1. DMT Bit Rate Maximization With Optimal Time Domain Equalizer Filter Bank Architecture *M. Milošević, **L. F. C. Pessoa, *B. L. Evans and *R. Baldick * Electrical and Computer Engineering Department The University of Texas at Austin ** Motorola, Inc, NCSG/SPS Austin, TX 36 th Asilomar IEEE Conference on Signals, Systems and Computers Nov 3-6, 2002, Pacific Grove, CA

2. MPEB Asilomar’02 2 P/S QAM decoder invert channel = frequency domain equalizer Serial-to- Parallel (S\P) QAM encoder mirror data and N -IFFT add Cyclic Prefix( CP) Digital-to- Analog Converter + transmit filter N -FFT and remove mirrored data S/P remove CP TRANSMITTER RECEIVER N/2 subchannelsN samples N/2 subchannels TEQ time domain equalizer receive filter + Analog-to- Digital Converter channel Basic Architecture: DMT Transceiver Bits 00101 Parallel-to- Serial (P\S) noise

3. MPEB Asilomar’02 3 DMT Symbol CP: Cyclic Prefix N samplesv samples CP s y m b o l ( i ) s y m b o l ( i+1) copy D/A + transmit filter Inverse FFT

4. MPEB Asilomar’02 4 ISI and ICI in DMT Channel is longer than cyclic prefix (CP)+1 –Adjacent symbols interfere (ISI) –Subchannel are no longer orthogonal (ICI) TEQ mitigates the problem by shortening the channel –No symbol at demodulator contains contributions of other symbols –Cyclic prefix converts linear convolution into circular –Symbol  channel  FFT(symbol) x FFT(channel) –Division by the FFT(channel) can undo linear time-invariant frequency distortion in the channel

5. MPEB Asilomar’02 5 Channel Impairments and TEQ Design Conventional ADSL TEQ design –Mitigate inter-symbol interference at the TEQ output Proposed ADSL TEQ design - Maximize data rate –Inter-symbol interference at the output of the demodulator (FFT) –Near-end crosstalk (NEXT) –Design with respect to digital noise floor (DNF) –White noise in the channel (colored by TEQ) Other impairments present in an ADSL system –Impulse noise –Near-end echo –Far-end echo (of concern in voice-band communication) –Phase and frequency content distortion (compensated by FEQ)

6. MPEB Asilomar’02 6 Proposed TEQ Design Method Maximize bit rate at the demodulator (FFT) output instead of TEQ output Incorporate more sources of distortion into design framework Expected contributions –Model SNR at output of the FFT demodulator –Data Rate Optimal Time Domain Per-Tone TEQ Filter Bank Algorithm (TEQFB) –Data Rate Maximization Single TEQ Design Results

7. MPEB Asilomar’02 7 Model SNR at Output of Demodulator Desired signal in k th frequency bin at FFT output is DFT of circular convolution of channel and symbol – is desired symbol circulant convolution matrix for delay  –H is channel convolution matrix –q k is k th column vector of N DFT matrix Received signal in k th frequency bin at FFT output – is actual convolution matrix (includes contributions from previous, current, and next symbol) –G (*) is convolution matrix of sources of noise or interference

8. MPEB Asilomar’02 8 Model SNR at Output of Demodulator Proposed SNR model at the demodulator output After some algebra, we can rewrite the SNR model as  dig – Digital noise floor (depends on number of bits in DSP) (*) H – Hermitian (conjugate transpose)

9. MPEB Asilomar’02 9 Bits per symbol as a nonlinear function of equalizer taps. –Multimodal for more than two-tap w. –Nonlinear due to log and. –Requires integer maximization. –A k and B k are Hermitian symmetric. Unconstrained optimization problem: Model SNR at Output of Demodulator

10. MPEB Asilomar’02 10 Per channel maximization: find optimal TEQ for every k subchannel in the set of used subchannels I Generalized eigenvalue problem Bank of optimal TEQ filters Data Rate Optimal Time Domain Per-tone TEQ Filter Bank (TEQFB) Algorithm

11. MPEB Asilomar’02 11 Frequency Domain Equalizer Goertzel Filter Block TEQ Filter Bank TEQ Filter Bank Architecture w1w1 w2w2 w N/2-1 G1G1 G2G2 G N/2-1 Received Signal x={x 1,…x N ) FEQ 1 FEQ 2 FEQ N/2-1 y1y1 y2y2 y N/2-1 Y1Y1 Y2Y2 Y N/2-1

12. MPEB Asilomar’02 12 TEQFB Computational Complexity Creating matrices A k and B k ~ N O (M 2 N) Up to N/2 solutions of symmetric-definite problems –Using Rayleigh quotient iteration Single TEQReal MACsWords/Sym TEQMf s 2M2M FFT2Nlog 2 Nf sym 4N4N FEQ2Nf sym 2N2N TEQFBReal MACsWords/Sym TEQ FBN/2Mf s M(1+N/2) Goertzel FBN(f s +f sym )4N4N FEQ2Nf sym 2N2N PTEReal MACsWords/Sym FFT2Nlog 2 Nf sym 4N+2 Combiner2NMf sym (M+1)N N= 512, =32, M  2, f s = 2.204 MHz, f sym =4 kHz InitializationData Transmission

13. MPEB Asilomar’02 13 Find a single TEQ that performs as well as the optimal TEQ filter bank. –Solution may not exist, may be unique, or may not be unique. –Maximizing b (w) more tractable than maximizing b DMT int (w). –b (w) is still non-linear, multimodal with sharp peaks. Data Rate Maximization Single TEQ Design

14. MPEB Asilomar’02 14 Find a root of gradient of b (w) corresponding to a local maximum closest to the initial point –Parameterize problem to make it easier to find desired root. –Use non-linear programming –Find a good initial guess at the vector of equalizer taps w – one choice is the best performing TEQ FIR in TEQFB. –No guarantee of optimality –Simulation results are good compared to methods we looked at Data Rate Maximization Single TEQ Design

15. MPEB Asilomar’02 15 Measurement of the SNR in subchannel k –S = 1000 symbols –Every subchannel in a symbol loaded with a random 2-bit constellation point X k i, passed through the channel, TEQ block and FEQ block (where applicable) to obtain Y k i Bit rate reported is then Simulation Results

16. MPEB Asilomar’02 16 Effect of TEQ Size on Bit Rate Data rates achieved for different number of TEQ taps, M N = 512, = 32, input power = 23.93 dBm, AWGN power = -140 dBm/Hz, and NEXT modeled as 49 disturbers. Accuracy of bit rate:  60 kbps. (a) CSA loop 2(b) CSA loop 7

17. MPEB Asilomar’02 17 Effect of Transmission Delay on Bit Rate Data rates achieved as a function of  for CSA loop 1. N = 512, = 32, input power = 23.93 dBm, AWGN power = -140 dBm/Hz, and NEXT modeled as 49 disturbers. Accuracy of bit rate:  60 kbps.

18. MPEB Asilomar’02 18 We evaluate TEQFB, proposed single TEQ, MBR, Min- ISI, LS PTE, MMSE-UTC and MMSE-UEC for CSA loops 1-8 Results presented in a table –Each row entry –Final row entry Simulation Results

19. MPEB Asilomar’02 19 TEQ Design Methods - Comparison CSA loop LS PTE New TEQ Min-ISIMBRMSSNR MMSE- UEC MMSE- UTC 199.5%99.6%97.5%97.3% 95.0%86.3%84.4% 299.5%99.6%97.3%97.0% 96.5%87.2%85.8% 399.6%99.5%97.3%97.8% 97.0%83.9%83.0% 499.1%99.3%98.2%98.1% 95.4%81.9%81.5% 599.5%99.6%97.2%97.7% 97.1%88.6%88.9% 699.4%99.5%98.3%97.7% 96.4%82.7%79.8% 799.6%98.8%96.3% 96.7%75.75%78.4% 899.2%98.7%97.5%97.4% 97.5%82.6%83.6% Avg.99.4% 99.3%97.5%97.4% 96.4%83.6%83.2% CSA – carrier serving area, MBR – Maximum Bit Rate, Min-ISI – Minimum InterSymbol Interference TEQ Design, LS PTE – Least-squares Per-Tone Equalizer, MMSE – Minimum Mean Square Error, UTC – Unit Tap Constraint, UEC – Unit Energy Constraint

20. MPEB Asilomar’02 20 TEQFB Data Rates Highest data rates in Mbps achieved by TEQFB for TEQ lengths 2-32, input power = 23.93 dBm CSA loopTEQFB 111.417 Mbps 212.680 Mbps 310.995 Mbps 411.288 Mbps 511.470 Mbps 610.861 Mbps 710.752 Mbps 8 9.615 Mbps

21. Backup Slides Milos Milosevic Lucio F. C. Pessoa Brian L. Evans Ross Baldick

22. MPEB Asilomar’02 22 Bit/symbol for a 2-tap TEQ

23. MPEB Asilomar’02 23 Bit/symbol for a 3-tap TEQ

24. MPEB Asilomar’02 24 CSA Loops Configuration of eight standard carrier serving loops (CSA). Numbers represent length in feet/ gauge. Vertical lines represent bridge taps. From Guner, Evans and Kiaei, “Equalization For DMT To Maximize bit Rate”.

25. MPEB Asilomar’02 25 Selected Previous TEQ Design Methods Minimize mean squared error –Minimize mean squared error (MMSE) method [Chow & Cioffi, 1992] –Geometric SNR method [Al-Dhahir & Cioffi, 1996] Minimize energy outside of shortened channel response –Maximum Shortening SNR method [Melsa, Younce & Rohrs, 1996] –Divide-and-Conquer methods – Equalization achieved via a cascade of two tap filters [Lu, Evans & Clark, 2000] –Minimum ISI method - Near-maximum bit rate at TEQ output [Arslan, Evans & Kiaei, 2001] –Maximum Bit Rate (MBR) - Maximize bit rate at TEQ output [Arslan, Evans & Kiaei, 2001] Per-tone equalization –Frequency domain per-tone equalizer [Acker, Leus, Moonen, van der Wiel & Pollet, 2001]

26. MPEB Asilomar’02 26 Used to calculate single DFT point Denote with y k (n) as the signal emanating from TEQ making up TEQFB Then, the corresponding single point DFT Y k is: where G k (-1) = G k (-2) = 0 and n={0,1,…,N} Goertzel Filters