1. Contents Communications Theory Parallel vs. serial transmission
Transmission Capacity (Shannon)
Error detection and correction
Communications Media
Optical fibers
Coaxial cables
Twisted pairs
Wireless

2. Error Detection Example Belgian Bank Account Numbers
Bank account number structure
Bank identification : 3 digits
Account number : 7 digits
Error detection : 2 digits
The ten first digits modulo 97 are appended for error detection purposes.
This algorithm allows detection of all single digit errors
Example :
MOD 97 = 08
MOD 97 = 01

3. Error detection and correction
Length of messages :
Informative message:
Redundancy:
# Messages send:
# Messages received:
Hamming Distance (X-Y):
k + r <= LMax
k bits
r bits, f(inf.mess.)
2 k
2 k+r
i=1
|Xi-Yi|
k+r

4. Error detecting codes 00 11 k = 1; r = 1; red.bit = inf.bit. 00 01 1110
Hd = 2 00 11
Single bit errors are detected if
hamming distance between legitimate messages > 1.
No guessing is possible as erroneous messages are
at equal distances from several correct ones.

5. Error correcting codes
k = 1; r = 2; red.bits = inf.bit.
001
011
111
101
000
010
110
100
Hd = 3
000
111
Hamming distance between legitimate messages > 2.
This implies that each erroneous message is closer
to one correct message than to any other.

6. Error correcting codes for single bit error correction
Required Overhead
for single bit error correction
k+r < 2r
information
redundancy
Overhead 1
<= 4
<= 11
<= 26
<= 57
<= 120
<= 247 2 3 4 5 6 7 8
200 %
75 %
36 %
19 %
11 %
6 %
3 %

7. Error Correction Error detecting codes
Correction by retransmission of erroneous blocks
If few errors, very low overhead
Most common approach to error correction in data communications
Error correcting codes
Very high overhead with short data blocks
Longer data blocks can have multiple errors
Used when retransmission impossible or impractical
Also used when error rate rather high.
Error correcting codes for long blocks, with multiple errors exist and are used (trellis encoding)