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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.


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.


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.


channel equal is at ion

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