Maximizing Data Rate of Discrete Multitone SystemsMaximizing Data Rate of Discrete Multitone SystemsUsing Time Domain Equalization DesignUsing Time Domain Equalization Design
Miloš Milošević
Committee Members
Prof. Ross Baldick
Prof. Gustavo de Veciana
Prof. Brian L. Evans (advisor)
Prof. Edward J. Powers
Prof. Robert A. van de Geijn
Ph.D. Defense
2
OutlineOutline• 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
3
Broadband Access TechnologiesBroadband 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
• 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)
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
4
Digital Subscriber Line (DSL) StandardsDigital Subscriber Line (DSL) Standards• Dedicated link
over copper twisted pair
• “Last mile”
• Widely deployed: North America, West. Europe, South Korea (35M lines)
• In US cable leads2 : 1 industry3 : 1 consumer
xDSL Modulation Data Rate BandHDSL Single 1544 kbps (N.A.)
2320 kbps (Europe)
2 x 1168 kbps (Europe)
3 x 784 kbps (Europe)
193 kHz
580 kHz
292 kHz
196 kHz
SDSL Single 1.544 kbps <386 kHz
ADSL (1998)
Multicarrier
<256 tones
6144 (8192) kbps down
786 (640) kbps up
1104 MHz
ADSL Lite
(1998)
Multicarrier
<128 tones
1536 kbps down
512 kbps up
552 kHz
VDSL
(2003)
Single or Multicarrier <4092 tones
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
5
DSL Broadband AccessDSL Broadband Access
ATM - Asynchronous Transfer ModeDMT - Discrete MultitoneDSLAM - Digital Subscriber Line Access MultiplexerLAN – Local Area NetworkPSTN - Public Switched Telephone Network
Splitter
DMT Modem
Telephone
Wireless Modem
Wireless Modem
Home Hub
Local Area Network
Home Wireless LAN
Splitter
Voice Switch
PSTN
DSLAM
ATM Switch
Internet
Router
Set-top box
Customer Premises
Central Office
downstream
upstreamPC
6
OutlineOutline• 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
7
Multicarrier ModulationMulticarrier 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
Serial-to-Parallel
Converter
M bits
mn bits
Encoding
Encoding
Encoding
m2 bits
m1 bits
f1
f2
fn
To physical medium
Bit rate is M fsymbol bits/s
Transmit filter
-fx fx
8
Discrete Multitone Transmitter Discrete Multitone Transmitter S
eria
l-to-
Par
alle
l
QA
Men
cod
er
Mirr
or d
ata
and
N-I
FF
T
Add Cyclic Prefix
Digital-to-Analog Converter +
Transmit Filter
N/2 subchannels(complex-valued)
Bits
00101
Par
alle
l-to-
Ser
ial
To Physical Medium
N coefficients(real-valued) N + coefficients
copy
I
Q iX
00101
sym
bo
l
sym
bo
lC
P
symbolCP
CP: Cyclic PrefixFFT: Fast Fourier TransformQAM: Quadrature Amplitude Modulation : cyclic prefix length
9
Channel and NoiseChannel and Noise• Channel model
– Finite impulse response (FIR) filter
– Additive noise sources
• Channel noise sources– White noise
– Near-end echo
– Near-end crosstalk (NEXT)
– Intersymbol interference (ISI)
• 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
Channel Equalizer
White Noise, ISI, NEXT, Echo, Quantization Error
Digital Noise Floor
Input
Output
10
InterferenceInterference• 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
• Solution: Use channel shortening filter
Tx Symbol Tx Symbol Tx Symbol
Rx Symbol Rx Symbol Rx Symbol
* channel
=
CP
Tx Symbol Tx Symbol Tx Symbol
Rx Symbol Rx Symbol Rx Symbol
* channel
=* filter
11
Channel Shortening FilterChannel Shortening Filter• Called time-domain equalizer (generally an FIR filter)
• 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
Channel impulse response
Shortened channel impulse response
Transmission delay
12
TEQtime
domain equalizer
QAMdecoder
Frequencydomain
equalizer= invert channel
N-FFTand
removemirrored
data
Discrete Multitone Receiver Discrete Multitone Receiver
Remove Cyclic Prefix
Receive Filter+Analog-to-Digital
Converter
N/2 subchannels
Bits
00101
Ser
ial-t
o-P
aral
lel
From Physical Medium
Par
alle
l-to-
Ser
ial
N coefficientsN + coefficients
ADSL downstream upstream 32 4 N 512 64
13
z-
h + w
b
-x y e
n
+
• Minimize E{eTe}
Error: e = x*b - y*w
Equalized channel: h*w
Pick channel delay and length of
b to shorten length of h*w
Minimum mean squared error
solution satisfies:
• DisadvantagesDeep notches in shortened channel
frequency response Long equalizer reduces bit rateDoes not consider bit rate or noise
Minimum Mean Squared Error MethodMinimum Mean Squared Error Method
yyxy RwRb TT
|DFT{h*w}|
Virtual path
Chow & Cioffi, 1992
14
Maximum Shortening SNR MethodMaximum Shortening SNR Method• Minimize energy leakage outside shortened channel length
• Disadvantages– Does not consider bit rate
or channel noise
– Long equalizer reduces bit rate
– Requires generalizedeigenvalue solution orCholesky decomposition
– Cannot shape TEQ accordingto frequency domain needs
Yellow – leads to Hwall
Gray – leads to Hwin
sample number
Channel h (blue line)
1 s.t. min winTwin
Twall
Twall
T wHHwwHHww
wHwHHwwhh wallwinshort *
Melsa, Younce & Rohrs, 1996
DistortionSignal
15
Minimum ISI MethodMinimum 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
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
1 s.t. min winTwin
Twall
2/
1
H
,
,Twall
T
wHHwwHqqHww
N
kk
kn
kxk S
S
Subchannel SNR
Arslan, Kiaei & Evans, 2000
16
Dual-path Time Domain EqualizerDual-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
• Time domain equalizers are designed using sub-optimal methods
FEQ – Frequency domain equalizer
TEQ 1
TEQ 2
FFT
FFT
Subchannel SNR
Comparison
FEQReceived Signal
Ding, Redfern & Evans, 2002
17
Per-tone EqualizerPer-tone Equalizer• Transfers time
domain equalizer operations to frequency domain
• Combined complex multi-tap equalizer
• Each tone (subchannel) equalized separately
SlidingN-point
FFT
yN+M-1
yN+M-2
y0
Z1
Z2
w1,0 w1,1 wi,M-1
0
w2,0 w2,1 w2,M-1
0
wN/2,0 wN/2,1 wN/2,M-1
0 ZN/2
N+M-1
N/2
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
Acker, Leus, Moonen, van der Wiel & Pollet, 2001
18
OutlineOutline• 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
19
Interference-free Symbol at FFT OutputInterference-free Symbol at FFT Output• FFT of circular convolution of channel and discrete
multitone symbol in kth subchannel
is the desired subsymbol in subchannel k at FFT output 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
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
wHUqY circHD kk
circU
DkY
kkk D wGGGGGHUqY ADCEchoFEXTNEXTWhiteISIHR
ISIU
Contribution #1
20
Model SNR at Output of DemodulatorModel SNR at Output of Demodulator• Proposed subchannel SNR model at demodulator output
– Ratio of quadratic functions in equalizer coefficients w
• Bits per frame as a nonlinear function of equalizer taps.
– 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
wBw
wAw
YYYY
YYw
T
T
k
k
kkkk
kkk ~
~
)]()E[(
])E[()(SNR
DRHDR
DHD
Contribution #1
Ik k
k
Ik k
kbwBw
wAwww
T
T
22int log
SNR1log
21
Data Rate Optimal Filter BankData Rate Optimal Filter Bank• Find optimal time domain equalizer for every subchannel
• Generalized eigenvalue problem
• Bit rate of bank of optimal time domain equalizer filters
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
2intopt log
wBw
wAw
Contribution #2
22
TEQ Filter Bank
Filter Bank Equalizer ArchitectureFilter Bank Equalizer Architecture
Goertzel Filter Bank
G0
G1
GN/2-1
y0
y1
yN/2-1
Y0
Y1
YN/2-1
Frequency Domain
Equalizer
FEQ0
FEQ1
FEQN/2-1
Z0
Z1
ZN/2-1
w0
w1
wN/2-1
x
CP
CP
CP
Received frame
TEQinput DFT output
Contribution #2
23
• 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
Filter Bank SummaryFilter Bank Summary
Contribution #2
24
• Find single FIR that performs as well as the filter bank
• Maximizing b(w) more tractable than maximizing bint(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
Data Rate Maximization Single FIR DesignData Rate Maximization Single FIR Design
Ik k
kbwBw
wAww
T
T
2log
Contribution #3
25
• 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
– H() is a convex, non-increasing
function of vector – Solution reached when H() = 0
– Solution corresponds to local maximum closest to initial point
Data Rate Maximization Single FIR DesignData Rate Maximization Single FIR Design
wBAwλ
ww IkkkkkH λrmax)( T
1,2
wwBw
wAww
wAww
T
T
T
kk
kk
kkr
SNR)(
2log
2)(
Contribution #3
26
Equalizer Implementation ComplexityEqualizer Implementation Complexity
• Per tone equalizer and single FIR similar complexity
• Filter bank has high complexity
• Example shownN = 512
fsymbol = 4 kHz
fs=2.208 MHz
M = 3
= 32
Subsystem Multiply/adds*
Words/symbol
Single FIR
FIR 6.6e6 6
FFT 36.9e6 2048
FEQ 4.1e6 1024
Total 46.7e6 3078
FilterBank
FIR 1700e6 771
Goertzel 1000e6 2048
FEQ 4.1e6 1024
Total 2704.1e6 3843
Per ToneEqualizer
FFT 36.9e6 2112
Sliding FFT 8.2e6 512
Combiner 12.3e6 1024
Total 57.4e6 3648
fsymbol – Symbol rate fs – Sample rate M – Equalizer length * – Calculations assume N/2 data populated subchannels
27
Filter Bank Simulation ResultsFilter 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 searchCSA 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
28
Proposed vs. Other Equalization DesignsProposed 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
LS PTE – Least-squares Per-Tone Equalizer; UEC – Unit energy Constraint; UTC – Unit Tap Constraint
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%
29
Data Rate vs. Equalizer Filter LengthData Rate vs. Equalizer Filter Length• CSA loop 2 data rates for different equalizer filter lengths
– Standard ADSL parameters
– NEXT modeled as 49 disturbers
expanded
30
Spectrally Flat Equalizer ResponseSpectrally 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
Detail
Blue - Single FIR Red – Min-ISIGreen - MBR
MBR – Maximum Bit Rate Time Domain Equalizer Design
31
Data Rate vs. Transmission DelayData Rate vs. Transmission Delay• Transmission delay:
not known, TEQ design parameter
• MMSE: Bit rate does not change smoothly as function of delay
– Optimal delay not easily chosen prior to actual design
– Exhaustive search of delay values needed
• Single FIR: Bit rate changes smoothly as function of delay – Example: CSA loop 1 – “Sweet spot” increases with filter length– Optimal bit rate for range of delays
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
Bit
Rat
e (M
bp
s)
Transmission Delay
- M=3
- M=10
- M=30
32
ConclusionsConclusions• 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
33
Future WorkFuture 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
34
Publications in Discrete MultitonePublications 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.
35
Publications in Discrete MultitonePublications 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.