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1 Block Coding Messages are made up of k bits. Transmitted packets have n bits, n > k: k-data bits and r-redundant bits. n = k + r
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2 Modulo-2 Arithmetic Addition and subtraction are described by the logical exclusive-or operation.
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3 Modulo-2 Arithmetic (logical xor)
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4 Chapter 10 Error Detection and Correction Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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Interference Example causes of interference: Heat Noise due to interference (EM fields) Attenuation Distortion Interference causes bit errors.
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Bit Error Classifications Single bit error Burst error
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Single Bit Error Single bit error A one is interpreted as a zero (or vice-versa) Refers to only one bit modified in a specified transmission unit of data An uncommon type of error for serial data due to the duration of a bit being much less than the duration of interference.
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8 Figure 10.1 Single-bit error
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9 Burst Error More than one bit is damaged by interference.
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10 Figure 10.2 Burst error of length 8
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11 Error Detection Detection of errors is necessary to determine if data should be rejected. Possible responses to error detection include: Retransmission request Forward error correction (corrected on the receiving end) Forward correction saves bandwidth & time
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12 Redundancy Extra bits can be included with the data transmission to assist in the detection and correction of errors – For digital transmissions that implies using block-coding
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13 Redundant Bits
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14 Detection vs Correction Detection is easier than correction
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15 Data Coding Schemes Block coding – described in this course or – Convolution coding – this is covered in advanced Math or EE signal processing courses.
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16 Block Coding Data-words are made up of k bits. Codewords are made up of k data bits (a data- word) and r redundant bits. n = k + r Where n is the number of bits in a codeword
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17 Figure 10.5 Datawords and codewords in block coding
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18 Examples No block coding for 10base-T Ethernet, Manchester coding. 4B/5B block coding (Fast Ethernet), MLT-3 line coding. 8B/10B block coding (Gigabit Ethernet), PAM-5 line coding.
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20 Error Detection With Block Coding The receiver can detect an error if – The receiver has a list of valid code words – A received codeword is not a valid code word
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21 Table 10.2 A code for error correction (Example 10.3)
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22 Possible Transmission Outcomes A codeword is sent and received without incident A codeword is sent but is modified in transmission. The error is detected if the codeword is not in the valid codeword list. A codeword is sent but is modified in transmission. The error is not detected if the resulting new codeword is in the list of valid codewords.
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23 Error-Detecting Code Error detecting code can only detect the types of errors it was designed to detect. Other types of errors may go undetected.
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24 The Hamming Distance The Hamming Distance, d(x,y), between two words of the same size is the number of differences between corresponding bits. Examples d(000,011) = 2 d(10101,11110) = 3
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25 Let us find the Hamming distance between two pairs of words. 1. The Hamming distance d(000, 011) is 2 because Example 10.4 2. The Hamming distance d(10101, 11110) is 3 because
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26 The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words. Note
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10.27 Table 10.1 A code for error detection (Example 10.2)
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28 Find the minimum Hamming distance of the coding scheme in Table 10.1. Solution We first find all Hamming distances. Example 10.5 The d min in this case is 2.
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29 Table 10.2 A code for error correction (Example 10.3)
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30 Find the minimum Hamming distance of the coding scheme in Table 10.2. Solution We first find all the Hamming distances. The d min in this case is 3. Example 10.6
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31 To guarantee the detection of up to s errors in all cases, the minimum Hamming distance in a block code must be d min = s + 1. Note
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32 Hamming Code Notation What is the maximum number of detectable errors for each of the two previous coding schemes? – d-min = 2, s = – d-min = 3, s =
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33 Figure 10.8 Geometric concept for finding d min in error detection
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34 To guarantee correction of up to t errors in all cases, the minimum Hamming distance in a block code must be d min = 2t + 1. Note
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35 Figure 10.9 Geometric concept for finding d min in error correction
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How many errors can be corrected for the two example coding schemes: d-min = 2, t = d-min = 3, t =
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37 To guarantee correction of up to t errors in all cases, the minimum Hamming distance in a block code must be d min = 2t + 1. Note
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38 Example A code scheme has dmin = 5. – What is the maximum number of detectable errors? – What is the maximum number of correctable errors?
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39 Linear Block Codes A linear block code is a code where the logical exclusive-or of any two valid codewords creates another valid codeword.
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40 Linear Block Codes The min Hamming distance for a LBC is the minimum number of ones in a non-zero valid codeword.
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Exercise: For each of the next two examples, Compute the Hamming distance, the number of detectable errors, number of correctable errors Show that the code is linear
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42 Table 10.1 A code for error detection C(3,2)
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43 Table 10.2 A code for error correction C(5,2)
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44 Parity Check Count the number of ones in a data word. If the count is odd, the redundant bit is one If the count is even, the redundant bit is zero
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45 A simple parity-check code is a single-bit error-detecting code in which n = k + 1 with d min = 2. Note
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46 A simple parity-check code can detect an odd number of errors. Note
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47 Table 10.3 Simple parity-check code C(5, 4)
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48 How Useful is a Parity Check? Detecting any odd number of errors is pretty good, can we do better?
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49 2-Dimensional Parity Check It is possible to create a 2-D parity check code that detects and corrects errors.
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50 Figure 10.11 Two-dimensional parity-check code
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51 Figure 10.11 Two-dimensional parity-check code
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52 Figure 10.11 Two-dimensional parity-check code
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53 Figure 10.11 Two-dimensional parity-check code
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Interleaving ● By interleaving the columns into slots, it becomes possible to ● detect up to n-row errors. ● The example is 70%efficient. The efficiency can be improved by adding more rows.
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55 Cyclic Codes If a codeword is shifted cyclically, the result is another codeword. – (highest order bit becomes the lowest order bit) – Cyclic codes are linear codes
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C(7,4) Assume d-min = 3. Answer the following about table 10.6: ● What is the codeword size? ● What is the data-word size?
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57 Table 10.6 is this C(7, 4) cyclic?
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C(7,4) Is the table 10.6 code cyclic? Is it linear? What is d-min? How many detectible errors? How many correctible errors?
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Hamming Codes d-min >= 3 Minimum number of detectable errors: 2 Minimum number of correctable errors: 1 For C(n,k), n = 2^r – 1 r = n – k (number of redundant bits) k >= 3
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Which are Hamming Codes? C(7,4) C(7,3)
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Error Correction 1. Using CRC codes computing bit syndromes 2. Using interleaving with multiplexing. – Use a parity bit in each frame – Check for invalid code words (see example exercises)
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62 Checksum Adding the codewords together at the source and destination. If the sum at the source and destination match, there is a good chance that no errors occurred. Checksums are not as reliable as the CRC.
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63 Checksum Examples 1's complement 16 bit checksum used by the Internet
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Sender: Checksum Calculation 1.Divide into 16 bit unsigned words 2.Add the 16 bit words using 1’s complement addition. 3.Complement the total 4.Send all the above words.
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Receiver: Checksum Calculation 1. Divide into 16 bit words 1.Add the 16 bit words and the checksum value using 1’s complement arithmetic. 2.The complement of the total should be zero.
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Example: Sender 0x466f 0x726f 0x757a +_______ 0x12e58 partial sum 0x2e59 sum 0xd1a6 complement
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Example: Receiver 0x466f 0x726f 0x757a 0xd1a6 (sender’s checksum) +_______ 0x1fffe (partial sum) 0xffff (sum) 0x0000 (complement)
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