department of electrical and computer engineering …ertem/publications_files/it...1784 ieee...

1782 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 4, APRIL 2010

Wyner–Ziv Coding Over Broadcast Channels:Digital Schemes

Jayanth Nayak, Ertem Tuncel, Member, IEEE, and Deniz Gündüz, Member, IEEE

Abstract—This paper addresses lossy transmission of a commonsource over a broadcast channel when there is correlated sideinformation at the receivers, with emphasis on the quadraticGaussian and binary Hamming cases. A digital scheme thatcombines ideas from the lossless version of the problem, i.e.,Slepian–Wolf coding over broadcast channels, and dirty papercoding, is presented and analyzed. This scheme uses layeredcoding where the common layer information is intended forboth receivers and the refinement information is destined onlyfor one receiver. For the quadratic Gaussian case, a quantitycharacterizing the combined quality of each receiver is identifiedin terms of channel and side information parameters. It is shownthat it is more advantageous to send the refinement informationto the receiver with “better” combined quality. In the case whereall receivers have the same overall quality, the presented schemebecomes optimal. Unlike its lossless counterpart, however, theproblem eludes a complete characterization.

Index Terms—Broadcast channels, dirty paper coding,Slepian–Wolf coding, Wyner–Ziv coding.

I. INTRODUCTION

C ONSIDER a sensor network of nodes taking peri-odic measurements of a common phenomenon. We study

the communication scenario in which one of the sensors is re-quired to transmit its measurements to the other nodes over abroadcast channel. The receiver nodes are themselves equippedwith side information unavailable to the sender, e.g., measure-ments correlated with the sender’s data. This scenario, which isdepicted in Fig. 1, can be of interest either by itself or as part ofa larger scheme where all nodes are required to broadcast theirmeasurements to all the other nodes.

Finding the capacity of a broadcast channel is a longstandingopen problem, and thus, limitations of using separate source andchannel codes in this scenario may never be fully understood. Incontrast, a very simple joint source-channel coding strategy is

Manuscript received July 18, 2008; revised November 23, 2009. Current ver-sion published March 17, 2010. The material in this paper was presented in partat the 2008 Information Theory Workshop (ITW) Porto, Portugal .

J. Nayak was with the University of California, Riverside, CA. He is now withMayachitra, Inc., Santa Barbara, CA 93111 USA (e-mail: [email protected]).

E. Tuncel is with the Department of Electrical Engineering, University ofCalifornia, Riverside, CA 92521 USA (e-mail: [email protected]).

D. Gündüz was with Department of Electrical, Princeton University,Princeton, NJ 08544 USA, and Stanford University, Stanford, CA. He is nowwith the Centre Tecnológic de Telecomunicacions de Catalunya (CTTC),08860, Castelldefels, Barcelona, Spain (e-mail: [email protected];[email protected]).

Communicated by I. Kontoyiannis, Associate Editor for Shannon Theory.Color version of Figure 2 in this paper is available online at http://ieeexplore.

ieee.org.Digital Object Identifier 10.1109/TIT.2010.2040891

optimal for the special case of lossless coding [20]. More specif-ically, it was shown in [20] that in Slepian–Wolf coding overbroadcast channels (SWBC), as the lossless case was referred to,for a given source , side information , and a broad-cast channel , lossless transmission (in the Shannonsense) is possible with channel uses per source symbol if andonly if there exists a channel input distribution such that

(1)

for . In the optimal coding strategy, every typicalsource word is randomly mapped to a channel codeword

, where and are so that . If (1) is satisfied,there exists a channel codebook such that with high probability,there is a unique index for which is jointly typical withthe side information and is jointly typical with thechannel output simultaneously, at any receiver . This resultexhibits some striking features which are worth repeating here.

i) The optimal coding scheme is not separable in the clas-sical sense, but consists of separate components that per-form source and channel coding in a broader sense. Thisresults in the separation of source and channel variablesas in (1).

ii) If the broadcast channel is such that the same input distri-bution achieves capacity for all individual channels, then(1) implies that one can utilize all channels at full ca-pacity. Binary symmetric channels and Gaussian channelsare the widely known examples of this phenomenon.

iii) The optimal coding scheme does not explicitly involvebinning, which is commonly used in network informationtheory. Instead, with the simple coding strategy of [20],each channel can be thought of as performing its own bin-ning. More specifically, the channel output at eachreceiver can be viewed as corresponding to a virtual bin1

containing all source words that map to channelcodewords jointly typical with . In general, thevirtual bins can overlap and correct decoding is guaran-teed by the size of the bins, which is about .

In this paper, we consider the general lossy coding problem inwhich the reconstruction of the source at the receivers need notbe perfect. We shall refer to this problem setup as Wyner–Zivcoding over broadcast channels (WZBC). We present a codingscheme for this scenario and analyze its performance in thequadratic Gaussian and binary Hamming cases. This schemeuses ideas from SWBC [20] and dirty paper coding (DPC) [3],

1The bins can also be viewed as exponentially sized lists and a similarstrategy that interprets the decoding as the intersection of exponentiallysized lists was derived independently in [10] and [20]. Another alternativebinning-based coding scheme that achieves the same performance using blockMarkov encoding and backward decoding can be found in [7].

0018-9448/$26.00 © 2010 IEEE

Authorized licensed use limited to: Univ of Calif Riverside. Downloaded on June 10,2010 at 16:08:32 UTC from IEEE Xplore. Restrictions apply.

NAYAK et al.: WYNER–ZIV CODING OVER BROADCAST CHANNELS:DIGITAL SCHEMES 1783

Fig. 1. Block diagram for Wyner–Ziv coding over broadcast channels.

[6] as a starting point. The SWBC scheme is modified a) toallow quantization of the source, and b) to handle channel stateinformation (CSI) at the encoder by using DPC. The modifi-cation with DPC is then employed in a layered transmissionscheme with receivers, where there is common layer(CL) information destined for both receivers and refinementlayer (RL) information meant for only one of the receivers. Thechannel codewords corresponding to the two layers are super-posed and the resultant interference is mitigated using DPC. Weshall briefly discuss other possible layered schemes obtained byvarying the encoding and the decoding orders of the two layersand using successive coding or DPC to counteract the interfer-ence, although for the bandwidth matched Gaussian and binaryHamming cases, we observe that these variants perform worse.

DPC is used in this work in a manner quite different from theway it was used in [2], which concentrated on sending private in-formation to each receiver in a broadcast channel setting, wherethe information that forms the CSI and the information that isdirty paper coded are meant for different receivers. Therefore,although the DPC auxiliary codewords are decoded at one of thereceivers, unlike in our scheme, this is of no use to that receiver.For our problem, this difference leads to an additional interplayin the choice of channel random variables. The DPC techniquesin this work are most similar to those in [17], [21], where, as inour scheme, the CSI carries information about the source andhence decoding the DPC auxiliary codeword helps improve theperformance. However, our results indicate a unique feature ofDPC in the framework of WZBC. In particular, in our layeredscheme, the optimal Costa parameter for the quadratic Gaussianproblem turns out to be either 0 or 1. When it is 0, there is ef-fectively no DPC, and when it is 1, the auxiliary codeword isidentical to the channel input corrupted by the CSI. To the bestof our knowledge, although the latter choice is optimal for bi-nary symmetric channels, it has never been shown to be optimalfor a Gaussian channel in a scenario considered before.

When an appropriately defined “combined” channel and sideinformation quality is constant at each receiver, the new schemeis shown to be optimal in the quadratic Gaussian case. We alsoderive conditions for the same phenomenon to occur in the bi-nary Hamming case, although the expressions are not as elegantas in the quadratic Gaussian problem. Unlike in [20], however,the scheme that we derive is not always optimal. A simple al-ternative approach is to separate the source and channel coding.

Both Gaussian and binary symmetric broadcast channels are de-graded. Hence their capacity regions are known [4] and further,there is no loss of optimality in confining ourselves to two layersource coding schemes. The corresponding source and side in-formation pairs are also degraded. Although a full character-ization of the rate-distortion performance is available for thequadratic Gaussian case [18], only a partial characterization isavailable for the binary Hamming problem [16], [18]. In anycase, we obtain an achievable distortion tradeoff of separatesource and channel coding by combining the known rate-distor-tion results with the capacity results. For the quadratic Gaussianproblem, we show that our scheme always performs at least aswell as separate coding. The same phenomenon is numericallyobserved for the binary Hamming case.

For the two examples we consider, a second alternative is un-coded transmission if there is no bandwidth expansion or com-pression. This scheme is optimal in the absence of side infor-mation at the receivers in both the quadratic Gaussian and bi-nary Hamming cases. However, in the presence of side informa-tion, the optimality may break down. We show that, dependingon the quality of the side information, our scheme can indeedoutperform uncoded transmission as well. In particular, if thecombined quality criterion chooses the worse channel as the re-finement receiver (because it has much better side information),then our layered scheme outperforms uncoded transmission forthe quadratic Gaussian problem.

The paper is organized as follows. In Section II, we formallydefine the problem and present relevant past work. Our mainresults are presented in Sections III and IV, namely the exten-sions of the scheme in [20] that we develop for the lossy sce-nario. We then analyze a layered scheme in particular for thequadratic Gaussian and binary Hamming cases in Sections Vand VI, respectively. For these cases, we compare the derivedschemes with separate source and channel coding, and with un-coded transmission. Section VII concludes the paper by sum-marizing the results and pointing to future work.

II. BACKGROUND AND NOTATION

Let be randomvariables denoting a source with independent and iden-tically distributed (i.i.d.) realizations. Source is to betransmitted over a memoryless broadcast channel defined by

.



Decoder has access to side information in addition tothe channel output . Let single-letter distortion measures

be defined at each receiver, i.e.

for .

Definition 1: An code consists of anencoder

and decoders at each receiver

The rate of the code is channel uses per source symbol.

Definition 2: A distortion tuple is said to beachievable at a rational rate if for every , there existssuch that for all integers with , thereexists an code satisfying

where and denotes the channel outputcorresponding to .

In this paper, we present some general WZBC techniquesand derive the corresponding achievable distortion regions. Westudy the performance of these techniques for the followingcases.

• Quadratic Gaussian: All source and channel variables arereal-valued, and we use the notation to denote the vari-ance of any Gaussian random variable . The source andside information are jointly Gaussian and the channels areadditive white Gaussian, i.e., where isGaussian and is independent of . There is an inputpower constraint on the channel:

where . Without loss of generality, we as-sume that andwith and . Thus, is the meansquared-error in estimating from , or equivalently,from since . Reconstruction quality is mea-sured by squared-error distance: .

• Binary Hamming: All source and channel alphabets arebinary. The source is , where denotes theBernoulli distribution with . The channels are bi-nary symmetric with transition probabilities , i.e.,

where and and are inde-pendent with denoting modulo 2 addition (or the XORoperation). The side information sequences at the receiversare also noisy versions of the source corrupted by pas-sage through virtual binary symmetric channels; that is,

with and and areindependent. Reconstruction quality is measured by Ham-ming distance: .

The problems considered in [9], [14], [20] can all be seen asspecial cases of the WZBC problem. However, the quadraticGaussian and the binary Hamming cases with non-trivial sideinformation have never, to our knowledge, been analyzed be-fore. Nevertheless, separate source and channel coding and un-coded transmission are obvious strategies. We shall evaluate theperformance of these alternative strategies and present numer-ical comparisons with our proposed scheme.

A. Wyner–Ziv Coding Over Point-to-Point Channels

Before analyzing the WZBC problem in depth, we shallbriefly discuss known results for Wyner–Ziv coding over apoint-to-point channel, i.e., the case . Since ,we shall drop the subscripts that relate to the receiver. TheWyner–Ziv rate-distortion performance is characterized in [23]as

(2)

where is an auxiliary random variable, and the capacityof the channel is well-known (cf. [4]) to be

It is then straightforward to conclude that combining separatesource and channel codes yields the distortion

(3)

On the other hand, a converse result in [15] shows that evenby using joint source-channel codes, one cannot improve thedistortion performance further than (3).

We are further interested in the evaluation of ,as well as in the test channels achieving it, for the quadraticGaussian and binary Hamming cases. We will use similar testchannels in our WZBC schemes.

1) Quadratic Gaussian: It was shown in [22] that the optimalbackward test channel is given by

where and are independent Gaussians. For the rate we have2

(4)

The optimal reconstruction is a linear estimate

which yields the distortion

(5)

2All logarithms are base 2.



and therefore

(6)

2) Binary Hamming: It was implicitly shown in [23] that theoptimal auxiliary random variable is givenby

where are all independent, and are andwith and , respectively, and

is an erasure operator, i.e.,

This choice results in

(7)

where

with denoting the binary convolution, i.e.,, and denoting the binary entropy function, i.e.,

It is easy to show that when , is increasingin and decreasing in .

Since and, the corresponding optimal reconstruction function

boils down to a maximum likelihood estimator given by

or, and, and

The resultant distortion is given by

(8)

implying together with (7) that

(9)

where the extra constraint is imposed because is aprovably suboptimal choice. It also follows from the discussionin [23] that there exists a critical rate above which theoptimal test channel assumes and ,and below which it assumes and . Thereason why we discussed other values of above is because

we will use the test channel in its most general form in all WZBCschemes.

B. A Trivial Converse for the WZBC Problem

At each terminal, no WZBC scheme can achieve a distor-tion less than the minimum distortion achievable by ignoringthe other terminals. Thus

(10)

where is the capacity of channel . For the source-channelpairs we consider, (10) can be further specialized. For thequadratic Gaussian case, we obtain using (6) and

that

(11)

For the binary Hamming case, using (9) and ,the converse becomes

C. Separate Source and Channel Coding

For a general source and channel pair, the source and channelcoding problems are extremely challenging. The set of allachievable rate triples (common and two private rates) forgeneral broadcast channels are not known. The correspondingsource coding problem has not been explicitly considered inprevious work either. But there is considerable simplificationin the quadratic Gaussian and binary Hamming cases since thechannel and the side information are degraded in both cases:we can assume that one of the two Markov chains,or , holds (for arbitrary channel input ) for thechannel, and similarly either or holdsfor the source. The capacity region for degraded broadcastchannels is fully known. In fact, since any information sent tothe weaker channel can be decoded by the stronger channel,we can assume that no private information is sent to the weakerchannel. As a result, two layer source coding, which has beenconsidered in [16], [18], [19], is sufficiently general.

To be able to analyze and simul-taneously, we denote the random variables, rates, and distortionlevels associated with the ood channel by the subscript andthose associated with the ad one by , i.e., the channel variablesalways satisfy where is either 1 or 2 and takesthe other value.

Let denote the capacity region for channel uses3. Asshown in [1], [5], is the convex closure of allsuch that there exist a channel input and an auxiliary

3As a matter of fact, has a meaningful interpretation only when is aninteger. We use simply to refer to the rates in scaled by a commonfactor of .



random variable satisfying , thepower constraint (if any) , and

(12)

(13)

Let denote the region of rates that must be sentto each source decoder to enable the receivers to reconstruct thesource within the respective distortions and . A distor-tion pair is achievable by separate source and channelcoding with channel uses per source symbol if and only if

Note that we use cumulative rates at the good receiver.Despite the simplification brought by degraded side infor-

mation, there is no known complete single-letter characteriza-tion of for all sources and distortion measures when

. Let be defined as the convex closure ofall such that there exist source auxiliary random vari-ables with eitheror , and reconstruction functions

satisfying

(14)

for , together with

(15)

and

(16)

if , and

(17)

if . It was shown in [16] thatwhen . On the other hand, [18] showed

that even when , forthe quadratic Gaussian problem. For all other sources anddistortion measures, we only knowin general when . We shall present explicit ex-pressions for the complete tradeoff in the quadratic Gaussiancase in Section V and an achievable tradeoff for the binaryHamming case in Section VI.

D. Uncoded Transmission

In the bandwidth-matched case, i.e., when , if thesource and channel alphabets are compatible, uncoded transmis-sion is a possible strategy. For the quadratic Gaussian case, thedistortion achieved by uncoded transmission is given by

(18)

for . This, in turn, is also because the channel is the sameas the test channel up to a scaling factor. More specifically, when

is transmitted and corrupted by noise , one can write

with , where is an appropriatelyscaled version of the received signal and

Substituting this into (5) then yields (18). Comparing with (11),we note that (18) achieves only when orwhen , which, in turn, translate to trivial or zero

, respectively.For the binary Hamming case, this strategy achieves the dis-

tortion pair

(19)

for . That is because the channel is the same as thetest channel that achieves with . The distortionexpression in (19) then follows using (8). One can also show that(19) coincides with only when or .Once again, these respectively correspond to trivial and zero

.

III. A BASIC WZBC SCHEME

In this section, we present a basic coding scheme that we shallthen develop into the schemes that form the main contributionsof this paper. In what follows, we only present code construc-tions for discrete sources and channels. The constructions canbe extended to the continuous case in the usual manner. Ourcoding arguments rely heavily on the notion of typicality. Givena random variable defined over a discrete alphabet

the typical set at block length is defined as [12]

where denotes the number of times appears in .Our scheme, termed Common Description Scheme (CDS),

is a basic extension of the scheme in [20] where the sourceis first quantized before transmission over the channel. Eventhough our layered schemes are constructed for the case of

receivers, CDS can be utilized for any . Unlike in[20], where typical source words are placed in one-to-one corre-spondence with a channel codebook, the source words are firstmapped to quantized versions and it is these quantized versionsthat are mapped to the channel codebook. Like [20], there is noexplicit binning, but the channel performs virtual binning. Be-fore discussing the performance of the CDS, we shall present anextension of the CDS for a more general coding problem. Thisextension will be later utilized in our more sophisticated layeredscheme.

Suppose that there is CSI available solely at the encoder, i.e.,the broadcast channel is defined by the transition probability

and the CSI with some, where is some fixed distribution defined on the CSI

alphabet , is available non-causally at the encoder. Given asource and side information at the decoders , codes

and achievability of distortion pairs is definedas in the WZBC scenario except that the encoder now takes theform . The following theorem characterizes



the performance of an extension of the CDS, which we termCDS with DPC.

Theorem 1: A distortion pair is achievable atrate if there exist random variables ,and functions withand such that

(20)

(21)

for .Proof: The code construction is as follows. For fixed

, a source codebookis chosen from . A set of bins

, where each is chosen randomly at uniformfrom , is also constructed. Given a source wordand CSI , the encoder tries to find a pair such that

and .If it is unsuccessful, it declares an error. If it is suc-cessful, the channel input is drawn from the distribution

. At terminal , the decoder goesthrough all pairs until itfinds the first pair satisfying and

simultaneously. If there is nosuch pair, the decoder sets . Once is decided,coordinate-wise reconstruction is performed using withand .

We define the error events as

Using standard typicality arguments, it can be shown that forfixed , if

and

with as , then , ,and all vanish large enough . Similarly, it followsthat if

and

then for some as ,

which also vanishes for large thanks to (20). This completesthe proof.

Note that, if is a trivial random variable, independent ofthe channel, the scenario becomes the original WZBC setup andCDS with DPC becomes CDS. By equating and , we obtainthe following corollary that characterizes the performance of theCDS.

Corollary 1: A distortion tuple is achievableat rate for the WZBC problem if there exist random variables

, and functions withsuch that

(22)

(23)

for .

Corollary 2: The coding scheme in the proof of Theorem 1can also decode successfully.

Proof: Define

It then suffices to show that vanishes for large enough. Indeed, since ,

The assumption is not restrictive at all,because otherwise no information can be delivered to terminal

to begin with.

The significance of Corollary 2 is that decodingprovides information about the CSI . This information, inturn, will be very useful in our layered WZBC scheme wherethe CSI is self-imposed and related to the source itself.

Examining the proof of Theorem 1, we notice an apparentseparation between source and channel coding in that the sourceand channel codebooks are independently chosen. Furthermore,successful transmission is possible as long as the source codingrate for each terminal is less than the corresponding channelcoding rate for a common channel input. However, the decodingmust be jointly performed and neither scheme can be split intoseparate stand-alone source and channel codes. Nevertheless,



Fig. 2. Components of LDS: and are the first and second stage quantized source words. is binned and the bin index is channel codedto in the usual sense. , on the other hand, is mapped to using CDS with DPC, where serves as the CSI. The two channel codewordsare then superposed, resulting in . Decoding of is exactly as in CDS with DPC at both receivers. In decoding of , the refinement channel decodermakes use of both the channel output and the auxiliary code word to decode the bin index .

due to the quasi-independence of the source and channel code-books we shall refer to source codes and channel codes sepa-rately when we discuss layered WZBC schemes. This quasi-sep-aration was shown to be optimal for the SWBC problem and wastermed operational separation in [20].

IV. A LAYERED WZBC SCHEME

In this section, we focus on the case of receivers.In CDS, the same information is conveyed to both receivers.However, since the side information and channel characteristicsat the two receiving terminals can be very different, we mightbe able to improve the performance by layered coding, i.e., bynot only transmitting a common layer (CL) to both receivers butalso additionally transmitting a refinement layer (RL) to one ofthe two receivers. The resultant interference between the CL andRL can then be mitigated by successive decoding or by dirtypaper encoding. Since there are two receivers, we are focusingon coding with only two layers because intuitively, more layerstargeted for the same receiver can only degrade the performance.

Unless the better channel also has access to better side in-formation, it is not straightforward to decide which receivershould receive only the CL and which should additionally re-ceive the RL. We shall therefore refer to the decoders as theCL decoder and the RL decoder (which necessarily also de-codes the CL) instead of using the subscripts 1 and 2. For thequadratic Gaussian problem, we will later develop an analyt-ical decision tool. For all other sources and channels, one cancombine the distortion regions resulting from the two choices,namely, and and vice versa.

For ease of exposition, for a given choice of CL and RL de-coders, we also rename the source and channel random variablesby replacing the subscripts 1 and 2 by (for random variablescorresponding to the CL information or to the CL decoder) and

(for random variables corresponding to the RL information orto the receiver that decodes both CL and RL).

As mentioned earlier, the inclusion of an RL codewordchanges the effective channel observed while decoding the CL.It is on this modified channel that we send the CL using CDSor CDS with DPC, and the respective channel rate expressionsin (22) and (20) must be modified in a manner that we describein the following subsections where we also present the capacityof the effective channel for transmitting the RL. Each possibleorder of channel encoding and decoding (at the RL decoder)leads to a different scheme. We shall concentrate on the schemethat has the best performance among the four in the Gaussianand binary Hamming cases, deferring a discussion of the otherthree to Appendix A. In this scheme, illustrated in Fig. 2, theCL is coded using CDS with DPC with the RL codeword actingas CSI. We shall refer to this scheme as the Layered DescriptionScheme (LDS). We characterize the source and channel codingrates for LDS in the following. We will only sketch the proofsof the theorems, as they rely only on CDS with DPC, and otherstandard tools.

A. Source Coding Rates for LDS

The RL is transmitted by separate source and channel coding.In coding the source, we restrict our attention to systems wherethe communicated information satisfieswhere corresponds to the CL and is the RL. The source



coding rate for the RL is therefore (cf. [18]).This has to be less than the RL capacity. Due to the separabilityof the source and channel variables in the required inequalitieswe can say that a distortion pair is achievable if

Here, is the “capacity” region achieved by ei-ther LDS or any of its variations discussed in Appendix A,and is the set of all tripletsso that there exist and reconstruction functions

and satisfyingand

(24)

(25)

(26)

(27)

(28)

The subscripts and are used to emphasize transmission ofthe CL to receivers and , respectively. Similarly, the subscript

refers to transmission of RL to receiver .

B. Channel Coding Rates for LDS

The next theorem provides the effective channel rate regionfor LDS.

Theorem 2: Let be the union of allfor which there exist , , and with

and such that

(29)

(30)

(31)

Then .

Remark 1: The various random variables that appear in The-orem 2 have the following interpretation: and are thechannel outputs when the input is . and correspond tothe partial channel codewords that are superposed to form thechannel input. Finally is the auxiliary random variable usedin DPC with forming the CSI.

Remark 2: In LDS, a trivial together with reducesto CDS.

Proof: We construct an RL codebook with elements from. We then use the CDS with DPC construction with

the chosen RL codeword acting as CSI. It follows from The-orem 1 that the CL information can be successfully decoded(together with the auxiliary codeword ) at both receivers if(29) and(30) are satisfied. This way, the effective communica-tion system for transmission of RL becomes a channel withas input and the pair and as output. For reliable trans-mission, (31) is then sufficient.

V. PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN

PROBLEM

In this section, we analyze the distortion tradeoff of the LDSfor the quadratic Gaussian case. While CDS with DPC is devel-oped only as a tool to be used in layered WZBC codes, CDSitself is a legitimate WZBC strategy. We thus analyze its perfor-mance in some detail first before proceeding with LDS. It turnsout, somewhat surprisingly, that CDS may in fact be the optimalstrategy for an infinite family of source and channel parameters.Understanding the performance of CDS also gives insight intowhich receiver should be chosen as receiver , and which one asreceiver . We remind the reader that the variance of a Gaussianrandom variable will be denoted by .

A. CDS for the Quadratic Gaussian Problem

Using the test channel with Gaussian andwhere , and a Gaussian channel input , (22) becomes(cf.(4))

for . In other words,

By analyzing (5), it is clear that should be chosen so as toachieve the above inequality with equality. Substituting thatchoice in (5) yields

(32)

For all that achieve the minimum in (32), we have

Thus, as seen from (11), . This, in partic-ular, means that if

is a constant, CDS achieves the trivial converse and there is noneed for a layered WZBC scheme. Specialization of (32) to thecase is also of interest

(33)

In particular, all maximizing achieve. Thus, the trivial converse is achieved if is

a constant.

B. LDS for the Quadratic Gaussian Problem

For LDS, we begin by analyzing the channel coding per-formance and then the source coding performance in terms ofachievable channel rates. Then closely examining the channel



rate regions, we determine whether , oris more advantageous given , , , , , and . The

resultant expression when exhibits an interesting phe-nomenon which we will make use of in deriving closed formexpressions for the tradeoff in LDS.

1) Channel Coding Performance: For LDS, we choosechannel variables and as independent zero-mean Gaus-sians with variances and , respectively, with ,and use the superposition rule . Motivated byCosta’s construction for the auxiliary random variable , weset . Using (29)–(31), we obtain achievable

as

(34)

(35)

(36)

Here, (35) follows by replacing with in (34).2) Source Coding Performance: We choose the auxiliary

random variables so that and whereand are Gaussian random variables satisfying

and . This choice imposes the Markov chain, and implies with and .

Using (4), one can then conclude

(37)

(38)

(39)

For any achievable triplet , (37)–(39) can beused to find the corresponding best . More specifically,(37)–(39) and (24)–(26) together imply

(40)

(41)

Since we have from (5) that

(42)

it is easy to conclude that both (40) and (41) should be satisfiedwith equality to obtain the best , which becomes

(43)

(44)

where

(45)

Now, if

(46)

then . But in the LDS, we have, implying ,

regardless of the chosen parameters. Moreover, will be min-imized when (46) is satisfied with equality. Thus, it suffices toconsider only

(47)

because equality in (47) already gives . Wethus have

(48)

(49)

3) Choosing the Refinement Receiver: Note that settingreduces LDS to CDS. This is regardless of which receiver is

designated as or . This simple observation, along with thediscussion in Section V-A, leads to the following lemma.

Lemma 1: In order to maximize the performance of LDS, onemust set and so that

(50)



Remark 3: When , (50) translates to

(51)

Therefore, the product determines the combinedchannel and side information quality, so that the “better” re-ceiver is chosen to receive the RL information. Recall from thediscussion in Section V-A that if is constant, then infact there is no need for refinement, as CDS already achievesthe optimal performance.

Proof: When , i.e., when all the power is allocatedto the CL, LDS achieves the same performance as CDS. In par-ticular, it achieves the channel rate point

If (50) does not hold, then from (32), it follows that LDS alsoachieves and some . Now,if we set , it is obvious that cannot be lowered anyfurther. We claim that cannot be lowered either. Therefore,LDS would not be able to achieve a better than whatCDS achieves. On the other hand, sending the refinement toreceiver could potentially result in a better performance.

Towards proving the above claim, observe from (45) that itsuffices to show that neither nor can increase when

compared to the case . That, in turn, followsby closely examining the expressions for and inSection V-B.I. In particular, for LDS, both (34) and(35) will bemaximized by their corresponding optimal Costa parameters,i.e., by and by , respectively. This

results in andas the maximum possible values, which are strictly smaller than

and , respectively. Therefore, the proof is complete.

C. Performance Comparisons for the Bandwidth MatchedCase:

We first derive the closed-form tradeoff for LDS.

Lemma 2: A distortion pair is achievable usingLDS if and only if , where is theconvex hull of

(52)

for

where

and satisfies . More specifi-cally,

(53a)

when ,

(53b)

when , and finally

(53c)

when .

Remark 4: The cases andare not considered in (53) because

those are prohibited by the rule (51). The same rule also guar-antees .

As a byproduct of the proof, which is deferred to Appendix B,we observe that the Costa parameter is either 0 or 1, dependingon whether or , respectively. When it is0, we have . On the other hand, when , we have

. Thus, setting the auxiliary codewordto be the same as the channel input constitutes the optimalchoice. To the best of our knowledge, this choice, which is typ-ically encountered in DPC for binary symmetric channels, hasnever been obtained as the optimal choice involving Gaussianchannels.

We now compare LDS with other schemes for the WZBCproblem. The performance of uncoded transmission is governedby (18). The distortion trade-off of separate coding is given bythe following lemma, which is proved in Appendix C. Recallthat the subscripts and refer to good and bad channels, i.e.,the Markov chain holds for all channel inputs .

Lemma 3: For the quadratic Gaussian case with , thedistortion pair with is achiev-able using separate coding if and only ifwhere is the convex hull of

(54)



when , and

(55)

when .

We now discuss the relative performance of the variousschemes case by case.

1) It is obvious by comparing (54) and (52) that whenand , LDS obtains the exact same perfor-

mance as in separate source and channel coding (Note thatin this case). The case where there is no side

information, i.e., , falls under this categorysince the refinement information must go the receiver withthe better channel. Therefore we see that the purely dig-ital LDS is worse than the schemes analyzed in [14] in theabsence of side information. Preliminary results from com-bining LDS with hybrid analog/digital schemes as in [14]were presented in [8].This behavior is displayed in Fig. 3(d) and (e).As for uncoded transmission, it can be better than the dig-ital schemes. For example, consider the case

depicted in Fig. 3(e), which corresponds to no side infor-mation at the receivers. In this case, uncoded transmissionactually achieves the trivial converse, and therefore, is theoptimal strategy.

2) When and , it follows from (55) and(52) that a sufficient condition for superiority of LDS overseparate coding is given by

and is therefore granted since and. Moreover, equality is satisfied,

i.e., the two schemes have equal performance, only when. This behavior is exemplified in Fig. 3(b).

Even though is prohibited in this case, onecan consider and with arbitrarilysmall . Uncoded transmission is also superior to allthe digital schemes in this limiting case.

3) When and , since inthis case, we need to explicitly write the best for a given

for LDS. From (52), it follows that LDS can achieve

(56)

for . On the other hand,(55) implies that the minimum that can be achieved byseparate coding must necessarily satisfy

(57)Superiority of LDS over separate coding then easily fol-lows from (56) and (57). An example of this case is shownin Fig. 3(a).We next show that LDS always outperforms uncoded trans-mission in this case. In fact, uncoded transmission is evenworse than CDS. Since CDS achieves ,it suffices to compare the values. Comparing (18) and(33), this reduces to showing

or equivalently

But since , this is trivially true.4) In Fig. 3(f), we also include an example where

, i.e., where the combined channeland side information qualities are the same. CDS achievesthe trivial converse as discussed in Section V-A. Wealso observed that uncoded transmission may achieve adistortion pair below the best known digital tradeoff, asshown in Fig. 3(d) and (e). This was expected becauseit is well-known that the optimal scheme is uncodedtransmission when there is no side information at eitherreceiver, as is the case in Fig. 3(e). For cases other than

, one could roughly say that LDS isbetter than uncoded transmission when the quality of theside information is sufficiently high, although we do notcurrently have the analytical means for comparison.

VI. PERFORMANCE ANALYSIS FOR THE BINARY HAMMING

PROBLEM

In this section, we first analyze the CDS for the binary Ham-ming problem and show that it can be optimal in this case aswell. We then analyze the LDS and present numerical compar-isons of the LDS with separate coding and uncoded transmis-sion.

A. CDS for the Binary Hamming Problem

It follows from Corollary 1 and Equations (7) and (8) thatin the binary Hamming case, if there exists and

such that

(58)

for all , then

(59)

can be achieved by the CDS. Unlike in the quadratic Gaussiancase, the constraint (58) does not result in a single best valuefor and . Therefore, CDS produces a tradeoff of ’s ratherthan one best point.



Fig. 3. Performance comparison for Gaussian sources and channels. In (a)-(e), , and therefore the choice , is made. In addition,in (e), , implying that there is no side information at either receiver and hence uncoded transmission is optimal. In (f), makingCDS optimal.

As discussed at the end of Section II-B, the distortion-ratefunction is achieved either by and ,

or by and . The implication of this factto the CDS is the following:



Fig. 4. Auxiliary random variables for binary source coding. The edge labelsdenotes transition probabilities. We also use the convention that .

1) If are not identical, neither are , and thus weneed and some to attain all

simultaneously, i.e.

(60)

for all . When this happens, we must necessarily havei.e., does not depend on .

2) If for , and thus doesnot depend on , we need (and hence )so that the same test channel achievessimultaneously. But, this makes the problem trivial.

B. LDS for the Binary Hamming Problem

1) Source Coding Rates: To evaluate and , wefirst fix and with , where the testchannels are also confined to degraded versions of those thatachieve , as shown in Fig. 4 for the case

. More specifically,

where , and are all Bernoulli random variableswith parameters , and , respectively. To obtain aMarkov relation , it suffices to enforce and

. In that case, one can find andsuch that and , and can alternativelybe written as

where and are and , respectively.This results in

We next make channel variable choices and derive the re-sulting channel coding rates for LDS. Unlike in the quadraticGaussian case, there is no power allocation parameter to vary.However, we have freedom in choosing the distributions of

and as and , respectively, as well asin choosing the auxiliary random variable as eitheror . We use independent and and thesuperposition rule .

2) Channel Coding Rates: With , (29)–(31) become

(61)

But since is increasing in its second argument, we haveas the optimal value achieving

(62)

(63)

On the other hand, if , we obtain

(64)

(65)

(66)

C. Performance Comparisons for the Bandwidth MatchedCase:

Analytical performance comparisons prove more difficult forthe binary Hamming problem. Even the question of which re-ceiver should be designated as and which as is not straight-forward to answer. That is because 1) there is no power allo-cation parameter we can control, and 2) even CDS can pro-duce a curve which could achieve both and

, rather than a single best point.It is also not clear that our choice of source random vari-

ables are the best. As mentioned earlier, our main motivationin adopting the same test channel as in point-to-point coding forLDS is its simplicity. The alphabet size bounds in [16], [18],however, are much higher and therefore it might be possible tofurther improve the performance of LDS.

Using the same auxiliary random variables in separate codinggives us the following achievable result. We do not have a com-plete characterization of the distortion tradeoff.

Lemma 4: A distortion pair is achievable if thereexist variables and that satisfy

for , with

(67)



and

(68)

if , and

(69)

if .

The proof is presented in Appendix D.The performance of the various schemes for certain source-

channel pairs at rate is presented in Fig. 5. For LDS, theconvex hull of two curves is shown, where in oneand in the other . In Fig. 5(a)–(d), the parame-ters , , and are fixed so that (60) is satisfied for ,and is varying. As increases, the collective behavior of theschemes dramatically changes. In Fig. 5(a), is con-sistently the best choice among all schemes. As the quality ofthe second channel decreases, and reaches the point where (60)is also satisfied for , CDS becomes optimal, as shown inFig. 5(b). When is increased even further, as in Fig. 5(c),

becomes the better choice. When reachesthe point where the first receiver has access to both the betterchannel and the better side information, as in Fig. 5(d) and (e),separate coding and LDS become identical as in the quadraticGaussian case. However, uncoded transmission can still outper-form the LDS as shown in Fig. 5(e) for the case of trivial sideinformation. Finally, Fig. 5(f) exemplifies the interesting phe-nomenon mentioned above, where CDS (and LDS) produces acurve, rather than a point, which happens to be the best.

VII. CONCLUSIONS AND FUTURE WORK

We proposed a layered coding scheme for the WZBCproblem, and analyzed its distortion performance for thequadratic Gaussian and binary Hamming cases. Even thoughour scheme allows for arbitrary rate channel uses per sourcesymbol, the achievability regions are easiest to compute for

. In fact, for the quadratic Gaussian case, we were able toderive closed form expressions for the entire distortion tradeoffand show that our layered scheme is always at least as good as(in fact, except for one certain case, always better than) separatecoding. By numerical comparisons, we observed the samephenomenon for the binary Hamming case under the regimewhere all the test channels are constrained to be of the formwhich achieves the Wyner–Ziv rate-distortion function. Onthe other hand, our scheme may not always improve over theperformance of uncoded transmission. This is not surprising,since when there is no (or trivial) side information, it is knownthat uncoded transmission is optimal.

One can combine uncoded transmission with digital codingto enhance the performance of the latter. In fact, as we show ina preliminary version [8], the hybrid digital/analog scheme ismore than the sum of its parts and distortions outside the con-vexification of the digital and analog regions are achievable.

APPENDIX

A. Other Layered WZBC Schemes

The LDS that we focus on in this paper is only one of manypossible layered coding schemes based on CDS and CDS withDPC. We shall briefly discuss these schemes. In all schemes, thesource coding rates are the same as in LDS and only the channelcoding rates differ.

• Scheme 1: This scheme is the simplest extension of CDS.The CL is encoded as in CDS. The RL is encoded on topof the CL. At both decoders, the RL is a source of inter-ference while decoding the CL. Once the CL is decodedat the refinement receiver, its effect can be cancelled whiledecoding the RL. The acheivable channel rates are givenby the next theorem.

Theorem 3: Let be the union of all forwhich there exist in some auxiliary alphabet andwith such that

(70)

(71)

(72)

Then .Proof: Given random variables and such that

and (70)–(72) are satisfied, each in the CLchannel codebook is chosen uniformly and independently from

. Similarly, for each , codewords to be trans-mitted over the channel are chosen uniformly and independentlyfrom . It then follows from Corollary 1 that (70) and(71) are sufficient for successful decoding of both and

simultaneously at both decoders. It also follows fromstandard arguments that (72) is sufficient for reliable transmis-sion of additional information with rate to the refinementreceiver.

• Scheme 2: The CL is encoded as in Scheme 1. The RL,however, is sent using dirty paper coding with the CL code-word as encoder CSI, and is decoded first.


and such that

(73)

(74)

(75)

Then .Proof: Since RL is to be sent by separate source and

channel codes, the channel coding part can proceed as in stan-dard dirty-paper coding (cf. [6]), if (75) is satisfied. Note that asin Corollary 2, the auxiliary codeword can also be decodedin the process of decoding the RL. With high probability, thiscodeword is typical with the CL codeword in addition to

. Subsequently, for decoding the CL, the channel output atthe decoder can be taken to be a pair . Therefore,



Fig. 5. Performance comparison for binary sources and channels. In (a)-(d), , , and are fixed, and as increases, how all the schemes compare changes. In(e), uncoded transmission is optimal. In (f), CDS and consequently LDS is the best. It is also noteworthy that it touches both trivial converse bounds simultaneously.

as in Scheme 1, can be successfully decoded given that(73) and(74) hold.

• Scheme 3: The encoding is performed as in LDS, but thedecoding order is reversed. Since RL is decoded first at



the receiver, the CL codeword purely acts as noise. Butthe decoder then has access to the RL codeword. So forthat receiver, the CSI is also available at the decoder. Thefollowing theorem makes use of these observations.


and such that

(76)

(77)

(78)

Then .Proof: Since RL is both encoded and decoded first, (78) is

necessary and sufficient for successful decoding of . Onceis decoded, the channel between CL and receiver reduces

to one with input , output , and CSI . It thenfollows from Theorem 1 that (76) and (77) suffices for reliabletransmission of . Note that the right-hand side of (77) isequivalent to .

Partial analytical results regarding the relative performanceof the four schemes can be found in [11]. The performance ofScheme 1 is always superior to that of Scheme 2, regardless ofchannel statistics. Further, under the regime wherewhere is an appropriately defined addition operation andand are independent, Scheme 1 becomes a special case ofLDS. Thus, for both the quadratic Gaussian and the binary Ham-ming cases, LDS performs at least as well as Scheme 1. Finally,for the quadratic Gaussian case with matched bandwidths, LDSis superior to Scheme 3 also. Even in the binary Hamming case,extensive numerical evaluations failed to produce a single ex-ample where Scheme 3 outperforms LDS. For this reason, weshall not consider Schemes 1 through 3 further in this paper.

B. Proof of Lemma 2

It follows from (48) and (49) that by varying and , weobtain the tradeoff

(79)

(80)

where

We next fix , which, in turn, fixes as

(81)

and minimize , which reduces to maximizing . Sinceneither nor can be negative, we need both

and to be satisfied. The former requirement isguaranteed because we naturally limit ourselves to .

The latter, on the other hand, becomes vacuous since rewriting(47) gives

(82)

whose right-hand side is always less than or equal to 1.Now if , we always have since

Thus, among all choices of and which satisfy (81), the onethat potentially minimizes is and

That is because with this choice we have .It then remains to check (82), which can be written after somealgebra as

This is granted if and is equivalent to

(83)

if . The constraint (83), on the other hand, is in effectonly if

for otherwise, it is trivially satisfied because . Substi-tuting in (44) yields

On the other hand, if , it is more helpful to write

as this reveals . Thus, the optimal choiceof parameters is potentially and

provided this choice satisfies (82). Once again, after some al-gebra, that translates to

Substituting in (44) yields

Combining all the above results yields (52) and (53).

C. Proof of Lemma 3

The Gaussian broadcast channel capacity is achieved byGaussian and with (cf. [4]). Let



and control the power allocation betweenand . The source rate-distortion function is similarly

achieved by the test channel with for. For these choices, (12), (13), and (15)–(17) can be

combined to give the following characterization of achievabledistortions for general :

(84)

together with

(85)

if , and

(86)

if .The key to the proof is the observation that for optimal per-

formance, (84) needs to be satisfied with equality for any . Tosee this, assume that with satisfies (84) withstrict inequality for some . Then one can decreaseuntil equality is obtained in (84), and still satisfy (85) or (86),depending on whether or , respec-tively. That, in turn, follows because the right-hand side of eitherof (85) or (86) are decreasing in . Thus, if (84) is not tight, onecan keep the same while decreasing .

When , equality in (84) translates to

For the case , (85) then becomes the same as (54).Similarly, if , (86) implies

and

simultaneously, which is the same as (55).

D. Proof of Lemma 4

For the binary symmetric channel, is achieved byand with and indepen-

dent of . The parameter serves as a tradeoff between and. The conditions (12) and (13) then become (cf. [4])

(87)

(88)

For the source coding part, we evaluate onlywith the auxiliary random variables chosen as in Section VI-Bwhere subscripts and are to be replaced by and or byand .

These simple choices may potentially result in degradation ofthe separate coding performance, as the bounds on the alphabetsizes for and in [16], [18], [19] are much larger. However,our limited choice of can be justified in two ways: 1)to the best of our knowledge, there is no other choice known toachieve better rates, and 2) to be fair, we use the same choice inour joint source-channel coding schemes.

As in the quadratic Gaussian case, we can write

(89)

for . Combining (15), (87), and (89) yields (67).Similarly, combining (16), (17), (88), and (89), we obtain (68)when , and (69) when .

REFERENCES

[1] P. P. Bergmans, “Random coding theorem for broadcast channels withdegraded components,” IEEE Trans. Inf. Theory, vol. 19, pp. 197–207,Mar. 1973.

[2] G. Caire and S. Shamai (Shitz), “On the achievable throughput of amultiantenna Gaussian broadcast channel,” IEEE Trans. Inf. Theory,vol. 49, pp. 1691–1706, Jul. 2003.

[3] M. H. M. Costa, “Writing on dirty paper,” IEEE Trans. Inf. Theory, vol.29, pp. 439–441, May 1983.

[4] T. Cover and J. Thomas, Elements of Information Theory. New York:Wiley, 1991.

[5] R. G. Gallager, “Capacity and coding for degraded broadcast channels,”Probl. Peredach. Inform., vol. 10, no. 3, pp. 3–14, 1974.

[6] S. I. Gel’fand and M. S. Pinsker, “Coding for channels with randomparameters,” Probl. Contr. Inf. Theory, vol. 9, no. 1, pp. 19–31, 1980.

[7] D. Gündüz and E. Erkip, “Reliable cooperative source transmissionwith side information,” in Proc. IEEE Inf. Theory Workshop, Bergen,Norway, Jul. 2007.

[8] D. Gündüz, J. Nayak, and E. Tuncel, “Wyner-Ziv coding over broadcastchannels using hybrid digital/analog transmission,” in Proc. IEEE Int.Symp. Inf. Theory, Toronto, ON, Jul. 2008.

[9] G. Kramer and S. Shamai, “Capacity for classes of broadcast channelswith receiver side information,” in Proc. IEEE Inf. Theory Workshop,Lake Tahoe, CA, Sep. 2007.

[10] J. N. Laneman, E. Martinian, G. W. Wornell, and J. G. Apostolopoulos,“Source-channel diversity for parallel channels,” IEEE Trans. Inf.Theory, vol. 51, pp. 3518–3539, Oct. 2005.

[11] J. Nayak, D. Gündüz, and E. Tuncel, Digital Schemes for Wyner-ZivCoding Over Broadcast Channels, vol. ArXiv:0911.4167v1 [cs.IT].

[12] A. Orlitsky and J. R. Roche, “Coding for computing,” IEEE Trans. Inf.Theory, vol. 47, pp. 903–917, Mar. 2001.

[13] R. Puri, K. Ramchandran, and S. Pradhan, “On seamless digital up-grade of analog transmission systems using coding with side informa-tion,” in Proc. 40th Allerton Conf. Commun., Contr. Comput., Allerton,IL, Oct. 2002.

[14] Z. Reznic, M. Feder, and R. Zamir, “Distortion bounds for broad-casting with bandwidth expansion,” IEEE Trans. Inf. Theory, vol. 52,pp. 3778–3788, Aug. 2006.

[15] S. Shamai, S. Verdú, and R. Zamir, “Systematic lossy source/channelcoding,” IEEE Trans. Inf. Theory, vol. 44, pp. 564–579, Mar. 1998.

[16] Y. Steinberg and N. Merhav, “On successive refinement for theWyner-Ziv problem,” IEEE Trans. Inf. Theory, vol. 50, pp. 1636–1654,Aug. 2004.

[17] A. Sutivong, M. Chiang, T. M. Cover, and Y.-H. Kim, “Channel ca-pacity and state estimation for state-dependent Gaussian channels,”IEEE Trans. Inf. Theory, vol. 51, pp. 1486–1495, Apr. 2005.

[18] C. Tian and S. Diggavi, “Side-information scalable source coding,”IEEE Trans. Inf. Theory, vol. 54, pp. 5591–5608, Dec. 2008.



[19] C. Tian and S. Diggavi, “On multistage successive refinement forWyner-Ziv source coding with degraded side informations,” IEEETrans. Inf. Theory, vol. 53, pp. 2946–2960, Aug. 2007.

[20] E. Tuncel, “Slepian-Wolf coding over broadcast channels,” IEEETrans. Inf. Theory, vol. 52, pp. 1469–1482, Apr. 2006.

[21] M. P. Wilson, K. Narayanan, and G. Caire, “Joint source channelcoding with side information using hybrid digital analog codes,” IEEETrans. Inf. Theory, vol. ArXiv:0802.3851v1 [cs.IT], submitted forpublication.

[22] A. D. Wyner, “The rate-distortion function for source coding with sideinformation at the decoder-II: General sources,” Inf. Contr., vol. 38, pp.60–80, 1978.

[23] A. D. Wyner and J. Ziv, “The rate-distortion function for source codingwith side information at the decoder,” IEEE Trans. Inf. Theory, vol. 22,pp. 1–10, Jan. 1976.

Jayanth Nayak received the Ph.D. degree in electrical and computer engi-neering from the University of California, Santa Barbara, in 2005.

He is currently a Research Staff Member at Mayachitra, Inc., in Santa Bar-bara. His research interests include information theory, pattern recognition, andcomputer vision.

Ertem Tuncel (S’99–M’04) received the Ph.D. degree in electrical and com-puter engineering from University of California, Santa Barbara, in 2002.

In 2003, he joined the Department of Electrical Engineering, Universityof California, Riverside, where he is currently an Associate Professor. Hisresearch interests include rate-distortion theory, multiterminal source coding,joint source-channel coding, zero-error information theory, and content-basedretrieval in high-dimensional databases.

Dr. Tuncel received the National Science Foundation CAREER Award in2007.

Deniz Gündüz (S’02–M’08) received the B.S. degree in electrical and elec-tronics engineering from the Middle East Technical University in 2002 and theM.S. and Ph.D. degrees in electrical engineering from Polytechnic Institute ofNew York University (formerly Polytechnic University), Brooklyn, NY, in 2004and 2007, respectively.

He is currently a Research Associate at the Centre Tecnológic de Telecomuni-cacions de Catalunya (CTTC), Barcelona, Spain. Previously, he was a consultingAssistant Professor at the Department of Electrical Engineering, Stanford Uni-versity and a postdoctoral Research Associate at the Department of ElectricalEngineering, Princeton University. In 2004, he was a summer researcher in thelaboratory of information theory (LTHI) at EPFL in Lausanne, Switzerland.His research interests lie in the areas of communication theory and informa-tion theory with special emphasis on joint source-channel coding, cooperativecommunications and cross-layer design.

Dr. Gündüz is the recipient of the 2008 Alexander Hessel Award of Poly-technic University given to the best Ph.D. Dissertation, and a coauthor of thepaper that received the Best Student Paper Award at the 2007 IEEE Interna-tional Symposium on Information Theory.


department of electrical and computer engineering …ertem/publications_files/it...1784 ieee...

Documents