Equalization Ed Us At

7/31/2019 Equalization Ed Us At

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Equalization, Diversity, andEqualization, Diversity, and

Channel CodingChannel Coding

20


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Equalization, Diversity, and

Channel Coding

Three techniques are used independently orin tandem to improve receiver signal quality

Equalization compensates for ISI created bymultipath with time dispersive channels

(W>BC) Change the overall response to remove ISI


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Equalization, Diversity, and

Channel Coding

Diversity also compensates for fadingchannel impairments, and is usuallyimplemented by using two or more

receiving antennas Multiple received copies: Spatial

diversity, antenna polarization diversity,

frequency diversity, time diversity. Reduces the depth and duration of the

fades experienced by a receiver in a flatfading (narrowband) channel


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Channel Coding improves mobilecommunication link performance byadding redundant data bits in the

transmitted message Channel coding is used by the Rx to

detect or correct some (or all) of the

errors introduced by the channel (Postdetection technique)

Block code and convolutional code


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Equalization Techniques

The term equalization can be used to describe

any signal processing operation that minimizesISI

Two operation modes for an adaptive equalizer:

training and tracking Three factors affect the time spanning over

which an equalizer converges: equalizer

algorithm, equalizer structure and time rate of

change of the multipath radio channel TDMA wireless systems are particularly well

suited for equalizers


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Equalization Techniques

Symbol

Mapper

ISI

Channel

Equalizern

dndn

vnr

Decision

Device

nz


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Channel ResponseChannel Response

Equalizer is usually implemented at

baseband or at IF in a receiver

f*

(t): complex conjugate of f(t)nb(t): baseband noise at the input of

the equalizer

heq(t): impulse response of the

equalizer

)()()()( tbntftxty +=


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Block DiagramBlock Diagram


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EqualizationEqualization If the channel is frequency selective, the

equalizer enhances the frequency componentswith small amplitudes and attenuates the strong

frequencies in the received frequency response

For a time-varying channel, an adaptive

equalizer is needed to track the channelvariations

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

( ) ( ) 1

=

=

+=

=

fHfF

t

thtmthtftx

thtytd

eq

eqbeq

eq


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Basic Structure of Adaptive Equalizer

Traversal filter with N delay elements, N+1taps, and N+1 tunable complex weights

These weights are updated continuously by

an adaptive algorithm The adaptive algorithm is controlled by the

error signal ek


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Basic Structure of Adaptive Equalizer


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Minimize Estimation Error

Classical equalization theory : using training

sequence to minimize the cost function

E[e(k) e*(k)]

Recent techniques for adaptive algorithm : blind

algorithms Constant Modulus Algorithm (CMA, used for

constant envelope modulation)

Spectral Coherence Restoral Algorithm(SCORE, exploits spectral redundancy or

cyclostationarity in the Tx signal)


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DerivationDerivation Error signal

where

Mean square error

Expected MSE

where

k

T

kkk

T

kkkxxe yy ==

[ ]TNkkkkk y....yyy = 21y

[ ]TNkkkkk = ....21

k

T

kkk

T

kk

T

kkk xxe yyy 222

+=

[ ] [ ] TTkkxe pR 222

+== EE


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DerivationDerivation

[ ]

==

2

1

21

21

2

11

21

2

Nk

Nkk

Nkk

kNkkNkkNk

kkkkk

kkkkk

*

kk

y

....

yy

yy

....yyyyyy

................

....yyyyy

....yyyyy

EE yyR

[ ] [ ]TNkkkkkkkkkk yxyxyxyxyx == ....21EEp


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DerivationDerivation Optimum weight vector

Minimum mean square error (MMSE)

Minimizing the MSE tends to reduce the bit error

rate

Training Sequence then Data transmission within

each frame

pR1 =

[ ]= 2min E pRp1T

[ ] =2E

p

Training

Sequence Data transmissionTraining

Sequence Data transmission


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Classification of EqualizerClassification of Equalizer

if d(t) is not in the feedback path to adapt the equalizer, the

equalization is linear

if d(t) is fed back to change the subsequent outputs of the

equalizer, the equalization is nonlinear


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Linear transversal equalizer

LTE, made up of tapped delay lines


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ARMA Model (FIR, IIR)ARMA Model (FIR, IIR)


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Linear Transversal EqualizerLinear Transversal Equalizer


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Linear Transversal EqualizerLinear Transversal Equalizer

nk

N

Nn

*nk yCd

2

1

==

[ ]

dNe

NTe(n) TT

o

jo

+= 2t

2

)(F2E

)( tjeF

:frequency response of the channel

oN :noise spectral density


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Algorithm for Adaptive Equalization

The circuit complexity and processing timeincreases with the number of taps and delayelements

Three classic equalizer algorithms : zeroforcing (ZF), least mean squares (LMS), andrecursive least squares (RLS) algorithms


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Example

Y(k)= 0.0,0.1,1.0,-0.2,0.1 use ZF and find

the tap weights for a 3 tap equalizer.

D D

C-1 C0C1

dK

Yk


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ZF

X(0) x(-1) x(-2)

X(1) x(0) x(-1)

X(2) x(1) x(0)

C-1

C0

C1

=

0

1

0

1.0 0.1 0.0

-0.2 1.0 0.1

0.1 -.02 1.0

0

1

0

-1

=

C-1

C0

C1


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Solving

C-1 =-0.096, C0=0.96, C1=0.2

Find the equalized output for sampling instants of 2

d2 = c-1y3 + c0y2 +c1 y1 = -0.096x0 + 0.96x0.1+0.2x-0.2 = 0.056

d-2 = c-1y-1 + c0y-2 +c1 y-3 = -0.096 x 0.1 + 0.96 x 0 +0.2x0 = -0.0096

L tti Filt


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Lattice Filter


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Characteristics ofLattice Filter

Advantages

Numerical stability

Faster convergence

Unique structure allows the dynamicassignment of the most effective

length

Disadvantages

The structure is more complicated


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Nonlinear Equalization

Used in applications where the

channel distortion is too severe

Three effective methods

Decision Feedback Equalization (DFE)

Maximum Likelihood Symbol Detection

Maximum Likelihood SequenceEstimator (MLSE)


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Nonlinear Equalization--DFE Basic idea : once an information symbol

has been detected and decided upon, theISI that it induces on future symbols can be

estimated and substracted out before

detection of subsequent symbols

Can be realized in either the direct

transversal form or as a lattice filter

= = +=

32

1

N

1iikink

N

Nn

*

nk dFyCd

[ ] }])(F

[2

{E2

2

+

=

d

Ne

Nln

Texpe(n) T

To

Tj

o

min


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DFEDFE

P di ti DFE


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Predictive DFE

Predictive DFE (proposed by Belfiore and

Park) Consists of an FFF and an FBF, the latter

is called a noise predictor

Predictive DFE performs as well asconventional DFE as the limit in the number

of taps in FFF and the FBF approach

infinity The FBF in predictive DFE can also be

realized as a lattice structure

The RLS algorithm can be used to yield fast


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Predictive DFEPredictive DFE

MLSE


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MLSE

MLSE tests all possible data sequences

(rather than decoding each received symbol

by itself ), and chooses the data sequence

with the maximum probability as the output

Usually has a large computational

requirement

First proposed by Forney using a basic

MLSE estimator structure and implementingit with the Viterbi algorithm


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MLSEMLSE


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MLSEMLSE


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Performance measures for analgorithm

Rate of convergence

Misadjustment

Computational complexity

Numerical properties


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Factors dominate the choice of an

equalization structure and its algorithm

The cost of computing platformThe power budget

The radio propagation characteristics


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The speed of the mobile unit determinesthe channel fading rate and the Dopplerspread, which is related to the coherenttime of the channel directly

The choice of algorithm, and itscorresponding rate of convergence,depends on the channel data rate andcoherent time

The number of taps used in the equalizerdesign depends on the maximumexpected time delay spread of the channel


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The circuit complexity and processing timeincreases with the number of taps and delayelements

Three classic equalizer algorithms : zeroforcing (ZF), least mean squares (LMS), andrecursive least squares (RLS) algorithms


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Zero forcing (ZF) Algorithm

The equalizer coefficients are chosen to forcethe combined channel and the equalizer

response to zero at all but one of the NTspaced sample points.

The impulse Response of the equalizer is theinverse of the Channel

Hch (f) Heq (f) = 1, | f| < 1/2T

By letting the length of the equalizer to infinity, zero ISI at the output is obtained.


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Example

Y(k)= 0.0,0.1,1.0,-0.2,0.1 use ZF and find

the tap weights for 3 tap equalizer.

D D

C-1 C0C1

dK

Yk


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ZF

X(0) x(-1) x(-2)

X(1) x(0) x(-1)

X(2) x(1) x(0)

C-1

C0

C1

=

0

1

0

1.0 0.1 0.0

-0.2 1.0 0.1

0.1 -.02 1.0

0

1

0

-1

=

C-1

C0

C1


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L t M S Al ithL t M S Al ith


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Least Mean Square AlgorithmLeast Mean Square Algorithm Error signal

where

Mean square error

Expected MSE

where

k

T

kkk

T

kkkxxe yy ==

[ ]TNkkkkk y....yyy = 21y

[ ]TNkkkkk .... = 21

k

T

kkk

T

kk

T

kkk xxe yyy 222

+=

[ ] [ ] TTkkxe pR 222

+== EE



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[ ]

==

2

1

21

21

2

11

21

2

Nk

Nkk

Nkk

kNkkNkkNk

kkkkk

kkkkk

*

kk

y

....

yy

yy

....yyyyyy

................

....yyyyy

....yyyyy

EE yyR

[ ] [ ]TNkkkkkkkkkk yxyxyxyxyx == ....21EEp


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Least Mean Square Algorithm LMS is computed iteratively by

dk(n) =WnT(n)YN(n)

e(n )= xk(n)-dk(n)

WN(n+1) = WN(n) - e*k(n) YN(n)

0 < < 2 / I

I

= YN(n) Y

N

T (n)

Summary of algorithms


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Summary of algorithms


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Diversity Techniques

Requires no training overhead

Can provides significant link

improvement with little added cost Diversity decisions are made by the

Rx, and are unknown to the Tx

Diversity Techniques


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Diversity Techniques Diversity concept

If one radio path undergoes a deep fade,another independent path may have a strong

signal

By having more than one path to select from,

both the instantaneous and average SNRs atthe receiver may be improved, often by as

much as 20 dB to 30 dB

Diversity order How many independent copies

How many links to bring down the system

Diversity Example


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Diversity Example

Equalization Ed Us At

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