Transcript 幻灯片 1

Equalization
Equalization
Fig. Digital communication system using an adaptive equaliser at the receiver.
Equalization
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Equalization compensates for or mitigates inter-symbol
interference (ISI) created by multipaths in time dispersive
channels (frequency selective fading channels).
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Equalizer must be “adaptive”, since channels are time
varying.
Zero forcing equalizer
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Design from frequency domain viewpoint.
Zero forcing equalizer
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∴
must compensate for the channel distortion.
⇒ Inverse channel filter ⇒ completely eliminates ISI
caused by the channel ⇒ Zero Forcing equaliser, ⇒ ZF.
Zero forcing equalizer
Zero forcing equalizer
Fig. Pulses having a raised cosine spectrum
Zero forcing equalizer
Zero forcing equalizer
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Example:
A two-path channel with impulse response
The transfer function is
The inverse channel filter has the transfer function
Zero forcing equalizer
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Since DSP is generally adopted for automatic equalizers
⇒ it is convenient to use discrete time (sampled)
representation of signal.
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Received signal
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For simplicity, assume
say
Zero forcing equalizer
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Denote a T-time delay element by Z− 1, then
Zero forcing equalizer
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The transfer function of the inverse channel filter is
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This can be realized by a circuit known as the linear
transversal filter.
Zero forcing equalizer
Zero forcing equalizer
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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
Modeling of ISI channels
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Complex envelope of any modulated signal can be
expressed as
where ha(t) is the amplitude shaping pulse.
Modeling of ISI channels
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In general, ASK, PSK, and QAM are included, but most
FSK waveforms are not.
Received complex envelope is
where
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is channel impulse response.
Maximum likelihood receiver has impulse response
matched to f(t)
Modeling of ISI channels
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Output:
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where nb(t) is output noise and
Least Mean Square Equalizers
Fig. A basic equaliser during training
Least Mean Square Equalizers
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Minimization of the mean square error (MSE), ⇒ MMSE.
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Equalizer input
h(t): impulse response of tandem combination of transmit
filter, channel and receiver filter.
In the absence of noise and ISI
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The error due to noise and ISI at t=kT is given by
The error is
Least Mean Square Equalizers
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The MSE is
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In order to minimize
, we require
……
Least Mean Square Equalizers
Least Mean Square Equalizers
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The optimum tap coefficients are obtained as W = R−1 P.
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But this is solved on the knowledge of xk's, which are the
transmitted pilot data.
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A given sequence of xk's called a test signal, reference
signal or training signal is transmitted prior to the
information signal, (periodically).
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By detecting the training sequence, the adaptive
algorithm in the receiver is able to compute and update
the optimum wnk‘s -- until the next training sequence is
sent.
Least Mean Square Equalizers
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Example:
Determine the tap coefficients of a 2-tap MMSE for:
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Now, given that
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Least Mean Square Equalizers
Mean Square Error (MSE) for optimum
weights
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Let:
Mean Square Error (MSE) for optimum
weights
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Now, the optimum weight vector was obtained as
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Substituting this into the MSE formula above, we have
Mean Square Error (MSE) for optimum
weights
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Now, apply 3 matrix algebra rules:
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For any square matrix
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For any matrix product
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For any square matrix
Mean Square Error (MSE) for optimum
weights
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For the example
MSE for zero forcing equalizers
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Recall for ZF equalizer
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Assuming the same channel and noise as for the MMSE
equalizer
for MMSE
MSE for zero forcing equalizers
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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.
Diversity Techniques
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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.
Diversity Techniques
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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.
PDF of SNR for diversity systems
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Consider an M-branch space diversity system.
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Signal received at each branch has Rayleigh distribution.
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All branch signals are independent of one another.
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Assume the same mean signal and noise power ⇒ the
same mean SNR for all branches.
Instantaneous
PDF of SNR for diversity systems
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Probability that
threshold x is,
takes values less than some
Selection Diversity
Selection Diversity
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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,
Selection Diversity
Maximal Ratio Combining
Maximal Ratio Combining
is complex envelope of signal in the k-

th branch.
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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
Maximal Ratio Combining
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Assume time variation of gk (t) is much slower than that
of u(t) .
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Let nk(t) be the complex envelope of the additive
Gaussian noise in the k-th receiver (branch).
⇒ usually all k N are equal.
Maximal Ratio Combining
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Now define SNR of k-th branch as
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Now,
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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
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These factors are changed from instant to instant as the
branch signals change over the short term fading.
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How should
be chosen to achieve maximum
combiner output SNR at each instant?
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Assuming nk(t)’s are mutually independent (uncorrelated),
we have
Maximal Ratio Combining
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Instantaneous output SNR,
,
Maximal Ratio Combining
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Apply the Schwarz Inequality for complex valued
numbers.
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The equality holds if
arbitrary complex constant.
Let

for all k, where K is an
Maximal Ratio Combining
with equality holding if and only if

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, 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”.
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 Γ,
Convolutional Codes
Department of
Electrical Engineering
Wang Jin
Overview
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Background
Definition
Speciality
An Example
State Diagram
Code Trellis
Transfer Function
Summary
Assignment
Background
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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
Block codes Vs Convolutional Codes
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Block codes take k input bits and produce n
output bits, where k and n are large
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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
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Data passes through convolutional codes in a
continuous stream
Useful for low-latency communication
Definition
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A type of error-correction code in which
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each k-bit information symbol (each k-bit string) to
be encoded is transformed into an n-bit symbol,
where n>k
the transformation is a function of the last M
information symbols, where M is the constraint
length of the code
Speciality
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k bits are input, n bits are output
k and n are very small (usually k=1~3, n=2~6).
Frequently, we will see that k=1
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 k=1
An Example
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A simple rate k/n= 1/2 convolutional code
encoder (M=3)
+
Binary
information
Code digits
digits
Output
Input
+

The box represents one element of a serial
register
An Example (cont’d)
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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, g1=[1 1 1] and g2=[1 0 1]
An Example (cont’d)
+
Binary
information
Code digits
digits
Output
Input
+
t=0
x2 x1 x0
t=1
x3 x2 x1
t=2
x4 x3 x2
t=3
x5 x4 x3
When t=3, the
content of the
initial state (x2, x1,
x0 ) is missing.
To Determine the Output Codeword
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There are essentially two ways
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Only concentrate on state diagram approach
Contents of shift registers make up “state” of code:
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State diagram approach
Transform-domain approach
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
To Determine the Output Code
Word ---State Diagram
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Rate k/n=1/2 convolutional code encoder
(M=3)
+
Binary
information
digits
Code digits
D0 D1 D2
+
State
(recent M-1 digits)
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State is defined by the most (M-1)
message bits moves into the encoder
State Diagram (cont’d)
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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
State Diagram (cont’d)
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Easiest way to determine the state
diagram is to first determine the state table
as shown below
Input
(x3 )
0
1
0
1
0
1
0
1
Start state
(x2, x1)
00
00
01
01
10
10
11
11
Final state
(x3, x2)
00
10
00
10
01
11
01
11
Output
00
11
11
00
10
01
01
10
State Diagram (cont’d)
1/10
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1/01 means (for
example), that the
input binary digit to
the encoder was 1
and the
corresponding
codeword output is
01
11
1/01
0/01
0/10
10
01
1/00
1/11
0/11
00
0/00
Trellis Representation of
Convolutional Code
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State diagram is “unfolded” a function of time
Time indicated by movement towards right
Code Trellis
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It is simply another way of drawing the
state diagram
Code trellis for rate k/n=1/2 ,M=3
convolutional code shown below
00
1/11
01
Start state
10
0/11
01
1/00
Final state
0/10
10
1/01
0/01
11
00
0/00
1/10
11
Encoding Example Using Trellis
Diagram
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Trellis diagram, similar to state diagram, also
shows the evolution in time of the state
encoder
Consider the r=1/2, M=3 convolutional code
Encoding Example Using Trellis
Diagram State
0/00
00
00
0/11
01
1/11
01
1/00
0/10
10
10
0/01 1/01
11 1/10
Input data
0
1
Output
11
00
11
0
0
1
10
11
11
Distance Structure of a Convolutional
code
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The Hamming distance between any two distinct
code sequences
is the number of bits in
which they differ:
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The minimum free Hamming distance
of a
convolutional code is the smallest Hamming
distance separating any two distinct code
sequences:
The Transfer Function
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This is also known as the generating function
or the complete path enumerator.
Consider the r=1/2 , M=3 convolutional code
example and redraw the state diagram.
JND
d
JND
JND2
a0
JD
JD2
JD
c
b
JN
a1
The Transfer Function (Con’d)
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State “a” has been split into an initial state “a0”and 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:

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-----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
The Transfer Function (Con’d)
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Assuming a unity input, we can write the set of
equations
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By solving these equations,
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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.
Search for Good Codes
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We would like convolutional codes with large
free distance

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Must avoid “catastrophic codes”
Generators for best convolutional codes are
generally found via computer search
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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
Best Rate ½ Codes
M
3
Generators
(in Octal)
5
7
dfree
4
15
17
6
5
23
35
7
6
53
75
8
7
133
171
10
8
247
371
10
9
561
753
12
5
Best Rate 1/3 Codes
M
3
4
5
6
7
8
9
5
13
25
47
133
225
557
Generators
(in Octal)
7
15
33
53
145
331
663
dfree
7
17
37
75
171
367
711
8
10
12
13
15
16
18
Best Rate 2/3 Codes
M
2
17
Generators
(in Octal)
16
dfree
3
27
75
72
6
4
236
155
337
7
15
4
Summary
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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
Assignment

Question: Construct the state table and state
diagram for the encoder below.
Binary
information
+
digits
Input (k=1)
Code digits
Output (n=3)
+
+
THANK YOU