1. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2007 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring07/ecomms/ Lecture 9b March 21, 2007

2. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversityPlan Digital Baseband Communications Source Encoding Huffman Coding Error Control Coding Hamming Distance Error Detection Coding Parity Check Code Error Correction Coding Hamming Code

4. S. Mandayam/ ECOMMS/ECE Dept./Rowan University ECOMMS: Topics

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

6. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Source Encoding Why are we doing this? Analog Message A/D Converter Digital Source Encoder Source Symbols (0/1) Source Entropy Encoded Symbols (0/1) Source-Coded Symbol Entropy

7. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Source Encoding Requirements Decrease L av Unique decoding Instantaneous decoding

8. S. Mandayam/ ECOMMS/ECE Dept./Rowan University 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)ABCDEFGH P(j)0.10.180.40.050.060.10.070.04

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

10. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Error Control Coding 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 Error Detection (ARQ Technique) Error Correction (FEC Technique)

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

12. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Parity Check Codes P I4I4 I1I1 I2I2 I3I3 I5I5 I6I6 I7I7 P is set such that the total no. of bits in the code word is even or odd

13. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Remediation for Detected Errors: ARQ 1 Rx 1 Tx 2 3 3 23 ACK NACK Error detected!!!

14. S. Mandayam/ ECOMMS/ECE Dept./Rowan University (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 I1I1 P1P1 P2P2 P3P3 I2I2 I3I3 I4I4

15. S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Code Generator s1s1 s2s2 s3s3 I1I1 I2I2 I3I3 I4I4 C4C4 C1C1 C2C2 C3C3 C5C5 C6C6 C7C7 Encoding Info bits Code word Decoding Parity Check R4R4 R1R1 R2R2 R3R3 R5R5 R6R6 R7R7 Received code word Error position indicator I1I1 P1P1 P2P2 P3P3 I2I2 I3I3 I4I4 = Parity bits

16. S. Mandayam/ ECOMMS/ECE Dept./Rowan University (7, 4) Hamming Code Code Generator Matrix

17. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Encoder P1P1 P2P2 P3P3 I1I1 I2I2 I3I3 I4I4 + + + P 3 = I 4  I 2  I 1 P 2 = I 4  I 3  I 1 P 1 = I 4  I 3  I 2 Info bits Parity bits

18. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Syndrome Decoding Parity Check Matrix

19. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Decoder + R4R4 R1R1 R2R2 R3R3 R5R5 R6R6 R7R7 s1s1 s2s2 s3s3 + + Received code word Error Position Indicator s 3 = R 7  R 5  R 4  R 3 s 2 = R 7  R 6  R 4  R 2 s 1 = R 7  R 6  R 5  R 1

20. S. Mandayam/ ECOMMS/ECE Dept./Rowan University Hamming Decoder Error Position, e Error position indicator (syndrome) s No error

21. S. Mandayam/ ECOMMS/ECE Dept./Rowan UniversitySummary