equalization ed us at
TRANSCRIPT
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Equalization, Diversity, andEqualization, Diversity, and
Channel CodingChannel Coding
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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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Algorithm for Adaptive Equalization
Performance measures for analgorithm
Rate of convergence
Misadjustment
Computational complexity
Numerical properties
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Algorithm for Adaptive Equalization
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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Algorithm for Adaptive Equalization
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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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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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
DerivationDerivation
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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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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