Electrical Communications Systems ECE.09.433

1. Electrical Communications SystemsECE.09.433 Hamming Encoding Dr. Shreek Mandayam Electrical & Computer Engineering Rowan University

2. Plan • Digital Baseband Communications • Error Control Coding • Hamming distance • Parity check codes • Hamming Code

3. ECOMMS: Topics

4. 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

5. 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

6. 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

7. 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

8. 1 Tx 2 3 3 NACK ACK ACK 1 2 3 Rx Error detected!!! Remediation for Detected Errors: ARQ

9. 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

10. 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

11. (7, 4) Hamming Code Code Generator Matrix

12. 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

13. Syndrome Decoding Parity Check Matrix

14. 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

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

16. Summary