Minimum Intersymbol Interference Methods for Time Domain Equalizer Design

1. Minimum Intersymbol Interference Methods for Time Domain Equalizer Design R. K. Martin and C. R. Johnson, Jr. Cornell University Ithaca, NY 14853 USA {frodo,johnson}ece.cornell.edu M. Ding and B. L. Evans The University of Texas at Austin Austin, TX 78712-1084 USA {ming,bevans}@ece.utexas.edu December 2, 2003

2. Discrete Multitone Modulation copy copy s y m b o li+1 CP CP s y m b o li v samples N samples • 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 < CP length + 1 • Frequency equalization accomplished in DFT domain December 2, 2003

3. Two-Step Equalization n+1 channel impulse response effective channel impulse response • : transmission delayn: cyclic prefix lengthN: symbol/FFT length • 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 December 2, 2003

4. Minimum ISI TEQ Design [Arslan, Evans & Kiaei, 2001] n+1 channel impulse response effective channel impulse response • : transmission delayn: cyclic prefix length • 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 December 2, 2003

5. 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 • Ynot invertible in this case • Xinvertible only if all subchannel weights non-zero • High computational cost for delay optimization search • Both Hwin and Hwalldepend 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 December 2, 2003

6. 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 December 2, 2003

7. 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: December 2, 2003

8. 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: December 2, 2003

9. 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 December 2, 2003

10. 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/dmtteq/index.html December 2, 2003

11. BACKUP SLIDES December 2, 2003

12. Constrained Minimization of Iterative Min-ISI • Use the Lagrange multipliers • Iterative updates: • where Noted here X is Hermitian and Y is symmetric. December 2, 2003

13. Introduction Received bit stream Message bit stream Transmitter Channel Receiver Equalizer • 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 December 2, 2003