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
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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
noise
transmitter channel + equalizer detector
hc(n)
Ideal channel frequency response
Amplitude response A( f ) is constant
Phase response ( f ) is linear in f
Channel distortions
Intersymbol interference (ISI)
0 1 1.0 1.0 0.75
0.75
0.5
1 1
Additive noise
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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
Assuming no
duplicate poles
ai: complex amplitude
Problem: given a noisy observation of
channel impulse response h(n)
Estimate
Least-squares method to compute {ai}
from
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MATRIX PENCIL METHOD
[Hua & Sarkar, 1990]
Matrix pencil of matrices A and B is the set
of all matrices AB,
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 = H0†H1 († 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]
SVD
truncation
LRHA
Hankel Hankel Hankel
low-rank low-rank approximately
low-rank
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)
SVD
truncation
LRHA
partition
steps 3-4 in MP method
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
Pole 1 at 0.8464
Pole 2 at 0.7146 Pole 3 at 0.2108
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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
channel frequency
response
subchannel
etc.
Frequency
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COMBAT ISI IN DMT SYSTEMS
Add cyclic prefix (CP) to eliminate ISI
CP CP
i th symbol (i+1) th symbol
samples N samples samples N samples
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, 1973][Chow & Cioffi, 1992][Al-Dhahir & Cioffi, 1996]
h w
z - b
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 #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
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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]
h w
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
MSE = 0.0019 with
For each , maximum SSNR method requires
Multiplications:
Additions:
Divisions:
Delay search
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CONTRIBUTION #3:
DIVIDE-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 #3:
DC-TEQ-MINIMIZATION (UTC)
Objective function
At ith iteration, minimize Ji over gi
Closed-form solution
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CONTRIBUTION #3:
DC-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 #3:
DC-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 #3:
DC-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
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
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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
TEQ: Time-domain Equalizer
UNC: Unit Norm Constraint
UTC: Unit Tap Constraint
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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 #1:
PROPOSED MP METHODS
Modified MP method 1 (MMP1)
partition
SVD SVD
truncation truncation
LRHA LRHA
Steps 3-4 in MP method
Noise may corrupt and to lose the
connection
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CONTRIBUTION #1:
PROPOSED MP METHODS
Modified MP method 2 (MMP2)
partition
SVD SVD
truncation truncation
Joint
LRHA
partition
Step 3-4 in MP method
SVD truncation may destroy the connection
between Y0 and Y1
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COMPUTER SIMULATION
Data model
where
K=2, N=25, L=17, 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
R L
Z0
C
Characteristic impedance of the line
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SSNR VS. DATA RATE
CSA DSL channel 1
SSNR = 40 dB
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