dc2014 03 baseband transmission

8/12/2019 DC2014 03 Baseband Transmission

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Baseband TransmissionHa Hoang Kha, Ph.D

Ho Chi Minh City University of Technology

Email: [email protected]

Chapter


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Baseband Transmission 2 H. H. Kha, Ph.D.


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Content

1) Discrete PAM signals

2) Power Spectra of Discrete PAM Signals

3) InterSymbol Interference

4) Nyquists Criterion For Distortionless Baseband Binary

Transmission

5) Correlative Coding



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1. Discrete PAM Signals

The use of an appropriate for baseband

representation of digital is basic to itstransmission from a source to a destination

There are some different formats for therepresentation of the binary data sequence

Unipolar format (on-off signaling) Polar format

Bipolar format (also known as pseudoternarysignaling)

Manchester format (also known as biphase baseband

signaling)



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Discrete PAM Signal


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2. Power Spectra of Discrete PAM Signals

Data signaling rate (or data rate) is defined as

the rate, measured in bits per second (bps), at

which data are transmitted.

It is also common practice to refer to the data

signaling rate as the bit rate, denoted by

where Tb

is the bit duration

b

bT

R 1



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Power Spectra of Discrete PAM Signals

In contrast, the modulation rate is defined as the

rate at which signal level is changed, dependingon the nature of the format used to represent thedigital data

The modulation rate is measured in bauds or

symbol per secondFor an M-ary format (withMan integer power of

two) used to represent binary data, the symbolduration of the M-ary format is related to the bitduration Tb by

MTT b 2log



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Discrete amplitude-modulated pulse train may

be described as different realizations (sample

functions) of a random processX(t)

The coefficientAk is a discrete random variable

v(t) is basic pulse shape, centered at the origin, t = 0,

and normalized such that v(0) = 1

Tis the symbol duration

k

k kTtvAtX )(



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To proceed with the analysis, we model the

mechanism responsible for the generation of thesequence {Ak}, defining as a discrete stationaryrandom source

The source is characterized as having

ensemble-averaged autocorrelation function

whereEis the expectation operator

nkkA AAEnR )(



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The power spectral density of the discrete PAM

signalX(t) is given by

V(f) is the Fourier transform of the basic pulse v(t)

The values of the functions V(f) andRA(n) depend on

the type of discrete PAM signal being considered

)2exp()()(1

)( 2

nfTjnRfVT

fS AX



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NRZ Unipolar Format

Suppose that the 0s and 1s of a random binary

sequence occur with equal probability

For n = 0, we may write

21)()0( aAPAP kk

2)()()0()0(][

2222 aaAPaAPAE kkk



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NRZ Unipolar Format

Consider next the productAkAk-n for n 0

The autocorrelation functionRA(n) may be expressed as

follows

44

14

1032

2 aaAAE nkk 0n

4

2)(2

2

a

a

nRA

0

0

n

n



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NRZ Unipolar Format

For the basic pulse v(t), we have a rectangular pulse

of unit amplitude and duration Tb. The Fourier

transform of v(t) equals

The power spectral density of NRZ unipolar format

)(sin)( bb fTcTfV

nbb

bb

bX nfTjfTc

TafTc

TafS )2exp()(sin4)(sin4)(

22

22



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NRZ Unipolar Format

Use Poisons formula written in the form

We may simplify the expression for the power spectral

density SX(f) as

m bbn

bT

mf

TnfTj

1)2exp(

)(4)(sin4)(

2

2

2

fa

fTcTa

fS bbX



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NRZ Polar Format

Similar to that described for the unipolar format, we findthat

The basic pulse v(t) for the polar format is the same asthat for unipolar format

The power spectral density of the NRZ polar format is

0

)(2a

nRA0

0

n

n

)(sin)( 22 bbX fTcTafS



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NRZ Bipolar Format

The successive 1s in the bipolar format be assigned

pulses of alternating polarity

The bipolar format has three level: a, 0, -a

Assume that the 1s and 0s in the input binary data

occur with equal probability, we find the respective

probabilities of occurrence of these level are

4

1

210

41

aAP

AP

aAP

k

k

k



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NRZ Bipolar Format

For n = 0, we may write

For n = 1, the dibit represented by the sequence

(AkAk-1) can assume only four possible forms:

(0,0), (0,1), (1,0), (1,1). Hence we may write

2

002

2222 aaAPaAPaAPaAE kkkk

441

4

1032

2

1

aaAAE

kk



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NRZ Bipolar Format

For n > 1, we find that

For the NRZ Bipolar format, we have

0nkkAAE

0

4

2

)( 2

2

a

a

nRA

otherwise

1

0

n

n



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NRZ Bipolar Format

The basic pulse v(t) for the NRZ bipolar format

has its Fourier transform as in previous cases

The power spectral density of the NRZ bipolar

format is given

)2exp(2exp(

42)(sin)(

222

bbbbX fTjfTjaa

fTcTfS

)(sin)(sin

)2cos(1)(sin2222

22

bbb

bb

b

fTfTcTa

fTfTc

Ta



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Manchester Format

In Manchester format, the input binary data consists ofindependent, equally likely symbol

The autocorrelation functionRA(n) for the Manchester format is

the same as for the NRZ polar format

0

)(2a

nRA0

0

n

n



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Manchester Format

The basic pulse v(t) for the Manchester formatconsists of a doublet pulse of unit amplitude and total

duration Tb.The Fourier transform of the pulse equals

The power spectral density of the Manchester format

is given

2sin2sin)( bb

b

fTfT

cjTfV

2sin

2sin)( 222 bbbX

fTfTcTafS



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3. InterSymbol Interference

Consider basic elements of a baseband binary PAM

system The input signal consists of a binary data sequence {bk} with a

bit duration of Tb seconds

This sequence is applied to a pulse generator, producing the

discrete PAM signal

v(t) denotes the basic pulse, normalize such that v(0) = 1

The coefficient akdepends on the input data and the type of

format used The waveformx(t) represents one realization of the random

processX(t)

k

bk kTtvatx )(



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InterSymbol Interference



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The receiving filter output may be written as

is scaling factor

The pulsep(t) is normalized such that

k

bk kTtpaty )(

1)0( p



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The outputy(t) is produced in response to binary data

waveform applied to the input of the transmitting filter.Especially, the pulse is response of the cascade

connection of the transmitting filter, the channel, and the

receiving filter, which is produced by the pulse v(t) applied

to the input of this cascade connection

P(f) and V(f) are Fourier transform ofp(t) and v(t)

)(tp

)()()()()( fHfHfHfVfP RCT



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The receiving filter outputy(t) is sampled at time ti = iTb

The first term is produced by the ith transmitted bit.

The second term represents the residual effect of all

other transmitted bits on the decoding of the ith bit; thisresidual effect is called intersymbol interference (ISI)

k

bbki kTiTpaty )(

ik

k

bbki kTiTpaa


4 Nyquists Criterion For Distortionless


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4. Nyquist s Criterion For Distortionless

Baseband Binary Transmission

Typically, the transfer function of the channel

and the transmitted pulse shape are specified,and the problem is to determine the transferfunctions of the transmitting and receiving filtersso as to reconstruct the transmitted datasequence {bk}

The receiver does this by extracting and thendecoding the corresponding sequence ofweights, {ak}, from the outputy(t).

Except for a scaling factor,y(t) is determined by

the akand the received pulsep(t)


Nyquists Criterion For Distortionless


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Nyquist s Criterion For Distortionless


The extraction involves sampling the outputy(t)

at some time t = iTb

The decoding requires that the weighted pulse

contribution akp(iTb-kTb) for k = i be free form ISI

due to the overlapping tails of all other weightedpulse contributions represented by k i


giai nen



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Nyquist s Criterion For Distortionless


This, in turn, require that we control the received

pulsep(t), as shown by

where, by normalization,p(0) = 1

0

1bb kTiTp ki

ki




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Nyquist s Criterion For DistortionlessBaseband Binary Transmission

The receiver output

Which implies zero intersymbol interference (ISI)

This condition assures perfect reception in theabsence of noise

ii aty




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Consider the sequence of samples {p(nTb)},

where n = 0, 1, 2,

Sampling in the time domain produces

periodicity in frequency domain

WhereRb = 1/Tb is the bit rate

P(f) is the Fourier transform of an infinite periodic sequence of

delta functions of period Tb, and whose strengths are weighted

by the respective sample values ofp(t)

n

bb nRfPRfP )(




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That is

where m = ik.

Impose the condition of zero ISI on the samplevalues ofp(t)

dtftjmTtmTpfP bb 2exp)()()(

dtftjtpfP 2exp)()0()(

)0(p




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Sincep(0) = 1, by normalization, the condition for

zero ISI is sastisfied if

Nyquist criterion for distortionless baseband

transmission

bn

b TnRfP




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Ideal solution

A frequency functionP(f), occupying the narrowest band, isobtained by permitting only one nonzero component in the seriesfor eachfin the range extending fromB0 toB0, whereB0denotes half the bit rate

We specifyP(f)

Hence, signal waveform that produces zero ISI is defined by thesinc function

2

0bRB

00 22

1)(

B

frect

BfP


tB

tBtp

0

0

2

2sin)(

tBc 02sin


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Ideal solution



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Ideal solution




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There are two practical difficulties that make it

an undesirable objective for system design: It requires that the amplitude characteristic ofP(f) be

flat formB0 toB0 and zero elsewhere. This isphysically unrealizable because of the abrupttransitions at B

0 The function p(t) decreases as 1/|t| for large |t|,

resulting in a slow rate of decay. This is caused bythe discontinuity ofP(f) at B0. Accordingly, there ispractically no margin of error in sampling times in the

receiver




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Practical solution

We may overcome the practical difficulties posed by the idealsolution by extending the bandwidth fromB0 = Rb/2 to an

adjustable value betweenB0 and 2B0

In doing so, we permit three components as shown by

0

002

122)(

BBfpBfpfP

00 BfB




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Practical solution

A particular form ofP(f) that embodies many desirablefeatures is constructed by a raised cosine spectrum

Rolloff factor

0

22cos1

4

1

2

1

)(10

1

0

0

fB

ff

B

B

fP

10

101

1

2

2

fBf

fBff

ff

0

11B

f



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Practical solution



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Practical solution

The time responsep(t), that is, the inverse Fourier

transform ofP(f), is defined

A more general relationship between required

bandwidth and symbol transmission rate involves the

roll-off factor

22

0

2

00

161

2cos)2(sin)(

tB

tBtBctp


)1(2 010 BfBB

5 C l ti C di


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5. Correlative Coding

It is possible to achieve a bit rate of 2B0 per second in a

channel of bandwidthB0 Hertz by adding intersymbolinterference to the transmitted signal in a controlled manner

Such schemes are called correlative coding or partial-

response signaling schemes

The design of these schemes is based on the premise thatsince intersymbol interference introduced into the

transmitted signal is known, its effect can be compensated

at the receiver.

Correlative coding may be regarded as a practical means ofachieving the theoretical maximum signaling rate of 2Bo per

second in a bandwidth ofB0 hertz


C l ti C di


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Correlative Coding

Duobinary signaling

Consider a binary input sequence {bk} consisting of

uncorrelated binary digits each having duration Tbseconds, with symbol 1 represented by a pulse of

amplitude +1 volt, and symbol 0 by a pulse of

amplitude -1 volt This sequence is applied to duobinary encoder, it is

converted into a three-level output, namely -2, 0, and

+2 volts


Correlative Coding


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Correlative Coding

Duobinary signaling


Correlative Coding


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Correlative Coding

Duobinary signaling The digit ck at the duobinary coder output is the

sum of the resent binary digit bkand its previousvalue bk-1

One of the effects of the transformation is tochange the input sequence {bk} of uncorrelatedbinary digits into a sequence {ck} of correlateddigits

This correlation between the adjacent transmittedlevels may be viewed as introducing ISI into thetransmitted signal

1 kkk bbc


Correlative Coding


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Correlative Coding

Duobinary signaling

The overall transfer function of this filter connected in cascadewith the ideal channelHc(f) is

( ) ( ) 1 exp 2C bH f H f j fT

bbC

bbbC

fTjfTfH

fTjfTjfTjfH

expcos)(2

expexpexp1)(


Correlative Coding


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Correlative Coding

Duobinary signaling

For the ideal channel of bandwidthB0 = R b/2, we have

The overall frequency response has the form of a

half-cycle cosine function

0

1)(fHC

otherwise

2bRf

0

expcos2

)( bb fTjfT

fH

otherwise

2bRf


Correlative Coding


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Correlative Coding

Duobinary signaling

The corresponding value of the impulse response consists of twosinc pulse, time-displaced by Tb seconds

bb

bb

b

b

TTt

TTt

Tt

Ttth

sinsin)(

tTt

TtT

TTt

Tt

Tt

Tt

b

bb

bb

b

b

b

sin

sinsin

2



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Duobinary signaling

C l ti C di


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Correlative Coding

Duobinary signaling


Correlative Coding


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Correlative Coding

Duobinary signaling

The original data {bk} may be detected from theduobinary-coded sequence {ck} by subtracting theprevious decoded binary digit from the currentlyreceived digit ck

It is apparent that if ck is received without error and ifalso the previous estimate at time t = (k-1)Tbcorresponds to a correct decision, then the current

estimate will be correct too

1

kkk bcb

kb

1

kb


Correlative Coding


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Correlative Coding

Duobinary signaling Practical solution

Use precoder before the duobinary coding to avoid errorpropagation

The precoder operation performed on the input binary sequence

{bk} converts it into another sequence {ak} defined by

1 kkk aba


Correlative Coding


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Correlative Coding

Duobinary signaling Practical solution

The resulting precoder output {ak} is applied to the duobinarycoder

The sequence {ck} is related to {ak} as follows

1 kkk aac


Correlative Coding


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Correlative Coding


Correlative Coding


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Correlative Coding

Illustrating doubinary coding

Decision rule

volt1if1

volt1if0

k

k

kcsymbol

csymbolb


6 Eye Pattern


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6. Eye Pattern

One way to study ISI in a PCM or data

transmission system experimentally is to applythe received wave to the vertical deflectionplates of an oscilloscope an to apply a sawtoothwave at the transmitted symbol rate R = 1/T tothe horizontal deflection plates

The waveforms in successive symbol intervalsare thereby translated into one interval on theoscilloscope display

The resulting display is called an eye pattern


Eye Pattern


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Eye Pattern


Eye Pattern


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Eye Pattern

The width of the eye opening defines the time

interval over which the received wave can besampled without error form ISI. It is apparentthat the preferred time for sampling is the instantof time at which the eye is opened widest

The sensitivity of the system to timing error isdetermined by the rate of closure of the eye asthe sampling time is varied

The height of the eye opening, at a specifiedsampling time, defines the margin over noise


Eye Pattern


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Eye Pattern


Homework


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Homework

Problems: 4.1, 4.2, 4.3

Problems: 4.7, 4.8, 4.9 Problems: 4.16, 4.18, 4.19

Problems: 4.21, 4.25, 4.26

Textbook:

Simon Haykin, Commun icat ion System, 4th Edition,

John Wiley & Son, Inc. , 2001.

dc2014 03 baseband transmission

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