a brief history of coding theory - matthieu bloch's...
TRANSCRIPT
© M
atth
ieu
Bloc
h 20
10
A (Very) Brief History of Coding Theory
1948-present
© M
atth
ieu
Bloc
h 20
10
Coding is Everywhere
• Magnetic recording
• Optical recording
• Wireless/wireline communication systems
• Network communications systems
© M
atth
ieu
Bloc
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Types of Coding
• Acknowledge/Request (ARQ)
• TCP/IP: retransmit packet until received
• Forward Error Correction (FEC)
• DVD, cellular standards
• Hybrid ARQ: ARQ+FEC
© M
atth
ieu
Bloc
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System Abstraction
“noisy”(Wireless link, Telephone line, Network link,
DVD)
Digital datavoice, music,
movie
© M
atth
ieu
Bloc
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Coding ?
• Decoding algorithms with reasonable complexity
• Efficient encoding
• Provable properties
• Design to achieve given performance
© M
atth
ieu
Bloc
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Gaussian Channel
message codeword coded symbol
receivedsymbol
received word
decoded message
001 0011010 1011011 001
N0
+Eb
-Eb
-Eb +Eb
k bits n bits
QPSK
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atth
ieu
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• Objective: bit error rate of 10-5
• Power efficiency:
• Spectral efficiency:
Performance of a Code
Eb
N0
ρ � R
W
© M
atth
ieu
Bloc
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1948
Eb
N0≥ 2ρ − 1
ρ
Claude E. Shannon1916-2001
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
© M
atth
ieu
Bloc
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1948Cutoff rate
“practical coding limit”
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)©
Mat
thie
u Bl
och
2010
1948Uncoded QPSK
Benchmark for 10-5
error rate
Eb
N0=
ρ =
9.6 dB
2.0-2 0 2 4 6 8 10 12
0.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
© M
atth
ieu
Bloc
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1958Hamming codes
binary linear block codes
Hamming (n,k)
# message bits
# codeword bits
decreasespectral
efficiency
coding gain
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)©
Mat
thie
u Bl
och
2010
1958Golay codes
Reed-Muller codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
© M
atth
ieu
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1968BCH and RS
codes
non-binary linear block codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)©
Mat
thie
u Bl
och
2010
1968Soft-decoding
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
1969 Mariner Mission to Mars
1.78 dB
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atth
ieu
Bloc
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1968Low-density Parity-Check
Convolutional codes
“Graph-based” codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
1968 Pioneer “Weather Network”
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atth
ieu
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1978Viterbi decoder
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
) Maximum-likelihood decoding on a trellis with 2m
states
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atth
ieu
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1978Concatenated
codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)©
Mat
thie
u Bl
och
2010
1988Big Viterbi
Decoder (BVD)
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
1989 Galileo Mission to Jupiter
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atth
ieu
Bloc
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1998Cellular
standards
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)©
Mat
thie
u Bl
och
2010
1998Soft-decoding
algorithms
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
© M
atth
ieu
Bloc
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1998Turbo-Codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
Iterative decoding
© M
atth
ieu
Bloc
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1998LDPC codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)
© M
atth
ieu
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2008Polar codes
-2 0 2 4 6 8 10 120.0
0.5
1.0
1.5
2.0
Power efficiency (dB)
Sp
ect
ral
Effi
cie
ncy
(b
/2D
)©
Mat
thie
u Bl
och
2010
What’s missing ?
• Increasing power efficiency without losing spectral efficiency
• signal set expansion
• ECE 6604 - Digital Communications
• Understanding fundamental limits
• ECE 6606 - Information Theory
© M
atth
ieu
Bloc
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2018 ?
• Modern code designs
• Golden code, Raptor codes, Polar codes
• New applications
• network coding, quantum coding
• Still many design issues
• high-speed decoding, wireless channels