Joint Source-Channel Coding at the Application Layer for Parallel Gaussian Sources

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by  Ozgun Y. Bursalioglu

Joint Source-Channel Coding at the Application
Layer for Parallel Gaussian Sources
Ozgun Y. Bursalioglu, Giuseppe Caire Maria Fresia, H. Vincent Poor
Ming Hsieh Department of Electrical Engineering Department of Electronic Electrical Engineering
University of Southern California Princeton University
Email: {bursalio, caire}@usc.edu Email: {mfresia, poor}@princeton.edu
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Abstract—Inthispaperthemulticastingofindependentparal- losses since packets contain sequence numbers, and therefore
0
lel Gaussian sources over a binary erasure broadcasted channel can treat missing packets as erasures.
0
is considered. Multiresolution embedded quantizer and layered
2 Motivated by the above scenario, we focus on efficient
joint source-channel coding schemes are used in order to serve
Joint Source-Channel Coding (JSCC) for the transmission of
n simultaneously several users at different channel capacities.
a The convex nature of the rate-distortion function, computed by a source over the BEBC. We refer to this approach as JSCC
J means of reverse water-filling,allows usto solve relevant convex “at the application layer” since we disregard the underlying

6 optimization problems corresponding to different performance physical channels, in contrast to most JSCC approaches pre-
1 criteria.Then,layeredjointsource-channelcodesareconstructed sentedintheliterature(e.g.,[6]andreferencestherein).Inthis
based on the concatenation of embedded scalar quantizers with
paperweextendourapproachforJSCCofanindependentand
] binary rateless encoders.
T identically distributed (i.i.d.) Gaussian source over the BEBC
.I I. INTRODUCTION proposed in [3] to the case of parallel Gaussian sources.
s The parallel Gaussian sources model is a good first-order
c Multimedia streaming over heterogeneous digital networks approximation of the output to a linear transform operating a
[
is one of the fastest growing applications both in terms of subband decomposition of some natural source. For example,

1 traffic demands and in terms of market potential. A conven- the JPEG2000 image coder [7] transforms the original image
v
tional approach consists of establishing individual streaming by using a discrete wavelet transform (DWT) and the first-
6
sessions from the server to each users. This may be very order marginal statistics of the transform coefficients in the
9
inefficient when many users wish to receive the same content differentsubbandsmay be approximatedby Gaussian random
3
2 (e.g.,inmobiletelevisionorvideoondemandapplications).A variables with different variances.
. definite advantage of analog broadcasting systems is sending Building on our previous works [3] and [8], we use multi-
1
0 simultaneouslythesamesignaltoapotentiallyunlimitednum- resolution (embedded) quantization and layered JSCC. The
9 ber of receivers, with possibly differentreconstructionquality allocationof source layersto differentusers can be optimized
0 that depends on the channel conditions. Analog transmission according to different criteria, as discussed in Section III.
:
v finds its theoretical justification in the fact that a Gaussian Performingclose to the theoreticallimits requires coding rate
i sourceunderthe quadraticdistortionmeasure is “matched”to adaptationwith veryfine granularity.This is accomplishedby
X
an Additive White Gaussian Noise (AWGN) channel[1]. Un- using“rateless”raptorencoders[9],whichareabletoproduce
r
a fortunately,thislucky“matching”conditiondoesnotgenerally any arbitrary number of coded symbols with a single, low
apply to a heterogeneous digital network (e.g., the Internet). complexity, encoding machine. The key idea of our practical
Such a network, which may be formed by wireline backbone JSCCschemeisthat(see[10]and[11])therate-distortion(R-
infrastructure combined perhaps with wireless component to D) limit for a smooth source over a symmetric channel can
the end users, is characterized by a large variability of the be closely approached by using scalar quantization followed
“capacity” from the media server to the individual users. byalinearencodingfunctionthatmapsdirectlytheredundant
The protocol stack is basically fixed. Therefore multimedia quantizationindicesintochannelinputsymbols.Inthelimitof
streaming applications are typically designed as “overlay” arbitrarily large block length and unlimited coding/decoding
systems, that is, they act at the application layer by using complexity, this approach provably achieves the quantizer
a fixed transport mechanism already built-in to the network. distortion DQ at bandwidth ratio H/C (channel uses per
In this paper, following [2], [3], [4], we model the transport source sample) where C denotes the channel capacity and H
mechanism as a binary erasure broadcast channel (BEBC). denotes the entropy rate of the quantization indices, modeled
This model captures the essential behavior of a network in as a discrete source. For a well-behaved source, the rate H
whichtheserversendsastreamofpacketswithoutend-to-end achievedbyentropy-codedquantizationisclosetotheoptimal
flow control (e.g., using User Datagram Protocol, UDP [5]) rate R(DQ) (where R(·) denotes the R-D function of the
andthese may belost becauseoferrorsor bufferoverflowsin source) [12], [13]. Hence, linear encoding of quantization
the network. The receivers have perfect knowledge of packet indices performs close to optimal. The main advantage of the
scheme,though,appearsatfinite blocklengthandlowencod- This region is achieved by time-sharing between the vertices
ing/decoding complexity. In fact, our approach eliminates the of the region’s dominant face, defined by the hyperplane
needforan explicitentropycodingstage after quantization.It L Rl =1.TheBEBCbelongstotheclassofstochastically
l=1 Cl
is well-known that standard entropy coding is ill-conditioned degraded broadcast channels [18]. This implies that any
P
with respect to residual channel decoding errors. In contrast, message to user l can be also decoded by all users j > l
our scheme reconstructs directly the quantization symbols (i.e., by the users with better channels).
using a “soft-bit” approach,i.e., using the estimated posterior The considered source model consists of s independent
log-likelihoodratios(LLRs)generatedbyaBeliefPropagation Gaussian source “components”. A source block of length k
(BP) iterative decoder that incorporates the a priori statistics is denoted by S ∈ Rs×k, where its ith row, denoted by
of the source (details can be found in [10], [11], [8] and S , hasi.i.d.elementsS ∼N(0,σ2), wherewe assume
(i,:) (i,j) i
[3]).Inthisway,the“catastrophic”behaviorofentropy-coded σ2 ≥···≥σ2. Note that S contains s×k source symbols.
1 s
quantization is greatly reduced. The encoder maps S into a channel input codeword X =
As noted above, we implement linear encoding by using f(k)(S), where f(k) : Rs×k → {0,1}n is the encoding
raptorcodes[9].Recently,raptorcodeshavebeenstandardized function. At each l-th user decoder, the received channel
as application layer Forward Error Correction (FEC) coding output Y is mapped into a reconstructed source array S =
l l
for Multimedia Broadcast/Multicast Services (MBMS) within g(k)(Y ), where g(k) : {0,1,e}n → Rs×k is a decoding
l l l
3GPP [14]. We hasten to say that our application is very function.The Mean SquareError (MSE) distortionfor thbe lth
differentfrom this standard. In 3GPP [14], the “static broad- decoderandtheith sourcecomponentisD(k) = 1E[kS −
i,l k (i,:)
casting” [15] of a common content file is considered, where S k2],wheretheexpectationiswithrespecttothejointk-
(i,:),l
each user gathers channel observations until it has received
dimensionalprobabilitydistributionof(S ,S )induced
non-erased symbols such that the whole file can be perfectly (i,:) (i,:),l
bby the source, by the channel erasures and by the coding
decoded.Usersmayhavedifferentdecodingdelaysdepending
scheme.Theaveragetotaldistortionatthelth dbecoderisgiven
ontheirerasureprobabilities.Onthecontrary,inthisworkwe
by D(k) = 1 s D(k).
consider a truly “real-time” multicasting where the decoding l s i=1 i,l
TheR-DfunctionofSisgivenbythe“reversewaterfilling”
delay is the same for all users, but each user reconstructs the P
parametric form [18]:
sourceatapossiblydifferentdistortionlevel,dependingonits
own channel capacity. Our scheme can be naturally applied 1 s 1 σ2 1 s
to static broadcasting (e.g., video on demand applications) R(D)= s 2max 0,log2 γi , D = s min{γ,σi2}.
by using it in conjunction with well-known protocols such Xi=1 n o Xi=1
as harmonic broadcasting [16], [17]. However, we do not For each pair (R,D) there exist a pair (γ,m) such that
investigate the details of this application in this paper. The D = 1{γm+ s σ2}, and R= 1 m 1log (σ2/γ),
s j=m+1 j s j=1 2 2 j
numericalresults of Section V show that the proposedcoding with γ < σ2 . Similar to a single Gaussian source [19],
scheme can achieve end-to-end distortion performance very it is easy tomPsee that parallel GaussiaPn sources are also
close to the theoreticallimits with finite block lengthand low successively refinable under MSE distortion. In particular,
encoding/decodingcomplexity. letting (R ,D ) and (R ,D ) denote two points on the R-
1 1 2 2
D function with D <D , and letting (m ,γ ) and (m ,γ )
II. BEBC AND PARALLELGAUSSIAN SOURCES 2 1 1 1 2 2
the corresponding parameters of the reverse waterfilling for-
The Binary Erasure Broadcast Channel (BEBC) has in- mula, the rate increment for each i-th source component is
put alphabet {0,1}, output alphabet {0,1,e}, (“e” denoting givenby max{0, 1 log (σ2/γ )}−max{0, 1 log (σ2/γ )},
2s 2 i 2 2s 2 i 1
erasure), and is defined by L channel transition probabilities corresponding to a total rate increment per source symbol
P(l)(y|x)=1−ǫ fory =x,P(l)(e|x)=ǫ andP(l)(y|x)=0
l l
for y 6=x,e. Without loss of generality, we let ǫ ≥...≥ǫ 1 1 γ m2 1 σ2
1 L ∆ =∆ m log 1 + log i , (2)
EanradsudreenCotheanbnyelC(lBE=C)1.−Thǫislcthhaenncaeplasecritvyesoafstahesiml-pthleBminoadreyl R s " 1(cid:18)2 2 γ2(cid:19) i=Xm1+12 2 γ2#
for a multicast network, in which an arbitrarily large number where the first term correspondsto the extra quantization bits
of users are grouped into L classes, each characterized by a forthem sourcecomponentsthatare(coarsely)quantizedto
1
different channel capacity. For simplicity, we shall refer to achievedistortionD , andthe secondterm correspondstothe
1
each class as a “user” since, in a multicast scenario, all users additional quantization bits to quantize additional m −m
2 1
belonging to the same class are indistinguishable and achieve source components that are discarded at distortion D , but
1
the same performance. need to be encoded to achieve distortion D .
2
It is well-known that, under mild conditions on the erasure An ideal (R-D achieving) multiresolution source code pro-
statistics the capacity region of the BEBC is given by [2] duces L layers for each source component at rates R′ , for
i,l
i = 1,...,s and l = 1,...,L (some layers may have zero
L
Rl rate). The L layers are encoded by a broadcast code for the
C = (R ,...,R ):R ≥0, ≤1, l∈{1,...,L} .
1 L l
n Xl=1 Cl (o1) mBEuBltiCresaotlurtaitoenslRaye=red(cRo1d,i.n.g.,scRhLe)me∈foCr.thFeigi.th1coshmopwosnetnhte.
Inspired by the bit-per-pixel convention in image compres-
10
sion literature, we define bandwidth expansion factor b as the 9 Dlopt
ratio of the number of channel uses to the number of source Dl
8
symbols, i.e. b = n/(ks). For a binary-input channel such 7
as the BEBC, b corresponds precisely to the ratio of (source- 6
channel) coded bits per source symbol. Then for each layer 5
l, we have 1s si=1Ri′,l = bRl and hence user l achieves the 4
distortion 3
P
2
l s l
1
D =R−1 R′ =R−1 b R . (3) 1
l s i,l  j 0
j=1i=1 j=1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
XX X C
   
III. OPTIMIZATIONPROBLEMS Fig. 2. Dlopt for capacity values ranging from 0.1 to 0.9. The Dl point
A layered source-channelcoding scheme can be optimized (stars) represent the solutions of the MMDP problem for the capacities
C = [0.3645, 0.81, 0.9] when 4 parallel sources with variances σ2 =
according to various criteria. Here we propose the following
[50, 12, 8, 5]areconsidered.
alternatives:
Min Bandwidth (MB): For given target distortions d ≥
1 Again, in [3] we treated the case s=1 that can be cast as
··· ≥ d , we wish to minimize b. Using (3) and (1), the
L a simple linear program.In the generalcase s>1, we obtain
problem can be formulated as
the followingconvexproblemwith respect to the variablesα,
minimize b {Rl} and {Di,l} :
l
minimize α
subject to b R ≥R(d ), ∀l
j l
L
R
Xj=1 subject to l ≤1, R ≥0, ∀l
l
L Rl ≤1, R ≥0, ∀l (4) Xl=1 Cl
C l 0≤D ≤σ2, ∀i,l
l=1 l i,l i
Letting R(d0) = 0, we eXasily see that a necessary condition 1 s 1log σi2 ≤b l R , ∀l
for optimality is that the inequalities bRl ≥R(dl)−R(dl−1) s i=1 2 2 Di,l j=1 j
must hold with equality for all l =1,...,L. Replacing these sX X
into the capacity region constraint, we obtain the solution D ≤αDopt, ∀ l (7)
i,l l
b⋆ = L R(dl)−R(dl−1). (5) In Fig. 2, we plotXiD=1opt for capacity values ranging from
Cl l
Xl=1 0.1to0.9.ThepointsDl representthesolutionoftheMMDP
Min Weighted Total Distortion (MWTD): For given b and optimizationproblemforthecapacitiesandthesetofvariances
non-negativeweights{wl},we wishto minimize Ll=1wlDl. we use in Section V for the simulations. For each considered
In [3],thecases=1issolvedinclosedform.Thegeneral capacity, the distortion penalty of each user can observed by
cases>1treatedhereismoredifficult.However,Pwecancast comparing the curve Dopt with the distortion points D .
l l
MWTD as a convexoptimization problem with respect to the
variables{R}and{D }thatcanbesolvedbystandardtools IV. JOINT SOURCE-CHANNEL CODING FORTHEBEBC
l i,l
byincorporatingthereversewaterfillingsolutionoftheR(D) In this section we present some code design examples that
into the problem itself. We have approach the theoretical limits obtained in (5), (6) and (7).
L s Generalizing the scheme proposed in [3], the layered JSCC
1
minimize w D scheme for parallelGaussian sourcesis based on the concate-
l i,l
s
l=1 i=1 ! nation of scalar embedded quantizers with linear encoding
X X
L functions, implemented by the same basic raptor encoding
R
subject to l ≤1, Rl ≥0, ∀l “machine”. We use an embedded scalar quantizer Q : R →
C
l=1 l {0,1}N, optimizedfora Gaussiansource∼N(0,1).In order
X
0≤Di,l ≤σi2, ∀i,l to quantize the i-th source component, the source symbols
1 s 1 σ2 l are scaled by 1/σi and the inverse scaling of the quantizer
log i ≤b R , ∀l (6) reconstruction points is applied at the decoder.
s 2 2 D j
Xi=1 i,l Xj=1 Let U(i) = Q(Si,:/σi) denote the sequence of quanti-
Min-MaxDistortionPenalty(MMDP):Forgivenb,wewish zation indices, formatted as an N × k binary array. The
tominimizemaxl∈LDl/Dlopt,whereDlopt =R−1(bCl)isthe j-th row of U(i), denoted by Uj(i,:), is referred to as the
individual R-D bound for user l. the jth “bit-plane”. Without loss of generality, we let U(i)
1,:
Fig.1. Amultiresolutionsourceencodercreates3layerswithDi,1>Di,2>Di,3forthreeclassesofuserswithcapacities C1≤C2≤C3.Thechannel
encoder-decoder pairatrateC isshownasblocks CE(C)andCD(C),respectively.
reflectingthebit-planes.Thedetailsofthisapproximationshall
60
be given later on. For the time being, it is sufficient to say
50 thattheresultofthisfirst designstep isa bit-planeallocation,
defined by the integers {p }, such that p is the number of
40 i,l i,l
bit-planes of the i-th source componentthat must be used for
SNR30 source reconstruction at the l-th decoder (we define pi,0 = 0
SNR (R’) for notation convenience).
20 rd
SNR (R’) Then,foreachl,weformthel-thlayerchannelcodewordby
eq
10 SNRΓ (R’) collectingall thequantizationbit-planeswithindicespi,l−1 <
j ≤p for all source components i=1,...,s, and mapping
00 2 4 6 8 10 themii,nldependently onto binary codeword x via a linear
i,j
R’
raptor encoder. Building on our results in [8], we use a
systematic binary raptor encoder such that the source bit-
Fig.3. SNRvs.Rate,foraGaussiansourcewithσ=1,obtainedbytheR-
Dfunction(SNRrd(R′)),bythesetofR-Dpointsoftheembeddedquantizer planes form the systematic part, and the produced codeword
bwyithDcQo(oRrd′i)na=tesΓ2`−P2piR=′1wHhie,rDeΓQ(=p)1´.5fo,r(SpN=RΓ1(,R..′).),.10 (SNReq(R′))and cdoertareilssp).onWdestuosethtehe“pfaorliltoyw”isnygmdbeoglrsee(sdeeist[r9i]b,u[t1io1n],[[98]]:, [3] for
denote the sign bit-plane, and U(i),...,U(i) denote the Ω(x) = 0.008x+0.494x2+0.166x3+0.073x4+0.083x5
2,: N,:
magnitude bit-planes with decreasing order of significance. +0.056x8+0.037x9+0.056x19+0.025x65+0.003x66.
ThequantizeroutputU(i) formsadiscretememorylesssource,
with entropy rate H(i) = 1H(U(i)) (in bits/source symbol). Finally, the si=1pi,L codewords x1,1,...,xs,ps,L are trans-
This can be decomposed aks H(i) = N H(i), where the mitted in sePquence. This corresponds to the time-sharing
j=1 j nature of the BEBC capacity region. We note here that the
conditional entropy rates of the bit-planes are denoted by
H(i) = 1H(U(i)|U(i),...,U(i) ),foPrj =1,...,N.TheR- verysame schemecanbe appliedwith minormodificationsto
j k j,: 1,: j−1,: any underlying L-user broadcast channel, if one is willing to
Dfunctionachievedbyembeddedscalarquantizationfollowed
accept the performance penalty of using time-sharing instead
by entropy coding of the quantization indices is given by the
of an optimal broadcast coding strategy.
set of R-D points with coordinates pj=1Hj(i),DQ,i(p) , Ateachlth decoder,allcodewordsfrom(i,1)to(i,pi,l)for
for p = 1,...,N, where the quan(cid:16)tizer distortion can b(cid:17)e all source components i = 1,...,s are sequentially decoded
P
approximatedby the followingfunctionDQ,i(r)≈Γσi22−2r, usingthesuccessivemulti-stagedecoderdescribedin[8],[11].
where Γ is a multiplicative penalty factor [12], [13] and for Each (i,j)-th stage of the multistage decoder is an iterative
the ith source r= pj=1Hj(i) is the total rate of p bit-planes. Belief Propagation decoder that incorporates the information
We define reconstruction signal-to-noise-ratio SNR=∆ correspondingtotheaprioriprobabilitydistributionofthebit-
P
10log σ2.The SNR-R curve of the embedded scalar quan- planesymbolsandtheposteriorlog-likelihoodratiosproduced
10 D
tizerusedinthisworkisshowninFig.3forσ =1.Wenotice by the decoders at previous stages, (i,1)...,(i,j −1). We
that the familiar 6dB per quantization bit improvementin the omit the detailed description of the decoder because of space
reconstruction SNR is achieved within a gap Γ≈1.5. limitation.Ifthecodeiswelldesigned,thedistortionachieved
The code design proceedsas follows: first, we approximate at decoderl is essentially equal to the quantizationdistortion,
the solution of the desired optimization problem (e.g., MB, given by Dl ≈ Γs si=1σi22−2Ppj=i,1lHj(i).
MWTP, MMDP said above) by taking into account that the In the rest of this section we discuss the details of the bit-
P
actual quantizer R-D function has gap Γ and that the source plane allocation to the layers and of the choice of the coding
codingratesmustbequantizedintoatmostN discretevalues, rates. Let {D⋆ } denote the set of distortions resulting from
i,l
TABLEI TABLEII
MMDP,b=2.5 MWTD,b=8.75
bmin=2.58 S(1,:) S(2,:) S(3,:) S(4,:) bmin=8.83 S(1,:) S(2,:) S(3,:) S(4,:)
D∗ 9.8 9.8 8 5 D∗ 0.22 0.22 0.22 0.22
:,1 :,1
D∗ 1.74 1.74 1.74 1.74 D∗ 0.09 0.09 0.09 0.09
:,2 :,2
D∗ 1.27 1.27 1.27 1.27 D∗ 0.09 0.09 0.09 0.09
:,3 :,3
MMDP,Simulation Result MWTD,Simulation Result
ˆbJSCC =3.59 S(1,:) S(2,:) S(3,:) S(4,:) ˆbJSCC =10.75 S(1,:) S(2,:) S(3,:) S(4,:)
p:,1 3 0 0 0 p:,1 5 4 4 4
D:,1 5.27 12 8 5 D:,1 0.33 0.32 0.21 0.13
p:,2 4 3 2 2 p:,2 6 5 5 4
D:,2 1.32 1.26 3.46 2.16 D:,2 0.08 0.08 0.05 0.13
p:,3 4 3 3 2 p:,3 6 5 5 4
D:,3 1.32 1.26 0.84 2.16 D:,3 0.08 0.08 0.05 0.13
the solution of one of the optimization problems of Section [2] R.L.UrbankeandA.D.Wyner,“Packetizing fortheerasurebroadcast
III. We choose pi,l such that Γσi22−2Ppj=i,1lHj(i) ≈Di⋆,l. cInhfaonrnmealtwiointhTahneoinrtyerannedtaSptaptliisctaitciso,nP,”oirntlaPnrdo,c.MInEt,.1C9o9n7f,.Cp.om93b.inatorics,
Since p must be an integer, this step requires some ad-
i,l [3] O. Y. Bursalioglu, M. Fresia, G. Caire, and H. V. Poor, “Joint source-
hoc approximation. Recalling that the bit-planes with indices channelcodingattheapplication layer,”inProc.IEEEDataCompres-
opfi,lc−o1d<ewjor≤d xpi,l misugstivbeendbeycoded by user l, the block length [4] sAio.nFC. oDnafenrae,ncRe., SGnoowwabiikradr,,URT.,2P0a0la9n.ki, B. Hassibi, and M. Effros,
i,j “Capacity of wireless erasure networks,” IEEE Trans. Inform. Theory,
vol.52,no.3,pp.789–804, Mar.2006.
kHj(i) [5] J. F. Kurose and W. K. Ross, Computer Networking: A Top-Down
n = (8)
i,j Approach(4thEdition). Reading: AddisonWesley,2007.
C −δ
l l [6] U. Mittal and N. Phamdo, “Hybrid digital-analog (HDA) joint source-
where δ denotes a suitable “gap-to-capacity”(or “overhead”) channel codes for broadcasting and robust communications,” IEEE
l
Trans.Inform.Theory,vol.48,no.5,pp.1082–1102, May2002.
of the raptor code (see [9]). The overallbandwidth expansion
[7] D. S. Taubman and M. W. Marcellin, JPEG2000: Image Compression
factor of the scheme is given by Fundamentals,Standards,andPractices. Norwell:KluwerAcademics
Publishers, 2002.
1 L s pi,l 1 L s pi,l H(i) [8] O. Y. Bursalioglu, M. Fresia, G. Caire, and H. V. Poor, “Lossy joint
j
b= ni,j = . source-channel coding using raptor codes,” Int. Journal of Digital
sk s C −δ
Xl=1Xi=1j=pXi,l−1+1 Xl=1Xi=1j=pXi,l−1+1 l l [9] MA.ulStihmoekdroiallaBhrio,a“dRcaapsttionrg,covdoel.s,”ArItEicEleEIDTra1n2s4.6I8n5fo,r1m8.pTahgeeos,ry2,0v0o8l.. 52,
b
V. RESULTS no.6,pp.2551–2567,June2006.
[10] M. Fresia and G. Caire, “Combined error protection and compression
Inthissection,wereportafewsimulationresultsforsource with turbocodes forimage transmission usingaJPEG2000-like archi-
block length k = 10000, with L = 3 users with capacities tecture,” inProc.IEEEInt.Conf. Image Proc.,Atlanta, GA,2006,pp.
821–824.
C = [0.3645, 0.81, 0.9] and s = 4 source components with
[11] ——,“Apractical approach tolossyjointsource-channel coding,”Jan.
variances σ2 =[50, 12, 8, 5]. 2007,submittedtoIEEETrans.Inform.Theory.
For the MMDP optimization scenario, we fix b = 2.5. [12] H. Gish and J. N. Pierce, “Asymptotically efficient quantizing,” IEEE
The optimal values of D∗ obtained by solving (7) are given Trans.Inform.Theory,vol.14,no.5,pp.676–683,Sep.1968.
i,l
in Tab. I (top). The allocation of bit-planes to layers are [13] J.Ziv,“Onuniversalquantization,”IEEETrans.Inform.Theory,vol.31,
no.3,pp.344–347,May1985.
done as described earlier. The resulting D values and the
i,l [14] M.Luby,T.Gasiba,T.Stockhammer,andM.Watson,“Reliable multi-
bit-plane allocation variables p ’s are reported in Tab. I
i,l media download delivery in cellular broadcast networks,” IEEETrans.
(bottom). As a sanity check, if we use the values D of
i,l Broadcasting, vol.53,no.1,pp.235–246,March2007.
Tab. I (bottom) to obtain target distortions dl’s in the MB [15] N. Shulman and M. Feder, “Static broadcasting,” in Proc. IEEE Int.
optimization problem, we find that the required bandwidth Symp.Inform.Theory,Sorrento, Italy,2000.
expansion factor is bmin = 2.58. Hence, the rounding of [16] L. S. Juhn and L. M. Tseng, “Harmonic broadcasting for video-on-
the optimal distortion values has only a marginal effect on demandservice,”IEEETrans.Broadcasting,vol.43,no.3,pp.268–271,
the overall theoretical bandwidth efficiency. The actual linear Sep.1997.
coding and soft-bit reconstructionJSCC scheme, after careful [17] J. F. Paris, S. W. Carter, and D. D. E. Long, “Efficient broadcasting
protocolsforvideoondemand,”inProc.Symp.Modeling,Analysisand
optimizationoftheraptorcoderedundancyoverhead,achieves
SimulationofComputerandTelecommunicationSystems,Houston,TX,
the desired distortions at ˆbJSCC = 3.59. Similar results are 1998,pp.127–132.
reported for the MWTD optimization scenario, where we fix [18] T.CoverandJ.Thomas,Elements ofInformationTheory. NewYork:
b = 8.75. The solution of (6) is reported in Tab. II (top) and Wiley,1991.
the actual code performance,obtained by simulation, is given [19] W. H. R. Equitz and T. M. Cover, “Successive refinement of informa-
in Tab. II (bottom). tion,” IEEE Trans. Inform. Theory, vol. 37, no. 2, pp. 269–275, Mar.
1991.
REFERENCES
[1] M.Gastpar,B.Rimoldi,andM.Vetterli,“Tocodeornottocode:lossy
source-channel communication revisited,” IEEETrans.Inform.Theory,
vol.49,no.5,pp.1147–1158, May2003.