Channel Equalization To Achieve High Bit Rates In

Discrete Multitone Modulation Systems

Ming DingPh.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

OutlineOutline

• 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

SN

R 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-pointInverse

FFT

X1

X2

X1*

x1

x2

x3

xNX2*

XN/2

XN/2-1*

X0

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 vN

N

N samplesv samples

CP CPs y m b o l i s y m b o l ( i+1)

copy copy

6

Bit Loading in DMT

• Number of bits allocated to ith subchannel

– SNRi is SNR in subchannel i

i is SNR gap to channel capacity

• Turn off subchannels that cannotsupport 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

iib

SNR1log2

rate symbol iibR

i = 9.8 dB in uncoded DMT ADSL/VDSL system

Symbol rate is 4 kHz inDMT ADSL/VDSL system

7

ADSL TransceiverData Transmission Subsystem

reversefunction

QAM decisiondevice

(Viterbi)

N/2complex multiply

units

superframescramble,encode,

interleavetone order

QAMmapping(Trellis)

mirrordataand

N-IFFT

add cyclic prefix

P/SD/A +

transmit filter

N-FFTand

removemirrored

data

S/Premove

cyclic prefix

TRANSMITTER

RECEIVER

N/2 subchannels N real samples

N real samplesN/2 subchannels

time domain

equalizer

receive filter

+A/D

channel

ATM

StructureEqualizer alConvention

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

ADSL G.DMT Values Down

stream Up

stream 32 4

N 512 64

9

Linear Equalizer Structures

10

Equalizer Training Complexity

• Periodic 4-QAM training sequence– No cyclic prefix

– Constant transmit power spectrum Sx

• Receiver monitors additive noise power spectrum Sn

Multiplications& Additions

Memory

(Words)

Single TEQ O(Lw3) Lw

TEQ Filter Bank O(Lw2N2) N/2 Lw

Per Tone Equalizer O(Lw2N + LwN2) N Lw

Complex Filter Bank O(Lw2N + LwN2) N Lw

Example ADSL Parameters

FFT Size N = 512TEQ Length Lw = 17

[Martin, Vanbleu, Ding et al. 2004]

11

OutlineOutline

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 minimizeproduct of generalizedRayleigh quotients

• For M = 1, solution is generalizedeigenvector of the matrix pair(B, A) corresponding to smallestgeneralized eigenvalue

• For M > 1, solution is not well-understood– Various searching methods exist to find a local optimum

M

j jT

jT

opt

1

minargwBw

wAww

w

1 subject to

min

Bww

Awww

T

T

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

1

1

v

yxyyxyxx

IB

RRRRA

TT 1 yyxyRRbw

wB

GHGHwhh

wA

DHDHwhh

TTTwin

Twin

TTTwall

Twall

h + w

z- b-

xk

yk ek

nk

+

1 subject to min BwwAwww

TT

hwin

hwall

Adepends

on

A and Bdepend

on

Channel TEQ

14

Single Quotient Cases

• Minimum Intersymbol Interference [Arslan et al. 2000]

• Minimum Delay Spread [Schur et al. 2001]Modified Maximum Shortening SNR with distance weighting

GHGHB

DHqqDHA

TT

i

Hi

in

ixi

TT

S

S

,

, Generalization of MaximumShortening SNR method withfrequency weighting

HHB

HHA

T

T ckdiag

])[( 2

channel taps

d1

d2

k =0, 1, 2,…, N-1 c: center of mass

1 subject to min BwwAwww

TT

+1

15

Multiple Filters (each with a Single Quotient)

• Per-tone equalization [Acker et al. 2001]– Generalized eigenvalue problem for each tone i

– Received frame (CP + symbol) is y and ith FFT coefficient is Yi

• Time domain equalizer bank [Milosevic et al. 2002]

])([])[(

])[(

*iii

Hii

iH

ii

XEXE

E

B

A

HQQHB

ILL

QRQ

HLLHHUUHA

Hcirci

circi

Ti

LwTDNFHnoise

innoise

wallHii

Twallwall

Hii

Twalli

i

][

][

)(2

1

2

2,2,1,1,

where

][ 11

Niii

Lii

yy

Yw

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. 2001]– ISI + AWGN, Quasi-Newton algorithm

• Maximum Data Rate [Milosevic et al. 2002]– ISI + Cross-talk + Echo + digital noise floor

– Almogy and Levin iteration

• Bitrate Maximizing [Vanbleu et al. 2003]– Eventually all possible noises and interference resources

– Recursive Gauss-Newton update

i i iT

iT

i

i

i

ibwAw

wBwSNR1log2

17

OutlineOutline

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 energyA = HT DT D H converges asymptotically to doubly symmetric HT H

• Minimum Mean Square Error TEQTarget impulse response is symmetric/skew symmetric

A becomes a doubly symmetric matrix

1356

3245

5423

6531

51.0

48.0

48.0

51.0

48.0

51.0

51.0

48.0

32.0

63.0

63.0

32.0

63.0

32.0

32.0

63.0

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-

h + w

b-

xk

yk ek

nk

+

20

Contribution #1 Symmetric TEQ design

• Implementation: instead of finding eigenvector of Lw Lw

matrix, find eigenvector of matrix– Some matrix operations ~ O(Lw

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

2/2/ ww LL

2

)1(delay group

wL

0 2 4 6 8 10 12 14 16 18-300

-200

-100

0

100

200

300

400

500

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 to

fixed-point computation

– High computational cost when performing

delay optimization (A and B depend on Δ )

Bww

Aww

GHwGHw

DHWqqDHw

wT

T

TTTi

Hi

in

ixi

TTT

S

S

J

,

,

)(

22

Contribution #2 Improving Minimum ISI Method

• Define new cost function

– : weighting value for subchannel i

– HT H is always positive definite and invertible

• Suitable for arbitrary length TEQ design

• Reduces computational cost when performing delay optimization

HwHw

DHwqqDHw

w

TTi

Hiii

TTT a

J

power Total

power ISI Weighted)(

ia

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 forsupporting 2 bits in subchannel

iin

ixi

Hiii S

SSNR where

,

, qq

)10/(,103 gap

xpresetn

SS

presetnS ,inS ,

ADSL fixesSx = -40 dBm/Hz

gap = 9.8 dBDuring 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 w0, and iteratively calculate wk, using deterministic gradient search

i

i

Hiii

TTT DHqqDHAHHB ,TAAA ˆ

kT

kkkkk wAwBwwAww ˆˆˆ 1

2maxmax )1)()(ˆ(

1

BA

[Chatterjee, et. al 1997]

Division-free iteration

1)(

05.0)ˆ(02.0

channels test ADSL Standard

max

max

B

A

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={r1,

…rN}

DelaysGoertzel Filters

Complex Equalizers

w1

w2

wN/2-1

G1

G2

GN/2-1

y1

y2

yN/2-1

X1

X2

XN/2-1

1

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

][][ 1ii

HHii

Hi XEE qYYqqYw

iii XXe ˆ

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 Examples

TEQ 1

TEQ 2

PFFT

PFFT

PathSel.

FEQ

PFFT: Partial FFT

28

OutlineOutline

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 andComplex TEQ Filter Bank Equalizers

Simulation ParametersTEQ length 17Cyclic prefix 32 samplesFFT size (N) 512 samplesCoding gain 5 dBMargin 6 dBInput power 23 dBmNoise PSD -140 dBm/HzCrosstalk noise 5 ISDNRF interference 6 AM stationsChannels Carrier Serving

Area Loops 1-8

Testing 1000 symbols1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

Carrier Serving Area loop number

Bit

rate

(M

bp

s)

MIN-ISIDual-pathTEQFBPERTONECTEQFB

30

Proposed Symmetric TEQ Design Methods

Simulation ParametersTEQ length 17Cyclic prefix 32 samplesFFT size (N) 512 samplesCoding gain 5 dBMargin 6 dBInput power 23 dBmNoise PSD -140 dBm/HzCrosstalk noise 5 ISDNRF interference 6 AM stationsChannels Carrier Serving

Area Loops 1-8Testing 1000 symbols1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

Carrier Serving Area loop number

Bit

rate

(M

bp

s)

MMSESYM-MMSEMSSNRSYM-MSSNRMin-ISISYM-Min-ISI

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 symbols5 10 15 20 25 30

1.7

1.75

1.8

1.85

1.9

1.95

2x 10

6

TEQ length

aver

age

ache

ivab

le b

it ra

te

Matched Filter BoundMin-ISIIter-Min-ISIIter-Min-ISI-on/off

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. 297-301, Jan. 2004.

– 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. 1279-1290, Dec. 1, 2003.

– B. Farhang-Boroujeny and M. Ding, ``Design Methods for Time Domain Equalizer in DMT Transceivers'', IEEE Transactions on Communications, vol. 49 Issue: 3, pp. 554 -562, 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. 29 - 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. 1-5 2003, vol. 4, pp. 2146-2150, 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. 97-100, 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. 2573-2576, Orlando, FL.

36

Backup Slides

37

Overview of ADSL Technology

ADSL Modem

Telephone

TV

PC

BroadbandNetwork

NarrowbandNetwork

ADSL Modem

POTS Splitters

telephoneline

Central Office Subscriber End

Fax

1.5 - 8 Mbps

16 - 640 kbps

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 T1.413-1995

• June 1999, ITU-T SG 15 approves G.992.1 (G.dmt) standard for full rate ADSL

– Tone: subchannel

– Value format: Downstream/Upstream

– BER: Bit Error Rate

item value item value

No. of Tones 256/32 FFT size (N) 512/64

CP Length () 32/4 Tone Width 4.3 KHz

Symbol Rate 4 kHz Sampling Rate 2.208 MHz

Target BER 10-7 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]

Transmitter ChannelReceiver

TEQInformationsource

Informationsink

41

MBR TEQ Designs[Arslan, Evans & Kiaei, 2000]

• A subchannel SNR definition:

• Maximize nonlinear function to obtain the optimal TEQ

2

,

2

,

2

,

power ISIpower noise

power signalSNR

ISIix

noisein

signalix

i

ii

i

HSHS

HS

Fwq

DHwq

GHwq

Hi

noisei

Hi

ISIi

Hi

signali

H

H

H

wBw

wAw

DHwqFwq

GHwq

iT

iT

Hiix

Hiin

Hiix

iSS

S

2

,

2

,

2

,SNR

2/

12 )

11 (log

N

i iT

iT

DMTbwBw

wAw

42

Pertone Equalizer[Acker, Leus, Moonen, van de Wiel & Pollet, 2001]

• Output of conventional equalizer structure for tone i

Zi = Di rowi(QN ) R w

Di is the complex value of one-tap FEQ for tone i

QN 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

Zi = Di rowi(QN ) R w = rowi(QN R) ( w Di )

• A multi-tap FEQ for tone i combines TEQ and FEQ operations. The output is

Zi = rowi(QN R) wi

43

Performance Comparison for TEQ vs Pertone

The performance gapfor any single tone isnot universally wide. In tones associated with higher SNR, theimprovement of pertone tends to besignificant. For othertones, 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

Yww

Xww

GHwGHw

DHWqqDHw

wT

T

TTT

i

Hi

in

ixi

TTT

S

S

J

,

,

)(

1 subject to min YwwXwww

TT

min

1

qYw

Topt

wXwY ~1

1 1

TYXYC

qmin 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

wwYwX 11 ~

YX 1

YX 1

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

elements

ones 1zeros

00,,1,1,0,0diag

N

G

GID

)()2()1(

0)0()1(

00)0(

wLNhNhNh

hh

h

H

TNNjNiji ee

N/)1(2/21

1 q

48

Invertibility of X

H

i

Hiii UUqqZ

HN

N

N qqqqqqZ

212

1

21

000

00

00

00

ZDHDHX TT

Hiii qq is obviously a rank 1 matrix.

Conclusion: X is invertible if and only if all is are non-zero.

49

Goertzel Filters

• The N-point DFT of a length N sequence x(l):

• Define

Noticed

• A recursive DFT computation scheme:

1

0

1

0

1

0

)()()()(N

l

N

l

N

l

lNkN

klN

kNN

klNk WlxWlxWWlxX

)()()()(0

)( nuWlxWlxny knN

n

l

lnkNk

Nnk nykX

)()(

0)( and 0)1(with

0 ,)1()()(

Nxy

NnnyWnxny

k

kk

Nk

50

More definitions

)()2()1(

)()()1(

)1()1()(

w

w

w

LNyNyNy

Lyyy

Lyyy

R

TwLwww )1()1()0( w

51

Second Order Conditions of J

• Hessian

TJ XXw )(2

ZDHDHX TT

HN

N

N qqqqqqZ

212

1

21

000

00

00

00

All is are non-negative ) ,diag( N1 HDUΦ

ΦΦX

T

is positive-semidefinite.

52

Constrained Minimization of Iterative Min-ISI

• Use the Lagrange multipliers

• Iterative updates:

• where

1 subject to min YwwXwww

TT

wYYXX

YwwXwww

w )()(

)1(),(TT

TT

L

L

kT

kT

kk wYYXXww )()(ˆ 1

)(2

1k

TTkk wXXw

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

RqqR

qRw

Hii

Ti

Ti

i E

XE~

i

Hii

THi

iiHii

Ti

Hixi

E

XEXESJ

wRqqRw

wRqqRw~~

~~ *

0~)~( *

wRqqRqRw

Hii

Ti

Tii EXEJ

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 + wxk

yk rk

nk

w

B

HHwhh

w

A

HHwhh

winTwin

Twin

Twin

wallTwall

Twall

Twall

1 s.t. )(min BwwAwww

TT

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.

Yww

Xww

wHHw

wHqqHw

wT

T

winT

winT

iwall

Hi

in

ixi

Twall

T

S

S

J

,

,

)(

1 subject to min YwwXwww

TT

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

SlidingN-Point

FFT(Lw-frame)

N+

N+

N+z-1

z-1

z-1

y

N + Lw – 1channels

W1,1W1,0 W1,2 W1,Lw-

1

WN/2,0 WN/2,1 WN/2,2 WN/2,Lw-1

FEQ is a linear combinerof up to N/2 Lw-tap FEQs

N-FFTTEQ FEQiy zi

z1

zN/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 improvedGuarantee 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 30 - 40] – TEQ 2 addresses the upstream bandwidth [subcarrier >40]

58

New SNR model after FEQ

• Define SNR at the ith FEQ output as

: Transmitted power of ith subchannel

: Transmitted complex-valued QAM symbol on ith subchannel

: Output of ith FEQ, estimated QAM symbol on ith subchannel

• The practical ADSL systems use flat subchannel power allocation

2

,

ˆSNR

ii

ixi

XXE

S

ixS ,

iX

iX̂

xix SS ,

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!

matrix data received theis

*

RVww

Uww

wRqqRw

wRqqRRqqRw

iTi

iTi

iHii

TTi

itHi

Ti

Hii

Tx

Ti

i E

XEXEESJ

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

w1

w2

wN/2-1

G1

G2

GN/2-1

Received Signal R={r1,

…rN)

FEQ1

FEQ2

FEQN/2-1

y1

y2

yN/2-1

Y1

Y2

YN/2-1

1X̂

2X̂

12/ˆ

NX

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

m

i

n

j

ijFa

1 1

2A

)Trace( HF

AAA

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-

h + w

b

-xk

yk ek

nk

+

1 subject to min | bbbRbb

Tyx

T TT 1** yyxyRRbw

bRb

bRRRRb

yxT

yxyyxyxxT

ke

|

1

2

][

E MSE

Poor bit rateperformance

64

SNR Gap

• Channel capacity in bits per 2-dimensional symbol

• SNR gap: excessive SNR needed to achieve capacity

)SNR1(log drec'2 C

drec'2

SNR1logb

dBin 77.4gain codingmargin

21

e

eN

PQ

neighboursnearest ofNumber :

error symbol of Prob. :

e

e

N

P

65

Cholesky Decomposition

• 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

ALLLA H

1 1

HBABC wBv H

vvBwwCvvAww TTTT and

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 Di per tone

• Time domain filter bank plus FEQ

• Per-tone equalizer

• Complex time domain filter bank

complex tap-1 FEQ: matrix DFT :

signal received : real tap-multi TEQ :

rowˆ

i

Niii

D

DX

Q

Yw

YwQ

iww

iiD Dw~

iiD ww ~

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

HHT

HHT

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

55-85

Achieved Bit Rate

Path 1: 2.5080 Mbps

Dual Path: 2.6020 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 12000 feet

loop 1

loop 2

loop 7

loop 5

loop 8

loop 6

loop 4

loop 3

5900/26 1800/26600/26

3000/26 700/24 350/26700/26

3000/26650/26

12000/24

10700/24800/24

9000/26

5800/26 150/24 1200/26 300/24 300/261200/26

550/26 6250/26 800/26400/26 800/26

2200/26 700/26 1500/26 500/26 600/24 3050/2650/24 50/24 50/2650/24 100/24

200 300 400 500 600 700 800 900 1000 1100-100

-90

-80

-70

-60

-50

-40

-30

Frequency (kHz)

Mag

nitu

de R

espo

nse

(dB

)

CSA Loop 1CSA Loop 2CSA Loop 3CSA Loop 4

70

Symmetric design

• Even length TEQ

• Odd length TEQ

vJJAJAJAAv

Jv

v

AA

AAJvv

][

22122111

2221

1211

T

TT

v

AJAA

JAAJJAJAJAAv

Jv

v

AAA

AAA

AAA

Jvv

222311

321233311311

333231

232221

131211

T

TT