Electrical Communications Systems ECE.09.331 Spring 2007 Lecture 10a March 27, 2007 Shreekanth Mandayam ECE Department Rowan University http://engineering.rowan.edu/~shreek/spring07/ecomms/

Plan Pulse Amplitude Modulation Error Control Coding Hamming Distance Error Detection Coding Parity Check Code Error Correction Coding Hamming Code Pulse Amplitude Modulation Sampling Quantization TDM Standards T1 (DS-1) Line North American TDMA Hierarchy Line Encoding

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 Rx Tx 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

ECOMMS: Topics

Digital Communications Transceiver Anti- aliasing Filter Data Encryption Encoder Error Control Encoder Sampling Channel/ Line Encoder Quantization Source Encoder 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

Pulse Code Modulation (PCM) Standard for all digital communications Analog Signal Sample & Hold Digital Bitstream Quantize Encode PAM PCM

Pulse Amplitude Modulation: Flat-top Sampling -0.5 0.5 1 1.5 2 2.5 3 3.5 4 -6 -4 -2 6 8 time Analog PAM Matlab Demo: pam.m

Quantization Uniform Non-Uniform m-Law A-Law Mid-rise Mid-tread Matlab Demos: quantization.m pcm.m Mid-rise Mid-tread Mid-rise Mid-tread Matlab Demos: companding.m pcm_mucompand.m pcm_mucompand2.m pcm_acompand.m pcm_acompand2.m

Digital Communications Transceiver Anti- aliasing Filter Data Encryption Encoder Error Control Encoder Sampling Channel/ Line Encoder Quantization Source Encoder Modulator MUX 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

Time Division Multiplexing Message 1 Quantize & Encode Message 2 Channel Message N Message 1 LPF Decode Message 2 LPF Message N LPF

T1-Line TDM Format 1 Frame = (24 Channels + F) = 125 ms Channel 1 1 8 7 6 5 4 3 2 Channel 2 1 8 7 6 5 4 3 2 Channel 24 1 8 7 6 5 4 3 2 F 193 bits Frame Sync Bit Even Frames: 0 0 1 1 1 0 Odd Frames: 1 0 1 0 1 0 Every Sixth Frame LSB (8th bit) is a Signaling bit Bipolar R-Z Line Encoding

North American TDMA Hierarchy 64 kbs 1 VF DS-0 1 1 7 DS-2 6.312 Mbs 96 VF 3rd M U X 1 6 DS-3 44.736 Mbs 672 VF 4th M U X 1 2 DS-4 274.176 Mbs 4032 VF 5th M U X 1 4 DS-1 1.44 Mbs 24 VF 2nd M U X 1st M U X 24 DS-5 560.160 Mbs 8064 VF Figure 3-40, p. 207 in Couch Table 3-8, p. 208 in Couch Table 3-9, p. 209 in Couch

Digital Communications Transceiver Anti- aliasing Filter Data Encryption Encoder Error Control Encoder Sampling Channel/ Line Encoder Quantization Source Encoder 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

Line Encoding Couch, p. 162 Couch, p. 157

Summary