Lecture 4
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Error Detection
-
Forward Error Correction (Hamming Code)
-
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1- Error Detection
1.1 Parity check
1.2 Two dimensional parity check
1.3 Checksum
1.4 Cyclic redundancy check
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1.1 Parity check
- Appends a parity bit to the end of the data.
10110010 0 Even
10110010 1 Odd
- It can detect all single bit errors.
-It
can also detect burst errors, if the number of
bits in error is odd.
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1.2 Two Dimensional Parity
Check
-Parity check bits are calculated for each
- equivalent to a simple parity check bit.
-Parity
-Both
row, which is
check bits are also calculated for all columns.
are sent along with the data.
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1.2 Two Dimensional Parity
Check
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1.3 Checksum
- At the sender’s end, the data is divided into segments.
-The segments are added using ones complement
arithmetic to get the sum.
-The sum is complemented to get the checksum which is
sent along with the data segments.
-At
the receiver, the received segments are added using
ones complement arithmetic to get the sum.
-If the complement of the sum is zero, the received data
is accepted.
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1.3 Checksum
Ex: Find the checksum at the sender and receiver for the following
sequence: 10110011 10101011 01011010 11010101
10110011
10101011
__________
01011110
1
_______________
01011111
01011010
_______________
10111001
11010101
______________
10001110
1
_____________
sum: 10001111
Checksum: 01110000
10110011
10101011
__________
01011110
1
_______________
01011111
01011010
_______________
10111001
11010101
______________
10001110
1
_____________
10001111
01110000
________________
Sum: 11111111
Checksum: 00000000
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1.4 Cyclic Redundancy Check
-The sender generates a bit sequence known as a frame
check sequence (FCS), so that the resulting frame
consisting of the original data followed by this FCS.
- FCS is the remainder which obtained from dividing the
original data M(X) by generator polynomial G(x).
- The receiver divides the incoming frame by G(X), if there
is no remainder the received data has no errors.
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1.4 Cyclic Redundancy Check
EX. M ( X )  X 3  X , G( X )  X 3  X  1
G( X )  X 3  X  1  1* X 3  0 * X 2  1* X 1  1* X 0
G( X )  1011
( X 3  X ) * X 3  X 6  X 4  1010000
1001
1001
1011
1010000
1011
________
0010
0000
_________
0100
0000
__________
1000
1011
__________
FCS 0 1 1
1011
1010011
1011
________
0010
0000
_________
0101
0000
__________
1011
1011
__________
000
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1.4 Cyclic Redundancy Check
EX . Use Linear FeedbackShift Re gister to implementCRC process
If
M ( X )  X 3  X , G( X )  X 3  X  1
( X 3  X 0 ) * X 3  X 6  X 3  1001000
C3
C1
C0
1010000
+
+
Initial
1
0
1
000
001
010
101
0
001
0
0
0
010
100
011
FCS
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2- Forward Error Correction
-Forward Error correction (FEC) which based on the receiver only is used mainly when
retransmissions cannot be requested, e.g., simplex links.
-Backward error correction which based on retransmission of the frame when an error is
detected is commonly used.
Hamming Code:
• It can correct any single bit error
• Bits in power of 2 positions (1,2,4,8,) are check bits, the rest are m data bits
• Check bit forces the parity of some collection of bit “1” to be even
• To see which check bits the data bit in position k contributes to, rewrite k as a
sum of powers of 2.
11=1+2+8, 29=1+4+8+16
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Hamming Code
EX: If a 12-bit hamming code 100110111000 arrives at receiver, is there any bit
error? If so, which bit is wrong?
1 2 3=2+1 4 5=4+1 6=4+2 7=4+2+1 8 9=8+1 10=8+2 11=8+2+1 12=8+4
10 0
11
0
1
11
0
0
0
Check Checked Data bits
bit
Bits Sequence
1
3 5 7 9 11
101110
2
3 6 7 10 11
0 0 0 1 0 0 (No)
4
5 6 7 12
11010
8
9 10 11 12
11000
(Yes)
(No)
(Yes)
Hence bit 6 = 2+4 is wrong, so correct hamming code
should be 100111111000
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