Fault Detection Unit Active ●● imary Module I econfiguration Unit Primary Module N Inputs ●自● Spare Module I Spare Module M FIGURE 93.3 Hybrid redundancy appro The encoding operation is the process of determining the corresponding code word for a particular data item. other words, the encoding process takes an original data item and represents it as a code word using the rules of the code. The decoding operation is the process of recovering the original data from the code word. In other words, the decoding process takes a code word and determines the data that it represents It is possible to create a binary code for which the valid code words are a subset of the total number of possible combinations of ls and Os. If the code words are formed correctly, errors introduced into a code word will force it to lie in the range of illegal, or invalid, code words, and the error can be detected. This is the basic concept of the error detecting codes. The basic concept of the error correcting code is that the code word is structured such that it is possible to determine the correct code word from the corrupted, or erroneous, code A fundamental concept in the characterization of codes is the Hamming distance[Hamming, 1950]. The Hamming distance between any two binary words is the number of bit positions in which the two words differ. For example, the binary words 0000 and 0001 differ in only one position and therefore have a Hamming distance of 1. The binary words 0000 and 0101, however, differ in two positions; consequently, their Hamming distance is 2. Clearly, if two words have a Hamming distance of 1, it is possible to change one word into the other simply by modifying one bit in one of the words. If, however, two words differ in two bit positions, it is impossible to transform one word into the other by changing one bit in one of the word The Hamming distance gives insight into the requirements of error detecting codes and error correcting codes. We define the distance of a code as the minimum Hamming distance between any two valid code word a binary code has a distance of two, then any single-bit error introduced into a code word will result in the erroneous word being an invalid code word because all valid code words differ in at least two bit positions. If a code has a distance of 3, then any single-bit error or any double-bit error will result in the erroneous word eing an invalid code word because all valid code words differ in at least three positions. However, a code distance of 3 allows any single-bit error to be corrected, if it is desired to do so, because the erroneous word with a single-bit error will be a Hamming distance of 1 from the correct code word and at least a Hamming e 2000 by CRC Press LLC© 2000 by CRC Press LLC The encoding operation is the process of determining the corresponding code word for a particular data item. In other words, the encoding process takes an original data item and represents it as a code word using the rules of the code. The decoding operation is the process of recovering the original data from the code word. In other words, the decoding process takes a code word and determines the data that it represents. It is possible to create a binary code for which the valid code words are a subset of the total number of possible combinations of 1s and 0s. If the code words are formed correctly, errors introduced into a code word will force it to lie in the range of illegal, or invalid, code words, and the error can be detected. This is the basic concept of the error detecting codes. The basic concept of the error correcting code is that the code word is structured such that it is possible to determine the correct code word from the corrupted, or erroneous, code word. A fundamental concept in the characterization of codes is the Hamming distance [Hamming, 1950]. The Hamming distance between any two binary words is the number of bit positions in which the two words differ. For example, the binary words 0000 and 0001 differ in only one position and therefore have a Hamming distance of 1. The binary words 0000 and 0101, however, differ in two positions; consequently, their Hamming distance is 2. Clearly, if two words have a Hamming distance of 1, it is possible to change one word into the other simply by modifying one bit in one of the words. If, however, two words differ in two bit positions, it is impossible to transform one word into the other by changing one bit in one of the words. The Hamming distance gives insight into the requirements of error detecting codes and error correcting codes. We define the distance of a code as the minimum Hamming distance between any two valid code words. If a binary code has a distance of two, then any single-bit error introduced into a code word will result in the erroneous word being an invalid code word because all valid code words differ in at least two bit positions. If a code has a distance of 3, then any single-bit error or any double-bit error will result in the erroneous word being an invalid code word because all valid code words differ in at least three positions. However, a code distance of 3 allows any single-bit error to be corrected, if it is desired to do so, because the erroneous word with a single-bit error will be a Hamming distance of 1 from the correct code word and at least a Hamming FIGURE 93.3 Hybrid redundancy approach. Primary Module 1 Fault Detection Unit Active Unit Outputs Disagreement Detection Inputs Reconfiguration Unit Voter Output Primary Module N Spare Module 1 Spare Module M