Channel Equalisation Graham C. Goodwin Day 5: Lecture 4 17th - - PowerPoint PPT Presentation

Channel Equalisation

Graham C. Goodwin Day 5: Lecture 4 17th September 2004 International Summer School Grenoble, France

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Introduction

In the previous lecture, we used the Channel Equalisation problem of Telecommunication as a motivating example. Here we further explore this application.

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The Problem

We transmit data (drawn for a finite alphabet – say ±1) over a communication channel. During transmission, the data is corrupted by (i) dispersion due to the channel (i.e., neighbouring symbols interfer) (ii) noise

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Removal of Inter-Symbol Interference in Digital Communications

vk noise Communications Channel Digital Data Received Data uk yk

1 1 k k d k d k d k

y g u g u g u ν

− − − − −

= + + + +

l l

K

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Express in State Space Form

1 2 k k k k d

u u x u

− − − −

⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

l

M

1 k k k k k k

x Ax Bu y Cx ν

+ =

+ = +

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{ 0

1 1

1 1 ; 1 [0 ]

d

A B C g g

− +

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ =

l l

K K O M M K K 1 4 24 3

Problem: Given {yk} – what is {uk} ?

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Special feature of our case: uk ∈ Finite Set Use a Rolling Horizon constrained state estimator. Note: Closed Form solutions available as for the control problem – particularly simple for N = 1.

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Special Case; N = 1, R → 0

| 1 1 1| 1 | 1

ˆ { } ˆ ˆ

ˆ constrained to (finite alphabet)

N d N N N d N N d N g

u q y g u g u

u

− Ω − − − − − −

= Θ ⎡ ⎤ Θ = − − ⎢ ⎥ ⎣ ⎦

Ω

l l

K

where This optimal Receding Horizon solution is actually used extensively in practice. (Called Decision Feedback Equalizer)

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Diagrammatic Form

1/g0 N/L G′(q) Decision Feedback Equalizer

Recall that this circuit was introduced on: Day 1: Lecture 2. We now see that it is a special case of receding horizon finite alphabet estimation.

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Example 1

Here we recall the results presented in the second lecture on Day 1.

1 2

1.7 0.72

k k k k k

y u u u n

− −

= − + +

5 10 15 20 25 −4 −3 −2 −1 1 2 3 4

k uk , ˆ uk

Figure: Data uk (circle-solid line) and estimate ˆ uk (triangle-solid line) using the DFE. Noise variance: σ2 = 0.1.

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5 10 15 20 25 −4 −3 −2 −1 1 2 3 4

k uk , ˆ uk

Figure: Data uk (circle-solid line) and estimate ˆ uk (triangle-solid line) using the DFE. Noise variance: σ2 = 0.2.

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5 10 15 20 25 −4 −3 −2 −1 1 2 3 4

k uk , ˆ uk

Figure: Data uk (circle-solid line) and estimate ˆ uk (triangle-solid line) using the moving horizon two-step estimator. Noise variance: σ2 = 0.2.

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8 Example 2

Consider an FIR channel described by H(z) = 1 + 2z−1 + 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

• f 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 2 4 6 8 10 12 14 10

−2

10

−1

10 Output Signal to Noise Ratio (dB) Probability of Symbol Error L1 = 2, L2 = 3 L1 = 1, L2 = 2 DFE Direct Quantization

Figure: Bit error rates of the communication systems simulated.

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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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