This document discusses Cyclic Redundancy Check (CRC), a technique used to detect errors in digital data during transmission or storage. CRC works by calculating a checksum based on the remainder of binary long division of the transmitted data divided by a fixed, predetermined polynomial. The sender appends the CRC checksum to the end of the message before transmission. The receiver re-calculates the CRC and checks if it matches, to detect any errors introduced during transmission. Examples are provided to demonstrate how CRC encoding and decoding works using different generator polynomials.

1. CRC
Cyclic Redundancy Check
Prof. Chintan Patel
Chintan.patel@marwadieducation.edu.in

2. Introduction
• Most powerful and Easy to implement technique
• Checksum calculation is based on summation while CRC is based on
binary division
• Cyclic Redundancy check bits are appended at the end of data unit.
• If a K bit message is to be transmitted, the transmitter generates an r-bit
sequence called as FCS(Frame check sequence).
• So k+r bits are actually being transmitted.
• Generator Polynomial : it is a predetermined number of length r+1
which is used to generate FCS bits
• NOTE : Generator polynomial is decided by sender and receiver by
their mutual understanding.
Prof. Chintan Patel

3. • Procedure at sender side :
1. Determine size of original massage (k bits)
2. Establish Generator Polynomial (r + 1 bits).
3. Append r zeros with original message [x = (k (Message)+ r (zeros))]
4. Divide this x by the generator polynomial.
5. Append remainder(also considered as a FCS of r bits) with original
message k.
6. Transmit this data.
• Procedure at Receiver side :
1. Receive k + r bits
2. Establish Generator Polynomial(r+1 bits)
3. Divide this bits by generator polynomial.
4. If remainder is all bits 0 , no error in transmission
Prof. Chintan Patel

4. CRC Sender and Receiver
Prof. Chintan Patel

5. Concepts………………..
1. How to represent a binary sequence using polynomial, and how to represent
polynomial using binary sequence?
7
+ 0x
– M1 = 0010,1101  M1(x) = 0x
6
+ 1x
5
+ 0x
4
+ 1x
3
+ 1x
2
+ 0x
1
+
0
– 1x
5
+ x
= x
3
+ x
2
+ 1
2. How to subtract (or add) two polynomials?
– Represent the polynomials using binary sequences
– Perform bit-wise XOR between the two sequences
– Convert the XOR result back to polynomial
7
+ x
• M1(x) – M2(x)  M1 ⊕ M2 = 00101101 ⊕ 10000100 = 10101001 x
5
+ x
3
+ 1
Prof. Chintan Patel

6. Polynomial Division
• Dividing a polynomial with another one of lower degree is similar
to normal polynomial division with “subtract” simple (XOR).
Prof. Chintan Patel

7. Example 1
• Given information
– Generator polynomial = X3 + x + 1
– Data part = X3 + x2 + 1
• What will be FCS Value ?
• What will be final bits transmitted by sender?.
Prof. Chintan Patel

8. Solution
• Convert polynomial to binary
– Generator polynomial = 1011 (r + 1 = 4 bits)
– Original Message = 1101
• Append 3(r bits), 0’S with original message.
• So it will be 1101000
• Divide this sequence by 1011
• Reminder or FCS Value = 001.
• Append this reminder with original message. So it will be 1101001,
which will be transmitted by sender
Prof. Chintan Patel

9. Example 2
• Use CRC with general polynomial x3 + 1 to encode
the value: 10101111
Prof. Chintan Patel

10. Example 3
• Obtain the 4-bit CRC code word for the data bit sequence
10011011100 (leftmost bit is the least significant) using the
generator polynomial = 10101
Prof. Chintan Patel

11. Solution
Prof. Chintan Patel

12. Practice Examples
1 P : 1001
D : 101110
2 P : 1101
D : 10010110101
3 P : 1011
D : 11010011101100
Prof. Chintan Patel

13. Prof. Chintan Patel