Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Parity-check codeSlide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Hamming CodeSlide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68CRC using Digital Logic - ParametersCRC using Digital LogicExampleExample: G(X) = X5+X4+X2+1 (pre determined generator)Slide 73Slide 74Cyclic Code AnalysisSlide 76Slide 77Slide 78Slide 79Slide 80Slide 81Slide 82Slide 83Slide 84Slide 85Burst ErrorsSlide 87Slide 88Slide 89Slide 90Slide 91Slide 92Slide 93Slide 94Slide 95Slide 96Slide 97Slide 98Slide 99Slide 100Slide 101Slide 102Slide 10310.1Chapter 10Error Detection and CorrectionCopyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.10.2Data can be corrupted during transmission.Some applications require that errors be detected and corrected.Note10.310-1 INTRODUCTION10-1 INTRODUCTIONLet us first discuss some issues related, directly or Let us first discuss some issues related, directly or indirectly, to error detection and correction.indirectly, to error detection and correction.Types of ErrorsRedundancyDetection Versus CorrectionForward Error Correction Versus RetransmissionCodingModular ArithmeticTopics discussed in this section:Topics discussed in this section:10.4In a single-bit error, only 1 bit in the data unit has changed.Note10.5Figure 10.1 Single-bit error10.6A burst error means that 2 or more bits in the data unit have changed.Note10.7Figure 10.2 Burst error of length 810.8To detect or correct errors, we need to send extra (redundant) bits with data.Note10.9Figure 10.3 The structure of encoder and decoder10.10In this book, we concentrate on block codes; we leave convolution codes to advanced texts.Note10.11In modulo-N arithmetic, we use only the integers in the range 0 to N −1, inclusive.Note10.12Figure 10.4 XORing of two single bits or two words10.1310-2 BLOCK CODING10-2 BLOCK CODINGIn block coding, we divide our message into blocks, In block coding, we divide our message into blocks, each of k bits, called each of k bits, called datawordsdatawords. We add r redundant . We add r redundant bits to each block to make the length n = k + r. The bits to each block to make the length n = k + r. The resulting n-bit blocks are called resulting n-bit blocks are called codewordscodewords..Error DetectionError CorrectionHamming DistanceMinimum Hamming DistanceTopics discussed in this section:Topics discussed in this section:10.14Figure 10.5 Datawords and codewords in block coding10.15The 4B/5B block coding discussed in Chapter 4 is a good example of this type of coding. In this coding scheme, k = 4 and n = 5. As we saw, we have 2k = 16 datawords and 2n = 32 codewords. We saw that 16 out of 32 codewords are used for message transfer and the rest are either used for other purposes or unused.Example 10.110.16Figure 10.6 Process of error detection in block coding10.17Let us assume that k = 2 and n = 3. Table 10.1 shows the list of datawords and codewords. Later, we will see how to derive a codeword from a dataword. Assume the sender encodes the dataword 01 as 011 andsends it to the receiver. Consider the following cases:1. The receiver receives 011. It is a valid codeword. The receiver extracts the dataword 01 from it.Example 10.210.182. 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.Example 10.2 (continued)10.19Table 10.1 A code for error detection (Example 10.2)10.20An error-detecting code can detect only the types of errors for which it is designed; other types of errors may remain undetected.Note10.21Figure 10.7 Structure of encoder and decoder in error correction10.22Let us add more redundant bits to Example 10.2 to see if the receiver can correct an error without knowing what was actually sent. We add 3 redundant bits to the 2-bit dataword to make 5-bit codewords. Table 10.2 shows the datawords and codewords. 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.Example 10.310.231. 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.2. By the same reasoning, the original codeword cannot be the third or fourth one in the table.3. 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. The receiver replaces 01001 with 01011 and consults the table to find the dataword 01.Example 10.3 (continued)10.24Table 10.2 A code for error correction (Example 10.3)10.25The Hamming distance between two words is the number of differences between corresponding bits.Note10.26Let us find the Hamming distance between two pairs of words.1. The Hamming distance d(000, 011) is 2 because Example 10.42. The Hamming distance d(10101, 11110) is 3 because10.27The minimum Hamming distance is the smallest Hamming distance between all possible pairs in a set of words.Note10.28Find the minimum Hamming distance of the coding scheme in Table 10.1.SolutionWe first find all Hamming distances.Example 10.5The dmin in this case is 2.10.29Find the minimum Hamming distance of the coding scheme in Table 10.2.SolutionWe first find all the Hamming distances.The dmin in this case is 3.Example 10.610.30To guarantee