Kha DC03 Equalization

Chapter 3

Equalization

Chapter 3

qua at oHa Hoang Kha, Ph.DHo Chi Minh City University of TechnologyEmail: hhkha@hcmut edu vnEmail: [email protected]

Content

1) Introduction 2) Fundamentals of equalization2) Fundamentals of equalization3) Linear equalizer

Zero-forcingZero forcing MMSE equalizer

4) Adaptive equalizer5) Nonlinear equalizer

Decision feedback equalizer (DFE)

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

1. Introduction

Basic block diagram of communication system

Information Pulse Trans

X(t)X(t)HHTT(f)(f) HHcc(f)(f)

Informationsource

Pulse generator

Trans.filter Channel

++Channel noiseChannel noise

ReceiverA/D

n(t)n(t)

DigitalDigital

++y(t)

filterA/DDigital Digital ProcessingProcessing

HHRR(f)(f)

Basic Communication System

TT R iR i

HHTT(f)(f) HHRR(f)(f)HHcc(f(f))y(t)TransTrans

filterfilter channelchannel ReceiverReceiverfilterfilter

y(t)ak

( ) ( ) ( )k d bk

y t a h t t kT n t= − − +∑The received Signal is the transmitted signal, convolved with the channel and added with AWGN

( )y a a h k m T n+ +⎡ ⎤⎣ ⎦∑ ( )k k m b km k

y a a h k m T n≠

= + − +⎡ ⎤⎣ ⎦∑

ISIISI II tt SS b lb l II t ft fISI ISI -- IInter nter SSymbol ymbol IInterferencenterference

Inter-symbol interference

Baseband system model

1a 2aTx filter Channel

)(tn

)(tr Rx. filterDetector

ky

kTt =

{ }ˆka{ }ka1 2

3aT T

)()(fHth

t

t

)()(fHth

r

r

)()(fHth

c

c

Equivalent model

)(tn3

Equivalent system y( )tDetector

ky { }ˆka{ }ka1a 2a

)(th

)(ˆ tn

DetectorkTt =

3aT T

)( fH

filtered noise)()()()( fHfHfHfH )()()()( fHfHfHfH rct=


Nyquist bandwidth constraint

Nyquist bandwidth constraint:The theoretical minimum required system bandwidth to detect R [symbols/s] without ISI is R /2 [Hz]detect Rs [symbols/s] without ISI is Rs/2 [Hz]. Equivalently, a system with bandwidth B0=1/2T=Rs/2 [Hz] can support a maximum transmission rate of 2B0=1/T=Rs[ b l / ] ith t ISI[symbols/s] without ISI.

01 2 [symbol/s/Hz]s sR RB= ≤ ⇒ ≥

Bandwidth efficiency, Rb/B0 [bits/s/Hz] :

00

[ y ]2 2T B

b 0An important measure in DCs representing data throughput per hertz of bandwidth.Showing how efficiently the bandwidth resources are usedShowing how efficiently the bandwidth resources are used by signaling techniques.


Ideal Nyquist pulse (filter)

)( fH )/i ()( Ttth

Ideal Nyquist filter Ideal Nyquist pulse

T

)( fH )/sinc()( Ttth =1

f t0 T T2T−T2−0

T21

T21− f t0 T T2TT20

1B0 2B

T=


Nyquist pulses (filters)

Nyquist pulses (filters):Pulses (filters) which results in no ISI at the sampling ( ) p gtime.

Nyquist filter: Its transfer function in frequency domain is obtained by convolving a rectangular function with any real even-symmetric frequency functionsymmetric frequency function

Nyquist pulse: Its shape can be represented by a sinc(t/T) functionIts shape can be represented by a sinc(t/T) function multiply by another time function.

Example of Nyquist filters: Raised-Cosine filter


Pulse shaping to reduce ISI

Goals and trade-off in pulse-shapingReduce ISIEfficient bandwidth utilizationRobustness to timing error (small side lobes)

H. H. Kha, Ph.D. 9Channel Equalization

The raised cosine filter

Raised-Cosine FilterA Nyquist pulse (No ISI at the sampling time)

10

1 f o r | |2

f fB

⎧ <⎪⎪⎪

2 11 0 1

0 0 1

0 1

| | f1( ) c o s f o r | | 22 40 f o r | | 2

fH f f f B fB B f

f B f

π⎪ ⎡ ⎤−⎪= < < −⎨ ⎢ ⎥−⎣ ⎦⎪⎪ > −⎪⎪⎪⎩

00 2 2 2

cos[2 ]( ) sinc(2 )1 16

B th t B tB t

παα

=

Excess bandwidth: 1f Roll-off factor 11 fB

α = −

01 16 B tα−

1f0B0 1α≤ ≤


The Raised cosine filter – cont’d

|)(||)(| fHfH RC= )()( thth RC=

15.0=r

0=r1 1

0=r5.0=r

1=r1=r0.5 0.5

T21

T43

T1

T43−

T21−

T1− 0 0 T T2 T3T−T2−T3−

0Baseband B (1 ) Bα= +


Equalizing filter

Baseband system model1a d( )t

Tx filter Channel

)(tn

)(tr Rx. filterDetector

ˆkd

kTt =

{ }ka

2a 3aT )(

)(fHth

t

t

)()(fHth

r

r

)()(fHth

c

c

∑ −k

k kTta )(δ Equalizer

)()(fHth

e

e

d( )t

)(tn3

)()()()( fHfHfHfH rct=Equivalent model

Equivalent systemDetector

kd)(th

)()()()( ffff rct

1a∑ −k

k kTta )(δ y( )t Equalizer)(the

{ }kad( )t

)(ˆ tn

DetectorkTt =

)( fH

filtered (colored) noise

2a 3aT )( fHe

e

( ))()()(ˆ thtntn r∗=


Pulse shaping and equalization to remove ISI

)()()()()( fHfHfHfHfH =No ISI at the sampling time

Square-Root Raised Cosine (SRRC) filter and Equalizer

)()()()()(RC fHfHfHfHfH erct=

Square Root Raised Cosine (SRRC) filter and Equalizer

)()()()(

)()()(RC

fHfHfHfH

fHfHfH rt=Taking care of ISI caused by tr filter)()()()( SRRCRC fHfHfHfH tr === caused by tr. filter

)(1)(fH

fHe = Taking care of ISI )( fHc caused by channel


Equalization: Channel is a LTI Filter

ISI due to filtering effect of the communications channel (e.g. wireless channels)

Channels behave like band-limited filters

)()()( fjfHfH θ )()()( fjcc

cefHfH θ=

Non-constant amplitude Non-linear phase

Amplitude distortion Phase distortion


Multipaths: Power-Delay Profile

Pow

er

path-2path-3

path-1

multi-path propagation

Path Delay

path-2path-3

path 1

th 3

path-1

Base Station (BS)Mobile Station (MS)

path-3

Channel Impulse Response: Channel amplitude |h| correlated at delays τ. Each “tap” value @ kTs Rayleigh distributed

(actually the sum of several sub-paths)

ISI due to Multi-Path Fading

Transmitted signal:

Received Signals:Line-of-sight:

Reflected:

Th b l ddThe symbols add up on the channelDistortion!

Delays


2. Fundamentals of equalizer

A linear equalizer effectively inverts the channel.

Equalizer

Heq(f)≈ 1 H (f)

Channel Hc(f)

n(t)

eq( )Hc(f)Hc(f)

The linear equalizer is usually implemented as a tapped

On a channel with deep spectral nulls, this equalizer enhances the noise (note: both signal and noise pass thru eq )

delay line

noise. (note: both signal and noise pass thru eq.)

poor performance on frequency-selective fading channelsp p q y g


Equalization Techniques

• The term equalization can be used to describe any signal processing operation that minimizes ISIsignal processing operation that minimizes ISI

• Two operation modes for an adaptive equalizer: training and tracking. g g

•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•TDMA wireless systems are particularly well suited for equalizers.


Equalizer Types


3. Linear equalizer

Two typical linear equalizers

Zero-forcing equalizer

Minimum mean square error (MMSE) equalizer


Zero-forcing equalizer

Equalizer

Heq(f)≈ 1 Channel H (f)

n(t)

y(t)d(t) d(t)^

Z f i (ZF) li

( ) ( ) 1c eqH z H z =

Heq(f)≈Hc(f)Hc(f)

Zero-forcing (ZF) equalizer:

The filter taps are adjusted such that the equalizer output is forced to be zero at N sample points except that equalizer output =1 at the desired signal :

1 0k =⎧Adjust 1 0ˆ0 1, 2,...,k

kd

k N=⎧

= ⎨ = ± ± ±⎩{ }Ni i Nw

=−

Adjust


Equalization by transversal filtering

Transversal filter: A weighted tap delayed line that reduces the effect of ISI b dj t t f th filt tby proper adjustment of the filter taps.

ˆ N

k i k ii N

d w y −=−

= ∑ 0, 1,...,k N= ± ±

sTk Ny +sT sT sT

Nw− 1Nw− + 1Nw − Nw

∑ˆkd∑ kd

Coeff. adjustmentadjustment


ZF-equalizer in matrix form

ˆNd−⎡ ⎤

⎢ ⎥0 1 2 2Ny y y y− − −⎡ ⎤

⎢ ⎥Nw−⎡ ⎤

⎢ ⎥

ˆ =d Yw

0ˆ ˆ

N

d

⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥

d

1 0 2 2 1

2 1 0 2 2

N

N

y y y yy y y y

− − +

− +

⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥

Y 0w

⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥

w

ˆNd

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

2 2 1 2 2 0N N Ny y y y− −

⎢ ⎥⎢ ⎥⎣ ⎦ Nw

⎢ ⎥⎢ ⎥⎣ ⎦

Note that the y is the channel impulse response and Y isNote that the yk is the channel impulse response and Y is known at the receiver

1 0k =⎧We seek the weight vectors w such that 1 0ˆ0 1, 2,...,k

kd

k N=⎧

= ⎨ = ± ± ±⎩1 ˆ−=w Y d=w Y d


Example: Zero Forcing Solution

Determine wi for a 3 tap equalizer (N=1) from the pulse response (i.e. a training pulse) using the zero forcing solution.

[ ]{ }k 2 1 0k ± ±Given:

Zero forcing:

[ ]y{ } 0.0,0.2,0.9, 0.3,0.1k = −ˆ [0,1,0]d =

2, 1,0k = ± ±

Then: 0 1 2 1 1

1 0 1 0 0

2 1 0 1 1

0 0.9 0.2 0.01 0.3 0.9 0.20 0.1 0.3 0.9

y y y w wy y y w wy y y w w

− − − −

−

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

Carrying out the matrix multiplication and solving the simultaneous equations, or using , get:1 ˆ−=w Y d

w-1 = -0.214, w0 = 0.963, w1 = 0.345


Minimum MSE equalizer

Transversal filter with N delay yelements, N+1 taps, and N+1 tunable complextunable complex weights These weights are updated continuously by an adaptivean adaptive algorithm controlled by the error signal eerror signal ek


MMSE equalizer

T Tk k k k k k ke x x= − = −y w w y

[ ]1 2 .... Tk k k k k Ny y y y− − −=y

Error signalwhere

[ ]0 1 2 .... Tk k k k Nkω ω ω ω=w

2 2 w 2T T Tk k k k k k k ke x x= + −w y y y wSquare error k k k k k k k ky y y

2 2 2T Tk ke E xξ ⎡ ⎤ ⎡ ⎤= = + −⎣ ⎦⎣ ⎦ w Rw p wE

q

Expected MSEwhere input correlation matrix

21 2

211 1 1 2

....

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

k k Nk k k k k

k k NT k k k k kk k

y yy y y y yy yy y y y y

E

−− −

− −− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤= =⎣ ⎦ ⎢ ⎥⎢ ⎥

R y yE E

where input correlation matrix

21 2

.... .... .... ........ k Nk N k k N k k N k yy y y y y y −− − − − −

⎢ ⎥⎢ ⎥⎣ ⎦

[ ] [ ]Tand cross-correlation vector p with the desired signal xk

[ ] [ ]1 2 .... Tk k k k k k k k k k Nx x y x y x y x y− − −= =p yE E


Optimum Weights

Optimum weight vector 1ˆ −w R p

Minimum mean square error (MMSE)

=w R p

q ( )2

minξ E xκ⎡ ⎤= −⎣ ⎦ pRp 1T −

2E x⎡ ⎤= −⎣ ⎦Τ ˆp w

Minimizing the MSE tends to reduce the bit error rate

E xκ⎡ ⎤= ⎣ ⎦ p w


Example

Determine the tap coefficients of a 2-tap MMSE for:

Now, given that


4. Algorithm for Adaptive Equalization

Least mean square (LMS) algorithm:

New weight=previous weights+constant x previous error x current input vectors

Previous error=previous desired output-previous actual output

New weight=previous weights+constant x previous error x current input vectors

LMS Algorithm: Update the coefficients according to the error. ˆ ( ) ( ) ( )T

k N Nd n n n=w y

*

ˆ( ) ( ) ( )

( 1) ( ) ( ) ( )k k ke n x n d n

n n e n nα

= −

+ = −w w y

where and λi is the ith eigvalues of of

( 1) ( ) ( ) ( )N N k Nn n e n nα+ =w w y

10 2 /

N

ii

α λ=

< < ∑correlation matrix R


5. Decision Feedback Equalizer (DFE)

Forward

n(t)

x(t)

DFE

+ x(t)^Hc(f) Forward

Filter

( )

Feedback Filt

+-

x(t)

The DFE determines the ISI from the previously detected symbols and subtracts it from the incoming symbols. This equalizer does not suffer from

Filter

subtracts it from the incoming symbols. This equalizer does not suffer from noise enhancement because it estimates the channel rather than inverting it.

The DFE has better performance than the linear equalizer in aThe DFE has better performance than the linear equalizer in a frequency-selective fading channel.

The DFE is subject to error propagation if decisions are made incorrectly The DFE does not work well for low SNRincorrectly. The DFE does not work well for low SNR


Homework


Homework 2


Homework 3


Kha DC03 Equalization

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