kha dc02 baseband transmission

Chapter 2

Baseband Transmission

Chapter 2

aseba d a s ss oHa Hoang Kha, Ph.DHo Chi Minh City University of TechnologyEmail: hhkha@hcmut edu vnEmail: [email protected]

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

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


1. Discrete PAM Signals

The use of an appropriate for baseband representation of digital is basic to its transmission from a source to a destinationtransmission from a source to a destinationThere are some different formats for the representation of the binary data sequence

Unipolar format (on-off signaling)Polar formatBipolar format (also known as pseudoternaryBipolar format (also known as pseudoternarysignaling)Manchester format (also known as biphase baseband signaling)s g a g)


Discrete PAM Signal

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 p ( p )which data are transmitted.It is also common practice to refer to the data signaling rate as the bit rate, denoted by

bR1

=

where Tb is the bit durationb

b T


Power Spectra of Discrete PAM Signals

In contrast, the modulation rate is defined as the rate at which signal level is changed, depending

th t f th f t d t t thon the nature of the format used to represent the digital dataThe modulation rate is measured in bauds orThe modulation rate is measured in bauds or symbol per secondFor an M-ary format (with M an integer power of y ( g ptwo) used to represent binary data, the symbol duration of the M-ary format is related to the bit duration T byduration Tb by

MTT b 2log=



Discrete amplitude-modulated pulse train may be described as different realizations (sample ( pfunctions) of a random process X(t)

( )

= kTtvAtX )(

The coefficient Ak is a discrete random variable

( )=

=k

k kTtvAtX )(

v(t) is basic pulse shape, centered at the origin, t = 0, and normalized such that v(0) = 1T is the symbol durationT is the symbol duration



To proceed with the analysis, we model the mechanism responsible for the generation of the

{A } d fi i di t t tisequence {Ak}, defining as a discrete stationary random sourceThe source is characterized as havingThe source is characterized as having ensemble-averaged autocorrelation function

[ ]where E is the expectation operator

[ ]nkkA AAEnR =)(p p



The power spectral density of the discrete PAM signal X(t) is given byg ( ) g y

= )2exp()()(1)( 2 nfTjnRfVT

fS AX

V(f) is the Fourier transform of the basic pulse v(t)The values of the functions V(f) and RA(n) depend on the type of discrete PAM signal being considered



NRZ Unipolar Format

Suppose that the 0s and 1s of a random binary sequence occur with equal probability

21)()0( ==== aAPAP kk

For n = 0, we may write

2222 a

2)()()0()0(][ 222 aaAPaAPAE kkk ==+==



NRZ Unipolar FormatConsider next the product AkAk-n for n 0

[ ] ( )( ) ( )( )44

14

1032

2 aaAAE nkk =+= 0n

The autocorrelation function RA(n) may be expressed as f ll

( ) ( )444

follows

2

2a 0=n

=

4

2)( 2anRA

0n



NRZ Unipolar FormatFor the basic pulse v(t), we have a rectangular pulse p ( ) g pof unit amplitude and duration Tb. The Fourier transform of v(t) equals

The power spectral density of NRZ unipolar format

)(sin)( bb fTcTfV =

The power spectral density of NRZ unipolar format

+= bbb

bb

X nfTjfTcTa

fTcTa

fS )2exp()(sin)(sin)( 22

22

=

+n

bbbX nfTjfTcfTcfS )2exp()(sin4)(sin

4)(



NRZ Unipolar FormatUse Poisons formula written in the form

=

=

=

m bbnb T

mfT

nfTj 1)2exp(

We may simplify the expression for the power spectral d it (f)density SX(f) as

)()(sin)(2

22

fafTcTafS b += )(4

)(sin4

)( ffTcfS bX +=



NRZ Polar FormatSimilar to that described for the unipolar format, we find thatthat

=0

)(2a

nRA 00

=

nn

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

)(sin)( 22 bbX fTcTafS =



NRZ Bipolar FormatThe successive 1s in the bipolar format be assigned pulses of alternating polarity The bipolar format has three level: a, 0, -aA th t th 1 d 0 i th i t bi d tAssume 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

( )

( ) 14

1== aAP k

( )

( ) 412

10

==

==

aAP

AP

k

k



NRZ Bipolar FormatFor n = 0, we may write

For 1 the dibit represented by the sequence

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

002

2222 aaAPaAPaAPaAE kkkk ==+=+==

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( , ), ( , ), ( , ), ( , ) y

[ ] ( )( ) ( )( )44

14

1032

21

aaAAE kk =+= 444



NRZ Bipolar FormatFor n > 1, we find that

[ ] 0=nkk AAE

For the NRZ Bipolar format, we have

= 42

)( 22

aa

nRA 10=

=nn

0 otherwise



NRZ Bipolar FormatThe basic pulse v(t) for the NRZ bipolar format has its Fourier transform as in previous casesThe power spectral density of the NRZ bipolar f t i iformat is given

( )

+= )2exp(2exp()(sin)(22

2bbbbX fTjfTj

aafTcTfS ( )

+ )2exp(2exp(42

)(sin)( bbbbX fTjfTjfTcTfS

[ ])2cos(1)(sin2

22

bbb fTfTcTa = [ ]

)(sin)(sin2

222bbb

bb

fTfTcTa =



Manchester FormatIn Manchester format, the input binary data consists of i d d t ll lik l b lindependent, equally likely symbolThe autocorrelation function RA(n) for the Manchester format is the same as for the NRZ polar format

2a 0=n

=0

)(a

nRA 00

nn



Manchester FormatThe basic pulse v(t) for the Manchester format consists of a doublet pulse of unit amplitude and total duration Tb.The Fourier transform of the pulse equals

fTfT

The power spectral density of the Manchester format

=

2sin

2sin)( bbb

fTfTcjTfV

p p yis given

fTfT

=

2sin

2sin)( 222 bbbX

fTfTcTafS


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 secondsThis sequence is applied to a pulse generator, producing the discrete PAM signal

( )

=

=k

bk kTtvatx )(

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

k

o at used The waveform x(t) represents one realization of the random

process X(t)


InterSymbol Interference



The receiving filter output may be written as

( )

k)(

is scaling factor

( )=

=k

bk kTtpaty )(

is scaling factorThe pulse p(t) is normalized such that

1)0( =p

1)0( =p



The output y(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)(tpEspecially, 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

)(tp

to the input of this cascade connection

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

P(f) and V(f) are Fourier transform of p(t) and v(t)

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



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

( )

( )=

=k

bbki kTiTpaty )(

( )

+= bbki kTiTpaa

The first term is produced by the ith transmitted bit

( )=ik

kbbki

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; this g ;residual effect is called intersymbol interference (ISI)


4. Nyquists Criterion For DistortionlessBaseband Binary Transmission

Typically, the transfer function of the channel and the transmitted pulse shape are specified, and the problem is to determine the transferand the problem is to determine the transfer functions of the transmitting and receiving filters so as to reconstruct the transmitted data sequence {b }sequence {bk}The receiver does this by extracting and then decoding the corresponding sequence of g p g qweights, {ak}, from the output y(t).Except for a scaling factor, y(t) is determined by the ak and the received pulse p(t)the ak and the received pulse p(t)


Nyquists Criterion For DistortionlessBaseband Binary Transmission

The extraction involves sampling the output y(t)at some time t = iTbbThe 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 weighted pulse contributions represented by k i



This, in turn, require that we control the received pulse p(t), as shown byp p( ) y

( )

=01

bb kTiTp kiki

=

where, by normalization, p(0) = 1



The receiver output

( )

Which implies zero intersymbol interference (ISI)

( ) ii aty =

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



Consider the sequence of samples {p(nTb)},where n = 0, 1, 2, Sampling in the time domain produces periodicity in frequency domain

( )

=

=n

bb nRfPRfP )(

Where Rb = 1/Tb is the bit rate

n

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 of p(t)



That is

[ ] ( )dfjTTfP 2)()()(

where m = i k.

[ ] ( )dtftjmTtmTpfP bb 2exp)()()( = where m i k.

Impose the condition of zero ISI on the sample values of p(t)p( )

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

)0(p=



Since p(0) = 1, by normalization, the condition for zero ISI is sastisfied if

( ) bn

b TnRfP =

=

Nyquist criterion for distortionless baseband transmissiontransmission



Ideal solutionA frequency function P(f), occupying the narrowest band, is obtained by permitting only one nonzero component in the seriesobtained by permitting only one nonzero component in the series for each f in the range extending from B0 to B0, where B0denotes half the bit rate

bRB

We specify P(f)20

bB =

1)( ff

Hence, signal waveform that produces zero ISI is defined by the

=

00 221)(

Bfrect

BfP

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

( )tB

tBtp 02

2sin)( = ( )tBc 02sin=


tB02


Ideal solution



There are two practical difficulties that make it an undesirable objective for system design:

It requires that the amplitude characteristic of P(f) be flat form B0 to B0 and zero elsewhere. This is physically unrealizable because of the abrupt p y y ptransitions at B0The function p(t) decreases as 1/|t| for large |t|,resulting in a slow rate of decay. This is caused byresulting in a slow rate of decay. This is caused by the discontinuity of P(f) at B0. Accordingly, there is practically no margin of error in sampling times in the receiverreceiver



Practical solutionWe may overcome the practical difficulties posed by the ideal

l ti b t di th b d idth f B R /2 tsolution by extending the bandwidth from B0 = Rb/2 to an adjustable value between B0 and 2B0In doing so, we permit three components as shown by

( ) ( )0

00 2122)(B

BfpBfpfP =+++00 BfB

02B



Practical solutionA particular form of P(f) that embodies many desirable f t i t t d b i d i tfeatures is constructed by a raised cosine spectrum

1

( )

+=

22cos1

41

2

)(10

1

0

0

fBff

B

B

fP

101

1

2 fBff

ff

InterSymbol InterferenceInterSymbol Interference

Practical solution


Practical solutionThe time response p(t), that is, the inverse Fourier p p( ) ,transform of P(f), is defined

( )02cos)2(sin)( tBtBctp

A more general relationship between required bandwidth and symbol transmission rate involves the

( )22

02

00 161

)2(sin)(tB

tBctp

=

bandwidth and symbol transmission rate involves the roll-off factor

)1(2 010 +== BfBB


5. Correlative Coding

It is possible to achieve a bit rate of 2B0 per second in a channel of bandwidth B0 Hertz by adding intersymbolinterference to the transmitted signal in a controlled mannerSuch schemes are called correlative coding or partial-response signaling schemesresponse signaling schemesThe design of these schemes is based on the premise that since 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 ofCorrelative coding may be regarded as a practical means of achieving the theoretical maximum signaling rate of 2Bo per second in a bandwidth of B0 hertz


Correlative Coding

Duobinary signalingConsider a binary input sequence {bk} consisting of y p q { k} guncorrelated binary digits each having duration Tbseconds, with symbol 1 represented by a pulse of amplitude +1 volt and symbol 0 by a pulse ofamplitude +1 volt, and symbol 0 by a pulse of amplitude -1 voltThis sequence is applied to duobinary encoder, it is converted into a three-level output, namely -2, 0, and +2 volts


Correlative Coding

Duobinary signaling


Correlative Coding

Duobinary signalingThe digit ck at the duobinary coder output is theThe digit ck at the duobinary coder output is the sum of the resent binary digit bk and its previous value bk-1

bb

One of the effects of the transformation is to change the input sequence {b } of uncorrelated

1+= kkk bbc

change the input sequence {bk} of uncorrelated binary digits into a sequence {ck} of correlated digitsThis correlation between the adjacent transmittedThis correlation between the adjacent transmitted levels may be viewed as introducing ISI into the transmitted signal


Correlative Coding

Duobinary signalingThe overall transfer function of this filter connected in cascade

ith th id l h l H (f) iwith the ideal channel Hc(f) is

( )[ ]bC fTjfHfH += exp1)()( ( )[ ]bC fjff p)()(

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

bbbC

fTjfTfHfTjfTjfTjfH

=++=

expcos)(2expexpexp1)(

( ) ( )bbC fTjfTfH = expcos)(2


Correlative Coding

Duobinary signalingFor the ideal channel of bandwidth B0 = R b/2, we have

=01

)( fH Cotherwise

2bRf

The overall frequency response has the form of a half-cycle cosine functionhalf cycle cosine function

( ) ( )

=expcos2

)( bbfTjfT

fH 2bRf

= 0

)( fH otherwise


Correlative Coding

Duobinary signalingThe corresponding value of the impulse response consists of two i l ti di l d b T dsinc pulse, time-displaced by Tb seconds

( ) ( )[ ]( )

bbb

TTTTt

TTt

th

+= sinsin

)( ( ) bbb TTtTt )(

( ) [ ]( ) TT

TtT

Tt bb = sinsin ( )( )

( )tTtTtT

TTtTt

bb

bbb

=

sin2

( )tTt b


Duobinary signaling

Correlative Coding

Duobinary signaling


Correlative Coding

Duobinary signalingThe original data {bk} may be detected from the d bi d d { } b bt ti thduobinary-coded sequence {ck} by subtracting the previous decoded binary digit from the currently received digit ck

It is apparent that if c is received without error and if1

= kkk bcb

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

1

kb

estimate will be correct tookb


Correlative Coding

Duobinary signaling Practical solutionUse precoder before the duobinary coding to avoid error

tipropagationThe precoder operation performed on the input binary sequence {bk} converts it into another sequence {ak} defined by

1= kkk aba


Correlative Coding

Duobinary signaling Practical solutionThe resulting precoder output {ak} is applied to the duobinary

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

1+= kkk aac


Correlative Coding


Correlative Coding

Illustrating doubinary coding

Decision rule

= volt1 if 1 volt1 if 0

k

kk csymbol

csymbolb


6. Eye Pattern

One way to study ISI in a PCM or data transmission system experimentally is to apply the received wave to the vertical deflectionthe received wave to the vertical deflection plates of an oscilloscope an to apply a sawtoothwave at the transmitted symbol rate R = 1/T to the horizontal deflection platesthe horizontal deflection platesThe waveforms in successive symbol intervals are thereby translated into one interval on the yoscilloscope displayThe resulting display is called an eye pattern


Eye Pattern


Eye Pattern

The width of the eye opening defines the time interval over which the received wave can be sampled without error form ISI It is apparentsampled without error form ISI. It is apparent that the preferred time for sampling is the instant of time at which the eye is opened widestTh iti it f th t t ti i iThe sensitivity of the system to timing error is determined by the rate of closure of the eye as the sampling time is variedp gThe height of the eye opening, at a specified sampling time, defines the margin over noise


Eye Pattern


Homework

Problems: 4.1, 4.2, 4.3Problems: 4.7, 4.8, 4.9Problems: 4.16, 4.18, 4.19Problems: 4.21, 4.25, 4.26

Textbook:

Simon Haykin, Communication System, 4th Edition, John Wiley & Son, Inc. , 2001.y , ,


kha dc02 baseband transmission

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