Digital Communication and Error Correcting Codes

1. Timothy J. SchulzProfessor and Chair Digital Communication and Error Correcting Codes Engineering Exploration Fall, 2004

2. Digital Data • ASCII Text A 01000001 B 01000010 C 01000011 D 01000100 E 01000101 F 01000110 . . . . . .

3. Digital Sampling 000010001000001011011011010000111110111111111 011 010 001 000 111 110 101 100

4. Digital Communication • Example: Frequency Shift Keying (FSK) • Transmit a tone with a frequency determined by each bit:

5. Digital Channels Binary Symmetric Channel 1-p 0 0 p p 1 1 1-p Error probability: p

6. Error Correcting Codes 3 channel bits per 1 information bit: rate = 1/3 encode book decode book

7. Error Correcting Codes information bits channel code received bits decoded bits 5 channel errors; 1 information error

8. Error Correcting Codes • An error will only be made if the channel makes 2 or three errors on a block of 3 channel bits situation probability ccc no errors(1-p)(1-p)(1-p) = 1-3p+3p2-p3cce one error (1-p)(1-p)(p) = p-2p2+p3 cec one error (1-p)(p)(1-p) = p-2p2+p3 cee two errors (1-p)(p)(p) = p2-p3 ecc one error (p)(1-p)(1-p) = p-2p2+p3 ece two errors (p)(1-p)(p) = p2-p3 eec two errors (p)(p)(1-p) = p2-p3 eee three errors (p)(p)(p) = p3 error probability = 3p2 – 2p3

9. 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.