Channel Equalization To Achieve High Bit Rates In Discrete Multitone Modulation Systems Ming Ding Ph.D. Defense Committee members Prof. Ross Baldick Prof. Melba M. Crawford Prof. Brian L. Evans (Advisor) Prof. Robert W. Heath, Jr. Prof. Edward J. Powers April 21, 2004

2 Outline Introduction Unification of Discrete Multitone (DMT) Equalization –Common Mathematical Framework –Case Studies Contributions in DMT Equalization Methods –Symmetric Design –Minimum Intersymbol Interference Methods –Filter Bank Equalization Simulation Results Conclusions

3 Multicarrier Modulation Divide wideband channel into narrowband subchannels –Subchannel is approximately flat DMT is baseband muliticarrier modulation method –Band partition based on fast Fourier transform (FFT) –Line code for asymmetric digital subscribe line (ADSL) and very- high speed digital subscriber line standards subchannel frequency SNR carrier channel Subchannel width = bandwidth/ no. of subcarriers

4 DMT Transmission Quadrature Amplitude Modulation (QAM) constellation mapping in each subchannel Composed of N/2 complex-valued subsymbols Mirror and conjugate subsymbols to obtain real-valued inverse FFT output N-point Inverse FFT X1X1 X2X2 X1*X1* x1x1 x2x2 x3x3 xNxN X2*X2* X N/2 X N/2- 1 * X0X0 one symbol of N real-valued samples N/2 subsymbols (one subsymbol per carrier)

5 Cyclic Prefix (CP) Prepended to each DMT symbol –Serves as guard time to combat intersymbol interference (ISI) –Converts linear convolution of transmitted symbol and channel impulse response into circular convolution –FFT of circular convolution is product of FFTs Allows receiver to remove ISI if cyclic prefix length +1 is greater than length of channel impulse response Reduces throughput by a factor of N samplesv samples CP s y m b o l i s y m b o l ( i+1) copy

6 Bit Loading in DMT Number of bits allocated to i th subchannel –SNR i is SNR in subchannel i –  i is SNR gap to channel capacitySNR gap Turn off subchannels that cannot support minimum number of bits Bit rate Channels with length longer than cyclic prefix cause ISI –Significantly lowers SNR and bit rate Channel equalization essential for combating ISI  i = 9.8 dB in uncoded DMT ADSL/VDSL system Symbol rate is 4 kHz in DMT ADSL/VDSL system

7 ADSL Transceiver Data Transmission Subsystem reverse function QAM decision device (Viterbi) N/2 complex multiply units superframe scramble, encode, interleave tone order QAM mapping (Trellis) mirror data and N -IFFT add cyclic prefix P/S D/A + transmit filter N -FFT and remove mirrored data S/P remove cyclic prefix TRANSMITTER RECEIVER N/2 subchannelsN real samples N/2 subchannels time domain equalizer receive filter + A/D channel ATM

8 Conventional Two-Step Equalization Channel modeled as finite impulse response filter plus additive noise Time domain equalizer (TEQ) –Finite impulse response filter –Shortens channel impulse response to be at most  1 samples –Converts linear convolution to circular Frequency domain equalizer (FEQ) –Single division per subchannel (tone) –Compensate for amplitude/phase distortions Design objectives –High bit rates at fixed bit error rate –Low implementation complexity channel impulse response effective channel impulse response   : transmission delay : cyclic prefix length

9 Linear Equalizer Structures

10 Equalizer Training Complexity Periodic 4-QAM training sequence –No cyclic prefix –Constant transmit power spectrum S x Receiver monitors additive noise power spectrum S n Multiplications & Additions Memory (Words) Single TEQ O(L w 3 )LwLw TEQ Filter Bank O(L w 2 N 2 )N/2 L w Per Tone Equalizer O(L w 2 N + L w N 2 )N L w Complex Filter Bank O(L w 2 N + L w N 2 )N L w Example ADSL Parameters FFT Size N = 512 TEQ Length L w = 17 [Martin, Vanbleu, Ding et al. 2004]

11 Outline Introduction  Unification of DMT Equalization –Common Mathematical Framework –Case Studies Contributions in DMT Equalization Methods –Symmetric Design –Minimum Intersymbol Interference Methods –Filter Bank Equalization Simulation Results Conclusions

12 Unification of Equalizer Design Algorithms Most algorithms minimize product of generalized Rayleigh quotients For M = 1, solution is generalized eigenvector of the matrix pair (B, A) corresponding to smallest generalized eigenvalue For M > 1, solution is not well-understood –Various searching methods exist to find a local optimum

13 Single Quotient Cases Minimum Mean Square Error [Chow & Cioffi, 1992] –Minimizes squared error between output of TEQ w and output of virtual target impulse response filter b Maximum Shortening SNR [Melsa et al. 1996] –Channel convolution matrix H h + w z-z- b - xkxk ykyk ekek nknk + h win h wall A depends on  A and B depend on  ChannelTEQ

14 Single Quotient Cases Minimum Intersymbol Interference [ Arslan et al ] Minimum Delay Spread [ Schur et al ] Modified Maximum Shortening SNR with distance weighting Generalization of Maximum Shortening SNR method with frequency weighting channel taps d1d1 d2d2 k =0, 1, 2,…, N-1 c: center of mass +1

15 Multiple Filters (each with a Single Quotient) Per-tone equalization [ Acker et al ] –Generalized eigenvalue problem for each tone i –Received frame (CP + symbol) is y and ith FFT coefficient is Y i Time domain equalizer bank [ Milosevic et al ]

16 Product of Quotients Bit rate Maximum Geometric SNR [ Al-Dhahir et al. 1995] –Additive white Gaussian Noise (AWGN), Sequential Quadratic Programming Maximum Bit Rate [ Arslan et al ] –ISI + AWGN, Quasi-Newton algorithm Maximum Data Rate [ Milosevic et al ] –ISI + Cross-talk + Echo + digital noise floor –Almogy and Levin iteration Bitrate Maximizing [ Vanbleu et al ] –Eventually all possible noises and interference resources –Recursive Gauss-Newton update

17 Outline Introduction Unification of DMT Equalization –Common Mathematical Framework –Case Studies  Contributions in DMT Equalization Methods –Symmetric Design –Minimum Intersymbol Interference Methods –Filter Bank Equalization Simulation Results Conclusions

18 Contribution #1 Infinite Length TEQ Results Eigenvectors of a doubly symmetric matrix Maximum Shortening SNR TEQ with unit energy A = H T D T D H converges asymptotically to doubly symmetric H T H Minimum Mean Square Error TEQ Target impulse response is symmetric/skew symmetric A becomes a doubly symmetric matrix

19 Contribution #1 Observation of Long TEQ Designs Minimum Mean Square Error TEQs –Target impulse response is approximately symmetric Maximum Shortening SNR TEQs –A and B are almost doubly symmetric –w becomes almost perfectly symmetric Minimum Intersymbol Interference TEQs –Same as Maximum Shortening SNR case Can exploit symmetry in TEQ designs –Force TEQ to be symmetric –Compute half of TEQ coefficients –Apply symmetry z-z- h + w b - xkxk ykyk ekek nknk +

20 Contribution #1 Symmetric TEQ design Implementation: instead of finding eigenvector of L w  L w matrix, find eigenvector of matrix –Some matrix operations ~ O(L w 3 )) Phase response of symmetric TEQ is linear –Phase response fixed when given TEQ length –No amplitude scaling needed for 4-QAM –Enables design of FEQ in parallel

21 Contribution #2 Minimum ISI Method Advantages –Push ISI to unused subchannels or subchannels with lower SNR –Practical real-time implementation on digital signal processors Disadvantages –TEQs longer than + 1 taps B is not invertible method fails –Cholesky decomposition sensitive todecomposition fixed-point computation –High computational cost when performing delay optimization (A and B depend on Δ )

22 Contribution #2 Improving Minimum ISI Method Define new cost function – : weighting value for subchannel i –H T H is always positive definite and invertible Suitable for arbitrary length TEQ design Reduces computational cost when performing delay optimization Does not depend on Δ

23 Contribution #2 Quantized Frequency Weighting Min-ISI weighting in each subchannel is On-off quantization –Compare noise power with threshold –Choose zero weights in subchannels with larger-than-threshold noise power –Choose unit weights in other subchannels –Choose threshold as noise power for supporting 2 bits in subchannel ADSL fixes S x = -40 dBm/Hz  gap = 9.8 dB During training

24 Contribution #2 Iterative Minimum ISI Method 1.Obtain weighting values for subchannel i 2.Pre-compute and 3.Choose step size  4.Start with non-zero initial guess w 0, and iteratively calculate w k, using deterministic gradient search [Chatterjee, et. al 1997] Division-free iteration Method avoids Cholesky decomposition and directly calculates generalized eigenvector associated with minimum eigenvalue

25 Contribution #3 Complex Filter Bank Equalization Move all FEQ operations to time domain Combine with TEQ to obtain multi-tap complex-valued FIR filter bank Received Signal R={r 1,…r N } Delays Goertzel Filters Complex Equalizers w1w1 w2w2 w N/2-1 G1G1 G2G2 G N/2-1 y1y1 y2y2 y N/2-1 X1X1 X2X2 X N/2-1 11  2 2  N/2-1

26 Contribution #3 Design of Filter Bank For each subchannel, define at FEQ output –Classical MMSE solution for TEQ for each subchannel Quadratic cost function leads to iterative implementation use deterministic steepest descent search Different delays can be introduced on each subchannel Introduce different TEQ length to each subchannel Upper bound on achievable bit rate performance

27 Contribution #3 Dual-path TEQ Each path exploits a different TEQ aiming at optimize over a different subset of data-carrying subchannels Advantages –Less frequency selectivity makes equalization easier –Achieve higher data rates than conventional structure at relatively low implementation cost ExamplesExamples TEQ 1 TEQ 2 PFFT Path Sel. FEQ PFFT: Partial FFT

28 Outline Introduction Unification of DMT Equalization –Common Mathematical Framework –Case Studies Contributions in DMT Equalization Methods –Symmetric Design –Minimum Intersymbol Interference Methods –Filter Bank Equalization  Simulation Results Conclusions

29 Proposed Dual-Path and Complex TEQ Filter Bank Equalizers Simulation Parameters TEQ length 17 Cyclic prefix 32 samples FFT size (N) 512 samples Coding gain 5 dB Margin 6 dB Input power 23 dBm Noise PSD -140 dBm/Hz Crosstalk noise 5 ISDN RF interference 6 AM stations Channels Carrier Serving Area Loops 1-8Carrier Testing 1000 symbols

30 Proposed Symmetric TEQ Design Methods Simulation Parameters TEQ length 17 Cyclic prefix 32 samples FFT size (N) 512 samples Coding gain 5 dB Margin 6 dB Input power 23 dBm Noise PSD -140 dBm/Hz Crosstalk noise 5 ISDN RF interference 6 AM stations Channels Carrier Serving Area Loops 1-8 Testing 1000 symbols

31 Proposed Iterative Minimum ISI Method Simulation Parameters TEQ length 3-32 Cyclic prefix 32 samples FFT size (N) 512 samples Coding gain 5 dB Margin 6 dB Input power 23 dBm Noise PSD -140 dBm/Hz Crosstalk noise 24 HDSL RF interference none Channels Carrier Serving Area Loop average Testing 1000 symbols Mbps

32 Conclusions Unification and evaluation of existing methods Design methods for conventional equalizer structures –Symmetric methods reduce complexity by order of magnitude –Modified Minimum ISI method simplifies delay optimization –Iterative Minimum ISI method applicable to any generalized eigendecomposition method and suitable for fixed-point realization Filter bank equalization structures –Complex filter bank benchmarks achievable bit rate –Dual path achieves best tradeoff of bit rate vs. training complexity and allows VLSI design reuse of a conventional equalizer Deliverables –MATLAB discrete multitone equalization toolbox –Analysis of Advanced Signal Technology ADSL measurements

33 Future topics Effect of channel estimation error on bit rate performance –Channel estimation based on frequency domain zero-forcing –Perturbation bounds on generalized eigenvector computation Minimum phase equalizer design –Minimum group delay, energy delay and phase lag –Reduced TEQ length compare to linear phase design –Efficient designs use a linear phase design as a start point Upstream transmission Equalization in multi-input multi-output case –Multiple lines are grouped in cable –Future DSL systems deployed with central unit

34 Publications in DMT Journal Papers –M. Ding, B. L. Evans, ``Effect of Channel Estimation Error on Bit Rate Performance in a Multicarrier Transceiver’’, IEEE Transactions on Signal Processing, to be submitted. –R. K. Martin, K. Vanbleu, M. Ding, G. Ysebaert, M. Milosevic, B. L. Evans, M. Moonen, and C. R. Johnson, Jr., ``Multicarrier Equalization: Unification and Evaluation. Part I: Optimal Designs'', IEEE Transactions on Signal Processing, submitted. –R. K. Martin, K. Vanbleu, M. Ding, G. Ysebaert, M. Milosevic, B. L. Evans, M. Moonen, and C. R. Johnson, Jr., ``Multicarrier Equalization: Unification and Evaluation. Part II: Implementation Issues and Performance Comparisons'', IEEE Transactions on Signal Processing, submitted. –R. K. Martin, M. Ding, B. L. Evans, and C. R. Johnson, Jr, ``Infinite Length Results and Design Implications for Time-Domain Equalizers'', IEEE Trans. on Signal Processing, vol. 52, no. 1, pp , Jan –R. K. Martin, M. Ding, B. L. Evans, and C. R. Johnson, Jr, ``Efficient Channel Shortening Equalizer Design '', EURASIP Journal on Applied Signal Processing, vol. 2003, no. 13, pp , Dec. 1, –B. Farhang-Boroujeny and M. Ding, ``Design Methods for Time Domain Equalizer in DMT Transceivers'', IEEE Transactions on Communications, vol. 49 Issue: 3, pp , March 2001.

35 Publications in DMT Conference Papers –M. Ding, Z. Shen, B. L. Evans, ``An Achievable Performance Bound for Discrete Multitone Systems” Proc. IEEE Globecom Conf., Nov Dec. 3, 2004, Dallas, USA, submitted. –M. Ding, B. L. Evans, R. K. Martin, and C. R. Johnson, Jr, ``Minimum Intersymbol Interference Methods for Time Domain Equalizer Design'', Proc. IEEE Globecom Conf., Dec , vol. 4, pp , San Francisco, CA, USA. –R. K. Martin, C. R. Johnson, Jr, M. Ding, and B. L. Evans, ``Infinite Length Results for Channel Shortening Equalizers '', Proc. IEEE Int. Work. on Signal Processing Advances in Wireless Communications, June 15-18, 2003, Rome, Italy, accepted for publication. –R. K. Martin, C. R. Johnson, Jr, M. Ding, and B. L. Evans, ``Exploiting Symmetry in Channel Shortening Equalizers '', Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, April 6-10, 2003, vol. V, pp , Hong Kong, China. –M. Ding, A. J. Redfern, and B. L. Evans, ``A Dual-path TEQ Structure for DMT- ADSL Systems'', Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, May 13-17, 2002, vol. III, pp , Orlando, FL.

36 Backup Slides

37 Overview of ADSL Technology

38 Bi-directional Transmission in ADSL ADSL modems divide the available bandwidth in one of two ways -- Frequency Division Multiplexing (FDM) or Echo Cancellation. –FDM assigns one band for upstream data and another band for downstream data. –Echo Cancellation assigns the upstream band to over-lap the downstream, and separates the two by means of local echo cancellation.

39 ADSL Specifications T1E1.4 group developed ANSI Standard T June 1999, ITU-T SG 15 approves G (G.dmt) standard for full rate ADSL –Tone: subchannel –Value format: Downstream/Upstream –BER: Bit Error Rate itemvalueitemvalue No. of Tones256/32FFT size (N)512/64 CP Length ( ) 32/4Tone Width4.3 KHz Symbol Rate4 kHzSampling Rate2.208 MHz Target BER10 -7 SNR Gap (  ) 9.8 dB

40 Conventional Channel Shortening Methods Design single finite impulse response (FIR) filter to convolve with the channel such that the combined impulse response has only + 1 non-zero values This filter is called time domain equalizer (TEQ) Major TEQ design methods implemented in real-time fixed-point DSP –Minimum Mean-Squared Error design (MMSE) [Stanford 1992] –Maximum Shortening SNR design (MSSNR) [Tellabs 1997] –Minimum Intersymbol Interference design (Min-ISI [UT 1999] TransmitterChannel Receiver TEQ Information source Information sink

41 MBR TEQ Designs [Arslan, Evans & Kiaei, 2000] A subchannel SNR definition: Maximize nonlinear function to obtain the optimal TEQ

42 Pertone Equalizer [Acker, Leus, Moonen, van de Wiel & Pollet, 2001] Output of conventional equalizer structure for tone i Z i = D i row i (Q N ) R w D i is the complex value of one-tap FEQ for tone i Q N is the N  N complex-valued DFT matrix R is an N  T real-valued Toeplitz matrix of received samples w is a T  1 column vector of real-valued TEQ taps Rearrange computation of output for tone i Z i = D i row i (Q N ) R w = row i (Q N R) ( w D i ) A multi-tap FEQ for tone i combines TEQ and FEQ operations. The output is Z i = row i (Q N R) w i

43 Performance Comparison for TEQ vs Pertone The performance gap for any single tone is not universally wide. In tones associated with higher SNR, the improvement of per tone tends to be significant. For other tones, the improvement is insignificant.

44 Min-ISI: Revisited Min-ISI method minimizes the ratio of a weighted sum of the ISI power over the sum of desired signal power within a target window. Solution under the condition that Y is invertible A practical solution using Cholesky decomposition under a stronger condition: Y is positive definite q min is the eigenvector corresponding to minimum eigenvalue of C

45 Under the condition Y is not invertible, but X is invertible: The optimum Min-ISI TEQ is the eigenvector corresponding to the maximum eigenvalue of Practical Solution: –Use Power Method to iteratively compute the dominant eigenvalue and eigenvector of Alternative Solution of Min-ISI

46 Delay Optimization in Min-ISI design Min-ISI needs to perform delay optimization to find the optimum transmission delay  to maximizes the bit rate performance. Exhaustive searching over all possible  is required since no other approaches available. For each , we should solve the Min-ISI problem to find the optimum TEQ. To save the computation cost: –A fast algorithm to implement matrix multiplication [Wu, Arslan, & Evans 2000] –An efficient algorithm to minimize the redundant computations between successive  s. [Martin, Ding, Evans & Johnson 2003]

47 Matrices Definitions

48 Invertibility of X is obviously a rank 1 matrix. Conclusion: X is invertible if and only if all  i s are non-zero.

49 Goertzel Filters The N-point DFT of a length N sequence x(l): Define Noticed A recursive DFT computation scheme:

50 More definitions

51 Second Order Conditions of J Hessian All  i s are non-negative is positive-semidefinite.

52 Constrained Minimization of Iterative Min-ISI Use the Lagrange multipliers Iterative updates: where Noted here X is Hermitian and Y is symmetric.

53 Optimum Complex Filter Bank Solution The cost function is Take conjugate derivative of the cost function and equate to zero: The optimum solution is

54 MSSNR TEQ Design Maximum Shortening SNR (MSSNR) TEQ: Choose w to minimize energy outside window of desired length Design problem: Disadvantages: –Doesn’t consider noise –Doesn’t maximize subchannel SNR –Longer TEQ start killing subcarriers h + w xkxk ykyk rkrk nknk

55 Min-ISI TEQ Design Generalize MSSNR with frequency weighting Y is the same matrix as B in MSSNR design Convert to a constrained minimization problem: Optimum Solution is generalized eigenvector of matrix pencil (X,Y) corresponding to the minimum eigenvalue. In practice we need Cholesky decomposition to solve it.

56 Per-tone Equalizers Move the TEQ operations to the frequency domain and combine with the FEQ to obtain a multitap FEQ for each subchannel Sliding N-Point FFT ( L w -frame ) N+ z -1 y N + L w – 1 channels W 1,1 W 1,0 W 1,2 W 1,Lw- 1 W N/2,0 W N/2,1 W N/2,2 W N/2,Lw-1 FEQ is a linear combiner of up to N/2 L w -tap FEQs N-FFT TEQ FEQ i y zizi z1z1 z N/2 TEQ-FEQ Stucture Pertone Structure

57 Real Dual TEQ Implementations Make good ones better: –Path 1: TEQ optimizes some measure of performance over the entire bandwidth –Path 2: TEQ optimizes the subchannels within a preset window of frequencies (with highest SNRs)  Generally those subchannels have higher potential to be improved  Guarantee a higher bit rate than single TEQ case Make dead ones alive [Warke, Redfern, Sestok & Ali 2002]  In some cases, good subchannels are killed due to receiver operations (such as subcarriers close to the transition band) –TEQ 1 takes care of the transition band [subcarrier ] –TEQ 2 addresses the upstream bandwidth [subcarrier >40]

58 New SNR model after FEQ Define SNR at the i th FEQ output as : Transmitted power of i th subchannel : Transmitted complex-valued QAM symbol on i th subchannel : Output of i th FEQ, estimated QAM symbol on i th subchannel The practical ADSL systems use flat subchannel power allocation

59 Optimization based on the proposed SNR model A cost function based on the SNR model: We know how to solve the same Rayleigh Quotient minimization problem! Adaptive algorithm based on stochastic gradient method can be applied to each tone to design the filter bank!

60 Time-Domain Per Tone TEQ Filter Bank Find optimum TEQ to maximize SNR after FFT for every subchannel in use Pick the best one as data rate maximum TEQ Frequency Equalizers Goertzel Filters TEQ Filter Bank w1w1 w2w2 w N/2-1 G1G1 G2G2 G N/2-1 Received Signal R={r 1,…r N ) FEQ 1 FEQ 2 FEQ N/2-1 y1y1 y2y2 y N/2-1 Y1Y1 Y2Y2 Y N/2-1

61 Frobenius norm The Frobenius norm, is matrix norm of an matrix defined as the square root of the sum of the absolute squares of its elements, It is also equal to the square root of the matrix trace of

62 Methods with Frequency Control ADSL Transmission are partially bandwidth-occupied –Frequency Division Multiplexing –Unused subcarriers (Bad SNR or coexistence of other applications) Many methods are targeted to full bandwidth only –MMSE, MSSNR, MDS, etc –May not be optimum for partially bandwidth occupied case Methods with frequency control are suitable –Multi-tones partition Optimum design: Min-ISI, MBR, MGSNR, MDR, BM, etc Sub-optimum: Tones grouping –Tone-wise: Per-tone, Filter bank

63 MMSE TEQ Design MMSE TEQ minimizes the squared error between TEQ output and the output of a virtual target impulse response (TIR) filter. Disadvantages: –Doesn’t maximize subchannel SNR –Longer TEQ start killing subcarriers z-z- h + w b - xkxk ykyk ekek nknk + Poor bit rate performance

64 SNR Gap Channel capacity in bits per 2-dimensional symbol SNR gap: excessive SNR needed to achieve capacity

65 Cholesky DecompositionDecomposition If A is a symmetric (Hermitian) positive definite matrix, there exists a non-singular lower triangular L with positive real diagonal entries such that Cholesky Decomposition can be used to convert a generalized eigenvalue problem into a normal one

66 Alternative Structure The demodulated signal at the FEQ output Design freedom is limited in the TEQ-FEQ structure –All tones share same TEQ w –All taps of TEQ share same complex multiplier D i per tone Time domain filter bank plus FEQ Per-tone equalizer Complex time domain filter bank

67 Contribution #1 Infinite Length TEQ Results TIR for a MMSE TEQ has all zeros on the unit circle –A becomes a symmetric Toeplitz matrix –Eigenvector of A has all zeros on the unit circle TIR for a MMSE TEQ will be symmetric/skew symmetric –A also becomes a doubly symmetric matrix –Eigenvectors of A will be either symmetric or skew symmetric A MSSNR TEQ will be symmetric/skew symmetric – is doubly symmetric –Infinite length case: A converges to asymptotically Can exploit symmetry in TEQ designs

68 Dual Path TEQ performance Simulation Parameters TEQ length 17 AWGN PSD -140 dBm/Hz Crosstalk noise 24 ISDN FDM filter 5th order IIR Test Loop ANSI-13 Second path only optimizes tones Achieved Bit Rate Path 1: Mbps Dual Path: Mbps 4% improvement in bit rate

69 Carrier Serving Area Loops Served by a digital loop carrier, which multiplexes hundreds of analog lines into one high-speed digital trunk Limited to feet

70 Symmetric design Even length TEQ Odd length TEQ