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1
Programa de Maestría en Gestión de Tecnologías de Información y
Comunicación (M-GTIC)
Nombre del Curso:
Comunicaciones Digitales
Instructor:
Marvin Arias Olivas, PhD
E-mail: [email protected],
Universidad Nacional de Ingeniería (UNI)
Managua, Nicaragua
UNIDAD III: Transmisión de Canales Limitados en Ancho de Banda
Contenido:
• Introducción
• Inter-symbol interference
• Linear Equalizers
• Decision-feedback equalizers
• Maximum-likelihood sequence estimation
• Summary
2
Digital Comunication System
3
Esquema Tipico de un Sistema de Comunicación ( Fuente: B. Sklar)
Introducción
Esquema Básico de un Sistema de Comunicación Inalámbrico.
Source
Coding
Channel
CodingMultiplex Modulate
Multiple
Access
RF and
Antennas
Source
Decoding
Channel
DecodingDemultiplex Demodulate
Multiple
Access
Antennas
and RF
Wireless
Channel
T r a n s m i t t e r
R e c e i v e r
Information
source
Information
sink
4
5
Comparison of various technologies
2011 M. Arias
6
Comparison of various technologies
7
Inter-Symbol interference- Background
8
Modeling of channel impulse response
9
Modeling of channel impulse response
10
The discrete-time channel model
11
Channel estimation
Training sequence
Transmit signal
Channel impulse
response
Delayed and
attenuated
echoes of Tx
signal
correlation
Estimated
Impulse
response
Region for measurement of impulse response
12
Linear Equalizer
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Linear Equalizer/ Principle
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Zero- forcing equalizer
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Zero- forcing equalizer
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MSE- equalizer
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Decision- feedback equalizer
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Decision- feedback equalizer
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Decision- feedback equalizer
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Zero- forcing DFE equalizer
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MSE- Decision- feedback equalizer
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Maximum- Likelihood Sequence
Estimation (MLSE)
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Maximum- Likelihood Sequence
Estimation (MLSE)/ Principle
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MLSE/ Principle
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The Viterbi - equalizer
26
The Viterbi – equalizer (2)
27
Channel coding
Distribution of low-quality bits
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Channel coding – Block interleaver
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Channel coding – Block interleaver
Matlab/ Example LDPC_enc_qpsk
2011 M. Arias
30
%Transmit an LDPC-encoded, QPSK-modulated bit stream through an AWGN
%channel, then demodulate, decode, and count errors.
hEnc = comm.LDPCEncoder;
hMod = comm.PSKModulator(4, 'BitInput',true);
hChan = comm.AWGNChannel(...
'NoiseMethod','Signal to noise ratio (SNR)','SNR',1);
hDemod = comm.PSKDemodulator(4, 'BitOutput',true,...
'DecisionMethod','Approximate log-likelihood ratio', ...
'Variance', 1/10^(hChan.SNR/10));
hDec = comm.LDPCDecoder;
hError = comm.ErrorRate;
for counter = 1:10
data = logical(randi([0 1], 32400, 1));
encodedData = step(hEnc, data);
modSignal = step(hMod, encodedData);
receivedSignal = step(hChan, modSignal);
demodSignal = step(hDemod, receivedSignal);
receivedBits = step(hDec, demodSignal);
errorStats = step(hError, data, receivedBits);
end
fprintf('Error rate = %1.2f\nNumber of errors = %d\n', errorStats(1), errorStats(2))
Matlab/ Example/ conv_enc_8dpsk
2011 M. Arias
31
%Transmit a convolutionally encoded 8-DPSK-modulated bit stream through an
%AWGN channel. Then, demodulate, decode using a Viterbi decoder, and count
%errors.hConEnc = comm.ConvolutionalEncoder;
hMod = comm.DPSKModulator('BitInput',true);
hChan = comm.AWGNChannel('NoiseMethod', ...
'Signal to noise ratio (SNR)', 'SNR',10);
hDemod = comm.DPSKDemodulator('BitOutput',true);
hDec = comm.ViterbiDecoder('InputFormat','Hard');
% Delay in bits is TracebackDepth times the number of bits per symbol
delay = hDec.TracebackDepth*...
log2(hDec.TrellisStructure.numInputSymbols);
hError = comm.ErrorRate('ComputationDelay',3,'ReceiveDelay',delay);
for counter = 1:20
data = randi([0 1],30,1);
encodedData = step(hConEnc, data);
modSignal = step(hMod, encodedData);
receivedSignal = step(hChan, modSignal);
demodSignal = step(hDemod, receivedSignal);
receivedBits = step(hDec, demodSignal);
errorStats = step(hError, data, receivedBits);
end
fprintf('Error rate = %f\nNumber of errors = %d\n', ...
errorStats(1), errorStats(2))
Summary
2011 M. Arias
32
Linear equalizers suffer from noise enhancement.
Decision-feedback equalizers (DFEs) use decisions on data
to remove part of ISI, allowing the linear equalizer part to be
less “powerful”and there by suffer less from noise
enhancement..
Incorrect decisions can cause error-propagation in DFEs,
since an incorrect decision may add ISI instead of removing
it.
Decoding of convolution codes is efficiently done with the
Viterbi algorithm.
Summary (2)
2011 M. Arias
33
Maximum-likelihood sequence estimation (MLSE) is optimal
in the sense of having the lowest probability of detecting the
wrong sequence.
Brut-force MLSE is prohibitively complex.
The Viterbi-equalizer (detector) implements the MLSE with
considerable lower complexity.
In fading channels we need interleaving in order to break
up fading dips (but causes delay).
UNIDAD III : transmisión en canales limitados en ancho de Banda
Contenido
• Introducción
• Adaptive Equalization
• Adaptive Linear Equalizers
• Adaptive Decision-feedback equalizers
• Summary
2011 M. Arias 34
35
Inter-Symbol interference- Background
Introduction
– Intersymbol Interference (ISI)
– Noise
Channel
Noise
desired signal ISI noise
2011 M. Arias
Introduction
The purpose of an equalizer is to reduce the ISI as much as
possible to maximize the probability of correct decisions
Channel
Noise
Equalizer
2011 M. Arias
Adaptive Equalization• Band-limited channels distort digital signals and introduce ISI , i.e., every
received sample becomes
Three different techniques can be used to compensate for this ISI
Maximum likelihood sequence estimation detector
Linear equalization
Decision-feedback equalization
All these techniques assume a known characteristics at the receiver
The channel is usually not known at the receiver and is time-varying.
Algorithms that automatically adjust the equalizer coefficients and adaptively compensate for time variation of the channel are needed.
0
1
L
k k n k n k
n
v f I f I n
Adaptive Linear Equalizer
• A linear equalizer can be designed based on:
Minimizing the peak distortion at the equalizer output
An adaptive linear equalizer based on this criterion is called The zero-Forcing Algorithm
Minimizing the mean-square error at the equalizer output
1 1
, 0 , 0
K L K L
n j n j
n K n n K n j
D c q c f
Adaptive Linear Equalizer
An adaptive linear equalizer based on this criterion is called The Least-Mean Square (LMS) Algorithm
2
2
2 *
or in a matrix form
, , 0,
0, otherwise, otherwise
K
k k j k j
j K
lj n l
lj lj
J E E I c v
C
x l j l j L f L l
o
The Zero-forcing AlgorithmThe output of the equalizer is given by
The distortion D(c) at the equalizer output is minimized by forcing the
equalizer response to the following:
Considering the following crosscorrelation
Imposing the condition:
The zero-forcin criterion is satisfied, i.e.,
0ˆ
K
k j k j k n n j k j
j K n k j
I c v q I q I c n
1, 0
0, 1n
nq
n K
* *ˆ
=
k k j k k k j
j j
E I E I I I
q
* 0k k jE I
1, 0
0, 1n
nq
n K
The Zero-forcing Algorithm
2011 M. Arias
The zero-forcing algorithm can be implemented as follows:
Initial training: Initial channel coefficinets are obtained by transmitting a
known training sequence of the same length or more than the equalizer
length.
The equalizer coefficients can be adjusted through a recursive algorithm
Decision-Directed Mode of adaptation: After the training period the
algorithm uses the actual estimates of the information data
( 1) ( ) *
( ) * ,
is a scale factor that controls the rate of adjustments.
ˆ
k k
j j k k j
k
j k k j
k k k
c c E I
c I K j K
I I
( 1) ( ) * ,
ˆ
k k
j j k k j
k k k
c c I K j K
I I
%%
%%
43
Adaptive Zero- forcing equalizer
44
Adaptive Zero- forcing equalizer
The Least Mean Squares (LMS)Algorithm
• This algorithm is based on minimizing the mean
square which consist of solving the set of linear
equations
• Two possible solutions:
• Inverting the covariance matrix Г.
C
The Least Mean Squares (LMS)Algorithm
1
1
* *
- Use an iterative procedure
, 0,1,2,3,...
is a small positive number that controls convergentce,
is the gradientvector
... ...
k k k
k
k k k k k k
T
k k K k k K
C
C C G k
G
G C E V V
V v v v
The Least Mean Squares (LMS)Algorithm
• Iterative procedure reduces to the following equations
*
1
0
, 0,1,2,3,...
Convergence is reached at some , i.e.,
=0 ,
no further change occurs in the tap weights.
k k k k
ko
C C V k
k k
G
Converge of Adaptive Linear Equalization
• The convergence of adaptive linear equalizers is governed by
the step-size parameter Δ
• For the LMS algorithm, convergence is ensure if Δ satisfies
• The convergence rate is rather dependent on the ratio
is small, Δ can be selected so as to achieve
rapid convergence
• If is large, the convergence rate will be slow.
max
max
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is the largest eigenvalue of .
max min/
max min/
Adaptive Equalization
• The object is to adapt the coefficients to minimize the noise and intersymbol interference (depending on the type of equalizer) at the output.
• The adaptation of the equalizer is driven by an error signal.
The aim is to minimize:
2
kJ E e
EqualizerChannel
+
Error
signal
2011 M. Arias
Adaptive Filter Block Diagram
Adaptive Filter Block Diagram
d(n) Desired
y(n)
e(n)
+
-x(n)
Filter InputAdaptive Filter
e(n) Error Output
Filter Output
( ) ( ) ( )e n d n y n
Adaptive Filter Equation
• The Adaptive Filter is a Finite Impulse Response Filter (FIR), with N variable coefficients w.
0
( ) ( ) ( )N
k
k
y n w n x n k
The LMS Equation
• The Least Mean Squares Algorithm (LMS) updates each
coefficient on a sample-by-sample basis based on the error
e(n).
• This equation minimises the power in the error e(n).
kw (n 1) ( ) ( ) ( )k kw n e n x n
The Least Mean Squares (LMS)Algorithm
• The value of Δ (delta) is critical.
• If Δ is too small, the filter reacts slowly.
• If Δ is too large, the filter resolution is poor.
• The selected value of Δ is a compromise.
Adaptive EqualizationThere are two modes that adaptive equalizers work;
• Decision Directed Mode:
The receiver decisions are used to generate the error signal. Decision directed equalizer adjustment is effective in tracking slow variations in the channel response. However, this approach is not effective during initial acqusition .
• Training Mode:
To make equalizer suitable in the initial acqusition duration, a training signal is needed. In this mode of operation, the transmitter generates a data symbol sequence known to the receiver.
Once an agreed time has elapsed, the slicer output is used as a training signal and the actual data transmission begins.
55
Adaptive Linear Equalizer
56
Matlab/Example/ Training Sequence
%The following code illustrates how to use equalize with a training sequence. The
%training %sequence in this case is just the beginning of the transmitted message.
% Set up parameters and signals.
M = 4; % Alphabet size for modulation
msg = randi([0 M-1],1500,1); % Random message
hMod = comm.QPSKModulator('PhaseOffset',0);
modmsg = step(hMod,msg); % Modulate using QPSK.
trainlen = 500; % Length of training sequence
chan = [.986; .845; .237; .123+.31i]; % Channel coefficients
filtmsg = filter(chan,1,modmsg); % Introduce channel distortion.
% Equalize the received signal.
eq1 = lineareq(8, lms(0.01)); % Create an equalizer object.
eq1.SigConst = step(hMod,(0:M-1)')'; % Set signal constellation.
[symbolest,yd] = equalize(eq1,filtmsg,modmsg(1:trainlen));
% Equalize.
57
Matlab/Example/ Training Sequence% Plot signals.
h = scatterplot(filtmsg,1,trainlen,'bx'); hold on;
scatterplot(symbolest,1,trainlen,'g.',h);
scatterplot(eq1.SigConst,1,0,'k*',h);
legend('Filtered signal','Equalized signal',...
'Ideal signal constellation');
hold off;
% Compute error rates with and without equalization.
hDemod = comm.QPSKDemodulator('PhaseOffset',0);
demodmsg_noeq = step(hDemod,filtmsg); % Demodulate unequalized signal.
demodmsg = step(hDemod,yd); % Demodulate detected signal from equalizer.
hErrorCalc = comm.ErrorRate; % ErrorRate calculator
ser_noEq = step(hErrorCalc, ...
msg(trainlen+1:end), demodmsg_noeq(trainlen+1:end));
reset(hErrorCalc)
ser_Eq = step(hErrorCalc, msg(trainlen+1:end),demodmsg(trainlen+1:end));
disp('Symbol error rates with and without equalizer:')
disp([ser_Eq(1) ser_noEq(1)])
58
Adaptive Decision-Feedback EqualizerIn this case the equalizer output is given by
Both the zero-forcing algorithm and the LMS can be applied
the same way:
2
1
0
"
1
ˆK
k j k j j k j
j K j
I c v c I
%
1 2
*
1
*
1
1
For the LMS algorithm
, 0,1,2,3,...
and in a decision-directed mode,
, 0,1,2,3,...
with
V= ... ...
ˆ,
k k k k
k k k k
T
k K k k k K
k k k k k k
C C V k
C C V k
v v I I
I I I I
)
%%
60
Adaptive Decision- feedback equalizer
61
Adaptive Decision- feedback equalizer
62
The Viterbi - equalizer
63
The Viterbi – equalizer (2)
Adaptive Channel Estimator
2011 M. Arias
64
A MLSE detector requires channel state information in
computing the branch metrics
The channel coefficients fn can be adjusted using an adaptive
algorithm having the same structure as that used in linear
equalization.
By letting
We can write
For starup operation, a short training sequence can be sent to
perform the initial tap coefficients adjustments. Decision-
directed mode is used then after,
2
"
0
ˆ ˆL
k n n
n
v f I
ˆk k kv v
( 1) *
"ˆ ˆ , 0,1,...,k k
n n k kf f I n L %
65
Adaptive Channel Estimator for MLSE
Summary
2011 M. Arias
66
Zero-forcing and MSE Equalizations
A training sequence is required for initially adjusting the equalizer
coefficients
Reduce the system efficiency
Self Recovering (Blind) Equalization
These equalizers do not require any training sequence.
There are three types of blind equalizers:
Steepest descent approach for coefficient adaptation
Higher order statistics of the received signal for channel
estimation
Maximum likelihood criterion
• Channel estimation on average over data sequences
• Joint channel and data estimation
References
Digital Communications. John G. Proakis and
Masoud Salehi. Fifth Edition 2008, Editorial: McGraw-
Hill, ISBN 978-0—07-295716-7. (Chap. 9)
Contemporary Communication Systems using
Matlab , John G. Proakis and Masoud Salehi.
Second Edition 2004, Editorial Cengage Learning
(Chapter 8).
Digital Communications: Fundamentals and
Applications. Bernard Sklar. Second Edition, 2001.
ISBN: 978- 0130847881. (Chap. 8).
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