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Electrical Communications Systems ECE.09.331 Spring 2007
Lecture 9a March 20, 2007 Shreekanth Mandayam ECE Department Rowan University
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Plan Digital Communications Introduction
Digital Communications Transceiver (CODEC/MODEM) Digital Baseband Communications Source Encoding Huffman Coding Error Control Coding Hamming Distance Error Detection Coding Parity Check Code Error Correction Coding Hamming Code
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ECOMMS: Topics
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Digital Communications
Some Milestones Claude Shannon, 1948 X.25 (Telephony) IEEE (Ethernet) ARPANET, 1969 IEEE (FDDI) ISO-OSI 7-layer Network Reference Model CDMA GSM VOIP SIP protocols.com
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Digital Communications: Rationale
Information Theory: What is the fundamental limit on the compression and refinement of information generated by the source? What is the fundamental limit on the transmission rate of information over a noisy channel? How do we approach these limits?
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Principle 1 0 1 0……… Digital message 1 1 0 0 Digital Analog code
……… Digital code Analog message modulate AM Sinusoidal carrier FM PM AM & PM
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Digital Communication Paradigms
Multiplexer Message 2 Message 3 Message 1 1 2 3 S Demultiplexer Circuit Switching Sync bits Packet Switching Header bits Packetizing Message 2 Message 3 Message 1 1 H 2 3 Depacket-izing
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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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Source Encoding Why are we doing this? Encoder Source Encoded Symbols
Analog Message A/D Converter Digital Source Encoder Source Symbols (0/1) Source Entropy Encoded Symbols (0/1) Source-Coded Symbol Entropy Why are we doing this?
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Source Encoding Requirements
Decrease Lav Unique decoding Instantaneous decoding
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Huffman Coding 2-Step Process Reduction Splitting Example
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) A B C D E F G H P(j)
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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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Summary
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