Electrical Communications Systems ECE.09.331 Spring 2007

1. Electrical Communications SystemsECE.09.331Spring 2007 Lecture 9bMarch 21, 2007 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring07/ecomms/

2. Henry M. Rowan Speaker Series Date: TODAY!!! Time: 1:00-2:00 pm Place: Rowan Hall Auditorium EXTRA CREDIT FOR ECOMMS  Dr. Kristina M. Johnson “Advances in modern optics and photonics” Dr. Kristina M. Johnson, Dean of the Pratt School of Engineering at Duke University, will present a lecture titled “Advances in modern optics and photonics.”

3. Plan • Digital Baseband Communications • Source Encoding • Huffman Coding • Error Control Coding • Hamming Distance • Error Detection Coding • Parity Check Code • Error Correction Coding • Hamming Code

4. ECOMMS: Topics

5. Digital Communications Transceiver Anti- aliasing Filter Error Control Encoder Data Encryption Encoder Channel/ Line Encoder Source Encoder Sampling Quantization Modulator MUX ADC Analog i/p CODEC MODEM Multiple access channel Analog o/p Error Control Decoder Data Encryption Decoder Source Decoder Audio Amp Reconstruction/ DAC Equalization / Decision Circuits Demod-ulator DEMUX

6. Analog Message A/D Converter Source Encoder Digital Source Source Encoding • Why are we doing this? Source Symbols (0/1) Source Entropy Encoded Symbols (0/1) Source-Coded Symbol Entropy

7. Source Encoding Requirements • Decrease Lav • Unique decoding • Instantaneous decoding

8. Huffman Coding 2-Step Process • Reduction • List symbols in descending order of probability • Reduce the two least probable symbols into one symbol equal to their combined probability • Reorder in descending order of probability at each stage • Repeat until only two symbols remain • Splitting • Assign 0 and 1 to the final two symbols remaining and work backwards • Expand code at each split by appending a 0 or 1 to each code word • Example m(j) A B C D E F G H P(j) 0.1 0.18 0.4 0.05 0.06 0.1 0.07 0.04

9. Digital Communications Transceiver Anti- aliasing Filter Error Control Encoder Data Encryption Encoder Channel/ Line Encoder Source Encoder Sampling Quantization Modulator MUX ADC Analog i/p CODEC MODEM Multiple access channel Analog o/p Error Control Decoder Data Encryption Decoder Source Decoder Audio Amp Reconstruction/ DAC Equalization / Decision Circuits Demod-ulator DEMUX

10. Error Control Coding Error Detection (ARQ Technique) Error Correction (FEC Technique) • Hamming Distance The number of locations (bits) at which two code words differ • Theorem 1 A code with a Hamming distance of d >= t+1 can detect t errors in the received code word • Theorem 2 A code with a Hamming distance of 2t+1 <= d <= 2t+2 can detect and correct t errors in the received code word

11. Block Coder n encoded bits k information bits k n-k Information bits Parity bits Add Redundancy!! n-bit codeword Error Control Codes Block Codes (memoryless) Convolutional Codes (with memory) Principle Will not discuss! (n, k) systematic block code

12. I6 I4 I7 I5 I3 I2 I1 P Parity Check Codes P is set such that the total no. of bits in the code word is even or odd

13. 1 Rx 2 3 3 NACK ACK ACK 1 2 3 Tx Error detected!!! Remediation for Detected Errors: ARQ

14. I3 I1 I4 I2 P3 P2 P1 (7, 4) Hamming Code • Single error detection and correction code • Hamming distance, d = 3 • Fits into a general category of coding techniques called BCH codes • Employs a • Code Generator Matrix • Syndrome Decoding Technique

15. I3 I3 I1 I1 I4 I4 I2 I2 P3 s3 s2 P2 P1 s1 R6 C6 R4 C4 C7 R7 C5 R5 C3 R3 C2 R2 R1 C1 (7, 4) Hamming Code Encoding Parity bits Code Generator = Info bits Code word Decoding Parity Check Received code word Error position indicator

16. (7, 4) Hamming Code Code Generator Matrix

17. I3 I1 I4 I2 P3 P2 P1 + + + Hamming Encoder Info bits Parity bits P3 = I4 I2 I1 P2 = I4 I3 I1 P1 = I4 I3 I2

18. Syndrome Decoding Parity Check Matrix

19. s3 s2 s1 R6 R4 R7 R5 R3 R2 R1 Hamming Decoder Received code word Error Position Indicator + + + s3 = R7 R5 R4  R3 s2 = R7 R6 R4  R2 s1 = R7 R6 R5  R1

20. Hamming Decoder Error position indicator (syndrome) s Error Position, e No error

21. Summary