Transcript Equalization for Discrete Multitone Transceivers

Equalization for
Discrete Multitone Transceivers
Güner Arslan
Ph.D. Defense
Committee
Prof. Ross Baldick
Prof. Alan C. Bovik
Prof. Brian L. Evans, advisor
Prof. Joydeep Ghosh
Dr. Sayfe Kiaei
Prof. Edward J. Powers
Outline
• Introduction to high-speed wireline digital communications
• Problem: Increase ADSL transceiver bit rate by increasing
performance of the time-domain equalizer (TEQ) in the receiver
• Contribution #1: New model for equalized channel
• Contribution #2: Optimal channel capacity TEQ
• Contribution #3: Closed-form near-optimal TEQ
• Simulation results
• Summary and future work
Receive
bit stream
Transmit
bit stream
Transmitter
Channel
TEQ
Receiver
2
Introduction
Residential Applications
Application
Internet
High Definition TV
Broadcast Video
Video Phone
Database Access
On-line Directory; Yellow Pages
Shop-at-Home
Video Games
Downstream
Upstream
Rate (kb/s)
Rate (kb/s)
3,000
384
24,000
0
6,000
0
1,500
1,500
384
9
384
9
1,500
64
1,500
1,500
Willingness
to pay
High
High
Low
High
High
Low
Low
Medium
Demand
Potential
Medium
Medium
High
Medium
Medium
High
Medium
Medium
Willingness
to pay
High
High
Medium
Medium
High
High
High
Medium
Demand
Potential
High
Medium
Medium
Low
Low
Low
Low
High
Business Applications
Application
Internet
Remote Office
LAN Interconnection
Financial News
Video Phone
Video Conference
Supercomputing, CAD
On-line Directory; Yellow Pages
Downstream
Upstream
Rate (kb/s)
Rate (kb/s)
3,000
384
6,000
1,500
10,000
10,000
1,500
9
1,500
1,500
3,000
3,000
45,000
45,000
384
9
3
Standards for High-Speed Digital Communications
Standard Meaning
Integrated Services
ISDN
Digital Network
T-Carrier One
T1
(requires two pairs)
High-Speed Digital
HDSL
Subscriber Line
(requires two pairs)
HDSL2 Single Line HDSL
G.DMT
G.Lite
VDSL
Asymmetric Digital
Subscriber Line
Splitterless
Asymmetric Digital
Subscriber Line
Very High-Speed
Digital Subscriber
Line (proposed)
Data Rate
Mode
Applications
144 kbps Symmetric Internet Access, Voice, Pair
Gain (2 channels)
1.544 Mbps Symmetric Enterprise, Expansion,
Internet Service
1.544 Mbps Symmetric Pair Gain (12 channels),
Internet Access, T1/E1
replacement
1.544 Mbps Symmetric Same as HDSL except pair
gain is 24 channels
1.5 to 8 Mbps
Down
Internet Access, Digital
16 to 640 kbps
Up
Video
32 to 1500 kbps
Down
Internet Access, Digital
32 to 512 kbps
Up
Video
13 to 52 Mbps
1.5 to 6 Mbps
Down
Up
Internet Access, Digital
Video
Courtesy of Shawn McCaslin (Cicada Semiconductor, Austin, TX)
4
Intersymbol Interference (ISI)
2.1
111
1
.7
.4
Channel
Transmit
signal
.1
– Impulse response is an impulse
– Frequency response is flat
=
*
-1
1.7
1
• Ideal channel
Received
signal
Threshold
at zero
11 1 1
Detected
signal
• Non-ideal channel causes ISI
– Channel memory
– Magnitude and phase variation
• Received symbol is weighted
sum of neighboring symbols
– Weights are determined by
channel impulse response
5
Combat ISI with Equalization
• Problem: Channel frequency response is not flat
• Solution: Use equalizer to flatten channel frequency response
Magnitude (dB)
• Zero-forcing equalizer
Zero-forcing
Equalizer
frequency
response
MMSE
Equalizer
frequency
response
– Inverts channel
– Flattens frequency response
– Amplifies noise
• Minimum mean squared
error (MMSE) equalizer
Channel
frequency
response
– Optimizes trade-off between
noise amplification and ISI
• Decision-feedback equalizer
Frequency
– Increases complexity
– Propagates error
6
Multicarrier Modulation
• Divide broadband channel into many narrowband subchannels
• No intersymbol interference (ISI) in subchannels if channel
gain is constant in every subchannel
• Discrete Multitone (DMT) modulation
magnitude
– Multicarrier modulation based on fast Fourier transform (FFT)
– Standardized for ADSL and proposed for VDSL
channel
frequency
response
carrier
subchannel
frequency
7
Multicarrier Modulation
• Advantages
– Efficient use of bandwidth without full channel equalization
– Robust against impulsive noise and narrowband interference
– Dynamic rate adaptation
• Disadvantages
– Transmitter: High signal peak-to-average power ratio
– Receiver: Sensitive to frequency and phase offset in carriers
• Active areas of research
–
–
–
–
Pulse shapes of subchannels (orthogonal, efficient realization)
Channel equalizer design (increase capacity, reduce complexity)
Synchronization (timing recovery, symbol synchronization)
Bit loading (allocation of bits in each subchannel)
8
Eliminating ISI in DMT
copy
copy
CP
v samples
s y m b o l i
N samples
CP
s y m b o l ( i+1)
CP: Cyclic Prefix
• Convolve stream of samples with channel
– Symbols are spread out in time
– No ISI if channel length is shorter than v+1 samples
ADSL Standard
v
N
32
512
• Symbols are distorted in frequency
– Cyclic prefix converts linear convolution into circular convolution
– Division in FFT domain can undo distortion if channel length is
less than v+1 samples
• Time domain equalizer shortens channel length
• Frequency domain equalizer inverts channel frequency response
9
Discrete Multitone Transmitter and Receiver
N/2 subchannels
serial to
parallel
QAM
encoder
N subchannels
mirror
data
and
N-IFFT
(N = 512 for ADSL)
add
parallel
cyclic
to serial
prefix
DAC
and
transmit
filter
TRANSMITTER
channel
RECEIVER
N/2 subchannels
parallel
QAM
to serial decoder
invert
channel
=
frequency
domain
equalizer
N subchannels
N-FFT
and
remove
mirrored
data
serial remove TEQ
to
cyclic
time
parallel prefix domain
equalizer
receive
filter
and
ADC
10
Problem Definition and Contributions
• Problem:
– Find a TEQ design method that maximizes channel capacity at
the TEQ output
• Proposed solution
–
–
–
–
Decompose equalized channel into signal, noise, and ISI paths
Model subchannel SNR based on this decomposition
Write channel capacity as a function of TEQ taps
Develop design methods to maximize channel capacity
• Contributions
– A new model for subchannel SNR
– Optimal maximum channel capacity (MCC) TEQ design method
– Near-optimal minimum ISI TEQ design method
11
Minimum Mean Squared Error (MMSE) Method
Chow, Cioffi, 1992
nk
xk
yk
h
+
z-
w
b
rk
ek
- +
zk
• Minimize mean squared error E{ek} where ek=bk- - hk*wk
• Chose length of bk to shorten length of hk*wk
• Disadvantages
– Does not consider channel capacity
– Zeros low SNR bands
– Deep notches in equalizer frequency response
12
Maximum Shortening SNR (MSSNR) Method
Melsa, Younce, Rohrs, 1996
• For each possible position of a window of +1 samples,
 energy inside window after TEQ 

max SSNR in dB  max 10 log 10 
w
w
 energy outside window after TEQ 
h
w
• Disadvantages
– Does not consider channel capacity
– Requires Cholesky decomposition
and eigenvector calculation
– Does not take channel noise into
account

13
Capacity of a Multicarrier Channel
• Each subchannel modeled as white Gaussian noise channel
2
 SNR i 
Matched Filter
S x ,i H i


bDMT   log 2 1 
SNR i 



 
Bound (MFB)
S n ,i
i 1

S x ,i and S n ,i : Signal and noise power in i th subchannel
N /2
H i : Channel gain in i th subchannel and  : SNR gap
• Define geometric SNR
SNR geom
  N / 2  SNR i  2 / N

    1 
 1


  i 1 






• Channel capacity of a multicarrier channel
N
 SNR geom 
bDMT  log 2 1 

2



14
Maximum Geometric SNR Method
Al-Dhahir, Cioffi, 1996
• Maximize approximate geometric SNR
nk
rk
xk
w - + ek
h +
z-
• Disadvantages
b
zk


SNR geom   SNR i 
 i 1

Bi  H i Wi
N /2
SNR i 
– Subchannel SNR definition ignores ISI
– Objective function ignores interdependence of b and w
2
L(b ) 
N
N /2
 ln Bi
i 1
2/ N
S x Bi
S n ,i Wi
2
2
H i : channel gain in i th subchannel
2
Wi : equalizer gain in i th subchannel
Bi : target gain in i th subchannel
– Requires solution of nonlinear constrained optimization problem
– Based on MMSE method: same drawbacks as MMSE method
– Ad-hoc parameter MSEmax has to be tuned for different channels
15
Comparison of Existing Methods
Method
MMSE
MSSNR
Geometric
Advantages

Maximize channel capacity

Minimize ISI
Low-medium
Bit Rate
High
Low-medium
Disadvantages

Nonlinear optimization
Computationally complexity
Artificial constraints
Low

Ad-hoc parameters
Low-pass frequency response
Unrealistic assumptions

Medium
High




16
Contribution #1
New Subchannel Model: Motivating Example
• Received signal Delay  y1  h~1a4
~ ~
y

x

h
n
–
~
– h
 w h
CP
– x is transmitted
signal
• Symbols a b
• Symbol length
– N=4
CP
~
• Length of h
– L=4
• Cyclic prefix
– v=1
• Delay
– =1
Tail
 y  ~
 2   h~1a1
 y3  h1a2
  ~
 y4   h1a3
 y5  h~ a
   ~1 4
 y6   h1b4
 y    h~ b
 7   ~1 1
 y8   h1b2
 y  ~
 9   h~1b3
 y10   h1b4
  
 y11  
 y12  
  
 y13  
ISI









~
h2 a4
~
h2 a1
~
h2 a2
~
h2 a3
~
h2 a4
~
h2b4
~
h2b1
~
h2b2
~
h2b3
~
h2b4
~
 h3 a4
~
 h3 a1
~
 h3 a2
~
 h3 a3
~
 h3 a4
~
 h3b4
~
 h3b1
~
 h3b2
~
 h3b3
~
h3b4
signal









  n~1 
 ~ 
  n2 
  n~3 
~  ~ 
h4 a4   n4 
~
h4 a1   n~5 
~  ~ 
h4 a2   n6 
~  ~ 
h4 a3   n7
~ 
~
h4 a4   n8 
~   
h4b4   n~9 
~  ~
n10 
h4b1
~  ~ 
h4b2   n11 
~
h4b3  n~12 
~  ~ 
h4b4   n13 
ISI
noise
17
Contribution #1
Proposed Subchannel SNR Model
• Partition equalized channel into
signal path, ISI path, noise path
nk
xk
xk
h
h
yk
+
w
w
x
rk
h
w
x
h
hkISI
hknoise  wk
~
hk  hk  wk
Signal
gk
xk
1   k    
gk  
otherwise
0
ISI
gk
1-gk
1
nk
w
~
h k g k
~
 h k 1  g k 
signal
k
...
noise
  1   1
k
18
Contribution #1
Subchannel SNR Definition
• SNR in i th subchannel is defined as
signal 2
S x ,i H i
signal power
SNR i 

noise power  ISI power S H noise 2  S H ISI
n ,i
x ,i
i
i
2
H isignal : gain of hksignal in subchannel i
H iISI : gain of hkISI in subchannel i
H inoise : gain of hknoise in subchannel i
S x ,i : transmitt ed signal power in subchannel i
S n,i : channel noise power in subchannel i
19
Contribution #2
Optimal Maximum Channel Capacity (MCC) TEQ
• Channel capacity as a nonlinear function of equalizer taps
2
H
H isignal  q iH GHw
S x ,i q i GHw
wT Ai w
SNRi 
2
2 
T
ISI
H
H
H
w
Bi w
H i  q i DHw
S n,i qi Fw  S x ,i q i DHw
H inoise  q iH Fw
GT
Ai =
FT
Bi =
qi
Sn,i
qi
H
HT
qi
DT
F
+
HT
qi
Sx,i
qi
H
G
H
Sx,i
qi
H
D
H
• Maximize nonlinear function to obtain the optimal TEQ
N /2
 1 wT Ai w 

bDMT   log 2 1 
T
i 1
  w Bi w 
• Good performance measure for any TEQ design method
• Not an efficient TEQ design method in computational sense
20
Contribution #2
MCC TEQ vs. Geometric TEQ
Method
MCC
Geometric
Advantages
Maximize channel capacity
Minimize ISI
Bit rate



optimal
Low-medium
Disadvantages
Low-pass frequency response
Computationally complex
Artificial constraints
Ad-hoc parameters
Nonlinear optimization
Unrealistic assumptions








21
Contribution #3
Near-optimal Minimum-ISI (min-ISI) TEQ
• ISI power in ith subchannel is
ISIi  S x ,i q DHw
H
i
2
• Minimize ISI power as a frequency weighted sum of subchannel ISI
T
ISI

K
q
DHw

w
Xw
 i  i
H
i
i
2
i
• Constrain signal path gain to one to prevent all-zero solution
| h signal |2 | GHw |2  wT Yw  1
• Solution is a generalized eigenvector of X and Y
• Possible weightings
K i  S x ,i
Ki  S x,i | Hi |2 / Sn,i
• Performance virtually equal to that of the optimal method
• Generalizes MSSNR method by weighting in frequency domain
22
Contribution #3
Min-ISI TEQ vs. MSSNR TEQ
Method
Min-ISI
MSSNR
Advantages
Maximize channel capacity
Minimize ISI
Frequency domain weighting
Bit rate


high
Disadvantages
Computationally complex
very high

high
high
• Min-ISI weights ISI power with the SNR
– Residual ISI power should be placed in high noise frequency bands
1
1
 0.09
 0.1 SNR50 
signal power
10  1
10
SNR i 
noise power  ISI power
1
1
SNR 2 
 10 SNR 2 
 0.9
0.1
0.1  1
23
SNR 50 
Contribution #3
Efficient Implementations of Min-ISI Method
• Generalized eigenvalue problem can solved with generalized
power iteration:
Xw k 1  Yw k
• Recursively calculate diagonal elements of X and Y from first
Method
Capacity MACs
column [Wu, Arslan, Evans, 2000]
Original
99.6%
132,896
Recursive
99.5%
44,432
Row-rotation
99.5%
25,872
No-weighting
97.8%
10,064
24
Bit Rate vs. Number of TEQ Taps
• Min-ISI, MCC, and MSSNR
perform close to Matched Filter
Bound (MFB) even with small
TEQ sizes
• Geometric and MMSE TEQ
require 20 taps to achieve 90%
of MFB performance
• Geometric TEQ gives little
improvement over MMSE TEQ
• Two-tap min-ISI TEQ
outperforms 21-tap MMSE TEQ
TEQ taps
cyclic prefix ()
FFT size (N)
coding gain
margin
input power
noise power
crosstalk noise
32
512
4.2 dB
6 dB
14 dBm
-113 dBm/Hz,
10 ADSL disturbers
25
Bit Rate vs. Number of TEQ Taps
• Min-ISI and MCC give virtually
same performance
• Min-ISI and MCC outperform
MSSNR by 2%
• 9 taps is enough for best
performance for min-ISI, MCC,
and MSSNR TEQs
• No performance gain for more
than 9 taps
TEQ taps
cyclic prefix ()
FFT size (N)
coding gain
margin
input power
noise power
crosstalk noise
32
512
4.2 dB
6 dB
14 dBm
-113 dBm/Hz,
10 ADSL disturbers
26
Bit Rate vs. Cyclic Prefix Size
• Min-ISI, MCC, and MSSNR
perform close to MFB
• Geometric and MMSE TEQ
require cyclic prefix of 30
samples
• Geometric TEQ gives worse
performance for short cyclic
prefix
• Performance drops because
cyclic prefix does not carry
new information
• MSSNR does not work for
cyclic prefix smaller than the
number of TEQ taps
TEQ taps
FFT size (N)
coding gain
margin
input power
noise power
crosstalk noise
17
512
4.2 dB
6 dB
14 dBm
-113 dBm/Hz,
10 ADSL disturbers
27
Simulation Results
• Min-ISI, MCC, and MSSNR
require cyclic prefix of 17
samples to hit matched filter
bound performance
• Geometric and MMSE TEQs do
not work with 2 taps even with a
cyclic prefix of 32 samples
• Geometric TEQ gives lower
performance for small cyclic
prefix length
• Min-ISI TEQ with 3-sample
cyclic prefix outperforms
MMSE TEQ with 32-sample
cyclic prefix
TEQ taps
FFT size (N)
coding gain
margin
input power
noise power
crosstalk noise
2
512
4.2 dB
6 dB
14 dBm
-113 dBm/Hz,
10 ADSL disturbers
28
Simulation Results for 17-tap TEQ
Achievable percentage of MFB bit rate
Loop
MMSE
Geo
1
2
3
4
5
6
7
8
93%
97%
94%
91%
93%
93%
93%
93%
93%
97%
95%
92%
93%
93%
94%
94%
Cyclic prefix length of
FFT size (N)
Coding gain
Margin
MSSNR Min-ISI
99%
98%
98%
100%
100%
99%
100%
100%
32
512
4.2 dB
6 dB
100%
100%
100%
100%
100%
100%
100%
100%
Mbps
MCC
MFB
100%
100%
100%
100%
100%
100%
100%
100%
1.07
1.15
1.12
0.92
1.09
0.99
1.06
1.06
Input power
Noise power
Crosstalk noise
14 dBm
-113 dBm/Hz
10 ADSL disturbers
29
Simulation Results for Two-Tap TEQ
Mbps
Achievable percentage of MFB bit rate
Loop
MMSE
Geo
1
2
3
4
5
6
7
8
29%
31%
33%
22%
29%
29%
27%
28%
30%
30%
30%
23%
29%
29%
26%
29%
Cyclic prefix length of
FFT size (N)
Coding gain
Margin
MSSNR Min-ISI
99%
98%
99%
99%
99%
99%
99%
98%
32
512
4.2 dB
6 dB
99%
99%
100%
99%
100%
99%
99%
99%
MCC
MFB
99%
99%
100%
99%
100%
99%
99%
99%
1.07
1.15
1.12
0.92
1.09
0.99
1.06
1.06
Input power
Noise power
Crosstalk noise
14 dBm
-113 dBm/Hz
10 ADSL disturbers
30
Summary
• Design TEQ to maximize channel capacity
– No previous method truly maximizes channel capacity
• New subchannel SNR model
– Partitions the equalized channel into signal, noise, and ISI paths
– Enables to write channel capacity as a function of equalizer taps
• New maximum channel capacity TEQ design method
– Good benchmark for all design methods
– Requires nonlinear optimization
• New minimum-ISI design method
– Virtually same performance as the optimal method
– Fast implementation using recursive calculations
31
MATLAB DMTTEQ Toolbox
• Toolbox features ten TEQ design methods
• Available at http://signal.ece.utexas.edu/~arslan/dmtteq/
32
Future Research
• End-to-end optimization of channel capacity
– Joint optimization of bit loading and TEQ
– On-line adaptation of TEQ taps to track changes in channel
• Analysis of TEQ design methods
– Effect of analog transmit/receive filters and A/D and D/A converters
– Analyze performance under channel estimation errors
– Fixed-point analysis
• Extension to MCC and min-ISI methods
– Taking into account the noise floor
– Modifications to subchannel SNR model
– Optimal frequency domain weighting in min-ISI method
33
Capacity of Additive White Gaussian Noise Channel
• Maximum theoretical capacity of an additive white Gaussian
noise channel (no inter-symbol interference) is
C  log 2 1  SNR bits/symbo l
• Maximum achievable capacity can be defined as
 SNR 
m  log 2 1 
 bits/symbo l
 

• : SNR gap between theoretical and practical capacity
– Modulation method
– Coding gain
– Probability of error
– Margin for unaccounted distortions
34
Publications
•
G. Arslan, B. L. Evans, and S. Kiaei, ``Equalization for Discrete Multitone
Transceivers to Maximize Channel Capacity'', IEEE Trans. on Signal Processing,
submitted on April 17, 2000.
•
B. Lu, L. D. Clark, G. Arslan, and B. L. Evans, ``Discrete Multitone Equalization
Using Matrix Pencil and Divide-and-Conquer Methods'', IEEE Trans. on Signal
Processing, submitted on May 30, 2000.
•
J. Wu, G. Arslan, and B. L. Evans, ``Efficient Matrix Multiplication Methods to
Implement a Near-Optimum Channel Shortening Method for Discrete Multitone
Transceivers'', IEEE Asilomar Conf. on Signals, Systems, and Computers, Oct. 29 Nov. 1, 2000, Pacific Grove, CA.
•
B. Lu, L. D. Clark, G. Arslan, and B. L. Evans, ``Fast Time-Domain Equalization
for Discrete Multitone Modulation Systems'', IEEE Digital Signal Processing
Workshop, Oct. 15-18, 2000, Hunt, TX.
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G. Arslan, B. L. Evans, and S. Kiaei, ``Optimum Channel Shortening for
Multicarrier Transceivers'', IEEE Int. Conf. on Acoustics, Speech, and Signal
Processing, Jun. 5-9, 2000, vol. 5, pp. 2965-2968, Istanbul, Turkey.
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Acronyms
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ADC: Analog digital converter
ADSL: Asymmetric DSL
CAD: Computer aided design
CP: Cyclic prefix
DAC: Digital-analog converter
DMT: Discrete multitone
DSL: Digital subscriber line
FFT: Fast Fourier transform
HDSL: High-speed DSL
IFFT: Inverse FFT
ISDN: Integrated service digital
network
ISI: Intersymbol interference
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LAN: Local area network
MCC: Maximum channel capacity
MFB: Matched filter bound
min-ISI: Minimum ISI
MMSE: Minimum MSE
MSE: Mean squared error
MSSNR: Maximum SSNR
QAM: Quadrature amplitude
modulation
SNR: Signal-to-noise ratio
SSNR: shortening SNR
TEQ: Time domain equalizer
VDSL: Very-high-speed DSL
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