5. Kha-DC03-Equalization.pdf

8/10/2019 5. Kha-DC03-Equalization.pdf

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EqualizationHa Hoang Kha, Ph.D

Ho Chi Minh City University of Technology

Email: [email protected]

Chapter 3


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Content

1. Introduction

Multipath fading

ISI

2. Discrete-time channel model

3. Linear equalizer

Zero-forcing equalizer

MMSE equalizer

4. Nonlinear equalizer ZF decision feedback equalizer

MMSE decision feedback equalizer

Equalization 2 H. H. Kha, Ph.D.


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ISI due to Multi-Path Fading

Transmitted signal:

Received Signals:

Line-of-sight:

Reflected:

The symbols add up onthe channel

Distortion!

Delays



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Performance of multipath channels

First, compare 1-tap (i.e. f lat) Rayleigh-fading channelvs AWGN.

i.e. y = hx + w vs y = x + w

Note: all multipaths with random attenuation/phasesare aggregated into 1-tap

Next consider f requency select iv i ty, i.e. multi-tap,

broadband channel, with multi-paths Effect: ISI

Equalization techniques for ISI & complexities



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Flat Fading (Rayleigh) vs AWGN



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ISI/Freq. Selective Channel

Typical BER vs. S/N curves

S/N

BER

Frequency-selective channel(no equalization)

Flat fading channel

Gaussianchannel

(no fading)

Frequency selective fading

irreducible BERfloor!!!



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Freq. selective channel with equalization

Typical BER vs. S/N curves

S/N

BER

Flat fading channel

Gaussianchannel

(no fading)

Diversity (e.g. multipath diversity)

Frequency-selective channel

(with equalization)

improvedperformance



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Rayleigh Flat Fading Channel

BPSK: Coherent detection.

Conditional on h,

Averaged over h,

at high SNR.

Looks like

AWGN, butpeneeds to be

unconditioned

To get a much

poorer scaling


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2. Discrete time model for ISI channels

{bm} is a data sequence.



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Examples of waveforms

The output of the impulse modulator

L: the length of the data sequence

T: symbol duration



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Noise is assumed to be zero-mean white Gaussian with

double-sided spectral density

and autocorrelation function

The received signal after receiver filtering

a modified transmitter filter includes the transmitter filter

and channel

Channel output without noise



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Discrete-time CIR:

The sampled signal of y(t) at every T seconds is

given by

Sampled noise

Discrete channel



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Given

Example of discrete channel

Find the discrete-time CIR

Find the sampled noise nl



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With the discrete-time CIR channel

can be found

Solution



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The sampled noise is given by

Solution

The variance of noise is given by

It can be shown that



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The discrete-time CIR is a causal impulse response

of length P, i.e.,

The discrete-time channel

The received signal in the discrete-time domain is

given by



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Suppose that is the desired signal

ISI problem

The received signal can be re written as

is a delay.

Due to the ISI terms, the desired signal cannot

be clearly observed from the received signals.

It is necessary to eliminate the ISI terms to extract

the desired signal.



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It is desired when the noise is ignored.


In z-domain, it is required

The linear equalizer is called the zero-forcing (ZF) equalizer because the ISI is forced to be

zero.



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The noise after ZF equalization is


If has nulls (in frequency response), the

variance of noise can be infinity.

Another disadvantage of the ZF equalization is thatZF equalizer has an infinite impulse response for

finite-length channels.


m t ph c ng su tca nhiu

c p ng xung d i t i v c ng cho k nhc p ng xung hu hn


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Assume that is independent identically distributed

(iid) and , .

Minimum mean square error linear equalizer

The error is defined as

The desired signal

The MSE is given by

Rewritten in matrix form as

M: the length of the LE.



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MMSE equalizer: Orthogonal principle

The error should be uncorrected with at

the optimality, i.e.,

where and

At the optimality, the MMSE is given by


sai s kh ng tng quanvi tn hiu ng vo

-->


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MMSE equalizer: derivative

Note that the MSE cost function is a quadratic

function of g.

The minimum is obtained from

This is the same condition found from the

orthogonality principle.



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4. Nonlinear equalizer: DFE

Structure of the decision feedback equalizer

DFE



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Zero-forcing DFE

The convolution of gm and hp is given by

M: length of FFF and P is the length of channel



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Zero-forcing DFE

Output of the FFF is given by

To estimate at time l

Assume are available.



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Zero-forcing DFE

The ISI due to these past symbols can be

eliminated

is the impulse response of the FBF

denotes the detected symbols of

If the decision are correct , we chose


oi t n p n isi c a c c i u tr c


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Zero-forcing DFE

In matrix form

When


mong mu n ch thuc cm


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Solution



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S


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MMSE DFE

ZE DFE only attempts to remove the ISI, the noise

can be enhanced

The MSE is given by


E l MMSE DFE


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Example: MMSE DFE


S l ti


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Solution


S l ti


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Solution


S l ti


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Solution


E l Ch l ith f ll


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Example: Channels with frequency nulls


E l BER f MMSE LE


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Example: BER for MMSE LE


khng dngb linear

E l BER f MMSE DFE


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Example: BER for MMSE DFE


dng b linear

5 Adapti e linear eq ali er


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5. Adaptive linear equalizer

Adaptive equalizers can be considered as practical

approaches.

They do not require second-order statistics of

signals.

A training sequence is used to find the equalization

for the LE or DFE.


Iterative approaches: steepest descent algorithm


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Iterative approaches: steepest descent algorithm

The output of the LE is given by

Assume that an LE is causal and has a finite length of M.

The MSE as a function of g is defined by


p ng xung

Steepest descent algorithmphng php lp


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Steepest descent algorithm

Suppose that is an initial vector

The gradient is defined as

The next vector which may yield a smaller than

can given by

Steepest descent Constant step size


phng php lp

nu o hm g0 >0 --> ang tng--> tr xung



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A recursion toward the minimum is given by

k: iteration index

The iteration is terminated when the SD direction

becomes zeros, i.e.,




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If is large, the recursive

diverges and never findsthe minimum.

If is too small, it would

take too many iterations

to converge.

It is important to

determine the value of

such that the recursioncan converge at a fast

rate.


u ln

Convergence analysis of the SD algorithm


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Note that

We consider the different vector

Egeindecomposition of is given by


sai s bclp th k



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Let . Then it follows that

represents the mth element of

We can fined the following property




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The sufficient condition for convergence is

is the maximum eigenvalue

Since

another practical sufficient condition is


Ry = E(yyT)

Least mean square (LMS) approach


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The SD algorithm can overcome the matrix

inversion of the MMSE approach.However, it has not been overcome the need for

second-order statistics.

The least mean square (LMS) algorithm is anapproximation of the SD algorithm.

Recall the MSE




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The SD algorithm can be represented by

where

If is replaced by without the expectationwe can obtain the LMS approach


t Ul = Sl - gT.Yl

yl: tn hiu u vo b cnbng thi im l

ko cn quan tmknh truyn

Adaptive decision feedback equalizers


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Adaptive decision feedback equalizers

We the training sequence is available, we can use

the correct symbols in the DFE.The MSE of the DFE is written as

where

The adaptive DFE finds the equalization vector

from the and

The LMS algorithm can used for the adaptive DFE.


g: b cn b ng thunf: feedback

`tn hiu ng vo

Home work 1


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Home work 1

Given the discrete-time channel model as fig

bn={-1, +1} denotes the information bits

channel impulse response h=[0.227, 0.460, 0.688, 0.460, 0.227]

Wn is AWGN with zero mean and variance N0/2

Design and simulation ZF and ZF-DFE, and for

SNR=[5, 10, 15, 20, 25, 30]

Plot BER vs SNR

Channel hn

dnnb n

Equalizer filter

oise wn


dB

3. contemporary communication system usingmatlab

adaptive equalizer for isi ch

t chn m, chn chi u d i

Home work 2


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Home work 2


bn={-1, +1} denotes the information bits

channel impulse response h=[0.227, 0.460, 0.688, 0.460, 0.227]


Design and simulation MMSE and MMSE-DFE,

and for SNR=[5, 10, 15, 20, 25, 30]

Plot BER vs SNR

Channel hn

dnnb n

Equalizer filter

oise wn


Home work 3


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Home work 3


bn={-1, +1} denotes the information bits channel impulse response h=[0.227, 0.460, 0.688, 0.460, 0.227]


Design and simulation: LMS algorithm for MMSE

and MMSE-DFE for SNR=[5, 10, 15, 20, 25, 30]

Plot BER vs SNR, and the convergence for

different

Channel hn

dnnb n

Equalizer filter

oise wn

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5. Kha-DC03-Equalization.pdf