1. UT Austin 1 Biao Lu 1 WIRELINE CHANNEL ESTIMATION AND EQUALIZATION Ph.D. Defense Biao Lu Embedded Signal Processing Laboratory The University of Texas at Austin Committee Members Prof. Brian L. Evans Prof. Alan C. Bovik Prof. Joydeep Ghosh Prof. Risto Miikkulainen Dr. Lloyd D. Clark

2. UT Austin 2 Biao Lu 2 OUTLINE  Wireline channel equalization  Wireline channel estimation  Channel modeling  Matrix pencil methods  Contribution #1: modified matrix pencil methods for channel estimation  Discrete multitone modulation  Minimum mean squared error equalizer  Contribution #2: matrix pencil equalizer  Maximum shortening SNR equalizer  Contribution #3: fast implementation »Divide-and-conquer methods »Heuristic search  Summary and future research

3. UT Austin 3 Biao Lu 3 WIRELINE CHANNEL EQUALIZATION transmitterchannel equalizer detector noise + hc(n)hc(n)  Wireline digital communication system  Ideal channel frequency response  Amplitude response A( f ) is constant  Phase response  ( f ) is linear in f  Channel distortions  Intersymbol interference (ISI)  Additive noise 0 1 1.0 0.75 1.0 0.75 0.5 1

4. UT Austin 4 Biao Lu 4 COMBATTING ISI IN WIRELINE CHANNELS  Channel equalizer response H eq ( f ) compensates for channel distortion  Equalizers may compensate for  Frequency distortion: e.g. ripples  Nonlinear phase  Long impulse response  Channels may have  Spectral nulls  Nonlinear distortion, e.g. harmonic distortion  Goal: Design time-domain equalizers  Shorten channel impulse response  Reduce intersymbol interference

5. UT Austin 5 Biao Lu 5 OUTLINE  Wireline channel equalization  Wireline channel estimation  Channel modeling  Matrix pencil methods  Contribution #1: modified matrix pencil methods for channel estimation  Discrete multitone modulation  Minimum mean squared error equalizer  Contribution #2: matrix pencil equalizer  Maximum shortening SNR equalizer  Contribution #3: fast implementation »Divide-and-conquer methods »Heuristic search  Summary and future research

6. UT Austin 6 Biao Lu 6 WIRELINE CHANNEL ESTIMATION  Problem: Given N samples of the received signal, estimate channel impulse response  Training-based: transmitted signal known  Blind: transmitted signal unknown  Time-domain channel estimation methods  Least-squares [Crozier, Falconer & Mahmoud, 1996]  Singular value decomposition (SVD) [Barton & Tufts, 1989; Lindskog & Tidestav, 1999]  Frequency-domain channel estimation  Discrete Fourier transform [Tellambura, Parker & Barton, 1998; Chen & Mitra, 2000]  Discrete cosine transform [Sang & Yeh 1993; Merched & Sayed, 2000]

7. UT Austin 7 Biao Lu 7 WIRELINE CHANNEL ESTIMATION  Broadband channel impulse responses have long tails  Model channel as infinite impulse response (IIR) filter  Transfer function with K poles

8. UT Austin 8 Biao Lu 8 WIRELINE CHANNEL ESTIMATION  All-pole portion of an IIR filter  Problem: given a noisy observation of channel impulse response h(n)  Estimate  Least-squares method to compute {a i } from a i : complex amplitude Assuming no duplicate poles

9. UT Austin 9 Biao Lu 9 MATRIX PENCIL METHOD [Hua & Sarkar, 1990]  Matrix pencil of matrices A and B is the set of all matrices A  B,    Noise-free case: N samples of h(n)  L is the pencil parameter (K  L  N  K)  H, H 0 and H 1 are Hankel and low rank, where rank is K.

10. UT Austin 10 Biao Lu 10 MATRIX PENCIL METHOD [Hua & Sarkar, 1990]  Noise-free data 1. Form matrices H, H 0 and H 1 2. Calculate C = H 0 † H 1 ( † is pseudoinverse) 3. K non-zero eigenvalues of C are  Noisy data 1. Form matrices Y, Y 0 and Y 1 2. Calculate : rank-K SVD truncated pseudoinverse : rank-K SVD truncated approximation »v i and u i are left and right singular vectors »  i is i th largest singular value 3. Calculate 4. K non-zero eigenvalues of C are

11. UT Austin 11 Biao Lu 11 LOW-RANK HANKEL APPROXIMATION  Problem in noisy data case  Noise destroys rank deficiency  SVD truncation restores rank deficiency, but destroys Hankel structure  Low-rank Hankel approximation (LRHA) [Cadzow, Sun & Xu, 1988]  Replaces each matrix cross-diagonal with average of cross-diagonal elements  Restores low rank after SVD truncation  Iteratively apply SVD truncation and LRHA [Cadzow, Sun & Xu, 1988]  Modified Kumaresan-Tufts method (MKT) uses LRHA instead of SVD truncation [Razavilar, Yi & Liu, 1996] Hankel low-rank Hankel low-rank SVD truncation LRHA Hankel approximately low-rank

12. UT Austin 12 Biao Lu 12 CONTRIBUTION #1: PROPOSED MATRIX PENCIL METHODS  Modified MP methods 1 and 2 in dissertation  Modified MP method 3 (MMP3)  Maintain relationship between partitioned matrices SVD truncation steps 3-4 in MP method LRHA partition

13. UT Austin 13 Biao Lu 13 COMPUTER SIMULATION  Channel [Al-Dhahir, Sayed & Cioffi, 1997]  Zeros at 1.0275 and  0.4921  Poles at 0.8464, 0.7146, and 0.2108  Parameters for matrix pencil methods  K = 3, N = 25, L = 17  Additive Gaussian noise with variance   SNR varied from 0 to 30 dB at 2 dB steps  500 runs for each SNR value  Performance measure

14. UT Austin 14 Biao Lu 14 COMPUTER SIMULATION Pole 1 at 0.8464 Pole 2 at 0.7146 Pole 3 at 0.2108

15. UT Austin 15 Biao Lu 15 OUTLINE  Wireline channel equalization  Wireline channel estimation  Channel modeling  Matrix pencil methods  Contribution #1: modified matrix pencil methods for channel estimation  Discrete multitone modulation  Minimum mean squared error equalizer  Contribution #2: matrix pencil equalizer  Maximum shortening SNR equalizer  Contribution #3: fast implementation »Divide-and-conquer methods »Heuristic search  Summary and future research

16. UT Austin 16 Biao Lu 16 MULTICARRIER MODULATION  Divide frequency band into subchannels  Each subchannel is ideally ISI free  Based on fast Fourier transform (FFT)  Orthogonal frequency division multiplexing  Discrete multitone (DMT) modulation  ADSL standards use DMT: ANSI 1.413, G.DMT and G.lite etc. Magnitude Frequency channel frequency response subchannel

17. UT Austin 17 Biao Lu 17 COMBAT ISI IN DMT SYSTEMS  Add cyclic prefix (CP) to eliminate ISI  Problem: Reduces throughput by factor of  ADSL standards use time-domain equalizer (TEQ) to shorten effective channel to ( +1) samples  Goal: TEQ design during ADSL initialization  Low implementation complexity  “Acceptable” performance CP samples i th symbol N samples (i+1) th symbol N samples samples

18. UT Austin 18 Biao Lu 18 MINIMUM MSE METHOD  MMSE method [Falconer & Magee, 1973][Chow & Cioffi, 1992][Al-Dhahir & Cioffi, 1996]  Constraints to avoid trivial solution  Unit tap constraint:  Unit norm constraint:  ADSL parameters: L h = 512, N w = 21, = 32,   L h + N w - - 2  Computational cost for a candidate delay   Inversion of N w  N w matrix  Eigenvalue decomposition of N w  N w matrix (or power method) hw z -  b

19. UT Austin 19 Biao Lu 19 CONTRIBUTION #2: MATRIX PENCIL TEQ  From MMSE TEQ  MMSE TEQ cancels poles  Matrix pencil (MP) TEQ  Estimate pole locations using a matrix pencil method on »Channel impulse response »Received signal — blind channel shortening  Set TEQ zeros at pole locations

20. UT Austin 20 Biao Lu 20 MAXIMUM SHORTENING SNR METHOD  Maximum shortening SNR (SSNR) method: minimize energy outside a window of ( +1) samples [Melsa, Younce & Rohrs, 1996]  Simplify solution by constraining  Computational cost at each candidate delay   Inversion of N w  N w matrix  Cholesky decomposition of N w  N w matrix  Eigenvalue decomposition of N w  N w matrix (or power method) hw 

21. UT Austin 21 Biao Lu 21 MOTIVATION  MMSE method minimizes MSE both inside and outside window of ( +1) samples  For each , maximum SSNR method requires  Multiplications:  Additions:  Divisions:  Delay search MSE = 0.0019 with

22. UT Austin 22 Biao Lu 22 CONTRIBUTION #3: DIVIDE-AND-CONQUER TEQ  Divide N w TEQ taps into (N w - 1) two-tap filters in cascade  The i th two-tap filter is initialized as  Unit tap constraint (UTC)  Unit norm constraint (UNC)  Calculate g i or  i using a greedy approach  Minimize : Divide-and-conquer TEQ minimization  Minimize energy in h wall : Divide-and conquer TEQ cancellation  Convolve two-tap filters to obtain TEQ

23. UT Austin 23 Biao Lu 23 CONTRIBUTION #3: DC-TEQ-MINIMIZATION (UTC)  Objective function  At i th iteration, minimize J i over g i  Closed-form solution

24. UT Austin 24 Biao Lu 24 CONTRIBUTION #3: DC-TEQ-CANCELLATION (UTC)  Objective function to cancel energy in h wall  At i th iteration, minimize J i over g i  Closed-form solution

25. UT Austin 25 Biao Lu 25 CONTRIBUTION #3: DC-TEQ-MINIMIZATION (UNC)  Each two-tap filter  At i th iteration, minimize J i over  i  Calculate  i in the same way as g i for DC- TEQ-minimization (UTC)

26. UT Austin 26 Biao Lu 26 CONTRIBUTION #3: DC-TEQ-CANCELLATION (UNC)  Each two-tap filter  At i th iteration, minimize J i over  i  Closed-form solution

27. UT Austin 27 Biao Lu 27 COMPUTATIONAL COMPLEXITY  Computational complexity for each candidate  for G.DMT ADSL L h = 512, = 32, N w = 21  Divide-and-conquer TEQ design methods vs. maximum SSNR method  Reduce multiplications and additions by a factor of 2 or 3  Reduce divisions by a factor of 7 or 22  Reduce memory by a factor of 3  Avoids matrix inversion, and eigenvalue and Cholesky decompositions

28. UT Austin 28 Biao Lu 28 KNOWN CHANNEL Dedicated data channel Carrier-Serving-Area (CSA) ADSL channel 1

29. UT Austin 29 Biao Lu 29 UNKNOWN CHANNEL Dedicated data channel Carrier-Serving-Area (CSA) ADSL channel 1

30. UT Austin 30 Biao Lu 30 HEURISTIC SEARCH DELAY   Estimate optimal delay  before computing TEQ taps  Computational cost for each   Multiplications:  Additions:  Divisions: 1  Reduce computational complexity of TEQ design for ADSL by a factor of 500 over exhaustive search

31. UT Austin 31 Biao Lu 31 HEURISTIC SEARCH  Maximum SSNR method for CSA DSL channel 1 DC-TEQ-cancellation (UTC) for CSA DSL channel 1

32. UT Austin 32 Biao Lu 32 SUMMARY  Channel estimation by matrix pencil methods  New methods to estimate channel poles by applying low-rank Hankel approximation to multiple matrices [Lu, Wei, Evans & Bovik, 1998]  Time-domain equalizer  channel shortening  Matrix pencil TEQ [Lu, Clark, Arslan & Evans, 2000] »From known channel impulse response »From received signal: blind channel shortening  Reduce computational cost [Lu, Clark, Arslan & Evans, 2000] »Divide-and-conquer TEQ minimization method »Divide-and-conquer TEQ cancellation method »Heuristic search for delay  Other contributions: cascade two neural networks to form a channel equalizer [Lu & Evans, 1999]  Multilayer perceptron to suppress noise  Radial basis function network to equalize the channel

33. UT Austin 33 Biao Lu 33 FUTURE RESEARCH  Discrete multitone systems  Maximize channel capacity »Optimize channel capacity at TEQ output »Jointly optimize a TEQ with other blocks  Frequency–domain equalizers  TEQ to shorten time-varying channels »Fast and accurate channel estimation »Convert time-varying channels to additive white Gaussian noise channel  Reduce computational complexity  Fast training for neural networks  Parallelize matrix pencil method

34. UT Austin 34 Biao Lu 34 ABBREVIATIONS  ADSL: Asymmetrical Digital Subscriber Line  CP: Cyclic Prefix  CSA: Carrier-Serving Area  DC: Divide-and-Conquer  DMT: Discrete Multitone  DSL Digital Subscriber Line  FFT: Fast Fourier Transform  IIR: Infinite Impulse Response  ISI: Intersymbol Interference  LRHA: Low-Rank Hankel Approximation  MKT: Modified Kumaresan-Tufts  MLP: Multilayer Perceptron  MMP: Modified Matrix Pencil  MMSE: Minimum Mean Squared Error  MP: Matrix Pencil  RBF: Radial Basis Function  SNR: Signal-to-Noise Ratio  SSNR: Shortening Signal-to-Noise Ratio  SVD: Singular Value Decomposition  TEQ: Time-domain Equalizer  UNC: Unit Norm Constraint  UTC: Unit Tap Constraint

35. UT Austin 35 Biao Lu 35 NEURAL NETWORK EQUALIZERS  Equalization is a classification problem  Feedforward neural network equalizers  Multilayer perceptron (MLP) equalizer »Has to be trained several times »Reduces additive uncorrelated noise  Radial basis function (RBF) equalizer »The number of hidden units increases exponentially with the number of inputs »Adapts to local patterns in data  Cascade MLP and RBF networks  Use MLP to suppress noise  Use RBF to perform equalization

36. UT Austin 36 Biao Lu 36 PROBLEMS FROM NN EQUALIZER  Computational cost: training NN takes time  Number of symbols used in training [Mulgrew, 1996] where M : number of constellations L h : length of channel impulse response N in : number of neurons in the input layer e.g., M = 4, L h = 8, N in = 3 means that number of symbols = 1,048,576  Channel length is unknown  Goals  Estimate channel impulse response — L h can be known  Shorten channel impulse response to be less than L h

37. UT Austin 37 Biao Lu 37 BACKUP INFORMATION  Derivation from H ap (z) to h ap (n)

38. UT Austin 38 Biao Lu 38 KUMARESAN-TUFTS (KT) AND MODIFIED KT METHOD  KT-method: noisy data 1. Form matrix 2. Solve 3. Form 4. Calculate zeros of B(z) 5. All the zeros outside unit circle gives  Modified KT (MKT) method: apply LRHA to matrix A before step 2

39. UT Austin 39 Biao Lu 39 COMPARISON BETWEEN MMP3 AND MKT  Common procedures  Iterative LRHA  SVD-truncated pseudoinverse  MMP3 only  Matrix partition  Eigenvalue decomposition  MKT only  Solve equation

40. UT Austin 40 Biao Lu 40 CONTRIBUTION #1: PROPOSED MP METHODS  Modified MP method 1 (MMP1)  Noise may corrupt and to lose the connection partition Steps 3-4 in MP method LRHA SVD truncation LRHA SVD truncation

41. UT Austin 41 Biao Lu 41 CONTRIBUTION #1: PROPOSED MP METHODS  Modified MP method 2 (MMP2)  SVD truncation may destroy the connection between Y 0 and Y 1 SVD truncation Joint LRHA partition SVD truncation partition Step 3-4 in MP method

42. UT Austin 42 Biao Lu 42 COMPUTER SIMULATION  Data model where  K=2, N=25, L=17, A 1 = A 2 = 1  p i = -d i + j2  f i, i = 1, 2 where d 1 = 0.2 and d 2 = 0.1, f 1 = 0.42 and f 2 = 0.52  w(n) is complex zero-mean white Gaussian noise with variance  2  Signal-to-noise ratio (SNR)  SNR varied from 5 to 25 dB at 2 dB step  500 runs for each SNR value  Performance measure

43. UT Austin 43 Biao Lu 43 ESTIMATION OF DAMPING FACTORS  d 1 = 0.2  d 2 = 0.1

44. UT Austin 44 Biao Lu 44 ESTIMATION OF FREQUENCIES  f 1 = 0.42  f 2 = 0.52

45. UT Austin 45 Biao Lu 45 PREVIOUS WORK  Maximum channel capacity  Based on geometric SNR »Nonlinear optimization techniques [Al-Dhahir & Cioffi, 1996, 1997] »Projection onto convex sets [Lashkarian & Kiaei, 1999]  Based on model of signal, noise, ISI paths [Arslan, Evans & Kiaei, 2000] »Equivalent to maximum SSNR when input signal power distribution is constant over frequency

46. UT Austin 46 Biao Lu 46 COMPUTER SIMULATION  Simulation parameters

47. UT Austin 47 Biao Lu 47 FREQUENCY RESPONSE OF A TRANSMISSION LINE  Model as a RC circuit  Characteristic impedance of the line RL C Z0Z0

48. UT Austin 48 Biao Lu 48 SSNR VS. DATA RATE  CSA DSL channel 1 SSNR = 40 dB