lecture5.ppt

4/5/00
p. 1
Postacademic Course on
Telecommunications
Module-3 Transmission Marc Moonen
Lecture-5 Equalization K.U.Leuven/ESAT-SISTA
Module-3 : Transmission
Lecture-5 (4/5/00)
Marc Moonen
Dept. E.E./ESAT, K.U.Leuven
marc.moonen@esat.kuleuven.ac.be
www.esat.kuleuven.ac.be/sista/~moonen/

Telecommunications
4/5/00
p. 2
Lecture-5 Equalization K.U.Leuven-ESAT/SISTA
Prelude
Comments on lectures being too fast/technical
* I assume comments are representative for (+/-)whole group
* Audience = always right, so some action needed….
To my own defense :-)
* Want to give an impression/summary of what today’s
transmission techniques are like (`box full of mathematics
& signal processing’, see Lecture-1).
Ex: GSM has channel identification (Lecture-6), Viterbi (Lecture-4),...
* Try & tell the story about the maths, i.o. math. derivation.
* Compare with textbooks, consult with colleagues working in
transmission...

Telecommunications
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p. 3
Prelude
Good news
* New start (I): Will summarize Lectures (1-2-)3-4.
-only 6 formulas-
* New start (II) : Starting point for Lectures 5-6 is 1 (simple)
input-output model/formula (for Tx+channel+Rx).
* Lectures 3-4-5-6 = basic dig.comms principles, from then
on focus on specific systems, DMT (e.g. ADSL), CDMA
(e.g. 3G mobile), ...
Bad news :
* Some formulas left (transmission without formulas = fraud)
* Need your effort !
* Be specific about the further (math) problems you may have.

Telecommunications
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p. 4
Lecture-5 : Equalization
Problem Statement :
• Optimal receiver structure consists of
* Whitened Matched Filter (WMF) front-end
(= matched filter + symbol-rate sampler + `pre-cursor
equalizer’ filter)
* Maximum Likelihood Sequence Estimator (MLSE),
(instead of simple memory-less decision device)
• Problem: Complexity of Viterbi Algorithm (MLSE)
• Solution: Use equalization filter + memory-less
decision device (instead of MLSE)...

Telecommunications
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p. 5
Lecture-5: Equalization - Overview
• Summary of Lectures (1-2-)3-4
Transmission of 1 symbol :
Matched Filter (MF) front-end
Transmission of a symbol sequence :
Whitened Matched Filter (WMF) front-end & MLSE (Viterbi)
• Zero-forcing Equalization
Linear filters
Decision feedback equalizers
• MMSE Equalization
• Fractionally Spaced Equalizers

Telecommunications
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p. 6
Summary of Lectures (1-2-)3-4
Channel Model:
Continuous-time channel
=Linear filter channel + additive white Gaussian noise (AWGN)
(symbols)
k
a
k
â
n(t)
+
AWGN
transmitter receiver (to be defined)
h(t)
channel
...
?
?

Telecommunications
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p. 7
Transmitter:
* Constellations (linear modulation):
n bits -> 1 symbol (PAM/QAM/PSK/..)
* Transmit filter p(t) :
receiver (to be defined)
...
s
k E
a .
r(t)
k
â
transmit
pulse
s(t)
n(t)
p(t) +
AWGN
transmitter
h(t)
channel
?
 

k
s
k
s kT
t
p
a
E
t
s )
(
.
.
)
(
k
a

Telecommunications
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p. 8
Transmitter:
-> piecewise constant p(t) (`sample & hold’) gives s(t) with
infinite bandwidth, so not the greatest choice for p(t)..
-> p(t) usually chosen as a (perfect) low-pass filter (e.g. RRC)
s
k E
a .
transmit
pulse
s(t)
p(t)
transmitter
discrete-time
symbol sequence
continuous-time
transmit signal
t
p(t)
t
Example

Telecommunications
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p. 9
Receiver:
In Lecture-3, a receiver structure was postulated (front-end
filter + symbol-rate sampler + memory-less decision
device). For transmission of 1 symbol, it was found that the
front-end filter should be `matched’ to the received pulse.
0
â
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
s
E
a .
0
transmit
pulse
p(t)
transmitter
h(t)
channel
0
u

Telecommunications
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p. 10
Receiver: In Lecture-4, optimal receiver design was
based on a minimum distance criterion :
• Transmitted signal is
• Received signal
• p’(t)=p(t)*h(t)=transmitted pulse, filtered by channel
  

k
s
k
s
a
a
a dt
kT
t
p
a
E
t
r
K
2
ˆ
,...,
ˆ
,
ˆ |
)
(
'
.
ˆ
.
)
(
|
min 1
0
 

k
s
k
s kT
t
p
a
E
t
s )
(
.
.
)
(
)
(
)
(
'
.
.
)
( t
n
kT
t
p
a
E
t
r
k
s
k
s 

 

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p. 11
Receiver: In Lecture-4, it was found that for transmission
of 1 symbol, the receiver structure of Lecture 3 is indeed
optimal !
2
0
0
0
ˆ ˆ
).
.
(
min 0
a
g
E
u s
a 
0
â
p’(-t)*
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
s
E
a .
0
transmit
pulse
p(t)
transmitter
h(t)
channel
sample at t=0
p’(t)=p(t)*h(t)
0
u

Telecommunications
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p. 12
• Receiver: For transmission of a symbol sequence, the
optimal receiver structure is...
k
â
p’(-t)*
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
s
k E
a .
transmit
pulse
p(t)
transmitter
h(t)
channel
sample at t=k.Ts
k
u
 






 
 

 
K
k
k
k
l
l
k
K
k
K
l
k
s
a
a u
a
a
g
a
E
K
1
*
1 1
*
ˆ
,...,
ˆ .
ˆ
2
ˆ
.
.
ˆ
.
min 0

Telecommunications
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p. 13
Receiver:
• This receiver structure is remarkable, for it is
based on symbol-rate sampling (=usually below
Nyquist-rate sampling), which appears to be
allowable if preceded by a matched-filter front-end.
• Criterion for decision device is too complicated.
Need for a simpler criterion/procedure...

Telecommunications
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p. 14
Receiver: 1st simplification by insertion of an additional
(magic) filter (after sampler).
* Filter = `pre-cursor equalizer’ (see below)
* Complete front-end = `Whitened matched filter’
k
â
p’(-t)*
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
s
k E
a .
transmit
pulse
p(t)
transmitter
h(t)
channel
k
u
1/L*(1/z*)
k
y
2
1 1
ˆ
,...,
ˆ .
ˆ
min 0  
 


K
m
K
k
k
m
k
m
a
a h
a
y
K

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p. 15
Receiver: The additional filter is `magic’ in that it turns the
complete transmitter-receiver chain into a simple input-
output model:
k
k
z
H
k
k
k
k
k
k
k
w
a
z
h
z
h
z
h
h
y
w
a
h
a
h
a
h
a
h
y


















.
...)
.
.
.
(
...
.
..
..
.
)
(
3
3
2
2
1
1
0
3
3
2
2
1
1
0




 




 

k
â
p’(-t)*
front-end
filter
1/Ts
receiver
n(t)
+
AWGN
s
k E
a .
transmit
pulse
p(t)
transmitter
h(t)
channel
k
u
1/L*(1/z*)
k
y

Telecommunications
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p. 16
Receiver: The additional filter is `magic’ in that it turns the
complete transmitter-receiver chain into a simple input-
output model:
= additive white Gaussian noise
means interference from future
(`pre-cursor) symbols has been cancelled, hence only
interference from past (`post-cursor’) symbols remains
k
k
k
k
k
k w
a
h
a
h
a
h
a
h
y 




 

 ...
.
.
.
. 3
3
2
2
1
1
0
k
w
0
...
2
1 

 
 h
h

Telecommunications
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p. 17
Receiver: Based on the input-output model
one can compute the transmitted symbol sequence as
A recursive procedure for this = Viterbi Algorithm
Problem = complexity proportional to M^N !
(N=channel-length=number of non-zero taps in H(z) )
k
k
k
k
k
k w
a
h
a
h
a
h
a
h
y 




 

 ...
.
..
..
. 3
3
2
2
1
1
0
2
1 1
ˆ
,...,
ˆ .
ˆ
min 0  
 


K
m
K
k
k
m
k
m
a
a h
a
y
K

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p. 18
Problem statement (revisited)
• Cheap alternative for MLSE/Viterbi ?
• Solution: equalization filter + memory-less
decision device (`slicer’)
Linear filters
Non-linear filters (decision feedback)
• Complexity : linear in number filter taps
• Performance : with channel coding, approaches
MLSE performance

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p. 19
Preliminaries (I)
• Our starting point will be the input-output model for
transmitter + channel + receiver whitened matched filter
front-end
k
k
k
k
k
k w
a
h
a
h
a
h
a
h
y 




 

 ...
.
.
.
. 3
3
2
2
1
1
0

1
h

3
h
0
h

2
h
k
a 3

k
a
2

k
a
1

k
a
k
y
k
w

Telecommunications
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p. 20
Preliminaries (II)
• PS: z-transform is `shorthand notation’ for discrete-time
signals…
…and for input/output behavior of discrete-time systems
)
(
)
(
).
(
)
(
hence
...
.
.
.
. 3
3
2
2
1
1
0
z
W
z
A
z
H
z
Y
w
a
h
a
h
a
h
a
h
y k
k
k
k
k
k







 


....
.
.
.
.
)
(
....
.
.
.
.
)
(
2
2
1
1
0
0
0
2
2
1
1
0
0
0
























z
h
z
h
z
h
z
h
z
H
z
a
z
a
z
a
z
a
z
A
i
i
i
i
i
i
)
(z
A
H(z)
)
(z
W
)
(z
Y

Telecommunications
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p. 21
Preliminaries (III)
• PS: if a different receiver front-end is used (e.g. MF
instead of WMF, or …), a similar model holds
for which equalizers can be designed in a similar fashion...
k
k
k
k
k
k
k w
a
h
a
h
a
h
a
h
a
h
y ~
...
.
~
.
~
.
~
.
~
.
~
... 2
2
1
1
0
1
1
2
2 






 






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p. 22
Preliminaries (IV)
PS: properties/advantages of the WMF front end
• additive noise = white (colored in general model)
• H(z) does not have anti-causal taps
pps: anti-causal taps originate, e.g., from transmit filter design (RRC,
etc.). practical implementation based on causal filters + delays...
• H(z) `minimum-phase’ :
=`stable’ zeroes, hence (causal) inverse exists &
stable
= energy of the impulse response maximally concentrated
in the early samples
k
w
0
...
2
1 

 
 h
h
)
(
1
z
H 

Telecommunications
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p. 23
Preliminaries (V)
• `Equalization’: compensate for channel distortion.
Resulting signal fed into memory-less decision device.
• In this Lecture :
- channel distortion model assumed to be known
- no constraints on the complexity of the
equalization filter (number of filter taps)
• Assumptions relaxed in Lecture 6

NOISE
ISI
3
3
2
2
1
1
0 ...
.
.
.
. k
k
k
k
k
k w
a
h
a
h
a
h
a
h
y 




 






 




 


Telecommunications
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p. 24
Zero-forcing & MMSE Equalizers
2 classes :
Zero-forcing (ZF) equalizers
eliminate inter-symbol-interference (ISI) at the
slicer input
Minimum mean-square error (MMSE) equalizers
tradeoff between minimizing ISI and minimizing
noise at the slicer input

NOISE
ISI
3
3
2
2
1
1
0 ...
.
.
.
. k
k
k
k
k
k w
a
h
a
h
a
h
a
h
y 




 






 




 


Telecommunications
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p. 25
Zero-forcing Equalizers
Zero-forcing Linear Equalizer (LE) :
- equalization filter is inverse of H(z)
- decision device (`slicer’)
• Problem : noise enhancement ( C(z).W(z) large)
)
(
)
( 1
z
H
z
C 

H(z)
)
(z
W
)
(z
Y
C(z)
)
(z
A )
(
ˆ z
A

Telecommunications
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p. 26
Zero-forcing Linear Equalizer (LE) :
- ps: under the constraint of zero-ISI at the slicer
input, the LE with whitened matched filter front-end
is optimal in that it minimizes the noise at the slicer
input
- pps: if a different front-end is used, H(z) may have
unstable zeros (non-minimum-phase), hence may
be `difficult’ to invert.

Telecommunications
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p. 27
Zero-forcing Non-linear Equalizer
Decision Feedback Equalization (DFE) :
- derivation based on `alternative’ inverse of H(z) :
(ps: this is possible if H(z) has , which is
another property of the WMF model)
- now move slicer inside the feedback loop :
)
(z
Y
1-H(z)
H(z)
)
(z
W
)
(z
A )
(
ˆ z
A
1
0 
h

Telecommunications
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p. 28
moving slicer inside the feedback loop has…
- beneficial effect on noise: noise is removed that
would otherwise circulate back through the loop
- beneficial effect on stability of the feedback loop:
output of the slicer is always bounded, hence
feedback loop always stable
Performance intermediate between MLSE and linear equaliz.
)
(z
Y
D(z)
H(z)
)
(z
W
)
(z
A
)
(
ˆ z
A
)
(
1
)
( z
H
z
D 


Telecommunications
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p. 29
Decision Feedback equalization (DFE) :
- general DFE structure
C(z): `pre-cursor’ equalizer
(eliminates ISI from future symbols)
D(z): `post-cursor’ equalizer
(eliminates ISI from past symbols)
)
(z
Y
)
(z
A
H(z)
)
(z
W
C(z) )
(
ˆ z
A
D(z)

Telecommunications
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p. 30
Decision Feedback equalization (DFE) :
- Problem : Error propagation
Decision errors at the output of the slicer cause a
corrupted estimate of the postcursor ISI.
Hence a single error causes a reduction of the noise
margin for a number of future decisions.
Results in increased bit-error rate.
)
(z
Y
H(z)
)
(z
W
)
(z
A
C(z) )
(
ˆ z
A
D(z)

Telecommunications
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p. 31
`Figure of merit’
• receiver with higher `figure of merit’ has lower error
probability
• is `matched filter bound’ (transmission of 1 symbol)
• DFE-performance lower than MLSE-performance, as DFE
relies on only the first channel impulse response sample
(eliminating all other ‘s), while MLSE uses energy of all
taps . DFE benefits from minimum-phase property (cfr.
supra, p.20)
MF
MLSE
DFE
LE 


 


MF

0
h
i
h
i
h

Telecommunications
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p. 32
MMSE Equalizers
• Zero-forcing equalizers: minimize noise at
slicer input under zero-ISI constraint
• Generalize the criterion of optimality to allow
for residual ISI at the slicer & reduce noise
variance at the slicer
=Minimum mean-square error equalizers

Telecommunications
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p. 33
MMSE Equalizers
MMSE Linear Equalizer (LE) :
- combined minimization of ISI and noise leads to
2
*
*
*
*
*
*
*
*
)
1
(
).
(
)
1
(
)
(
)
1
(
).
(
).
(
)
1
(
).
(
)
(
n
W
A
A
z
H
z
H
z
H
z
S
z
H
z
H
z
S
z
H
z
S
z
C





H(z)
)
(z
W
)
(z
Y
C(z)
)
(z
A )
(
ˆ z
A

Telecommunications
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p. 34
MMSE Equalizers
- signal power spectrum (normalized)
- noise power spectrum (white)
- for zero noise power -> zero-forcing
- (in the nominator) is a discrete-time matched filter,
often `difficult’ to realize in practice
(stable poles in H(z) introduce anticausal MF)
2
*
*
*
*
*
*
*
*
)
1
(
).
(
)
1
(
)
(
)
1
(
).
(
).
(
)
1
(
).
(
)
(
W
W
A
A
z
H
z
H
z
H
z
S
z
H
z
H
z
S
z
H
z
S
z
C






1
)
(z
SA

 2
)
( W
W z
S 
)
(
)
( 1
z
H
z
C 

)
1
( *
*
z
H

Telecommunications
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p. 35
MMSE Equalizers
MMSE Decision Feedback Equalizer :
• MMSE-LE has correlated `slicer errors’
(=difference between slicer in- and output)
• MSE may be further reduced by incorporating a `whitening’
filter (prediction filter) E(z) for the slicer errors
• E(z)=1 -> linear equalizer
• Theory & formulas : see textbooks
)
(z
Y
H(z)
)
(z
W
)
(z
A
C(z)E(z) )
(
ˆ z
A
1-E(z)

Telecommunications
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p. 36
Fractionally Spaced Equalizers
Motivation:
• All equalizers (up till now) based on (whitened) matched
filter front-end, i.e. with symbol-rate sampling, preceded by
an (analog) front-end filter matched to the received pulse
p’(t)=p(t)*h(t).
• Symbol-rate sampling = below Nyquist-rate sampling
(aliasing!). Hence matched filter is crucial for performance !
• MF front-end requires analog filter, adapted to channel
h(t), hence difficult to realize...
• A fortiori: what if channel h(t) is unknown ?
• Synchronization problem : correct sampling phase is
crucial for performance !

Telecommunications
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p. 37
• Fractionally spaced equalizers are based on Nyquist-rate
sampling, usually 2 x symbol-rate sampling (if excess
bandwidth < 100%).
• Nyquist-rate sampling also provides sufficient statistics,
hence provides appropriate front-end for optimal receivers.
• Sampler preceded by fixed (i.e. channel independent)
analog anti-aliasing (e.g. ideal low-pass) front-end filter.
• `Matched filter’ is moved to digital domain (after sampler).
• Avoids synchronization problem associated with MF
front-end.

Telecommunications
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p. 38
• Input-output model for fractionally spaced equalization :
`symbol rate’ samples :
`intermediate’ samples :
• may be viewed as 1-input/2-outputs system
k
k
k
k
k w
a
h
a
h
a
h
y ~
...
.
~
.
~
.
~
... 2
2
1
1
0 




 

2
/
1
2
2
/
5
1
2
/
3
2
/
1
2
/
1
~
...
.
~
.
~
.
~
... 


 




 k
k
k
k
k w
a
h
a
h
a
h
y

Telecommunications
4/5/00
p. 39
• Discrete-time matched filter + Equalizer (LE) :
• Fractionally spaced equalizer (LE) :
)
(
ˆ z
A
1/2Ts
2
MF(z) C(z)
equalizer
)
(t
r
C(z) )
(
ˆ z
A
1/2Ts
2
Fractionally spaced equalizer
)
(t
r
F(f)
F(f)

Telecommunications
4/5/00
p. 40
• Fractionally spaced equalizer (DFE):
• Theory & formulas : see textbooks & Lecture 6
C(z) )
(
ˆ z
A
D(z)
1/2Ts
2
)
(t
r
F(f)

Telecommunications
4/5/00
p. 41
Conclusions
• Cheaper alternatives to MLSE, based on
equalization filters + memoryless decision
device (slicer)
• Symbol-rate equalizers :
-LE versus DFE
-zero-forcing versus MMSE
-optimal with matched filter front-end, but several
assumptions underlying this structure are often
violated in practice
• Fractionally spaced equalizers (see also Lecture-6)

Telecommunications
4/5/00
p. 42
Assignment 3.1
• Symbol-rate zero-forcing linear equalizer has
i.e. a finite impulse response (`all-zeroes’) filter
is turned into an infinite impulse response filter
• Investigate this statement for the case of fractionally spaced
equalization, for a simple channel model
and discover that there exist finite-impulse response inverses in this
case. This represents a significant advantage in practice. Investigate
the minimal filter length for the zero-forcing equalization filter.
)
(
)
( 1
z
H
z
C 

2
2
1
1
0 .
.
)
( 



 z
h
z
h
h
z
H
)
.
.
/(
1
)
( 2
2
1
1
0




 z
h
z
h
h
z
C
2
2
/
5
1
2
/
3
2
/
1
2
/
1
2
2
1
1
0
.
.
.
.
.
.











k
k
k
k
k
k
k
k
a
h
a
h
a
h
y
a
h
a
h
a
h
y

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