Module-3 : Transmission Lecture-5 (4/5/00)

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/

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

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.

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)...

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

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

Channel Model:

Continuous-time channel

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

(symbols) a

k

aˆ

k

n(t) +

AWGN

transmitter receiver (to be defined) h(t)

channel

...

? ?

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

Transmitter:

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

receiver (to be defined) ...

s

k E

a .

r(t)

aˆk

transmit pulse

s(t)

n(t)

p(t) +

AWGN transmitter

h(t)

channel ?

 ^

 E

_s

a

_k

p t kT

_s

t

s ( ) . . ( )

a

k

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

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

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

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.

ˆa0

front-end filter

1/Ts

receiver n(t)

+ AWGN

Es

a .₀

transmit pulse

p(t)

transmitter

h(t)

channel

u0

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

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

r t E a p t kT dt

K

2 ,...,ˆ

,ˆ

ˆ

| ( ) . ˆ . ' ( ) |

min

0 1

 ^



k

s k

s

a p t kT

E t

s ( ) . . ( )

) ( )

( ' . .

)

( t E a p t kT n t

r

k

s k

s

 

 

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

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

ˆ 0

( . ). ˆ

min

_a0

u  E

_s

g a

ˆa0

p’(-t)*

front-end filter

1/Ts

receiver n(t)

+ AWGN

Es

a .₀

transmit pulse

p(t)

transmitter

h(t)

channel

sample at t=0 p’(t)=p(t)*h(t)

u0

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

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

aˆk

p’(-t)*

front-end filter

1/Ts n(t)

+ AWGN

s

k E

a .

transmit pulse

p(t) h(t)

sample at t=k.Ts

uk

  _



 



    

 

 

K k

k k l

l k K

k

K l

k s

a

a

E a g a a u

K 1

*

1 1

* ,...,ˆ

ˆ

. ˆ . . ˆ 2 ˆ .

min

0

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

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...

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

Receiver: 1st simplification by insertion of an additional (magic) filter (after sampler).

* Filter = `pre-cursor equalizer’ (see below)

* Complete front-end = `Whitened matched filter’

aˆk

p’(-t)*

front-end filter

1/Ts n(t)

+ AWGN

s

k E

a .

transmit pulse

p(t) h(t)

uk

1/L*(1/z*) yk

2

1 1

,...,ˆ

ˆ

ˆ .

min

0

 





 

K m

K k

k m k m

a

a

y a h

K

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

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









 









 



aˆk

p’(-t)*

front-end filter

1/Ts

receiver n(t)

+ AWGN

s

k E

a .

transmit pulse

p(t)

transmitter

h(t)

channel

uk

1/L*(1/z*) yk

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

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

h a h a h a h a w

y 

₀

. 

₁

.

_1



₂

.

_2



₃

.

_3

 ... 

w

k

0

2

...

1



_

 



h

h

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

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

h a h a h a h a w

y 

₀

. 

₁

..

_1



₂

..

_2



₃

.

_3

 ... 

2

1 1

,...,ˆ

ˆ

ˆ .

min

0

 





 

K m

K k

k m k

m a

a

y a h

K

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

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

h a h a h a h a w

y 

₀

. 

₁

.

_1



₂

.

_2



₃

.

_3

 ... 



h

1



h

3

h

0



h

2

a

k

a

_k1

a

_k2

a

_k3

y

k

w

k

Preliminaries (II)

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

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

) ( )

( ).

( )

(

hence

...

. .

.

.

_{1 1 2 2 3 3}

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) ^{Y (z )}

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

h a h a h a h a h a w

y ~ . ... ~

~ .

~ .

~ .

~ .

... 

_{2 2}



_{1 1}



₀



_{1 1}



_{2 2}

 



_{     }

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

w

k

0

2

...

1



_

 



h

h

)

1

( z

H

^

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

h a h a h a h a w

y  

_



_



_

 









 









 



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

NOISE



ISI

3 3

2 2

1 1

0

.

_k

.

_k

.

_k

.

_k

...

_k

k

h a h a h a h a w

y  

_



_



_

 









 









 



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) )

( )

( z H

¹

z C 

^

H(z)

) (z W

) (z Y

C(z) )

(z

A A ˆ z ( )

Zero-forcing Equalizers

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.

Zero-forcing Equalizers

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 A ˆ z ( )

0

 1

h

Zero-forcing Equalizers

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

) (z Y

D(z) H(z)

) (z W )

(z A

) ˆ z ( A

) ( 1

)

( z H z

D  

Zero-forcing Equalizers

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) ^{A ˆ z ( )}

D(z)

Zero-forcing Equalizers

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) ^{A ˆ z ( )}

D(z)

Zero-forcing Equalizers

`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

h

0

h

i

h

i

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

MMSE Equalizers

MMSE Linear Equalizer (LE) :

- combined minimization of ISI and noise leads to

2

*

*

*

*

*

*

*

*

1 ) ( ).

(

1 ) ( )

( 1 )

( ).

( ).

(

1 ) ( ).

( )

(

n W

A

A

H z z H

H z z

z S H

z H z S

H z z S z

C











H(z)

) (z W

) (z Y

C(z) )

(z

A A ˆ z ( )

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

H z z H

H z z

z S H

z H z S

H z z S z

C













 1 )

(z S

_A





²

)

(

_W

W

z

S 

) ( )

( z H

¹

z C 

^

1 ) (

_*

*

H z

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) ^{A ˆ z ( )}

1-E(z)

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

Fractionally Spaced Equalizers

• 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.

Fractionally Spaced Equalizers

• 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

h a h a h a w

y ~ . ... ~

~ .

~ .

... 

₀



_{1 1}



_{2 2}

 



_{ }

2 / 1 2

2 / 5 1

2 / 3 2

/ 1 2

/

1

~ . ... ~

~ .

~ .

...

_{  }



 

_k



_k



_k

 

_k

k

h a h a h a w

y

Fractionally Spaced Equalizers

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

• Fractionally spaced equalizer (LE) :

) ˆ z ( 1/2Ts A

2

MF(z)

C(z)

equalizer

) (t

r C(z) ^{A ˆ z ( )}

1/2Ts

2

Fractionally spaced equalizer )

(t r

F(f)

F(f)

Fractionally Spaced Equalizers

• Fractionally spaced equalizer (DFE):

• Theory & formulas : see textbooks & Lecture 6

C(z) ^{A ˆ z ( )}

D(z) 1/2Ts

2 )

(t r

F(f)

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)

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

) ( )

( z H

¹

z

C 

^

2 2

1 1

0

. .

)

( z  h  h z

^

 h z

^

H

) .

. /(

1 )

( z  h

₀

 h

₁

z

^1

 h

₂

z

^2

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