baseband transmission.pdf

7/27/2019 Baseband transmission.pdf

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BASEBAND TRANSMISSION


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Discrete PAM Signals

The use of an appropriate for basebandrepresentation of digital is basic to itstransmission from a source to a destination

There are some different formats for the

representation of the binary data sequence Unipolar format (on-off signaling) Polar format Bipolar format (also known as pseudoternary

signaling) Manchester format (also known as biphase

baseband signaling)


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Discrete PAM Signal


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Power Spectra of DiscretePAM Signals

Data signaling rate (or data rate) is definedas the rate, measured in bits per second(bps), at which data are transmitted.

It is also common practice to refer to thedata signaling rate as the bit rate, denoted by

where Tb is the bit duration

bb T

R1

=


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In contrast, the modulation rate is defined asthe rate at which signal level is changed,depending on the nature of the format usedto represent the digital data

The modulation rate is measured in bauds orsymbol per second

For an M-ary format (withMan integerpower of two) used to represent binary data,the symbol duration of the M-ary format isrelated to the bit duration Tb by

MTT b 2log=


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Discrete amplitude-modulated pulse train maybe described as different realizations (samplefunctions) of a random process X(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 themechanism responsible for the generation ofthe sequence {Ak}, defining as a discretestationary random source

The source is characterized as havingensemble-averaged autocorrelation function

whereEis the expectation operator[ ]nkkA AAEnR =)(


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The power spectral density of the discretePAM 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 beingconsidered

= )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 write2

1)()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 beexpressed as follows

[ ] ( )( ) ( )( ) 44141032

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. TheFourier transform ofv(t) equals

The power spectral density of NRZ unipolar format

)(sin)( bb fTcTfV =

=

+=n

bbb

bb

X nfTjfTcTa

fTcTa

fS )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 thepower spectral density S

X(f) as

=

=

= m bbnb

T

mfTnfTj

1)2exp(

)(4

)(sin4

)(2

22

fa

fTcTa

fS bb

X +=


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NRZ Polar Format Similar to that described for the unipolar

format, we find that

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

the same as that for unipolar format The power spectral density of the NRZ

polar format is

=0

)(2

anRA 00

=

nn

)(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 therespective probabilities of occurrence of these

level are( )

( )

( ) 41

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 thesequence (AkAk-1) can assume only four

possible forms: (0,0), (0,1), (1,0), (1,1). Hencewe may write

[ ] ( ) ( ) ( ) ( ) ( ) ( ) 200

22222 a

aAPaAPaAPaAEkkkk

==+=+==

[ ] ( )( ) ( )( )44

1

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

[ ] 0=nkk

AAE

=

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 bipolarformat is given

( )

+= )2exp(2exp(

42

)(sin)(22

2

bbbbX fTjfTjaa

fTcTfS

[ ]

)(sin)(sin

)2cos(1)(sin2

222

22

bbb

bbb

fTfTcTa

fTfTcTa

=

=


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Manchester Format In Manchester format, the input binary

data consists of independent, 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 format

consists of a doublet pulse of unit amplitude and

total duration Tb.The Fourier transform of thepulse equals

The power spectral density of the Manchesterformat is given

=

2sin

2sin)( bbb

fTfTcjTfV

=

2sin2sin)(

222 bb

bX

fTfT

cTafS


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InterSymbol Interference

Consider basic elements of a baseband binary PAMsystem

The input signal consists of a binary data sequence {bk} witha bit duration ofTb seconds

This sequence is applied to a pulse generator, producing thediscrete PAM signal

v(t) denotes the basic pulse, normalize such that v(0) = 1 The coefficient ak depends on the input data and the type of

format used

The waveformx(t) represents one realization of the randomprocessX(t)

( )

=

=k

bk kTtvatx )(


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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 binarydata waveform applied to the input of thetransmitting filter. Especially, the pulse isresponse of the cascade connection of the

transmitting filter, the channel, and the receivingfilter, which is produced by the pulse v(t) applied tothe 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 othertransmitted bits on the decoding of the ith bit; this residualeffect is called intersymbol interference (ISI)

( )

=

=k

bbki kTiTpaty )(

( )

=

+=

ikk

bbki kTiTpaa


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Nyquists Criterion For DistortionlessBaseband Binary Transmission

Typically, the transfer function of the channeland the transmitted pulse shape arespecified, and the problem is to determinethe transfer functions of the transmitting and

receiving filters so as to reconstruct thetransmitted data sequence {bk} The receiver does this by extracting and then

decoding the corresponding sequence of

weights, {ak}, from the outputy(t). Except for a scaling factor,y(t) is determined

by the ak and the received pulsep(t)


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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 freeform ISI due to the overlapping tails of allother weighted pulse contributions

represented by k

i


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This, in turn, require that we control thereceived pulsep(t), as shown by

where, by normalization,p(0) = 1

( )

=0

1bb kTiTp

ki

ki

=


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The receiver output

Which implies zero intersymbol interference (ISI) This condition assures perfect reception in the

absence 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 areweighted by the respective sample values ofp(t)

( )

=

=n

bb nRfPRfP )(


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That is

where m = i k.

Impose the condition of zero ISI on thesample values ofp(t)

[ ] ( )dtftjmTtmTpfP bb 2exp)()()( =

( )dtftjtpfP 2exp)()0()( =

)0(p=


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Sincep(0) = 1, by normalization, the conditionfor zero ISI is sastisfied if

Nyquist criterion for distortionless

baseband transmission

( ) bn

b TnRfP =

=


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Ideal solutionA frequency functionP(f), occupying the

narrowest band, is obtained by permittingonly one nonzero component in the seriesfor eachfin the range extending fromB0toB0, whereB0 denotes half the bit rate

We specifyP(f)

20

bRB =

=

00 22

1)(

B

frect

BfP


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Ideal solution Hence, signal waveform that produces zero

ISI is defined by the sinc function

( )tB

tBtp

0

0

2

2sin)(

=

( )tBc 02sin=


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Ideal solution


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There are two practical difficulties that makeit an undesirable objective for system design: It requires that the amplitude characteristic ofP(f)

be flat formB0 toB0 and zero elsewhere. This is

physically unrealizable because of the abrupttransitions at B0

The function p(t) decreases as 1/|t| for large |t|,resulting in a slow rate of decay. This is caused by

the discontinuity ofP(f) at B0. Accordingly, thereis practically no margin of error in sampling timesin the receiver


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Practical solution We may overcome the practical difficulties

posed by the ideal solution by extending

the bandwidth fromB0 = Rb/2 to anadjustable 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 desirable

features is constructed by a raised cosine spectrum

Rolloff factor

( )

+=

0

22cos1

4

1

21

)(10

1

0

0

fB

ff

B

B

fP

10

101

1

2

2

fBf

fBff

ff


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

transform ofP(f), is defined

( )220

20

01612cos)2(sin)(

tBtBtBctp

=


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Practical solution


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Correlative Coding

It is possible to achieve a bit rate of2B0

per secondin a channel of bandwidthB0 Hertz by addingintersymbol interference to the transmitted signal in acontrolled manner

Such schemes are called correlative coding or partial-

response signaling schemes The design of these schemes is based on the premise

that since intersymbol interference introduced intothe transmitted signal is known, its effect can be

compensated at the receiver. Correlative coding may be regarded as a practical

means of achieving the theoretical maximumsignaling rate of2Bo per second in a bandwidth ofB0hertz


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Correlative Coding

Duobinary signaling Consider a binary input sequence {bk} consisting

of uncorrelated binary digits each having durationTb seconds, with symbol 1 represented by a pulse

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

This sequence is applied to duobinary encoder, it

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


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Correlative Coding

Duobinary signaling


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Correlative Coding

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

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

One of the effects of the transformation is tochange the input sequence {bk} of uncorrelated

binary digits into a sequence {ck} pf correlateddigits This correlation between the adjacent transmitted

levels may be viewed as introducing ISI into thetransmitted signal

1+= kkk bbc


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Correlative Coding

Duobinary signaling The overall transfer function of this filter

connected in cascade with the ideal

channelHc(f) is( )[ ]bC fTjfHfH += exp1)()(

( ) ( )[ ] ( )

( ) ( )bbC

bbbC

fTjfTfH

fTjfTjfTjfH

=

++=

expcos)(2

expexpexp1)(


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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 ahalf-cycle cosine function

= 0

1

)( fHC otherwise

2bRf

( ) ( )

=0

expcos2)(

bb fTjfTfH

otherwise

2bRf


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Correlative Coding

Duobinary signaling The corresponding value of the impulse

response consists of two sinc pulse, time-

displaced by Tb seconds( ) ( )[ ]

( ) bbbb

b

b

TTt

TTt

Tt

Ttth

+=

sinsin)(

( ) [ ]( )

( )

( )tTt

TtT

TTtTt

TtTt

b

bb

bb

b

b

b

=

=

sin

sinsin

2


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Correlative Coding

Duobinary signaling


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Correlative Coding

Duobinary signaling The original data {bk} may be detected from the

duobinary-coded sequence {ck} by subtracting theprevious decoded binary digit from the currently

received digit ck

It is apparent that ifck is received without error

and if also the previous estimate at time t = (k-1)Tb corresponds to a correct decision, then thecurrent estimate will be correct too

1

= kkk bcb

kb

1

kb


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Eye Pattern

One way to study ISI in a PCM or datatransmission system experimentally is toapply the received wave to the verticaldeflection plates of an oscilloscope an toapply a sawtooth wave at the transmittedsymbol rate R = 1/T to the horizontaldeflection plates

The waveforms in successive symbol intervals

are thereby translated into one interval onthe oscilloscope display The resulting display is called an eye pattern


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Eye Pattern


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Eye Pattern

The width of the eye opening defines thetime interval over which the received wavecan be sampled without error form ISI. It isapparent that the preferred time for samplingis the instant of time at which the eye isopened widest

The sensitivity of the system to timing error isdetermined by the rate of closure of the eye

as the sampling time is varied The height of the eye opening, at a specified

sampling time, defines the margin over noise


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Eye Pattern

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