Lecture5

Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven/ESAT-SISTA

4/5/00p. 1

Postacademic Course on Telecommunications

Module-3 : Transmission

Lecture-5 (4/5/00)

Marc Moonen

Dept. E.E./ESAT, K.U.Leuven

[email protected]

www.esat.kuleuven.ac.be/sista/~moonen/


Module-3 Transmission Marc MoonenLecture-5 Equalization K.U.Leuven-ESAT/SISTA

4/5/00p. 2

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...



4/5/00p. 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.



4/5/00p. 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)...



4/5/00p. 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



4/5/00p. 6

Summary of Lectures (1-2-)3-4

Channel Model:

Continuous-time channel

=Linear filter channel + additive white Gaussian noise (AWGN)

(symbols) kaka

n(t)

+

AWGN

transmitter receiver (to be defined)

h(t)

channel

...

??



4/5/00p. 7


Transmitter:

* Constellations (linear modulation): n bits -> 1 symbol (PAM/QAM/PSK/..)

* Transmit filter p(t) :

receiver (to be defined)

...

sk Ea .

r(t)

ka

transmit

pulse

s(t)

n(t)

p(t) +

AWGNtransmitter

h(t)

channel?

k

sks kTtpaEts )(..)(

ka



4/5/00p. 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)

sk Ea .

transmit

pulse

s(t)

p(t)

transmitter

discrete-timesymbol sequence

continuous-timetransmit signal

tp(t)

t

Example



4/5/00p. 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.

0afront-end

filter

1/Ts

receiver

n(t)

+

AWGN

sEa .0

transmit

pulse

p(t)

transmitter

h(t)

channel

0u



4/5/00p. 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

sksaaa dtkTtpaEtrK

2ˆ,...,ˆ,ˆ |)('.ˆ.)(|min

10

k

sks kTtpaEts )(..)(

)()('..)( tnkTtpaEtrk

sks



4/5/00p. 11


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

2

000ˆ ˆ)..(min0

agEu sa

0ap’(-t)*

front-end

filter

1/Ts

receiver

n(t)

+

AWGN

sEa .0

transmit

pulse

p(t)

transmitter

h(t)

channel

sample at t=0p’(t)=p(t)*h(t)

0u



4/5/00p. 12


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

ka

p’(-t)*

front-end

filter

1/Ts

receiver

n(t)

+

AWGN

sk Ea .

transmit

pulse

p(t)

transmitter

h(t)

channelsample at t=k.Ts

ku

K

kkkllk

K

k

K

lksaa uaagaE

K1

*

1 1

*ˆ,...,ˆ .ˆ2ˆ..ˆ.min

0



4/5/00p. 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...



4/5/00p. 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’

ka

p’(-t)*front-end

filter

1/Ts

receiver

n(t)

+

AWGN

sk Ea .

transmit

pulse

p(t)

transmitter

h(t)

channel

ku

1/L*(1/z*)

ky

2

1 1ˆ,...,ˆ .ˆmin

0

K

m

K

kkmkmaa hay

K



4/5/00p. 15


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

kk

zH

k

kkkkkk

wazhzhzhhy

wahahahahy

....)...(

.........

)(

33

22

110

3322110

ka

p’(-t)*front-end

filter

1/Ts

receiver

n(t)

+

AWGN

sk Ea .

transmit

pulse

p(t)

transmitter

h(t)

channel

ku

1/L*(1/z*)

ky



4/5/00p. 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

kkkkkk wahahahahy ....... 3322110

kw

0...21 hh



4/5/00p. 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) )

kkkkkk wahahahahy ......... 3322110

2

1 1ˆ,...,ˆ .ˆmin

0

K

m

K

kkmkmaa hay

K



4/5/00p. 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



4/5/00p. 19

Preliminaries (I)

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

kkkkkk wahahahahy ....... 3322110

1h

3h0h

2h

ka 3ka2ka1ka

kykw



4/5/00p. 20

Preliminaries (II)

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

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

)()().()(

hence

....... 3322110

zWzAzHzY

wahahahahy kkkkkk

........)(

........)(

22

11

00

0

22

11

00

0

zhzhzhzhzH

zazazazazA

i

ii

i

ii

)(zAH(z)

)(zW

)(zY



4/5/00p. 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...

kkkkkkk wahahahahahy ~....~

.~

.~

.~

.~

... 221101122



4/5/00p. 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

kw

0...21 hh

)(1 zH



4/5/00p. 23

Preliminaries (V)

• Èqualization’: 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

NOISEISI

3322110 ....... kkkkkk wahahahahy



4/5/00p. 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

NOISEISI

3322110 ....... kkkkkk wahahahahy



4/5/00p. 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 zHzC

H(z)

)(zW

)(zY

C(z)

)(zA )(ˆ zA



4/5/00p. 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.



4/5/00p. 27


Zero-forcing Non-linear Equalizer

Decision Feedback Equalization (DFE) :

- derivation based on àlternative’ 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 :

)(zY

1-H(z)

H(z)

)(zW

)(zA )(ˆ zA

10 h



4/5/00p. 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.

)(zY

D(z)

H(z)

)(zW

)(zA

)(ˆ zA

)(1)( zHzD



4/5/00p. 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)

)(zY)(zA

H(z)

)(zW

C(z) )(ˆ zA

D(z)



4/5/00p. 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.

)(zY

H(z)

)(zW

)(zA

C(z) )(ˆ zA

D(z)



4/5/00p. 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)

MFMLSEDFELE

MF

0hih

ih



4/5/00p. 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



4/5/00p. 33

MMSE Equalizers

MMSE Linear Equalizer (LE) :

- combined minimization of ISI and noise leads to

2*

*

**

**

**

)1

().(

)1

(

)()1

().().(

)1

().()(

nWA

A

zHzH

zH

zSz

HzHzS

zHzS

zC

H(z)

)(zW

)(zY

C(z)

)(zA )(ˆ zA



4/5/00p. 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

().()(

WWA

A

zHzH

zH

zSz

HzHzS

zHzS

zC

1)(zSA 2)( WW zS

)()( 1 zHzC )

1(

**

zH



4/5/00p. 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

)(zY

H(z)

)(zW

)(zA

C(z)E(z) )(ˆ zA

1-E(z)



4/5/00p. 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 !



4/5/00p. 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.



4/5/00p. 38


• Input-output model for fractionally spaced equalization :

`symbol rate’ samples :

ìntermediate’ samples :

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

kkkkk wahahahy ~....~

.~

.~

... 22110

2/122/512/32/12/1~....

~.

~.

~... kkkkk wahahahy



4/5/00p. 39


• Discrete-time matched filter + Equalizer (LE) :

• Fractionally spaced equalizer (LE) :

)(ˆ zA1/2Ts

2MF(z) C(z)

equalizer

)(trC(z) )(ˆ zA

1/2Ts2

Fractionally spaced equalizer

)(trF(f)

F(f)



4/5/00p. 40


• Fractionally spaced equalizer (DFE):

• Theory & formulas : see textbooks & Lecture 6

C(z) )(ˆ zA

D(z)

1/2Ts2

)(trF(f)



4/5/00p. 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)



4/5/00p. 42

Assignment 3.1

• Symbol-rate zero-forcing linear equalizer has

i.e. a finite impulse response (àll-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 zHzC

22

110 ..)( zhzhhzH

)../(1)( 22

110

zhzhhzC

22/512/32/12/1

22110

...

...

kkkk

kkkk

ahahahy

ahahahy

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