Equalization Department of Electrical Engineering Wang Jin

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Equalization Department of Electrical
EngineeringWang Jin
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Equalization
Fig. Digital communication system using an
adaptive equaliser at the receiver.
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Equalization

• Equalization compensates for or mitigates
  inter-symbol interference (ISI) created by
  multipaths in time dispersive channels (frequency
  selective fading channels).
• Equalizer must be adaptive, since channels are
  time varying.

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Zero forcing equalizer

• Design from frequency domain viewpoint.

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Zero forcing equalizer

• ? must compensate for the channel
  distortion.

? Inverse channel filter ? completely eliminates
ISI caused by the channel ? Zero Forcing
equaliser, ? ZF.
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Zero forcing equalizer
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Zero forcing equalizer
Fig. Pulses having a raised cosine spectrum
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Zero forcing equalizer
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Zero forcing equalizer

• Example
• A two-path channel with impulse response
• The transfer function is
• The inverse channel filter has the transfer
  function

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Zero forcing equalizer

• Since DSP is generally adopted for automatic
  equalizers ? it is convenient to use discrete
  time (sampled) representation of signal.
• Received signal
• For simplicity, assume
  say

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Zero forcing equalizer

• Denote a T-time delay element by Z- 1, then

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Zero forcing equalizer

• The transfer function of the inverse channel
  filter is
• This can be realized by a circuit known as the
  linear transversal filter.

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Zero forcing equalizer
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Zero forcing equalizer

• The exact ZF equalizer is of infinite length but
  usually implemented by a truncated (finite)
  length approximation.
• For , a 2-tap version of
  the ZF equalizer has coefficients

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Modeling of ISI channels

• Complex envelope of any modulated signal can be
  expressed as
• where ha(t) is the amplitude shaping pulse.

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Modeling of ISI channels

• In general, ASK, PSK, and QAM are included, but
  most FSK waveforms are not.
• Received complex envelope is
• where is
  channel impulse response.
• Maximum likelihood receiver has impulse response
• matched to f(t)

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Modeling of ISI channels

• Output
• where nb(t) is output noise and

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Least Mean Square Equalizers
Fig. A basic equaliser during training
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Least Mean Square Equalizers

• Minimization of the mean square error (MSE), ?
  MMSE.
• Equalizer input
• h(t) impulse response of tandem combination
  of transmit filter, channel and receiver filter.
• In the absence of noise and ISI
• The error due to noise and ISI at tkT is given
  by
• The error is

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Least Mean Square Equalizers

• The MSE is
• In order to minimize , we require

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Least Mean Square Equalizers
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Least Mean Square Equalizers

• The optimum tap coefficients are obtained as W
  R-1 P.
• But this is solved on the knowledge of xk's,
  which are the transmitted pilot data.
• A given sequence of xk's called a test signal,
  reference signal or training signal is
  transmitted prior to the information signal,
  (periodically).
• By detecting the training sequence, the adaptive
  algorithm in the receiver is able to compute and
  update the optimum wnks -- until the next
  training sequence is sent.

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Least Mean Square Equalizers

• Example
• Determine the tap coefficients of a 2-tap MMSE
  for
• Now, given that

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Least Mean Square Equalizers
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Mean Square Error (MSE) for optimum weights

• Let

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Mean Square Error (MSE) for optimum weights

• Now, the optimum weight vector was obtained as
• Substituting this into the MSE formula above, we
  have

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Mean Square Error (MSE) for optimum weights

• Now, apply 3 matrix algebra rules
• For any square matrix
• For any matrix product
• For any square matrix

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Mean Square Error (MSE) for optimum weights

• For the example

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MSE for zero forcing equalizers

• Recall for ZF equalizer
• Assuming the same channel and noise as for the
  MMSE equalizer

for MMSE
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MSE for zero forcing equalizers

• The ZF equalizer is an inverse filter ? it
  amplifies noise at frequencies where the channel
  transfer function has high attenuation.
• The LMS algorithm tends to find optimum tap
  coefficients compromising between the effects of
  ISI and noise power increase, while the ZF
  equalizer design does not take noise into
  account.

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Diversity Techniques

• Mitigates fading effects by using multiple
  received signals which experienced different
  fading conditions.
• Space diversity With multiple antennas.
• Polarization diversity Using differently
  polarized waves.
• Frequency diversity With multiple frequencies.
• Time diversity By transmission of the same
  signal in different times.
• Angle diversity Using directive antenna aimed at
  different directions.
• Signal combining methods.
• Maximal Ratio combining.

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Diversity Techniques

• Equal gain combining.
• Selection (switching) combining.
• Space diversity is classified into
  micro-diversity and macro-diversity.
• Micro-diversity Antennas are spaced closely to
  the order of a wavelength. Effective for fast
  fading where signal fades in a distance of the
  order of a wavelength.
• Macro (site) diversity Antennas are spaced wide
  enough to cope with the topographical conditions
  ( eg buildings, roads, terrain). Effective for
  shadowing, where signal fades due to the
  topographical obstructions.

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PDF of SNR for diversity systems

• Consider an M-branch space diversity system.
• Signal received at each branch has Rayleigh
  distribution.
• All branch signals are independent of one
  another.
• Assume the same mean signal and noise power ? the
  same mean SNR for all branches.
• Instantaneous

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PDF of SNR for diversity systems

• Probability that takes values less than some
  threshold x is,

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Selection Diversity
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Selection Diversity

• Branch selection unit selects the branch that has
  the largest SNR.
• Events in which the selector output SNR, , is
  less than some value, x,is exactly the set of
  events in which each is simultaneously below
  x.
• Since independent fading is assumed in each of
  the M branches,

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Selection Diversity
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Maximal Ratio Combining
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Maximal Ratio Combining

• is complex envelope
  of signal in the k-th branch.
• The complex equivalent low-pass signal u(t)
  containing the information is common to all
  branches.
• Assume u(t) normalized to unit mean square
  envelope such that

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Maximal Ratio Combining

• Assume time variation of gk (t) is much slower
  than that of u(t) .
• Let nk(t) be the complex envelope of the additive
  Gaussian noise in the k-th receiver (branch).
• ? usually all k N
  are equal.

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Maximal Ratio Combining

• Now define SNR of k-th branch as
• Now,
• Where are the complex combining weight
  factors.
• These factors are changed from instant to instant
  as the branch signals change over the short term
  fading.

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Maximal Ratio Combining

• These factors are changed from instant to instant
  as the branch signals change over the short term
  fading.
• How should be chosen to achieve maximum
  combiner output SNR at each instant?
• Assuming nk(t)s are mutually independent
  (uncorrelated), we have

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Maximal Ratio Combining

• Instantaneous output SNR, ,

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Maximal Ratio Combining

• Apply the Schwarz Inequality for complex valued
  numbers.
• The equality holds if for all k,
  where K is an arbitrary complex constant.
• Let

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Maximal Ratio Combining

• with equality holding if and only if
  , for each k.
• Optimum weight for each branch has magnitude
  proportional to the signal magnitude and
  inversely proportional to the branch noise power
  level, and has a phase, canceling out the signal
  (channel ) phase.
• This phase alignment allows coherent addition of
  branch signals ?co-phasing.

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Maximal Ratio Combining

• each has a chi-square
  distribution.
• is distributed as chi-square with 2M
  degrees of freedom.

• Average SNR, , is simply the sum of the
  individual
• for each branch, which is G,

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Convolutional CodesDepartment of Electrical
EngineeringWang Jin
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Overview

• Background
• Definition
• Speciality
• An Example
• State Diagram
• Code Trellis
• Transfer Function
• Summary
• Assignment

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Background

• Convolutional code is a kind of code using in
  digital communication systems
• Using in additive white Gaussian noise channel
• To improve the performance of radio and satellite
  communication systems
• Include two parts encoding and decoding

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Block codes Vs Convolutional Codes

• Block codes take k input bits and produce n
  output bits, where k and n are large
• There is no data dependency between blocks
• Useful for data communications
• Convolution codes take a small number of input
  bits and produce a small number of output bits
  each time period
• Data passes through convolutional codes in a
  continuous stream
• Useful for low-latency communication

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Definition

• A type of error-correction code in which
• each k-bit information symbol (each k-bit string)
  to be encoded is transformed into an n-bit
  symbol, where ngtk
• the transformation is a function of the last M
  information symbols, where M is the constraint
  length of the code

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Speciality

• k bits are input, n bits are output
• k and n are very small (usually k13, n26).
  Frequently, we will see that k1
• Output depends not only on current set of k input
  bits, but also on past input
• The constraint length M is defined as the
  number of shifts, over which a single message it
  can influence the encoder output
• Frequently, we will see that k1

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An Example

• A simple rate k/n 1/2 convolutional code encoder
  (M3)
• The box represents one element of a serial
  register

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An Example (contd)

• The content of the shift registers is shifted
  from left to right
• Plus sign represents modulo-2 (XOR) addition
• Output by encoder are multiplexed into serial
  binary digits
• For every binary digit enters the encoder, two
  code digits are output
• A generator sequence specifies the connections of
  a modulo-2 (XOR) adder to the encoder shift
  register.
• In this example, there are two generator
  sequences, g11 1 1 and g21 0 1

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An Example (contd)
t0
When t3, the content of the initial state (x2,
x1, x0 ) is missing.
t1
t2
t3
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To Determine the Output Codeword

• There are essentially two ways
• State diagram approach
• Transform-domain approach
• Only concentrate on state diagram approach
• Contents of shift registers make up state of
  code
• Most recent input is most significant bit of
  state
• Oldest input is least significant bit of state
• (this convention is sometimes reverse)
• Arcs connecting states represent allowable
  transitions
• Arcs are labeled with output bits transmitted
  during transition

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To Determine the Output Code Word ---State Diagram

• Rate k/n1/2 convolutional code encoder (M3)
• State is defined by the most (M-1) message bits
  moves into the encoder

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State Diagram (contd)

• There are four states 00, 01, 10, 11
  corresponding to the (M-1) bits
• Generally, assuming the encoder starts in the
  all-zero 00 state

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State Diagram (contd)

• Easiest way to determine the state diagram is to
  first determine the state table as shown below

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State Diagram (contd)

• 1/01 means (for example), that the input binary
  digit to the encoder was 1 and the corresponding
  codeword output is 01

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Trellis Representation of Convolutional Code

• State diagram is unfolded a function of time
• Time indicated by movement towards right

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Code Trellis

• It is simply another way of drawing the state
  diagram
• Code trellis for rate k/n1/2 ,M3 convolutional
  code shown below

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Encoding Example Using Trellis Diagram

• Trellis diagram, similar to state diagram, also
  shows the evolution in time of the state encoder
• Consider the r1/2, M3 convolutional code

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Encoding Example Using Trellis Diagram
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Distance Structure of a Convolutional code

• The Hamming distance between any two distinct
  code sequences is the number
  of bits in which they differ
• The minimum free Hamming distance of a
  convolutional code is the smallest Hamming
  distance separating any two distinct code
  sequences

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The Transfer Function

• This is also known as the generating function or
  the complete path enumerator.
• Consider the r1/2 , M3 convolutional code
  example and redraw the state diagram.

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The Transfer Function (Cond)

• State a has been split into an initial state
  a0and a final state a1
• We are interested in the number of paths that
  diverge from the all aero path at state a at
  some point in time and remerges with the all-zero
  path.
• Each branch transition is labeled with a term
  , where are all integers such that
• -----corresponds to the length of the branch
• -----Hamming weigh of the input zero for a 0
  input and one for a 1 input
• -----Hamming weight of the encoder output for
  that branch

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The Transfer Function (Cond)

• Assuming a unity input, we can write the set of
  equations
• By solving these equations,
• From the transfer function, there is one path at
  a Hamming distance of 5 from the all-zero path.
  This path is of length 3 branches and corresponds
  to a difference of one input information bit from
  the all zero path. Other terms can be interpreted
  similarly. The minimum distance is thus
  5.

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Search for Good Codes

• We would like convolutional codes with large free
  distance
• Must avoid catastrophic codes
• Generators for best convolutional codes are
  generally found via computer search
• Search is constrained to codes with regular
  structure
• Search is simplified because any permutation of
  identical generators is equivalent
• Search is simplified because of linearity

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Best Rate ½ Codes
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Best Rate 1/3 Codes
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Best Rate 2/3 Codes
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Summary

• What is convolutional code
• The transformation of a convolutional code
• We can represent convolutional codes as
  generators, block diagrams, state diagrams and
  trellis diagrams
• Convolutional codes are useful for real-time
  applications because they can be continuously
  encoded and decoded

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Assignment

• Question Construct the state table and state
  diagram for the encoder below.

Binary information digits
Code digits
Input (k1)
Output (n3)


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THANK YOU