11 Error Correction
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citations to reliable sources. Unsourced material may be challenged and removed. (August 2008) (Learn how and when to remove this template message) In information theory and coding theory with applications in computer science and telecommunication, error error detection and correction using hamming code example detection and correction or error control are techniques that enable reliable delivery of error correction exercises digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during
Error Correction In English
transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Contents 1 Definitions 2 History 3 Introduction 4
Error Correction Techniques
Implementation 5 Error detection schemes 5.1 Repetition codes 5.2 Parity bits 5.3 Checksums 5.4 Cyclic redundancy checks (CRCs) 5.5 Cryptographic hash functions 5.6 Error-correcting codes 6 Error correction 6.1 Automatic repeat request (ARQ) 6.2 Error-correcting code 6.3 Hybrid schemes 7 Applications 7.1 Internet 7.2 Deep-space telecommunications 7.3 Satellite broadcasting (DVB) 7.4 Data storage 7.5 Error-correcting memory 8 See also 9 References 10 Further reading 11 External links Definitions[edit] The hamming codes general definitions of the terms are as follows: Error detection is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. Error correction is the detection of errors and reconstruction of the original, error-free data. History[edit] The modern development of error-correcting codes in 1947 is due to Richard W. Hamming.[1] A description of Hamming's code appeared in Claude Shannon's A Mathematical Theory of Communication[2] and was quickly generalized by Marcel J. E. Golay.[3] Introduction[edit] The general idea for achieving error detection and correction is to add some redundancy (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message, and to recover data determined to be corrupted. Error-detection and correction schemes can be either systematic or non-systematic: In a systematic scheme, the transmitter sends the original data, and attaches a fixed number of check bits (or parity data), which are derived from the data bits by some deterministic algorithm. If only error detection is required, a receiver can simply apply the same algorithm to the received data bits and compare its output with the received check bits; if the values do not match, an error has occurred at some
game. Give the students time to realise they've made a mistake and try to correct it themselves. If they can't, maybe someone else can help them. If
Hamming Code Error Correction Technique
nobody can help then you can either step in and give the correct hamming code error correction example ppt form or make a note of it for later. As far as possible, correct mistakes anonymously. Do this by making hamming code error detection and correction pdf notes of students' mistakes as you monitor (see TT10 for further explanation) then putting them on the board later and give the students themselves the opportunity to correct them, in pairs or small groups. https://en.wikipedia.org/wiki/Error_detection_and_correction If no one knows the right answer, give it to them, but only as a last resort. Anonymous error correction is a kind way to deal with mistakes. It isn't important who made the mistake originally - the point is, can the students all correct it? I tend to doctor the mistakes so that even the perpetrator doesn't recognise them as his/her own. For example: Original error: http://www.tefl.net/teaching/teaching-tip_11.htm "I have been to Paris last year." = On the board: "I have been to London last week." Extra Info: Mistakes are good things and students need to know that they are. I explain like this: "Please make lots and lots of mistakes in my lessons - new mistakes, mind you, not the same old ones over and over. I like mistakes because we can all learn from them and because if you don't make any I won't have a job. If I find a student who doesn't make any mistakes in my lesson I will move that student to a higher level class because s/he obviously isn't learning anything at this level." Learning English is like learning to ride a bike - you fall off a lot, but you get the hang of it in the end. You will make a lot of mistakes but you will be able to communicate effectively in the end. Very few people become successful international cyclists and the chances are that even though you can ride a bike you are not a professional cyclist. Very few students reach mother-tongue (supposedly error-free) level but many students learn to communicate very well in Engl
limit for minimum number of check bits to do 1-bit error-correction. Bits of codeword are numbered: bit 1, bit 2, ..., bit n. Check bits are inserted at positions 1,2,4,8,.. (all powers of 2). The rest are the m data bits. Each check bit checks (as parity bit) a number of data bits. Each check bit checks a different collection of data http://www.computing.dcu.ie/~humphrys/Notes/Networks/data.hamming.html bits. Check bits only check data, not other check bits. Hamming Codes used in: Wireless comms, e.g. Fixed wireless broadband. High error rate. Need correction not detection. Any number can be written as sum of powers of 2 First note every number can be written in base 2 as a sum of powers of 2 multiplied by 0 or 1. i.e. As a simple sum of powers of 2. Number is sum of these: 1 2 4 8 16 Number: 1 x 2 x 3 x x 4 x 5 x x 6 x x 7 x x x 8 error correction x 9 x x 10 x x 11 x x x 12 x x 13 x x x 14 x x x 15 x x x x 16 x 17 x x 18 x x 19 x x x 20 x x 21 x x x 22 x x x 23 x x x x 24 x x 25 x x x 26 x x x 27 x x x x 28 x x x 29 x x x x 30 x x x x 31 x x x x x ... Scheme for check bits Now here is our scheme hamming code error for which bits each check bit checks: Checked by check bit: 1 2 4 8 16 Bit: 1 (not applicable - this is a check bit) 2 (n/a) 3 x x 4 (n/a) 5 x x 6 x x 7 x x x 8 (n/a) 9 x x 10 x x 11 x x x 12 x x 13 x x x 14 x x x 15 x x x x 16 (n/a) 17 x x 18 x x 19 x x x 20 x x 21 x x x 22 x x x 23 x x x x 24 x x 25 x x x 26 x x x 27 x x x x 28 x x x 29 x x x x 30 x x x x 31 x x x x x 32 (n/a) ... Check bit records odd or even parity of all the bits it covers, so any one-bit error in the data will lead to error in the check bit. Assume one-bit error: If any data bit bad, then multiple check bits will be bad (never just one check bit). Calculating the Hamming Code (check bits do even parity here) How it works 21 (as sum of powers of 2) = 1 + 4 + 16 Bit 21 is checked by check bits 1, 4 and 16. No other bit is checked by exactly these 3 check bits. If assume one-bit error, then if exactly these 3 check bits are bad, then we know that data bit 21 was bad and no other. Assume one-bit error: Error in a data bit: Will cause multiple errors in check bits. Will cause errors in exactly the check bits that correspond to the powers of 2 that the bit number can be written as a sum of. Er