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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,bevans}@ece.utexas.edu

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

channel

carrier

magnitude

subchannel

frequency

Subchannels are 4.3 kHz wide in ADSL and VDSL

- 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)

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

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

- 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

nk

yk

rk

xk

h

w

+

- 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

- 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.

- 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:

- 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/

- 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.