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Channel Equalisation
Graham C. Goodwin
Day 5: Lecture 4
17th September 2004
International Summer School
Grenoble, France
Centre for Complex DynamicSystems and Control
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Introduction
In the previous lecture, we used the
Equalisation problem of Telecommua motivating example.
Here we further explore this applicat
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The Problem
We transmit data (drawn for a finite a
say 1) over a communication chantransmission, the data is corrupted b
(i) dispersion due to the channel
(i.e., neighbouring symbols interfer)
(ii) noise
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Removal of Inter-Symbol Inter
in Digital Communications
vk noise
Communications
Channel
Digital
Data
Receiv
Datuk
0 1 1k k d k d k d y g u g u g u = + + + l lK
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Express in State Space Form
1
2
k
kk
k d
uu
x
u
=
l
M
1k k k x Ax Buy Cx
+ = ++
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{ 0
1 1
0 0 11 0;
1 0
[0 0 ]
d
A B
C g g
+
= =
= l
l
K K
O MM
K K14243
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Special feature of our case:
uk Finite Set
Use a Rolling Horizon constrained stateestimator.
Note: Closed Form solutions available
control problem particularly simple fo
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Special Case; N = 1, R 0
|
11 1| 1
0
{ }
constrained to (finite alph
N d N
N N d N N d g
u q
y g u g u
u
=
=
l lK
where
This optimal Receding Horizon solution is actu
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Diagrammatic Form
1/g0 N/L
G(q)
Decision Feedback Equalizer
Recall that this circuit was introduc
Day 1: Lecture 2.
E l 1
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Example 1
Here we recall the results presented in the
lecture on Day 1.
1 21.7 0.72k k k k k y u u u n = + +
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0 5 10 15 20 25
4
3
2
1
0
1
2
3
4
k
uk
,
uk
Figure: Data uk (circle-solid line) and estimate uk (triangle-solid line) using
the DFE. Noise variance: 2 = 0.2.
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0 5 10 15 20 25
4
3
2
1
0
1
2
3
4
k
uk
,
uk
Figure: Data uk (circle-solid line) and estimate uk (triangle-solid line) using
the moving horizon two-step estimator. Noise variance: 2 = 0.2.
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8 Example 2
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8 Example 2
Consider an FIR channel described by
H(z) = 1 + 2z1+ 2z
2. (1)
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In order to illustrate the performance of the multistep optimal
equaliser presented, we carry out simulations of this channel with
an input consisting of 10000 independent and equiprobable binary
digits drawn from the alphabet U = {1, 1}. The system is affected
by Gaussian noise with different variances.
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The following detection architectures are used: direct quantisation
of the channel output, decision feedback equalisation and moving
horizon estimation, with parameters (L1, L2) = (1, 2) and also with(L1, L2) = (2, 3).
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2 0 2 4 6 8 10 12 14102
101
100
Output Signal to Noise Ratio (dB)
ProbabilityofSymbolError
L1
= 2, L2
= 3
L1
= 1, L2
= 2
DFEDirect Quantization
Figure: Bit error rates of the communication systems simulated.
Centre for Complex DynamicSystems and Control
9 Conclusions
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9 Conclusions
In this lecture we have presented an approach that addresses
estimation problems where the decision variables are constrained
to belong to a finite alphabet.
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