Lecture5

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/

Postacademic Course on
Telecommunications

Module-3 Transmission
Lecture-5 Equalization

Marc Moonen
K.U.Leuven/ESAT-SISTA

4/5/00
p. 1

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


Marc Moonen
K.U.Leuven-ESAT/SISTA

4/5/00
p. 2

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


Marc Moonen

4/5/00
p. 3

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


Marc Moonen

4/5/00
p. 4

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


Marc Moonen

4/5/00
p. 5

Summary of Lectures (1-2-)3-4
Channel Model:
ak (symbols)

ˆ
ak
h(t)

?

transmitter

+
n(t)
AWGN

channel

...

?
receiver (to be defined)

Continuous-time channel
=Linear filter channel + additive white Gaussian noise (AWGN)
Telecommunications


Marc Moonen

4/5/00
p. 6

Transmitter:

r(t)

s(t)

ˆ
ak

ak . Es

p(t)

h(t)

...

transmit
pulse

transmitter

+
n(t)
AWGN

channel

?

receiver (to be defined)

* Constellations (linear modulation):
n bits -> 1 symbol a k (PAM/QAM/PSK/..)
* Transmit filter p(t) :

s(t )

Es .

ak . p(t kTs )
k

Telecommunications


Marc Moonen

4/5/00
p. 7

p(t)

Transmitter:

t

s(t)

Example

ak . Es

t

p(t)
discrete-time
symbol sequence

transmit
pulse

continuous-time
transmit signal

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)
Telecommunications


Marc Moonen

4/5/00
p. 8

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.
a0 . Es

p(t)

h(t)

transmit
pulse

transmitter
Telecommunications

1/Ts

+
n(t)
AWGN

channel

front-end
filter

u0

ˆ
a0

receiver
Marc Moonen

4/5/00
p. 9

Receiver: In Lecture-4, optimal receiver design was
based on a minimum distance criterion :

min a0 ,a1 ,...,aK | r (t )
ˆ ˆ
ˆ

ˆ
ak . p' (t kTs ) |2 dt

Es .
k

• Transmitted signal is

• Received signal

s(t )

Es .

ak . p(t kTs )
k

r (t )

Es .

ak . p' (t kTs ) n(t )
k

• p’(t)=p(t)*h(t)=transmitted pulse, filtered by channel
Telecommunications


Marc Moonen

4/5/00
p. 10

Receiver: In Lecture-4, it was found that for transmission
of 1 symbol, the receiver structure of Lecture 3 is indeed
optimal !

min a0 u0
ˆ
p’(t)=p(t)*h(t)

sample at t=0

a0 . Es

p(t)

h(t)

transmit
pulse

transmitter
Telecommunications

ˆ
( Es .g 0 ).a0

2

+
n(t)
AWGN

channel

1/Ts
p’(-t)*
u0
front-end
filter

ˆ
a0

receiver
Marc Moonen

4/5/00
p. 11

• Receiver: For transmission of a symbol sequence, the
optimal receiver structure is...
K

K

Es .
k 1 l 1

ak . Es

p(t)

h(t)

transmit
pulse

transmitter
Telecommunications

+
n(t)
AWGN

channel

2

ˆ*
ak .uk

uk

min a0 ,...,aK
ˆ
ˆ

ˆ*
ˆ
ak .g k l .al

K

ˆ
ak

k 1

1/Ts
p’(-t)*
front-end
filter
receiver

sample at t=k.Ts

Marc Moonen

4/5/00
p. 12

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


Marc Moonen

4/5/00
p. 13

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

min a0 ,...,aK
ˆ
ˆ

K

ym
m 1

2

ˆ
ak .hm

k

k 1

uk
ak . Es

p(t)
transmit
pulse

transmitter
Telecommunications

h(t)

+

n(t)
AWGN
channel

1/Ts
p’(-t)*
front-end
filter


yk

ˆ
ak

1/L*(1/z*)

receiver
Marc Moonen

4/5/00
p. 14

Receiver: The additional filter is `magic’ in that it turns the
complete transmitter-receiver chain into a simple inputoutput model:

yk

h0 .ak

h1..ak

h2 ..ak

yk

(h0 h1.z 1 h2 .z 2 h3 .z 3 ...).ak


1

2

h3 .ak

3

... wk
wk

H (z)

uk
ak . Es

p(t)
transmit
pulse

h(t)

1/Ts
p’(-t)*
front-end
n(t)
filter

ˆ
ak

+

AWGN
transmitter channel
Telecommunications

yk


1/L*(1/z*)

receiver
Marc Moonen

4/5/00
p. 15

Receiver: The additional filter is `magic’ in that it turns the
complete transmitter-receiver chain into a simple inputoutput model:

yk

h0 .ak

h1.ak

1

h2 .ak

2

h3 .ak

3

... wk

wk = additive white Gaussian noise
means interference from future
(`pre-cursor) symbols has been cancelled, hence only
interference from past (`post-cursor’) symbols remains

h1

h

Telecommunications

2

... 0


Marc Moonen

4/5/00
p. 16

Receiver: Based on the input-output model

yk

h0 .ak

h1..ak

h2 ..ak

1

2

h3 .ak

3

... wk

one can compute the transmitted symbol sequence as
K

min a0 ,...,aK
ˆ
ˆ

K

ym
m 1

2

ˆ
ak .hm

k

k 1

A recursive procedure for this = Viterbi Algorithm
Problem = complexity proportional to M^N !
(N=channel-length=number of non-zero taps in H(z) )
Telecommunications


Marc Moonen

4/5/00
p. 17

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
Telecommunications


Marc Moonen

4/5/00
p. 18

Preliminaries (I)
• Our starting point will be the input-output model for
transmitter + channel + receiver whitened matched filter
front-end

yk

h0 .ak

h1.ak

1

h2 .ak

ak

ak
h0

h1

2

h3 .ak
ak

1

h2

2

3

... wk
ak

h3

wk
Telecommunications

3

yk

Marc Moonen

4/5/00
p. 19

Preliminaries (II)
• PS: z-transform is `shorthand notation’ for discrete-time
signals…

A( z )

ai .z

i

a0 .z

0

a1.z

1

a2 . z

2

....

h0 .z

0

h1.z

1

h2 .z

2

....

i 0

H ( z)

hi .z

i

i 0

…and for input/output behavior of discrete-time systems

yk

h0 .ak

h1.ak

1

h2 .ak

hence
Y ( z ) H ( z ).A( z ) W ( z )
Telecommunications


2

h3 .ak
A(z )

3

... wk
Y (z )

H(z)
W (z )

Marc Moonen

4/5/00
p. 20

Preliminaries (III)

• PS: if a different receiver front-end is used (e.g. MF
instead of WMF, or …), a similar model holds

yk

~
... h 2 .ak

2

~
h 1.ak

1

~
~
h0 .ak h1.ak

1

~
h2 .ak

2

~
... wk

for which equalizers can be designed in a similar fashion...

Telecommunications


Marc Moonen

4/5/00
p. 21

Preliminaries (IV)
PS: properties/advantages of the WMF front end
• additive noise wk = white (colored in general model)
• H(z) does not have anti-causal taps h 1 h 2 ... 0
pps: anti-causal taps originate, e.g., from transmit filter design (RRC,
etc.). practical implementation based on causal filters + delays...

• H(z) `minimum-phase’ :
1

=`stable’ zeroes, hence (causal) inverse H ( z ) exists &
stable
= energy of the impulse response maximally concentrated
in the early samples
Telecommunications


Marc Moonen

4/5/00
p. 22

Preliminaries (V)
yk

h0 .ak

h1.ak 1 h2 .ak 2 h3 .ak 3 ...
 



wk


ISI

NOISE

• `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
Telecommunications


Marc Moonen

4/5/00
p. 23

Zero-forcing & MMSE Equalizers
yk

h0 .ak

h1.ak 1 h2 .ak 2 h3 .ak 3 ...
 



wk


ISI

NOISE

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
Telecommunications


Marc Moonen

4/5/00
p. 24

Zero-forcing Equalizers
Zero-forcing Linear Equalizer (LE) :
- equalization filter is inverse of H(z)
- decision device (`slicer’)
C ( z)
A(z )

H 1 ( z)
ˆ
A( z )

Y (z )
C(z)

H(z)

W (z )

• Problem : noise enhancement ( C(z).W(z) large)
Telecommunications


Marc Moonen

4/5/00
p. 25

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


Marc Moonen

4/5/00
p. 26

Zero-forcing Non-linear Equalizer
Decision Feedback Equalization (DFE) :
- derivation based on `alternative’ inverse of H(z) :
A(z )

ˆ
A( z )

Y (z )

H(z)
W (z )

1-H(z)

(ps: this is possible if H(z) has h0 1
another property of the WMF model)

, which is

- now move slicer inside the feedback loop :
Telecommunications


Marc Moonen

4/5/00
p. 27

A(z )

Y (z )
ˆ
A( z )

H(z)
W (z )

D(z)

D( z ) 1 H ( z )

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.
Telecommunications


Marc Moonen

4/5/00
p. 28

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)
A(z )

Y (z )

C(z)

H(z)
W (z )
Telecommunications

ˆ
A( z )


D(z)
Marc Moonen

4/5/00
p. 29

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.
A(z )

Y (z )

H(z)
W (z )
Telecommunications

ˆ
A( z )

C(z)


D(z)
Marc Moonen

4/5/00
p. 30

`Figure of merit’
LE

DFE

MLSE

MF

• 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 h0
(eliminating all other hi ‘s), while MLSE uses energy of all
taps hi . DFE benefits from minimum-phase property (cfr.
supra, p.20)
MF

Telecommunications


Marc Moonen

4/5/00
p. 31

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


Marc Moonen

4/5/00
p. 32

MMSE Equalizers
MMSE Linear Equalizer (LE) :
A(z )

ˆ
A( z )

Y (z )
C(z)

H(z)

W (z )

- combined minimization of ISI and noise leads to
1
)
*
z
* 1
S A ( z ).H ( z ).H ( * ) SW ( z )
z
S A ( z ).H * (

C ( z)

Telecommunications


1
)
*
z
* 1
H ( z ).H ( * )
z
H *(

Marc Moonen

2
n

4/5/00
p. 33

MMSE Equalizers
1
)
*
z
* 1
S A ( z ).H ( z ).H ( * ) SW ( z )
z
S A ( z ).H * (

C ( z)

-

1
)
*
z
* 1
H ( z ).H ( * )
z
H *(

2
W

S A (z ) 1

signal power spectrum (normalized)
2
SW ( z )
noise power spectrum (white)
W
1
for zero noise power -> zero-forcing C ( z ) H ( z )
* 1
H ( * ) (in the nominator) is a discrete-time matched filter,
z
often `difficult’ to realize in practice
(stable poles in H(z) introduce anticausal MF)

Telecommunications


Marc Moonen

4/5/00
p. 34

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
A(z )

Y (z )
ˆ
A( z )

C(z)E(z)

H(z)
W (z )

1-E(z)

• E(z)=1 -> linear equalizer
• Theory & formulas : see textbooks
Telecommunications


Marc Moonen

4/5/00
p. 35

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


Marc Moonen

4/5/00
p. 36

• 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


Marc Moonen

4/5/00
p. 37

• Input-output model for fractionally spaced equalization :
`symbol rate’ samples :

yk

~
... h0 .ak

~
h1.ak

1

~
h2 .ak

2

~
... wk

`intermediate’ samples :

yk

1/ 2

~
... h1/ 2 .ak

~
h3/ 2 .ak

1

~
h5 / 2 .ak

2

~
... wk

1/ 2

• may be viewed as 1-input/2-outputs system
Telecommunications


Marc Moonen

4/5/00
p. 38

• Discrete-time matched filter + Equalizer (LE) :
1/2Ts

r (t )

F(f)

MF(z)

2

C(z)

ˆ
A( z )

equalizer

• Fractionally spaced equalizer (LE) :
1/2Ts

r (t )

F(f)

C(z)

2

ˆ
A( z )

Fractionally spaced equalizer
Telecommunications


Marc Moonen

4/5/00
p. 39


• Fractionally spaced equalizer (DFE):
1/2Ts

r (t )

F(f)

C(z)

ˆ
A( z )

2
D(z)

• Theory & formulas : see textbooks & Lecture 6

Telecommunications


Marc Moonen

4/5/00
p. 40

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


Marc Moonen

4/5/00
p. 41

Assignment 3.1
• Symbol-rate zero-forcing linear equalizer has

H 1 ( z)

C ( z)

i.e. a finite impulse response (`all-zeroes’) filter

H ( z)

h0

h1.z

1

h2 .z

2

is turned into an infinite impulse response filter

C ( z ) 1 /(h0

1

h1.z

h2 .z 2 )

• Investigate this statement for the case of fractionally spaced
equalization, for a simple channel model

yk
yk

h0 .ak
1/ 2

h1.ak

h1/ 2 .ak

1

h2 .ak

h3 / 2 .ak

1

2

h5 / 2 .ak

2

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.
Telecommunications


Marc Moonen

4/5/00
p. 42

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