Equalization for ADSL Transceivers - University of Texas at ...

Equalization for ADSL Transceivers - University of Texas at ...

Equalizer Design to Maximize Bit Rate in ADSL Transceivers Prof. Brian L. Evans Dept. of Electrical and Comp. Eng. The University of Texas at Austin http://signal.ece.utexas.edu Last modified August 8, 2005 UT graduate students: Mr. Zukang Shen, Mr. Daifeng Wang, Mr. Ian Wong UT Ph.D. graduates: Dr. Gner Arslan (Silicon Labs), Dr. Biao Lu (Schlumberger), Dr. Ming Ding (Bandspeed), Dr. Milos Milosevic (Schlumberger) UT senior design students: Wade Berglund, Jerel Canales, David J. Love, Ketan Mandke, Scott Margo, Esther Resendiz, Jeff Wu Other collaborators: Dr. Lloyd D. Clark (Schlumberger), Prof. C. Richard Johnson, Jr. (Cornell), Prof. Sayfe Kiaei (ASU), Prof. Rick Martin (AFIT), Prof. Marc Moonen (KU Leuven), Dr. Lucio F. C. Pessoa (Motorola), Dr. Arthur J. Redfern (Texas Instruments) Introduction Interne t Digital Subscriber Line (DSL) Broadband Access DSLAM downstream Central Office DSL modem DSL modem upstream Voice Switch LPF

LPF Customer Premises Telephone Network DSLAM - Digital Subscriber Line Access Multiplexer LPF Lowpass Filter (passes voiceband frequencies) 2 Introduction Discrete Multitone (DMT) DSL Standards ADSL Asymmetric DSL Maximum data rates supported in G.DMT standard (ideal case) Echo cancelled: 14.94 Mbps downstream, 1.56 Mbps upstream Frequency division multiplexing (FDM): 13.38 Mbps downstream, 1.56 Mbps upstream Widespread deployment in US, Canada, Western Europe, and Hong Kong Central office providers only installing frequency-division multiplexed (FDM) ADSL:cable modem market G.DMT Asymmetric 1:2 in US & 2:1 worldwide ADSL+ 8 Mbps downstream min. ADSL2 doubles analog bandwidth VDSL Very High Rate DSL Asymmetric Faster G.DMT FDM ADSL 2m subcarriers m [8, 12] Symmetric: 13, 9, or 6 Mbps Optional 12-17 MHz band Data band Upstream subcarriers Downstream

subcarriers Target upstream rate Target downstream rate ADSL DMT VDSL 0.025 1.1 0.138 12 MHz MHz 32 256 256 2048/4096 1 Mbps 3 Mbps 8 Mbps 13/22 Mbps 3 Outline Multicarrier modulation Conventional equalizer training methods Minimum Mean Squared Error design [Stanford] Maximum Shortening Signal-to-Noise Ratio design Maximum Bit Rate design (optimal) [Tellabs] [UT Austin] Minimum Inter-symbol Interference design (near-optimal) [UT Austin]

Per-tone equalizer [Catholic University, Leuven, Belgium] Dual-path equalizer Conclusion [UT Austin] Message bit stream Transmitter Channel Equalizer Received bit stream Receiver 4 Multicarrier Modulation Single Carrier Modulation Ideal (non-distorting) channel over transmission band Flat magnitude response Linear phase response: delay is constant for all spectral components No intersymbol interference Impulse response for ideal channel over all frequencies nk Continuous time: g(t- Channel Equalizer y xk

rk ek k Discrete time: g[k- + w + h + Equalizer Shortens channel impulse response (time domain) Compensates for frequency distortion (frequency domain) Ideal Channel z- g Discretized Baseband System 5 Multicarrier Modulation Multicarrier Modulation Divide channel into narrowband subchannels No inter-symbol interference (ISI) in subchannels if constant gain within every subchannel and if ideal sampling Discrete multitone modulation pulse magnitude Baseband transmission Based on fast Fourier transform (FFT)

Standardized for ADSL and VDSL DTFT-1 sinc k c c channel sin c k k carrier subchannel Subchannels are 4.3 kHz wide in ADSL and VDSL frequency 6 Multicarrier Modulation Multicarrier Modulation by Inverse FFT Filter Bank e X1 g(t) g(t) X1 complex-valued j 2 f 2 t x

g(t) e X N /2 e x e X2 complex-valued j 2 f1 t + j 2 f N / 2 t x g(t) : pulse shaping filter Discrete time 1 k N x e X2 j 2 2

k N x real-valued e X N /2 j 2 j 2 + N /2 k N x Xi : ith subsymbol from encoder 7 Multicarrier Modulation Discrete Multitone Modulation Symbol N/2 subsymbols are in general complex-valued Quadrature ADSL uses 4-level Quadrature Amplitude Modulation (QAM) during training ADSL uses QAM of 22, 23, 24, , 215 levels during data transmission Xi In-phase Multicarrier modulation using inverse FFT Mirror and conjugate N/21 complex

subsymbols X0 X1 X2 XN/2 X2 * X1 * N-point Inverse Fast Fourier Transform x0 x1 x2 QAM Yields one symbol of N real-valued samples xN-1 e j t e j t 2 cos(t ) 8 Multicarrier Modulation Discrete Multitone Modulation Frame Frame is sent through D/A converter and transmitted Frame is the symbol with cyclic prefix prepended Cyclic prefix (CP) consists of last samples of the symbol copy copy

CP v samples s y m b o l i CP N samples CP reduces throughput by factor of s y m b o l N 16 N v 17 Linear convolution of frame with channel impulse response i+1 ADSL G.DMT Values Down Up stream stream 4 32 N 64 512 Is circular convolution if channel length is CP length plus one or shorter Circular convolution frequency-domain equalization in FFT domain Time-domain equalization to reduce effective channel length and ISI 9

Multicarrier Modulation Eliminating ISI in Discrete Multitone Modulation Time domain equalizer (TEQ) Finite impulse response (FIR) filter Effective channel impulse response: convolution of TEQ impulse response with channel impulse response channel impulse response effective channel impulse response Frequency domain equalizer (FEQ) Compensates magnitude/phase distortion of equalized channel by dividing each FFT coefficient by complex number Generally updated during data transmission ADSL G.DMT equalizer training Reverb: same symbol sent 1,024 to 1,536 times Medley: aperiodic pseudo-noise sequence of 16,384 symbols Receiver returns number SNR i bi log 1 of bits (0-15) to transmit i each subchannel i

: transmission delay : cyclic prefix length ADSL G.DMT Values Down Up stream stream 4 32 N 512 64 10 Multicarrier Modulation ADSL Transceiver: Data Transmission N/2 subchannels N real samples Bits 00110 S/P quadrature amplitude modulation (QAM) encoder mirror data and N-IFFT add cyclic prefix

D/A + transmit filter P/S TRANSMITTER channel RECEIVER N/2 subchannels P/S QAM demod invert channel = decoder frequency domain equalizer N real samples N-FFT and remove mirrored data remove S/P cyclic prefix time

domain equalizer (FIR filter) receive filter + A/D conventional ADSL equalizer structure 11 Outline Multicarrier modulation Conventional equalizer training methods Minimum Mean Squared Error design [Stanford] Maximum Shortening Signal-to-Noise Ratio design Maximum Bit Rate design (optimal) [Tellabs] [UT Austin] Minimum Inter-symbol Interference design (near-optimal) [UT Austin] Per-tone equalizer Dual-path equalizer Conclusion Message bit stream Transmitter Channel Equalizer

Received bit stream Receiver 12 Conventional Equalizer Minimum Mean Squared Error TEQ Design xk Channel h nk yk TEQ + w z- b rk ek - + bk- Minimize E{ek2} [Chow & Cioffi, 1992] Chose length of b (e.g. +1) to shorten length of h * w b is eigenvector of minimum eigenvalue of symmetric 1

channel-dependent matrix R R xx - R xy R yy R yx Minimum MSE when R yy w R xy b where w 0 Disadvantages Does not consider bit rate Deep notches in equalized frequency response Rxy is cross correlation matrix Why? 13 Conventional Equalizer Infinite Length MMSE TEQ Analysis As TEQ length goes to infinity, R becomes Toeplitz [Martin et al. 2003] Eigenvector of minimum eigenvalue of symmetric Toeplitz matrix has zeros on unit circle [Makhoul 1981] Zeros of target impulse response b on unit circle kills subchannels Finite length TEQ plot Each trace is a different zero of b Longer MMSE Distance of 32 zeros of b to unit circle averaged TEQ may be worse over 8 ADSL test channels for each TEQ length Zeros cluster at 0.01 and 10-4 from UC for TEQ lengths 32 and 100 14 Conventional Equalizer Maximum Shortening SNR TEQ Design Minimize energy leakage outside shortened channel length For each possible position of window [Melsa, Younce & Rohrs, 1996] energy inside window after TEQ

max SSNR in dB max 10 log10 w w energy outside window after TEQ h w Equivalent to noise-free MMSE TEQ Disadvantages channel impulse response effective channel impulse response Does not consider channel noise Does not consider bit rate Deep notches in equalized frequency response (zeros of target impulse response near unit circle kill subchannels) Requires Cholesky decomposition, which is computationally-intensive and does not allow TEQ lengths longer than cyclic prefix 15 Conventional Equalizer Maximum Shortening SNR TEQ Design Choose w to minimize energy outside window of desired length Locate window to capture maximum channel impulse response energy nk T T T xk

rk yk w + h h wall h wall w H wall H wall w wT Aw hTwin h win w T HTwin H win w w T Bw hwin, hwall : equalized channel within and outside the window Objective function is shortening SNR (SSNR) w T Bw subject to w T Bw 1 max SSNR max 10 log10 T w Aw w w Cholesky decomposition of B to find eigenvector for minimum generalized eigenvalue of A and B w opt B T 1 q min A C B 1

B T 1 q min : eigenvector of min eigenvalue of C 16 Conventional Equalizer Modeling Achievable Bit Rate Bit allocation bounded by subchannel SNRs: log(1 + SNRi / i) Model ith subchannel SNR [Arslan, Evans & Kiaei, 2001] signal power Used in Maximum SNR i Bit Rate Method noise power ISI power S x,i signal transfer function SNR i S n,i noise transfer function S x,i ISI transfer function S x ,i : transmitted signal power in subchannel i S n ,i : channel noise power in subchannel i Divide numerator and denominator of SNRi by noise power spectral density Sn,i Used in Minimum ISI Method S x ,i signal 2 Hi S n ,i SNR i 2 S x ,i ISI noise

Hi Hi S n ,i Conventional subchannel SNRi 2 17 Conventional Equalizer Maximum Bit Rate (MBR) TEQ Design Subchannel SNR as nonlinear function of equalizer taps w H isignal q iH GHw H iISI q iH DHw qi is ith row of DFT matrix Bi = qi Sn,i qiH S x ,i q GHw 2 wT Ai w SNR i 2 2 T H H w Bi w S n ,i q i Fw S x ,i q i DHw

H inoise q iH Fw FT H i Ai = HT GT DT F + HT qi qi Sx,i qiH Sx,i qiH G H D H Maximize nonlinear function of bits/symbol with respect to w N /2 1 wT Ai w Fractional bits

bDMT log2 ( 1 ) T for optimization w Bi w i 1 Good performance measure for comparison of TEQ design methods Not an efficient TEQ design method in computational sense 18 Conventional Equalizer Minimum-ISI (Min-ISI) TEQ Design Rewrite subchannel SNR S x ,i signal 2 Hi [Arslan, Evans & Kiaei, 2001] S n ,i SNR i ISI power weighted in S x ,i ISI 2 noise 2 frequency domain by Hi Hi S n ,i inverse of noise spectrum Generalize MSSNR method by weighting ISI in frequency 2 H T Minimize frequency weighted ISI K q DHw

w Xw i i i sum of subchannel ISI power i i Penalize ISI power in high conventional SNR subchannels: K i S x ,i / S n ,i Constrain signal path gain to one signal 2 2 T | h | | GHw | w Yw 1 to prevent all-zero solution for w Solution is eigenvector of minimum generalized eigenvalue of X and Y Iterative Min-ISI method [Ding et al. 2003] Avoids Cholesky decomposition by using adaptive filter theory Designs arbitrary length TEQs without loss in bit rate Overcomes disadvantages of Maximum SSNR method 19 Outline Multicarrier modulation Conventional equalizer training methods Minimum Mean Squared Error design Maximum Shortening Signal-to-Noise Ratio design Maximum Bit Rate design (optimal) Minimum Inter-symbol Interference design (near-optimal) Per-tone equalizer

[Catholic University, Leuven, Belgium] Dual-path equalizer Conclusion Message bit stream Transmitter Channel Equalizer Received bit stream Receiver 20 Per-Tone Equalizer Drawbacks to Using Single FIR Filter for TEQ Conventional equalizer N real samples time domain equalizer (FIR filter) remove cyclic S/P prefix

N/2 complex samples N-FFT and remove mirrored data invert channel = frequency domain equalizer Equalizes all tones in combined fashion: may limit bit rate Output of conventional equalizer for tone i computed using sequence of linear operations Zi = Di rowi(QN ) Y w Di is the complex scalar value of one-tap FEQ for tone i QN is the N N complex-valued FFT matrix Y is an N Lw real-valued Toeplitz matrix of received samples w is a Lw 1 column vector of real-valued TEQ taps Yw represents convolution 21 Per-Tone Equalizer Frequency-Domain Per Tone Equalizer Rewrite equalized FFT coefficient for each of N/2 tones [Van Acker, Leus, Moonen, van de Wiel, Pollet, 2001] Zi = Di rowi(QN ) Y w = rowi(QN Y) ( w Di ) = rowi(QN Y) wi Take sliding FFT to produce N Lw matrix product QN Y Design wi for each tone N+

z-1 N + Lw 1 channels Sliding N-Point FFT z-1 N+ y z-1 N+ (Lw-frame) W1,0 W1,1 W1,2 W1,Lw-1 WN/2,0 WN/2,1 WN/2,2 WN/2,Lw-1 FEQ is a linear combiner of up to N/2 Lw-tap FEQs 22 Outline Multicarrier modulation

Conventional equalizer training methods Minimum Mean Squared Error design Maximum Shortening Signal-to-Noise Ratio design Maximum Bit Rate design (optimal) Minimum Inter-symbol Interference design (near-optimal) Per-tone equalizer Dual-path equalizer Conclusion [UT Austin] Message bit stream Transmitter Channel Equalizer Received bit stream Receiver 23 Dual-Path Equalizer Dual-Path Time Domain Equalizer (DP-TEQ) [Ding, Redfern & Evans, 2002] First FIR TEQ equalizes entire available bandwidth Second FIR TEQ tailored for subset of subchannels Subchannels with higher SNR Subchannels difficult to equalize, e.g. at boundary of upstream and downstream channels in frequency-division multiplexed ADSL Minimum ISI method is good match for second FIR TEQ TEQ 1

TEQ 2 FFT FFT Path Selection for each Subchannel FEQ Path selection for each subchannel is fixed during training Up to 20% improvement in bit rate over MMSE TEQs Enables reuse of VLSI designs of conventional equalizers 24 Simulation Results Simulation Results for 17-Tap Equalizers Bit rate (Mbps) Parameters Cyclic prefix length 32 FFT size (N) 512 Coding gain (dB) 4.2 Margin (dB) 6 Input power (dBm) 23 Noise power (dBm/Hz) -140 Crosstalk noise 24 ISDN disturbers Carrier serving area (CSA) test loop Downstream

transmission Figure 1 in [Martin, Vanbleu, Ding, Ysebaert, Milosevic, Evans, Moonen & Johnson, Oct. 2005] UNC(b) means unit norm constraint on target impulse response b, i.e. || b || = 1 MDS is Maximum Delay Spread design method [Schur & Speidel, 2001] 25 Simulation Results Simulation Results for 17-Tap Equalizers Bit Rate (Mbps) Parameters Cyclic prefix length 32 FFT size (N) 512 Coding gain (dB) 4.2 Margin (dB) 6 Input power (dBm) 23 Noise power (dBm/Hz) -140 Crosstalk noise 24 ISDN disturbers Carrier Serving Area (CSA) Test Loop Downstream transmission Figure 3 in [Martin, Vanbleu, Ding, Ysebaert, Milosevic, Evans, Moonen & Johnson, Oct. 2005] MDR is Maximum Data Rate design method [Milosevic et al., 2002] BM-TEQ is Bit Rate Maximizing design method [Vanbleu et al., 2003] PTEQ is Per Tone Equalizer structure and design method [Acker et al., 2001] 26 Simulation Results

Computational Complexity in 10 log10(MACs) Estimated Computational Complexity Equalizer Design Algorithm MAC means a multiplication-accumulation operation 27 Simulation Results Bit rate (Mbps) Achievable Bit Rate vs. Delay Parameter Delay Parameter for CSA Test Loop 4 Large plateau of near-optimal delays (optimal choice requires search) One choice is to set the delay parameter equal to cyclic prefix length 28 Conclusion Contributions by Research Group New methods for single-path time-domain equalizer design Maximum Bit Rate method maximizes bit rate (upper bound) Minimum Inter-Symbol Interference method (real-time, fixed-point) Minimum Inter-Symbol Interference TEQ design method Generalizes Maximum Shortening SNR by frequency weighting ISI Improve bit rate in an ADSL transceiver by change of software only Implemented in real-time on three fixed-point digital signal processors: Motorola 56000, TI TMS320C6200 and TI TMS320C5000 http://www.ece.utexas.edu/~bevans/projects/adsl New dual-path time-domain equalizer Achieves bit rates between conventional and per tone equalizers Lower implementation complexity in training than per tone equalizers Enables reuse of ASIC designs

29 Conclusion Matlab DMTTEQ Toolbox 3.1 Single-path, dual-path, per-tone & TEQ filter bank equalizers Available at http://www.ece.utexas.edu/~bevans/projects/adsl/dmtteq/ 18 design methods 23 -140 various performance measures default parameters from G.DMT ADSL standard different graphical views 30 Backup Slides Introduction Applications of Broadband Access Residential Application Downstream Upstream Willing to pay rate (kb/s) rate (kb/s) 384 9

High Database Access 384 9 Low On-line directory; yellow pages 1,500 1,500 High Video Phone 1,500 64 Low Home Shopping 1,500 1,500 Medium Video Games 3,000 384 High Internet 6,000 0 Low Broadcast Video 24,000 0 High High definition TV Demand Potential Medium High Medium Medium Medium Medium High Medium

Business Application Downstream Upstream Willing to pay rate (kb/s) rate (kb/s) 384 9 Medium On-line directory; yellow pages 1,500 9 Medium Financial news 1,500 1,500 High Video phone 3,000 384 High Internet 3,000 3,000 High Video conference 6,000 1,500 High Remote office 10,000 10,000 Medium LAN interconnection 45,000 45,000 High Supercomputing, CAD Demand

Potential High Low Low High Low Medium Medium Low 32 Introduction Selected DSL Standards Standard Meaning ISDN Integrated Services Digital Network T1 T-Carrier One (requires two pairs) HDSL High-Speed Digital Subscriber Line (requires two pairs) HDSL2 Single Line HDSL G.Lite ADSL G.DMT ADSL VDSL Splitterless Asymmetric Digital Subscriber Line Asymmetric Digital Subscriber Line Very High-Speed Digital Subscriber Line (proposed)

Data Rate 144 kbps Mode Symmetric Applications Internet Access, Voice, Pair Gain (2 channels) 1.544 Mbps Symmetric Enterprise, Expansion, Internet Service 1.544 Mbps Symmetric Pair Gain (12 channels), Internet Access, T1/E1 replacement 1.544 Mbps Symmetric Same as HDSL except pair gain is 24 channels up to 1.5 Mbps Downstream Internet Access, Digital up to 512 kbps Upstream Video up to 10 Mbps Downstream Internet Access, Digital up to 1 Mbps Upstream Video up to 22 Mbps Downstream Internet Access, Digital up to 3 Mbps Upstream Video, Broadcast Video up to 13 Mbps Symmetric Courtesy of Shawn McCaslin (National Instruments, Austin, TX) 33 Introduction Discrete Multitone DSL Standards Discrete multitone (DMT) modulation uses multiple carriers ADSL Asymmetric DSL (G.DMT) Asymmetric: 8 Mbps downstream and 1 Mbps upstream Data band: 25 kHz 1.1 MHz Maximum data rates possible in standard (ideal case) Echo cancelled: 14.94 Mbps downstream, 1.56 Mbps upstream

Frequency division multiplexing: 13.38 Mbps downstream, 1.56 Mbps up Widespread deployment in US, Canada, Western Europe, Hong Kong Central office providers only installing frequency-division ADSL ADSL modems have about 1/3 of market, and cable modems have 2/3 VDSL Very High Rate DSL Asymmetric: either 22/3 or 13/3 Mbps downstream/upstream Symmetric: 13, 9, or 6 Mbps each direction Data band: 1 12 MHz DMT and single carrier modulation supported DMT VDSL essentially higher speed version of G.DMT ADSL 34 Introduction A Digital Communications System Message Source Encoder Modulator Transmitter Noise Channel Decoder Message Sink

Demodulator Receiver Encoder maps a group of message bits to data symbols Modulator maps these symbols to analog waveforms Demodulator maps received waveforms back to symbols Decoder maps the symbols back to binary message bits 35 Introduction Intersymbol Interference (ISI) 2.1 Ideal channel Impulse response is impulse Flat frequency response 1.7 111 1 Non-ideal channel Causes ISI Channel memory Magnitude and phase variation 1 * 1 .7 .4 .7 .1 =

Received Channel signal impulse -1 Transmitted response signal Threshold at zero is weighted Received symbol sum of neighboring symbols 11 1 1 1 Weights are determined by channel impulse response Detected signal 36 Introduction Combat ISI with Equalization Equalization because channel response is not flat Zero-forcing equalizer Inverts channel Flattens freq. response Amplifies noise MMSE equalizer Optimizes trade-off between noise amplification and ISI Zero-forcing equalizer frequency response

MMSE equalizer frequency response Channel frequency response Decision-feedback equalizer Increases complexity Propagates error 37 Introduction Cyclic Prefix Repeated symbol cyclic prefix * to be removed = equal 38 Multicarrier Modulation Open Issues for Multicarrier Modulation Advantages Efficient use of bandwidth without full channel equalization Robust against impulsive noise and narrowband interference Dynamic rate adaptation

Disadvantages Transmitter: High signal peak-to-average power ratio Receiver: Sensitive to frequency and phase offset in carriers Open issues Pulse shapes of subchannels (orthogonal, efficient realization) Channel equalizer design (increase bit rate, reduce complexity) Synchronization (timing recovery, symbol synchronization) Bit loading (allocation of bits in each subchannel) Echo cancellation 39 Conventional Equalizer TEQ Algorithm ADSL standards Set aside 1024 frames (~.25s) for TEQ estimation Reserved ~16,000 frames for channel and noise estimation for the purpose of SNR calculation TEQ is estimated before the SNR calculations Noise power and channel impulse response can be estimated before time slot reserved for TEQ if the TEQ algorithm needs that information 40 Conventional Equalizer Single-FIR Time-Domain Equalizer Design Methods All methods below perform optimization at TEQ output Minimizing the mean squared error Minimize mean squared error (MMSE) method [Chow & Cioffi, 1992] Geometric SNR method [Al-Dhahir & Cioffi, 1996]

Minimizing energy outside of shortened (equalized) channel impulse response Maximum Shortening SNR method [Melsa, Younce & Rohrs, 1996] Divide-and-conquer methods [Lu, Evans, Clark, 2000] Minimum ISI method [Arslan, Evans & Kiaei, 2000] Maximizing bit rate [Arslan, Evans & Kiaei, 2000] Implementation Geometric SNR is difficult to automate (requires human intervention) Maximum bit rate method needs nonlinear optimization solver Other methods implemented on fixed-point digital signal processors 41 Conventional Equalizer Minimum Mean Squared Error (MMSE) TEQ nk xk h yk rk w + b z- ek - + w w0 w1 wLw 1

b b0 b1 b T b 0 | bT | 0 Lh 1 bk- T T MSE {ek2 } b T R xxb 2b T R xy w w T R yy w minimum MSE is achieved only if bT R xy w T R yy MSE b T R xx R xy R yy1 R yx b b T R x|y b T Define R OT R x|y O then MSE b R b O selects the proper part out of Rx|y corresponding to the delay 42 Conventional Equalizer Near-optimal Minimum-ISI (Min-ISI) TEQ Design Generalizes MSSNR method by frequency weighting ISI 2 ISI power in ith subchannel is ISIi S x ,i q iH DHw Minimize ISI power as a frequency weighted sum of subchannel ISI H i

2 T ISI K q DHw w Xw i i i i Constrain signal path gain to one to prevent all-zero solution | h signal |2 | GHw |2 w T Yw 1 Solution is a generalized eigenvector of X and Y Possible weightings Amplify ISI objective function in subchannels with low noise power (high SNR) to put ISI in low SNR bins: Ki S x ,i S n ,i K i S x , i Set weighting to be constant in all subchannels (MSSNR): K i 1 Set weighting equal to input power spectrum: Performance virtually equal to MBR (optimal) method 43 Conventional Equalizer Efficient Implementations of Min-ISI Method

Generalized eigenvalue problem can solved with generalized power iteration: Xw k 1 Yw k Recursively calculate diagonal elements of X and Y from first column [Wu, Arslan, Evans, 2000] Method Bit Rate MACs Original 99.6% 132,896 Recursive 99.5% 44,432 Row-rotation 99.5% 25,872 No-weighting 97.8% 10,064 44 Conventional Equalizer Motivation for Divide-and-Conquer Methods Fast methods for implementing Maximum SSNR method Maximum SSNR Method For each , maximum SSNR method requires Multiplications: ( L 7 ) L 5 L2 25 L3 h w w

w 6 2 3 Additions: 5 3 25 ( Lh ) Lw L2w L3w 6 2 3 Divisions: L2w Exhaustive search for the optimal delay 0 Lh Lw 2 0 499 Divide Lw TEQ taps into (Lw - 1) two-tap filters in cascade Design first two-tap filter then second and so forth (greedy approach) Develop heuristic to estimate the optimal delay 45 Conventional Equalizer Divide-and-Conquer Approach The ith two-tap filter is initialized as either Unit tap constraint (UTC) 1 w i gi

Unit norm constraint (UNC) sin i w i cos i Calculate best gi or i by using a greedy approach either by Minimizing 1 (Divide-and-conquer TEQ minimization) SSNR Minimizing energy in hwall (Divide-and conquer TEQ cancellation) Convolve two-tap filters to obtain TEQ 46 Conventional Equalizer Divide-and-Conquer TEQ Minimization (UTC) At ith iteration, minimize Ji over gi a1,i 1 g i T a 2 ,i w i Aw i Ji T w i Bw i b1,i 1 g i b2,i a2 ,i 1

a3,i g i a1,i 2a2,i g i a3,i g i2 b2,i 1 b1,i 2b2,i g i b3,i g i2 b3,i g i a3,i b1,i a1,i b3,i D g i 1, 2 solution Closed-form 2 a3,i b2,i a2,i b3,i 2 a3,i b2,i a2,i b3,i D a3,i b1,i a1,i b3,i 4 a3,i b2,i a2,i b3,i a2,i b1,i a1,i b2,i 2 47 Conventional Equalizer Divide-and-Conquer TEQ Minimization (UNC) At ith iteration, minimize Ji over i a1,i sin i 1 i T a 2 ,i w i Aw i Ji T w i Bw i b1,i sin i 1 i b2,i a1,i 1 i a 2 ,i b1,i 1 i

b2,i where a 2 ,i 1 sin i a3,i i b2,i 1 sin i b3,i i a 2 ,i 1 a3,i i b2,i 1 b3,i i Calculate i in the same way as gi for UTC version of this method 1 sin i 1 w i sin i sin i

cos i cos i sin i i 48 Conventional Equalizer Divide-and-Conquer TEQ Cancellation (UTC) At ith iteration, minimize Ji over gi 2 ~ ~ ~T ~ J i h wallh wall hi 1 k g i hi 1 k 1 , kS S 1, 2, , , 2, , Lh~ i 1 Closed-form solution for the ith two-tap FIR filter g i ~ ~ hi 1 (k 1)hi 1 (k ) kS ~2 hi 1 (k 1) kS

49 Conventional Equalizer Divide-and-Conquer TEQ Cancellation (UNC) At ith iteration, minimize Ji over I 2 ~ ~ ~T ~ J i h wall h wall hi 1 k sin i hi 1 k 1 cos i , kS S 1, 2, , , 2,, Lh~ i 1 2 2 a a , cos 0.5 1 Closed-form sin i solution 0.5 1 2 i 2

2 2 a 4 b a 4 b ~2 ~2 ~ ~ a hi 1 k hi 1 k 1 , b hi 1 (k 1) hi 1 (k ) kS kS 50 Conventional Equalizer Computational Complexity Computational complexity for each candidate Method

Memory (words) Maximum SSNR DC-TEQ-minimization (UTC) DC-TEQ-cancellation (UNC) DC-TEQ-cancellation (UTC) 120379 118552 441 1899 53240 52980 60 563 42280 42160 20 555 41000 40880 20

554 G.DMT ADSL Lh = 512 = 32 Lw = 21 Divide-and-conquer methods vs. maximum SSNR method Reduces multiplications, additions, divisions, and memory No matrix calculations (saves on memory accesses) Avoids matrix inversion, and eigenvalue and Cholesky decompositions 51 Conventional Equalizer Heuristic Search for the Optimal Delay Estimate optimal delay before computing TEQ taps ratio arg max energy inside a window of original h energy outside a window of original h Total computational cost Multiplications: Lh Additions: Divisions: 3Lh 3 Lh Performance of heuristic vs. exhaustive search Reduce computational complexity by factor of 500 2% loss in SSNR for TEQ with four taps or more 8% loss in SSNR for two-tap TEQ 52 Conventional Equalizer

Comparison of Earlier Methods Method MMSE MSSNR Geometric Advantages Maximize bit rate Minimize ISI Bit Rate Low-medium High Low-medium Disadvantages Nonlinear optimization Computational complexity Artificial constraints Low Unrealistic assumptions High

Ad-hoc parameters Lowpass frequency response Medium 53 Conventional Equalizer MBR TEQ vs. Geometric TEQ Method MBR Geometric Advantages Maximize channel capacity Minimize ISI Bit rate optimal Low-medium Disadvantages Low-pass frequency response Computationally complex Artificial constraints Ad-hoc parameters Nonlinear optimization Unrealistic assumptions

54 Conventional Equalizer Min-ISI TEQ vs. MSSNR TEQ Method Min-ISI MSSNR Advantages Maximize channel capacity Minimize ISI Frequency domain weighting Bit rate high Disadvantages Computationally complex very high high high Min-ISI weights ISI power with the SNR Residual ISI power should be placed in high noise frequency bands

1 1 0.09 SNR 50 0.1 SNR 50 signal power 10 1 10 SNR i noise power ISI power 1 1 SNR 2 10 SNR 2 0.9 0.1 0.1 1 55 Conventional Equalizer Bit Rate vs. Cyclic Prefix (CP) Size Matched filter bound decreases because CP has no new information Min-ISI and MBR achieve bound with 16-sample CP Other design methods are erratic MGSNR better for 15-28 sample CPs TEQ taps (L ) 17 w FFT size (N) coding gain margin

512 4.2 dB 6 dB input power 23 dBm noise power -140 dBm/Hz crosstalk noise 8 ADSL disturbers 56 Conventional Equalizer Simulation Results Min-ISI, MBR, and MSSNR achieve matched filter bound owith CP of 27 samples Min-ISI with 13sample CP beats MMSE with 32sample CP MMSE is worst TEQ taps (Lw) FFT size (N) coding gain margin 3 512 4.2 dB 6 dB input power 23 dBm noise power -140 dBm/Hz crosstalk noise 8 ADSL disturbers 57 Per-Tone Equalizer

Bit Allocation Comparison AWG 26 Loop: 12000 ft + AWGN Equalizer Per Tone 5.7134 Mbps MBR 5.4666 Mbps MSSNR 5.2903 Mbps Min ISI 5.2586 Mbps ARMA 4.5479 Mbps MMSE 4.4052 Mbps Bit Rate Simulation NEXT from 24 DSL disturbers 32-tap equalizers: least squares training used for per-tone equalizer 58 Per-Tone Equalizer Subchannel SNR 59

Per-Tone Equalizer Frequency-Domain Per-Tone Equalizer Rearrange computation of FFT coefficient for tone i [Van Acker, Leus, Moonen, van de Wiel, Pollet, 2001] Zi = Di rowi(QN ) Y w = rowi(QN Y) ( w Di ) QN Y produces N Lw complex-valued matrix produced by sliding FFT Zi is inner product of ith row of QN Y (complex) and w Di (complex) TEQ has been moved into FEQ to create multi-tap FEQ as linear combiner After FFT demodulation, each tone equalized separately Equalize each carrier independently of other carriers (N/2 carriers) Maximize bit rate at output of FEQ by maximizing subchannel SNR Sliding FFT to produce N Lw matrix product QN Y Receive one ADSL frame (symbol + cyclic prefix) of N + samples Take FFT of first N samples to form the first column Advance one sample Take FFT of N samples to form the second column, etc. 60 Per-Tone Equalizer Per-Tone Equalizer: Implementation Complexity Conventional Real MACs Words TEQ Lw f s 2 Lw Sampling rate FFT

2 N log2(N) fsym 4N FEQ 4 Nu fsym 4 Nu Per Tone FFT Real MACs Words 2 N log2(N) fsym 4N+2 Sliding FFT 2 (Lw 1) N fsym Combiner Modified. Per Tone FFT 4 Lw Nu fsym Real MACs N Adds Value fs 2.208 MHz Symbol rate

fsym 4 kHz TEQ length Lw 3-32 Symbol length N 512 Subchannels used Nu 256 Cyclic prefix length 32 Words 4N (Lw 1) fsym 2 (Lw + 1) Nu fsym Symbol 2 (Lw + 1) Nu 2 N log2(N) fsym

Differencing Combiner Parameter Lw 1 2 Lw Nu 61 Dual-Path Equalizer Dual-Path TEQ (Simulated Channel) Optimized for subchannel 2-250 Optimized for subchannel 2-30 62 Motorola CopperGold ADSL Chip Announced in March 1998 5 million transistors, 144 pins, clocked at 55 MHz 1.5 W power consumption DMT processor consists Motorola MC56300 DSP core Several application specific ICs 512-point FFT 17-tap FIR filter for time-domain channel equalization based on MMSE method (20 bits precision per tap) DSP core and memory occupies about 1/3 of chip area 63

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