# L13 - Cyclic redundancy check for encoding and error...

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Cyclic redundancy check for encoding and error detection (continued) K-1 K-2 0 Write the data sequence (d , d , . ....... , d ) as K-1 K-2 0 M(X) = d X + d X ........ + d K-1 K-2 Transmission order is highest degree first. Computation of the c check bits is by polynomial division. The data polynomial is divided by a c-degree binary polynomial, P(X). The resulting check sequence is denoted R(X), and is of degree < c.The transmitted sequence can be represented as T(X) = X M(X) + R(X) c The check bits are in the lowest degree positions, transmitted last.

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Example: K=7, N=4. M(X) = X + X + X + X + 1 (Data is 1 1 1 0 1 0 1) 6542 P(X) = X + X + 1 43 X M(X) = X + X + X + X + X 4 1 09864 Calculate R(X) by long division - continue until a remainder polynomial, degree < 4, is found. X + X + X +X + X + 1 6432 X + X + 1 . X + X + X + X + X 1 X + X + X 10 9 6 X + X 84 X + X +X 874 X 7 X 763 X 63 X 652 532 X 54 X 432 + 1 Remainder R(X) -> X + X + 1 2 The total frame sequence is: 1 1 1 0 1 0 1 0 1 1 1 DATA CHECK
The total frame sequence is: 1 1 1 0 1 0 1 0 1 1 1 DATA CHECK ---------------> time order. Now let the circuit compute the checks: Note that the circuit gives the same answer for the checks as the long division.

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Error detection can be done with the same circuit Register initially all zeroes. If there are no errors, after the n bits are shifted in it should again be all zeroes: If the result is not zero, error must have occurred. If zero, error most likely has not occurred, though it is not certain.
Reasoning behind why the check works: Recall T(X) = X M(X) + R(X) c If no errors, after X M(X) marches into the c register, R(X) will appear there.

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L13 - Cyclic redundancy check for encoding and error...

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