Logic Functions and Equations by Christian Posthoff & Bernd Steinbach

Author:Christian Posthoff & Bernd Steinbach
Language: eng
Format: epub
ISBN: 9783030024208
Publisher: Springer International Publishing

Example 6.12

Let be given (x 0, x 1, x 2, x 3, x 4, x 5, x 6, x 7); then a parity bit can be added to these eight bits of a byte by means of

x 8 = 1 for an odd number of 1’s in the byte, x 8 = 0 for an even number of 1’s in the byte.

In each case the parity bit is set in such a way that the extended byte has an even number of 1’s (even parity). If an odd parity is required then

has to be used. In any case, the error in one single bit changes the parity, and on the receiver side a parity error will be indicated after the check of the parity. However, the information is not sufficient to find out the position that has been changed. Any single-bit error changes the parity. Even worse: two-bit errors will not be detected because the parity will not change.

But the general principle of error detection can already be seen—define a property of the code word that can change by transmission errors, and check the property on the receiver side. The price for the detection of these errors is “affordable”, the technical resources have to be increased by 12.5% (one more bit, error detection circuits). The 2-out-of-5-code offers similar possibilities. Error correction follows similar principles. Generally, we can assume that a “small” environment can be defined for the code words—small changes of the code word (possibly the change of one or two bits) leave the received word in this environment, and the original code word can be re-established on the receiver side. It is not difficult to understand the following.

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