unidad_3_piv_v

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],

[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

13

Linear Equalizer/ Principle

14

Zero- forcing equalizer

15

Zero- forcing equalizer

16

MSE- equalizer

17

Decision- feedback equalizer

18


19


20

Zero- forcing DFE equalizer

21

MSE- Decision- feedback equalizer

22

Maximum- Likelihood Sequence

Estimation (MLSE)

23

Maximum- Likelihood Sequence

Estimation (MLSE)/ Principle

24

MLSE/ Principle

25

The Viterbi - equalizer

26

The Viterbi – equalizer (2)

27

Channel coding

Distribution of low-quality bits

28

Channel coding – Block interleaver

29

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


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


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


• 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

20

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 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


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)])

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).

67

unidad_3_piv_v

Documents