Digital Communication and Error
Correcting Codes
Timothy J. Schulz
Professor and Chair
Engineering Exploration
Fall, 2004
Department of Electrical and Computer Engineering
Department of Electrical and Computer Engineering
Digital Data
• ASCII Text
A
B
C
D
E
F
.
.
.
01000001
01000010
01000011
01000100
01000101
01000110
.
.
.
Digital Coding for Error Correction
00101001110101101010101000
Department of Electrical and Computer Engineering
Digital Sampling
000010001000001011011011010000111110111111111
011
010
001
000
111
110
101
100
Digital Coding for Error Correction
00101001110101101010101000
Department of Electrical and Computer Engineering
Digital Communication
• Example: Frequency Shift Keying (FSK)
– Transmit a tone with a frequency determined by each bit:
s  t   b cos  2 f 0t   1  b cos  2 f1t 
Digital Coding for Error Correction
00101001110101101010101000
Department of Electrical and Computer Engineering
Digital Channels
Binary Symmetric Channel
0
1-p
0
p
p
1
1
1-p
Error probability: p
Digital Coding for Error Correction
00101001110101101010101000
Department of Electrical and Computer Engineering
Error Correcting Codes
3 channel bits per 1 information bit: rate = 1/3
encode book
information bits
channel bits
0
1
000
111
decode book
channel bits
information bits
000
001
010
011
100
101
110
111
0
0
0
1
0
1
1
1
Digital Coding for Error Correction
00101001110101101010101000
Department of Electrical and Computer Engineering
Error Correcting Codes
information bits
channel code
received bits
decoded bits
0
0
1
0
1
000 000 111 000 111
010 000 100 001 110
0
0
0
0
1
5 channel errors; 1 information error
Digital Coding for Error Correction
00101001110101101010101000
Department of Electrical and Computer Engineering
Error Correcting Codes
• An error will only be made if the channel makes 2 or
three errors on a block of 3 channel bits
ccc
cce
cec
cee
ecc
ece
eec
eee
0.5
probability
no errors
(1-p)(1-p)(1-p) =
one error
(1-p)(1-p)(p) =
one error
(1-p)(p)(1-p) =
two errors
(1-p)(p)(p) =
one error
(p)(1-p)(1-p) =
two errors
(p)(1-p)(p) =
two errors
(p)(p)(1-p) =
three errors
(p)(p)(p) =
error probability =
3p2
1-3p+3p2-p3
p-2p2+p3
p-2p2+p3
p2-p3
p-2p2+p3
p2-p3
p2-p3
p3
–
2p3
0.45
0.4
0.35
bit error probability
situation
0.3
0.25
0.2
0.15
0.1
0.05
0
Digital Coding for Error Correction
0
0.1
0.2
0.3
channel error probability
0.4
0.5
00101001110101101010101000
Department of Electrical and Computer Engineering
Error Correcting Codes
• Codes are characterized by the number of channel bits
(M) used for (N) information bits. This is called an N/M
code.
• An encode book has 2N entries, and each entry is an
M-bit codeword.
• A decode book has 2M entries, and each entry is an
N-bit information-bit sequence.
Digital Coding for Error Correction
00101001110101101010101000