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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=
Channel Equalization 5 H. H. Kha, Ph.D.
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.
Channel Equalization 6 H. H. Kha, Ph.D.
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=
Channel Equalization 7 H. H. Kha, Ph.D.
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
Channel Equalization 8 H. H. Kha, Ph.D.
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≤ ≤
Channel Equalization 10 H. H. Kha, Ph.D.
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= +
Channel Equalization 11 H. H. Kha, Ph.D.
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∗=
Channel Equalization 12 H. H. Kha, Ph.D.
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
Channel Equalization 13 H. H. Kha, Ph.D.
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
Channel Equalization 14 H. H. Kha, Ph.D.
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
Channel Equalization 16 H. H. Kha, Ph.D.
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
Channel Equalization 17 H. H. Kha, Ph.D.
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.
Channel Equalization 18 H. H. Kha, Ph.D.
Equalizer Types
Channel Equalization 19 H. H. Kha, Ph.D.
3. Linear equalizer
Two typical linear equalizers
Zero-forcing equalizer
Minimum mean square error (MMSE) equalizer
Channel Equalization 20 H. H. Kha, Ph.D.
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
Channel Equalization 21 H. H. Kha, Ph.D.
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
Channel Equalization 22 H. H. Kha, Ph.D.
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
Channel Equalization 23 H. H. Kha, Ph.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
Channel Equalization 24 H. H. Kha, Ph.D.
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
Channel Equalization 25 H. H. Kha, Ph.D.
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
Channel Equalization 26 H. H. Kha, Ph.D.
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
Channel Equalization 27 H. H. Kha, Ph.D.
Example
Determine the tap coefficients of a 2-tap MMSE for:
Now, given that
Channel Equalization 28 H. H. Kha, Ph.D.
Channel Equalization 29 H. H. Kha, Ph.D.
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
Channel Equalization 30 H. H. Kha, Ph.D.
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
Channel Equalization 31 H. H. Kha, Ph.D.
Homework
Channel Equalization 32 H. H. Kha, Ph.D.
Homework 2
Channel Equalization 33 H. H. Kha, Ph.D.
Homework 3
Channel Equalization 34 H. H. Kha, Ph.D.