Wyner-Ziv Coding over Broadcast Channels Using ... - IEEE Xplore

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Email: {jnayak,ertem}@ee.ucr.edu. Abstract—This paper deals with the design of coding schemes for transmitting a source over a broadcast channel when there.
ISIT 2008, Toronto, Canada, July 6 - 11, 2008

Wyner-Ziv Coding over Broadcast Channels Using Hybrid Digital/Analog Transmission Deniz Gunduz†, Jayanth Nayak∗ , Ertem Tuncel∗ † Princeton

University, Princeton, NJ E-mail: [email protected] ∗ University of California, Riverside, CA Email: {jnayak,ertem}@ee.ucr.edu Abstract—This paper deals with the design of coding schemes for transmitting a source over a broadcast channel when there is source side information at the receivers. Based on SlepianWolf coding over broadcast channels, three hybrid digital/analog schemes are proposed and their power-distortion tradeoff is investigated for Gaussian sources and Gaussian broadcast channels. All three transmit the same digital and analog information but with varying coding order. Although they are not provably optimal in general, they can significantly outperform uncoded transmission and separate source and channel coding.

I. I NTRODUCTION We discuss the problem of lossy transmission of a common source over a broadcast channel where each receiver has source side information unknown to the encoder. In a recent paper, the lossless version of this problem, which was termed Slepian-Wolf coding over broadcast channels (SWBC), was completely solved [5]. We similarly refer to the current problem as Wyner-Ziv coding over broadcast channels (WZBC). We present coding schemes for this problem when the rate is 1 channel use per source symbol, i.e., with no bandwidth compression/expansion, and explicitly analyze the performance of our schemes in the quadratic Gaussian case. Unlike in [5], however, we do not have a tight converse. We shall point that the quadratic Gaussian problem at rate 1 is trivial when there is no side information. By sending the source uncoded (which is the simplest possible strategy), one can achieve the optimal performance. However, in most cases, uncoded transmission is suboptimal for quadratic Gaussian WZBC. In fact, even for transmitting a Gaussian source over a point-to-point Gaussian channel, uncoded transmission is suboptimal when the side information is non-trivial. Another simple approach is to separate the source and the channel coding. The rate-distortion performance of successive refinement in the presence of side information was fully characterized in the quadratic Gaussian case in [4]. By comparing these rate-distortion results with the capacity results for the Gaussian broadcast channel [2], we obtain the set of all powerdistortion triples achievable by separate coding. We present three new hybrid digital/analog schemes, and investigate their power-distortion tradeoff. In all the three schemes, the source is quantized hierarchically in two stages, and the same set of resultant digital and analog information is transmitted by sharing the same available power: (i) output of the first stage as common information to both receivers, (ii) output of the second stage to the “refinement receiver,”

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and finally (iii) the quantization error in the first stage in an uncoded fashion to both receivers. What differs in the three schemes is the order of coding, i.e., which information “sees” which as noise, and treats which as channel state information (CSI) known at the encoder. Depending on the coding order of the common, refinement, and analog information, we refer to our schemes as C-A-R, A-C-R, or A-R-C. For the refinement information, the CSI is canceled using dirty paper coding as usual without any performance loss. However, for the transmission of the common information in A-R-C or A-CR, we first extend the dirty paper coding theorem [3] to the SWBC setting, and then utilize Costa’s construction [1] to characterize an achievable effective capacity region for this case. Unlike in the original SWBC problem, one cannot use both channels in full capacity in the dirty paper extension. The resultant expressions for the power-distortion tradeoff of the schemes do not easily allow for an analytical comparison. However, numerical evaluations show that none of the three schemes is universally better than the other two. Finally, in one special circumstance, all three schemes achieve a trivial outer bound by allocating all their power to the transmission of the common information. II. P RELIMINARIES ∞ Let {X, Y1 , Y2 }t=1 be independent copies of correlated Gaussians of zero mean and unit variance. Also let Yi = 2 ρi X + Ni with Ni ⊥ X and σN = 1 − ρ2i , i = 1, 2. Length-n i blocks of X are to be transmitted across a Gaussian broadcast channel where receiver 1 has access to side information Y1n and receiver 2 has access to side information Y2n . Let Win denote the additive white Gaussian noise at channel i with 2 variance σW . We assume without loss of generality that i 2 2 2 σN1 ≥ σN2 , but do not assume any ordering between σW 1 2 and σW2 . Consider first the simplest approach of uncoded transmission. √ For a fixed power level P , at each instant, we transmit U = P X. Receiver i computes the minimum mean squared error estimate of X given U + Wi and Yi . Using results from estimation theory, it is easy to show that the performance achieved by this scheme is given by   2 2 2 σN1 σW1 σN σ2 2 W2 (D1 , D2 ) = σ2 +σ2 P , σ2 +σ2 P W1

N1

W2

N2

Another simple approach is to use separate source and channel codes. Combining known source [4] and channel

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ISIT 2008, Toronto, Canada, July 6 - 11, 2008

2 2 coding [2] results, and assuming σW ≤ σW , (D1 , D2 ) is 1 2 achievable if for some α ∈ [0, 1],  2  4 (σ2 +αP )(P +σ2 ) σN σ2 σ 1 N2  max DN11 ,  2 < Wσ21 (αP +σ2 W) 2 2 2 2 D2 (σN −σN )D1 +σN σN 1

2

1

W1

2

Q 1n

D2

4 σN σ2 1 N2



2 −σ 2 )D +σ 2 σ 2 D2 (σN 1 N N N 1

2