Line Codes

Presentation in course TTT4130 Digital Communication Line codes and scrambling
Curriculum found in Barry, Lee, Messerschmitt Chapter 19 (19.1, 19.3 and 19.5)
IET, NTNU Digital Communication, Line codes and scrambling
Line coding
The purpose of a line code is to match the output signal to the channel for baseband transmission. The line coder consists of a mapping of bits into symbols (nonlinear) and a pulseshaping lter (linear).
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Properties of line codes

Power spectrum Timing recovery properties DC-balance (zero DC component)
Redundancy Linearity Polarity independence
Efciency and redundancy

Maximum bitrate: R = fb lb(L) (lb = log2) L: no of output symbol levels fb: symbol rate in symb/s B: actual bitrate in bit/s Code efciency: Redundancy: = B/R r = 1IET, NTNU Digital Communication, Line codes and scrambling 4
Example, 4B3T code

The 4B3T code is coding 4 input bits to 3 three-level symbols (24 = 16 combinations of input and 33 = 27 potential output
combinations).
Maximum bitrate: R = fb lb 3 Actual bitrate: Code efciency: Redundancy: B = fb 4/3 = R/B = 4/(3lb 3) = 0.84 r = 1- 0.84 = 0.16
Linear codes
Input symbols are ak = 1 and pulseshapes are dened in Fig 19.2
Biphase has zero DC component in each pulse RZ and NRZ may have a DC component
Biphase and RZ has transitions in each pulse ): good timing properties NRZ may have long sequences of +1 or -1 ): poor timing properties
Coded sequences
AMI code (Alternate Mark Inversion)

ak = 0 bk = 0 1 bk = 1 (alternating 1 and + 1)
The AMI code may be generated as shown in the gure:

bk = 0 : c k = c k 1 ak = 0 bk = 1: c k c k 1 ak = 1 c k changes between 0 and 1 each time bk = 1
+ zero DC component - may contain long sequences of 0 (poor timing recovery)

DC-balance
Most transmission systems will be a.c. coupled. In metallic cables a transformer is used to avoid overvoltages. A.c. coupling may also be used due to implementation aspects. A.c. coupling means that d.c. is ltered out in a high-pass lter as shown in Fig 19.1a. This causes a slowly varying InterSymbol Interference that is denoted baseline wander. A Nyquist channel + a high pass lter in cascade will give a model for ISI as shown in Fig 19.1b.
= 2RC = exp(-T/ )
Running digital sum

By assuming = 1, the overall time discrete impulse response may be approximated: 0 k <0
pk = p(kT) = 1 T/
am pk
m= k
k =0 k >0
m
Output signal at time k:
yk =
= ak
k 1
am = ak
m=
Sk
Running digital sum, RDS:
Sk =
m=
am
ISI(k) = T Sk
1
Intersymbol interference at time k:

max(ISI) = T min(Sk 1 ), max(ISI)
for independent ak
For a balanced line code RDS is bounded, and the DC content is zero.
Impulse response for DC cutoff,

simulation example illustrating baseline wander
Cosine rolloff 100% + H( f ) = j j +2 0.02
3 dB cutoff frequency: 0.02/T =T/(2 0.02)=7.96T
Eye curve for NRZ code with DC cutoff

Cosine rolloff 100% + j H( f ) = j + 2 0.02
NRZ is an unbalanced line code ): serious baseline wander
Eye curve for AMI code with DC cutoff
Cosine rolloff 100% + H( f ) = j j +2 0.02
AMI is a balanced line code ): small ISI
Power spectrum of a line code

Transmitted signal:
X(t) =
k=
ak g(t kT)
Power spectrum:
SX ( f ) =
1 Sa (e j 2 T
fT
) G( f )
Sa(z) : Time discrete power spectrum of sequence {ak}
Power spectrum NRZ code
Power spectrum Biphase
Power spectrum AMI code
Power spectrum for three simple codes
NB: All codes have equal average power
Comparison of simple codes

Code RZ NRZ Biphase AMI Balanced no no yes yes
Timing properties Minimum bandwidth
good poor good poor
1/(2T) 1/(2T) 1/T 1/(2T)
Ternary (3-level) codes

An kBnT block code is coding k bits into n 3-level symbols
One alternative of a 4B3T code
Binary codes
Binary codes are relevant for optical transmission. The number of balanced codewords of length n is given by:
N= n! (n /2)! (n /2)!
Corresponding codes are shown in the table
Due to low efciency, nBmB codes are used instead, where m = n +1 (e.g. 5B6B)
Partial response
ck = 1 yk
F(z)
F(z) = (1 - z-1)m (1 + z-1)n
Alternatives of partial response

Type partial response Dicode Duobinary Modied duobinary n= 0 1 1 m= 1 0 1 F(z)= 1-z-1 1+z-1 (1+z-1)(1-z-1)=1-z-2
1 z 1 = 1 exp( j2 fT) = 2 j exp( j fT) sin( fT), ) : zero at f = 0 1+ z 1 = 1+ exp( j2 fT) = 2 exp( j fT) cos( fT), ) : zero at f = 1/(2T)
Duobinary partial response

yk = ck + ck-1
ck -1 -1 1 1 ck-1 -1 1 -1 1 Output yk -2 0 0 2
For duobinary, a two-level signal is coded into three levels. Modied duobinary and dicode will also have three-level output for binary input. Other alternatives of partial response will have more output levels.
Spectral shaping by partial response
Decoding of partial response

?
ck - ck-1 ck-1 ck - ck-1 ck-1
ck - ck-1
Precoding of partial response (duobinary)

Precoder
bk = 0,1
dk = 0,1
Convert to bipolar
Duobinary coding
0 to -1 1 to +1
ck = 1 z-1
ak
3-level decision
bk
z-1 bk = 0 dk = dk bk = 1 dk dk
1 1
ck = ck ck ck
1 1
ak = c k + c k 1 = 2 bk = 0 ak = c k + c k 1 = 0 bk = 1
Decoding depends only on current received signal sample ): no error propagation. Precoding may be used also for modied duobinary and dicode.
Scrambling
Most transmission systems are based on an assumption of random input data ): statistically independent bits and p(0) = p(1) = 1/2 Practical input data often differ from this assumption and may for instance contain long strings of ones or zeroes Scrambling is a way to ensure approximately random data Scrambling is based on pseudorandom sequences generated by linear feedback shift registers.
Linear feedback shift register

Number of states in a shift register of length n: 2n The output sequence is periodic, and for a maximum length shift register the period is: 2n-1
Frame-synchronized scrambler
Assuming error free transmission:
bk = c k
xk = ck
x k = bk
xk
x k = bk
- The transmitter and receiver shift registers have to be synchronized + One bit error per transmission error
Self synchronized scrambler

Assuming error free transmission:
bk = c k
dk = c k
dk = bk
dk
dk = bk
+ Automatic synchronization - Minimum three bit errors per transmission error

Applications of scrambling
Echo cancelling
Two different scramblers, one for each direction of transmission
Synchronization Adaptive equalization Interference cancellation ...

IET, NTNU 33
Digital Communication, Line codes and scrambling
Maximum length shift registers
Table of maximum length shift registers

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Presentation in course TTT4130 Digital Communication Line codes and scrambling

IET, NTNU Digital Communication, Line codes and scrambling

Properties of line codes

Efciency and redundancy

Example, 4B3T code

IET, NTNU Digital Communication, Line codes and scrambling 7

AMI code (Alternate Mark Inversion)

The AMI code may be generated as shown in the gure:

+ zero DC component - may contain long sequences of 0 (poor timing recovery)

IET, NTNU Digital Communication, Line codes and scrambling 9

Running digital sum

Output signal at time k:

Running digital sum, RDS:

Intersymbol interference at time k:

Impulse response for DC cutoff,

Cosine rolloff 100% + H( f ) = j j +2 0.02

3 dB cutoff frequency: 0.02/T =T/(2 0.02)=7.96T

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Eye curve for NRZ code with DC cutoff

NRZ is an unbalanced line code ): serious baseline wander

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Eye curve for AMI code with DC cutoff

Cosine rolloff 100% + H( f ) = j j +2 0.02

AMI is a balanced line code ): small ISI

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Power spectrum of a line code

Sa(z) : Time discrete power spectrum of sequence {ak}

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Power spectrum NRZ code

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Power spectrum Biphase

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Power spectrum AMI code

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Power spectrum for three simple codes

NB: All codes have equal average power

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Comparison of simple codes

good poor good poor

1/(2T) 1/(2T) 1/T 1/(2T)

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Ternary (3-level) codes

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One alternative of a 4B3T code

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Corresponding codes are shown in the table

F(z) = (1 - z-1)m (1 + z-1)n

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Alternatives of partial response

Duobinary partial response

Spectral shaping by partial response

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Decoding of partial response

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Precoding of partial response (duobinary)

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Linear feedback shift register

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Self synchronized scrambler