Maximizing Data Rate of Discrete Multitone Systems
Using Time Domain Equalization Design
Miloš Milošević
Ph.D. Defense
Committee Members
Prof. Ross Baldick
Prof. Gustavo de Veciana
Prof. Brian L. Evans (advisor)
Prof. Edward J. Powers
Prof. Robert A. van de Geijn
Outline
• Broadband access technologies
• Background
– Multicarrier modulation
– Channel and noise
– Equalization
• Contributions
– Model subchannel SNR at multicarrier demodulator output
– Data rate optimal filter bank equalizer
– Data rate maximization finite impulse response equalizer
• Simulation results
• Conclusions and future work
2
Broadband Access Technologies
• Wireless Local Area Network
– Standardized in 1997
– 15M adaptors sold (2002)
– 4.4M access points sold (2002)
– Up to 54 Mbps data rate
– Data security issues
Standard
Modulation
Data Rate
Carrier
802.11
Single carrier
2 Mbps
2.4 GHz
802.11a
Multicarrier
54 Mbps
5.2 GHz
802.11b
Single carrier
11 Mbps
2.4 GHz
802.11g
Multicarrier
54 Mbps
2.4 GHz
• Cable Network
– Video broadcast since 1948
– Data service standardized 1998
– Shared coaxial cable medium: data security is an issue
– 42-850 MHz downstream (for broadcast), 5-42 MHz upstream
– Data Over Cable Service Interface Specifications 2.0 (2002)
• Downstream 6.4 MHz channel: up to 30.72 Mbps (shared)
• Upstream 6.4 MHz channel: up to 30.72 Mbps (shared)
3
Digital Subscriber Line (DSL) Standards
• Dedicated link
xDSL Modulation
over copper
HDSL Single
twisted pair
• “Last mile”
• Widely deployed: SDSL Single
North America, ADSL Multicarrier
(1998) <256 tones
West. Europe,
ADSL Multicarrier
South Korea
Lite
<128 tones
(35M lines)
(1998)
• In US cable leads VDSL Single or
(2003) Multicarrier
2 : 1 industry
<4092 tones
3 : 1 consumer
Data Rate
1544 kbps (N.A.)
2320 kbps (Europe)
2 x 1168 kbps (Europe)
3 x 784 kbps (Europe)
Band
193 kHz
580 kHz
292 kHz
196 kHz
 1.544 kbps
<386 kHz
 6144 (8192) kbps down
 786 (640) kbps up
1104 MHz
 1536 kbps down
 512 kbps up
552 kHz
 13 Mbps (N.A.) sym.
 22/3 Mbps (N.A.) asym.
 14.5 Mbps (N.A.) sym.
 23/4 Mbps (N.A.) asym.
12 MHz
(N.A.) - North America
4
DSL Broadband Access
Internet
Home Wireless LAN
Local Area
Network
Router
ATM
Switch
Home Hub
Wireless
Modem
DMT
Modem
DSLAM
downstream
Splitter
Wireless
Modem
Splitter
Set-top box
PC
upstream
Voice
Switch
PSTN
Central Office
Telephone
Customer Premises
ATM - Asynchronous Transfer Mode
DMT - Discrete Multitone
DSLAM - Digital Subscriber Line Access Multiplexer
LAN – Local Area Network
PSTN - Public Switched Telephone Network
5
Outline
•
•
Broadband access technologies
Background
–
–
–
•
Contributions
–
–
–
•
•
Multicarrier modulation
Channel and noise
Equalization
Model subchannel SNR at multicarrier demodulator output
Data rate optimal filter bank equalizer
Data rate maximization finite impulse response equalizer
Simulation results
Conclusions and future work
6
Multicarrier Modulation
• Frequency division multiplexing for transmission
• Carrier frequencies are spaced in regular increments up to
available system bandwidth
– Discrete multitone (DMT) modulation
– Orthogonal frequency division multiplexing
Transmit filter
m1 bits
M bits
Serial-toParallel
Converter
m2 bits
Encoding
To physical
medium
f1
Encoding
f2
mn bits
Encoding
Bit rate is M fsymbol bits/s
fn
-fx
fx
7
Parallel-to-Serial
Mirror data
and
N-IFFT
00101
QAM
encoder
Bits
Serial-toParallel
Discrete Multitone Transmitter
Add
Cyclic
Prefix
Digital-to-Analog
Converter +
Transmit Filter
N/2 subchannels
(complex-valued)
00101
I
CP: Cyclic Prefix
FFT: Fast Fourier Transform
QAM: Quadrature Amplitude Modulation
copy
symbol
Xi
CP
Q
symbol
N coefficients
(real-valued) N + n coefficients
CP
To Physical
Medium
symbol
n : cyclic prefix length
8
Channel and Noise
• Channel model
– Finite impulse response (FIR) filter
– Additive noise sources
White Noise, ISI, NEXT,
Echo, Quantization Error
• Channel noise sources
–
–
–
–
White noise
Channel
Input
Near-end echo
Near-end crosstalk (NEXT)
Intersymbol interference (ISI)
Output
Equalizer
Digital Noise
Floor
• Model other noise not introduced by the channel
– Analog-to-digital and digital-to-analog quantization error
– Digital noise floor introduced by finite precision arithmetic
9
Interference
• Intersymbol interference (ISI) occurs if channel impulse
response longer than cyclic prefix (CP) length + 1
– Received symbol is weighted sum of neighboring symbols
– Weights determined by channel impulse response
– Causes intercarrier interference
CP
Tx Symbol
=
Tx Symbol
Rx Symbol
Tx Symbol
Rx Symbol
*
channel
Rx Symbol
• Solution: Use channel shortening filter
Tx Symbol
*
filter
Tx Symbol
=
Tx Symbol
Rx Symbol
Rx Symbol
*
channel
Rx Symbol
10
Channel Shortening Filter
• Called time-domain equalizer (generally an FIR filter)
Channel
impulse
response
Transmission
delay
Shortened channel
impulse response

• If shortened channel length at most cyclic prefix length + 1
– symbol  channel  FFT(symbol) x FFT(channel)
– Division by FFT(channel) can undo linear time-invariant
frequency distortion in the channel
11
Frequency
domain
equalizer
= invert
channel
N-FFT
and
remove
mirrored
data
Remove
Cyclic
Prefix
Serial-to-Parallel
Discrete Multitone Receiver
TEQ
time
domain
equalizer
From Physical
Medium
N/2 subchannels
Parallel-toSerial
N coefficients
QAM
decoder
Receive Filter+
Analog-to-Digital
Converter
N + n coefficients
Bits
ADSL
00101
downstream upstream
4
32
64
512
n
N
12
Chow & Cioffi, 1992
Minimum Mean Squared Error Method
n
x
y
h
+
w
- +
• Minimize E{eTe}
e
Error: e = x*b - y*w
Equalized channel: h*w
b
z-
Virtual path
|DFT{h*w}|
Pick channel delay  and length of
b to shorten length of h*w
Minimum mean squared error
solution satisfies:
bT R xy  w T R yy
• Disadvantages
Deep notches in shortened channel
frequency response
Long equalizer reduces bit rate
Does not consider bit rate or noise
13
Melsa, Younce & Rohrs, 1996
Maximum Shortening SNR Method
• Minimize energy leakage outside shortened channel length
hshort  h * w  Hw  H winw  H wallw

T
min w H
w
• Disadvantages

Signal
Distortion
H wallw s.t. w H H winw  1
T
wall
– Does not consider bit rate
or channel noise
– Long equalizer reduces
bit rate
– Requires generalized
eigenvalue solution or
Cholesky decomposition
– Cannot shape TEQ according
to frequency domain needs
T
T
win
Channel h (blue line)
Yellow – leads to Hwall
Gray – leads to Hwin

sample number
14
Arslan, Kiaei & Evans, 2000
Minimum ISI Method
• Extends Maximum Shortening SNR method
– Adds frequency domain weighting of ISI
– Weight according to subchannel SNR; favors high SNR subchannels
– Does not minimize ISI in unused subchannels
• Minimizes weighted sum of subchannel ISI power under
constraint that power of signal is constant
 T T N / 2  S x ,k H 

min  w H wall   q k
q k H wallw  s.t. w T H Twin H win w  1

w 
S
k

1
n
,
k




Subchannel SNR
qk is kth column vector of N-length Discrete Fourier Transform matrix
(*)H is the Hermitian (conjugate transpose)
• Method is not optimal as it does not consider system bit
rate
15
Ding, Redfern & Evans, 2002
Dual-path Time Domain Equalizer
• Received signal passes through two parallel time domain
equalizers
– One time domain equalizer designed to minimize ISI over the
system bandwidth
– Other time domain equalizer designed for particular frequency
band, e.g. by using Minimum Intersymbol Interference method
TEQ 1
FFT
Subchannel
SNR
Comparison
Received
Signal
TEQ 2
FEQ
FFT
• Time domain equalizers are designed using sub-optimal
methods
FEQ – Frequency domain equalizer
16
Acker, Leus, Moonen, van der Wiel & Pollet, 2001
Per-tone Equalizer
• Transfers time
domain equalizer
operations to
frequency
domain
• Combined
complex multitap equalizer
• Each tone
(subchannel)
equalized
separately
yN+M-1
0
w1,0 w1,1
wi,M-1
Z1
yN+M-2
N/2
N+M-1
Sliding 0 w w
2,0
2,1
N-point
FFT
0
wN/2,0 wN/2,1
Z2
w2,M-1
Z
wN/2,M-1 N/2
y0
y – received symbol; M – subchannel equalizer length; w – complex equalizer;
Zk – received subsymbol in subchannel k; Sliding FFT - efficient implementation
of M fast Fourier transforms on M columns of convolution matrix of y with w
17
Outline
•
•
Broadband access technologies
Background
–
–
–
•
Contributions
–
–
–
•
•
Multicarrier modulation
Channel and noise
Equalization
Model subchannel SNR at multicarrier demodulator output
Data rate optimal filter bank equalizer
Data rate maximization finite impulse response equalizer
Simulation results
Conclusions and future work
18
Contribution #1
Interference-free Symbol at FFT Output
• FFT of circular convolution of channel and discrete
multitone symbol in kth subchannel

YkD  q Hk U circ
Hw
YkD is the desired subsymbol in subchannel k at FFT output

U circ
is desired symbol circular convolution matrix for delay 
H is channel convolution matrix
qk is kth column vector of N-length FFT matrix
• Received subsymbol in kth subchannel after FFT


YkR  q Hk U ISI H  G White  G NEXT  G FEXT  G Echo  G ADC w  Dk
U ISI is symbol convolution matrix (includes contributions from
previous, current, and next symbol)
G(*) is convolution matrix of source of noise or interference
Dk is digital noise floor, which is not affected by TEQ
19
Contribution #1
Model SNR at Output of Demodulator
• Proposed subchannel SNR model at demodulator~output
E[( YkD ) H YkD ]
wTAk w
SNR k (w ) 
 T~
R
D H
R
D
E[( Yk  Yk ) (Yk  Yk )] w B k w
– Ratio of quadratic functions in equalizer coefficients w
• Bits per frame as a nonlinear function of equalizer taps.
T








SNR
w
w
A k w 
int
k
   log 2  T

b w    log 2 1 
k
kI 

 kI 
 w B k w 
–
–
–
–
Multimodal for more than two-tap w
Nonlinear due to log and flooring operations
Requires integer maximization
Ak and Bk are Hermitian symmetric
• Maximizing bint is an unconstrained optimization problem
20
Contribution #2
Data Rate Optimal Filter Bank
• Find optimal time domain equalizer for every subchannel
w
opt
k

 w Tk A k w k
 arg max log 2  T
wk
 w k Bk w k


 w Tk A k w k
  arg max  T
wk

 w k Bk w k



• Generalized eigenvalue problem
opt
opt
opt
opt
w opt
satisfies
A
w

λ
B
w
for
λ
k
k
k
k
k
k
k  λ k for k
• Bit rate of bank of optimal time domain equalizer filters
int
opt
b
 
 

 w opt T A w opt 
  log 2  k T k k 
 w opt B w opt 
kI 
k
k 
 k

21
Contribution #2
Filter Bank Equalizer Architecture
y0
w0
w1
G0
CP
CP
Y0
y1
G1
Z0
FEQ0
Y1
FEQ1
Z1
x
Received
frame
wN/2-1 CP
input
yN/2-1
GN/2-1
TEQ Filter
Bank
Goertzel
Filter
Bank
TEQ
DFT
YN/2-1
FEQN/2-1
ZN/2-1
Frequency
Domain
Equalizer
output
22
Contribution #2
Filter Bank Summary
• Advantages
–
–
–
–
Provides a new achievable upper bound on bit rate performance
Single FIR can only perform at par or worse
Supports different subchannel transmission delays
Can modify frequency and phase offsets in multiple carriers by
adapting carrier frequencies of Goertzel filters
– Easily accommodates equalization of groups of tones with a
common filter with corresponding drop in complexity
• Disadvantages - computationally intensive
– Requires up to N/2 generalized eigenvalue solutions during
transceiver initialization
– Requires up to N/2 single FIR and as many Goertzel filters
23
Contribution #3
Data Rate Maximization Single FIR Design
• Find single FIR that performs as well as the filter bank
• Maximizing b(w) more tractable than maximizing bint(w)
 wTAk w 

bw    log 2  T
kI
 w Bk w 
• Maximizer of b(w) may be the maximizer of bint(w)
– Conjecture is that it holds true for 2- and 3-tap w
– Hope is that it holds for higher dimensions
• Maximizing sum of ratios is an open research problem
24
Contribution #3
Data Rate Maximization Single FIR Design
• Gradient-based optimization of b(w)
– Find gradient root corresponding to a local maximum
– Start with a good initial guess of equalizer taps w
– No guarantee of finding global maximum of b(w)
• Initial guess: filter bank FIR wkopt resulting in highest b(w)
• Parameterize problem to make it easier to find desired root
 T

H (λ )  max2 w  rk A k  λ k B k  w 
w , w 1 
kI

– H(l) is a convex, non-increasing
function of vector l
– Solution reached when H(l) = 0
– Solution corresponds to local
maximum closest to initial point
2
w T A k w log 2
wTAk w
lk ( w )  T
 SNR k w 
w Bk w
rk (w ) 
25
Equalizer Implementation Complexity
• Per tone equalizer
and single FIR
similar complexity
• Filter bank has
high complexity
• Example shown
N = 512
fsymbol = 4 kHz
fs=2.208 MHz
M=3
n= 32
Subsystem
Single
FIR
Filter
Bank
Multiply/adds* Words/
symbol
FIR
6.6e6
6
FFT
36.9e6
2048
FEQ
4.1e6
1024
Total
46.7e6
3078
FIR
1700e6
771
Goertzel
1000e6
2048
FEQ
4.1e6
1024
Total
2704.1e6
3843
FFT
36.9e6
2112
8.2e6
512
12.3e6
1024
57.4e6
3648
Per Tone Sliding FFT
Equalizer Combiner
fsymbol – Symbol rate
fs – Sample rate
Total
M – Equalizer length
* – Calculations assume N/2 data populated subchannels
26
Filter Bank Simulation Results
• Search to find filter length just before diminishing returns
– ADSL parameters except no constraints on bit allocation
– ADSL carrier serving area (CSA) lines used
• Optimal transmission delay found using line search
CSA loop
Data Rate
opt TEQ Size
1
11.417 Mbps
15
8
2
12.680 Mbps
22
12
3
10.995 Mbps
26
8
4
11.288 Mbps
35
6
5
11.470 Mbps
32
16
6
10.861 Mbps
20
8
7
10.752 Mbps
34
13
8
9.615 Mbps
35
11
27
Proposed vs. Other Equalization Designs
• Percentage of filter bank data rates for same filter length
– Each table entry averaged over TEQ lengths 2-32
– ADSL parameters with NEXT modeled as 49 ADSL disturbers
CSA loop Single FIR Min-ISI LS PTE MMSE-UEC MMSE-UTC
1
99.6%
97.5%
99.5%
86.3%
84.4%
2
99.6%
97.3%
99.5%
87.2%
85.8%
3
99.5%
97.3%
99.6%
83.9%
83.0%
4
99.3%
98.2%
99.1%
81.9%
81.5%
5
99.6%
97.2%
99.5%
88.6%
88.9%
6
99.5%
98.3%
99.4%
82.7%
79.8%
7
98.8%
96.3%
99.6%
75.8%
78.4%
8
98.7%
97.5%
99.2%
82.6%
83.6%
Average
99.3%
97.5%
99.4%
83.6%
83.2%
LS PTE – Least-squares Per-Tone Equalizer; UEC – Unit energy Constraint; UTC – Unit Tap Constraint
28
Data Rate vs. Equalizer Filter Length
• CSA loop 2 data rates for different equalizer filter lengths
– Standard ADSL parameters
– NEXT modeled as 49 disturbers
expanded
29
Spectrally Flat Equalizer Response
• Some design
methods
attempt to achieve
flatness using
empirical design
constraints
• Example: CSA
loop 4 SNR for
Single FIR, MBR
and Min-ISI
– MBR and Min-ISI
place nulls in
SNR (lowers data
rate)
– Proposed Single
FIR avoids nulls
Blue - Single FIR
Red – Min-ISI
Green - MBR
Detail
MBR – Maximum Bit Rate Time Domain Equalizer Design
30
Data Rate vs. Transmission Delay
– Optimal delay not
easily chosen prior to
actual design
– Exhaustive search of
delay values needed
12
- M=3
- M=10
- M=30
10
Bit Rate (Mbps)
• Transmission delay:
not known, TEQ design
parameter
• MMSE: Bit rate does
not change smoothly
as function of delay
8
6
4
2
0
0
10
20
30 40 50 60 70
Transmission Delay
• Single FIR: Bit rate
changes smoothly as function of delay
80
90 100
– Example: CSA loop 1 – “Sweet spot” increases with filter length
– Optimal bit rate for range of delays
31
Conclusions
• Subchannel SNR model noise sources not in other methods
– Crosstalk and echo
– Analog-to-digital conversion noise and digital noise floor
• Optimal time domain equalizer filter bank
– Bit rate in each subchannel maximized by separate TEQ filter
– Provides achievable upper bound on bit rate performance
– Available in freely distributable Discrete Multitone Time
Domain Equalizer Matlab Toolbox by Embedded Signal
Processing Laboratory (http://signal.ece.utexas.edu)
• Data maximization single time domain equalizer
– Achieves on average 99.3% of optimal filter bank performance
– Outperforms state of the art Min-ISI by 2% and MMSE by 15%
– Similar performance to least-squares per-tone equalizer
32
Future Work
•
•
•
Further research into architectures where equalizers are
assigned to spectral bands instead for each subchannel
Possibility of integrating time domain equalization with
the adjustment of Discrete Fourier Transform carrier
frequencies to maximize subchannel SNR
Adaptive and numerically inexpensive implementation of
Min-ISI method that removes TEQ length constraint of
the original method
33
Publications in Discrete Multitone
• Journal papers
–
–
–
–
M. Milosevic, L. F. C. Pessoa, B. L. Evans, and R. Baldick, “Optimal
time domain equalization design for maximizing data rate of discrete
multitone systems,” accepted for publication in IEEE Trans. On Signal
Proc.
M. Milosevic, T. Inoue, P. Molnar, and B. L. Evans, “Fast unbiased
echo canceller update during ADSL transmission,” to be published in
IEEE Trans. on Comm., April 2003.
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,” to be submitted to IEEE Trans. On
Signal Proc.
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,” to be submitted to IEEE Trans.
On Signal Proc.
34
Publications in Discrete Multitone
• Conference papers
–
–
M. Milosevic, L. F. C. Pessoa, and B. L. Evans, “Simultaneous
multichannel time domain equalizer design based on the maximum
composite shortening SNR,” in Proc. IEEE Asilomar Conf. on Sig.,
Sys., and Comp., vol. 2, pp. 1895-1899, Nov. 2002.
M. Milosevic, L. F. C. Pessoa, B. L. Evans, and R. Baldick, “Optimal
time domain equalization design for maximizing data rate of discrete
multitone systems,” in Proc. IEEE Asilomar Conf. on Sig., Sys., and
Comp., vol. 1, pp. 377-382, Nov. 2002.
35