154 Part 2 Lectures 1 and 2

154 Part 2 Lectures 1 and 2 - Error Control Coding:...

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Coding 1 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Error Control Coding: Introduction
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Coding 2 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Error Control Coding Basic Principles r Encoder takes k bit blocks of message bits and adds n-k error control bits and creates n bit code words; The redundant bite are used by the decoder to detect and or correct transmission errors; Code rate R = r b /r c Error control coding o Decreases E b /N 0 for the coded bits by k/n; i.e E c /N 0 = k/n(E b /N 0) b Increases BW by n/k ( or information bit rate reduced accordingly) o Decreases overall message P e compared to uncoded transmission if Encoder Mod/Channel/Demod Decoder { } k b { } m c { } m c ˆ { } k b ˆ Bit Rate r b Bit Rate r c Bit Rate r b Bit Rate r c Channel BER = = 0 2 ] ˆ [ N E Q q c c P c c m m r b < C D < C A Message BER ( 29 0 / 2 ) ˆ ( N E Q b b P P b k k e = = = k-bit blocks n-bit blocks Next Chart
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Coding 3 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Coded Versus Uncoded Transmission Assume R = 1/2 e e 0 0 2 c 0 0 2 e 2 2 c 2 b P to coding y with probabilit error bit message reduce To : Goal P 2 2 q ty probabaili error bit Channel 2 2 P ion) transmiss ty(Uncoded probabaili error bit Message 1 ; E bit " channel coded " per energy Signal E ion) transmiss d bit(Uncode message" " per energy Signal < = = = = < = = = = = N E Q N T A Q N E Q N T A Q R R T A T A T A c c b b b c b T b Coded Trans. Uncoded Trans. -A A BW ≈ r b BW ≈ r c = 2r b T c -A A Message bits r b bits/sec Channel r b bits/sec Message bits r b bits/sec Encoder Channel r c bits/sec Decoder
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Coding 4 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Error Control Coding Basic Principles r Coding schemes are compared on the basis of o Bandwidth expansion ( n/k ) o Coder/decoder complexity o Coding gain ( savings in power for given BER) o Types of codes: r Block codes (error detection or correction) o Convolutional codes(error correction) o Combinations (Concatenated; Turbo codes) o Methods of error control r Error detection and retransmission ( or discard bad frames or blocks) Used when a reverse channel, and retransmission are feasible r FEC: Forward error correction ( decoder detects AND attempts to correct errors) Used when reverse channel is not available and or retransmission is not feasible r Error detection yields better performance than FEC Coding gain Uncoded BER Coded BER E b /N 0 Log BER -1 -2 -3 -4 - 5 2dB 6dB 10dB
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Coding 5 1/1/2007 Information Theory and Coding © K. Sam Shanmugan, 2007 Assumptions about the Discrete Channel r Channel is symmetric and memoryless – errors are independent o Error Distribution: )] 1 ( log ) 1 ( log 1 [ 2 2 c c c c c D q q q q r C - - + + = r c 0 1 0 1 q c 1-q c 1-q c 1 ; 1 0 error] 1 than P[more ; 1 1 ] error 1 [ P ; 1 1 1 error] 0 [ P ,.... 2 , 1 , 0 , 1 bits] received of block a in errors [ P 1 << - = - - = = - = - - c c c c n c c c c n c j n c j c nq and q when q when nq ) q ( nq nq when nq ) q ( n j ) q ( ) (q j n n j % Simulation of BSC %Generate 1000 input bits: X = (rand(1,1000)<0.5) %Generate 1000 “error bits” with prob q E = (rand(1,1000)<q) % Generate channel output bits
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154 Part 2 Lectures 1 and 2 - Error Control Coding:...

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