Wyner–Ziv Coding Over Broadcast Channels: Digital Schemes

1. Wyner–Ziv Coding Over Broadcast Channels:Digital Schemes JayanthNayak, ErtemTuncel, Member, IEEE, and DenizGündüz, Member, IEEE

2. simplified  • DPC : dirty paper coding • CSI : channel state information • CL : common layer • RL : refinement layer • LDS : Layered Description scheme

3. 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

4. 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,

6. 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

7. 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).

8. 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.

9. The optimal coding scheme does not explicitly involve • binning,

10. 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).

11. We present a coding • scheme for this scenario and analyze its performance in the • quadratic Gaussian and binary Hamming cases.

12. 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.

13. 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

14. 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

15. single-letter distortion measures

17. 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

19. 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

20. 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

21. Quadratic Gaussian

22. Binary Hamming

23. 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

24. 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.

25. 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

26. Quadratic Gaussian(cont.) • which yields the distortion • and therefore

27. 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

28. Binary Hamming(cont.) • Where • with denoting the binary convolution, i.e., • , and denoting the binary entropy function, i.e.,

29. Binary Hamming(cont.)

30. 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

31. 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.

32. Uncoded Transmission

33. 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

34. A BASIC WZBC SCHEME

35. A BASIC WZBC SCHEME(cont.)

41. Corollary 1

42. Corollary 2

43. 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

44. 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.

45. 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.

47. Source Coding Rates for LDS

48. Channel Coding Rates for LDS

49. 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

50. PERFORMANCE ANALYSIS FOR THE QUADRATIC GAUSSIAN PROBLEM