Minimum Intersymbol Interference Methods for Time Domain Equalizer Design

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Minimum Intersymbol Interference Methods for
Time Domain Equalizer Design
M. Ding and B. L. Evans The University of Texas
at Austin Austin, TX 78712-1084
USA ming,bevans_at_ece.utexas.edu

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

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Discrete Multitone Modulation

• Wireline multicarrier modulation method
• Symbol real inverse FFT output samples
• Cyclic prefix (CP) is last n samples of symbol
• Linear convolution w/ channel impulse resp.
• Circular convolution if channel length lt CP
  length 1
• Frequency equalization accomplished in DFT domain

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Two-Step Equalization

• Time domain equalizer (TEQ)
• Channel and TEQ modeled as finite impulse
  response filters
• Cascade of channel and TEQ has response of at
  most n 1 samples
• Frequency domain equalizer
• Single division per subchannel
• Compensate for amplitude and phase distortions
• Training sequence

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Minimum ISI TEQ Design Arslan, Evans Kiaei,
2001

• Minimize frequency weightedISI energy w/r to TEQ
  taps w
• Hwin, Hwall channel in/outside window
• qi ith fast Fourier transform vector
• Eigenvector corresponding to minimum generalized
  eigenvalue of (X,Y)
• Cholesky decomposition of Y

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Minimum ISI TEQ Design Arslan, Evans Kiaei,
2001

• Advantages
• Pushes ISI to unused and low SNR subchannels
• Has real-time implementation in DSP software
• Disadvantages
• Inability to design TEQs longer than ? 1 taps
• Y not invertible in this case
• X invertible only if all subchannel weights
  non-zero
• High computational cost for delay optimization
  search
• Both Hwin and Hwall depend on delay D
• Cholesky decomposition needed for each delay D
• Cholesky decomposition sensitive to fixed-point
  computation TEQ limited to 15 taps on 16-bit DSP

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Extending Min-ISI TEQ Lengths

• Define new objective function
• weight for subchannel i, e.g. SNR in ith
  subchannel
• HTH always positive definite and invertible
• Suitable for arbitrary length TEQ design
• Reduces delay optimization search complexity

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Quantize Frequency Weighting

• Subchannel weight
• Sx,i transmit power in subchannel i
• Sn,i noise power in subchannel i
• On-off quantization removes multiplication
• Compare noise power with threshold
  
• Put zeros in those subchannels with
  larger-than-threshold noise power and ones in
  others
• One choice of threshold is noise power that only
  can support 2 bits in subchannel given
  transmitted power

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Iterative Min-ISI Method

• Obtain weighting values
• Precompute
• Decide step size ?, and precompute
• Compute non-zero initial guess w0 and iteratively
  calculate wk, using deterministic gradient search
• Gradient
• Update
• Normalization

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Simulation Results
Simulation Parameters Cyclic prefix 32
samples FFT size (N) 512 samples Coding gain
5 dB Margin 6 dBInput power
23 dBm Noise power -140 dBm/Hz Crosstalk noise
24 HDSL POTS splitter 5th order IIR
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Conclusion

• Reformulated objective function
• TEQs may have arbitrary length
• Orders of magnitude reductionin delay search
  complexity
• Iterative Min-ISI implementation
• Uses iterative gradient search
• Low complexity, avoids Cholesky decomposition
• Achieves comparable bit rate performance.
• Freely distributable discrete multitone equalizer
  Matlab toolbox 3.1 from UT Austin

http//www.ece.utexas.edu/bevans/projects/adsl/dm
tteq/index.html
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BACKUP SLIDES
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Constrained Minimization of Iterative Min-ISI

• Use the Lagrange multipliers
• Iterative updates
• where

Noted here X is Hermitian and Y is symmetric.
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Introduction

• Multicarrier wireline broadband communications to
  home and small businesses via xDSL
• Wireline systems fix bit error rate and vary bit
  rate
• Key to maximize bit rate is equalizer design
• Design equalizer to max. bit rate subject to
• Reducing intersymbol and intercarrier
  interference  
• Compensating channel frequency distortion