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Electrical Communications Systems ECE.09.331 Spring 2007
Lecture 10a March 27, 2007 Shreekanth Mandayam ECE Department Rowan University
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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
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ECOMMS: Topics
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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
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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
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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!!
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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
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Remediation for Detected Errors: ARQ
1 Rx Tx 2 3 ACK NACK Error detected!!!
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(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
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(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
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(7, 4) Hamming Code Code Generator Matrix
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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
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Syndrome Decoding Parity Check Matrix
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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
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Error position indicator
Hamming Decoder Error position indicator (syndrome) s Error Position, e No error
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ECOMMS: Topics
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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
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Pulse Code Modulation (PCM)
Standard for all digital communications Analog Signal Sample & Hold Digital Bitstream Quantize Encode PAM PCM
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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
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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
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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
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Time Division Multiplexing
Message 1 Quantize & Encode Message 2 Channel Message N Message 1 LPF Decode Message 2 LPF Message N LPF
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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: Odd Frames: Every Sixth Frame LSB (8th bit) is a Signaling bit Bipolar R-Z Line Encoding
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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 Mbs 672 VF 4th M U X 1 2 DS-4 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 Mbs 8064 VF Figure 3-40, p. 207 in Couch Table 3-8, p. 208 in Couch Table 3-9, p. 209 in Couch
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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
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Line Encoding Couch, p. 162 Couch, p. 157
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Summary
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