Modulation, Demodulation and Coding Coursesite.iugaza.edu.ps/ahdrouss/files/2015/09/slides31.pdf ·...

Lecture 3

Lecture 3 2

Example of M-ary PAM

-B

B

T

‘01’

3B

T

T

-3B

T

‘00’ ‘10’

‘1’

A.

T

‘0’

T

-A.

Assuming real time transmission and equal energy per

transmission data bit for binary-PAM and 4-ary PAM:

• 4-ary: T=2Tb and Binary: T=T

b

•

4-ary PAM

(rectangular pulse)‏

Binary PAM

(rectangular pulse)‏

‘11’

22 10BA

Lecture 3 3

Example of M-ary‏PAM‏…‏

0 Tb 2T

b 3T

b 4T

b 5T

b 6T

b

0 Ts 2T

s

0 T 2T 3T

2.2762 V 1.3657 V

1 1 0 1 0 1

0 T 2T 3T 4T 5T 6T

Rb=1/T

b=3/T

s

R=1/T=1/Tb=3/T

s

Rb=1/T

b=3/T

s

R=1/T=1/2Tb=3/2T

s=1.5/T

s

Lecture 3 4

Today we are going to talk about:

Receiver structure

Demodulation (and sampling)‏

Detection

First step for designing the receiver

Matched filter receiver

Correlator receiver

Lecture 3 5

Demodulation and detection

Major sources of errors:

Thermal noise (AWGN)‏ disturbs the signal in an additive fashion (Additive)

has flat spectral density for all frequencies of interest (White)‏

is modeled by Gaussian random process (Gaussian Noise)

Inter-Symbol Interference (ISI)‏ Due to the filtering effect of transmitter, channel and receiver, symbols‏are‏“smeared”.‏

Format Pulse

modulate

Bandpass

modulate

Format Detect Demod.

& sample

)(tsi)(tgiim

im̂ )(tr)(Tz

channel )(thc

)(tn

transmitted symbol

estimated symbol

Mi ,,1

M-ary modulation

Lecture 3 6

Example: Impact of the channel

Lecture 3 7

Example:‏Channel‏impact‏…

)75.0(5.0)()( Tttthc

Lecture 3 8

Figure : Three examples of filtering an ideal pulse. (a) Example 1: Good-fidelity output. (b) Example 2: Good-recognition output. (c) Example3: Poor-recognition output.

Lecture 3 9

Receiver tasks

Demodulation and sampling:

Waveform recovery and preparing the received

signal for detection:

Improving the signal power to the noise power (SNR)

using matched filter

Reducing ISI using equalizer

Sampling the recovered waveform

Detection:

Estimate the transmitted symbol based on the

received sample

Lecture 3 10

Receiver structure

Frequency

down-conversion

Receiving

filter

Equalizing

filter

Threshold

comparison

For bandpass signals Compensation for

channel induced ISI

Baseband pulse

(possibly distored)‏ Sample

(test statistic)‏ Baseband pulse

Received waveform

Step 1 – waveform to sample transformation Step 2 – decision making

)(tr)(Tz

im̂

Demodulate & Sample Detect

Lecture 3 11

Baseband and bandpass

Bandpass model of detection process is

equivalent to baseband model because:

The received bandpass waveform is first

transformed to a baseband waveform.

Equivalence theorem:

Performing bandpass linear signal processing followed by

heterodyning the signal to the baseband, yields the same

results as heterodyning the bandpass signal to the

baseband , followed by a baseband linear signal

processing.

12

Detection of Binary Signal in Gaussian Noise

For any binary channel, the transmitted signal over a symbol interval (0,T) is:

The received signal r(t) degraded by noise n(t) and possibly degraded by the impulse response of the channel hc(t), is

Where n(t) is assumed to be zero mean AWGN process

For ideal distortionless channel where hc(t) is an impulse function and convolution with hc(t) produces no degradation, r(t) can be represented as:

1

2

( ) 0 1( )

( ) 0 0i

s t t T for a binarys t

s t t T for a binary

2,1)()(*)()( itnthtstr ci

Ttitntstr i 02,1)()()(

13


The recovery of signal at the receiver consist of two parts

Filter

Reduces the effect of noise (as well as Tx induced ISI)

The output of the filter is sampled at t=T. This reduces the received signal to a single variable z(T) called the test statistics

Detector (or decision circuit)

Compares the z(T) to some threshold level 0 , i.e.,

where H1 and H2 are the two possible binary hypothesis

1

2

0( )

H

H

z T

Lecture 3 14

Model the received signal

Simplify the model:

Received signal in AWGN

)(thc)(tsi

)(tn

)(tr

)(tn

)(tr)(tsi

Ideal channels

)()( tthc

AWGN

AWGN

)()()()( tnthtstr ci

)()()( tntstr i

Receiver Functionality

15


The recovery of signal at the receiver consist of two parts:

1. Waveform-to-sample transformation (Blue Block)

Demodulator followed by a sampler

At the end of each symbol duration T, predetection point yields a sample z(T ), called test statistic

Where ai‏(T ) is the desired signal component,

and no(T ) is the noise component

2. Detection of symbol

Assume that input noise is a Gaussian random process and receiving filter is linear

0( ) ( ) ( ) 1,2iz T a T n T i

2

0

0

0

02

1exp

2

1)(

nnp

16

Then output is another Gaussian random process

Where 0 2 is the noise variance

The ratio of instantaneous signal power to average noise power , (S/N)T, at a time t=T, out of the sampler is:

Need to achieve maximum (S/N)T

2

11

00

1 1( | ) exp

22

z ap z s

2

22

00

1 1( | ) exp

22

z ap z s

2

0

2

i

T

a

N

S


17

The Matched Filter

Objective: To maximizes (S/N)T

Expressing signal ai(t) at filter output in terms of filter transfer function H(f ) (Inverse Fourier transform of the product H(f)S(f)).

where S(f) is the Fourier transform of input signal s(t)

Output noise power can be expressed as:

Expressing (S/N)T as:

dfefSfHta ftj

i

2)()()(

dffH

N 202

0 |)(|2

dffHN

dfefSfH

N

SfTj

T 20

22

|)(|2

)()(

18

Now according to Schwarz’s‏Inequality:

Equality holds if f1(x) = k f*2(x) where k is arbitrary constant and * indicates complex conjugate

Associate H(f) with f1(x) and S(f) ej2 fT with f2(x) to get:

Substituting to yield:

dffSdffHdfefSfH fTj222

2 )()()()(

dxxfdxxfdxxfxf2

2

2

1

2

21 )()()()(

dffSNN

S

T

2

0

)(2

The Matched Filter

19

Or and energy E of the input signal s(t):

Thus (S/N)T depends on input signal energy

and power spectral density of noise and

NOT on the particular shape of the waveform

Equality for holds for optimum filter transfer function H0(f)

such that:

For real valued s(t):

0

2max

N

E

N

S

T

dffSE2

)(

0

2max

N

E

N

S

T

fTjefkSfHfH 2

0 )(*)()(

fTjefkSth 21 )(*)(

( ) 0( )

0

ks T t t Th t

else where

The Matched Filter

20

The impulse response of a filter producing maximum output signal-to-noise ratio is the mirror image of message signal s(t), delayed by symbol time duration T.

The filter designed is called a MATCHED FILTER

Defined as:

a linear filter designed to provide the maximum signal-to-noise power ratio at its output for a given transmitted symbol waveform

( ) 0( )

0

ks T t t Th t

else where

The Matched Filter

21

Correlation realization of Matched filter

A filter that is matched to the waveform s(t), has an impulse response

h(t) is a delayed version of the mirror image of the original signal waveform

( ) 0( )

0

ks T t t Th t

else where

Signal Waveform Mirror image of signal waveform

Impulse response of matched filter

Lecture 3 22

Example of matched filter

T t T t T t 0 2T

)()()( thtsty opti 2A)(tsi )(thopt

T t T t T t 0 2T

)()()( thtsty opti 2A)(tsi )(thopt

T/2 3T/2 T/2 T T/2

2

2A

T

A

T

A

T

A

T

A

T

A

T

A

Lecture 3 23

Properties of the matched filter

The Fourier transform of a matched filter output with the matched signal as input is, except for a time delay factor, proportional to the ESD of the input signal.

The output signal of a matched filter is proportional to a shifted version of the autocorrelation function of the input signal to which the filter is matched.

The output SNR of a matched filter depends only on the ratio of the signal energy to the PSD of the white noise at the filter input.

Two matching conditions in the matched-filtering operation:

spectral phase matching that gives the desired output peak at time T.

spectral amplitude matching that gives optimum SNR to the peak value.

)2exp(|)(|)( 2 fTjfSfZ

sss ERTzTtRtz )0()()()(

2/max

0N

E

N

S s

T

24

This is a causal system

Recall that a system is causal if before an excitation is applied at time t = T, the response is zero for - < t < T

The signal waveform at the output of the matched filter is

Substituting h(t) to yield:

When t =T,

dthrthtrtzt

)()()(*)()(0

dtTsr

dtTsrtz

t

t

0

0

)(

)()()(

dsrtzT

)()()(0


25

The function of the correlator and matched filter are the same

Compare (a) and (b)

From (a)

T

dttstrtz0

)()()(

T

TtdsrTztz

0)()()()(


26

From (b)

But

At the sampling instant t = T, we have

This is the same result obtained in (a)

Hence

0'( ) ( )* ( ) ( ) ( ) ( ) ( )

t

z t r t h t r h t d r h t d

)()]([)()()( tTstTsthtTsth

t

dtTsrtz0

)()()('

0 0'( ) '( ) ( ) ( ) ( ) ( )

T T

t Tz t z T r s T T d r s d

)(')( TzTz

T

dsrTz0

)()()('


Lecture 3 27

Implementation of matched filter receiver

Mz

z

1

z)(tr

)(1 Tz)(

*

1 tTs

)(*

tTsM )(TzM

z

Bank of M matched filters

Matched filter output:

Observation

vector

)()( tTstrz ii Mi ,...,1

),...,,())(),...,(),(( 2121 MM zzzTzTzTz z

Lecture 3 28

Implementation of correlator receiver

dttstrz i

T

i )()(0

T

0

)(1 ts

T

0

)(ts M

Mz

z

1

z)(tr

)(1 Tz

)(TzM

z

Bank of M correlators

Correlators output:

Observation

vector

),...,,())(),...,(),(( 2121 MM zzzTzTzTz z

Mi ,...,1

Lecture 3 29

Implementation example of matched filter

receivers

2

1

z

z

z)(tr

)(1 Tz

)(2 Tz

z

Bank of 2 matched filters

T t

)(1 ts

T t

)(2 tsT

T 0

0

T

A

T

A

T

A

T

A

0

0

30

Detection

Matched filter reduces the received signal to a single variable z(T), after which the detection of symbol is carried out

The concept of maximum likelihood detector is based on Statistical Decision Theory

It allows us to

formulate the decision rule that operates on the data

optimize the detection criterion

1

2

0( )

H

H

z T

31

P[s1], P[s2] a priori probabilities

These probabilities are known before transmission

P[z]

probability of the received sample

p(z|s1), p(z|s2)

conditional pdf of received signal z, conditioned on the class si

P[s1|z], P[s2|z] a posteriori probabilities

After examining the sample, we make a refinement of our previous knowledge

P[s1|s2], P[s2|s1]

wrong decision (error)

P[s1|s1], P[s2|s2]

correct decision

Probabilities Review

32

Maximum Likelihood Ratio test and Maximum a posteriori (MAP) criterion:

If

else

Problem is that a posteriori probabilities are not known.

Solution:‏Use‏Bay’s‏theorem:

1 2 1( | ) ( | )p s z p s z H

1 1

2 2

1 1 2 21 1 2 2

)( | ) ( ) ( | ) (( | ) ( ) ( | ) ( )

( ) ( )

H H

H H

p z s P s p z s P sp z s P s p z s P s

P z P z

How to Choose the threshold?

2 1 2( | ) ( | )p s z p s z H

( | ) ( )( | )

( )

p z s P si ip s z

i p z

33

1

2

1 2

2 1

( )( | ) ( )

( )( | ) ( )

H

H

likelihood ratio test LRTp z s P s

L zp z s P s

When the two signals, s1(t) and s2(t), are equally likely, i.e., P(s2) = P(s1) = 0.5, then the decision rule becomes

This is known as maximum likelihood ratio test because we are selecting the hypothesis that corresponds to the signal with the maximum likelihood.

In terms of the Bayes criterion, it implies that the cost of both types of error is the same

1

2

1

2

max( | )

( ) 1( | )

H

H

likelihood ratio testp z s

L zp z s

MAP criterion

34

Substituting the pdfs

2

11 1

00

1 1: ( | ) exp

22

z aH p z s

2

22 2

00

1 1: ( | ) exp

22

z aH p z s

2

11 1

001

22

2

2 200

1 1exp

22( | )( ) 1 1

( | ) 1 1exp

22

z aH H

p z sL z

p z s z aH H

MAP criterion

35

Taking the log of both sides will give

1

2 2

1 2 1 2

2 2

0 0

2

( ) ( )ln{ ( )} 0

2

H

z a a a aL z

H

1

2 2

1 2 1 2 1 2 1 2

2 2 2

0 0 0

2

( ) ( ) ( )( )

2 2

H

z a a a a a a a a

H

1

2 2

1 2 1 2

2 2

0 0

2

( ) ( )exp 1

2

H

z a a a a

H

Hence:

MAP criterion

36

Hence

where z is the minimum error criterion and 0 is optimum threshold

For antipodal signal, s1(t) = - s2 (t) a1 = - a2

1

2

0 1 2 1 2

2

0 1 2

2

( )( )

2 ( )

H

a a a az

a a

H

1

1 20

2

( )

2

H

a az

H

1

2

0

H

z

H

MAP criterion

37

This means that if received signal was positive, s1 (t) was sent, else s2(t) was sent

MAP criterion

38


The output of the filtered sampled at T is a Gaussian random process

39

The impulse response of a filter producing maximum output signal-to-noise ratio is the mirror image of message signal s(t), delayed by symbol time duration T.

The filter designed is called a MATCHED FILTER and is given by:

( ) 0( )

0

ks T t t Th t

else where

Matched Filter and Correlation

40

Hence

where z is the minimum error criterion and 0 is optimum threshold

For antipodal signal, s1(t) = - s2 (t) a1 = - a2

1

1 20

2

( )

2

H

a az

H

1

2

0

H

z

H

Bay’s Decision Criterion and Maximum Likelihood

Detector

41

Probability of Error

Error will occur if

s1 is sent s2 is received

s2 is sent s1 is received

The total probability of error is the sum of the errors

0

2 1 1

1 1

( | ) ( | )

( | ) ( | )

P H s P e s

P e s p z s dz

0

1 2 2

2 2

( | ) ( | )

( | ) ( | )

P H s P e s

P e s p z s dz

2

1 1 2 2

1

2 1 1 1 2 2

( , ) ( | ) ( ) ( | ) ( )

( | ) ( ) ( | ) ( )

B i

i

P P e s P e s P s P e s P s

P H s P s P H s P s

42

If signals are equally probable

Numerically, PB is the area under the tail of either of the conditional distributions p(z|s1) or p(z|s2) and is given by:

2 1 1 1 2 2

2 1 1 2

( | ) ( ) ( | ) ( )

1( | ) ( | )

2

BP P H s P s P H s P s

P H s P H s

2 1 1 2 1 2

1( | ) ( | ) ( | )

2

by Symmetry

BP P H s P H s P H s

0 0

0

1 2 2

2

2

00

( | ) ( | )

1 1exp

22

BP P H s dz p z s dz

z adz


43

The above equation cannot be evaluated in closed form (Q-function)

Hence,

0

1 2

0

2

2

00

2

0

2

( )

2

1 1exp

22

( )

1exp

22

B

a a

z aP dz

z au

udu

1 2

0

.182

B

a aP Q equation B

21( ) exp

22

zQ z

z


44

Table for computing of Q-Functions


45

A vector View of Signals and Noise

N-dimensional orthonormal space characterized by N linearly independent basis function {ψj(t)}, where:

From a geometric point of view, each ψj(t) is mutually perpendicular to each of the other {ψj(t)} for j not equal to k.

ji

jidttt

T

ji if 0

if 1 )()(

0

46

Representation of any set of M energy signals { si(t) } as a linear combinations of N orthogonal basis functions where N M.

where:

N

j

jijiMi

Tttats

1 ,...,2,1

0)()(

Nj

Midtttsa j

T

iij,...,2,1

,...,2,1)()(

0


47

Therefore we can represent set of M energy signals {si(t) } as:

Representing (M=3) signals, with (N=2) orthonormal basis functions

Waveform energy:

N

j

ij

T N

j

jij

T

ii tattadttsE1

2

01

2

0

2 )( )]()([ )(

Miaaas iNiii ,...,2,1).......,,( 21


48

Question 1: Why use orthormal functions?

In many situations N is much smaller than M. Requiring few matched filters at the receiver.

Easy to calculate Euclidean distances

Compact representation for both baseband and passband systems.

Gram-Schmidt orthogonalization procedure.

Question 2: How to calculate orthormal

functions?


49

Examples


50

Examples (continued)


51

Generalized One Dimensional Signals

One Dimensional Signal Constellation

52

Binary Baseband Orthogonal Signals

Binary Antipodal Signals

Binary orthogonal Signals

Is a method of representing the symbol states of modulated bandpass signals in terms of their amplitude and phase

In other words, it is a geometric representation of signals

There are three types of binary signals:

Antipodal

Two signals are said to be antipodal if one signal is the negative of the other

The signal have equal energy with signal point on the real line

ON-OFF

Are one dimensional signals either ON or OFF with signaling points falling

on the real line 53

Constellation Diagram

)()( 01 tsts

54

With OOK, there are just 2 symbol states to map onto the constellation space

a(t) = 0 (no carrier amplitude, giving a point at the origin)

a(t) = A cos wct (giving a point on the positive horizontal axis at a distance A from the origin)

Orthogonal

Requires a two dimensional geometric representation since there are two linearly independent functions s1(t) and s0(t)

Constellation Diagram

55

Typically, the horizontal axis is taken as a reference for symbols that are Inphase with the carrier cos wct, and the vertical axis represents the Quadrature carrier component, sin wct

Error Probability of Binary Signals

Recall:

Where we have replaced a2 by a0.

18.2 0

01 Bequationaa

QPB

Error Probability of Binary Signal

56

To minimize PB, we need to maximize:

or

We have

Therefore,

02

2

01 )(

aa

0

01

aa

2

1 0

2

0 0 0

( ) 2

/ 2

d da a E E

N N

2

1 0 1 0

2

0 0 0 0

( ) 21 1

2 2 2 2

d da a a a E E

N N


57

TTT

T

d

tstsdttsdtts

dttstsE

001

2

00

2

01

2

001

)()(2)()(

)()(

02N

EQP d

B

The probability of bit error is given by:


58

The probability of bit error for antipodal signals:

The probability of bit error for orthogonal signals:

The probability of bit error for unipolar signals:

0

2

N

EQP b

B

02N

EQP b

B

0N

EQP b

B


59

Bipolar signals require a factor of 2 increase in energy compared to orthogonal signals

Since 10log102 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than orthogonal


60

Comparing BER Performance

For the same received signal to noise ratio, antipodal provides lower bit error rate than orthogonal

4

,

2

,

0

10x8.7

10x2.9

10/

antipodalB

orthogonalB

b

P

P

dBNEFor

61

Relation Between SNR (S/N) and Eb/N0

In analog communication the figure of merit used is the average signal power to average noise power ration or SNR.

In the previous few slides we have used the term Eb/N0 in the bit error calculations. How are the two related?

Eb can be written as STb and N0 is N/W. So we have:

Thus Eb/N0 can be thought of as normalized SNR.

Makes more sense when we have multi-level signaling.

0 /

b b

b

E ST S W

N N W N R

Modulation, Demodulation and Coding Coursesite.iugaza.edu.ps/ahdrouss/files/2015/09/slides31.pdf ·...

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