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Figure 1.3 Fault, error, and failure.
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Hard errors are caused by permanent faults; they therefore affect the system functions for a long period of time. This type of error is typically provoked by faults that appear as open or short anywhere on the chips, modules, cards, or boards. Hard errors are also called permanent errors. Soft errors, on the other hand, are caused by temporal faults, especially those resulting from external causes. Soft errors have a limited duration, meaning they interrupt system functions for a very short time period. The most likely sources of soft errors are radioactive particles and external noise. Alpha particles and cosmic particles [ZIEG96a, ZIEG96b, ZIEG96c, OGOR96, SRIN96] are the major contributors mentioned previously. Therefore soft errors are also called transient errors. The intermittent errors are provoked by intermittent faults. 1.2.2 Random Errors, Clustered Errors, and Their Mixed-Type Errors
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Multiple errors that occur randomly in time and / or space are called random errors. Error can occur in every bit position of a word with almost equal probability. The random type of error is unpredictable and is typically caused by white noise or external particles. Errors may cluster non-uniformly in a word, and these multiple errors may gather in particular and unpredictable positions in the word. Clustered errors include burst errors and byte errors. Burst errors occur typically in disks or tape memory. Byte errors are typically found in semiconductor memory. The difference is in the data-recording medium. In disk memory, the data are recorded on a continuous surface. In semiconductor memory, the data are stored in RAM chips, and a data fragment, called a byte, is read or stored in each chip. In disk or tape memory, defects or dust particles on the recording surface can cause burst errors to occur anywhere in the continuous recording medium.
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Clustered errors may occur in the two-dimensional matrix symbols as well as in the tape or disk memory of a continuous two-dimensional recording medium. In semiconductor memory, on the other hand, byte errors may occur in a fragment of readout data, namely in a single byte, corresponding to the faulty chip. This is because each chip is physically separated and independent, and therefore the presence of a fault in a chip does not extend to the adjacent chips. Figure 1.4 illustrates the different cases of random errors, byte errors, and burst errors. Another error model consists of mixed clustered and random errors in the operational phase. The clustered errors mentioned above are sometimes caused by physical faults due
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Random Errors
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External noise, particles, and permanent faults occurred randomly
Received data / Readout data
Byte Errors
. . .
. . .
Faulty chip
. . .
. . .
Memory chip with b-bit output Memory card (Package) Readout data
b b-Bit byte error
Burst Errors
Defects Continuous recording medium
Readout data
b b-Bit burst error
Figure 1.4 Models of random errors, byte errors, and burst errors.
to aging problems. However, systems and devices are more prone to damage from transient faults than from physical faults. Transient faults are source of random errors. Therefore, when a physical fault occurs during the operational phase, both types of error clustered and random must be taken into account. For example, in semiconductor memories with byte-organized RAM chips, the major types of errors are transient errors, (i.e., random bit errors) caused by a-particles or external noises. After some time in operation, byte errors will occur due to the aging of RAM chips. Therefore both bit errors and byte errors, meaning both random errors and permanent errors, may occur separately or simultaneously. A similar situation holds for transmission systems, where both random bit errors and burst errors can occur. 6 deals with the codes which control the mixed type of single-byte errors and random bit errors. 1.2.3 Symmetric Errors, Asymmetric Errors, and Unidirectional Errors
In binary systems the probability of errors that force 0 to 1, called 0-errors, is, in general, equal to those going from 1 to 0, called 1-errors. This class of errors is known as symmetric errors. When these errors occur with unequal probabilities, they are called asymmetric errors. In the binary asymmetric error model, only one type of error, either 0-errors or 1-errors, can occur, and the error type is known a priori. If both error types occur but are not mixed, then this class of errors is said to be unidirectional errors [BLAU93]. In binary systems these errors are caused by symmetric faults, asymmetric faults, or unidirectional faults. In nonbinary systems using numerals, 0; 1; 2; 3; . . . ; 9, or alpha-numeric symbols, asymmetric errors are the type that occur. That is, the probability of an error that forces one nonbinary symbol A to another symbol B is sometimes different from that of symbol A forced to yet another symbol C. For example, in handwritten character recognition systems, the probability of a 7 being mistaken for a 9 is much higher than that of a 7 being mistaken for a 4, or p 9j7 ) p 4j7 , where p BjA means probability of a symbol A being mistaken for another symbol B. This is because the numbers 7 and 9 are close in shape whereas 7 and 4 are not so similar. Likewise in keyboard input systems the symbols located on adjacent keys can be more easily mistyped. Figure 1:5 shows examples of these error models. In the asymmetric error model, the error graphs are not perfect and sometimes not bi-directional. On the other hand, in the symmetric nonbinary error model, they are perfect and bi-directional. If symbols are removed or added in a word, as is sometimes caused by human mistakes (i.e., human-made faults), this class of errors is called deletion errors or insertion errors, respectively.
7 0 7 9 1 2 3 8 6 4 5 4 1 0
8 5 2
9 6 3
7 4 1 0
8 5 2
9 6 3