Information Theory Chapter_7_PartI - Yazd University ECE 204 Information Theory and Coding Chapter 7 Error Correcting Codes Part I Jamshid Abouei ECE

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Chapter 7: Error Correcting Codes Part I ECE 204 Information Theory and Coding Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011 Yazd University
2 Outline Introduction Parity Check Codes Hamming Distance Yazd University Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011 Syndrome and Error Detection/Error Correction Application of Group Theory in Parity Check Codes Standard Array and Syndrome Decoding
3 Introduction Yazd University Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011 So far, we have restricted our attention to the source coding and the capacity . We will focus on the theory and practice of error correcting codes (ECCs) in communication systems corrupted with noise. We limit our attention to binary forward error correcting (FEC) block codes. Symbol alphabet consists of 0 and 1 , Receiver can correct a transmission error without asking the sender for more information or for a retransmission . Transmissions consist of a sequence of fixed length blocks , called codewords.
4 Introduction Yazd University Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011 In this chapter, we present two categories of the channel codes: Error detecting codes, According to the Shannon’s Theorem, “ there exist codes that allow transmission at any rate below channel capacity with an arbitrarily small probability of error. Problem 1: The proofs do not provide clue on “ how to construct” the codes ; Error correcting codes. Problem 2: In the Shannon’s Theorem, it is assumed that the codeword block lengths are infinity in order to achieve Problem 3: In real communication systems, the codeword block lengths are finite and we have an error in transmitting symbols.
5 Introduction Yazd University Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011 The main goal in this chapter: Find a good encoding scheme to detect (in the first step) the errors and then correct them. Assumptions: The channel is assumed as BSC with parameter p, Precise bounds on the error correcting ability of codes. The inputs to the channel is chosen from a set of binary codewords of length The codewords occur with equal probability , The Ideal-Observer (I.O.) decision scheme is used
6 Decision Rule Yazd University Jamshid Abouei, ECE 204 Information Theory and Coding, Yazd University, Iran, Spring 2011 Since an error may be made in any digit, the set of possible encoder output sequences is the set of all binary sequences of length n . Our first problem is to find the form of the I.O. decision scheme for such a code. Decision Rule: The channel output sequences are where we assume that the channel output could be any sequence due to the error in the channel.