DMT Bit Rate Maximization DMT Bit Rate Maximization With Optimal Time Domain EqualizerWith Optimal Time Domain Equalizer

Filter Bank ArchitectureFilter Bank Architecture

*M. Milošević, **L. F. C. Pessoa, *B. L. Evans and *R. Baldick

*Electrical and Computer Engineering Department

The University of Texas at Austin

**Motorola, Inc, NCSG/SPS

Austin, TX

36th Asilomar IEEE Conference on Signals, Systems and Computers

Nov 3-6, 2002, Pacific Grove, CA

2MPEB Asilomar’02

P/S QAMdecoder

invert channel

=frequency

domainequalizer

Ser

ial-t

o-P

aral

lel (

S\P

)

QAMencoder

mirrordataand

N-IFFT

add Cyclic Prefix(

CP)

Digital-to-Analog

Converter +transmit

filter

N-FFTand

removemirrored

data

S/Premove

CP

TRANSMITTER

RECEIVER

N/2 subchannels N samples

N samplesN/2 subchannels

TEQtime

domain equalizer

receive filter

+Analog-to-

Digital Converter

channel

Basic Architecture: DMT Transceiver Basic Architecture: DMT Transceiver

Bits

00101

Par

alle

l-to-

Ser

ial (

P\S

)

noise

3MPEB Asilomar’02

DMT SymbolDMT Symbol

CP: Cyclic Prefix

N samplesv samples

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

copy copy

D/A + transmit filter

ADSL downstream upstream

CP 32 4 N 512 64

Inverse FFT

4MPEB Asilomar’02

ISI and ICI in DMTISI and ICI in DMT

• Channel is longer than cyclic prefix (CP)+1– Adjacent symbols interfere (ISI)

– Subchannel are no longer orthogonal (ICI)

• TEQ mitigates the problem by shortening the channel– No symbol at demodulator contains contributions of other

symbols

– Cyclic prefix converts linear convolution into circular

– Symbol channel FFT(symbol) x FFT(channel)

– Division by the FFT(channel) can undo linear time-invariant frequency distortion in the channel

5MPEB Asilomar’02

Channel Impairments and TEQ DesignChannel Impairments and TEQ Design

• Conventional ADSL TEQ design– Mitigate inter-symbol interference at the TEQ output

• Proposed ADSL TEQ design - Maximize data rate– Inter-symbol interference at the output of the demodulator (FFT)

– Near-end crosstalk (NEXT)

– Design with respect to digital noise floor (DNF)

– White noise in the channel (colored by TEQ)

• Other impairments present in an ADSL system– Impulse noise

– Near-end echo

– Far-end echo (of concern in voice-band communication)

– Phase and frequency content distortion (compensated by FEQ)

6MPEB Asilomar’02

Proposed TEQ Design MethodProposed TEQ Design Method

• Maximize bit rate at the demodulator (FFT) output instead of TEQ output

• Incorporate more sources of distortion into design framework

• Expected contributions– Model SNR at output of the FFT demodulator

– Data Rate Optimal Time Domain Per-Tone TEQ Filter Bank Algorithm (TEQFB)

– Data Rate Maximization Single TEQ Design

• Results

7MPEB Asilomar’02

Model SNR at Output of DemodulatorModel SNR at Output of Demodulator

• Desired signal in kth frequency bin at FFT output is DFT of circular convolution of channel and symbol

– is desired symbol circulant convolution matrix for delay – H is channel convolution matrix

– qk is kth column vector of N DFT matrix

• Received signal in kth frequency bin at FFT output

– is actual convolution matrix (includes contributions from previous, current, and next symbol)

– G(*) is convolution matrix of sources of noise or interference

wGGGGHUqY echofextnextawgnisiHR kk

HwUqY circHDkk

isiU

circU

8MPEB Asilomar’02

Model SNR at Output of DemodulatorModel SNR at Output of Demodulator

• Proposed SNR model at the demodulator output

• After some algebra, we can rewrite the SNR model as

digDRHDR

DHDModel

)]()E[(

])E[()(SNR

kkkk

kkk YYYY

YYw

wBw

wAww

T

T

k

kk ~

~SNR Model

dig – Digital noise floor (depends on number of bits in DSP)

(*)H – Hermitian (conjugate transpose)

9MPEB Asilomar’02

• Bits per symbol as a nonlinear function of equalizer taps.

– Multimodal for more than two-tap w.

– Nonlinear due to log and .

– Requires integer maximization.

– Ak and Bk are Hermitian symmetric.

• Unconstrained optimization problem:

Model SNR at Output of DemodulatorModel SNR at Output of Demodulator

k k

k

k

kbwBw

wAwww

T

T

2

Model

2int log

SNR1log

DMT

*

ww

intoptint

DMTDMTmax bb

10MPEB Asilomar’02

• Per channel maximization: find optimal TEQ for every k subchannel in the set of used subchannels I

• Generalized eigenvalue problem

• Bank of optimal TEQ filters

Data Rate Optimal Time Domain Per-tone TEQ Filter Data Rate Optimal Time Domain Per-tone TEQ Filter Bank (TEQFB) AlgorithmBank (TEQFB) Algorithm

kkk

kkk

kkk

kkkk

kk wBw

wAw

wBw

wAww

wwT

T

T

T

2opt maxarglogmaxarg

kkkkkkkkk for λλfor λ satisfies optoptoptoptopt wBwAw

Ik kkk

kkkboptTopt

optTopt

2

optint logDMT

wBw

wAw

11MPEB Asilomar’02

Frequency Domain

Equalizer

Goertzel Filter Block

TEQ Filter Bank

TEQ Filter Bank ArchitectureTEQ Filter Bank Architecture

w1

w2

wN/2-1

G1

G2

GN/2-1

Received Signal x={x1,

…xN)

FEQ1

FEQ2

FEQN/2-1

y1

y2

yN/2-1

Y1

Y2

YN/2-1

12MPEB Asilomar’02

TEQFB Computational ComplexityTEQFB Computational Complexity

• Creating matrices Ak

and Bk ~ NO(M2N)

• Up to N/2 solutions of symmetric-definite problems– Using Rayleigh

quotient iteration

Single TEQ Real MACs Words/Sym

TEQ Mfs 2M

FFT 2Nlog2Nfsym 4N

FEQ 2Nfsym 2N

TEQFB Real MACs Words/Sym

TEQ FB N/2Mfs M(1+N/2)

Goertzel FB N(fs+fsym) 4N

FEQ 2Nfsym 2N

PTE Real MACs Words/Sym

FFT 2Nlog2Nfsym 4N+2Combiner 2NMfsym (M+1)N

MM

Miter

NR 412

322

3

N= 512, =32, M 2, fs= 2.204 MHz, fsym=4 kHz

Initialization Data Transmission

13MPEB Asilomar’02

• Find a single TEQ that performs as well as the optimal TEQ filter bank.– Solution may not exist, may be unique, or may not be unique.

– Maximizing b (w) more tractable than maximizing bDMTint(w).

– b (w) is still non-linear, multimodal with sharp peaks.

Data Rate Maximization Single TEQ DesignData Rate Maximization Single TEQ Design

k

kbw

wModel

2int

SNR1log

DMT

k

kbw

wModel

2

SNR1log

14MPEB Asilomar’02

• Find a root of gradient of b (w) corresponding to a local maximum closest to the initial point– Parameterize problem to make it easier to find desired root.

– Use non-linear programming

– Find a good initial guess at the vector of equalizer taps w – one choice is the best performing TEQ FIR in TEQFB.

– No guarantee of optimality

– Simulation results are good compared to methods we looked at

Data Rate Maximization Single TEQ DesignData Rate Maximization Single TEQ Design

15MPEB Asilomar’02

• Measurement of the SNR in subchannel k– S = 1000 symbols

– Every subchannel in a symbol loaded with a random 2-bit constellation point Xk

i, passed through the channel, TEQ block and FEQ block (where applicable) to obtain Yk

i

• Bit rate reported is then

Simulation ResultsSimulation Results

1

0

21

2log10SNR S

i

ik

ik

k

YXS

256

72

SNR1log

k

kb

16MPEB Asilomar’02

Effect of TEQ Size on Bit RateEffect of TEQ Size on Bit Rate

Data rates achieved for different number of TEQ taps, MN = 512, = 32, input power = 23.93 dBm, AWGN power = -140 dBm/Hz,

and NEXT modeled as 49 disturbers. Accuracy of bit rate: 60 kbps.

(a) CSA loop 2 (b) CSA loop 7

17MPEB Asilomar’02

Effect of Transmission Delay on Bit RateEffect of Transmission Delay on Bit Rate

Data rates achieved as a function of for CSA loop 1.N = 512, = 32, input power = 23.93 dBm, AWGN power = -140 dBm/Hz,

and NEXT modeled as 49 disturbers. Accuracy of bit rate: 60 kbps.

18MPEB Asilomar’02

• We evaluate TEQFB, proposed single TEQ, MBR, Min-ISI, LS PTE, MMSE-UTC and MMSE-UEC for CSA loops 1-8

• Results presented in a table– Each row entry

– Final row entry

Simulation ResultsSimulation Results

%100*),,(

),,(

31

1),(

32

2TEQFB

TEQFB

opt

opt

M

Alg

MCSAb

MCSAbAlgCSARowAvg

8

1

),(8

1)(

CSA

AlgCSARowAvgAlgAvg

19MPEB Asilomar’02

TEQ Design Methods - ComparisonTEQ Design Methods - Comparison

CSA loop

LS PTENew TEQ

Min-ISI MBR MSSNRMMSE-

UECMMSE-

UTC

1 99.5% 99.6% 97.5% 97.3% 95.0% 86.3% 84.4%

2 99.5% 99.6% 97.3% 97.0% 96.5% 87.2% 85.8%

3 99.6% 99.5% 97.3% 97.8% 97.0% 83.9% 83.0%

4 99.1% 99.3% 98.2% 98.1% 95.4% 81.9% 81.5%

5 99.5% 99.6% 97.2% 97.7% 97.1% 88.6% 88.9%

6 99.4% 99.5% 98.3% 97.7% 96.4% 82.7% 79.8%

7 99.6% 98.8% 96.3% 96.3% 96.7% 75.75% 78.4%

8 99.2% 98.7% 97.5% 97.4% 97.5% 82.6% 83.6%

Avg. 99.4% 99.3% 97.5% 97.4% 96.4% 83.6% 83.2%

CSA – carrier serving area, MBR – Maximum Bit Rate, Min-ISI – Minimum InterSymbol Interference TEQ Design, LS PTE – Least-squares Per-Tone Equalizer, MMSE –

Minimum Mean Square Error, UTC – Unit Tap Constraint, UEC – Unit Energy Constraint

20MPEB Asilomar’02

TEQFB Data RatesTEQFB Data Rates

Highest data rates in Mbps achieved by TEQFB for TEQ lengths 2-32, input power = 23.93 dBm

CSA loop TEQFB

1 11.417 Mbps

2 12.680 Mbps

3 10.995 Mbps

4 11.288 Mbps

5 11.470 Mbps

6 10.861 Mbps

7 10.752 Mbps

8 9.615 Mbps

Backup SlidesBackup Slides

Milos Milosevic

Lucio F. C. Pessoa

Brian L. Evans

Ross Baldick

22MPEB Asilomar’02

Bit/symbol for a 2-tap TEQBit/symbol for a 2-tap TEQ

23MPEB Asilomar’02

Bit/symbol for a 3-tap TEQBit/symbol for a 3-tap TEQ

24MPEB Asilomar’02

CSA LoopsCSA Loops

Configuration of eight standard carrier serving loops (CSA). Numbers represent length in feet/ gauge. Vertical lines represent bridge taps. From Guner, Evans and Kiaei, “Equalization For DMT To Maximize bit Rate”.

25MPEB Asilomar’02

Selected Previous TEQ Design MethodsSelected Previous TEQ Design Methods

• Minimize mean squared error– Minimize mean squared error (MMSE) method [Chow & Cioffi, 1992]

– Geometric SNR method [Al-Dhahir & Cioffi, 1996]

• Minimize energy outside of shortened channel response– Maximum Shortening SNR method [Melsa, Younce & Rohrs, 1996]

– Divide-and-Conquer methods – Equalization achieved via a cascade of two tap filters [Lu, Evans & Clark, 2000]

– Minimum ISI method - Near-maximum bit rate at TEQ output [Arslan, Evans & Kiaei, 2001]

– Maximum Bit Rate (MBR) - Maximize bit rate at TEQ output [Arslan, Evans & Kiaei, 2001]

• Per-tone equalization– Frequency domain per-tone equalizer [Acker, Leus, Moonen, van der Wiel

& Pollet, 2001]

26MPEB Asilomar’02

• Used to calculate single DFT point

• Denote with yk(n) as the signal emanating from TEQ making up TEQFB

• Then, the corresponding single point DFT Yk is:

where Gk (-1) = Gk (-2) = 0 and n={0,1,…,N}

Goertzel FiltersGoertzel Filters

N

kNGj

N

kNGNGY

nGnGN

knynG

kkkk

kkkk

2sin1

2cos1

212

cos2