CONSIDER a sensor network of nodes taking

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, Ertem Tuncel, Member, IEEE Abstract A new hybrid digital/analog scheme is proposed for lossy transmission of a Gaussian source over a bwidth-matched Gaussian broadcast channel with source side information available at each receiver. The proposed scheme combines two schemes that were previously shown to achieve optimal point-to-point distortion/power tradeoff simultaneously at all receivers under two distinct conditions stated in terms of channel side information quality parameters. For the two-receiver case, the combined scheme is shown to achieve the same kind of optimality for the entire region in the parameter space swiched between those two conditions. Crucial to this result is a new degree of freedom discovered 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 CONSIDER 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 each of the nodes has source side information (SSI) only available to that node. The lossless version of this problem was studied in [6] 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], performance of several purely digital schemes were analyzed. In this paper, we consider the bwidth-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, University of California, Riverside, CA 92521 USA (e-mail: yagao@ucr.edu; ertem.tuncel@ucr.edu). 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, transmission 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 description scheme (CDS), was proposed in [5] for general sources, distortion measures, bwidth expansion factors. CDS is also a simple scheme based on losslessly broadcasting the quantized source by utilizing the binning-free joint decoding technique developed in [6] for the SWBC problem. It was shown in [5] that for the bwidth-matched quadratic Gaussian case, CDS achieves the trivial outer bound when an appropriately defined combined channel side information quality is constant among the receivers. In [5], CDS was also extended to a dirty-paper setting (termed DPC-CDS), the channel state information (CSI) is available non-causally at the encoder, to a layered description scheme (LDS), which was shown to outperform separate source 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 bwidth-matched quadratic Gaussian case. When the CSI is trivial the scenario is extended to broadcast channels ( thus the problem becomes WZBC), the scheme is also shown to achieve the trivial outer bound when another combined channel 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 rom coding arguments instead of lattices, by the analog source is integrated into the auxiliary rom 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 combining the HDA-WZ scheme 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 side information quality parameters fall between those yielding constant combined quality with respect to the HDA-WZ scheme 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

GAO AND TUNCEL: WYNER-ZIV CODING OVER BROADCAST CHANNELS: HYBRID DIGITAL/ANALOG SCHEMES 5661 HDA-CDS, also to be potentially used as artificial CSI for both HDA-WZ CDS components. However, although the analog stream is very useful as a third signal component, we numerically observed that it is useless as CSI, though a rigorous proof seems extremely difficult. The paper is organized as follows. Section II defines the problem formally discusses related past work in detail. In Section III, a new degree of freedom in point-to-point transmission, which is crucial to our main result, is revealed. Then HDA-CDS is presented, its optimality in the region of interest is proved. The more general scheme combining analog transmission with HDA-CDS (termed AHC) is introduced, its performance is numerically compared with HDA-CDS, separate source channel coding, LDS, analog transmission. Section IV concludes the paper. II. BACKGROUND AND NOTATION Let be real-valued jointly Gaussian rom variables generated in an i.i.d. fashion from. The Gaussian source sequence is to be transmitted over a Gaussian broadcast channel Since separate source channel coding is optimal, as proved in [7], transmission is possible if only if, which translates to For the Gaussian WZBC problem, a trivial outer bound is obtained by letting each receiver achieve its minimum distortion without considering other receivers: There are several schemes for the WZBC problem achieving the trivial outer bound under different conditions, though a scheme optimal under all circumstances is not known. With the bwidth-matched case under consideration, the simplest scheme one can use is analog transmission of the source, in which the unit-variance source is scaled with to adapt to the power constraint of the channel, it achieves the performance (1), are the channel input, channel output at receiver, the corresponding i.i.d. additive white Gaussian channel noise. The channel has an input power constraint Source side information is available at receiver, with. Without loss of generality, are assumed to have unit variance thus the variance of is. Hereafter, to ease exposition, a bold font capital letter will denote the variance of the corresponding rom variable as in [5]. The reconstruction quality is measured with squared error distortion at each receiver. It is obvious that uncoded transmission achieves the outer bound (1) with equality if only if, for all. This, in turn, corresponds to the extreme situation that is independent of, hence is of very limited use to us. The performance of separate source channel coding is known since both the channel the side information are degraded. The explicit expressions of the distortion pair for were given in [5, Lemma 3] are included here for completeness. Without loss of generality, is assumed. The distortion pair with is achievable using separate coding if only if is the convex hull of when for any source block reconstruction block. In the special case of point-to-point transmission, or, the capacity of the Gaussian channel is 12 the minimum distortion achieved by Wyner-Ziv coding with rate is given by [9] 1 All logarithms in this paper are base 2. 2 The subscripts for the receivers are omitted since there is only one receiver. when. Another purely digital scheme, the common description scheme (CDS), was proposed in [5], which can be utilized for general sources/channels bwidth expansion factors, but we focus here only on the bwidth-matched case. As illustrated in Fig. 1, CDS compresses the source sequence to one of source codewords, say, which in turn is mapped into an independently generated channel word (the number of channel words is still ). At

5662 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011 Fig. 2. Illustration of HDA-WZ. The collection of codewords T serve as a source codebook at the encoder as a channel codebook at the decoder. Fig. 1. Illustration of the codebooks of a successful CDS transmission. The cross-hatched codewords are the actually used ones the double sided arrows denote joint typicality, as the arrows with a cross denote atypical pairs. At the receiver side, all the hatched codewords in the source codebook Z together form a virtual bin. receiver, the channel output will be jointly typical with channel codewords. When traced back to the source codebook, the corresponding source codewords form a virtual bin (which can overlap with other virtual bins, so does not correspond to binning in the strict sense). Using the side information, the actual source codeword in this bin is then disambiguated with success if only if 3 for all. For the quadratic Gaussian case we are interested in here, it was shown in [5] that CDS satisfies (1) with equality at all the receivers if only if It has to be pointed out that although CDS is governed by the same inequality (2) as in separate source channel coding, it is inherently different since the source encoder does not perform binning. Intuitively, the benefit of CDS is that, with multiple receivers, it allows a weak receiver with larger channel noise to make up with better side information by avoiding binning at the encoder side. As proposed in [5], CDS can also be used in a dirty-paper setting with a channel the channel state information (CSI) is also available non-causally at the encoder. By using Costa coding [1], it was shown that transmission is successful if only if for all, the auxiliary dirty-paper codewords are generated according to for some with independent. This extension was termed dirty-paper-coded CDS (DPC-CDS) 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 description scheme (LDS) for receivers by adding a refinement layer designated for the receiver with better combined channel side information quality, i.e., the one with the smaller. At the encoder side, the binning index of the refinement message is mapped to a channel codeword, which, in turn, is 3 The left-h side of (2) coincides with the source coding rate with explicit binning as in Wyner-Ziv coding, is the same as I (X ; Z )0I (Y ; Z ) due to the Markov chain Y 0 X 0 Z. (2) (3) used as artificial CSI at the CDS encoder. The receiver first decodes the common information transmitted via CDS then the additional information is decoded only at the refinement receiver. It was shown that LDS outperforms separate coding. We will compare the performance of our schemes with that of LDS since, to the best of our knowledge, it provides the best known performance in digital schemes. In [8], a hybrid digital/analog scheme, which will be referred to as the HDA-WZ scheme here, was proposed proved to be optimal for point-to-point transmission. In the HDA-WZ scheme, an auxiliary rom variable is defined as, a codebook of size is generated from typical, as shown in Fig. 2. The encoder then finds a codeword that is jointly typical with the source sequence, then sends the corresponding. At the receiver side, the unique that is jointly typical with is chosen, is estimated from,,. To guarantee successful decoding of the correct needs to be satisfied. When the effective source coding rate channel coding rate match, the resultant distortion becomes. This also implies that, in the WZBC scenario, the trivial outer bound (1) is achieved by HDA-WZ if only if That is because only in this case, the same choice of matches what the receivers need for optimality, as is apparent from (5). Although this condition was not explicitly mentioned in [8], it was in [4], which uses an equivalent modulo-lattice modulation instead. III. RESULTS In this section, we will first propose a new scheme for point-to-point transmission with SSI by combining separate source-channel coding with HDA-WZ, discuss how this scheme provides complete freedom in choosing the dirtypaper auxiliary codeword structure (through the choice of the coefficient in DPC) for any power allocation between the two streams. The point-to-point scheme will then be extended to the two-receiver broadcast scenario (WZBC) by simply replacing the separate coding block with DPC-CDS. Utilizing the extra (4) (5)

GAO AND TUNCEL: WYNER-ZIV CODING OVER BROADCAST CHANNELS: HYBRID DIGITAL/ANALOG SCHEMES 5663 Fig. 3. New hybrid scheme for point-to-point transmission with SSI, featuring the freedom of power allocation between the branches, for any power allocation, the freedom to choose the auxiliary rom codeword from a range. freedom in choosing the DPC coefficient, we will then show that this WZBC scheme, termed HDA-CDS, achieves the trivial outer bound whenever the system parameters fall into the region swiched between those for which CDS HDA-WZ achieve the same. Finally, we will add an analog layer to HDA-CDS, discuss the performance of the resultant scheme, termed AHC. A. A New Freedom in Point-to-Point Transmission A new scheme for point-to-point transmission with SSI will be proposed in this section. In [2], it was shown that for point-topoint transmission without any SSI or CSI, the optimum distortion can be achieved with a range of auxiliary rom variables (or equivalently, a range of ) for any power allocation between an analog stream a digital stream, by making the analog stream serve as artificial CSI for the digital one fully utilizing the decoded auxiliary dirty-paper codeword. This exps the freedom of designing the system to another degree in addition to the well-known freedom of power allocation between the digital analog streams. Inspired by the result in [2], we now propose to combine the HDA-WZ scheme in [8] separate coding in a similar way, as shown in Fig. 3. We will show that the same kind of freedom exists in this system when we make full use of the decoded dirty-paper codeword at the receiver. The source is encoded as usual with an optimal quantizer characterized by a backward test channel with, followed by binning. The total channel input power is split into for hybrid digital paths, respectively,. The quantization error is transmitted by HDA-WZ scheme as an analog source, the HDA auxiliary codeword is constructed by. The HDA channel codeword is fed as artificial CSI to the DPC channel encoder, which maps the bin index into a bin of auxiliary dirty-paper codewords characterized by, chooses the unique in the bin which is jointly typical with. The corresponding digital channel input is the channel output is. Just as in [2], is not confined to the optimal choice in [1], which is but rather, any feasible is considered. At the receiver side, the digital decoder operates first. To ensure that it successfully decodes (as well as ), we need (6) is the Wyner-Ziv source coding rate is the effective digital channel capacity. The HDA-WZ decoder has access to,. Defining as the effective side information for the source, we then need to decode successfully, follows from. Now we are ready to introduce the extra level of freedom of the scheme. Theorem 1: is achievable for any satisfying The proof is given in Appendix A. Remark 1: Theorem 1 suggests that for any power allocation between the two streams, the proposed scheme achieves the optimal distortion in the presence of SSI, for a range of auxiliary codeword given by (8), instead of only Costa s construction. Thus even when the effective channel is not used at its full capacity, one can still achieve the optimal distortion by fully utilizing the decoded auxiliary codeword. This result implies the freedom introduced in [2] is not an isolated case. Moreover, as shown in the next section, since both can be freely chosen for any receiver, under certain conditions, we are able to find a pair making the encoder optimal for both receivers. It is worth noting that a similar scheme can be found is input to the HDA-WZ encoder instead of. The HDA auxiliary codebook would now be constructed according to 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 digital transmission is still given by (6). To successfully decode, the counterpart of (7) becomes which reduces to (23) (in Appendix A) as well. Note that since the construction of contains, the right-h side of (9) can not be simplified to as before, the optimal estimate of is given by a linear combination of,. By rigorous calculation, it can be shown that, once again, optimal distortion is achieved when equalities hold in both (22) (23). We omit the algebra here as it is not essential for the purposes of this paper. Finally, this scheme also specializes to the scheme in [8, Sec. III-C] by letting. (7) (8) (9)

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 has the same freedom of power allocation construction of auxiliary rom variable. Fig. 5. For WZBC problem, HDA-CDS combines two separately optimal schemes achieves the trivial outer bound in the whole region between the conditions under which either scheme achieves optimum. B. A Basic Scheme: HDA-CDS We can replace the separate digital source channel encoders inside the dotted box in Fig. 3 with a DPC-CDS encoder, resulting in a point-to-point scheme shown in Fig. 4, termed HDA-CDS. We stick to the previous notation for consistency, though the reader needs to remember that it is an essentially different scheme. As before, the source is quantized with a backward test channel, the quantization error is transmitted by an HDA-WZ stream. The HDA auxiliary rom variable is constructed by is the HDA channel word. What is different is that no source binning is performed the DPC-CDS encoder directly maps the quantization index into a bin of dirty-paper auxiliary channel codewords. The encoder then uses the CSI to choose the right codeword inside the bin, as before. When there is only one receiver, the governing inequalities remain to be (6) (7) therefore the freedom in Theorem 1 persists. Inspired by the fact that CDS HDA-WZ achieve the trivial outer bound under different channel side information conditions, we apply HDA-CDS on the WZBC problem. Although the number of receivers,, can be arbitrary, we focus here on the two-receiver case, as shown in Fig. 5. As we shall see, HDA-CDS takes advantage of the new level of freedom revealed for point-to-point transmission, the combination turns out to be better than the sum of its parts. Without loss of generality, we assume that the second receiver has lower channel noise, i.e.,, the channel output at receiver is now given by. At each receiver, with the help of side information the received channel word, first are decoded. From (3), this is possible whenever (10) Fig. 6. For any given W > W, each point on the (N ; N )-plane corresponds to a system with parameters (W ; W ; N ; N ). As indicated by the two lines in the figure, the conditions for achieving the trivial outer bound for CDS the HDA-WZ scheme are W N = W N N (P + W )= N (P + W ), respectively. When N = N = 1, or the SSI is trivial at each receiver, analog transmission achieves the outer bound, as indicated by the dot. The shaded region, as stated in Theorem 2, indicates our scheme is achieving the outer bound. Then the HDA-WZ decoder uses, to decode,. This is possible if (11) as follows from (7). Finally, an MMSE estimate of using, is performed. Now, we can optimize over the quantization rate of the encoder to minimize the distortions. As mentioned before, when, one can simply set, with which our scheme reduces to CDS achieves the outer bound (1). Similarly, when, the choice reduces our scheme to pure HDA-WZ, the outer bound (1) is achieved. Since, these correspond to distinct lines in the -plane shown in Fig. 6. As stated by the following theorem, HDA-CDS achieves the outer bound (1) in the entire shaded region swiched between these two lines.

GAO AND TUNCEL: WYNER-ZIV CODING OVER BROADCAST CHANNELS: HYBRID DIGITAL/ANALOG SCHEMES 5665 Theorem 2: Given with, the hybrid 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 exploiting the extra level of freedom benefits of virtual binning, we are able to find an encoder which is optimal for both receivers when they have similar combined channel/ssi quality (so that the system falls in the shaded region in Fig. 6). In this case, each receiver can achieve its as if it is the only receiver, although the receivers have different channel noise SSI quality. As we show in the next theorem, HDA-CDS performs at least as well as LDS introduced in [5] even outside of the shaded region in Fig. 6. That is because we can simply achieve the same the performance of the LDS by replacing the separate source channel coding for the refinement layer of the LDS with HDA-WZ coding, mimicking LDS with a proper choice of corresponding. Theorem 3: HDA-CDS can achieve the same pairs as LDS when either or. The proof is given in Appendix C. Unfortunately, the complete tradeoff between proved very difficult to derive. However, we numerically observed that outside of the region, the achievable distortion region (after convexification) is always the convex hull of pure HDA-WZ LDS. See Figs. 8(a) 8(c) 8(f) for examples of this phenomenon. In summary, HDA-CDS achieves the trivial outer bound in the entire shaded region shown in Fig. 6, instead of only at the two linear boundaries in other regions, HDA-CDS is also shown to outperform the LDS in [5], which, in turn, is the best known digital scheme. C. The Main Scheme Since analog transmission itself is optimal for WZBC with trivial SSI at all the receivers, it will improve the performance if an analog stream is added to the HDA-CDS 4. We propose to add the analog component using dirty paper coding as shown in Fig. 7. The structure of the decoders remains the same as in HDA-CDS. We call this scheme the Analog-HDA-CDS scheme, or AHC for short. The source vector is quantized in the same way with the backward test channel the quantization error is transmitted by both HDA-WZ 4 Adding the analog stream will improve the performance at least for the case with trivial SSI at all the receivers, when the analog stream does not help, we can always set P =0to let AHC reduce to HDA-CDS. Fig. 7. Encoder of the AHC scheme. An extra analog stream is superimposed with dirty paper coding to incorporate the benefits of analog transmission. The structure of decoders is the same as HDA-CDS. scheme analog transmission. The analog stream is considered at the dirty-paper digital channel encoder as additional artificial CSI besides. The channel power is now split into three parts for HDA-WZ, analog, CDS. The quantization error is scaled by a constant so that. At the HDA-WZ encoder, the auxiliary codeword is again constructed by, while the auxiliary dirty paper codeword at the DPC-CDS encoder by. Both receivers first decode, requiring (12), the channel output is now. Note that the system parameters should be chosen so that the effective digital channel capacity [the right-h side of (12)] is non-negative. The next lemma translates (12) into how, should be chosen for any triplet. Lemma 1: For any power allocation, both receivers can decode if only if (13) (14) with. Also, need to satisfy (15) The proof is given in Appendix D. Remark 3: We note here that the ellipse (15) has the center with, it can be expressed in the stard form as in (16), found at the bottom of the page. (16)

5666 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011 Fig. 8. Performance comparison between AHC scheme, LDS, analog transmission, separate coding when P =1. In (a), N > N. In (b), N < N < N ; AHC scheme achieves the trivial outer bound. In (c) (d), N < N < N. In (e), there is no side information analog transmission is optimal. In (f), N < N. As for the decoding of, it can be accomplished, just as in HDA-CDS, at receiver if only if (17) which is the same as After some algebra, the condition (18) reduces to the quadratic form (19) (18)

GAO AND TUNCEL: WYNER-ZIV CODING OVER BROADCAST CHANNELS: HYBRID DIGITAL/ANALOG SCHEMES 5667 with. The discriminant of (19) is given by which is clearly non-negative. Now since, it immediately follows that (19) corresponds to an interval containing. At receiver, the MMSE estimate of is computed using,, depending on whether it can be decoded,, the corresponding distortion is given in the following lemma. Lemma 2: Define for, satisfies (13) with equality. For any, the distortion 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. Since HDA-CDS is a special case of AHC obtained simply by setting, it is obvious that in the region, Theorem 2 still holds. An example of this can be observed in Fig. 8(b). In the other regions, although analytical dissection is very difficult to obtain, based on extensive numerical simulations we arrived at the following conjectures: Conjecture 1: When, the analog stream is not helpful at all, i.e., optimal distortion performance is achieved when. Moreover, is always the optimal choice, thus as a consequence of Theorem 3, the performance of AHC 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 transmission 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 observed that always, i.e., a simple superposition scheme with only analog HDA-WZ streams suffices. As can be seen in Figs. 8(c) 8(d), the resultant performance is strictly better than those of HDA-CDS analog transmission alone. Conjecture 3: When, all three streams make some contribution the performance is better than those of HDA-CDS analog alone. In addition,, i.e., dirty paper coding is not necessary simple superposition is good enough. The performance is illustrated in Fig. 8(f), when, the scheme reduces to analog transmission, which is indicated by the point on the curve. Bringing all the conjectures together, one can conclude that the analog stream is useless as CSI, the AHC scheme can be simplified to the superposition of HDA-CDS the analog stream. That is because we observe either, or, or. IV. CONCLUSION For the bwidth-matched quadratic Gaussian WZBC problem, we proposed a new hybrid digital/analog coding scheme called AHC, demonstrated that it outperforms all previously known schemes. For the case of two receivers, AHC is analytically shown to achieve (without the help of the analog stream) the trivial outer bound for the entire region in the parameter space swiched between the optimality conditions for HDA-WZ CDS, the two building blocks. This result uses the new level of freedom we discovered for point-to-point transmission, namely, the freedom in choosing the auxiliary dirty-paper codeword in addition to the well known freedom of power allocation. Outside that region, we numerically observed that the AHC scheme reduces to a simple superposition of HDA-CDS the analog stream, thus dirty paper coding is not necessary between the two. APPENDIX A PROOF OF THEOREM 1 Inequality (6) is the same as which can be exped as finally reduces to (21) (22). A condition is also necessary to ensure that the effective capacity in (6) or the right-h side of (21) is non-negative, are given in (8). We observe that the range of feasible in (8) is exactly the same as that in [2], with playing the role of for analog power. Following similar arguments as in [2], this also implies that when, the range of becomes the entire real line the system becomes purely digital. On the other h, when, the scheme becomes equivalent to HDA-WZ. To see that, notice that the right-h side of (22) is maximized by for any the resultant expression after setting approaches 1 as. Inequality (7) is the same as

5668 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011 which reduces to respectively, as before. Now further define (23) The optimal estimate of is given by the optimal estimate of, which is obtained by,, satisfy When we consider both receivers, to ensure decoding, we need due to the principle of orthogonality. We thus have The square-error distortion is given by (24) (25) Further, at each receiver, the HDA auxiliary codeword can be decoded if only if. In addition to (27) (28), the range of the free parameter has to be confined to so that the effective capacity in (10) is non-negative. We refer to pairs satisfying this requirement as feasible. In the point-to-point version of our scheme, it is shown that for any feasible pair, the minimum distortion can be achieved if are chosen so as to satisfy (22) (23), i.e., the counterparts of (27) (28), with equality. This implies for the WZBC problem at h that if we can find a feasible pair so that (27) (28) are satisfied with equality simultaneously for with some choice of, then the trivial outer bound (1) can be achieved. This, in turn, requires simultaneously. Now, can be written as (29) (26) (25) (26) are satisfied by equality when equalities hold in (23) (22), respectively. APPENDIX B PROOF OF THEOREM 2 Let us immediately remark that the hypothesis of the theorem implies. Since (10) (11) have the same forms as (6) (7) in the point-to-point scenario, by defining Note that in the region of interest on the -plane. Similarly, implies from that or equivalently, from (29) that (30) for, they reduce to (27) We first ignore the dependency on, search for a triplet in the interior feasible set. Note that is guaranteed in the feasible set because of. Solving for in both (29) (30) for a fixed pair yields (28) (31)

GAO AND TUNCEL: WYNER-ZIV CODING OVER BROADCAST CHANNELS: HYBRID DIGITAL/ANALOG SCHEMES 5669 For every fixed then implies Of course, (36) is meaningful only if (32) (33) which follows after some algebra using. In summary, if we pick any In fact, (33) automatically follows from the required consistency between the two solutions of in (31). To see that, rewrite (31) as (34) which, in particular, implies using the non-negativity of the right-h side that together with from (31), we simultaneously satisfy with. The only thing that remains is to find a that is consistent with,. We have that Note that (31) because is also granted due to the second equality in Instead of solving above: directly, let us temporarily treat as free variables solve the linear system whenever. Thus, it suffices to find such that (32) (34) are simultaneously satisfied, find the corresponding using either formula in (31). Now, exping (34), we obtain By close inspection, one can actually show that as have varies, we (35) which is quadratic in for every fixed. The discriminant can be computed as with which is strictly positive due to the fact that. Thus, (35) has two positive roots for any. Denoting the left-h side of (35) as, it can be shown after some algebra that This implies that setting to the larger root of, we automatically satisfy. What remains to be found is then under what conditions on that root also satisfies (32). Rewriting (32) as Since are constants, this results in a line on the -plane. Also, since, respectively, imply, this line stretches through the entire interval. In fact, the slope of the line depends only on for given. Now, we show that (37) therefore, implying also that. Towards that end, rewrite (37) as explicitly computing the larger root of with as The quadratic form is minimized at Equation (32) translates to (36)

5670 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011 Also implying since. Going back to,a value consistent with can be found if only if (39) But imply that intersects with. In fact, there are always two intersections ( thus two pairs of satisfying ) except when, or equivalently when Note that (38) (39) are actually the same as (79) (80) in [5] with, our receiver 2 corresponds to the common receiver in [5] because it has worse combined channel SSI quality. When, it can be shown that always. If we set, we then have. When we choose, the first receiver decodes both streams, as the second one decodes only the CDS stream, the resultant distortion pairs are given by APPENDIX C PROOF OF THEOREM 3 The distortion at each receiver depends on whether is chosen so that can be decoded at that receiver. If, is decoded at receiver, hence the distortion is given by (24). If, on the other h,, then cannot be decoded, since both are independent of or, the optimal reconstruction of is solely given by, (40) (41) The distortion in this case is given by Note that this distortion is the same as in (24) with. That is because when, no information about is transmitted decoding does not help at all. When, let, thus. Itis easy to show that when Equation (40) (41) are again the same as in [5], this time with receiver 1 being the common receiver since. APPENDIX D PROOF OF LEMMA 1 The left-h side of (12) is as before, the right-h side is. It can also be shown that in this case, is automatically satisfied because. Since now, is always satisfied. If we choose, the first receiver is able to decode both the CDS stream the HDA stream while the second can only decode the CDS stream. The resultant achievable distortion pairs are given by is defined in (14). Taking both receivers into consideration, we thus need (38)

GAO AND TUNCEL: WYNER-ZIV CODING OVER BROADCAST CHANNELS: HYBRID DIGITAL/ANALOG SCHEMES 5671 In addition, to make sure the effective channel capacity, i.e., the right-h side of (12), is non-negative, need to satisfy (42) for which corresponds to the intersection of two ellipses on the plane. By close inspection, it is easy to see that the ellipse corresponding to channel 1 is always contained in that of channel 2, as. It then suffices to consider (42) for only, which is the same as (15). APPENDIX E PROOF OF LEMMA 2 When can be decoded, the MMSE estimate of is, the coefficients satisfy (43), as shown at the bottom of the page. We thus have (44), also shown at the bottom of the page,. The resultant distortion is (45) However, if does not satisfy (19), since can not be decoded, the MMSE estimate becomes, the coefficients satisfy (46), shown at the bottom of the page. We thus have (47), as shown at the bottom of the page, 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 occur when is decodable at receiver coincides with the distortion when is not decodable at all, just as noted in Appendix C for HDA-CDS, although the physical reason here is not as obvious. Comparing (45) (48) with (20), one can then see that the proof will be complete after showing that will be minimized when satisfies (13) with equality. But that easily follows by: (i) the fact that the right-h sides of both (45) (48) are decreasing in (for fixed ) (ii) by observing that the interval of satisfying (19) exps as increases. (43) (44) (46) (47)

5672 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011 ACKNOWLEDGMENT The authors would like to thank the reviewers the Associate Editor for their suggestions that greatly improved the paper in readability. REFERENCES [1] M. Costa, Writing on dirty paper, IEEE Trans. Inf. Theory, vol. 29, no. 3, pp. 439 441, May 1983. [2] Y. Gao E. Tuncel, New hybrid digital/analog schemes for transmission 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, E. Tuncel, Wyner-Ziv coding over broadcast channels using hybrid digital/analog transmission, in Proc. IEEE Int. Symp. Inf. Theory (ISIT 2008), Toronto, ON, Canada, Jul. 2008. [4] Y. Kochman R. Zamir, Joint Wyner-Ziv/dirty-paper coding by modulo-lattice modulation, IEEE Trans. Inf. Theory, vol. 55, no. 11, pp. 4878 4889, Nov. 2009. [5] J. Nayak, E. Tuncel, D. Gündüz, Wyner-Ziv coding over broadcast channels: digital schemes, IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1782 1799, Apr. 2010. [6] E. Tuncel, Slepian-Wolf coding over broadcast channels, IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1469 1482, Apr. 2006. [7] S. Shamai, S. Verdu, R. Zamir, Systematic lossy source/channel coding, IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 564 579, Mar. 1998. [8] M. Wilson, K. Narayanan, G. Caire, Joint source channel coding with side information using hybrid digital analog codes, IEEE Trans. Inf. Theory, vol. 56, no. 10, pp. 4922 4940, Oct. 2010. [9] A. D. Wyner J. Ziv, The rate-distortion function for source coding with side information at the decoder, IEEE Trans. Inf. Theory, vol. 22, no. 1, pp. 1 10, Jan. 1976. Yang Gao (S 09) received the B.S. degree in electrical engineering from Tsinghua University, China, in 2004, the M.S. degree from the Chinese Academy of Sciences in 2007. He is currently pursuing the Ph.D. degree at the University 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 computer engineering from University of California, Santa Barbara, in 2002. In 2003, he joined the Department of Electrical Engineering, University of California, Riverside, he is currently an Associate Professor. His research interests include rate-distortion theory, multiterminal source coding, joint sourcechannel coding, zero-error information theory, content-based retrieval in high-dimensional databases. Dr. Tuncel received the National Science Foundation CAREER Award in 2007.