1. On the Relation Between Time-Domain
Equalizers and Per-Tone Equalizers for
DMT-Based Systems
Koen Vanbleu, Geert Ysebaert, Gert Cuypers, Marc Moonen
Katholieke Universiteit Leuven, ESAT / SCD-SISTA, Belgium
IEEE Benelux Signal Processing Symposium
Hilvarenbeek, the Netherlands
April 16, 2004

2. On Equalization Alternatives and their
Relations for ADSL Modems
Koen Vanbleu, Geert Ysebaert, Gert Cuypers, Marc Moonen
Katholieke Universiteit Leuven, ESAT / SCD-SISTA, Belgium
IEEE Benelux Signal Processing Symposium
Hilvarenbeek, the Netherlands
April 16, 2004

3. Overview
• ADSL Basics
 What? Multicarrier modulation
 Transmitter/Receiver
• ADSL Equalizer Design
 Problem Description
 Current Equalizers (TD-MMSE)
 Bit rate Maximizing Equalizers
(FD-MMSE criterion)
3
 Relations

4. Introduction
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
Broadband communication over telephone line
 ADSL (Asymmetric Digital Subscriber Line)
 ADSL2 / ADSL2+ (Second-Generation ADSL)
 VDSL (Very high bit rate Digital Subscriber Line)
 Bit rate is function of the line length
Downstream
Central
Customer
Upstream
Down
ADSL
6 Mbps
640 kbps
3.7 km
1.1 MHz
ADSL2+
5
Up
Line length Frequency band
15 Mbps
1.5 Mpbs
1.8 km
2.2 MHz
VDSL
52 Mbps
2.3 Mbps
< 1 km
12 MHz

5. Multicarrier Modulation
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
•
Digital multicarrier modulation scheme:
Discrete Multitone (DMT)
e.g. ADSL
POTS
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
UP
4 25
DOWN
138
1104 f (kHz)
•
•
6
Assign different frequency bins to up- and
downstream directions
Traditional telephony (POTS) still available over
the same wire.

6. Discrete Multitone: Transmitter
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
bits
10
Data symbols (QAM)
Im
00
0
Re
11
ˆ
xk , n
01
2 bits
Cyclic Prefix
...
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
.
.
.
N-point
.
.
.
IDFT
CP
P/S
xl
Im
7
4 bits
...
Re
N / 2 +1
Block
transmission!
N
IDFT modulation
(Inverse Discrete Fourier Transform)

7. Why Equalization?
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
...
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
CP
ˆ
xk , n
N / 2 +1
.
.
.
.
.
.
N-point
IDFT
P/S
...
N
Transmitter
8
noise nl
channel
xl
yl
Why equalization?
“Invert” channel
distortion while not
boosting noise

8. Discrete Multitone: Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
noise
- Bit rate Maximizing
Equalizers
- Relations
CP
nl
Frequency Data symbols
Domain
Equalizer
bits
1 tap / tone
TEQ w
yl
h∗w
h
Time
Domain
Equalizer
S/P
.
.
. N-point
DFT
x
9
CP length + 1
.
.
.
Im
00
Re
01
11
2 bits
Im
~
dn
T taps
l
10
x
x
channel h
xl
...
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
DFT demodulation
Re
4 bits

9. DMT Equalization:
Problem Description
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
10
T taps
y
• ADSL Equalizer
l
Design
- Problem Description
- Current Equalizers
To
- Bit rate Maximizing
Equalizers
- Relations
TEQ w
CP
1 tap/tone
S/P
~
... N-point ... FEQ d ...
FE n
DFT
Q
Im
11
00
Re
01
2 bits
~ +e
xk ,n ~k ,n
maximize bit rate:
 SNRn (w ) 

b = ∑ bits on tone n = ∑ log 2 1 +


Γn
tones n
tones n


~ 2
E xk , n
Residual ISI/ICI
where SNRn ( w ) =
~ (w ) 2
E e k ,n
Noise (RFI/XT/etc.)
10
{
{
}
}
is hard with time-domain equalizer w

10. Current ADSL Equalizers
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
Channel shorteners
e.g. time-domain MMSE-TEQ design
noise nl
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
Channel h
xl
TIR b
delay
el
-
TIR = target impulse response of (CP-length+1)
h∗w
h
L −1
min ∑ y w − x b s.t. constraint on w
w, b
11
yl
TEQ w
l
CP length + 1
l =0
T
l
T
l
2
Constrained linear rate optimization! MMSE-TEQ
No bit least-squares based

11. Bit rate Maximizing Equalizers:
Per-tone equalization
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
 Per-tone
equalization
 BM-TEQ
- Relations
∆
Time
Domain N
Equalize
r
T taps
yl
TEQ w
∆
∆
x
↓ N +ν
x
.
.
.
↓ N +ν
.
.
. N-point
DFT
~
FEQ d n
↓ N +ν
y
12
x
k, w
~
~ = d F ((Y w ))
ˆ
Fn Y kw
xk , n
n n k 
k



yk , ,,w
yykkww
.
.
.
~
ˆ
xk , n

12. Bit rate Maximizing Equalizers:
Per-tone equalization
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
T −1
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
 Per-tone
equalization
 BM-TEQ
- Relations
∆
∆
N
...
∆
...
↓ N +ν
↓ N +ν
...
∆
yl
13
∆
S/P N + ν
↓
↓ N +ν
...
PTEQ ...
sliding
~
T
ˆ
xk , n
N-point
DFT
T –tap linear combiner
... N-point
DFT
w for each tone
n
↓ N +ν
~
~
~ = d F (Y w ) =( FFY )(wd )
ˆ
xk , n
( n n Yk ) n
k
n n
k


 

y k ,w
wn

13. Bit rate Maximizing Equalizers:
Per-tone equalization
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
 Per-tone
equalization
 BM-TEQ
- Relations
2
K −1
min ∑ ( Fn Yk )w n − ~k ,n
x
~
wn
k =0
wd n
• Least-squares criterion per tone: “design TEQ per tone”
• Optimizes the SNR per tone
→ Performs always better than a TEQ
• Efficient computation: exploit sliding DFT structure
→ similar complexity as TEQ
14

14. • ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
Bit rate Maximizing (BM)
Equalizers:
Time-domain equalization
}
2
 SNRn 
 where SNRn (w / w n ) =
max ∑ log 2 1 +

2
Γn 
tones n


E ~k , n ( w / w n )
e
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
 Per-tone
equalization
 BM-TEQ
- Relations
{
PTEQ: min
wn
2
K −1
wd n
min ∑
~
w ,d n
∑
k = 0 tones n
T taps
yl
2
~ (w ) = min ( F Y )w − ~
∑ ek ,n n
∑ n k ~n xk ,n
wn
k =0
k =0
K −1
BM-TEQ:
}
K −1
~
γ n d n ( Fn Yk )w − ~k ,n
x
with γ n
15
{
E ~k ,n
x
CP
TEQ
S/P
2
~
= f ( SNRn ) = g (ek ,n )
~
FE d n
Q
x
... N-point ...
...
DFT
x

15. Bit rate Maximizing Equalizers:
Time-domain equalization
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
 Per-tone
equalization
 BM-TEQ
- Relations
K −1
min ∑
~
w ,d n
k = 0 tones n
K −1
= min ∑
~
w ,d n
with
∑
∑
k = 0 tones n
~
~ (w, d )
γ n ek ,n
n
2
~
γ n d n ( Fn Yk )w − ~k ,n
x
2
~
~ (d , w ))
γ n = f ( SNRn ) = g (ek ,n n
Iteratively-reweighted
separable non-linear least squares-based
frequency-domain MMSE-TEQ design
16

16. Relation between ADSL equalizers
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
(Channel shortening) TD-MMSE-TEQ: constrained linear LS
Block transmission/CP/Multicarrier
FD-MMSE-TEQ: constrained linear LS or separable NL-LS
Bit rate maximization
BM-TEQ: iteratively reweighted separable NL-LS
Only 1 tone
17
PTEQ: linear LS
Remarkable correspondence
between generalized
eigenvalue problems

17. Relation between ADSL equalizers
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
18
TD-MMSE-TEQ
Real FD-MMSE-TEQ 1
Complex FD-MMSE-TEQ
1
Real FD-MMSE-TEQ 2
Complex FD-MMSE-TEQ
2
Real BM-TEQ
Complex BM-TEQ
Real PTEQ
Complex PTEQ

18. Relation between ADSL equalizers
• ADSL Basics
- Intro
- DMT Transmitter
- Why Equalization?
- DMT Receiver
• ADSL Equalizer
Design
- Problem Description
- Current Equalizers
- Bit rate Maximizing
Equalizers
- Relations
(Channel shortening) TD-MMSE-TEQ: constrained linear LS
Block transmission/CP/Multicarrier
FD-MMSE-TEQ: constrained linear LS or separable NL-LS
Bit rate maximization
BM-TEQ: iteratively reweighted separable NL-LS
Only 1 tone
19
PTEQ: linear LS
Remarkable correspondence
between generalized
eigenvalue problems