VIEWS: 19 PAGES: 64 POSTED ON: 3/30/2012 Public Domain
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 November 30, 2009 UT graduate students: Mr. Aditya Chopra, Mr. Yousof Mortazavi UT MS/PhD grads: Dr. Güner Arslan (ST-Ericcson), Dr. Biao Lu (OpenSpirit), Dr. Ming Ding (Broadcom), Dr. Milos Milosevic (Schlumberger), Mr. Alex Olson (Schlumberger), Dr. Zukang Shen (Datang Mobile), and Dr. Ian Wong (Nat. Inst.) UT senior design students: Mr. Wade Berglund, Mr. Jerel Canales, Mr. David J. Love, Mr. Ketan Mandke, Mr. Scott Margo, Ms. Esther Resendiz, Mr. Jeff Wu Other collaborators: Dr. Lloyd D. Clark, Prof. C. Richard Johnson, Jr. (Cornell), Prof. Sayfe Kiaei (ASU), Prof. Rick Martin (AFIT), Prof. Marc Moonen (KU Leuven), Dr. Lucio F. C. Pessoa (Freescale), Dr. Arthur J. Redfern (TI) Introduction Interne t Digital Subscriber Line (DSL) Broadband Access DSLAM Downstream (higher data rate) Service DSL DSL Provider modem modem Upstream Voice LPF LPF Switch Customer Premises Telephone DSLAM - Digital Subscriber Line Access Multiplexer Network 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 1:2 in US & 2:1 worldwide G.DMT Asymmetric ADSL DMT VDSL ADSL+ 8 Mbps downstream min. 0.025 – 1.1 0.138 – 12 Data band ADSL2 doubles analog bandwidth MHz MHz Upstream VDSL – Very High Rate DSL subcarriers 32 256 Asymmetric Downstream 256 2048/4096 Faster G.DMT FDM ADSL subcarriers 2m subcarriers m [8, 12] Target up- 1 Mbps 3 Mbps Symmetric: 13, 9, or 6 Mbps stream rate Target down- Optional 12-17 MHz band 8 Mbps 13/22 Mbps stream rate 3 Outline • Multicarrier modulation • Conventional equalizer training methods – Minimum Mean Squared Error design [Stanford] – Maximum Shortening Signal-to-Noise Ratio design [Tellabs] – Maximum Bit Rate design (optimal) [UT Austin] – Minimum Inter-symbol Interference design (near-optimal) [UT Austin] • Per-tone equalizer [Catholic University, Leuven, Belgium] • Dual-path equalizer [UT Austin] • Conclusion Message Received bit stream bit stream Equalizer Transmitter Channel 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 – Continuous time: g d(t-T) Channel nk Equalizer – Discrete time: g d[k-D] xk yk rk ek h + w + • Equalizer Training + - – Shortens channel sequence impulse response Ideal Channel (time domain) Receiver – Compensates for generates z-D g frequency distortion xk (frequency domain) 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 sampled • Discrete multitone modulation pulse sinc DTFT-1 – Baseband transmission – Based on fast Fourier transform (FFT) w k – Standardized for ADSL and VDSL -w w sin w c k ) c c channel k magnitude carrier subchannel Subchannels are 4.3 kHz wide in ADSL and VDSL frequency 6 Multicarrier Modulation Multicarrier Modulation by Inverse FFT Filter Bank j 2 f1 t complex-valued j 2 1 e N k e X1 g(t) x X1 x j 2 f 2 t complex-valued j 2 2 e N k e X2 g(t) x Discrete X2 x + time + j 2 f N / 2 t real-valued j 2 N /2 e N k e X N /2 g(t) x X N /2 x g(t) : pulse shaping filter 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 Xi Modulation (QAM) during training – ADSL uses QAM of 22, 23, 24, …, 215 levels In-phase during data transmission • Multicarrier modulation using inverse FFT QAM X0 x0 X1 x1 Mirror and X2 N-point x2 Yields one conjugate Inverse symbol of N N/2–1 complex XN/2 Fast real-valued subsymbols Fourier samples X2* Transform X1* xN-1 X i e j wi n X i*e- j wi n 2 X i cos( i n X i ) w 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 n samples of the symbol copy copy CP s y m b o l i CP s y m b o l i+1 v samples N samples ADSL G.DMT Values N 16 Down Up – CP reduces throughput by factor of stream stream N v 17 n 32 4 • Linear convolution of frame with N 64 512 channel impulse response – 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) n1 – Finite impulse response (FIR) filter channel impulse – Effective channel impulse response: response convolution of TEQ impulse response with channel impulse response effective • Frequency domain equalizer (FEQ) channel impulse – Compensates magnitude/phase distortion response of equalized channel by dividing each FFT coefficient by complex number D – Generally updated during data transmission D: transmission delay • ADSL G.DMT equalizer training n: cyclic prefix length – Reverb: same symbol sent 1,024 to 1,536 times ADSL G.DMT Values – Medley: aperiodic pseudo-noise sequence of Down Up 16,384 symbols for subchannel SNR estimation stream stream – Receiver returns number n 32 4 SNR i of bits (0-15) to transmit bi log 2 1 Γi N 512 64 each subchannel i 10 Multicarrier Transceivers ADSL Transceiver: Data Transmission N/2 subchannels N real samples 2.208 MHz quadrature mirror Bits amplitude add D/A + data S/P modulation cyclic P/S transmit 00110 and (QAM) prefix filter N-IFFT encoder TRANSMITTER each block programmed in lab and covered in one full lecture channel each block covered in one full lecture RECEIVER N/2 subchannels N real samples invert N-FFT time channel receive and remove domain QAM = filter P/S remove S/P cyclic equalizer decoder frequency + mirrored prefix (FIR domain A/D equalizer data filter) 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 [Tellabs] – Maximum Bit Rate design (optimal) [UT Austin] – Minimum Inter-symbol Interference design (near-optimal) [UT Austin] • Per-tone equalizer • Dual-path equalizer • Conclusion Message Received bit stream bit stream Equalizer Transmitter Channel Receiver 12 Conventional Equalizer Minimum Mean Squared Error TEQ Design nk Channel TEQ xk yk rk ek h + w - + z-D b bk-D • Minimize E{ek2} [Chow & Cioffi, 1992] – Chose length of b (e.g. n+1) to shorten length of h * w – b is eigenvector of minimum eigenvalue of symmetric Rxy is cross channel-dependent matrix R D R xx - R xy R -1 R yx yy correlation – Minimum MSE when R yy w R xy b where w 0 matrix • Disadvantages – Does not consider bit rate – Deep notches in equalized frequency response Why? 13 Conventional Equalizer Infinite Length MMSE TEQ Analysis • As TEQ length goes to infinity, RD 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 n subchannels • Finite length TEQ plot [Martin et al., 2004] – Each trace is a different zero of b – Distance of 32 zeros of b to unit circle averaged Longer MMSE over 8 ADSL test channels for each TEQ length TEQ may be worse – 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 maxSSNR in dB) max 10 log10 energy outside window after TEQ w w n1 h w channel impulse response • Equivalent to noise-free MMSE TEQ effective • Disadvantages channel – Does not consider channel noise impulse response – Does not consider bit rate – Deep notches in equalized frequency response D (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 h wall h wall w H wall H wall w wT Aw T T yk rk xk h + w hT h win wT HT H win w wT Bw win win hwin, hwall : equalized channel within and outside the window • Objective function is shortening SNR (SSNR) wT Bw maxSSNR) max 10 log10 T subject to wT Bw 1 w Aw w w Cholesky decomposition of B to find eigenvector for minimum generalized eigenvalue of A and B C B ) A -1 B T ) -1 wopt B)q T -1 min q min : eigenvector of min eigenvalue of C 16 Conventional Equalizer Modeling Achievable Bit Rate • Bit allocation bounded by subchannel SNRs: log(1 + SNRi / Gi) • Model ith subchannel SNR [Arslan, Evans & Kiaei, 2001] signal power Used in Maximum SNR i noise power ISI power Bit Rate Method S x,i signal transferfunction SNRi Sn,i noise transferfunction S x,i ISI transferfunction S x,i : transmitted signal power in subchanneli Sn,i : channel noise power in subchanneli S x ,i signal 2 • Divide numerator and Hi denominator of SNRi by noise SNR S n ,i i power spectral density Sn,i noise 2 S x ,i ISI 2 Hi Hi S n ,i Used in Minimum Conventional ISI Method subchannel SNRi 17 Conventional Equalizer Maximum Bit Rate (MBR) TEQ Design • Subchannel SNR as nonlinear function of equalizer taps w H isignal q iH GHw H 2 S x,i qi GHw wT Ai w SNRi 2 H i qi DHw 2 ISI H S n,i qi Fw S x,i qi DHw H H wT Bi w qi H inoise q iH Fw GT Sx,i H G Ai = HT qi H qi is ith row of DFT matrix q qi FT Sn,i Hi F DT Sx,i H D Bi = qi + HT qi H • Maximize nonlinear function of bits/symbol with respect to w N /2 1 wT A i w bDMT log 2 ( 1 Fractional bits ) i 1 G wT Bi w for optimization – 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 [Arslan, Evans & Kiaei, 2001] Hi S n ,i ISI power weighted in SNR i noise 2 S x ,i ISI 2 frequency domain by Hi Hi inverse of noise spectrum S n ,i • Generalize MSSNR method by weighting ISI in frequency ISIi Ki qi DHw wT Xw 2 – Minimize frequency weighted H 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 to prevent all-zero solution for w | h signal |2 | GHw |2 w T Yw 1 – 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 Received bit stream bit stream Equalizer Transmitter Channel Receiver 20 Per-Tone Equalizer Drawbacks to Using Single FIR Filter for TEQ • Conventional N real N/2 complex equalizer samples samples N-FFT invert time remove and channel domain equalizer cyclic S/P remove = frequency (FIR prefix mirrored domain filter) data 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 Yw represents Di is the complex scalar value of one-tap FEQ for tone i convolution 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 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+n z-1 W1,0 W1,1 W1,2 W1,Lw- Sliding 1 N-Point N + Lw – 1 channels z-1 FFT N+n (Lw-frame) WN/2,0 WN/2,1 WN/2,2 WN/2,Lw-1 z-1 y N+n 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 [UT Austin] • Conclusion Message Received bit stream bit stream Equalizer Transmitter Channel 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 FFT Path Selection for each FEQ Subchannel TEQ 2 FFT • 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 Parameters Cyclic prefix length 32 FFT size (N) 512 Coding gain (dB) 4.2 Margin (dB) 6 Bit rate (Mbps) Input power (dBm) 23 Noise power (dBm/Hz) -140 Crosstalk noise 24 ISDN disturbers Downstream Carrier serving area (CSA) test loop 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 Parameters Cyclic prefix length 32 FFT size (N) 512 Coding gain (dB) 4.2 Bit Rate (Mbps) Margin (dB) 6 Input power (dBm) 23 Noise power (dBm/Hz) -140 Crosstalk noise 24 ISDN disturbers Downstream Carrier Serving Area (CSA) Test Loop 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 Estimated Computational Complexity Computational Complexity in 10 log10(MACs) Equalizer Design Algorithm MAC means a multiplication-accumulation operation 27 Simulation Results Bit Rate vs. Training Complexity Tradeoffs 6.6 6.5 6.4 Bit Rate (Mbps) 6.3 SymMinSSNR SymMMSE 6.2 MinSSNR SymMinISI 6.1 MMSE MinISI 6 MDS DualPath 5.9 5.8 5.7 5.6 4.5 5 5.5 6 6.5 7 Training Complexity in log10(MACs) MAC means a multiplication-accumulation operation 28 Simulation Results Achievable Bit Rate vs. Delay Parameter Bit rate (Mbps) Delay Parameter D 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 29 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 30 Conclusion UT Austin 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 default parameters from G.DMT ADSL 23 -140 standard various different performance graphical measures views 31 Conclusion UT Austin ADSL2 Simulator 1.1 1. Simulation controls 2. Simulation state indicator 3. Description window 4. Simulation parameters 5. Performance indicators http://users.ece.utexas.edu/~bevans/projects/adsl/simulator/index.html 32 Backup Slides Introduction Applications of Broadband Access Residential Application Downstream Upstream Willing to pay Demand rate (kb/s) rate (kb/s) Potential Database Access 384 9 High Medium On-line directory; yellow pages 384 9 Low High Video Phone 1,500 1,500 High Medium Home Shopping 1,500 64 Low Medium Video Games 1,500 1,500 Medium Medium Internet 3,000 384 High Medium Broadcast Video 6,000 0 Low High High definition TV 24,000 0 High Medium Business Application Downstream Upstream Willing to pay Demand rate (kb/s) rate (kb/s) Potential On-line directory; yellow pages 384 9 Medium High Financial news 1,500 9 Medium Low Video phone 1,500 1,500 High Low Internet 3,000 384 High High Video conference 3,000 3,000 High Low Remote office 6,000 1,500 High Medium LAN interconnection 10,000 10,000 Medium Medium Supercomputing, CAD 45,000 45,000 High Low 34 Introduction Selected DSL Standards Standard Meaning Data Rate Mode Applications ISDN Integrated Services 144 kbps Symmetric Internet Access, Voice, Pair Digital Network Gain (2 channels) T1 T-Carrier One 1.544 Mbps Symmetric Enterprise, Expansion, (requires two pairs) Internet Service HDSL High-Speed Digital 1.544 Mbps Symmetric Pair Gain (12 channels), Subscriber Line Internet Access, T1/E1 (requires two pairs) replacement HDSL2 Single Line HDSL 1.544 Mbps Symmetric Same as HDSL except pair gain is 24 channels G.Lite Splitterless up to 1.5 Mbps Downstream Internet Access, Digital ADSL Asymmetric Digital up to 512 kbps Upstream Video Subscriber Line G.DMT Asymmetric Digital up to 10 Mbps Downstream Internet Access, Digital ADSL Subscriber Line up to 1 Mbps Upstream Video VDSL Very High-Speed up to 22 Mbps Downstream Internet Access, Digital Digital Subscriber up to 3 Mbps Upstream Video, Broadcast Video Line up to 13 Mbps Symmetric Courtesy of Shawn McCaslin (National Instruments, Austin, TX) 35 Introduction A Digital Communications System Message Message Encoder Decoder Source Noise Sink Modulator Channel Demodulator Transmitter 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 36 Introduction Intersymbol Interference (ISI) 2.1 • Ideal channel 1.7 – Impulse response is impulse 111 1 1 1 – Flat frequency response .7 .7 .4 .1 • Non-ideal channel * = – Causes ISI Channel Received – Channel memory -1 impulse signal – Magnitude and phase Transmitted response variation signal Threshold • Received symbol is weighted at zero sum of neighboring symbols 11 1 1 1 – Weights are determined by channel impulse response Detected signal 37 Introduction Combat ISI with Equalization • Equalization because channel response is not flat • Zero-forcing equalizer – Inverts channel – Flattens freq. response Zero-forcing MMSE equalizer equalizer – Amplifies noise frequency frequency response • MMSE equalizer response – Optimizes trade-off Channel frequency between noise response amplification and ISI • Decision-feedback equalizer – Increases complexity – Propagates error 38 Introduction Cyclic Prefix Repeated cyclic symbol prefix * to be removed = equal 39 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 40 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 41 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 42 Conventional Equalizer Minimum Mean Squared Error (MMSE) TEQ nk xk yk rk ek w w0 w1 wLw -1 ] T w b b0 b1 bn ] T h + - + z-D b bk-D ˆ b 0D | bT | 0Lh -D-n -1 ] T ˆ ˆ ˆ MSE {ek } bT R xxb - 2bT R xy w wT R yy w 2 minimum MSE is achieved only if bT R xy wT Ryy ˆ ˆ ˆ ] ˆ MSE bT R xx - R xy R -1 R yx b bT R x|yb yy Define RD OT Rx|yO then MSE b R Δb T O selects the proper part out of Rx|y corresponding to the delay D 43 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 ISIi Ki q DHw wT Xw H 2 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 S x ,i – Amplify ISI objective function in subchannels with low Ki noise power (high SNR) to put ISI in low SNR bins: S n ,i – Set weighting equal to input power spectrum: K i S x ,i – Set weighting to be constant in all subchannels (MSSNR): K i 1 • Performance virtually equal to MBR (optimal) method 44 Conventional Equalizer Efficient Implementations of Min-ISI Method • Generalized eigenvalue problem can solved with generalized power iteration: Xwk 1 Ywk • 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 45 Conventional Equalizer Motivation for Divide-and-Conquer Methods • Fast methods for implementing Maximum SSNR method • Maximum SSNR Method – For each D, 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 - L2 L3 w w 6 2 3 • Divisions: L2 w – Exhaustive search for the optimal delay D 0 D Lh Lw -n - 2 0 D 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 46 Conventional Equalizer Divide-and-Conquer Approach • The ith two-tap filter is initialized as either 1 – Unit tap constraint (UTC) wi gi sin i – Unit norm constraint (UNC) wi 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 47 Conventional Equalizer Divide-and-Conquer TEQ Minimization (UTC) • At ith iteration, minimize Ji over gi a1,i a2 ,i 1 1 gi ] T w i Aw i a2 , i a3,i g i a1,i 2a2,i g i a3,i g i2 Ji T w i Bw i b1,i b2,i 1 b1,i 2b2,i g i b3,i g i2 1 gi ] b2,i b3,i g i • Closed-form solution - a3,i b1,i - a1,i b3,i ) D g i 1, 2 ) 2a3,i b2,i - a2,i b3,i ) 2a3,i b2,i - a2,i b3,i ) D a3,i b1,i - a1,i b3,i ) - 4a3,i b2,i - a2,i b3,i )a2,i b1,i - a1,i b2,i ) 2 48 Conventional Equalizer Divide-and-Conquer TEQ Minimization (UNC) • At ith iteration, minimize Ji over i a1,i a2 ,i 1 sin i 1 i ]) sin i T w i Aw i a 2 ,i a3,i i Ji T w i Bw i b1,i b2,i 1 sin i 1 i ]) sin i b2,i b3,i i a1,i a2 ,i 1 1 i ] Calculate i in a2 ,i a3,i i the same way b1,i b2,i 1 as gi for UTC 1 i ] b3,i i version of this b2,i method sin i 1 1 • where wi sin i cos sin sin i cos i i i i 49 Conventional Equalizer Divide-and-Conquer TEQ Cancellation (UTC) • At ith iteration, minimize Ji over gi ) J i h wallh wall hi -1 k ) g i hi -1 k - 1) , ~T ~ ~ ~ 2 kS S 1, 2,, D, D n 2,, Lh ~ i -1 • Closed-form solution for the ith two-tap FIR filter ~ ~ hi -1 (k - 1)hi -1 (k ) g i - kS ~2 hi -1 (k - 1) kS 50 Conventional Equalizer Divide-and-Conquer TEQ Cancellation (UNC) • At ith iteration, minimize Ji over I ~T ~ J i h wallh wall hi -1 k )sin i hi -1 k - 1) cos i , ~ ~ 2 ) kS S 1, 2,, D, D n 2,, Lh ~ i -1 • Closed-form solution a2 a2 sin i 0.51 2 , cos 0.51 a 4b 2 i 2 a 4b 2 ) a hi -1 k ) - hi -1 k - 1) , b hi -1 (k - 1) hi -1 (k ) ~2 ~2 ~ ~ kS kS 51 Conventional Equalizer Computational Complexity • Computational complexity for each candidate D Method Memory G.DMT (words) ADSL Maximum 120379 118552 441 1899 Lh = 512 SSNR DC-TEQ-mini- 53240 52980 60 563 n = 32 mization (UTC) Lw = 21 DC-TEQ-can- 42280 42160 20 555 cellation (UNC) DC-TEQ-can- 41000 40880 20 554 cellation (UTC) • 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 52 Conventional Equalizer Heuristic Search for the Optimal Delay • Estimate optimal delay D before computing TEQ taps energy inside a window of original h D ratio arg max D energy outside a window of original h • Total computational cost – Multiplications: Lh – Additions: 3Lh - 3 – Divisions: 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 53 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 Low Medium High Artificial constraints Ad-hoc parameters Lowpass frequency response Unrealistic assumptions 54 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 55 Conventional Equalizer Min-ISI TEQ vs. MSSNR TEQ Method Min-ISI MSSNR Advantages Maximize channel capacity Minimize ISI Frequency domain weighting Bit rate high high Disadvantages Computationally complex very high high • Min-ISI weights ISI power with the SNR – Residual ISI power should be placed in high noise frequency bands 1 1 0.1 SNR50 SNR50 0.09 SNR i signal power 10 10 1 noise power ISI power 1 1 SNR 2 10 SNR2 0.9 0.1 0.1 1 56 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 (Lw) 17 FFT size (N) 512 input power 23 dBm coding gain 4.2 dB noise power -140 dBm/Hz margin 6 dB crosstalk noise 8 ADSL disturbers 57 Conventional Equalizer Simulation Results • Min-ISI, MBR, and MSSNR achieve matched filter bound owith CP of 27 samples • Min-ISI with 13- sample CP beats MMSE with 32- sample CP • MMSE is worst TEQ taps (Lw) 3 FFT size (N) 512 input power 23 dBm coding gain 4.2 dB noise power -140 dBm/Hz margin 6 dB crosstalk noise 8 ADSL disturbers 58 Per-Tone Equalizer Bit Allocation Comparison • AWG 26 Loop: 12000 ft + AWGN Equalizer Bit Rate 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 • Simulation – NEXT from 24 DSL disturbers – 32-tap equalizers: least squares training used for per-tone equalizer 59 Per-Tone Equalizer Subchannel SNR 60 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 + 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. 61 Per-Tone Equalizer Per-Tone Equalizer: Implementation Complexity Conventional Real MACs Words Parameter Symbol Value TEQ Lw fs 2 Lw Sampling rate fs 2.208 MHz FFT 2 N log2(N) fsym 4N Symbol rate fsym 4 kHz FEQ 4 Nu fsym 4 Nu TEQ length Lw 3-32 Symbol length N 512 Per Tone Real MACs Words Subchannels used Nu 256 FFT 2 N log2(N) fsym 4N+2n Cyclic prefix n 32 Sliding FFT 2 (Lw – 1) N fsym N length Combiner 4 Lw Nu fsym 2 (Lw + 1) Nu Modified. Real MACs Adds Words Per Tone FFT 2 N log2(N) fsym 4N Differencing (Lw – 1) fsym Lw – 1 Combiner 2 (Lw + 1) Nu fsym 2 Lw Nu 62 Dual-Path Equalizer Dual-Path TEQ (Simulated Channel) Optimized for subchannel 2-250 Optimized for subchannel 2-30 63 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 64