Line Coding and Its Spectral Densities New

8/2/2019 Line Coding and Its Spectral Densities New

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Line Codes and Their Spectra

Types of Line Codes

Comparison of Line Codes

PSD of Line Codes


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Line Codes in PCMLine Codes in PCM

The output of an ADC can be transmitted over aThe output of an ADC can be transmitted over abaseband channelbaseband channel..

The digital information must first be converted into a physical signal.The digital information must first be converted into a physical signal.

TheThephysical signal is called aphysical signal is called a line codeline code. Line coders use the terminology. Line coders use the terminology markmark

to mean binary one andto mean binary one andspacespace to mean binary zero.to mean binary zero.

ADC

PCM signal

Sample

Quantize

Analog

Input

Signal

Encode

Line

Code

X

XQ

Xk

x(t)


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(a) Punched Tape

A

-A0(c) Polar NRZ

A

0(d) Unipolar RZ

A

-A0(e) Bipolar RZ

BINARY DATA1 1 0 1 0 0 1

Mark(hole)

Mark(hole)

Mark(hole)

Mark(hole)

space space space

A

-A

0(f) Manchester NRZ

Binary Signaling Formats

VoltsA

Time

0(b) Unipolar NRZTb

Line codes


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Goals of Line CodingGoals of Line Coding

A line code is designed to meet several goals:A line code is designed to meet several goals:

SelfSelf--synchronization.synchronization. The ability toThe ability to recover timingrecover timing from the signal itself.from the signal itself.

Long series of ones and zeros could cause a problem.Long series of ones and zeros could cause a problem.

Low probability of bit error.Low probability of bit error. The receiver needs to be able to distinguish the waveform associatedThe receiver needs to be able to distinguish the waveform associated

with a markfrom the waveform associated with a space, even if therewith a markfrom the waveform associated with a space, even if thereis ais a considerable amount of noise and distortion in the channelconsiderable amount of noise and distortion in the channel..

Spectrum that is suitable for the channel.Spectrum that is suitable for the channel. In some casesIn some cases DC componentsDC components should be avoided if the channel has ashould be avoided if the channel has a

DC blocking capacitance.DC blocking capacitance.

The transmissionThe transmissionbandwidthbandwidth should be minimized.should be minimized.


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Line CoderLine Coder

The input to the line encoder is a sequence of values aThe input to the line encoder is a sequence of values akk that isthat is

a function of a data bit or an ADC output bit.a function of a data bit or an ADC output bit.

The output of the line encoder is a waveform:The output of the line encoder is a waveform:

WhereWhere pp((tt) is the) is the Pulse ShapePulse Shape andand TTbb is theis the BitPeriodBitPeriod

RRbb =1/=1/TTbb

The operational details of this function are set by the particularThe operational details of this function are set by the particular

type oftype ofline codeline code that is being used.that is being used.

( ) ( )k bk

x t a p t kTg

!g

!

Line Coder

ka

Digital

Data

Physical

Waveform

( ) ( )k bk

x t a p t kTg

!g

!


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Types of Line CodesTypes of Line Codes

Each line code is described by aEach line code is described by asymbol mapping functionsymbol mapping function aakk

and aand apulse shapepulse shape pp((tt):):

Categories of line codes:Categories of line codes: Symbol mapping functions (Symbol mapping functions (aakk).).

UnipolarUnipolar

PolarPolar

Bipolar(a.k.a. alternate markinversion, pseudoternary)Bipolar(a.k.a. alternate markinversion, pseudoternary)

Pulse shapesPulse shapes pp((tt).).

NRZ (NonreturnNRZ (Nonreturn--toto--zero)zero)

RZ (Return to Zero)RZ (Return to Zero)

Manchester(split phase)Manchester(split phase)

( ) ( )k bk

x t a p t kTg

!g

!


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Unipolar NRZ Line CodeUnipolar NRZ Line Code

TheThe unipolar nonreturnunipolar nonreturn--toto--zerozero line code is defined by theline code is defined by the

unipolar mapping:unipolar mapping:

where Xwhere Xkk is the kis the kthth data bit.data bit.

In addition, the pulse shape for unipolar NRZ is:In addition, the pulse shape for unipolar NRZ is:

Where TWhere Tbb is theis the bit periodbit period..

!

!

!

0when

1when

0

k

k

kX

XAa

( ) NRZ pulse shapeb

tp t

T

! 4

1 1110 0

bT bT2 bT3 bT40 bT5

A

Note the DC component

This means wasted power!

Hard to recover symbol timing

when long string of 0s or 1s.


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Unipolar RZ Line CodeUnipolar RZ Line Code

TheThe unipolar returnunipolar return--toto--zerozero line code has the same symbolline code has the same symbol

mapping but a different pulse shape than unipolar NRZ:mapping but a different pulse shape than unipolar NRZ:

!

!!

0when

1when

0

k

k

kX

XAa

( ) RZ pulse shape/ 2

b

tp t

T

! 4

1 1110 0

bT bT2 bT3 bT40 bT5

A

Long strings of 1s no longer a problem.

However strings of 0s still problem.Pulse of half the duration ofNRZ

requires twice the bandwidth!


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Polar Line CodesPolar Line Codes

Polar line codes use the antipodal mapping:Polar line codes use the antipodal mapping:

Polar NRZ uses NRZ pulse shape.Polar NRZ uses NRZ pulse shape.

Polar RZ uses RZ pulse shape.Polar RZ uses RZ pulse shape.

!

!

!

0when

1when

k

k

kX

X

A

Aa

1 1110 0A

A

A

Polar NRZ

Polar RZ

No DC component,

so more energy efficient.

Now we can handle

long strings of 0s, too.


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Manchester Line CodesManchester Line Codes

Manchester line codesManchester line codes use the antipodal mapping anduse the antipodal mapping and

the followingthe followingsplitsplit--phasephasepulse shape:pulse shape:

/ 4 / 4( )

/ 2 / 2

b b

b b

t T t Tp t

T T

! 4 4

)(tp

1 111 0

A

0

Easy synchronization and better spectral characteristics than polarRZ.


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Bipolar Line CodesBipolar Line Codes

WithWith bipolar line codesbipolar line codes a space is mapped to zero and a marka space is mapped to zero and a mark

is alternately mapped tois alternately mapped to --A and +A:A and +A:

Also calledAlso calledpseudoternarypseudoternary signalling andsignalling and alternate markalternate markinversion (AMI)inversion (AMI)..

Either RZ or NRZ pulse shape can be used.Either RZ or NRZ pulse shape can be used.

A

Bipolar (RZ)

1 111 00

A

0 when 0

when 1 and last mark A

when 1 and last mark A

k

k k

k

X

a A X

A X

!

! ! p ! p


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Comparison of Line CodesComparison of Line Codes

SelfSelf--synchronization:synchronization:

Manchester codes have built in timing information becauseManchester codes have built in timing information becausethey always have a zero crossing in the center of the pulse.they always have a zero crossing in the center of the pulse.

Polar RZ codes tend to be good because the signal levelPolar RZ codes tend to be good because the signal levelalways goes to zero for the second half of the pulse.always goes to zero for the second half of the pulse.

NRZ signals are not good for selfNRZ signals are not good for self--synchronization.synchronization.

Error probability:Error probability:

Polar codes perform better(are more energy efficient) thanPolar codes perform better(are more energy efficient) thanUnipolar or Bipolar codes.Unipolar or Bipolar codes.

Channel characteristics:Channel characteristics:

We need to find the PSD of the line codes to answer this ...We need to find the PSD of the line codes to answer this ...


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Power Spectra for Binary Line Codes

2

2

1

( )( ) ( ) Where { ( )} ( )

( ) ( ) The autocorrelation function of

and are levels of the data pulses at the ' th and ( )'th symbol positions

Pro

data

ba

sj kfT

s

ks

I

n n k

k

i i

i

n n

i

F f P f R k e f t F f

T

R k a a P

a a n n k

P

T

g

!g

!

! !

!

bility of having the th productn n ka ai

PSD can be calculated using the autocorrelation function:

A digital signal is represented by

f(t) - Symbol Pulse shape; Ts - Duration of one symbol;

Binary signaling : Ts= Tb , Multilevel signaling: Ts= lTb

PSD depends on:

(1) The pulse shape used

(2) Statistical properties of data expressed by the autocorrelation function

The PSD of a digital signal is given by:

( ) ( ) ; ( ) for unipolar NRZn s

n s

ts t a f t nT f t

T

g

!g

! ! ! 4


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PSD for Polar NRZ Signaling

Possible levels for the as : +A and -A

2

2 2 2

1

4

2 2

1

2

2

s

2

1 1(0) ( ) ( )

2 2

For 0, ( ) ( ) 1/ 4 ( )( )1/ 4 ( )( )1/ 4 ( ) 1/ 4 0

, 0 ( )

0, 0

sin( ) ( / ) ( ) P

n n i i

i

n nk ii

polar

bb b

kb s

kfTs

R a a P A A A

k R k a a P A A A A A A

A kR k

k

F ffTf t t T F f T f R k e

fT T

TT

T

!

!

g

!g

! ! !

{ ! ! !

! !

{

! 4 m ! !

2

2

Polar NRZ

sin( ) bb

b

fT P f A T

fT

T

T

!

1

and are the level of the pulses at the th and ( )th( ) ( sy) mbols

I

n n k i i n n k

i

n nR k a a P a ka !

!


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PSD for line codes

Unipolar NRZ

Polar NRZ

22

Uni. NRZ

sin 1( ) 1 ( )

4

b b

b b

A T fT P f f

fT T

TH

T

!

-

2

2

Polar NRZsin( ) bb

b

fT P f A T fTT

T !

Bit rate: R=1/Tb


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PSD for line codesUnipolarRZ

BipolarRZ

ManchesterNRZ

Uni. R

2

Z

2

sin / 2 1( ) 1 ( )

16 / 2

b b

nb b b

A T fT n P f f

fT T T

TH

T

g

!g

!

-

Bipolar2

RZ

2

2sin( ) sin4

b bb

b

A T fT P f fT fTT T

T

!

Manch. NR

2

Z2 2sin / 2( ) sin / 2

/ 2

bb b

b

fT P f A T fT

fT

TT

T

!

Bit rate: R=1/Tb


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NRZ EncodingNRZ Encoding

There are other variations of NRZ encoding; which include:There are other variations of NRZ encoding; which include:NRZNRZ--LL: [Non: [Non--ReturnReturn--toto--ZeroZero--Level]: In NRZLevel]: In NRZ--L encoding, the polarity ofL encoding, the polarity ofthe signal changes only when the incoming signal changes from a one to athe signal changes only when the incoming signal changes from a one to azero or from a zero to a one. NRZzero or from a zero to a one. NRZ--L method looks just like the NRZL method looks just like the NRZmethod, except for the first input one data bit. This is because NRZ doesmethod, except for the first input one data bit. This is because NRZ doesnot consider the first data bit to be a polarity change, where NRZnot consider the first data bit to be a polarity change, where NRZ--L does.L does.NRZNRZ--MM [Non[Non--ReturnReturn--toto--ZeroZero--Mark(NRZMark(NRZ--M) Encoding]: The polarity ofM) Encoding]: The polarity ofthe signal changes when the incoming signal is a one. An incoming zerothe signal changes when the incoming signal is a one. An incoming zerowould not change the polarity of the signal.would not change the polarity of the signal.NRZNRZ--SS [Non[Non--ReturnReturn--toto--ZeroZero--Space (NRZSpace (NRZ--S) Encoding]: Works just likeS) Encoding]: Works just likeNRZNRZ--M, except the signal changes when the incoming data signal is a zero,M, except the signal changes when the incoming data signal is a zero,not when the signal is a one.not when the signal is a one.

NRZ

NRZ--

CC [Non[Non--ReturnReturn--toto--ZeroZero--Change Encoding]: Same as NRZChange Encoding]: Same as NRZ--L.L.NRZINRZI [Non[Non--ReturnReturn--toto--ZeroZero--Inverted Encoding]: A '0' is encoded as noInverted Encoding]: A '0' is encoded as nochange in the level. However a '1' is encoded depending on the current statechange in the level. However a '1' is encoded depending on the current stateof the line. If the current state is '0' [low] the '1' will be encoded as a high, ifof the line. If the current state is '0' [low] the '1' will be encoded as a high, ifthe current state is '1' [high] the '1' will be encoded as a low. Used withthe current state is '1' [high] the '1' will be encoded as a low. Used withFDDIFDDI andand USBUSB for example.for example.

Line Coding and Its Spectral Densities New

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