104612_Pulse Amplitude Modulation (Synchronisation_Intersymbol Interference_Eye Diagrams)

8/6/2019 104612_Pulse Amplitude Modulation (Synchronisation_Intersymbol Interference_Eye Diagrams)

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Introduction

The purpose of the modulator is to convertdiscrete amplitude serial symbols (bits in a

binary system) akto analogue output pulses

which are sent over the channel. The demodulator reverses this process

Modulator Channel Demodulator

Serial data

symbols

ak

analogue

channel pulses

Recovered

data symbols


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Introduction

Possible approaches include

Pulse width modulation (PWM)

Pulse position modulation (PPM)

Pulse amplitude modulation (PAM)

We will only be considering PAM in these

lectures


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PAM

PAM is a general signalling techniquewhereby pulse amplitude is used to convey

the message

For example, the PAM pulses could be thesampled amplitude values of an analogue

signal

We are interested in digital PAM, where thepulse amplitudes are constrained to chosen

from a specific alphabet at the transmitter


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PAM Scheme

HC()

hC(t)

Symbol

clock

HT() hT(t)

Noise N()

Channel

+

Pulse

generator

ak Transmit

filter

=

=k

ks kTtatx )()(

=

=k

Tk kTthatx )()(

Receive

filter

HR(), hR(t)

Data

slicer

Recoveredsymbols

Recovered

clock

)()()( tvkTthatyk

k +=

=

Modulator

Demodulator


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PAM

In binary PAM, each symbol aktakes only

two values, say {A1andA

2}

In a multilevel, i.e., M-ary system, symbols

may take Mvalues {A1,A

2,... A

M}

Signalling period, T

Each transmitted pulse is given by)( kTtha Tk

Where hT(t) is the time domain pulse shape


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PAM

Filtering of impulse train in transmit filter

Transmit

Filter

==

k

Tk kTthatx )()(

==

k

ks kTtatx )()(

)(thT

)(txs )(tx


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PAM Clearly not a practical technique so

Use a practical input pulse shape, then filter to

realise the desired output pulse shape

Store a sampled pulse shape in a ROM and read outthrough a D/A converter

The transmitted signalx(t) passes through the

channelHC() and the receive filterHR(). The overall frequency response is

H() = HT() H

C() H

R()


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PAM Hence the signal at the receiver filter output is

)()()( tvkTthaty

k

k +=

=Where h(t) is the inverse Fourier transform ofH()

and v(t) is the noise signal at the receive filter

output

Data detection is now performed by the DataSlicer


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PAM- Data Detection Samplingy(t), usually at the optimum instant

t=nT+tdwhen the pulse magnitude is the greatest

yields

n

k

dkdn vtTknhatnTyy ++=+=

=

))(()(

Where vn=v(nT+td) is the sampled noise and td is the

time delay required for optimum sampling

yn is then compared with threshold(s) to determine

the recovered data symbols


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PAM- Data Detection

Data Slicer decision

threshold = 0V

0

Signal at data

slicer input,y(t)

Sample clock

Sampled signal,

yn=y(nT+td)

Ideal sample instants

at t= nT+td

0

TX data

TX symbol, ak

1 0 0 1 0

+A -A -A +A -A

Detected data 1 0 0 1 0

td


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Synchronisation We need to derive an accurate clock signal at

the receiver in order thaty(t) may be sampled at

the correct instant

Such a signal may be available directly (usually

not because of the waste involved in sending a

signal with no information content)

Usually, the sample clock has to be derived

directly from the received signal.


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Synchronisation The ability to extract a symbol timing clock

usually depends upon the presence of transitions

or zero crossings in the received signal.

Line coding aims to raise the number of such

occurrences to help the extraction process.

Unfortunately, simple line coding schemes often

do not give rise to transitions when long runs of

constant symbols are received.


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Synchronisation

Some line coding schemes give rise to a

spectral component at the symbol rate

A BPF or PLL can be used to extract this

component directly

Sometimes the received data has to be non-

linearly processed eg, squaring, to yield acomponent of the correct frequency.


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Intersymbol Interference

If the system impulse response h(t) extends overmore than 1 symbol period, symbols become

smeared into adjacent symbol periods

Known as intersymbol interference (ISI) The signal at the slicer input may be rewritten as

n

nk

dkdnn vtTknhathay +++=

))(()(

The first term depends only on the current symbol an

The summation is an interference term which

depends upon the surrounding symbols


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ExampleResponse h(t) is Resistor-Capacitor (R-C) first

order arrangement- Bit duration is T

For this example we will assume that a

binary 0 is sent as 0V.

Time (bit periods)0 2 4 6

amp

litud

e

0.5

1.0

Time (bit periods)0 2 4 6

amp

litu

de

0.5

1.0

Modulator input Slicer input

Binary 1 Binary 1


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The received pulse at the slicer now extends

over 4 bit periods giving rise to ISI.

The actual received signal is the

superposition of the individual pulses

time (bit periods)

0 2 4 6

amp

litu

de

0.5

1.0

1 1 0 0 1 0 0 1


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For the assumed data the signal at the slicerinput is,

Clearly the ease in making decisions is data

dependant

time (bit periods)0 2 4 6

amp

litu

de

0.5

1.0

Note non-zero values at ideal sample instants

corresponding with the transmission of binary 0s

1 1 0 0 1 0 0 1

Decision threshold


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Matlab generated plot showing pulse superposition(accurately)

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Decision

threshold

time (bit periods)

Received

signal

Individual

pulses

amp

litu

de


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Intersymbol Interference Sending a longer data sequence yields the

following received waveform at the slicer input

Decision

threshold

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Decision

threshold

(Also showing

individual pulses)


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Eye Diagrams Worst case error performance in noise can be

obtained by calculating the worst case ISI over allpossible combinations of input symbols.

A convenient way of measuring ISI is the eyediagram

Practically, this is done by displayingy(t) on ascope, which is triggered using the symbol clock

The overlaid pulses from all the different symbolperiods will lead to a criss-crossed display, withan eye in the middle


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Example R-C responseEye Diagram

Decision

threshold

Optimum sample instant

h = eye height

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

h


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Eye Diagrams The size of the eye opening, h (eye height)

determines the probability of making incorrect

decisions

The instant at which the max eye opening occurs

gives the sampling time td

The width of the eye indicates the resilience to

symbol timing errors

For M-ary transmission, there will be M-1 eyes


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Eye Diagrams

The generation of a representative eye

assumes the use of random data symbols

For simple channel pulse shapes withbinary symbols, the eye diagram may be

constructed manually by finding the worst

case 1 and worst case 0 andsuperimposing the two


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Nyquist Pulse Shaping It is possible to eliminate ISI at the sampling

instants by ensuring that the received pulses

satisfy the Nyquist pulse shaping criterion

We will assume that td=0, so the slicer input is

n

nk

knn vTknhahay ++=

))(()0(

If the received pulse is such that

=

=0for0

0for1)(

n

nnTh


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Nyquist Pulse Shaping Then

nnn vay +=

and so ISI is avoided

This condition is only achieved if

TT

kfH

k

=

+

=

That is the pulse spectrum, repeated at

intervals of the symbol rate sums to a

constant value Tfor all frequencies


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Why? Sample h(t) with a train of pulses at times kT

=

=k

s kTtthth )()()(

Consequently the spectrum ofhs(t) is

=k

s TkHT

H )2(1

)(

Remember for zero ISI

=

=0for0

0for1)(

n

nnTh


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Why?

Consequently hs(t)= (t)

The spectrum of (t)=1, therefore

1)2(1

)( == k

s TkHT

H

Substitutingf=/2 gives the Nyquist

pulse shaping criterion =k

TTkfH )(


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Nyquist Pulse Shaping

1/

1/ 0

T

2/ 2/ f

No pulse bandwidth less than 1/2Tcan

satisfy the criterion, eg,

Clearly, the repeated spectra do not sum to a constant value


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The minimum bandwidth pulse spectrum

H(f), ie, a rectangular spectral shape, has a

sinc pulse response in the time domain,


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Hard to design practical brick-wall filters,

consequently filters with smooth spectral

roll-off are preferred Pulses may take values fort


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Raised Cosine (RC) Fall-Off

Pulse Shaping Practically important pulse shapes which

satisfy the criterion are those with Raised

Cosine (RC) roll-off The pulse spectrum is given by

2121

210

)21(4

cos

21

)( 2

+

+

= TfT

Tf

TfT

TfT

fH

With, 0


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RC Pulse Shaping

The general RC function is as follows,H(f)

f(Hz)

T

0

T2

1+

T2

1

T2

1

T

1

2121

210

)21(4

cos

21

)(2

+

+

= TfT

Tf

TfT

TfT

fH


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RC Pulse Shaping The corresponding time domain pulse shape

is given by,

( )

=2

41

2cossin

)(t

t

tT

tT

th

Now allows a trade-off between bandwidth and the pulse decayrate

Sometimes is normalised as follows,

( )T

x

21

=


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RC Pulse Shaping With =0 (i.e.,x = 0) the spectrum of the filter is

rectangular and the time domain response is a sinc

pulse, that is,

TfTfH 21)( =

=

tT

tT

th

sin

)(

The time domain pulse has zero crossings at

intervals ofnTas desired (See plots forx = 0).


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RC P l Sh i


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RC Pulse ShapingNormalised Spectrum H(f)/T Pulse Shape h(t)

x =

0

x = 0.

5

x = 1

f*T t/T

l h i l


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RC Pulse Shaping- Example 1

Eye diagram

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6 7 8-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Pulse shape and received signal,x = 0 ( = 0)

l h i l


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RC Pulse Shaping- Example 2

Eye diagram

Pulse shape and received signal,x = 1 ( = 1/2T)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8-0.2

0

0.2

0.4

0.6

0.8

1

1.2


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RC Pulse Shaping- Example

The much wider eye opening forx = 1 gives

a much greater tolerance to inaccurate

sample clock timing The penalty is the much wider transmitted

bandwidth


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Probability of Error In the presence of noise, there will be a finite chance of

decision errors at the slicer output

The smaller the eye, the higher the chance that the noise will

cause an error. For a binary system a transmitted 1 could

be detected as a 0 and vice-versa

In a PAM system, the probability of error is,

Pe=Pr{A received symbol is incorrectly detected}

For a binary system,Pe is known as the bit error probability,or the bit error rate (BER)


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BER The received signal at the slicer is

nin vVy +=

Where Viis the received signal voltage and

Vi=Vo for a transmitted 0 or

Vi=V1 for a transmitted 1

With zero ISI and an overall unity gain, Vi=a

n,

the current transmitted binary symbol

Suppose the noise is Gaussian, with zero mean

and variance 2v


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BER

2

2

2

22

1)( v

nv

v

n evf

=

Wheref(vn) denotes the probability densityfunction (pdf), that is,

dxxfdxxvx n )(}Pr{ =+

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104612_Pulse Amplitude Modulation (Synchronisation_Intersymbol Interference_Eye Diagrams)

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