Transcript Chapter3_Lect5.ppt

Chapter 3:
Line Codes and Their Spectra
 Types of Line Codes
 Comparison of Line Codes
 PSD of Line Codes
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
Eeng 360 1
Line Codes in PCM

The output of an ADC can be transmitted over a baseband channel.
• The digital information must first be converted into a physical signal.
• The physical signal is called a line code. Line coders use the terminology mark
to mean binary one and space to mean binary zero.
Analog
Input
Signal
Sample
X
Quantize
ADC
XQ
Encode
Xk
Line
Code
x(t)
PCM signal
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Line codes
1
(a) Punched Tape
Mark
(hole)
Volts
(b) Unipolar NRZ
1
Mark
(hole)
0
space
1
0
0
Mark space space
(hole)
1
BINARY DATA
Mark
(hole)
A
0
Tb
Time
A
(c) Polar NRZ
0
-A
A
(d) Unipolar RZ
0
A
(e) Bipolar RZ
0
-A
A
(f) Manchester NRZ 0
-A
Binary Signaling Formats
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Goals of Line Coding

A line code is designed to meet several goals:

Self-synchronization.
• The ability to recover timing from the signal itself.
• Long series of ones and zeros could cause a problem.

Low probability of bit error.
• The receiver needs to be able to distinguish the waveform associated
with a mark from the waveform associated with a space, even if there
is a considerable amount of noise and distortion in the channel.

Spectrum that is suitable for the channel.
• In some cases DC components should be avoided if the channel has a
DC blocking capacitance.
• The transmission bandwidth should be minimized.
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Line Coder

Digital
Data a k
Line Coder

Physical x(t ) 
ak p (t  kTb )
Waveform
k 
 The input to the line encoder is a sequence of values ak that is
a function of a data bit or an ADC output bit.
 The output of the line encoder is a waveform:
x(t ) 

a
k 
k
p (t  kTb )
 Where p(t) is the Pulse Shape and Tb is the Bit Period
 Tb =Ts/n for n bit quantizer (and no parity bits).
 Rb =1/Tb=nfs for n bit quantizer (and no parity bits).
 The operational details of this function are set by the particular
type of line code that is being used.
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Types of Line Codes
 Each line code is described by a symbol mapping function ak
and a pulse shape p(t):
x(t ) 

a
k 
k
p (t  kTb )
 Categories of line codes:

Symbol mapping functions (ak).
• Unipolar
• Polar
• Bipolar (a.k.a. alternate mark inversion, pseudoternary)

Pulse shapes p(t).
• NRZ (Nonreturn-to-zero)
• RZ (Return to Zero)
• Manchester (split phase)
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Unipolar NRZ Line Code

The unipolar nonreturn-to-zero line code is defined by the
unipolar mapping:
 A when X k  1
ak  
 0 when X k  0


where Xk is the kth data bit.
In addition, the pulse shape for unipolar NRZ is:
 t 
p(t )     NRZ pulse shape
 Tb 

Where Tb is the bit period.
Note the DC component
This means wasted power!
1
Hard to recover symbol timing
when long string of 0s or 1s.
0
1
1
0
1
A
0
Tb
2Tb
3Tb
4Tb
5Tb
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Unipolar RZ Line Code

The unipolar return-to-zero line code has the same symbol
mapping but a different pulse shape than unipolar NRZ:
 A when X k  1
ak  
 0 when X k  0
 t 
p(t )   
 RZ pulse shape
 Tb / 2 
Long strings of 1’s no longer a problem.
However strings of 0’s still problem.
Pulse of half the duration of NRZ
requires twice the bandwidth!
1
0
1
1
0
1
A
0
Tb
2Tb
3Tb
4Tb
5Tb
Eeng 360 8
Polar Line Codes

Polar line codes use the antipodal mapping:
 A when X k  1
ak  
  A when X k  0


Polar NRZ uses NRZ pulse shape.
Polar RZ uses RZ pulse shape.
No DC component,
so more energy efficient.
1
0
1
1
0
1
A
Polar NRZ
Now we can handle
long strings of 0’s, too.
A
Polar RZ
A
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Manchester Line Codes

Manchester line codes use the antipodal mapping and
the following split-phase pulse shape:
p (t )
 t  Tb / 4 
 t  Tb / 4 
p(t )   





T
/
2
T
/
2
 b

 b

1
0
1
1
0
1
A
• Easy synchronization and better spectral characteristics than polar RZ.
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Bipolar Line Codes

With bipolar line codes a space is mapped to zero and a mark
is alternately mapped to -A and +A:
0

ak   A
 A

•
•
when X k  0
when X k  1 and last mark   A
when X k  1 and last mark   A
Also called pseudoternary signalling and alternate mark
inversion (AMI).
Either RZ or NRZ pulse shape can be used.
A
1
0
1
1
0
1
Bipolar (RZ)
A
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Comparison of Line Codes
 Self-synchronization:



Manchester codes have built in timing information because
they always have a zero crossing in the center of the pulse.
Polar RZ codes tend to be good because the signal level
always goes to zero for the second half of the pulse.
NRZ signals are not good for self-synchronization.
 Error probability:

Polar codes perform better (are more energy efficient) than
Unipolar or Bipolar codes.
 Channel characteristics:

We need to find the PSD of the line codes to answer this ...
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Power Spectra for Binary Line Codes
 PSD can be calculated using the autocorrelation function:
 A digital signal is represented by
s(t ) 

a
n 
f(t) - Symbol Pulse shape;
Binary signaling : Ts= Tb ,
n
f (t  nTs ) ;
 t 
f (t )     for unipolar NRZ
 Ts 
Ts - Duration of one symbol;
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:
F( f )
Ps ( f ) 
Ts
2

 R ( k )e
 j 2 kfTs
Where { f (t )}  F ( f )
k 
I
R(k )   (an an  k )i Pi
The autocorrelation function of data
i 1
an and an  k are levels of the data pulses at the n ' th and (n  k )'th symbol positions
Pi Probability of having the ith an an  k product
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PSD for Polar NRZ Signaling
Possible levels for the a’s : +A and -A
I
R(k )   (an an  k )i Pi
i 1
an and an  k are the level of the pulses at the nth and (n  k )th symbols
2
R(0)   (an an )i Pi  A2
i 1
1
1
 ( A) 2  A2
2
2
4
For k  0, R(k )   (an an  k ) Pi  A21/ 4  (  A)( A)1/ 4  ( A)(  A)1/ 4  (  A) 21/ 4  0
i 1
 A2 , k  0
 R polar (k )  
0, k  0
sin  fTb
f (t )   (t / Tb )  F ( f )  Tb
 fTb
Ps  f  
F f 
2
Ts
 sin  fTb 
PPolar NRZ ( f )  A Tb 


fT
b




R k  e
2 kfTs
k 
2
2
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PSD for line codes
Unipolar NRZ
2

A2Tb  sin  fTb   1
PUni. NRZ ( f ) 
1


(
f
)

 

4   fTb   Tb

Polar NRZ
 sin  fTb 
PPolar NRZ ( f )  A2Tb 


fT
b


2
Bit rate: R=1/Tb
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PSD for line codes
Unipolar RZ
A2Tb  sin  fTb / 2 
PUni. RZ ( f ) 


16   fTb / 2 
Bipolar RZ
PBipolar RZ ( f ) 
2
 1
1 
 Tb

n 
  ( f  T )
n 
b

2
A Tb  sin  fTb 
2

 sin  fTb 
4   fTb 
2
Manchester
NRZ
2
 sin  fTb / 2 
2
PManch. NRZ ( f )  A2Tb 
 sin  fTb / 2 
  fTb / 2 
Bit rate: R=1/Tb
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