wyner-ziv coding over broadcast channels: hybrid digital/analog schemes

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5660 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011 Wyner-Ziv Coding Over Broadcast Channels: Hybrid Digital/Analog Schemes Yang Gao, Student Member, IEEE, and Ertem Tuncel, Member, IEEE Abstract—A new hybrid digital/analog scheme is proposed for lossy transmission of a Gaussian source over a bandwidth-matched Gaussian broadcast channel with source side information avail- able at each receiver. The proposed scheme combines two schemes that were previously shown to achieve optimal point-to-point dis- tortion/power tradeoff simultaneously at all receivers under two distinct conditions stated in terms of channel and side informa- tion quality parameters. For the two-receiver case, the combined scheme is shown to achieve the same kind of optimality for the en- tire region in the parameter space sandwiched between those two conditions. Crucial to this result is a new degree of freedom discov- ered in designing point-to-point hybrid digital/analog schemes with side information. When superimposed with analog transmission, the proposed scheme outperforms all previously known schemes even outside the optimality region in the parameter space. Index Terms—Broadcast channels, Costa coding, hybrid digital/ analog coding, joint source-channel coding, writing on dirty paper, Wyner-Ziv coding. I. INTRODUCTION C ONSIDER a sensor network of nodes taking periodic measurements of a common phenomenon. One node transmits its measurement to the other nodes over a broadcast channel and each of the nodes has source side information (SSI) only available to that node. The loss- less version of this problem was studied in [6] and termed Slepian-Wolf coding over broadcast channels (SWBC). The more general lossy version was referred to as Wyner-Ziv coding over broadcast channels (WZBC) in [5], and performance of several purely digital schemes were analyzed. In this paper, we consider the bandwidth-matched quadratic Gaussian case of the WZBC problem with emphasis on receivers. Even in this special case, there is no known scheme that is optimal under all circumstances. However, there are several competitive schemes, some of which, under certain conditions, achieve a trivial outer bound: the minimum distortion that point-to-point transmission would achieve at each individual receiver for the given power level. Of course, achieving this outer bound immediately implies optimality. Manuscript received May 21, 2010; revised March 19, 2011; accepted April 14, 2011. Date of current version August 31, 2011. This work was supported in part by the National Science Foundation CAREER Grant CCF-0643695. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Austin, TX, July 2010. The authors are with the Department of Electrical Engineering, Uni- versity of California, Riverside, CA 92521 USA (e-mail: [email protected]; [email protected]). Communicated by I. Kontoyiannis, Associate Editor for Shannon Theory. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2011.2162266 The simplest such scheme is analog, i.e., uncoded, transmis- sion of the source, in which the source is scaled to accommodate the power level of the channel. The condition under which the outer bound is achieved in this case is that the side information at each receiver be trivial, i.e., independent of the source. At the other end of the spectrum, a fully digital joint source-channel coding algorithm, termed the common descrip- tion scheme (CDS), was proposed in [5] for general sources, distortion measures, and bandwidth expansion factors. CDS is also a simple scheme based on losslessly broadcasting the quantized source by utilizing the binning-free joint de- coding technique developed in [6] for the SWBC problem. It was shown in [5] that for the bandwidth-matched quadratic Gaussian case, CDS achieves the trivial outer bound when an appropriately defined “combined” channel and side information quality is constant among the receivers. In [5], CDS was also extended to a dirty-paper setting (termed DPC-CDS), where the channel state information (CSI) is available non-causally at the encoder, and to a layered description scheme (LDS), which was shown to outperform separate source and channel coding. Finally, based on the same techniques, [3] introduced several hybrid digital/analog (HDA) schemes which outperform analog transmission as well as separate coding. In [4], for point-to-point Wyner-Ziv/dirty-paper coding, a scheme using modulo-lattice modulation was proved to be optimal for the bandwidth-matched quadratic Gaussian case. When the CSI is trivial and the scenario is extended to broadcast channels (and thus the problem becomes WZBC), the scheme is also shown to achieve the trivial outer bound when another combined channel and side information quality, i.e., defined differently from [5], is constant among the receivers. Later on, [8, Sec. III-B] proposed a closely related scheme with random coding arguments instead of lattices, whereby the analog source is integrated into the auxiliary random codeword. Not surprisingly, when extended to broadcast channels, this scheme achieves the trivial outer bound under the same condition as that in [4], though it is not explicitly mentioned in [8]. In the sequel, we will refer to the scheme in [8] as the HDA-WZ scheme. In this paper, we first present a basic WZBC scheme com- bining the HDA-WZ scheme and DPC-CDS, which we term HDA-CDS. We prove that by making full use of the decoded auxiliary dirty-paper codeword, our scheme achieves the trivial outer bound when the channel and side information quality parameters fall between those yielding constant combined quality with respect to the HDA-WZ scheme and CDS. We also show that HDA-CDS outperforms LDS for any set of system parameters. To take advantage of analog transmission, especially when the side information is weak, we added an analog stream onto 0018-9448/$26.00 © 2011 IEEE

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Page 1: Wyner-Ziv Coding Over Broadcast Channels: Hybrid Digital/Analog Schemes

5660 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

Wyner-Ziv Coding Over Broadcast Channels:Hybrid Digital/Analog SchemesYang Gao, Student Member, IEEE, and Ertem Tuncel, Member, IEEE

Abstract—A new hybrid digital/analog scheme is proposed forlossy transmission of a Gaussian source over a bandwidth-matchedGaussian broadcast channel with source side information avail-able at each receiver. The proposed scheme combines two schemesthat were previously shown to achieve optimal point-to-point dis-tortion/power tradeoff simultaneously at all receivers under twodistinct conditions stated in terms of channel and side informa-tion quality parameters. For the two-receiver case, the combinedscheme is shown to achieve the same kind of optimality for the en-tire region in the parameter space sandwiched between those twoconditions. Crucial to this result is a new degree of freedom discov-ered in designing point-to-point hybrid digital/analog schemes withside information. When superimposed with analog transmission,the proposed scheme outperforms all previously known schemeseven outside the optimality region in the parameter space.

Index Terms—Broadcast channels, Costa coding, hybrid digital/analog coding, joint source-channel coding, writing on dirty paper,Wyner-Ziv coding.

I. INTRODUCTION

C ONSIDER a sensor network of nodes takingperiodic measurements of a common phenomenon.

One node transmits its measurement to the other nodesover a broadcast channel and each of the nodes has sourceside information (SSI) only available to that node. The loss-less version of this problem was studied in [6] and termedSlepian-Wolf coding over broadcast channels (SWBC). Themore general lossy version was referred to as Wyner-Ziv codingover broadcast channels (WZBC) in [5], and performance ofseveral purely digital schemes were analyzed. In this paper, weconsider the bandwidth-matched quadratic Gaussian case ofthe WZBC problem with emphasis on receivers. Evenin this special case, there is no known scheme that is optimalunder all circumstances. However, there are several competitiveschemes, some of which, under certain conditions, achieve atrivial outer bound: the minimum distortion that point-to-pointtransmission would achieve at each individual receiver forthe given power level. Of course, achieving this outer boundimmediately implies optimality.

Manuscript received May 21, 2010; revised March 19, 2011; accepted April14, 2011. Date of current version August 31, 2011. This work was supportedin part by the National Science Foundation CAREER Grant CCF-0643695. Thematerial in this paper was presented in part at the IEEE International Symposiumon Information Theory, Austin, TX, July 2010.

The authors are with the Department of Electrical Engineering, Uni-versity of California, Riverside, CA 92521 USA (e-mail: [email protected];[email protected]).

Communicated by I. Kontoyiannis, Associate Editor for Shannon Theory.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2011.2162266

The simplest such scheme is analog, i.e., uncoded, transmis-sion of the source, in which the source is scaled to accommodatethe power level of the channel. The condition under which theouter bound is achieved in this case is that the side informationat each receiver be trivial, i.e., independent of the source.

At the other end of the spectrum, a fully digital jointsource-channel coding algorithm, termed the common descrip-tion scheme (CDS), was proposed in [5] for general sources,distortion measures, and bandwidth expansion factors. CDSis also a simple scheme based on losslessly broadcastingthe quantized source by utilizing the binning-free joint de-coding technique developed in [6] for the SWBC problem. Itwas shown in [5] that for the bandwidth-matched quadraticGaussian case, CDS achieves the trivial outer bound when anappropriately defined “combined” channel and side informationquality is constant among the receivers. In [5], CDS was alsoextended to a dirty-paper setting (termed DPC-CDS), wherethe channel state information (CSI) is available non-causally atthe encoder, and to a layered description scheme (LDS), whichwas shown to outperform separate source and channel coding.Finally, based on the same techniques, [3] introduced severalhybrid digital/analog (HDA) schemes which outperform analogtransmission as well as separate coding.

In [4], for point-to-point Wyner-Ziv/dirty-paper coding, ascheme using modulo-lattice modulation was proved to beoptimal for the bandwidth-matched quadratic Gaussian case.When the CSI is trivial and the scenario is extended to broadcastchannels (and thus the problem becomes WZBC), the schemeis also shown to achieve the trivial outer bound when anothercombined channel and side information quality, i.e., defineddifferently from [5], is constant among the receivers. Later on,[8, Sec. III-B] proposed a closely related scheme with randomcoding arguments instead of lattices, whereby the analogsource is integrated into the auxiliary random codeword. Notsurprisingly, when extended to broadcast channels, this schemeachieves the trivial outer bound under the same condition as thatin [4], though it is not explicitly mentioned in [8]. In the sequel,we will refer to the scheme in [8] as the HDA-WZ scheme.

In this paper, we first present a basic WZBC scheme com-bining the HDA-WZ scheme and DPC-CDS, which we termHDA-CDS. We prove that by making full use of the decodedauxiliary dirty-paper codeword, our scheme achieves the trivialouter bound when the channel and side information qualityparameters fall between those yielding constant combinedquality with respect to the HDA-WZ scheme and CDS. We alsoshow that HDA-CDS outperforms LDS for any set of systemparameters.

To take advantage of analog transmission, especially whenthe side information is weak, we added an analog stream onto

0018-9448/$26.00 © 2011 IEEE

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HDA-CDS, also to be potentially used as artificial CSI for bothHDA-WZ and CDS components. However, although the analogstream is very useful as a third signal component, we numeri-cally observed that it is useless as CSI, though a rigorous proofseems extremely difficult.

The paper is organized as follows. Section II defines theproblem formally and discusses related past work in detail. InSection III, a new degree of freedom in point-to-point trans-mission, which is crucial to our main result, is revealed. ThenHDA-CDS is presented, and its optimality in the region ofinterest is proved. The more general scheme combining analogtransmission with HDA-CDS (termed AHC) is introduced,and its performance is numerically compared with HDA-CDS,separate source and channel coding, LDS, and analog transmis-sion. Section IV concludes the paper.

II. BACKGROUND AND NOTATION

Let be real-valued jointly Gaussianrandom variables generated in an i.i.d. fashion from

. The Gaussian source sequence is to betransmitted over a Gaussian broadcast channel

where , and are the channel input, channel output atreceiver , and the corresponding i.i.d. additive white Gaussianchannel noise. The channel has an input power constraint

Source side information is available at receiver , where

with and . Without loss of generality,and are assumed to have unit variance andthus the variance of is . Hereafter, to easeexposition, a bold font capital letter will denote the variance ofthe corresponding random variable as in [5]. The reconstructionquality is measured with squared error distortion

for any source block and reconstruction block .In the special case of point-to-point transmission, or ,

the capacity of the Gaussian channel is1 2

and the minimum distortion achieved by Wyner-Ziv coding withrate is given by [9]

1All logarithms in this paper are base 2.2The subscripts for the receivers are omitted since there is only one receiver.

Since separate source and channel coding is optimal, as provedin [7], transmission is possible if and only if , whichtranslates to

For the Gaussian WZBC problem, a trivial outer bound isobtained by letting each receiver achieve its minimum distortionwithout considering other receivers:

(1)

There are several schemes for the WZBC problem achievingthe trivial outer bound under different conditions, though ascheme optimal under all circumstances is not known.

With the bandwidth-matched case under consideration, thesimplest scheme one can use is analog transmission of thesource , in which the unit-variance source is scaled with

to adapt to the power constraint of the channel, and itachieves the performance

at each receiver. It is obvious that uncoded transmissionachieves the outer bound (1) with equality if and only if

, for all . This, in turn, corresponds tothe extreme situation that is independent of , and hence isof very limited use to us.

The performance of separate source and channel coding isknown since both the channel and the side information are de-graded. The explicit expressions of the distortion pair forwere given in [5, Lemma 3] and are included here for complete-ness. Without loss of generality, is assumed. Thedistortion pair with is achievableusing separate coding if and only if where

is the convex hull of

when and

when .Another purely digital scheme, the common description

scheme (CDS), was proposed in [5], which can be utilized forgeneral sources/channels and bandwidth expansion factors,but we focus here only on the bandwidth-matched case. Asillustrated in Fig. 1, CDS compresses the source sequenceto one of source codewords, say , which inturn is mapped into an independently generated channel word

(the number of channel words is still ). At

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Fig. 1. Illustration of the codebooks of a successful CDS transmission. Thecross-hatched codewords are the actually used ones and the double sided arrowsdenote joint typicality, whereas the arrows with a cross denote atypical pairs. Atthe receiver side, all the hatched codewords in the source codebook� togetherform a virtual bin.

receiver , the channel output will be jointly typical withchannel codewords. When traced back

to the source codebook, the corresponding source codewordsform a virtual bin (which can overlap with other virtual bins,so does not correspond to binning in the strict sense). Using theside information , the actual source codeword in this binis then disambiguated with success if and only if3

(2)

for all . For the quadratic Gaussian case we are interested inhere, it was shown in [5] that CDS satisfies (1) with equality atall the receivers if and only if

It has to be pointed out that although CDS is governed by thesame inequality (2) as in separate source and channel coding, itis inherently different since the source encoder does not performbinning. Intuitively, the benefit of CDS is that, with multiplereceivers, it allows a weak receiver with larger channel noiseto make up with better side information by avoiding binning atthe encoder side. As proposed in [5], CDS can also be used ina dirty-paper setting with a channelwhere the channel state information (CSI) is also availablenon-causally at the encoder. By using Costa coding [1], it wasshown that transmission is successful if and only if

(3)

for all , where the auxiliary dirty-paper codewords are gen-erated according to for some with indepen-dent and . This extension was termed dirty-paper-codedCDS (DPC-CDS) and it also creates virtual bins at the receivers.DPC-CDS is an important building block of our schemes.

In [5], CDS was also extended to the layered descriptionscheme (LDS) for receivers by adding a refinementlayer designated for the receiver with better combined channeland side information quality, i.e., the one with the smaller

. At the encoder side, the binning index of the refinementmessage is mapped to a channel codeword, which, in turn, is

3The left-hand side of (2) coincides with the source coding rate with explicitbinning as in Wyner-Ziv coding, and is the same as ������� ��� ��� due tothe Markov chain � �� � � .

Fig. 2. Illustration of HDA-WZ. The collection of codewords � serve as asource codebook at the encoder and as a channel codebook at the decoder.

used as artificial CSI at the CDS encoder. The receiver firstdecodes the common information transmitted via CDS and thenthe additional information is decoded only at the refinementreceiver. It was shown that LDS outperforms separate coding.We will compare the performance of our schemes with that ofLDS since, to the best of our knowledge, it provides the bestknown performance in digital schemes.

In [8], a hybrid digital/analog scheme, which will be re-ferred to as the HDA-WZ scheme here, was proposed andproved to be optimal for point-to-point transmission. In theHDA-WZ scheme, an auxiliary random variable is definedas , and a codebook of size is generatedfrom typical , as shown in Fig. 2. The encoder then finds acodeword that is jointly typical with the source sequence

, and then sends the corresponding . Atthe receiver side, the unique that is jointly typical withand is chosen, and is estimated from , , and .To guarantee successful decoding of the correct

(4)

needs to be satisfied. When

(5)

the effective source coding rate and channel coding ratematch, and the resultant distortion becomes .

This also implies that, in the WZBC scenario, the trivial outerbound (1) is achieved by HDA-WZ if and only if

That is because only in this case, the same choice of matcheswhat the receivers need for optimality, as is apparent from (5).Although this condition was not explicitly mentioned in [8], itwas in [4], which uses an equivalent modulo-lattice modulationinstead.

III. RESULTS

In this section, we will first propose a new scheme forpoint-to-point transmission with SSI by combining separatesource-channel coding with HDA-WZ, and discuss how thisscheme provides complete freedom in choosing the dirty-paper auxiliary codeword structure (through the choice of thecoefficient in DPC) for any power allocation between the twostreams. The point-to-point scheme will then be extended to thetwo-receiver broadcast scenario (WZBC) by simply replacingthe separate coding block with DPC-CDS. Utilizing the extra

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Fig. 3. New hybrid scheme for point-to-point transmission with SSI, featuring the freedom of power allocation between the branches and, for any power allocation,the freedom to choose the auxiliary random codeword from a range.

freedom in choosing the DPC coefficient , we will thenshow that this WZBC scheme, termed HDA-CDS, achievesthe trivial outer bound whenever the system parameters fallinto the region sandwiched between those for which CDS andHDA-WZ achieve the same. Finally, we will add an analoglayer to HDA-CDS, and discuss the performance of the resul-tant scheme, termed AHC.

A. A New Freedom in Point-to-Point Transmission

A new scheme for point-to-point transmission with SSI willbe proposed in this section. In [2], it was shown that for point-to-point transmission without any SSI or CSI, the optimum distor-tion can be achieved with a range of auxiliary random variables(or equivalently, a range of ) for any power allocation betweenan analog stream and a digital stream, by making the analogstream serve as artificial CSI for the digital one and fully uti-lizing the decoded auxiliary dirty-paper codeword. This expandsthe freedom of designing the system to another degree in addi-tion to the well-known freedom of power allocation between thedigital and analog streams.

Inspired by the result in [2], we now propose to combine theHDA-WZ scheme in [8] and separate coding in a similar way,as shown in Fig. 3. We will show that the same kind of freedomexists in this system when we make full use of the decodeddirty-paper codeword at the receiver. The source is encoded asusual with an optimal quantizer characterized by a backward testchannel with , followed by binning. Thetotal channel input power is split into andfor hybrid and digital paths, respectively, whereand . The quantization error is transmitted byHDA-WZ scheme as an analog source, where the HDA auxiliarycodeword is constructed by . The HDA channelcodeword is fed as artificial CSI to the DPC channel encoder,which maps the bin index into a bin of auxiliary dirty-papercodewords characterized by , and choosesthe unique in the bin which is jointly typical with . Thecorresponding digital channel input is and thechannel output is . Just as in [2], is notconfined to the “optimal” choice in [1], which is

but rather, any feasible is considered.At the receiver side, the digital decoder operates first. To en-

sure that it successfully decodes (as well as ), we need

(6)

where is the Wyner-Ziv source coding rate andis the effective digital channel capacity.

The HDA-WZ decoder has access to , and .Defining as the effective side infor-mation for the “source” , we then need

(7)

to decode successfully, where follows from .Now we are ready to introduce the extra level of freedom of

the scheme.

Theorem 1: is achievable for any satisfyingand where

(8)

The proof is given in Appendix A.

Remark 1: Theorem 1 suggests that for any power alloca-tion between the two streams, the proposed scheme achieves theoptimal distortion in the presence of SSI, for a range of auxil-iary codeword given by (8), instead of only Costa’s construc-tion. Thus even when the effective channel is not used at its fullcapacity, one can still achieve the optimal distortion by fullyutilizing the decoded auxiliary codeword. This result impliesthe freedom introduced in [2] is not an isolated case. Moreover,as shown in the next section, since both and can be freelychosen for any receiver, under certain conditions, we are able tofind a pair making the encoder optimal for both receivers.

It is worth noting that a similar scheme can be found whereis input to the HDA-WZ encoder instead of . The HDA

auxiliary codebook would now be constructed according toand the digital dirty paper auxiliary codeword

would be constructed the same way as in the above scheme.In this alternative scheme, the condition for successful digitaltransmission is still given by (6). To successfully decode ,the counterpart of (7) becomes

(9)

which reduces to (23) (in Appendix A) as well. Note that sincethe construction of contains , the right-hand side of (9) cannot be simplified to as before, and the optimalestimate of is given by a linear combination of ,and . By rigorous calculation, it can be shown that, once again,optimal distortion is achieved when equalities hold in both(22) and (23). We omit the algebra here as it is not essential forthe purposes of this paper. Finally, this scheme also specializesto the scheme in [8, Sec. III-C] by letting .

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5664 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

Fig. 4. HDA-CDS for one receiver achieves the optimum distortion without binning and has the same freedom of power allocation and construction of auxiliaryrandom variable.

Fig. 5. For WZBC problem, HDA-CDS combines two separately optimal schemes and achieves the trivial outer bound in the whole region between the conditionsunder which either scheme achieves optimum.

B. A Basic Scheme: HDA-CDS

We can replace the separate digital source and channel en-coders inside the dotted box in Fig. 3 with a DPC-CDS encoder,resulting in a point-to-point scheme shown in Fig. 4, termedHDA-CDS.

We stick to the previous notation for consistency, thoughthe reader needs to remember that it is an essentially differentscheme. As before, the source is quantized with a backwardtest channel , and the quantization erroris transmitted by an HDA-WZ stream. The HDA auxiliaryrandom variable is constructed by and is theHDA channel word. What is different is that no source binningis performed and the DPC-CDS encoder directly maps thequantization index into a bin of dirty-paper auxiliary channelcodewords. The encoder then uses the CSI to choose theright codeword inside the bin, as before.

When there is only one receiver, the governing inequalitiesremain to be (6) and (7) and therefore the freedom in Theorem1 persists.

Inspired by the fact that CDS and HDA-WZ achieve the trivialouter bound under different channel and side information con-ditions, we apply HDA-CDS on the WZBC problem. Althoughthe number of receivers, , can be arbitrary, we focus hereon the two-receiver case, as shown in Fig. 5. As we shall see,HDA-CDS takes advantage of the new level of freedom revealedfor point-to-point transmission, and the combination turns out tobe better than the sum of its parts.

Without loss of generality, we assume that the second receiverhas lower channel noise, i.e., , and the channeloutput at receiver is now given by

. At each receiver , with the help of side informationand the received channel word , first and are de-

coded. From (3), this is possible whenever

(10)

Fig. 6. For any given� �� , each point on the �� �� �-plane corre-sponds to a system with parameters �� �� �� �� �. As indicated by thetwo lines in the figure, the conditions for achieving the trivial outer bound forCDS and the HDA-WZ scheme are� � �� � and� �� �� � �� �� �� �, respectively. When � � � � �, or the SSI is trivial ateach receiver, analog transmission achieves the outer bound, as indicated by thedot. The shaded region, as stated in Theorem 2, indicates where our scheme isachieving the outer bound.

Then the HDA-WZ decoder uses , and to decode ,where . This is possible if

(11)

as follows from (7).Finally, an MMSE estimate of using ,

and is performed.Now, we can optimize over and the quantization rate

of the encoder to minimize the distortions. As mentioned before,when , one can simply set , with whichour scheme reduces to CDS and achieves the outer bound (1).Similarly, when , the choice

reduces our scheme to pure HDA-WZ, and the outerbound (1) is achieved. Since , these correspond todistinct lines in the -plane shown in Fig. 6. As statedby the following theorem, HDA-CDS achieves the outer bound(1) in the entire shaded region sandwiched between these twolines.

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Theorem 2: Given and with , the hy-brid scheme HDA-CDS achieves the outer bound (1) for all

pairs satisfying

The proof is given in Appendix B.

Remark 2: The intuition of Theorem 2 is that by exploitingthe extra level of freedom and benefits of virtual binning, weare able to find an encoder which is optimal for both receiverswhen they have “similar” combined channel/SSI quality (so thatthe system falls in the shaded region in Fig. 6). In this case, eachreceiver can achieve its as if it is the only receiver, althoughthe receivers have different channel noise and SSI quality.

As we show in the next theorem, HDA-CDS performs at leastas well as LDS introduced in [5] even outside of the shadedregion in Fig. 6. That is because we can simply achieve the samethe performance of the LDS by replacing the separate sourceand channel coding for the refinement layer of the LDS withHDA-WZ coding, and mimicking LDS with a proper choice of

and corresponding .

Theorem 3: HDA-CDS can achieve the same pairsas LDS when either or .

The proof is given in Appendix C.Unfortunately, the complete tradeoff between and

proved very difficult to derive. However, we numerically ob-served that outside of the region ,the achievable distortion region (after convexification) is alwaysthe convex hull of pure HDA-WZ and LDS. See Figs. 8(a) and8(c)–8(f) for examples of this phenomenon.

In summary, HDA-CDS achieves the trivial outer bound inthe entire shaded region shown in Fig. 6, instead of only at thetwo linear boundaries and in other regions, and HDA-CDS isalso shown to outperform the LDS in [5], which, in turn, is thebest known digital scheme.

C. The Main Scheme

Since analog transmission itself is optimal for WZBC withtrivial SSI at all the receivers, it will improve the performanceif an analog stream is added to the HDA-CDS4. We propose toadd the analog component using dirty paper coding as shownin Fig. 7. The structure of the decoders remains the same asin HDA-CDS. We call this scheme the Analog-HDA-CDSscheme, or AHC for short. The source vector is quantizedin the same way with the backward test channeland the quantization error is transmitted by both HDA-WZ

4Adding the analog stream will improve the performance at least for the casewith trivial SSI at all the receivers, and when the analog stream does not help,we can always set � � � to let AHC reduce to HDA-CDS.

Fig. 7. Encoder of the AHC scheme. An extra analog stream is superimposedwith dirty paper coding to incorporate the benefits of analog transmission. Thestructure of decoders is the same as HDA-CDS.

scheme and analog transmission. The analog stream is consid-ered at the dirty-paper digital channel encoder as additionalartificial CSI besides . The channel power is now splitinto three parts for HDA-WZ, analog,and CDS. The quantization error is scaled by a constantso that . At the HDA-WZ encoder, the auxiliarycodeword is again constructed by , while theauxiliary dirty paper codeword at the DPC-CDS encoder by

.Both receivers first decode , requiring

(12)

, where the channel output is now. Note that the system parameters should be chosen so

that the effective digital channel capacity [the right-hand side of(12)] is non-negative.

The next lemma translates (12) into how , and shouldbe chosen for any triplet.

Lemma 1: For any power allocation , both re-ceivers can decode if and only if

(13)

where

(14)

with . Also, and need to satisfy

(15)

The proof is given in Appendix D.

Remark 3: We note here that the ellipse (15) has the centerwith , and it can be expressed

in the standard form as in (16), found at the bottom of the page.

(16)

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Fig. 8. Performance comparison between AHC scheme, LDS, analog transmission, and separate coding when � � �. In (a),� � � . In (b), � �

� � � � and AHC scheme achieves the trivial outer bound. In (c) and (d), � � � � � . In (e), there is no side information and analogtransmission is optimal. In (f), � � � .

As for the decoding of , it can be accomplished, just as inHDA-CDS, at receiver if and only if

(17)

which is the same as

(18)

After some algebra, the condition (18) reduces to the quadraticform

(19)

where

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with . The discriminant of (19) is given by

which is clearly non-negative. Now since and, it immediately follows that (19) corresponds to an interval

containing .At receiver , the MMSE estimate of is computed using

, and, depending on whether it can be decoded,, and the corresponding distortion is given in the following

lemma.

Lemma 2: Define

for , where satisfies (13) with equality. For any , thedistortion at receiver is then given by

otherwise. (20)

The proof is given in Appendix E.We now discuss the performance of the AHC scheme for each

aforementioned region on the -plane separately. SinceHDA-CDS is a special case of AHC obtained simply by setting

, it is obvious that in the region where, Theorem 2 still holds. An example of this can be

observed in Fig. 8(b). In the other regions, although analyticaldissection is very difficult to obtain, based on extensive numer-ical simulations we arrived at the following conjectures:

Conjecture 1: When , the analog stream is nothelpful at all, i.e., optimal distortion performance is achievedwhen . Moreover, is always the optimal choice,and thus as a consequence of Theorem 3, the performance ofAHC coincides with that of LDS.

An example of this phenomenon is shown in Fig. 8(a).As a sanity check, this conjecture implies that analog trans-

mission alone can never outperform LDS in this region on the-plane. But this is indeed the case, as was analytically

shown in [5, SectionV-C].

Conjecture 2: When , it is observedthat always, i.e., a simple superposition scheme withonly analog and HDA-WZ streams suffices.

As can be seen in Figs. 8(c) and 8(d), the resultant perfor-mance is strictly better than those of HDA-CDS and analogtransmission alone.

Conjecture 3: When , all three streams makesome contribution and the performance is better than those ofHDA-CDS and analog alone. In addition, , i.e.,dirty paper coding is not necessary and simple superposition isgood enough.

The performance is illustrated in Fig. 8(f), and when ,the scheme reduces to analog transmission, which is indicatedby the point on the curve.

Bringing all the conjectures together, one can conclude thatthe analog stream is useless as CSI, and the AHC scheme canbe simplified to the superposition of HDA-CDS and the analogstream. That is because we observe either , or ,or .

IV. CONCLUSION

For the bandwidth-matched quadratic Gaussian WZBCproblem, we proposed a new hybrid digital/analog codingscheme called AHC, and demonstrated that it outperforms allpreviously known schemes. For the case of two receivers, AHCis analytically shown to achieve (without the help of the analogstream) the trivial outer bound for the entire region in theparameter space sandwiched between the optimality conditionsfor HDA-WZ and CDS, the two building blocks. This resultuses the new level of freedom we discovered for point-to-pointtransmission, namely, the freedom in choosing the auxiliarydirty-paper codeword in addition to the well known freedom ofpower allocation. Outside that region, we numerically observedthat the AHC scheme reduces to a simple superposition ofHDA-CDS and the analog stream, and thus dirty paper codingis not necessary between the two.

APPENDIX APROOF OF THEOREM 1

Inequality (6) is the same as

which can be expanded as

(21)

and finally reduces to

(22)

where . A conditionis also necessary to ensure that the effective capacity in (6) orthe right-hand side of (21) is non-negative, where are givenin (8).

We observe that the range of feasible in (8) is exactly thesame as that in [2], with playing the role of foranalog power. Following similar arguments as in [2], this alsoimplies that when , the range of becomes the entirereal line and the system becomes purely digital. On the otherhand, when , and the scheme becomes equiva-lent to HDA-WZ. To see that, notice that the right-hand side of(22) is maximized by for any and the resultantexpression after setting approaches 1 as .

Inequality (7) is the same as

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which reduces to

(23)

The optimal estimate of is given by the optimal estimateof , which is obtained by

, where , and satisfy

due to the principle of orthogonality. We thus have

The square-error distortion is given by

(24)

(25)

(26)

where (25) and (26) are satisfied by equality when equalitieshold in (23) and (22), respectively.

APPENDIX BPROOF OF THEOREM 2

Let us immediately remark that the hypothesis of the theoremimplies .

Since (10) and (11) have the same forms as (6) and (7) in thepoint-to-point scenario, by defining

for , they reduce to

(27)

and

(28)

respectively, as before. Now further define

and

When we consider both receivers, to ensure decoding and, we need

Further, at each receiver, the HDA auxiliary codeword canbe decoded if and only if .

In addition to (27) and (28), the range of the free parameterhas to be confined to so that the effective capacity

in (10) is non-negative. We refer to pairssatisfying this requirement as feasible.

In the point-to-point version of our scheme, it is shown thatfor any feasible pair , the minimum distortion canbe achieved if and are chosen so as to satisfy (22) and (23),i.e., the counterparts of (27) and (28), with equality. This impliesfor the WZBC problem at hand that if we can find a feasible pair

so that (27) and (28) are satisfied with equality simultane-ously for with some choice of and , then the trivialouter bound (1) can be achieved. This, in turn, requiresand simultaneously.

Now, can be written as

(29)

where

Note that in the region of interest on the-plane.

Similarly, implies from that

or equivalently, from (29) that

(30)

We first ignore the dependency on , and search for a tripletin the “interior” feasible set and

. Note that is guaranteed in the feasibleset because of . Solving for in both (29) and (30)for a fixed pair yields

(31)

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For every fixed then implies

(32)

and

(33)

In fact, (33) automatically follows from the required consistencybetween the two solutions of in (31). To see that, rewrite (31)as

(34)

which, in particular, implies using the non-negativity of theright-hand side and that

Note that is also granted due to the second equality in(31) because

whenever . Thus, it suffices to findsuch that (32) and (34) are simultaneously satisfied, and findthe corresponding using either formula in (31).

Now, expanding (34), we obtain

(35)which is quadratic in for every fixed . The discriminant canbe computed as

which is strictly positive due to the fact that .Thus, (35) has two positive roots for any . Denoting theleft-hand side of (35) as , it can be shown after somealgebra that

This implies that setting to the larger root of ,we automatically satisfy . What remains to be found isthen under what conditions on that root also satisfies (32).Rewriting (32) as

and explicitly computing the larger root of aswith

Equation (32) translates to

(36)

Of course, (36) is meaningful only if

which follows after some algebra using .In summary, if we pick any

together with and from (31), we simultaneouslysatisfy and with and

.The only thing that remains is to find a that is consistent

with , and . We have that

Instead of solving directly, let us temporarily treatand as free variables and solve the linear systemabove:

By close inspection, one can actually show that as varies, wehave

with

and

Since and are constants, this results in a line on the-plane. Also, since and ,

respectively, imply and , this line stretchesthrough the entire interval . In fact, the slope of theline depends only on for given .

Now, we show that

(37)

and therefore , implying also that . Towardsthat end, rewrite (37) as where

The quadratic form is minimized at

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5670 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

Also

implying since .Going back to and , a value consis-

tent with and can be found if and only if

But and imply that intersectswith . In fact, there are always two inter-sections (and thus two pairs of satisfying and

) except when , or equivalently when

APPENDIX CPROOF OF THEOREM 3

The distortion at each receiver depends on whether ischosen so that can be decoded at that receiver. If ,

is decoded at receiver , and hence the distortion is givenby (24). If, on the other hand, , then cannot be de-coded, and since both and are independent of or , theoptimal reconstruction of is solely given by ,where

The distortion in this case is given by

Note that this distortion is the same as in (24) with . Thatis because when , no information about is transmittedand decoding does not help at all.

When , let , and thus . It iseasy to show that when

. It can also be shown that in this case,is automatically satisfied because . Since now

and , is alwayssatisfied. If we choose , the first receiver is able to decodeboth the CDS stream and the HDA stream while the second canonly decode the CDS stream. The resultant achievable distortionpairs are given by

(38)

and

(39)

Note that (38) and (39) are actually the same as (79) and (80) in[5] with , where our receiver 2 corresponds to the commonreceiver in [5] because it has worse combined channel and SSIquality.

When , it can be shown that al-ways. If we set , we then have and

. When we choose , the first receiver decodes bothstreams, whereas the second one decodes only the CDS stream,and the resultant distortion pairs are given by

(40)

and

(41)

Equation (40) and (41) are again the same as and in[5], this time with and receiver 1 being the commonreceiver since .

APPENDIX DPROOF OF LEMMA 1

The left-hand side of (12) is

as before, and the right-hand side is

where is defined in (14).Taking both receivers into consideration, we thus need

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In addition, to make sure the effective channel capacity, i.e.,the right-hand side of (12), is non-negative, and need tosatisfy

(42)

for which corresponds to the intersection of two el-lipses on the plane. By close inspection, it is easy tosee that the ellipse corresponding to channel 1 is always con-tained in that of channel 2, as . It then suffices toconsider (42) for only, which is the same as (15).

APPENDIX EPROOF OF LEMMA 2

When can be decoded, the MMSE estimate of is, where the coefficients satisfy

(43), as shown at the bottom of the page. We thus have (44),also shown at the bottom of the page, where

and . The resultant distortion is

(45)

However, if does not satisfy (19), since can not be decoded,the MMSE estimate becomes ,where the coefficients satisfy (46), shown at the bottom of thepage. We thus have (47), as shown at the bottom of the page,and the distortion is then given by

(48)

Now, we observe that substituting (the always feasible)in (45) yields (48). In fact, since maximizes

in (45), this implies that the worst distortion that can occurwhen is decodable at receiver coincides with the distortionwhen is not decodable at all, just as noted in Appendix C forHDA-CDS, although the physical reason here is not as obvious.

Comparing (45) and (48) with (20), one can then see that theproof will be complete after showing that will be minimizedwhen satisfies (13) with equality. But that easily follows by:(i) the fact that the right-hand sides of both (45) and (48) are de-creasing in (for fixed ) and (ii) by observing that the intervalof satisfying (19) expands as increases.

(43)

(44)

(46)

(47)

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5672 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

ACKNOWLEDGMENT

The authors would like to thank the reviewers and the Asso-ciate Editor for their suggestions that greatly improved the paperin readability.

REFERENCES

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[2] Y. Gao and E. Tuncel, “New hybrid digital/analog schemes for trans-mission of a Gaussian source over a Gaussian channel,” IEEE Trans.Inf. Theory, vol. 56, no. 12, pp. 6014–6019, Dec. 2010.

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

[4] Y. Kochman and R. Zamir, “Joint Wyner-Ziv/dirty-paper coding bymodulo-lattice modulation,” IEEE Trans. Inf. Theory, vol. 55, no. 11,pp. 4878–4889, Nov. 2009.

[5] J. Nayak, E. Tuncel, and D. Gündüz, “Wyner-Ziv coding over broadcastchannels: digital schemes,” IEEE Trans. Inf. Theory, vol. 56, no. 4, pp.1782–1799, Apr. 2010.

[6] E. Tuncel, “Slepian-Wolf coding over broadcast channels,” IEEETrans. Inf. Theory, vol. 52, no. 4, pp. 1469–1482, Apr. 2006.

[7] S. Shamai, S. Verdu, and R. Zamir, “Systematic lossy source/channelcoding,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 564–579, Mar.1998.

[8] M. Wilson, K. Narayanan, and G. Caire, “Joint source channel codingwith side information using hybrid digital analog codes,” IEEE Trans.Inf. Theory, vol. 56, no. 10, pp. 4922–4940, Oct. 2010.

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Yang Gao (S’09) received the B.S. degree in electrical engineering fromTsinghua University, China, in 2004, and the M.S. degree from the ChineseAcademy of Sciences in 2007. He is currently pursuing the Ph.D. degree at theUniversity of California, Riverside, under the supervision of Prof. E. Tuncel.His research focuses on joint source-channel coding.

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. In2003, he joined the Department of Electrical Engineering, University of Cali-fornia, Riverside, where he is currently an Associate Professor. His research in-terests include rate-distortion theory, multiterminal source coding, joint source-channel coding, zero-error information theory, and content-based retrieval inhigh-dimensional databases.

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