Electrical Communications Systems ECE.09.433 Hamming Encoding Dr. Shreek Mandayam Electrical & Computer Engineering Rowan University

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

ECOMMS: Topics

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

Error Control Coding Error Detection Error Correction (ARQ Technique) (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 2t+1 <= d <= 2t+2 can detect and correct t errors in the received code word

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

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

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

(7, 4) Hamming Code I1 P1 P2 P3 I2 I3 I4 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

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

(7, 4) Hamming Code Code Generator Matrix

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

Syndrome Decoding Parity Check Matrix

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

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

Summary