4. kha dc02 baseband transmission

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Baseband TransmissionHa Hoang Kha, Ph.D

Ho Chi Minh City University of Technology

Email: [email protected]

Chapter 2



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



Content

1) Discrete PAM signals

2) Power Spectra of Discrete PAM Signals

3) InterSymbol Interference

4) Nyquist’s 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 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)




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

which data are transmitted.

It is also common practice to refer to the data

signaling rate as the bit rate, denoted by

where T b

is the bit duration

b

b

T R

1




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 (with M an integer power of

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

M T T b 2log





Discrete amplitude-modulated pulse train may

be described as different realizations (sample

functions) of a random process X(t)

The coefficient Ak is a discrete random variable

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

and normalized such that v(0) = 1

T is the symbol duration

k

k kT t v At X )(





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

where E is the expectation operator

nk k A

A A E n R

)(





The power spectral density of the discrete PAM

signal X(t) is given by

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

The values of the functions V(f) and R A(n) depend on

the type of discrete PAM signal being considered

)2exp()()(1

)( 2

nfT jn R f V T

f S A X





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( a A P A P k k

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

2222 aa A P a A P A E

k k k


= Sum(x^2*f(x))




NRZ Unipolar Format

Consider next the product Ak Ak-n for n≠ 0

The autocorrelation function R A(n) may be expressed as

follows

44

14

1032

2 aa A A E nk k

0n

4

2)(2

2

a

a

n R A

0

0

n

n





NRZ Unipolar Format

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

of unit amplitude and duration T b. The Fourier

transform of v(t) equals

The power spectral density of NRZ unipolar format

)(sin)( bb fT cT f V

n

bbb

bb

X nfT j fT cT a

fT cT a

f S )2exp()(sin4)(sin4)( 2

22

2





NRZ Unipolar Format

Use Poison’s formula written in the form

We may simplify the expression for the power spectral

density S X (f) as

m bbn

bT

m f

T nfT j

1)2exp(

)(4)(sin4)(

2

2

2

f a

fT cT a

f S bb X





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

)(2

an R

A0

0

n

n

)(sin)( 22

bb X fT cT a f S





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

a A P

A P

a A P

k

k

k





NRZ Bipolar Format

For n = 0, we may write

For n = 1, the dibit represented by the sequence

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

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

2

00

2

2222 aa A P a A P a A P a A E

k k k k

44

1

4

103

2

2

1

aa A A E

k k





NRZ Bipolar Format

For n > 1, we find that

For the NRZ Bipolar format, we have

0nk k A A E

0

4

2

)( 2

2

a

a

n R A

otherwise

1

0

n

n





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

bbbb X fT j fT jaa

fT cT f S

)(sin)(sin

)2cos(1)(sin2222

22

bbb

bb

b

fT fT cT a

fT fT c

T a





Manchester Format

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

The autocorrelation function R A(n) for the Manchester format is

the same as for the NRZ polar format

0

)(2

an R

A

0

0

n

n





Manchester Format

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

duration T b.The Fourier transform of the pulse equals

The power spectral density of the Manchester format

is given

2sin2sin)(

bb

b

fT fT

c jT f V

2sin

2sin)( 222 bb

b X fT fT cT a f S




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 T b 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 ak depends on the input data and the type of

format used• The waveform x(t) represents one realization of the random

process X(t)

k

bk kT t vat x )(




InterSymbol Interference


Thi t k Ht v Hr sao cho ko g y ra ch ng phP(t) là kết quả sau khi qua kênh truyền <-- đkiện




The receiving filter output may be written as

is scaling factor

The pulse p(t) is normalized such that

k

bk kT t pat y )(

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

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 of p(t) and v(t)

)(t p

)()()()()( f H f H f H f V f P RC T





The receiving filter output y(t) is sampled at time t i = iT b

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

bbk i kT iT pat y )(

ik

k

bbk i kT iT paa


t n p n giao t oa

4 Nyquist’s Criterion For Distortionless



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 output y(t).

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

the ak and the received pulse p(t)


Nyquist’s Criterion For Distortionless



Nyquist s Criterion For Distortionless


The extraction involves sampling the output y(t)

at some time t = iT b

The decoding requires that the weighted pulse

contribution ak p(iT b-kT b ) for k = i be free form ISI

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







This, in turn, require that we control the received

pulse p(t), as shown by

where, by normalization, p(0) = 1

0

1bb kT iT p

k i

k i


Đi u ki n mongmuốn






The receiver output

Which implies zero intersymbol interference (ISI)

This condition assures perfect reception in theabsence of noise

ii

at y







Consider the sequence of samples { p(nT b )},

where n = 0, ±1, ±2, …

Sampling in the time domain produces

periodicity in frequency domain

Where Rb = 1/T b is the bit rate

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

delta functions of period T b, and whose strengths are weighted

by the respective sample values of p(t)

n

bb nR f P R f P )(







That is

where m = i – k.

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

dt ft jmT t mT p f P bb

2exp)()()(

dt ft jt p f P 2exp)()0()(

)0( p





Nyquist s Criterion For DistortionlessBaseband Binary Transmission

Since p(0) = 1, by normalization, the condition for

zero ISI is sastisfied if

Nyquist criterion for distortionless baseband

transmission

b

n

b T nR f P


Ph t 1 bit --> 1 xung





Ideal solution

A frequency function P(f), occupying the narrowest band, isobtained by permitting only one nonzero component in the seriesfor each f in the range extending from – B0 to B0, where B0

denotes half the bit rate

We specify P(f)

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

2

0

b R

B

00 22

1)(

B

f rect

B f P


t B

t Bt p

0

0

2

2sin)(

t Bc 02sin




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 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 of P(f) at ± B0. Accordingly, there ispractically no margin of error in sampling times in the

receiver






Practical solution

We may overcome the practical difficulties posed by the idealsolution by extending the bandwidth from B0 = Rb /2 to an

adjustable value between B0 and 2B0

In doing so, we permit three components as shown by

0

002

122)(

B B f p B f p f P

00 B f B






Practical solution

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

• Rolloff factor

0

22cos1

4

1

2

1

)(10

1

0

0

f B

f f

B

B

f P

10

101

1

2

2

f B f

f B f f

f f

0

11 B

f





Practical solutionĐ nh đ i giữa t c đ suyhao và độ rộng băng thông





Practical solution

The time response p(t), that is, the inverse Fourier

transform of P(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)(

t B

t Bt Bct p


)1(2 010 B f B B

5 C l ti C di



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 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 of B0 hertz


M a tương quan

C l ti C di



Correlative Coding

Duobinary signaling

Consider a binary input sequence {bk } consisting of

uncorrelated binary digits each having duration Tb

seconds, 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



Correlative Coding

Duobinary signaling


Correlative Coding



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 uncorrelatedbinary digits into a sequence {ck } of correlateddigits

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

1

k k k bbc


Correlative Coding



Correlative Coding

Duobinary signaling

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

bC fT j f H f H exp1)()(

bbC

bbbC

fT j fT f H

fT j fT j fT j f H

expcos)(2

expexpexp1)(


2

Correlative Coding



Correlative Coding

Duobinary signaling

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

The overall frequency response has the form of a

half-cycle cosine function

0

1)( f H C

otherwise

2b R f

0

expcos2

)( bb fT j fT

f H

otherwise

2b R f


Correlative Coding



Correlative Coding

Duobinary signaling

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

bb

bb

b

b

T T t

T T t

T t

T t t h

sinsin)(

t T t

T t T

T T t

T t

T t

T t

b

bb

bb

b

b

b

sin

sinsin

2




Duobinary signaling

C l ti C di



Correlative Coding

Duobinary signaling


Correlative Coding



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)T bcorresponds to a correct decision, then the current

estimate will be correct too

1

ˆˆ

k k k bcb

k bˆ

1

ˆk

b


Correlative Coding



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

k k k aba


Correlative Coding



Correlative Coding

Duobinary signaling – Practical solution

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

The sequence {c k } is related to {ak } as follows

1 k k k aac


Correlative Coding



Correlative Coding


Correlative Coding



Correlative Coding

Decision rule

volt1if 1

volt1if 0

k

k

k c symbol

c symbol b


6 Eye Pattern



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



Eye Pattern


Eye Pattern



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



Eye Pattern


Homework



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, Communication System , 4th Edition,

John Wiley & Son, Inc. , 2001.

4. kha dc02 baseband transmission

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