1. 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,johnson}ece.cornell.edu
M.Ding and B. L. Evans
The University of Texas at Austin
Dept. of Electrical and
Computer Engineering
Austin, TX , USA

2. 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)
subchannel
frequency
magnitude
carrier
channel
Subchannels are 4.3 kHz wide in ADSL and VDSL

3. ADSL Transceiver (ITU Structure)
P/S
QAM demod decoder
Up to N/2
1 - tap
FEQs
S/P
quadrature amplitude modulation (QAM) encoder
mirror
data
and
N-IFFT
add cyclic prefix
D/A +
transmit filter
N-FFT
remove
mirrored
remove cyclic prefix
TRANSMITTER
RECEIVER
N/2 subchannels
N real samples
time domain equalizer
(TEQ)
receive filter +
A/D
channel
Bits

4. Cyclic Prefix Combats ISI with TEQ
N samples
v samples
CP: Cyclic Prefix CP
s y m b o l i
s y m b o l ( i+1)
copy
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
channel
Shortened channel

5. 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. h + w xk yk rk nk
hwin, hwall : equalized channel within and outside the window

6. 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.

7. 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  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
CSA 1
Mbps
Mbps
10.39%
CSA 5
Mbps
Mbps
2.64%
CSA 2
Mbps
Mbps
4.02%
CSA 6
Mbps
Mbps
1.79%
CSA 3
Mbps
Mbps
0.12%
CSA 7
Mbps
Mbps
0.89%
CSA 4
Mbps
Mbps
2.27%
CSA 8
Mbps
9.956 Mbps
3.28%

8. 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

9. 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.