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DMT Bit Rate Maximization With Optimal Time Domain Equalizer Filter Bank Architecture

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36th Asilomar IEEE Conference on Signals, Systems and Computers### DMT Bit Rate Maximization With Optimal Time Domain EqualizerFilter Bank Architecture

Nov 3-6, 2002, Pacific Grove, CA

*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

Basic Architecture: DMT Transceiver

N/2 subchannels

N samples

Serial-to-Parallel (S\P)

QAM

encoder

mirror

data

and

N-IFFT

add

Cyclic Prefix(CP)

Digital-to-Analog Converter +

transmit filter

Parallel-to-Serial (P\S)

Bits

00101

TRANSMITTER

channel

noise

RECEIVER

N/2 subchannels

N samples

receive filter

+

Analog-to-Digital Converter

P/S

TEQ

time domain equalizer

QAM

decoder

N-FFT

and

remove

mirrored

data

S/P

remove

CP

invert channel

=

frequency

domain

equalizer

v samples

N samples

s y m b o l ( i+1)

CP

CP

s y m b o l ( i )

copy

copy

DMT SymbolInverse FFT

D/A + transmit filter

CP: Cyclic Prefix

ISI 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

Channel 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)

Proposed 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

Model 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 D
- H is channel convolution matrix
- qkis 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

Model SNR at Output of Demodulator

- Proposed SNR model at the demodulator output
- After some algebra, we can rewrite the SNR model as

- adig – Digital noise floor (depends on number of bits in DSP)
- (*)H – Hermitian (conjugate transpose)

Model SNR at Output of Demodulator

- 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:

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

- 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

G1

w1

FEQ1

G2

w2

FEQ2

GN/2-1

wN/2-1

FEQN/2-1

TEQ Filter Bank Architecturey1

Y1

TEQ Filter Bank

Goertzel Filter Block

Frequency Domain Equalizer

y2

Y2

Received Signal x={x1,…xN)

yN/2-1

YN/2-1

TEQFB Computational Complexity

- Creating matrices Ak and Bk ~ NO(M2N)
- Up to N/2 solutions of symmetric-definite problems
- Using Rayleigh quotient iteration

Initialization

Data Transmission

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

Data Rate Maximization Single TEQ Design

- 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 Design

- 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

Simulation Results

- Measurement of the SNR in subchannel k
- S = 1000 symbols
- Every subchannel in a symbol loaded with a random 2-bit constellation point Xki, passed through the channel, TEQ block and FEQ block (where applicable) to obtain Yki
- Bit rate reported is then

Effect of TEQ Size on Bit Rate

Data rates achieved for different number of TEQ taps, MN = 512, n= 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

Effect of Transmission Delay on Bit Rate

Data rates achieved as a function of D for CSA loop 1.N = 512, n= 32, input power = 23.93 dBm, AWGN power = -140 dBm/Hz, and NEXT modeled as 49 disturbers. Accuracy of bit rate: 60 kbps.

Simulation Results

- 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

TEQ Design Methods - Comparison

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

TEQFB Data Rates

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

CSA 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”.

Selected 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]

Goertzel Filters

- 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}

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