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L13_error_detection_correction

# L13_error_detection_correction - Detecting and Correcting...

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6.02 Spring 2008 Detecting and Correcting Errors, Slide 1 Detecting and Correcting Errors Codewords and Hamming Distance Error Detection: parity Single-bit Error Correction Burst Error Correction Framing 6.02 Spring 2008 Detecting and Correcting Errors, Slide 2 There’s good news and bad news… The good news: Our digital modulation scheme usually allows us to recover the original signal despite small amplitude errors introduced by the components and channel. An example of the digital abstraction doing its job! The bad news: larger amplitude errors (hopefully infrequent) that change the signal irretrievably. These show up as bit errors in our digital data stream.

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6.02 Spring 2008 Detecting and Correcting Errors, Slide 3 Channel coding Our plan to deal with bit errors: We’ll add redundant information to the transmitted bit stream (a process called channel coding ) so that we can detect errors at the receiver. Ideally we’d like to correct commonly occurring errors, e.g., error bursts of bounded length. Otherwise, we should detect uncorrectable errors and use, say, retransmission to deal with the problem. Digital Transmitter Digital Receiver Channel Coding Error Correction Message bitstream bitstream with redundant information used for dealing with errors redundant bitstream possibly with errors Recovered message bitstream 6.02 Spring 2008 Detecting and Correcting Errors, Slide 4 More good news, bad news… Good news: theoretically it is possible to transmit information without error at any rate below the limiting rate C Bad news: the proof doesn’t tell how to construct the appropriate error-correcting code for a given R and C! Shannon’s Noisy Channel Coding Theorem: Given a noisy channel with channel capacity C then for any ! > 0 and R < C, for large enough N, there exists a code of length N and rate ! R and a decoding algorithm, such that the maximal probability of block error is " ! .
6.02 Spring 2008 Detecting and Correcting Errors, Slide 5 Error detection and correction Suppose we wanted to reliably transmit the result of a single coin flip: Further suppose that during transmission a single-bit error occurs, i.e., a single “0” is turned into a “1” or a “1” is turned into a “0”. 0 1 “heads” “tails” Heads: “0” Tails: “1” This is a prototype of the “bit” coin for the new information economy. Value = 12.5¢ 6.02 Spring 2008 Detecting and Correcting Errors, Slide 6 Hamming Distance (Richard Hamming, 1950) HAMMING DISTANCE: The number of digit positions in which the corresponding digits of two encodings of the same length are different The Hamming distance between a valid binary code word and the same code word with single-bit error is 1.

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L13_error_detection_correction - Detecting and Correcting...

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