4/5/00 p. 1 postacademic course on telecommunications module-3 transmission marc moonen lecture-6...

Module-3 Transmission Marc MoonenLecture-6 Adaptive Equalization K.U.Leuven/ESAT-SISTA

4/5/00p. 1

Postacademic Course on Telecommunications

Module-3 : Transmission

Lecture-6 (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-6 Adaptive Equalization K.U.Leuven-ESAT/SISTA

4/5/00p. 2

Lecture 6 : Adaptive Equalization

Problem Statement :• Equalizers of Lecture-5 assume perfect

knowledge of channel distortion (impulse response h(t)) and possibly also noise characteristics (variance/color)

• What if channel is unknown or time-varying (e.g. mobile communications)... ?

• Channel model identification and/or (direct) equalizer design based on training sequences (and/or decision directed operation)



4/5/00p. 3

Lecture 6 : Adaptive Equalization -Overview

• Equalizers design subject to complexity constraint (=finite number of filter taps)

• Training sequence based direct equalizer design• Training sequence based channel identification• Recursive/adaptive algorithms

LMS (1965), RLS, Fast RLS

• Blind Equalization• Postscript: Adaptive filters in digital communications



4/5/00p. 4

Complexity Constrained Equalizer Design

• In Lecture-5, equalizer design ignores complexity issues (filter lengths,..)

• If (=practical approach) the number of equalizer filter coefficients (`taps’) is fixed, then what would be an optimal equalizer ?

• MMSE criterion based approach

zero-forcing criterion generally not compatible with complexity constraint. (+ noise enhancement)



4/5/00p. 5


Example : linear equalizer design

• complexity constraint : (3 taps)

• MMSE-LE equalizer is such that the slicer input is as close as possible (in expected value, E{.}) to transmitted symbol :

H(z)

)(zW

)(zY

C(z)

)(zA )(ˆ zA

22

110 ..)( zczcczC

2

22110,, ...min210 kkkkccc ycycycaE



4/5/00p. 6


Solution is given by Wiener Filter Theory:

….ignore formula!

2

2

1

0

21,, .min210

c

c

c

yyyaE kkkkccc

),(......

*

2

1

1

21

*

2

1

2

1

0

nik

k

k

k

kkk

k

k

k

opt

hfa

y

y

y

Eyyy

y

y

y

E

c

c

c



4/5/00p. 7


• Formula allows to compute MMSE equalizer from channel coefficients, noise variance, etc.

• Similar formulas for DFE, fractionally spaced equalizers,…• Conclusion : Necessary theory available• Wiener Filter theory = basis for adaptive filter theory,

see below.

• Here: immediately move on to training sequence based equalizer design, which may be viewed as a`deterministic version’ of the above (with true symbol/sample values instead of expected values).



4/5/00p. 8

Training sequence based equalizer design

• If the channel is unknown and/or time-varying, a fixed sequence of symbols (`training sequence’) may be transmitted for channel `probing’.

• example : GSM -> 26 training bits in each burst of 148 bits (=17% `overhead’)

• In the receiver, based on the knowledge of the training sequence, the channel model is identified and/or an equalizer is designed accordingly.



4/5/00p. 9


• Assume simple channel model (linear filter +AWGN)

• Assume transmitted training sequence is

• Received samples are

• Optimal (`least squares’) linear equalizer

)(zAH(z)

)(zW

)(zY

10321 ,...,,, aaaa

10321 ,...,,, yyyy

10

3

2

22110,, ...min21k

kkkkccc ycycycao

com

pare

to p

age

5 !



4/5/00p. 10


• In matrix notation this is .…

...remember matrix algebra? (`overdetermined set of equations’)

2

10

5

4

3

2

1

0

8910

345

234

123

,,,

:

.

::

min210

bA

ccc

a

a

a

a

c

c

c

yyy

yyy

yyy

yyy



4/5/00p. 11


• `Least Squares’ (LS) solution is .…

)..().( 1

2

1

0

bAAA

c

c

cHH

opt

com

pare

to p

age

6 !



4/5/00p. 12


• PS: possibly incorporate `delay optimization’ :

check delay within a range, and then pick one that gives smallest error norm

2

10

5

4

3

2

1

0

8910

345

234

123

,,,

:

.

::

min210

bA

ccc

a

a

a

a

c

c

c

yyy

yyy

yyy

yyy



4/5/00p. 13


• Similar least squares problem for fractionally spaced eq..…

...optimal solution

2

10

5

4

3

2

2/3

1

2/1

0

82/1692/1910

32/742/95

22/532/74

12/322/53

,,,,

:

.

:::

min22/312/10

bA

ccccc

a

a

a

a

c

c

c

c

c

yyyyy

yyyyy

yyyyy

yyyyy

)..().( 1 bAAA HH



4/5/00p. 14


• Similar least squares problem for DFE..…

…optimal solution

2

10

5

4

3

1

0

2

1

0

8981010

34345

23234

12123

,,,,

:

.

:::

min10210

bA

ddccc

a

a

a

a

d

d

c

c

c

aayyy

aayyy

aayyy

aayyy

)..().( 1 bAAA HH



4/5/00p. 15

Training sequence based channel identification

• Alternatively, the training sequence may be used to estimate a channel model, from which then an optimal equalizer (see Lecture-5) is computed (or by means of which an MLSE receiver is designed (ex: GSM))



4/5/00p. 16


• Assume simple channel model (linear filter +AWGN)

• Assume transmitted training sequence is

• Received samples are

• Optimal (`least squares’) channel model is

)(zAH(z)

)(zW

)(zY

10321 ,...,,, aaaa

10321 ,...,,, yyyy

10

3

2

22110,, ...min21k

kkkkhhh ahahahyo

com

pare

to p

age

9 !



4/5/00p. 17


• In matrix notation this is .…

…optimal solution

2

10

5

4

3

2

1

0

8910

345

234

123

,,

:

.

::

min210

bA

hhh

y

y

y

y

h

h

h

aaa

aaa

aaa

aaa

)..().( 1 bAAA HH



4/5/00p. 18


• Conclusion : MMSE-optimal equalizer design (LE, DFE, FS) or channel identification may be reduced to solving an overdetermined set of linear equations A.x=b in the least squares sense

where the optimal solution is always given as

• `Fast algorithms’ available (e.g. Levinson, Schur), that exploit matrix structure (`constant along diagonals’)

• In practice, sometimes iterative procedures (e.g. steepest descent) are used to find the optimal solution

(a la LMS, see below).

)..().( 1 bAAAx HHopt

2.min bxAx



4/5/00p. 19

Recursive/Adaptive Algorithms

• Up till now we considered `batch processing’ : at the end of the training sequence, the complete batch of data is processed…

• Is it possible to process data on a `per-sample’ basis, i.e. process samples as they come in?

• Answer=Yes : Àdaptive Filters’

References :

S. Haykin, Àdaptive filter theory’, Prentice-Hall 1996.

M. Moonen & I. Proudler : Ìntroduction to adaptive filtering’,

free copy @ www-address.



4/5/00p. 20


• Starting point is the (common) least squares problem (=overdetermined set of equations)

• Whenever new samples come in, a new row (=equation) is added to the underlying set of equations, and so the optimal solution vector x may be re-computed

• Most adaptive filtering algorithms have the following form

2.min bxAx

kk ya ,

n vector)(correctio 1 kk xx



4/5/00p. 21


Example : Least-Mean-Squares (LMS) Algorithm (Widrow 1965)

(=channel identification example of p.17)

• is step-size parameter, controls adaptation speed.

If too large -> divergence. Need for proper tuning !

2

1

ERROR PRIORI-A

12

1

0

21

12

1

0

2

1

0

.).(.

k

k

k

k

kkkk

kka

a

a

h

h

h

aaay

h

h

h

h

h

h



4/5/00p. 22


Example : Least-Mean-Squares (LMS) Algorithm

• LMS is a `stochastic gradient algorithm’, i.e. steepest descent algorithm for the least squares problem, with instantaneous estimates of the gradient.

• LMS (and variants) are by far the most popular algorithms in practical systems. Reason = simple (to understand & to implement)

• Complexity = O(N), where N is the number of filter taps (=dimension of x).

• Disadvantage : often (too) slow convergence (e.g. 1000 training symbols)



4/5/00p. 23


Example : `Normalized’ LMS

• Normalize step-size parameter, i.e. use

• For guaranteed convergence :

hence simpler tuning

2

22

12

2

121 .

kkk

k

k

k

kkk

aaa

a

a

a

aaa

20



4/5/00p. 24


Recursive Least Squares (RLS) algorithms :• Also of the form

but now exact update for the solution vector (unlike LMS)• Fast convergence (unlike LMS)• Complexity is O(N^2), where N is the number of filter

coefficients (dimension of x), which is often too much for practical systems

• Formulas : see textbooks

n vector)(correctio 1 kk xx



4/5/00p. 25


`Fast’ Recursive Least Squares (RLS) algorithms:

• Reduce complexity of RLS algorithm by exploiting special properties (structure) of the involved matrices (cfr. supra: `constant along the diagonals’)

• Convergence = RLS convergence !• Complexity is O(N), where N is the number of filter

coefficients (dimension of x), which approaches LMS-complexity.

• Great algorithms, but hardly used in practice :-(

• Formulas : see textbooks



4/5/00p. 26

Blind Equalization

• Problem Statement : channel identification or equalizer initialization based on

channel outputs only, i.e. without having to transmit a training sequence ??

• LMS-type algorithms : (constant modulus, Godard,…) simple but slow convergence (>1000 training symbols)

Reference : S. Haykin (ed.), `Blind deconvolution’, Prentice-Hall 1994

• Algorithms based on higher-order statistics• Algorithms based on `2nd-order’ statistics or

deterministic properties : fast, but mostly complex

Reference : vast recent literature (IEEE Tr. SP,...)



4/5/00p. 27

Postscript

Adaptive Filters in Digital Communications

Adaptive filters are used in dig.comms. systems for

-equalization (cfr. supra)

-channel identification (cfr supra)

-echo cancellation

-interference suppression

-etc..



4/5/00p. 28

Postscript

Basis for adaptive filter theory is

Wiener filter theory• Prototype Wiener filtering scheme :



4/5/00p. 29

Postscript

Prototype adaptive filtering scheme :

• 2 operations: filtering + adaptation



4/5/00p. 30

Postscript

Adaptive filters for channel identification :



4/5/00p. 31

Postscript

Adaptive filters for channel identification :• line echo cancellation in a telephone network



4/5/00p. 32

Postscript

Adaptive filters for channel identification :• echo cancellation in full-duplex modems



4/5/00p. 33

Postscript

Adaptive filters for channel identification :• acoustic echo cancellation for conferencing



4/5/00p. 34

Postscript

Adaptive filters for channel identification :• hands-free telephony



4/5/00p. 35

Postscript

Adaptive filters for channel equalization :



4/5/00p. 36

Postscript

Adaptive filters for channel equalization :

decision-directed operation



4/5/00p. 37

Postscript

Adaptive filters for channel equalization and interference cancellation (see also Lecture-10)



4/5/00p. 38

Conclusions

• Training sequence based channel identification and/or equalization :

Least squares optimization criterion provides common

framework/solution procedure for LE, DFE, fractionally

spaced equalization,..

• Recursive/adaptive implementation -simple & cheap (but slow) : LMS

-fast (but sometimes too expensive) : RLS, Fast RLS



4/5/00p. 39

Assignment 3.2

• Return to the zero-forcing fractionally spaced equalizer of assignment 3.1.

• Run to your favorite computer & simulation program (e.g. Matlab, Simulink,…) & simulate a transmitter/channel/receiver system as follows:

• Transmitter : random 2-PAM training symbols +1,-1• Channel : choose (random) values for the hi’s in the model. No additive

noise.• Receiver : NLMS-based adaptive zero-forcing equalizer.

Select appropriate filter length (see Assignment 3.1). • Experiment with the step-size parameter, and observe convergence

behavior. • Experiment with shorter and longer equalizer filter lengths.

22/512/32/12/1

22110

...

...

kkkk

kkkk

ahahahy

ahahahy

4/5/00 p. 1 postacademic course on telecommunications module-3 transmission marc moonen lecture-6...

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