EXPLOITING SYMMETRY IN TIME-DOMAIN EQUALIZERS

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EXPLOITING SYMMETRY IN TIME-DOMAIN EQUALIZERS

• R. K. Martin and C. R. Johnson, Jr.
• Cornell University
• School of Electrical and
• Computer Engineering
• Ithaca, NY 14853 USA
• frodo,johnsonece.cornell.edu

M. Ding and B. L. Evans The University of Texas
at Austin Dept. of Electrical and Computer
Engineering Austin, TX 78712-1084
USA ming,bevans_at_ece.utexas.edu
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Introduction

• Discrete Multitone (DMT) Modulation
• Multicarrier Divide Channel into Narrow band
  subchannels
• Band partition is based on fast Fourier transform
  (FFT)
• Standardized for Asymmetric Digital Subscriber
  Lines (ADSL)

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ADSL Transceiver (ITU Structure)
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Cyclic Prefix Combats ISI with TEQ

• CP provides guard time between successive symbols
• We use finite impulse response
• (FIR) filter called a time domain
• equalizer to shorten the channel
• impulse response to be no longer than cyclic
  prefix length

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MSSNR TEQ Design

• Maximum Shortening SNR (MSSNR) TEQ Choose w to
  minimize energy outside window of desired length
• The design problem is stated as
• The solution will be the generalized eigenvector
  corresponding to the largest eigenvalue of matrix
  pencil (B, A)

hwin, hwall equalized channel within and
outside the window
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Symmetry in MSSNR designs

• Fact eigenvectors of a doubly-symmetric matrix
  are symmetric or skew-symmetric.
• MSSNR solution
• A and B are almost doubly symmetric
• For long TEQ lengths, w becomes almost perfectly
  symmetric
• MSSNR for Unit Norm TEQ (MSSNR-UNT) solution
• A is almost doubly symmetric
• In the limit, the eigenvector of A converge to
  the eigenvector of HTH, which has symmetric or
  skew-symmetric eigenvectors.

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Symmetric MSSNR TEQ design

• Idea force the TEQ to be symmetric, and only
  compute half of the coefficients.
• Implementation instead of finding an eigenvector
  of an Lteq ? Lteq matrix, we only need to find an
  eigenvector of an
  matrix.
• The phase of a perfectly symmetric TEQ is linear,
• Achievable bit rates

Loop MSSNR SYM-MSSNR Loss Loop MSSNR SYM-MSSNR Loss
CSA 1 12.187 Mbps 10.921 Mbps 10.39 CSA 5 12.120 Mbps 11.800 Mbps 2.64
CSA 2 13.016 Mbps 12.493 Mbps 4.02 CSA 6 10.995 Mbps 10.798 Mbps 1.79
CSA 3 11.543 Mbps 11.529 Mbps 0.12 CSA 7 10.978 Mbps 10.880 Mbps 0.89
CSA 4 11.696 Mbps 11.431 Mbps 2.27 CSA 8 10.294 Mbps 9.956 Mbps 3.28
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Matlab DMTTEQ Toolbox 3.1

• The symmetric design has been implemented in
  DMTTEQ toolbox.

• Toolbox is a test platform
• for TEQ design and
• performance evaluation.
• Most popular algorithms
• are included in the toolbox
• Graphical User Interface
• easy to customize your
• own design.

Available at http//www.ece.utexas.edu/bevans/pro
jects/adsl/dmtteq/
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Conclusions

• Infinite length MMSNR TEQs with a unit norm
  constraint are exactly symmetric, while finite
  length MSSNR TEQs are approximately symmetric.
• A symmetric MSSNR TEQ only has one fourth of FIR
  implementation complexity.
• Symmetric design enables frequency domain
  equalizer and TEQ to be trained in parallel.
• Symmetric design exhibits only a small loss in
  the bit rate over non-symmetric MSSNR TEQs.
• Symmetric design doubles the length of the TEQ
  that can be designed in fixed point arithmetic.