LDPC Codes for Fading Channels: Two Strategies

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LDPC Codes for Fading ChannelsTwo Strategies

• Xiaowei Jin, Teng Li, Tom Fuja, and Oliver
  Collins
• Department of Electrical Engineering
• University of Notre Dame

This work has been supported by the US Army under
contract DAAD16-02-C-0057.
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Overview

• Background on block fading channels.
• Two strategies for LDPC codes in the context of
  block fading channels
• A Successive Decoding Receiver
• An Iterative Receiver
• A comparison of the two strategies under a delay
  constraint.

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Block Fading Channels

• Also known as the block interference channel
  introduced by McEliece and Stark in 1984.
• The channel stays constant for T channel uses
  then changes and stays constant for another T
  channel uses, etc.
• T coherence time of the channel
• A pretty good model for many systems
• Multi-carrier systems such as OFDM.
• Frequency-hopped spread spectrum.
• Slow fading channels.
• Our assumption The channel changes quickly
  relative to the error control structure used.

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Block Fading Channels

• An example with T6

BSC-constituted
BPSK/AWGN -constituted
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Some Related Work

• Worthen and Stark
• LDPC Codes for Fading Channels with Memory
• 1998 Allerton Conference
• Analysis/design of LDPC codes for block fading
  channels with two constituent BPSK-modulated AWGN
  channels chosen independently with probability p
  and 1-p.
• Eckford, Kschischang, and Pasupathy
• Analysis of LDPC Codes for the Gilbert-Elliot
  Channel
• IEEE Transactions on Information Theory, November
  2005.
• Designing Good LDPC Codes for Markov-Modulated
  Channels
• ISIT 2004
• Analysis/design of LDPC codes for channels in
  which the sequence of channel states forms a
  Markov chain. (Like a BSC-constituted BFC with
  block size h1.)
• Wijacksungsithi and Winick
• Joint Channel-State Estimation and Decoding of
  LDPC Codes on the Two-State Noiseless/Useless BSC
  Block Interference Channel
• Transactions on Communications, April 2005.
• Analysis/design of LDPC codes for BSC-constituted
  block fading channel with i.i.d. blocks, each
  with crossover probability 0 or ½.

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Block Fading Channel

• Our particular non-coherent model
• where
• xik e 1, -1 is the BPSK-modulated kth bit of
  the ith channel realization
• ci is the sequence of fading coefficients
  i.e. an i.i.d. sequence of complex Gaussian
  random variables with mean zero and variance one
  per dimension
• ni,k is (i.i.d.) Gaussian noise
• yik is the received symbol corresponding to xik.

yi,k ci xik nik for i1,2, and k1,2, T,
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Block Fading Channels

• Two Strategies for Reliable Communication
• A successive-decoding receiver based on decision
  feedback
• Uses interleaving to create parallel memoryless
  channels.
• Each decoders output is used as a pilot for the
  next decoder.
• An iterative receiver employing a joint
  channel/code graph
• Carries out channel estimation and data decoding
  iteratively.
• Each decoder iteration provides an improved
  channel estimate each channel estimate enhances
  the performance of the decoder.

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A Successive-Decoding Receiver

• Basic Idea
• Use an interleaver to decompose the block fading
  channel with coherence time T into a bank of T
  parallel channels.
• Use T codewords from appropriately chosen codes.
• GLOBECOM 05
• The Universality of LDPC Codes on Correlated
  Fading Channels with Decision Feedback Based
  Receivers
• First J codes are pilots (R0) used to estimate
  channel state information.
• Last K T-J codes are the same code an LDPC
  codes designed for channel with known CSI.

GLOBECOM 05 Design

• T coherence time K J
• Bits are transmitted row-by-row, top-to-bottom.
• Bits are decoded column-by-column, left-to-right.

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A Successive-Decoding Receiver

• The Design Considered Here
• Use T codes with T different rates.
• Set rate of code used on kth sub-channel equal to
  the capacity of that sub-channel as indicated by
  the chain rule of mutual information i.e.,

LDPC Codeword 2
LDPC Codeword 1
LDPC Codeword 3
LDPC Codeword T

• Bits are transmitted row-by-row, top-to-bottom.
• Bits are decoded column-by-column, left-to-right.

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An Iterative Receiver

• This approach
• Create a single graphical model that incorporates
  both the block fading nature of the channel and
  the structure of a single LDPC codeword.
• Use that graphical model for iterative channel
  estimation and decoding.

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An Iterative Receiver

• Message Passing on the Joint Graph
• Message from T to V (and vice versa) is a
  log-likelihood ratio associated with a code bit.
• Message from S to T is the PDF of the state,
  given the pilots. (Here, pilots includes
  pseudo-pilots judged sufficiently trustworthy
  based on the decoded data.)
• Message from T to S is the PDF of y given the
  channel state.

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Regarding Delay

• For the successive decoding receiver
• Delay incurred is BT bits, where
• B blocklength of constituent codes.
• T coherence time of channel in bits
• For the iterative decoder
• Delay in incurred is B bits, where B
  blocklength of code.
• So for a given delay constraint, the iterative
  decoder can use longer LDPC codewords.

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Simulations for T5

• Successive Decoding
• R1 0
• R2 0.4948
• R3 0.5643
• R4 0.5917
• R5 0.6058
• So overall rate is RS 0.4513
• Iterative Decoding
• Code rate is R 0.5641
• So overall rate (including 20 pilots) is RI
  0.4513.

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Simulations for T10

• Successive Decoding
• R1 0, R2 0.5177
• R3 0.5869, R4 0.6109
• R5 0.6229, R6 0.6302
• R7 0.6364, R8 0.6397
• R9 0.6430, R10 0.6453
• So overall rate is RS 0.5533
• Iterative Decoding
• Code rate is R 0.6148
• So overall rate (including 10 pilots) is RI
  0.5533.

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Conclusions

• For short-to-moderate delay constraints
• The shorter blocklengths imposed by the
  successive decoding structure lead to error
  propagation through the decision feedback
  mechanism.
• The longer blocklengths made possible by the
  iterative structure provide better performance.
• When longer delays are acceptable
• The decisions made at each step of the successive
  decoding algorithm can be made reliable with high
  probability.
• The successive decoding structure is optimal in
  the sense that it preserves mutual information at
  each step assuming prior decisions are correct.
• So the successive decoding structure wins.