channel capacity. techniques to reduce errors in digital communication systems automatic repeat...
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Channel Capacity
Information Measure
Information in message j
bitsP
Ij
j
1log2
jP Probability of transmitting the jth message Average information:
m
j
m
j
jjj PPIPH
1 2
1log
)/(/ sbitsTHR Source Rate (T is the time required to send a message)
Channel Capacity
NSBC 1log
2
If CR Shannon demonstrated that for signal plus white noise, it is possible to build a system with 0
eP If R < C
Coding, Channel Coding
Convolutional Codes
Techniques to reduce errors in digital communication systems
Techniques to reduce errors in digital communication systems
Automatic repeat request (ARC)Forward error correction (FEC) Channel coding as
opposed to source codingARC is typical of computer systems.FEC is typical of large transmission delay systems.
We will concentrate in FEC.
Codes Block Codes are maps of k input binary symbols into n
binary symbols n>k (memoryless) nkR / typically from ¼ to 7/8 with k>3
Convolutional codes are employed in memory systems where the current k binary symbols are combined with information from previous symbols to produce vkn binary symbols. nkR / Typically from ¼ to 7/8 1<k<8 and 2<v<60.
R is the code rate different from bit rate or information rate
Block Codes
Definitions Hamming weight number of binary 1 bits in a word Hamming distance “d” number of positions in which two words differs.
If 1 tsd s errors can be detected and t errors corrected. If s= t; d>2t+1
rkppppiiii ........
321321 typical, but not unique, transmission format
Hamming codes correct 1 error, s=t=1 d>3 mkn mm 12,12, . Posibble values (7,4), (15,11), (31,26)
There are many other possibilities for block codes like cyclic codes
Convolutional Codes
xhy i
jj
i
0
http://en.wikipedia.org/wiki/Convolutional_code
Rate 1/3 non-recursive, non-systematic convolutional encoder with constraint length 3
http://en.wikipedia.org/wiki/Convolutional_code
Sixteen-state recursive systematic convolutional (RSC) code.
http://en.wikipedia.org/wiki/Convolutional_code
Rate 1/2 8-state recursive systematic convolutional encoder.
Convolutional Codes
Convolutional Codes
Encoded bits in reverse order
http://home.netcom.com/~chip.f/viterbi/algrthms2.html
t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17State 002 0 0 2 3 3 3 3 4 1 3 4 3 3 2 2 4 5 2State 012 3 1 2 2 3 1 4 4 1 4 2 3 4 4 2State 102 2 0 2 1 3 3 4 3 1 4 1 4 3 3 2State 112 3 4 2 1 1 3 4 4 3 4 2 3 4 4
00 11 11 00 01 10 01 11 11 10 00 00 11 00 11 10 11
t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17State 002 0State 012
State 102
State 112
00 11 11 00 01 10 01 11 11 10 00 00 11 00 11 10 11
t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17State 002 0 2 3 3 3 3 4 1 3 4 3 3 2 2 4 5 2State 012 3 1 2 2 3 1 4 4 1 4 2 3 4 4 2State 102 2 0 2 1 3 3 4 3 1 4 1 4 3 3 2State 112 3 1 2 1 1 3 4 4 3 4 2 3 4 4
00 11 11 00 01 10 01 11 11 10 00 00 11 00 11 10 11
t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17State 002 0 2 3 3 3 3 4 1 3 4 3 3 2 2 4 5 2State 012 3 1 2 2 3 1 4 4 1 4 2 3 4 4 2State 102 2 0 2 1 3 3 4 3 1 4 1 4 3 3 2State 112 3 1 2 1 1 3 4 4 3 4 2 3 4 4
00 11 11 00 01 10 01 11 11 10 00 00 11 00 11 10 11
t = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17State 002 0 2 3 3 3 3 4 1 3 4 3 3 2 2 4 5 2State 012 3 1 2 2 3 1 4 4 1 4 2 3 4 4 2State 102 2 0 2 1 3 3 4 3 1 4 1 4 3 3 2State 112 3 1 2 1 1 3 4 4 3 4 2 3 4 4
00 11 11 00 01 10 01 11 11 10 00 00 11 00 11 10 11
0 1 0 1 1 1 0 0 1 0 1 0 0 0 1 0 0