digital communication (formating)

8/19/2019 Digital Communication (Formating)

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Digital CommunicationSystemsLecture-2


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ormatting


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Example 1:

In ASCII alphabets, numbers, and symbols are encoded using a 7-bit code

A total of 2 7 = 128 different characters can be represented usinga 7-bit unique ASCII code (see ASCII Table, Fig ! "#


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ormattingTransmit and Receive Formatting

Transition from information source → digital symbols → information sink


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Character Coding (Textual Information)

A te$tual information is a sequence of alphanumeric characters Alphanumeric and symbolic information are encoded into digital bitsusing one of se%eral standard formats, e g, ASCII, &'C IC


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Transmission of Analog Signals

Structure of igital Communication Transmitter

!nalog to igital Conversion


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Sampling

Sampling is the processes of con%erting continuous-time analog

signal, x a(t), into a discrete-time signal by ta)ing the *samples+ atdiscrete-time inter%alsSampling analog signals ma)es them discrete in time but stillcontinuous %aluedIf done properly ( Nyquist theorem is satisfied#, sampling does notintroduce distortion

Sampled values:The %alue of the function at the sampling points

Sampling intervalThe time that separates sampling points (inter%al b . samples#, T s

If the signal is slo.ly %arying, then fe.er samples per second .illbe required than if the .a%eform is rapidly %aryingSo, the optimum sampling rate depends on the maximumfrequency component present in the signal


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!nalog"to"digital conversion is (basically# a ! step process

SamplingCon%ert from continuous-time analog signal x a(t) to discrete-time continuous %alue signal x(n)

Is obtained by ta)ing the *samples+ of x a(t) at discrete-timeinter%als, T s

#uanti$ationCon%ert from discrete-time continuous %alued signal to discretetime discrete %alued signal


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Sampling

Sampling Rate (or sampling fre%uenc& f s ):The rate at .hich the signal is sampled, e$pressed as thenumber of samples per second (reciprocal of the samplinginter%al#, 1/T s = f s

'&%uist Sampling Theorem (or '&%uist Criterion):If the sampling is performed at a proper rate, no info is lost aboutthe original signal and it can be properly reconstructed later onStatement

*If a signal is sampled at a rate at least, but not e$actly equal tot.ice the ma$ frequency component of the .a%eform, then the.a%eform can be e$actly reconstructed from the samples.ithout any distortion+

max2 s f f ≥


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Ideal Sampling ( or Impulse Sampling)

Therefore, .e ha%e

Ta)e Fourier Transform (frequency con%olution#

1( ) ( ) e s jn t s

n s

x t x t T

ω ∞

=−∞

= ÷ ∑

{ }1 1( ) ( )* ( )* s s jn t jn t sn n s s

X f X f e X f eT T

ω ω ∞ ∞

=−∞ =−∞

= ℑ = ℑ ∑ ∑

1( ) ( )* ( ),

2 s

s s sn s

X f X f f nf f T

ω δ

π

∞

=−∞= − =∑

1 1( ) ( ) ( ) s s

n n s s s

n X f X f nf X f

T T T

∞ ∞

=−∞ =−∞= − = −∑ ∑


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Sampling

If R s < 2B , aliasing (o%erlapping of the spectra# results

If signal is not strictly bandlimited, then it must be passed throughLow Pass Filter (/0F# before sampling

Fundamental Rule of Sampling ('&%uist Criterion)The %alue of the sampling frequency f s must be greater than

t.ice the highest signal frequency f ma$ of the signalT&pes of samplingIdeal Sampling1atural SamplingFlat-Top Sampling


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Is accomplished by the multiplication of the signal x(t) by the uniform

train of impulses (comb function#Consider the instantaneous sampling of the analog signal x(t)

Train of impulse functions select sample %alues at regular inter%als

Fourier Series representation

( ) ( ) ( ) s sn

x t x t t nT δ ∞

=−∞= −∑

1 2( ) , s jn t s s

n n s s

t nT eT T

ω π δ ω ∞ ∞

=−∞ =−∞− = =∑ ∑


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This sho.s that the Fourier Transform of the sampled signal is theFourier Transform of the original signal at rate of 1/T s


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As long as f s> 2f m,no o%erlap of repeated replicas X(f - n/T s ) .ill

occur in X s(f)2inimum Sampling Condition

Sampling Theorem: A finite energy function x(t) can be completelyreconstructed from its sampled value x(nTs) with

provided that =>

2 s m m s m f f f f f − > ⇒ >

2 ( )sin2

( ) ( )( )

s

s s s

n s

f t nT T

x t T x nT t nT

π

π

∞

=−∞

− = −

∑

( ) sin (2 ( )) s s s sn

T x nT c f t nT ∞

=−∞= −∑ 1 1

2 s s mT

f f = ≤


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This means that the output is simply the replication of the originalsignal at discrete inter%als, e g


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T s is called the Nyquist interval It is the longest time inter%al that canbe used for sampling a bandlimited signal and still allo.reconstruction of the signal at the recei%er .ithout distortion


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Practical Sampling

In practice .e cannot perform ideal sampling

It is practically difficult to create a train of impulsesThus a non-ideal approach to sampling must be used3e can appro$imate a train of impulses using a train of %ery thinrectangular pulses

1ote

Fourier Transform of impulse train is another impulse train

Con%olution .ith an impulse train is a shifting operation

( ) s pn

t nT x t

τ

∞

=−∞

− = Π ÷ ∑


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Natural SamplingIf .e multiply x(t) by a trainof rectangular pulses x (t),

.e obtain a gated .a%eformthat appro$imates the idealsampled .a%eform, )no.nas natural sampling orgating (s ee Figure ! 4#

( ) ( ) ( ) s p x t x t x t =2( ) s j nf t n

n

x t c e π ∞

=−∞= ∑( ) [ ( ) ( ) s p X f x t x t = ℑ

2[ ( ) s j nf t nn

c x t e π ∞=−∞

= ℑ∑[ n s

n

c X f nf ∞

=−∞= −∑


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&ach pulse in x (t) has .idth T s and amplitude 1/T s

The top of each pulse follo.s the %ariation of the signal beingsampled X s (f) is the replication of X(f) periodically e%ery f s !"

X s (f) is .eighted by # n ← Fourier Series Coeffiecient

The problem .ith a natural sampled .a%eform is that the tops of thesample pulses are not flatIt is not compatible .ith a digital system since the amplitude of eachsample has infinite number of possible %alues

Another technique )no.n as flat to! sam!ling is used to alle%iatethis problem


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lat-Top Sampling

5ere, the pulse is held to a constant height for the .holesample periodFlat top sampling is obtained by the con%olution of the signalobtained after ideal sampling .ith a unity amplituderectangular pulse, (t)This technique is used to reali6e "am!le#and#$old (S 5#operationIn S 5, input signal is continuously sampled and then the%alue is held for as long as it ta)es to for the A to acquire

its %alue


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Flat top sampling (Time omain#

!( ) ( ) ( ) x t x t t δ =( ) !( )* ( ) s x t x t p t =

( )* ( ) ( ) ( )* ( ) ( ) sn

p t x t t p t x t t nT δ δ ∞

=−∞

= = − ∑


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Ta)ing the Fourier Transform .ill result to

.here $(f) is a sinc function

( ) [ ( ) s s X f x t =ℑ( ) ( ) ( ) s

n

P f x t t nT δ ∞

=−∞

= ℑ − ∑1( ) ( ) * ( ) s

n s

P f X f f nf T

δ ∞

=−∞ = ℑ − ∑

1( ) ( ) s

n s

P f X f nf T

∞

=−∞= −

∑


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Flat top sampling (Frequency omain#

Flat top sampling becomes identical to ideal sampling as the.idth of the pulses become shorter


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Recovering the Analog Signal

ne .ay of reco%ering the original signal from sampled signal X s(f) isto pass it through a /o. 0ass Filter (/0F# as sho.n belo.

If f s > 2B then .e reco%er x(t) e%actly

&lse .e run into some problems and signalis not fully reco%ered


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ndersampling and !liasingIf the .a%eform is undersam!led (i e fs < 2B # then there .ill be

s!ectral overla! in the sampled signal

The signal at the output of the filter .ill be

different from the original signal spectrum

This is the outcome of aliasing 8

This implies that .hene%er the sampling condition is not met, anirre%ersible o%erlap of the spectral replicas is produced


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This could be due to

1% x(t) containing higher frequency than .ere e$pected! An error in calculating the sampling rate 9nder normal conditions, undersampling of signals causing aliasing isnot recommended


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Solution 1: !nti"!liasing !nalog Filter

All physically reali6able signals are not completely bandlimitedIf there is a significant amount of energy in frequencies abo%ehalf the sampling frequency (f s !#, aliasing .ill occur

Aliasing can be pre%ented by first passing the analog signalthrough an anti"aliasing filter (also called a prefilter # before

sampling is performedThe anti-aliasing filter is simply a /0F .ith cutoff frequencyequal to half the sample rate


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Aliasing is pre%ented by forcing the band.idth of the sampledsignal to satisfy the requirement of the Sampling Theorem


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Solution : *ver Sampling and Filtering in the igitalomain

The signal is passed through a lo. performance (less costly#analog lo.-pass filter to limit the band.idthSample the resulting signal at a high sampling frequencyThe digital samples are then processed by a highperformance digital filter and do.n sample the resultingsignal


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Summary f Sampling

Ideal Sampling(or Impulse Sampling)

'atural Sampling(or +ating)

Flat"Top Sampling

For all sampling techni%uesIf fs > 2B t&en 'e can rec er x(t) exact*y If fs < 2B # s!ectral overla!!ing )no.n as a liasing .ill occur

( ) ( ) ( ) ( ) ( )

( ) ( )

s sn

s sn

x t x t x t x t t nT

x nT t nT

δ δ

δ

∞

=−∞∞

=−∞

= = −

= −

∑∑

2

( ) ( ) ( ) ( ) s j nf t

s p nn x t x t x t x t c e

π ∞

=−∞= = ∑( ) !( )* ( ) ( ) ( ) * ( ) s s

n

x t x t p t x t t nT p t δ ∞

=−∞

= = −

∑


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! " a p l e #$

Consider the analog signal x(t) gi%en by

3hat is the 1yquist rate for this signal:

! " a p l e % $Consider the analog signal x a(t) gi%en by

3hat is the 1yquist rate for this signal:3hat is the discrete time signal obtained after sampling, if

f s=+ samples s3hat is the analog signal x(t) that can be reconstructed from thesampled %alues:

( ) "cos(#$ ) 1$$sin("$$ ) cos(1$$ ) x t t t t π π π

= + −

( ) "cos 2$$$ #sin %$$$ cos12$$$a x t t t t π π π = + +


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Practical Sampling Rates

Speech- Telephone quality speech has a band.idth of ; )56

(actually "


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Pulse &ode 'odulation (P&')

0ulse Code 2odulation refers to a digital baseband signal that isgenerated directly from the quanti6er outputSometimes the term 0C2 is used interchangeably .ith quanti6ation


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See Figure ! =? (0age 4


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!dvantages of -C.:Re*ati e*y inex ensi e

asi*y mu*ti *exe. 0C2 .a%eforms from differentsources can be transmitted o%er a common digitalchannel (T 2#

asi*y re enerate. useful for long-distance

communication, e g telephone'etter noise performance than analog systemSignals may be stored and time-scaled efficiently (e g ,satellite communication#&fficient codes are readily a%ailable

isadvantage:@equires .ider band.idth than analog signals


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% Sources of &orruption in the sampled*+uanti,ed and transmitted pulses

"am!ling and &uanti'ation (ffectsuanti6ation (Branularity# 1oise @esults .hen

quanti6ation le%els are not finely spaced apart enoughto accurately appro$imate input signal resulting intruncation or rounding error

uanti6er Saturation or %erload 1oise @esults .heninput signal is larger in magnitude than highestquanti6ation le%el resulting in clipping of the signalTiming itter &rror caused by a shift in the sampler

position Can be isolated .ith stable cloc) reference)hannel (ffects

Channel 1oiseIntersymbol Interference (ISI#


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&he level of 'uanti ation noise is dependent on how close any particular sample is to one of the L levels in the converter

For a speech input, this quanti6ation error resembles a noise-li)e disturbance at the output of a AC con%erter

Signal to #uanti$ation 'oise Ratio


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niform .uanti,ation

A quanti6er .ith equal quanti6ation le%el is a 9niform uanti6er

&ach sample is appro$imated .ithin a quantile inter%al9niform quanti6ers are optimal .hen the input distribution isuniform

i e .hen all %alues .ithin the range are equallyli)ely

2ost A CDs are implemented using uniform quanti6ers&rror of a uniform quanti6er is bounded by 2 2

q qe− < ≤


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The mean-squared %alue (noise %ariance# of the quanti6ation erroris gi%en by

2 2 22 2 2

2 2 2

1 1( )2 q q q

q q q

e p e de e de e deq q

σ − − −

= =∫ ∫ ∫ ÷

=

" 2

2

21" 12

q

q

qeq −= =

Signal to #uanti$ation 'oise Ratio


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The pea) po.er of the analog signal (normali6ed to = hms #can bee$pressed as

Therefore the Signal to uati6ation 1oise @atio is gi%en by

22 2 2

2 1 pp p V V L q P ÷ ÷ ÷ ÷

= = =

2 2

2*

122"q

L qq

SNR L==


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.here 0 E ! n is the number of quanti6ation le%els for the con%erter(n is the number of bits#

Since 0 = 2 n, R = 2 2n or in decibels

ppV

Lq=

21$log (2 ) %

1$

nS n dB

N dB

÷ ÷

= =

If q is the step si6e, then the ma$imum quanti6ation error that canoccur in the sampled output of an A con%erter is q


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Nonuniform .uanti,ationNonuniform quanti'ers ha%e unequally spaced le%els

The spacing can be chosen to optimi6e the Signal-to-1oise @atio

for a particular type of signalIt is characteri6ed by

ariable step si6euanti6er si6e depend on signal si6e


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2any signals such as speech ha%e a nonuniform distribution

See Figure on ne$t page (Fig ! =7#

/asic principle is to use more le%els at regions .ith large probabilitydensity function (pdf#

Concentrate quanti6ation le%els in areas of largest pdf

r use fine quanti6ation (small step si6e# for .ea) signals andcoarse quanti6ation (large step si6e# for strong signals


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Statistics of speech Signal Amplitudes

Figure 2.17: Statistical distribution of single talker speech signalmagnitudes !age "1#


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Nonuniform +uanti,ation usingcompanding

Companding is a method of reducing the number of bits required in A C .hile achie%ing an equi%alent dynamic range or S 1@

In order to impro%e the resolution of .ea) signals .ithin a con%erter,and hence enhance the S 1@, the weak signals need to beenlarged , or the quanti'ation ste! si'e decreased , but only for the.ea) signals'ut str n si na*s can potentially be re.uce. .ithout significantlydegrading the S 1@ or alternati%ely increasing quanti6ation step si6eThe compression process at the transmitter must be matched .ith anequi%alent e$pansion process at the recei%er


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The signal belo. sho.s the effect of compression, .here theamplitude of one of the signals is compressed

After compression, input to the quanti6er .ill ha%e a more uniformdistribution after sampling

At the recei%er, the signal is

e$panded by an in%erseoperation

The process of C 2pressingand e$0A1 I1B the signal iscalled com!anding

Companding is a techniqueused to reduce the number ofbits required in A C or AC.hile achie%ing comparableS 1@


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'asically, companding introduces a nonlinearity into the signal

This maps a nonuniform distribution into something that moreclosely resembles a uniform distribution A standard A C .ith uniform spacing bet.een le%els can be usedafter the compandor (or compander#The companding operation is in%erted at the recei%er

There are in fact t.o standard logarithm based compandingtechniques

9S standard called *#law com!anding

&uropean standard called +#law com!anding


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Input/ utput Relationship of &ompander

/ogarithmic e$pression , = log - is the most commonly

used compander This reduces the dynamic range of ,


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Types of &ompanding $%a& 'ompanding Standard (orth ) South *merica+and ,apan#

.here$ and y represent the input and output %oltagesµ is a constant number determined by e$perimentIn the 9 S , telephone lines uses companding .ith µ = 2++

Samples ; )56 speech .a%eform at 4,


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A-Law Companding Standard (Europe,China, Russia, Asia, Africa)

.here

$ and y represent the input and output %oltages + = 0 2 + is a constant number determined by e$periment

maxmax

max

maxmax

max

+ ++ + 1

sgn( ), $(1 )

( )+ +

1 log 1 + +sgn( ), 1

(1 log )

e

e

x A x x

y x A x A

y x x

A x x y x

A A x

< ≤+=

+ ÷ < ≤+


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Pulse 'odulationRecall that analog signals can 0e represented 0& a se%uence of discretesamples (output of sampler)

-ulse .odulation results hen some characteristic of the pulse (amplitude2idth or position) is varied in correspondence ith the data signal

T o T&pes:-ulse !mplitude .odulation (-!.)

The amplitude of the periodic pulse train is varied in proportion to thesample values of the analog signal

-ulse Time .odulationEncodes the sample values into the time axis of the digital signal-ulse 3idth .odulation (-3.)

Constant amplitude2 idth varied in proportion to the signal-ulse uration .odulation (- .)

sample values of the analog aveform are used in determining theidth of the pulse signal


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P&' 0aveform TypesThe output of the A con%erter is a set of binary bits'ut binary bits are Gust abstract entities that ha%e no physical definition

3e use pulses to con%ey a bit of information, e g ,

In order to transmit the bits o%er a physical channel they must betransformed into a physical .a%eform

+ line coder or baseband binary transmitter transforms a stream of bitsinto a physical .a%eform suitable for transmission o%er a channel/ine coders use the terminology mark for *=+ and s!ace to mean *


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There are many types of .a%eforms 3hy: → performance criteria8

&ach line code type ha%e merits and demeritsThe choice of .a%eform depends on operating characteristics of asystem such as

2odulation-demodulation requirements'and.idth requirementSynchroni6ation requirement@ecei%er comple$ity, etc ,


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+oals of 4ine Coding (qualities to loo5 for) A line code is designed to meet one or more of the follo.ing goals

Self"s&nchroni$ationThe ability to reco%er timing from the signal itself

That is, self-cloc)ing (self-synchroni6ation# - ease of cloc) loc)or signal reco%ery for symbol synchroni6ation

/ong series of ones and 6eros could cause a problem

4o pro0a0ilit& of 0it error @ecei%er needs to be able to distinguish the .a%eform associated.ith a mar5 from the .a%eform associated .ith a s ace'&@ performance

relati%e immunity to noise&rror detection capability

enhances lo. probability of error


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Spectrum Suita0le for the channelSpectrum matching of the channel

e g presence or absence of C le%el

In some cases C components should be a%oidedThe transmission band.idth should be minimi6ed

-o er Spectral ensit&0articularly its %alue at 6ero

0S of code should be negligible at the frequency near 6eroTransmission /and idth

Should be as small as possibleTransparenc&

The property that any arbitrary symbol or bit pattern can betransmitted and recei%ed, i e , all possible data sequence shouldbe faithfully reproducible

1i & d


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1ine &oder

The input to the line encoder isthe output of the A con%erteror a sequence of %alues a n thatis a function of the data bitThe output of the line encoderis a .a%eform

.here f(t) is the pulse shape and T 6 is the bit period (T 6=T s /n for nbit quanti6er#

This means that each line code is described by a symbol mappingfunction a n and pulse shape f(t)

etails of this operation are set by the type of line code that isbeing used

( ) ( )n bn

s t a f t nT ∞

=−∞= −∑

S f ' 2 1i & d


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Summary of 'a2or 1ine &odes

Categories of 4ine Codes-olar - Send pulse or negati%e of pulse

nipolar - Send pulse or a </ipolar (a%5%a% a*ternate mar5 in ersi n, seu. ternary #

@epresent = by alternating signed pulses+enerali$ed -ulse Shapes

'R6 -0ulse lasts entire bit period0olar 1@H'ipolar 1@H

R6 - @eturn to Hero - pulse lasts Gust half of bit period0olar @H'ipolar @H

.anchester 4ine CodeSend a !- φ pulse for either = (high → lo.# or < (lo. → high#Includes rising and falling edge in each pulse1o C component


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3hen the category and the generali6ed shapes are combined, .e ha%ethe follo.ing

-olar 'R6:

3ireless, radio, and satellite applications primarily use 0olar1@H because band.idth is preciousnipolar 'R6

Turn the pulse 1 for a =D, lea%e the pulse FF for a


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/ipolar R6 A unipolar line code, e$cept no. .e alternatebet.een positi%e and negati%e pulses to send a =D

Alternating li)e this eliminates the C componentThis is desirable for many channels that cannottransmit the C components

+enerali$ed +rouping

1on-@eturn-to-Hero 1@H-/, 1@H-2 1@H-S@eturn-to-Hero 9nipolar, 'ipolar, A2I0hase-Coded 6i-f -/, 6i-f -2, 6i-f -S, 2iller, elay2odulation

2ultile%el 'inary dicode, doubinary

1ote There are many other %ariations of line codes (see Fig ! !!,page 4< for more#


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nipolar 'R6 4ine Codenipolar non"return"to"$ero ('R6) line code is defined by

unipolar mapping

In addition, the pulse shape for unipolar 1@H is.here T 6 is the bit period

, 1$, $

nn

n

A when X awhen X + == =

3here J n is the n th data bit

( ) , -./ ulse 0hapeb

t f t

T

= Π ÷


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-ipolar %ine 'odesith bipolar line codes a space is mapped to ero and a

mar is alternately mapped to 3A and 4A

It is also called pseudoternar& signaling or alternate mar5 inversion(!.I)

&ither @H or 1@H pulse shape can be used

, when 1 and last mar

, when 1 and last mar

$, when $

n

n n

n

A X A

a A X A

X

+ = → −= − = → +

=


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anchester %ine 'odes Manchester line codes use the antipodal mapping andthe following split-phase pulse shape5

( )

2 2

b b

b b

T T t t

f t T T

+ − ÷ ÷= Π − Π ÷ ÷ ÷ ÷

Summary of 1ine &odes


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Summary of 1ine &odes


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&omparison of 1ine &odes


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&omparison of 1ine &odes

Self"s&nchroni$ation

2anchester codes ha%e built in timing information because theyal.ays ha%e a 6ero crossing in the center of the pulse0olar @H codes tend to be good because the signal le%el al.aysgoes to 6ero for the second half of the pulse1@H signals are not good for self-synchroni6ation

Error pro0a0ilit&0olar codes perform better (are more energy efficient# than9nipolar or 'ipolar codes

Channel characteristics

3e need to find the po.er spectral density (0S # of the linecodes to compare the line codes in terms of the channelcharacteristics

&omparisons of 1ine &odes


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&omparisons of 1ine &odes

ifferent pulse shapes are usedto control the spectrum of the transmitted signal (no C %alue,band.idth, etc #guarantee transitions e%ery symbol inter%al to assist in symbol timingreco%ery

17 -o er Spectral ensit& of 4ine Codes (see Fig7 7 82 -age 9 ) After line coding, the pulses may be filtered or shaped to furtherimpro%e there properties such as

Spectral efficiencyImmunity to Intersymbol Interference

istinction bet.een /ine Coding and 0ulse Shaping is not easy7 C Component and /and idth

C Components9nipolar 1@H, polar 1@H, and unipolar @H all ha%e C components'ipolar @H and 2anchester 1@H do not ha%e C components

First 'ull /and idth


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First ull /and idth9nipolar 1@H, polar 1@H, and bipolar all ha%e =st null band.idths ofR6 = 1/T69nipolar @H has =st null '3 of 2R6

2anchester 1@H also has =st null '3 of 2R6 , although thespectrum becomes %ery lo. at 1%3R6

3eneration of 1ine &odes


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3eneration of 1ine &odes

The FI@ filter reali6es the different pulse shapes

'aseband modulation .ith arbitrary pulse shapes can bedetected bycorrelation detector matched filter detector (this is the most common detector#

4its per P&' 5ord and '-ary 'odulation


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Section 7;7


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2 2

22

+ + + +2

12

2

1log log (#$) %2

6$$$ 6$$$ 1%

6$$$ 12$$$ seclog ( )

pp

pp l pp

qe pV e

V V Lq q L

L p

l l p

fs Rs

R R symb!ls

≤ =

= = = ≥

≥ ⇒ ≥ = ÷ = = =

= = =

Solution to Pro6lem % #7

digital communication (formating)

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