UT Austin1

Biao Lu 1

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

UT Austin2

Biao Lu 2

OUTLINE

Wireline channel equalization Wireline channel estimation

Channel modelingMatrix pencil methodsContribution #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

UT Austin3

Biao Lu 3

WIRELINE CHANNEL EQUALIZATION

fjc efAfH

transmitter channel equalizer detector

noise

nm ns nw ny ns nm

+hc(n)

Wireline digital communication system

Ideal channel frequency response Amplitude response A( f ) is constantPhase response ( f ) is linear in f

Channel distortions Intersymbol interference (ISI)

Additive noise

0 1 1.00.75

1.0 0.75 0.5

1 1

UT Austin4

Biao Lu 4

COMBATTING ISI IN WIRELINE CHANNELS

Channel equalizer response Heq( f ) compensates for channel distortion

Equalizers may compensate forFrequency distortion: e.g. ripplesNonlinear phaseLong impulse response

Channels may haveSpectral nullsNonlinear distortion, e.g. harmonic

distortion Goal: Design time-domain equalizers

Shorten channel impulse responseReduce intersymbol interference

1 ceq fHfH

UT Austin5

Biao Lu 5

OUTLINE

Wireline channel equalization Wireline channel estimation

Channel modelingMatrix pencil methodsContribution #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

UT Austin6

Biao Lu 6

WIRELINE CHANNEL ESTIMATION

Problem: Given N samples of the received signal, estimate channel impulse response Training-based: transmitted signal knownBlind: transmitted signal unknown

Time-domain channel estimation methodsLeast-squares [Crozier, Falconer & Mahmoud, 1996]

Singular value decomposition (SVD) [Barton & Tufts, 1989; Lindskog & Tidestav, 1999]

Frequency-domain channel estimationDiscrete Fourier transform [Tellambura, Parker & Barton, 1998; Chen & Mitra, 2000]

Discrete cosine transform [Sang & Yeh 1993; Merched & Sayed, 2000]

UT Austin7

Biao Lu 7

WIRELINE CHANNEL ESTIMATION

Broadband channel impulse responses have long tails

Model channel as infinite impulse response (IIR) filterTransfer function with K poles

iii fjdp 2

K

i

p ze

zB

zA

zBzH

i

1

11

UT Austin8

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 {ai}

from

1,,1,0 ),()()( Nnnwnhny

nueanhK

i

npi

i

1

ap

K

i

p zezH

i

1

1ap

1

1

ai: complex amplitude

Kie ip ,,2 ,1,

Kie ip ,,2 ,1,

Assuming no duplicate poles

UT Austin9

Biao Lu 9

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.

)1()(1211

21432

1321

1210

LLNNNLNLNLN

LL

LL

LL

hhhhh

hhhhhhhhhh

hhhhh

H

1H

0H

UT Austin10

Biao Lu 10

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

K

i

Hiii

Hii

K

i i 11

† ˆ and 1ˆ vuYuvY 10

†ˆ0Y

1Y

Kie ip ,,2 ,1 ,

10 YYC ˆˆ †

Kie ip ,,2 ,1 ,

UT Austin11

Biao Lu 11

LOW-RANK HANKEL APPROXIMATION

Problem in noisy data caseNoise destroys rank deficiencySVD 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

AAA

UT Austin12

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

Y

SVD truncation

0Y 1Y

steps 3-4 in MP method

YLRHA

Ypartition

UT Austin13

Biao Lu 13

COMPUTER SIMULATION

321

21

1275.02757.03501.11

5056.05354.01

zzz

zz

zA

zBzH

Channel [Al-Dhahir, Sayed & Cioffi, 1997]

Zeros at 1.0275 and 0.4921Poles at 0.8464, 0.7146, and 0.2108

Parameters for matrix pencil methodsK = 3, N = 25, L = 17

Additive Gaussian noise with variance

SNR varied from 0 to 30 dB at 2 dB steps500 runs for each SNR value

Performance measure

210 2

1log 10SNR

2

10 original estimated MSE ,MSE

1log10 E

UT Austin14

Biao Lu 14

COMPUTER SIMULATION

Pole 1 at 0.8464

Pole 2 at 0.7146 Pole 3 at 0.2108

UT Austin15

Biao Lu 15

OUTLINE

Wireline channel equalization Wireline channel estimation

Channel modelingMatrix pencil methodsContribution #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

UT Austin16

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.

Mag

nitu

de

Frequency

channel frequency response

subchannel

UT Austin17

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 initializationLow implementation complexity“Acceptable” performance

vN

N

CP CP

samples i th symbolN samples

(i+1) th symbolN samples samples

UT Austin18

Biao Lu 18

MINIMUM MSE METHOD

MMSE method [Falconer & Magee, 1973][Chow & Cioffi, 1992][Al-Dhahir & Cioffi, 1996]

Constraints to avoid trivial solutionUnit 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)

h w

z - b

kxkn

ky ke

2 MSE

)()1()0(

)()1(

kT

kT

T

Tw

E

bbb

Nww

xbyw

b

w

, ,1 ,0 ,1)( iib

1or 1 wb

bRRRRb

RwRb

yxyyxyxx

yyxy

MSE 1

T

TT

UT Austin19

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

zW

zBzH

UT Austin20

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 Nw Nw matrix Cholesky decomposition of Nw Nw matrix Eigenvalue decomposition of Nw Nw

matrix (or power method)

h wkx

knky

Aww

Bww

BwwwHHwhh

AwwwHHwhh

T

T

TTTT

TTTT

10

winwinwinwin

wallwallwallwall

log10SSNR

1BwwT

wallh

winh

UT Austin21

Biao Lu 21

MOTIVATION

MMSE method minimizes MSE both inside and outside window of (+1) samples

For each , maximum SSNR method requiresMultiplications:

Additions:

Divisions: Delay search

32

3

25

2

5

6

7wwwh NNNL

32

3

25

2

3

6

5wwwh NNNL

2wN

499020 wh NL

T9080.0,4189.0

dB 27.13SSNR

1067.6MSE 9wall

w T6913.0,7226.0

dB 90.29SSNR

1083.6MSE 11wall

w

MSE = 0.0019 with 1w

UT Austin22

Biao Lu 22

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 approachMinimize : Divide-and-conquer TEQ

minimizationMinimize energy in hwall: Divide-and

conquer TEQ cancellation Convolve two-tap filters to obtain TEQ

ii g

1w

SSNR

1

i

ii

cos

sinw

UT Austin23

Biao Lu 23

CONTRIBUTION #3:DC-TEQ-MINIMIZATION (UTC)

Objective function

At ith iteration, minimize Ji over gi

Closed-form solution

Bww

AwwT

T

J SSNR

1

2

,3,2,1

2,3,2,1

,3,2

,2,1

,3,2

,2,1

2

2

11

11

iiiii

iiiii

iii

iii

iii

iii

iTi

iTi

i gbgbb

gagaa

gbb

bbg

gaa

aag

J

Bww

Aww

iiiiiiiiiiii

iiiiiiii

iiiii

babababababaD

baba

D

baba

babag

,2,1,1,2,3,2,2,32

,3,1,1,3

,3,2,2,3,3,2,2,3

,3,1,1,32,1

4

22

UT Austin24

Biao Lu 24

CONTRIBUTION #3:DC-TEQ-CANCELLATION (UTC)

Objective function to cancel energy in hwall

At ith iteration, minimize Ji over gi

Closed-form solution

wallwallhhTJ

1~

2

11wallwall

,,2 ,,,2 ,1

,1~~~~

ih

Skiii

Ti

LS

khgkhJ

hh

Ski

Skii

ikh

khkhg

)1(~

)(~

)1(~

21

11

UT Austin25

Biao Lu 25

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)

ii

iii

i

ii

1

sinsincos

1sin

cos

sinw

iii

iii

iii

iii

ii

ii

iiii

ii

ii

iiii

iTi

iTi

i

bb

bb

aa

aa

bb

bb

aa

aa

J

11

11

1sin1sin

1sin1sin

,3,2

,2,1

,3,2

,2,1

,3,2

,2,1

,3,2

,2,1

Bww

Aww

UT Austin26

Biao Lu 26

CONTRIBUTION #3:DC-TEQ-CANCELLATION (UNC)

Each two-tap filter

At ith iteration, minimize Ji over i

Closed-form solution

i

ii

cos

sinw

1~

2

11wallwall

,,2 ,,,2 ,1

,cos1~

sin~~~

ih

Skiiii

Ti

LS

khkhJ

hh

Skii

Skii

ii

khkhbkhkha

ba

a

ba

a

)(~

)1(~

,1~~

415.0cos,

415.0sin

1121

21

22

2

22

2

UT Austin27

Biao Lu 27

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 methodReduce multiplications and additions by a

factor of 2 or 3Reduce divisions by a factor of 7 or 22Reduce memory by a factor of 3Avoids matrix inversion, and eigenvalue

and Cholesky decompositions

Method Memory(words)

MaximumSSNR

120379 118552 441 1899

DC-TEQ-mini-mization(UTC)

53240 52980 60 563

DC-TEQ-can-cellation(UNC)

42280 42160 20 555

DC-TEQ-can-cellation(UTC)

41000 40880 20 554

UT Austin28

Biao Lu 28

KNOWN CHANNEL

Dedicated data channel

Carrier-Serving-Area (CSA) ADSL channel 1

UT Austin29

Biao Lu 29

UNKNOWN CHANNEL

Dedicated data channel

Carrier-Serving-Area (CSA) ADSL channel 1

UT Austin30

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

h

h

original of windowa outsideenergy

original of windowa insideenergy maxargratio

2hLhL

UT Austin31

Biao Lu 31

HEURISTIC SEARCH

Maximum SSNR method for CSA DSL channel 1

DC-TEQ-cancellation (UTC) for CSA DSL channel 1

UT Austin32

Biao Lu 32

SUMMARY

Channel estimation by matrix pencil methodsNew methods to estimate channel poles by

applying low-rank Hankel approximation to multiple matrices [Lu, Wei, Evans & Bovik, 1998]

Time-domain equalizer channel shorteningMatrix 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

UT Austin33

Biao Lu 33

FUTURE RESEARCH

Discrete multitone systemsMaximize channel capacity

» Optimize channel capacity at TEQ output

» Jointly optimize a TEQ with other blocks

Frequency–domain equalizersTEQ to shorten time-varying channels

» Fast and accurate channel estimation

» Convert time-varying channels to additive white Gaussian noise channel

Reduce computational complexityFast training for neural networksParallelize matrix pencil method

UT Austin34

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

UT Austin35

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 networksUse MLP to suppress noiseUse RBF to perform equalization

UT Austin36

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

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

GoalsEstimate channel impulse response —

Lh can be knownShorten channel impulse response to be

less than Lh

1 inh NLM

UT Austin37

Biao Lu 37

BACKUP INFORMATION

Derivation from Hap(z) to hap(n)

K

i

npi

K

i p

iK

i

p

i

ii

eanh

ze

a

zezH

1ap

1 1

1

1ap

11

1

UT Austin38

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

1

1

0

2

1

11

132

21

LNy

y

y

Lb

b

b

NyLNyLNy

Lyyy

Lyyy

A

Liib ,,2 ,1 ,

LzLbzbzbzB 21 211

Lie ip , 2, ,1 ,*

UT Austin39

Biao Lu 39

COMPARISON BETWEEN MMP3 AND MKT

Common proceduresIterative LRHASVD-truncated pseudoinverse

MMP3 onlyMatrix partitionEigenvalue decomposition

MKT onlySolve equation

UT Austin40

Biao Lu 40

CONTRIBUTION #1:PROPOSED MP METHODS

Modified MP method 1 (MMP1)

Noise may corrupt and to lose the connection

0Y 1Y

partition

Y

0Y

0Y 1Y

0Y 1Y

Steps 3-4 in MP method

1Y

LRHA

SVD truncation

LRHA

SVD truncation

UT Austin41

Biao Lu 41

CONTRIBUTION #1:PROPOSED MP METHODS

Modified MP method 2 (MMP2)

SVD truncation may destroy the connection between Y0 and Y1

SVD truncation

Y

0Y 1Y

0Y

0Y

Joint LRHA

Y

partition

SVD truncation

1Y

partition

Y

Step 3-4 in MP method

1Y

UT Austin42

Biao Lu 42

COMPUTER SIMULATION

Data model

whereK=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 step500 runs for each SNR value

Performance measure

1,,1 ,0

),()()()(1

Nn

nweAnwnhny npK

ii

i

2102

1log 10

SNR

210 original - estimated MSE ,

MSE

1log10 E

UT Austin43

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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 capacityBased 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

lbits/symbo SNR

1log geom2DMT

Nb

UT Austin46

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COMPUTER SIMULATION

Simulation parameters

Parameters DedicatedChannel

CSA DSLchannels 1-8

Sampling rate 300 kHz 2.208 MHz

Samples per symbol 512 512

Power of AWG noise 200 dBm/Hz 113 dBm/Hz

Transmitter power 1 W 1 W

Transmittertermination resistance

50 50

FFT size 512 512

Cyclic prefix 16 32

Bit error rate 10-7 10-7

Field margin 6 dB 6 dB

Coding gain 0 dB 0 dB

Number of runs 50 50

UT Austin47

Biao Lu 47

FREQUENCY RESPONSE OF A TRANSMISSION LINE

Model as a RC circuit

Characteristic impedance of the line

R L

C

Z0

fLjZL 2

fCjZC 2

1

fL

Rj

C

LZ

410

UT Austin48

Biao Lu 48

SSNR VS. DATA RATE

CSA DSL channel 1

SSNR = 40 dB