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 78712-1084 USA {ming,bevans}@ece.utexas.edu

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) subchannel frequency magnitude carrier channel Subchannels are 4.3 kHz wide in ADSL

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

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

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

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 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 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%

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 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.