LDPCCodes

# LDPCCodes - ECE 259B Probabilistic Coding ECE 259B...

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2/23/10 1 Lecture 15 Introduction to Low-Density Parity Check (LDPC) Codes © Copyright 2009 by Paul H. Siegel © Copyright 2009 by Paul H. Siegel ECE 259B Probabilistic Coding ECE 259B Probabilistic Coding

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2/ 24/09 2 LDPC Codes Outline Shannon’s Channel Coding Theorem Error-Correcting Codes – State-of-the-Art LDPC Code Basics Encoding Decoding LDPC Code Design Asymptotic performance analysis Design optimization
2/ 24/09 3 LDPC Codes Outline EXIT Chart Analysis Applications Binary Erasure Channel Binary Symmetric Channel AWGN Channel Rayleigh Fading Channel Partial-Response Channel Basic References

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2/ 24/09 4 LDPC Codes A Noisy Communication System INFORMATION SOURCE TRANSMITTER RECEIVER DESTINATION MESSAGE SIGNAL RECEIVED SIGNAL MESSAGE NOISE SOURCE CHANNEL
2/ 24/09 5 LDPC Codes Channels 00 1 1 1- p 1- p p p • Binary erasure channel BEC( ε ) • Binary symmetric channel BSC( p ) 0 0 1 1 ? ε ε 1- ε 1- ε

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2/ 24/09 6 LDPC Codes More Channels • Additive white Gaussian noise channel AWGN P P ) 1 | ( y f ) 1 | ( y f
2/ 24/09 7 LDPC Codes Shannon Capacity Every communication channel is characterized by a single number C , called the channel capacity. It is possible to transmit information over this channel reliably (with probability of error 0) if and only if: C R def < = use channel bits n informatio #

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2/ 24/09 8 LDPC Codes Channels and Capacities 00 1 1 1- p 1- p p p • Binary erasure channel BEC( ε ) • Binary symmetric channel BSC( p ) 1- ε 0 0 1 1 ? ε ε 1- ε ε 1 = C ) ( 1 2 p H C = ) 1 ( log ) 1 ( log ) ( 2 2 2 p p p p p H = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2/ 24/09 9 LDPC Codes More Channels and Capacities • Additive white Gaussian noise channel AWGN + = 2 2 2 1 1 log σ P C P P ) 0 | ( y f ) 1 | ( y f -10 -8 -6 -4 -2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

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2/ 24/09 10 LDPC Codes Coding We use a code to communicate over the noisy channel. k x x x , , , 2 1 = x k x x x ˆ , , ˆ , ˆ ˆ 2 1 = x Code rate: n k R = Source Encoder Decoder Sink Channel n c c c , , , 2 1 = c n y y y , , , 2 1 = y