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# source - Source Coding with Decoder Side Information...

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Source Coding with Decoder Side Information Presenter: Shivaprasad Kotagiri. Date: 27 September 2006. Reading: Elements of Information theory by Cover & Thomas [1, Secs 14.8 and 14.9] and Wyner-ziv paper [2] 1 Introduction In this report, we study source coding problems with decoder side information (SI). Source coding problems with decoder side information are a special case of distributed source coding problems. We outline the relation between various distributed source coding problems in Table 1 with the help of Figure 1. For reference, we also include the standard single-source lossless and lossy compression problems in Table 1. We discuss source coding problems with decoder SI in which an encoder maps a random vector x n 1 from a random source x to w ∈ W = { 1 , 2 , . . . , M } , and a decoder, provided with w and the side information correlated with x n , computes an estimate of x n 1 subject to a fidelity criterion 1 . In general, the correlated information that is available at the decoder is called as side information. A B Encoder-X Encoder-Y Decoder x n y n w x w y Output Figure 1: Block diagram of distributed source coding. The basic idea of source coding with decoder SI is to encode the conditional un- certainity in x n given the side information. In this report, we study the rate at which a source can be compressed wiht the side information at the decoder to achieve some specified distortion D x under a given distortion measure d ( · , · ). Let us denote the rate of codes used in Encoder-X and Encoder-Y as R x and R y , respectively. In this paper, we are interested in two source coding problems when switch B is closed in Figure 1. In these problems, y n is correlated with x n , the side information available to the decoder 1 In this report, we use the following notation: x -random variable, x -sample value of rv x , x n = ( x 1 , x 2 , . . . , x n )- n length random vector, x n -sample vector of x n , X - the alphabet set from which x takes vales, In general, we omit the subscripts of probability distribution functions (pdfs), i.e., p ( x ) = p x ( x )

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Type of coding R y (A,B) ( D x , D y ) Dec. O/p Reference Lossless source coding (of x n ) N/A (0 , 0) (0, N/A) ˆ x [1] Lossy source coding (SC) (of x n ) N/A (0 , 0) ( 0, N/A) ˆ x [1] Slepian-Wolf coding H ( y ) (0 , 1) (0,0) x n , ˆ y n ) [1, 3] Lossy multi-terminal source coding H ( y ) (0 , 1) ( 0 , 0) x n , ˆ y n ) [4] Conditional Rate-Distortion H ( y ) (1 , 1) ( , ) ˆ x n [5] Lossless SC with Lossy SI H ( y ) (0 , 1) (0 , ) ˆ x n [1, 6] Lossy SC with Lossless SI H ( y ) (0 , 1) ( 0 , ) ˆ x n [1, 2] Table 1: Comparison of several problems obtained via different switch configurations in Figure 1. Column 2 indicates whether lossy or lossless side information is available at the decoder. Column 3 indicates which switches are closed. If switch variable is 0 , then it is open. Otherwise, it is closed. Column 4 indicates distortion constraints on the decoder estimates. Column 5 indicates whether the decoder is concerned with esimating only one source or both. Appropriate references are mentioned in Column 7.
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