chap10..................................ppt
- 1. Chapter 10 Error Detection and Correction 1. Introduction 2. Block Coding 3. Checksum
- 2. Type of Errors Type of Errors An electromagnetic signal is subject to interference from heat, magnetism, and other forms of electricity Single-bit error: 0 1 or 1 0 Burst error: 2 or more bits have changed 10-2
- 3. Single-Bit Error Single-Bit Error Only one bit of a given data unit is changed The least likely type of error in serial transmission Single-bit error can happen in parallel transmission Dr. Md. Abdur Razzaque Green Networking Research Group Dept. of Computer Science and Engineering, University of Dhaka 10-3
- 4. Burst Error Burst Error Two or more bits in the data unit have changed Burst error does not necessarily mean that the errors occur in consecutive bits Most likely to happen in a serial transmission Number of bits affected depends on the data rate and duration of noise Dr. Md. Abdur Razzaque Green Networking Research Group Dept. of Computer Science and Engineering, University of Dhaka 10-4
- 5. Redundancy Redundancy Error detection uses the concept of redundancy, which means adding extra (redundant) bits for detecting errors at the destination 10-5
- 6. Error Control Error Control Detection Versus Correction Detection: error ? yes or no Correction: Need to know the exact number of bits that are corrupted, and their location in the message Forward Error Correction Versus Retransmission Retransmission (resending) : Backward error correction Coding for redundancy Block coding: discussed in our textbook Convolution coding 10-6
- 7. Modular Arithmetic Modular Arithmetic In modulo-N arithmetic, we use only the integers in the range 0 to N- 1, inclusive. Adding: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 Subtracting: 0 – 0 = 0 0 – 1 = 1 1 – 0 = 1 1 – 1 = 0 XORing of two single bits or two words 10-7
- 8. Block Coding Block Coding Divide the message into blocks, each of Divide the message into blocks, each of k k bits, called bits, called datawords. datawords. Add Add r r redundant bits to each block to make the length n = k + r. The redundant bits to each block to make the length n = k + r. The resulting n-bit blocks are called resulting n-bit blocks are called codewords codewords Example: Example: 4B/5B block coding k = 4 and n = 5. 2k = 16 datawords and 2n = 32 codewords. 10-8
- 9. Error Detection in Block Coding Error Detection in Block Coding Example: Assume that k = 2 and n = 3 (Table 10.1) 10-9
- 10. Error Detection: Example Error Detection: Example Assume the sender encodes the dataword 01 as 011 and sends it to the receiver. Consider the following cases: 1. The receiver receives 011 which is a valid codeword. The receiver extracts the dataword 01 from it. 2. The codeword is corrupted during transmission, and 111 is received. This is not a valid codeword and is discarded. 3. The codeword is corrupted during transmission, and 000 is received. This is a valid codeword. The receiver incorrectly extracts the dataword 00. Two corrupted bits have made the error undetectable. An error-detecting code can detect only the types of errors for which it is designed; other types of errors may remain undetected 10-10
- 11. Error Correction in Block Coding Error Correction in Block Coding Example: Assume that k = 2 and r = 3 n = 5 (Table 10.2) 10-11
- 12. Error Correction: Example Error Correction: Example Assume the dataword is 01. The sender creates the codeword 01011. The codeword is corrupted during transmission, and 01001 is received. First, the receiver finds that the received codeword is not in the table. This means an error has occurred. The receiver, assuming that there is only 1 bit corrupted, uses the following strategy to guess the correct dataword 1. Comparing the received codeword with the first codeword in the table (01001 versus 00000), the receiver decides that the first codeword is not the one that was sent because there are two different bits. (the same for third or fourth one in the table) 2.. The original codeword must be the second one in the table because this is the only one that differs from the received codeword by 1 bit. 10-12
- 13. Hamming Distance Hamming Distance The Hamming distance between two words is the number of differences between corresponding bits. Example: Hamming distance d(10101, 11110) is 3 The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words Example for Table 10.1 - dmin = 2 10-13
- 14. Hamming Distance Hamming Distance Three parameters to define the coding schemes Codeword size n Dataword size k The minimum Hamming distance dmin Coding scheme C(n, k) with a separate expression for dmin Hamming distance and error Hamming distance between the received codeword and the sent codeword is the number of bits that are corrupted Minimum distance for error detection To guarantee the detection of up to s errors in all cases, the minimum Hamming distance in a block code must be dmin = s + 1. 10-14
- 15. Minimum Hamming Distance: Minimum Hamming Distance: Example Example The minimum Hamming distance in Table 10.1 is 2. This code guarantees detection of only a single error. For example, if the third codeword (101) is sent and one error occurs, the received codeword does not match any valid codeword. If two errors occur, however, the received codeword may match a valid codeword and the errors are not detected. In Table 10.2, it has dmin = 3. This code can detect up to two errors. When any of the valid codewords is sent, two errors create a codeword which is not in the table of valid codewords. The receiver cannot be fooled. However, some combinations of three errors change a valid codeword to another valid codeword. The receiver accepts the received codeword and the errors are undetected. 10-15
- 16. Checksum Checksum Tendency is to replace the checksum with a CRC Not as strong as CRC in error-checking capability One’s complement arithmetic We can represent unsigned numbers between 0 and 2n – 1 using only n bits If the number has more than n bits, the extra leftmost bits need to be added to the n rightmost bits (wrapping) A negative number can be represented by inverting all bits. It is the same as subtracting the number from 2n – 1 10-16
- 17. Checksum: Example Checksum: Example The sender initializes the checksum to 0 and adds all data items and the checksum. However, 36 cannot be expressed in 4 bits. The extra two bits are wrapped and added with the sum to create the wrapped sum value 6. The sum is then complemented, resulting in the checksum value 9 (15 − 6 = 9). 10-17
- 18. Internet Checksum Internet Checksum Sender site: 1. The message is divided into 16-bit words. 2. The value of the checksum word is set to 0. 3. All words including the checksum are added using one’s complement addition. 4. The sum is complemented and becomes the checksum. 5. The checksum is sent with the data. Receiver site: 1. The message (including checksum) is divided into 16-bit words. 2. All words are added using one’s complement addition. 3. The sum is complemented and becomes the new checksum. 4. If the value of checksum is 0, the message is accepted; otherwise, it is rejected. 10-18
- 19. Internet Checksum: Example Internet Checksum: Example 10-19