7. Equalization, Diversity, Channel Coding

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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
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• 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

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7.1 Introduction
7.1 Introduction
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• 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

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• 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

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• 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

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• 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

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• 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

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7.2 Fundamentals of Equalization
7.2 Fundamentals of Equalization
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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
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(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

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• (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

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• 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

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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
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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

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7.3 Training Adaptive Linear Equalizer
7.3 Training a Generic Adaptive Linear
Equalizer
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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

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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

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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

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• 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

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• (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

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• 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)

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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
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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)

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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

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

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Finding ?min ? use gradient, ? of 7.17 if R is
non-singular ? ?min occurs when wk are such that
?? 0
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• MMSE derivation assumes two things that are not
  practical
• y(k) is stationary
• nb(t) 0

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7.4 Equalizers in Receivers
7.4 Equalizers in Receivers
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• 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)

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• 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
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• 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

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7.5 Survey of Equalization Techniques
7.5 Survey of Equalization Techniques
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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
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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
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• 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

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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))
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• (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
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Conversion of IIR structure to difference equation
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7.2 Fundamentals of Equalization
7.6 Linear Equalizers
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7.6 Linear Equalizers
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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
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Structure of LTE Digital FIR Filter
N1 2 N2 2 N 5
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Linear Transversal Equalizer (LTE) Structure has
MMSE given by Pro89

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

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• 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)

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Lattice Equalizer Structure
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Evaluation of Lattice Equalier

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

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• 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

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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)
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• 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)

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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

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• 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

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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

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Minimum MSE for Direct Transversal DFE given by
?DFE
Minimum MSE for Linear Transversal Equalizer
given by ?LTE
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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)

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

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• 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

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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

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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

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• 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

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• 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

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• 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
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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
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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

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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
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• (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

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• 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

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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

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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
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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)
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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

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equalizer output given by

• wN tap gain vector

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when 7.33 is satisifed ? MMSE is given by Jopt
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• 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
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• 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

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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

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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

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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

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• 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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from 7.43 it is possible to derive recursive
equation for RNN(n)
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Kalman RLS Algorithm
(1) initialization step w(0) k(0) x(0) 0
R-1(0) ? INN, ? positive constant
(2) recursively compute w(n)
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• ? 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

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7.8.4 Summary of Algorithms
7.8.4 Summary of Algorithms
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• 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
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7.9 Fractionally Space Equalizers (FSE)
7.9 Fractionally Space Equalizers (FSE)
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7.9 Fractionally Space Equalizers (FSE)

• taps spaced at Ts for LTE, DFE, MLSE
• optimum receiver for AWGN channel consists of
  match filter sampled
• periodically at Ts
• - match filter occurs prior to equalizer
• - match filter matched to corrupt signal from
  AWGN
• with channel distortion match filter must be
  matched to channel
• corrupted signal
• - practically channel response is unknown
• - thus optimum match filter must be adaptively
  estimated
• sub-optimal solution e.g. match filter matched
  to transmitted signal
• pulse x(k)
• - can result in significant degradation
• - extremely sensitive to sample timing

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• FSE samples incoming signal ? Nyquist rate
• compensates for channel distortion before
  aliasing effects occur
• - aliasing effects due to symbol rate sampling
• can compensate for timing delay for any
  arbitrary timing phase
• effectively incorporates match filter
  equalizer into single structure
• non-linear MLSE techniques gaining popularity in
  wireless systems