WIRELINE CHANNEL ESTIMATION AND EQUALIZATION

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WIRELINE CHANNEL ESTIMATION ANDEQUALIZATION

• 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

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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

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WIRELINE CHANNEL EQUALIZATION

• 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

noise
transmitter
channel
detector
equalizer

hc(n)
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COMBATTING ISI IN WIRELINE CHANNELS

• Channel equalizer response Heq( 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

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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

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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

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WIRELINE CHANNEL ESTIMATION

• Broadband channel impulse responses have long
  tails
• Model channel as infinite impulse response (IIR)
  filter
• Transfer function with K poles
• 
  
• 
  

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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 ai from

Assuming no duplicate poles
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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, H0 and H1 are Hankel and low rank, where rank
  is K.

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MATRIX PENCIL METHOD Hua Sarkar, 1990

• Noise-free data
• 1. Form matrices H, H0 and H1
• 2. Calculate C H0H1 ( is pseudoinverse)
• 3. K non-zero eigenvalues of C are
• Noisy data
• 1. Form matrices Y, Y0 and Y1
• 2. Calculate
• rank-K SVD truncated pseudoinverse
• rank-K SVD truncated approximation
• vi and ui are left and right singular vectors
• ?i is ith largest singular value
• 3. Calculate
• 4. K non-zero eigenvalues of C are

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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

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CONTRIBUTION 1 PROPOSED MATRIX PENCIL METHODS

• Modified MP methods 1 and 2 in dissertation
• Modified MP method 3 (MMP3)
• Maintain relationship between partitioned
  matrices

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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

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COMPUTER SIMULATION
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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

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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

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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

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MINIMUM MSE METHOD

• MMSE method
• Falconer Magee, 1973Chow Cioffi,
  1992Al-Dhahir Cioffi, 1996
• Constraints to avoid trivial solution
• Unit tap constraint
• Unit norm constraint
• ADSL parameters Lh 512, Nw 21,
• ? 32, ? ? Lh Nw - ? - 2
• Computational cost for a candidate delay ?
• Inversion of Nw ? Nw matrix
• Eigenvalue decomposition of Nw ? Nw matrix (or
  power method)

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CONTRIBUTION 2MATRIX 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

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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 Nw ? Nw matrix
• Cholesky decomposition of Nw ? Nw matrix
• Eigenvalue decomposition of Nw ? Nw matrix (or
  power method)

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MOTIVATION

• MMSE method minimizes MSE both inside and outside
  window of (?1) samples
• For each ?, maximum SSNR method requires
• Multiplications
• Additions
• Divisions
• Delay search

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CONTRIBUTION 3DIVIDE-AND-CONQUER TEQ

• Divide Nw TEQ taps into (Nw - 1) two-tap filters
  in cascade
• The ith two-tap filter is initialized as
• Unit tap constraint (UTC)
• Unit norm constraint (UNC)
• Calculate gi or ?i using a greedy approach
• Minimize Divide-and-conquer TEQ
  minimization
• Minimize energy in hwall Divide-and conquer TEQ
  cancellation
• Convolve two-tap filters to obtain TEQ

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CONTRIBUTION 3DC-TEQ-MINIMIZATION (UTC)

• Objective function
• At ith iteration, minimize Ji over gi
• Closed-form solution

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CONTRIBUTION 3DC-TEQ-CANCELLATION (UTC)

• Objective function to cancel energy in hwall
• At ith iteration, minimize Ji over gi
• Closed-form solution

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CONTRIBUTION 3DC-TEQ-MINIMIZATION (UNC)

• Each two-tap filter
• At ith iteration, minimize Ji over ?i
• Calculate ?i in the same way as gi for
  DC-TEQ-minimization (UTC)

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CONTRIBUTION 3DC-TEQ-CANCELLATION (UNC)

• Each two-tap filter
• At ith iteration, minimize Ji over ?i
• Closed-form solution

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COMPUTATIONAL COMPLEXITY

• Computational complexity for each candidate ? for
  G.DMT ADSL
• Lh 512, ? 32, Nw 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

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KNOWN CHANNEL
Dedicated data channel
Carrier-Serving-Area (CSA) ADSL channel 1
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UNKNOWN CHANNEL
Dedicated data channel
Carrier-Serving-Area (CSA) ADSL channel 1
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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

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HEURISTIC SEARCH ?
Maximum SSNR method for CSA DSL channel 1
DC-TEQ-cancellation (UTC) for CSA DSL channel 1
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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

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FUTURE RESEARCH

• Discrete multitone systems
• Maximize channel capacity
• Optimize channel capacity at TEQ output
• Jointly optimize a TEQ with other blocks
• Frequencydomain 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

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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

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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

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PROBLEMS FROM NN EQUALIZER

• Computational cost training NN takes time
• Number of symbols used in training Mulgrew,
  1996
• where
• M number of constellations
• Lh length of channel impulse response
• Nin number of neurons in the input layer
• e.g., M 4, Lh 8, Nin 3 means that
• number of symbols 1,048,576
• Channel length is unknown
• Goals
• Estimate channel impulse response
• Lh can be known
• Shorten channel impulse response to be less than
  Lh

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BACKUP INFORMATION

• Derivation from Hap(z) to hap(n)

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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

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COMPARISON BETWEEN MMP3 AND MKT

• Common procedures
• Iterative LRHA
• SVD-truncated pseudoinverse
• MMP3 only
• Matrix partition
• Eigenvalue decomposition
• MKT only
• Solve equation

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CONTRIBUTION 1PROPOSED MP METHODS

• Modified MP method 1 (MMP1)
• Noise may corrupt and to lose the
  connection

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CONTRIBUTION 1PROPOSED MP METHODS

• Modified MP method 2 (MMP2)
• SVD truncation may destroy the connection between
  Y0 and Y1

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COMPUTER SIMULATION

• Data model
• where
• K2, N25, L17, A1 A2 1
• pi -di j2? fi , i 1, 2
• where d1 0.2 and d2 0.1,
• f1 0.42 and f2 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

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ESTIMATION OF DAMPING FACTORS

• d1 0.2
• d2 0.1

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ESTIMATION OF FREQUENCIES

• f1 0.42
• f2 0.52

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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

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COMPUTER SIMULATION

• Simulation parameters

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FREQUENCY RESPONSE OF A TRANSMISSION LINE

• Model as a RC circuit
• Characteristic impedance of the line

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SSNR VS. DATA RATE

• CSA DSL channel 1