---

The effect of errors on a CRC message

---

• Preliminaries
  • Recall from the CRC decoding procedure:

    Let M be CRC protected message Let G be the generator polynomial If the message M is received with no error, then: -------- G / M ..... Remainder = 0000...000

  ---

  ---

  ---

  ---

• The error polynomial
  • Error polynomial:

    Error polynomial = the polynomial that represents the bit errors in the received message If a bit is received correctly: The corresponding bit in the error polynomial = 0       If a bit is received incorrectly: The corresponding bit in the error polynomial = 1

    ---

    ---

  • Example:

    Transmitted message = 10001100 Received message = 11101100 |||||||| VVVVVVVV Error polynomial = 01100000

    ---

    ---

  • How to determine the error polynomial:

    Error = ReceivedMsg XOR TransmittedMsg

  ---

  ---

  ---

  ---

• The CRC computation and the error polynomial
  • Claim:

    The CRC remainder computed using: Received Message / generator polynomial is equal to the CRC remainder computed using: Error / generator polynomial

  • Proof:

    Error = ReceivedMsg XOR TransmittedMsg Therefore: Error % GP = (ReceivedMsg XOR TransmittedMsg) % GP (GP = Gen Poly) = (ReceivedMsg % GP) XOR (TransmittedMsg % GP) // TransmittedMsg % GP = 0 !!! = (ReceivedMsg % GP) XOR 0 = ReceivedMsg % GP

    ---

    ---

  • Example: (previous example)

    Message with bit errors: Generator polynomial: 1101 (Math. notation: 1×x3 + 1×x2 + 0×x1 + + 1×x0) N = 4 (Generator polynomial has 4 terms, or bits in "1001") Sender: (from a previous example) Message: 10001 CRC protected message: 10001100 1. Received message = 11101100 2. Check CRC by dividing "received message" by generator polynomial: 00011 ---------------- 1101 / 11101100 1101 ---- (XOR) 01111 1101 ---- (XOR) 1000 1101 ---- (XOR) 0101 <---- remainder = 101 ≠ 000 !!! 3. Received message is assumed to be corrupted Bit errors have been detected !!! We can obtain the same CRC remainder as follows: 1. Received message = 11101100 = 10001100 + 01100000 ^ | "error polynomial" 2. Check CRC by dividing the error polynomial by generator polynomial: 1001 ---------------- 1101 / 01100000 1101 ---- 1000 1101 ----- 101 <---- same remainder = 101 !!!

  ---

  ---

  ---

  ---

• When can the CRC method detect error ?
  • Recall that:

    The CRC method has detect (some) bit errors if: The remainder of the check procedure ≠ 0000..0000         In other words: if ( ReceivedMsg % GenPolynome ≠ 0000...0000 ) then: bit errors has been detected

    ---

    ---

  • From above:

    ReceivedMsg % GenPolynome = Error % GenPolynome !!!

    Therefore:

    if ( Error % GenPolynome ≠ 0000...0000 ) then: bit errors has been detected

    In other words:

    If the error polynomial is not divisible by the generator polynomial, then: The bit error(s) will be detected by the CRC method (Because the remainder of the division is ≠ 000...0000 !!!) Otherwise: The bit error(s) will not be detected by the CRC method (Because the remainder of the division is = 000...0000 !!!)

    ---

    ---

    ---

  • Example:

    Generator polynomial: 1101 (Math. notation: 1×x3 + 1×x2 + 0×x1 + + 1×x0) N = 4 (Generator polynomial has 4 terms, or bits in "1001") Sender: (from a previous example) Message: 10001 CRC protected message: 10001100 1. Received message = 11100100 = 10001100 XOR 01101000 2. Check CRC by dividing "received message" by generator polynomial: 00011 ---------------- 1101 / 11100100 1101|||| ----v||| 01101|| 1101|| ----vv 0000 3. Received message is assumed to be CORRECT Bit errors have NOT been detected !!!