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## Wyner–Ziv Coding Over Broadcast Channels: Digital Schemes

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**Wyner–Ziv Coding Over Broadcast Channels:Digital Schemes**JayanthNayak, ErtemTuncel, Member, IEEE, and DenizGündüz, Member, IEEE**simplified**• DPC : dirty paper coding • CSI : channel state information • CL : common layer • RL : refinement layer • LDS : Layered Description scheme**Outline**• Introduction • Background and notation • A basic WZBC scheme • A layered WZBC scheme • Source coding rates for LDS • PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN PROBLEM • PERFORMANCE ANALYSIS FOR THE BINARY HAMMING PROBLEM • CONCLUSIONS AND FUTURE WORK**We study the communication scenario in which one of the**sensors is required to transmit its measurements to the other nodes over a broadcast channel. • The receiver nodes are themselves equipped with side information unavailable to the sender,**lossless transmission (in the Shannon**• sense) is possible with channel uses per source symbol if and • only if there exists a channel input distribution such that**striking features**• The optimal coding scheme is not separable in the clas- • sical sense, but consists of separate components that per- • form source and channel coding in a broader sense. This • results in the separation of source and channel variables • as in (1).**If the broadcast channel is such that the same input distri-**• bution achieves capacity for all individual channels, then • (1) implies that one can utilize all channels at full ca- • pacity. Binary symmetric channels and Gaussian channels • are the widely known examples of this phenomenon.**The optimal coding scheme does not explicitly involve**• binning,**In this paper, we consider the general lossy coding problem**in • which the reconstruction of the source at the receivers need not • be perfect. We shall refer to this problem setup as Wyner–Ziv • coding over broadcast channels (WZBC).**We present a coding**• scheme for this scenario and analyze its performance in the • quadratic Gaussian and binary Hamming cases.**Dirty paper coding**• In telecommunications, dirty paper coding (DPC) is a technique for efficient transmission of digitaldata through a channel that is subject to some interference that is known to the transmitter. The technique consists of precoding the data so as to cancel the effect of the interference.**Outline**• Introduction • Background and notation • A basic WZBC scheme • A layered WZBC scheme • Source coding rates for LDS • PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN PROBLEM • PERFORMANCE ANALYSIS FOR THE BINARY HAMMING PROBLEM • CONCLUSIONS AND FUTURE WORK**Background and notation**• be random variables denoting a source with independent and identically distributed (i.i.d.) realizations. • Source X is to be transmitted over a memoryless broadcast channel deﬁned by • Decoder k has access to side information in addition to the channel output**Deﬁnition 1**• Deﬁnition 1: An code consists of an encoder • and decoders at each receiver • The rate of the code is channel uses per source symbol**Deﬁnition 2**• A distortion tuple is said to be achievableat a rational rate • if for every , there exists such that for all integers with , there exists an code satisfying**In this paper, we present some general WZBC techniquesand**derive the corresponding achievable distortion regions. Westudy the performance of these techniques for the followingcases. • Quadratic Gaussian • Binary Hamming**Wyner–Ziv Coding Over Point-to-Point Channels**• the case . Since ,we shall drop the subscripts that relate to the receiver. TheWyner–Zivrate-distortion performance is characterized as • is an auxiliary random variable,and the capacityof the channel is well-known to be**It is then straightforward to conclude that combining**separatesource and channel codes yields the distortion • On the other hand, a converse result in [15] shows that evenby using joint source-channel codes, one cannot improve thedistortion performance further than (3). • We are further interested in the evaluation of ,as well as in the test channels achieving it, for the quadraticGaussian and binary Hamming cases.**Quadratic Gaussian**• It was shown in [22] that the optimalbackward test channel is given by • where and are independent Gaussians. For the rate we have • The optimal reconstruction is a linear estimate**Quadratic Gaussian(cont.)**• which yields the distortion • and therefore**Binary Hamming**• It was implicitly shown in [23] that the optimal auxiliary random variable is given by • where are all independent, and are and with and , respectively, and is an erasure operator, i.e., • This choice results in**Binary Hamming(cont.)**• Where • with denoting the binary convolution, i.e., • , and denoting the binary entropy function, i.e.,**A Trivial Converse for the WZBC Problem**• At each terminal, no WZBC scheme can achieve a distortion less than the minimum distortion achievable by ignoring the other terminals. Thus**Separate Source and Channel Coding**• there is considerable simpliﬁcation in the quadratic Gaussian and binary Hamming cases since the channel and the side information are degraded in both cases: we can assume that one of the two Markov chains • or holds (for arbitrary channel input ) for the channel • or holds for the source.**Outline**• Introduction • Background and notation • A basic WZBC scheme • A layered WZBC scheme • Source coding rates for LDS • PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN PROBLEM • PERFORMANCE ANALYSIS FOR THE BINARY HAMMING PROBLEM • CONCLUSIONS AND FUTURE WORK**Outline**• Introduction • Background and notation • A basic WZBC scheme • A layered WZBC scheme • Source coding rates for LDS • PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN PROBLEM • PERFORMANCE ANALYSIS FOR THE BINARY HAMMING PROBLEM • CONCLUSIONS AND FUTURE WORK**A LAYERED WZBC SCHEME**• the side information and channel characteristicsat the two receiving terminals can be very different, we mightbe able to improve the performance by layered coding, • i.e., bynot only transmitting a common layer (CL) to both receivers butalso additionally transmitting a reﬁnement layer (RL) to one ofthe two receivers. • Since there are two receivers, we are focusingon coding with only two layers because intuitively, more layerstargeted for the same receiver can only degrade the performance.**A LAYERED WZBC SCHEME(cont.)**• CL : c • RL : r • In this scheme, illustrated in Fig. 2, the CL is coded using CDS with DPC with the RL codeword acting as CSI.**Outline**• Introduction • Background and notation • A basic WZBC scheme • A layered WZBC scheme • Source coding rates for LDS • PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN PROBLEM • PERFORMANCE ANALYSIS FOR THE BINARY HAMMING PROBLEM • CONCLUSIONS AND FUTURE WORK