7. Equalization, Diversity, Channel Coding

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7. Equalization, Diversity, Channel Coding

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7.1 Introduction 7.2 Fundamentals of Equalization 7.3 Training a Generic Adaptive Linear Equalizer 7.4 Equalizers in Receivers 7.5 Survey of Equalization Techniques – PowerPoint PPT presentation

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Title: 7. Equalization, Diversity, Channel Coding

1
7. Equalization, Diversity, Channel Coding
7.1 Introduction 7.2 Fundamentals of
Equalization 7.3 Training a Generic Adaptive
Linear Equalizer 7.4 Equalizers in Receivers 7.5
Survey of Equalization Techniques 7.6 Linear
Equalizers 7.7 Non-Linear Equalization 7.7.1
Decision Feedback Equalization (DFE) 7.7.2
Maximum Likelihood Sequence Equalizer (MLSE) 7.8
Algorithms for Adaptive Equalization 7.8.1 Zero
Forcing (ZF) Algorithm 7.8.2 Least Mean Square
(LMS) 7.8.3 Recursive Least Squares Algorithm
(RLS) 7.8.4 Summary of Algorithms
2

7. Equalization, Diversity, Channel Coding
radio channel is dynamic due to
(i) multipath propagation
(ii) doppler spread
effects have strong impact on BER of all
modulation techniques
channel impairments cause signal to distort
fade
significantly more than to AWGN channel
? signal processing techniques improve link
performance

3
7.1 Introduction
7.1 Introduction
4

7.1 Introduction
different techniques can improve link performance
without
altering air interface
increasing transmit power or bandwidth
1.Equalization used to counter ISI (time
dispersion)
2. Diversity used to reduce depth duration of
fades due to motion
3. Channel Coding coded bits improve small-scale
link performance

1. Equalization compensates for ISI created by
multipath
if modulation bandwidth gt coherence bandwidth ?
ISI results
causing signal distortion
equalizer compensate for
- average range of channel amplitude
- delay characteristics
must be adaptive with unknown dynamic
time-varying channel

2. Diversity compensates impairments of fading
channel
e.g. use 2 or more receive antennas
(1) transmit diversity (used in 3G)
BST transmits replica signals separated by
frequency
spatially separated antennas
(2) spatial diversity (most common)
multiple receive antennas spaced to achieve
uncorrelated fading
one antennas see peak while other see nulls
receiver selects strongest signal
(3) antenna polarization diversity
(4) frequency diversity
(5) time diversity CDMA RAKE receivers

3. Channel Coding
during instantaneous (short) fades ? still able
to recover data
coding performed at baseband transmitter segment
(premodulation)
- channel coder encodes user bit stream
- coded message is modulated transmitted
coding is often independent of modulation scheme
newer techniques combine coding, diversity,
modulation
- Trellis coding
- OFDM
- space-time processing
? achieve large coding gains without expanding
bandwidth

Receiver uses coded data to detect or correct
percentage of errors
incoming signal is first demodulated and bit
stream is recovered
Decoding is performed post detection
Coding lowers effective data rate (requires more
bandwidth )
3 General Types of Codes
(i) Block Codes (Reed-Solomon)
(ii) Convolutional Codes
(iii) Turbo Codes
types vary widely in terms of cost, complexity,
effectiveness

9
7.2 Fundamentals of Equalization
7.2 Fundamentals of Equalization
10
7.2 Fundamentals of Equalization

Intersymbol Interference (ISI) is caused by
multipath
results in signal distortion
occurs in time dispersive, frequency selective
fading
(bandlimited) channels

Equalization is a method of overcoming ISI
broadly refers to any signal processing that
minimizes ISI
adaptive equalizers can cancel interference
provide diversity
- mobile fading channel is random time varying
- adaptive equalizers track time varying channel
characteristics

Adaptive Equalizers have two Operating Modes
(1) training (2) tracking
11
(1) training adaptive equalizer

i. send training sequence of known fixed-length
bit pattern
typically a pseudo random binary signal
designed to permit acquisition of filter
coefficients in worst case
e.g. maximum velocity, deepest fades, longest
time delays
ii. receivers equalizer recovers training
sequence
adapts settings to minimize BER
recursive algorithm evaluates channel
estimates filter coefficients
filter compensates for multipath in the channel
iii. convergence training obtains near optimal
filter coefficients

time duration (delay) to achieve convergence
depends on
equalizing algorithm used
equalizer structure
multipath channels time rate of change

(2) tracking in an adaptive equalizer
continually track and adjust filter coefficients
as data is received
adjustments compensate for time-varying channel
data can be encoded (channel coded) for better
performance

periodic retraining required to maintain
effective ISI cancellation
effective when user data is segmented into short
blocks (time-slots)
TDMA systems use short time-slots ? well suited
for equalizers
training sequence usually sent at start of each
short block
training sequence is not changed

Implementation of equalizers is usually at
baseband or IF
baseband complex envelope expression can be used
to represent
bandpass waveforms
channel response
demodulated signal
adaptive equalizer algorithms usually simulated
implemented at
baseband

14
Communication System with Adaptive Equalizer
f(t) combined complex baseband impulse response of transmitter, channel, receivers RF/IF sections
heq(t) impulse response of equalizer
x(t) initial base band signal
y(t) input to equalizer
nb(t) baseband noise at equalizer input
e(t) equalizer prediction error
d(t) reconstructed data
equalizer output
15
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16
Frequency Domain expression of

Equalizers Goal is to satisfy 7.4 7.5
makes combined transmitter, channel, receiver ?
all-pass channel

Thus ideal equalizer is inverse filter of channel
transfer function
provides flat composite received response with
linear phase response
for time varying channel ? 7.5 can be
approximately satisfied
for frequency selective channel it is required
to
- amplify frequency components with small
amplitudes
- attenuate frequency components with large
amplitudes

17
7.3 Training Adaptive Linear Equalizer
7.3 Training a Generic Adaptive Linear
Equalizer
18
7.3 Training a Generic Adaptive Linear Equalizer

an adaptive equalizer is a time varying filter
parameters are constantly re-tuned at discrete
time intervals

Transversal Filter equalizer (TFE) common
adaptive equalizer
structure
single equalizer input y(k), is a random process
that depends on
(i) instantaneous state of radio channel noise
(ii) transmitter receiver
(iii) original data, x(t)

TFE structure
N delay elements
N1 taps
N1 weights, wk (tunable complex multipliers)
- k position location in equalizer structure
e(k) error signal used for feedback control

19
Typical Adaptive Algorithm
(i) use e(k) to minimize some cost function (ii)
update wks on nth sample (n 1,2,) to
iteratively reduce cost function

wk1 updated weight derived from
wk current weight
e(k) error
y equalizer input vector

error, e(k) derived from d(k) and
d(k) scale replica of x(t) or represents known
property of x(t)
x(k) original transmitted baseband signal
equalizer output

20
e.g. use least mean squares (LMS) iteratively
search for near optimum wks
wk1 wk Ke(k)y

K constant adjusted to control variation in
successive weights

convergence phase repeat process rapidly and
evaluate e(k)

if e(k) lt threshold ? system is converged -
hold wks constant
if e(k) gt threshold ? initiate new training
sequence

Equalization Algorithms
(1) Classical Equalization Theory computes cost
function between
desired signal equalizer output
Mean Square Error (MSE), is most common approach
MSE Ee(k),e(k), is the auto-correlation of
e(k)
periodically transmit known training sequence
known copy of training sequence is required at
equalizer output
e.g. dk is set equal to xk and is known apriori
detect training sequence and compute cost
function
minimize cost function by adjusting wks
repeat until next training sequence is sent

(2) Modern Adaptive Equalization Algorithms
Blind Algorithms no training sequence required
for convergence
uses property restoral techniques of x(t)
becoming more important

Constant Modulus Algorithm (CMA)
used in constant envelope modulation
forces wks to maintain constant envelope on
received signal

Spectral Coherence Restoral Algorithm (SCORE)
exploits spectral redundancy in transmitted
signal

Linear Equalizer using adaptive algorithm
N 3 delay elements
4 taps and 4 weights

Training Mode

d(k) is either
set equal to transmitted signal x(k)
represents known property of x(k)

24
Mathematical Model of Equalizer
using known training sequence? d(k) x(k) error
signal given by
7.11
7.12
by substitution
e(k) x(k) - ykTwk
25
Computing Mean Square Error (MSE)
Compute square of error at kth sample
7.13

filter weights, wk are not included in time
average - assume they
have converged to optimum value
simplifying 7.14 would be trivial if x(k) and
y(k) are independent
however, y(k) should be correlated to d(k)
x(k)

26
Let p cross correlation vector between
y(k)
7.15

Let R input correlation matrix (or input
covariance matrix)
diagonal contains mean square values of y(k-i)
cross terms specify autocorrelation terms
resulting from delayed
samples of input signals

27
By Substitution from 7.15 7.16 into 7.14 ? MSE
can be written as
28

MSE in 7.17 is multidimensional function
with 2 tap weights (w0,w1) ? MSE is paraboloid
function
w0, w1 are plotted on horizontal axis
? is plotted on vertical axis
3 or more tap weights ? hyperparaboloid
in all cases ? error function is concave
upwards- minimum
may be found

29
Finding ?min ? use gradient, ? of 7.17 if R is
non-singular ? ?min occurs when wk are such that
?? 0
30

MMSE derivation assumes two things that are not
practical
y(k) is stationary
nb(t) 0

31
7.4 Equalizers in Receivers
7.4 Equalizers in Receivers
32

7.4 Equalizers in Receivers
7.3 describes equalizer fundamentals notation
7.4 describes role of equalizer in wireless link
Practical Received Signal always includes noise,
nb(t)
in 7.4 it was assumed nb(t) 0
in practice nb(t) ? 0 ? practical equalizers
are non-ideal

residual ISI small tracking errors always exist
instantaneous combined frequency response isnt
always flat
results in finite prediction error, e(n)

Assume digital system with
T sampling interval
tn nT time of nth sampling interval, n
0,1,2,

MSE is given by Ee(n)2 expected value
(ensemble average) of e(n)2
34

MSE, Ee(n)2 is an important indicator of
equalizer performance
better equalizers have smaller MSE
time-average can be used if e(n) is ergodic
- practically, ergodicity is impossible to
determine
- time-averages are used in practice

Minimizing MSE ? reduces BER
(i) assume e(n) is Gaussian distributed with
mean 0
(ii) Ee(n)2 is variance (or power) of e(n)
minimizing Ee(n)2 ? less chance of perturbing
training
sequence, d(n) x(n)
more likely decision device will detect x(n)?
thus
mean probability of error is smaller

better to minimize instantaneous probability of
error instead mean e(n)
generally results in non-linear equations
more difficult to solve in real-time

35
7.5 Survey of Equalization Techniques
7.5 Survey of Equalization Techniques
36
7.5 Survey of Equalization Techniques

decision making device generally processes,
, an analog equalizer
output
device determines the value of d(t), the
incoming digital bit stream
to determine d(t), device applies either
- slicing operation
- threshold operation (non-linear operation)

linear equalizer d(t) isnt used in equalizers
adaptive feedback non-linear equalizer d(t) is
used in feedback path
many filter structures are used in equalizer
implementation for each structure, there are
numerous algorithms used to adapt equalizer
37
types
structures
adaptive algorithm
equalizer
non-linear
linear
ML symbol detector
MLSE
DFE
Transversal Channel Estimator
transveral
transversal
lattice
lattice
0-forcing, LMS, RLS Fast RLS, Sq. Root RLS
LMS, RLS, Fast RLS Sq. Roor RLS
LMS, RLS, Fast RLS Sq. Roor RLS
Gradient RLS
Gradient RLS
38

1. Linear Transversal Equalizer (LTE) most
common equalizer structure
tapped delay lines spaced at symbol period, Ts
delay elements transfer function given by z-1
or exp(-jwTs)
delay elements have unity gain

(i) Finite Impulse Response (FIR) is the
simplest LTE
only feed-forward taps
transfer function polynomial in z-1
many zeros
poles only at z 0

39
FIR LTE Structure showing weights, processor
w0k(y(k) nb(k)) w1k(y(k-1) nb(k-1))
w2k(y(k-2) nb(k-2)) w3k(y(k-3) nb(k-3))
40

(ii) Infinite Impulse Response (IIR)
feed-forward feedback taps
transfer function is rational function in z-1
tend to be unstable in channels where strongest
pulse arrives after
an echo pulse (leading echoes) - rarely used

LTE with feed-forward feedback taps
41
Conversion of IIR structure to difference equation
42
7.2 Fundamentals of Equalization
7.6 Linear Equalizers
43
7.6 Linear Equalizers
44

cn complex conjugate of filter coefficients
y(i) input received sampled at time t0
iT t0 equalizer start time N N1 N2 1
number of taps N1 number of feeds in forward
position of equalizer N2 number of feeds in
reverse position of equalizer
45
Structure of LTE Digital FIR Filter
N1 2 N2 2 N 5
46
Linear Transversal Equalizer (LTE) Structure has
MMSE given by Pro89

Fexp(jwT) channels frequency response
N0 noise power spectral density

2. Lattice Filter
input y(k) is transformed into N intermediate
signals classified as
i. fn(k) forward signals
ii. bn(k) backward signal
intermediate signals are used
as input into tap multipliers
to calculate update coefficients
each stage characterized by recursive equations
for fn(k), bn(k)

48
Lattice Equalizer Structure
49
Evaluation of Lattice Equalier

Kn(k) reflection coefficient for nth stage of
lattice
bn backward error signals input into tap
weights

advantages of lattice equalizer
numerical stability
faster conversion
unique structure allows dynamic effective length
- if channel is not time dispersive ? use minimal
number of stages
- if channel dispersion increases ? increase
number of stages used
- adjustments in equalizer length made during
run-time
more complicated than LTE

51
7.7 Non-Linear Equalization
7.7 Non-Linear Equalization 7.7.1 Decision
Feedback Equalization (DFE) 7.7.2 Maximum
Likelihood Sequence Equalizer (MLSE)
52

7.7 Non-Linear Equalization
common in practical wireless systems
used with severe channel distortion
linear equalizers dont perform well in channels
with deep
spectral nulls
- tries to compensate for distortion ? high gain
in the vicinity
of spectral nulls
- high gain enhances noise at spectral nulls
3 effective non-linear methods used in may 2G
3G systems
1. Decision Feedback Equalization (DFE)
2. Maximum Likelihood Symbol Detection (MLSD)
3. Maximum Likelihood Symbol Estimation (MLSE)

53
7.7.1 Decision Feedback Equalization (DFE) basic
idea i. detect information symbol and pass it
through the decision device ii. after detection
- estimate ISI induced into future symbols
iii. subtract estimated ISI from detection of
future symbols DFE realization can be either
direct transversal form or lattice filters

Direct Transversal Form of DFE consists of FFF
and FBF filter
FFF feed forward filter with N1N21 taps
FBF feedback filter with N3 taps
- driven by detectors output (decision
threshold)
- filter coefficients adjusted based on past
detected symbol
- goal is to cancel ISI on current symbol

DFE structure
y(k - N2 i) are past equalizer inputs used
for FFF segment
y(k N1 i) are delayed equalizer inputs for
FFF segement

55
DFE output given by
Operation of Direct Transversal DFE

d(k), d(k-1), d(k-N3) feed back
y(k-i), y(k), y(ki) equalizer inputs

56
Minimum MSE for Direct Transversal DFE given by
?DFE
Minimum MSE for Linear Transversal Equalizer
given by ?LTE
57
Let FejwT)2denote channel response

LTE works well with flat fading deteriorates
with
severe distortion
spectral nulls
non-minimum phase channel - strongest energy
arrives after 1st
arriving signal components (aka obstructed
channel)

Lattice implementation of DFE is equivalent to
transversal DFE
feed-forward filter length N1
feedback filter length N2
N1 gt N2

Predictive DFE consists of FFF and FBF
FBF is driven by the input sequence given by
d(k)
FFF output
where d(k) detector output
FBF is called noise ISI predictor
- predicts noise residual ISI contained in
signal at FFF output
- subtracts this from d(k) after some feedback
delay
Predictive DFE performs as well as conventional
DFE as number of
taps in FFF and FBF approach infinity
Predictive Equalizer can also be realized as
lattice equalizer
RLS lattice algorithm can be used for fast
convergence

60
Predictive DFE
y(k) kth received signal sample d(k) output
decision e(k) error signal d(k) output
prediction

FBF input
e(k) compare input into decision device with
decision output, d(k)
compare FFF output with decision output
FFF inputs
compare FFF output and decision output
compare input into decision device with decision
output

61
7.7.2 Maximum Likelihood Sequence Equalizer (MLSE)
MSE based equalizers are optimum with respect to
minimum probability of symbol error (Ps) -
drawback assumes channel doesnt introduce
amplitude distortion - not practical for mobile
wireless channel
MLSE Class of equalizers surfaced as result of
MSE limitations - Optimum nearly Optimum
non-linear structures - Various forms of
maximum likelihood receiver structures exist -
channel impulse response simulator is part of the
algorithm

MLSE Operation is computationally intensive
does not decode each received symbol by itself
inputs several possible symbols tests all
possible data sequences
selects most likely

MLSE problem ? state estimation problem of
discrete-time finite
state machine
radio channel discrete-time finite state
machine with channel
coefficients fk
radio channel state is estimated by receiver at
any time using
L most recent input samples
M size of symbol alphabet
ML possible channel states
current state estimated by receiver
receiver uses ML trellis to model channel over
time
- Viterbi algorithm tracks channel state by paths
through trellis
- At stage k - a rank ordering ML most probable
sequences given
terminating in most recent L symbols

MLSE was 1st proposed by Forney-78
set up basic MLSE estimator structure
implemented with Viterbi algorithm
Viterbi algorithm is MLSE of state sequence of
finite state
Markov process observed in memoryless noise
successfully implemented in equalizers for
mobile wireless systems

MLSE based on DFE
optimal in the sense that it minimizes
probability of sequence error
requires knowledge of
i. channel characteristics to compute metrics
for making decision
ii. statistical distribution of noise corrupting
the signal
distribution of noise determines metrics form
for optimum
demodulation of received signal

an Estimated Data Sequence
match filter operates on continuous signal MLSE
channel estimator rely on discrete samples
65
7.8 Algorithms for Adaptive Equalization
7.8 Algorithms for Adaptive Equalization 7.8.1
Zero Forcing (ZF) Algorithm 7.8.2 Least Mean
Square (LMS) 7.8.3 Recursive Least Squares
Algorithm (RLS) 7.8.4 Summary of Algorithms
66
7.8 Algorithms for Adaptive Equalization

a wide range of algorithms exist to compensate
for unknown
time-varying channel
algorithms are used to
i. update equalizer coefficients
ii. track channel variations
we describe practical design issues outline 3
basic algorithms for
adaptive equalization
(i) zero-forcing (ZF)
(ii) least mean square (LMS)
(iii) recursive least squares (RLS)
- somewhat primitive by todays wireless
standards
- offers fundamental insight into design
operation
algorithms derived for LTE can be extended to
other structures
algorithm design is beyond scope of this class

67
Performance Measure of Algorithms
(1) Rate of Convergence (RoC) (2) Misadjustment
(3) Computational Complexity (4) Numerical
Properties Inaccuracies (5) practical cost
power issues (6) radio channel characteristics
(1) Rate of Convergence (RoC) iterations needed
for converge to optimal solution in response
to stationary inputs - fast RoC allows rapid
adaptation to stationary environment of
unknown statistics
(3) Computational Complexity number of
operations required to complete 1
iteration of the algorithm
68

(4) Numerical Properties inaccuracies produced
by round-off noise
representation errors in digital format
(floating point, shifting)
errors can influence stability of the algorithm

(5) Practical Issues in the choice of equalizer
structure algorithms
equalizers must justify cost, including relative
cost of computing platform
power budget-battery drain radio propagation
characteristics

(6) Radio Channel Characteristics Intended Use
mobiles speed ? determines channel fading rate
doppler spread
doppler spread directly related to channels
coherence time, Tc
data rate Tc directly impact algorithms
choice convergence rate
maximum expected time-delay spread dictates
number of taps
equalizer can only equalize over delay interval
? maximum delay
in filter structure

e.g. assume
the delay element of an equalizer has a maximum
10us delay
equalizer has 5 taps with 4 delay elements
? maximum delay spread than can be equalized
10us? 4 40us
- if transmission has multipath delay spread gt
40us ? cant be
equalized
- circuit complexity processing time increases
with number of
taps delay elements
- must know maximum number of delay elements
before selecting
equalizer structure

70
e.g. assume USDC sytem with fc 900MHz and max
symbol rate, Rs 24k symbols/sec let mobiles
velocity be v 80km/hr

c. coherence over TDMA slot ? slot duration must
be lt 6.34ms
max number of symbols per slot given by Nb
RsTc (24k ? 6.34) 154
equalizer updated after each slot

71
7.8.1 Zero Forcing (ZF) Algorithm
assume tapped delay line filter with N taps
delayed by T weights cns

cns selected to force samples of hch(t)?heq(t)
to 0 at all but 1 sample point
- N sample points each delayed by Ts (symbol
duration)
- hch(t) ? heq(t) convolved equalizer channel
impulse response
let n increase without bound ? obtain infinite
length equalizer with zero
ISI at ouput

Nyquist Criterion must be satisfied by combined
channel response Hch(f)Heq(f)
1, f lt 1/2Ts
7.28
Heq(f) frequency response of equalizer that is
periodic with 1/Ts Hch (f) folded channel
frequency response
72
infinite length ZF filter is inverse filter -
inverts folded response of channel - practically
length of ZF filter is truncated - may
excessively amplify noise at channel nulls not
used in mobile systems - performs well for
static channels with high SNR (local wired phone
lines
y(k-1)
y(k-2)
y(k)
y(k-4)
y(k-3)
d(k)
73
7.8.2 Least Mean Square (LMS)
LMS seeks to compute minimum mean square error,
?min

more robust than ZF algorithm for adaptive
equalizers
criterion minimization of MSE or desired
actual outputs
e(k) prediction error given by

x(k) original transmitted baseband signal d(k)
x(k) ? known training sequence transmitted

74
equalizer output given by

wN tap gain vector

75
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76
when 7.33 is satisifed ? MMSE is given by Jopt
77

Practically minimization of MSE (?min) is done
recursively
Can use stochastic gradient algorithm (commonly
called LMS algorithm)
requires 2N1 operations per iteration
simplest equalization algorithm
filter coefficients determined by 7.36a-c

n variable denoting sequence of iterations k
time instant N number of delay stages in
equalizer ? step size controls convergence
rate stability
78

LMS equalizer properties
maximizes signal-to-distortion ratio within
constraints of of equalizers
filter length, N as a performance constraint
if time-dispersion characteristics of y(k) gt
propagation-delay through
equalizer ? equalizer is not able to reduce
distortion
convergence rate is slow only one parameter,
step size, ? to control
adaptation rate

79
to prevent unstable adaptation ? ? bounded by
? can be controlled by total input power to avoid
instability

LMS has slow convergence rate
especially when ?max gtgt?min ? ?is has large
spread

80
7.8.3 Recursive Least Squares Algorithm (RLS)

LMS has a slow convergence rate-uses statistical
approach
RLS adaptive signal processing
faster convergence
more complex
additional parameters
based on least squares approach

81
Least Square Error based on time average
Hay86, Pro91
cumulative squared error of new tap gains on all
old data given by
yN(i) y(i), y(i-1),,y(i-N1)T

? weight factor that places emphasis on recent
data, ? 1
e(i,n) error based on wN(n) , used to test old
data, yN(i), at time i
e(i,n) complex conjugate of error
yN(i) data input vector at time i
wN(n) new tap gain vector at time n

RLS solution ? determine wN(n) such that J(n) is
minimized
J(n) cumulative squared error
wN(n) equalizers tap gain vector
RLS uses all previous data to test new wN(n)
? spreads more weight on most recent data
- J(n) tends to forget old data in non-stationary
environment
- stationary channel ? ? set 1

To obtain minimum J(n) ? solve gradient of J(n)
0
7.41
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84
from 7.43 it is possible to derive recursive
equation for RNN(n)
85
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86
Kalman RLS Algorithm
(1) initialization step w(0) k(0) x(0) 0
R-1(0) ? INN, ? positive constant
(2) recursively compute w(n)
87

? weight that determines tracking ability of
RLS equalizer
for time invariant channel ? set ? 1
normal range 0.8 lt ? lt1.0
smaller ? ? better tracking
too small ? ? unstable
value of ? has no influence on rate of
convergence

88
7.8.4 Summary of Algorithms
7.8.4 Summary of Algorithms
89

7.8.4 Summary of Algorithms
Several variations of LMS RLS algorithms for
adaptive equalizer
RLS algorithms
have similar convergence properties tracking
performance
better than LMS
high computational complexity
complex programming structures
some are tend to be unstable
Fast Transversal algorithm (FTF)
requires least computation of RLS class of
algorithms
can rescue a variable to avoid instability
rescue techniques are difficult for widely
varying mobile channel,
thus FTF is not widely used in these channels

90
Algorithm ? Operations pros cons
LMS Gradient DFE 2N1 simple slow convergence poor tracking
Kalman RLS 2.5N24.5N fast convergence good tracking computationally complex
FTF 7N14 fast convergence good tracking simple computation complex programming unstable (rescue)
Gradient Lattice 13N-8 stable simple computation flexible structure complex programming inferior performance
Gradient Lattice DFE 13N133N2-36 simple computation complex programming
Fast Kalman DFE 20N5 fast convergence good tracking complex programming computation not low unstable
Sq. Root RLS DFE 1.5N26.5N better numerical properties complex computation
91
7.9 Fractionally Space Equalizers (FSE)
7.9 Fractionally Space Equalizers (FSE)
92
7.9 Fractionally Space Equalizers (FSE)