b)Loss
Uncoded transmission fromSDRupperboundasafunctionofρx.
25
S−K mapping
DQLC
20 ρ =0.95 the distance to the upper boundis largest for DQLC when ρx
B) x is around0.9, and that DQLC and uncodedtransmission both
d
R (15 reach the upper bound in the limit ρx →1. Note that DQLC
D performsbetter than uncodedtransmissionfor mostρx values
S at 40dB SNR.
10
ρ =0. If there is a demand for equal transmit power from each
x Robustness encoder, one can still use DQLC with timesharing. I.e. each
plot
5 encoderdescribedhere is used oneach source 1/3 of the time
(1/M in general). As shown for the M = 2 case in [3], a
0 further backoff from the bound compared to that in Fig. 7
0 10 20 30 40 50
must then be expected.
Channel SNR (dB)
Fig. 7. Performance of DQLC compared to the SDR upper bound, S-K V. SUMMARY AND EXTENSIONS
mappings and uncoded transmission. The black dots shows the design SNR
fortherobustnessplots forDQLC. In this paper, a distributed delay-free and low complexity
joint source-to-channel mapping was proposed and bounds
σ2/D∗, where D∗ is the optimal distortion given in (3). were derived for transmission of a multivariate Gaussian over
x
When ρ = 0, DQLC drops around 2.5-3.5dB from the a Gaussian MAC. Both linear and nonlinear mappings were
x
upperbound,whileitdropsaround4to7dBwhenρ =0.95. analyzed. A linear mapping (uncoded transmission) achieves
x
The loss at high SNR (50dB) corresponds well with the the performance upper bound within a certain range of low
calculatedestimate shownin Fig. 6(a)when ρ =0.95,while channel SNR. A nonlinear mapping, DQLC, was introduced
x
the calculated estimate is a bit pessimistic when ρ = 0. to improve on uncoded transmission at higher SNR. A col-
x
Fortunately, DQLC gets somewhat closer to the bound as laborative scheme (Shannon-Kotel’nikov mapping) was also
the SNR drops. There is also a backoff to the S-K map- introduced to provide an approximate bound for zero delay
ping of around 1 to 2 dB. This is because S-K mappings schemes.
avoid threshold effects as well as the “loss accumulation” DQLC does not achieve the performance upper bound, but
mentioned in Section III-D. Interestingly, DQLC improves leaves a certain gap which value depends on both correlation
with increasing ρ without changing the basic encoder and and the number of sources. However, DQLC constitutes a
x
decoder structure, only the different parameters needs to be constant gap to the bound in any case as SNR and its
→ ∞
adapted. The improvement is significant, around 5 to 7 dB received fidelity (SDR) improves with increasing correlation
when ρ goes from 0 to 0.95. DQLC is also robust against without changing the basic encoder and decoder structure.
x
variationsinnoiselevel.NotethatDQLCoutperformuncoded DQLC therefore outperforms uncoded transmission as the
transmission for SNR> 7dB when ρ = 0 and SNR> 28dB SNR gets high enough in any case. Unfortunately, the loss
x
when ρ =0.95. to the bound increases somewhat with the number of sources
x
The gain from increasing correlation as a function of ρ (something the collaborative S-K mappings does not), but is
x
is shown in Fig. 8(a) for DQLC and uncoded transmission fortunatelyboundedtoa finite valueasthenumberofsources
at 40 dB channel SNR. Note that the gain for DQLC is not goestoinfinity.Optimizationandsimulationfor3sourcesalso
significant before ρ > 0.7, whereas the gain gets large showed that the gap to the bound decreases somewhat as the
x
≈
when ρ 1 (around37.5dB). Uncodedtransmission shows SNR drops.
x
→
anevengreatergain,whichisnaturalsinceitgoesfrombeing It is also important to note that DQLC can be applied for
highly sub-optimal when ρ = 0 to achieve the bound for anyunimodalsource(andchannel)distributionandoptimized
x
all SNR when ρ = 1. The gap to the performance upper using the same method as presented in this paper.
x
bound as a function of ρ is plotted for DQLC and uncoded Futureresearchshouldaim atfinding,ifpossible,a scheme
x
transmission in Fig. 8(b), for 40dB channel SNR. Note that wherethelosstothebounddoesnotincreasewiththenumber