Exploiting symmetry in time domain equalizers
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EXPLOITING SYMMETRY IN TIME-DOMAIN EQUALIZERS. M.Ding and B. L. Evans The University of Texas at Austin Dept. of Electrical and Computer Engineering Austin, TX 78712-1084, USA {ming,[email protected] R. K. Martin and C. R. Johnson, Jr. Cornell University School of Electrical and

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EXPLOITING SYMMETRY IN TIME-DOMAIN EQUALIZERS

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Exploiting symmetry in time domain equalizers

EXPLOITING SYMMETRY IN TIME-DOMAIN EQUALIZERS

M.Ding and B. L. Evans

The University of Texas at Austin

Dept. of Electrical and

Computer Engineering

Austin, TX 78712-1084, USA

{ming,[email protected]

R. K. Martin and C. R. Johnson, Jr.

Cornell University

School of Electrical and

Computer Engineering

Ithaca, NY 14853, USA

{frodo,johnson}ece.cornell.edu


Introduction

channel

carrier

magnitude

subchannel

frequency

Subchannels are 4.3 kHz wide in ADSL and VDSL

Introduction

  • Discrete Multitone Modulation (DMT)

    • Multicarrier: Divide Channel into Narrow band subchannels

    • Band partition is based on fast Fourier transform (FFT)

    • Standardized for Asymmetric Digital Subscriber Line (ADSL)


Adsl transceiver itu structure

N/2 subchannels

N real samples

S/P

quadrature amplitude modulation (QAM) encoder

mirror

data

and

N-IFFT

add cyclic prefix

P/S

D/A +

transmit filter

Bits

TRANSMITTER

channel

RECEIVER

N/2 subchannels

N real samples

P/S

time domain equalizer

(TEQ)

QAM demoddecoder

N-FFT

and

remove

mirrored

data

S/P

remove cyclic prefix

receive filter

+

A/D

Up to N/2

1 - tap

FEQs

ADSL Transceiver (ITU Structure)


Cyclic prefix combats isi with teq

copy

copy

s y m b o l ( i+1)

CP

CP

s y m b o li

CP: Cyclic Prefix

v samples

N samples

channel

Shortened channel

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


Mssnr and mmse solutions

nk

yk

rk

xk

h

w

+

MSSNR and MMSE Solutions

  • 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 eigenvecotr corresponding to the largest eigenvalue of matrix pencil (B, A)

  • Minimum Mean Square Error (MMSE) solution for a white input is the generalized eigenvecotor corresponding to the largest eigenvalue of matrix pencil (B, A+Rn), where Rn is the autocorrelation matrix of noise.

hwin, hwall : equalized channel within and outside the window


Symmetry in mssnr designs

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.


Symmetric mssnrteq design

Symmetric MSSNRTEQ design

  • Idea: force the TEQ to be symmetric, and only compute half of the coefficients.

  • Implementation: instead of finding an eigenvector of an Lteq  Lteqmatrix, we only need to find an eigenvector of an matrix.

  • The phase of a perfectly symmetric TEQ is linear,

  • Achievable bit rates:


Matlab dmtteq toolbox 3 1

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/projects/adsl/dmtteq/


Conclusions

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, enables frequency domain equalizer and TEQ to be trained in parallel, and 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.


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