3F4 Line Coding

1
3F4 Line Coding

• Dr. I. J. Wassell

2
Introduction

• Line coding is the procedure used to convert an
  incoming bit stream, bk to symbols ak which are
  then sent as pulses akhT(t-kT) on to the channel
• The line coding technique affect the properties
  of the transmitted signal
• Desirable properties are
• Self Synchronisation. There should be sufficient
  information in the transitions and zero-crossings
  to permit symbol timing clock regeneration

3
Introduction

• Spectrum suited to the channel. The PSD of the
  transmitted signal should be compatible with the
  channel frequency response Hc(w), eg,
• Many channels cannot pass dc (zero frequency)
  owing to ac coupling
• Lowpass response limits the ability to carry high
  frequencies

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Line Codes for Copper Cables

• Following a qualitative introduction to some
  practical line codes, we will look at a framework
  for their quantitative analysis
• Unipolar Binary
• binary symbols transmitted
• pulse weightings, ak0 or A Volts, eg, 0 and
  1
• no guarantee of transitions (for timing
  regeneration)
• transmitted signal has non-zero mean, so the
  channel must be able to pass dc. If not, errors
  will occur with long runs of 1s in the data

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Unipolar Binary
A
0
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Line Codes for Copper Cables

• Polar Binary (do not confuse with bipolar)
• binary symbols transmitted
• pulse weightings, ak-A or A Volts, eg, 0 and
  1
• no guarantee of transitions (for timing
  regeneration)
• transmitted signal is zero mean (provided data is
  equiprobable), but there is still a high PSD at
  low frequencies, therefore still a problem with
  channels which do not pass dc

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Polar Binary
A
0
-A
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Line Codes for Copper Cables

• Bipolar (or Alternate Mark Inversion (AMI))
• ternary (3 level) symbols transmitted
• pulse weightings, ak0 or either -A or A Volts,
  eg, 0 or 1, that is,
• 0 sent as ak0
• 1 sent as ak-A or A alternately
• no guarantee of transitions (for timing
  regeneration)
• signal mean is always zero (independent of data)
• Spectral null at dc means there is no longer a
  problem if channel does not pass dc
• 1 binary symbol sent as 1 ternary symbol, 1B1T
  code

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Bipolar (AMI)
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Line Codes for Copper Cables

• High Density Bipolar (HDBn)
• modified bipolar codes which guarantee
  transitions despite runs of zeros.
• thus substitute runs of more than n zeros
• an alternative name is BnZS
• in this case substitute runs of n or more zeros
• HDB3 is a popular code
• note HDB3 is equivalent to B4ZS

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HDB3

• Any run of 4 zeros is replaced by the special
  pattern B00D
• D is sent as /- A, such that successive Ds have
  alternating polarity (1st D is arbitrary)
• B is sent as 0 or /- A. Select B such that the
  next D violates the AMI alternating polarity
  result, ie send B as 0 if D violates the polarity
  rule. Otherwise send BD to force a violation of
  the alternating polarity rule

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HDB3

• Equivalently,
• B is sent as 0 if there has been an odd number of
  input 1s since the last special sequence B00D
• B is sent as /- A if there has been an even
  number of input 1s since the last special
  sequence B00D
• All other symbols obey the AMI rules
• The scheme allows the unique detection of the
  special sequences, since polarity violations
  correspond with the D symbols.
• An overall mean of zero is achieved

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HDB3
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HDB3

• There are never more than 3 consecutive zeros, so
  plenty of edges for timing regeneration
• The channel is not required to pass dc
• The transmit power requirement is a little
  greater than AMI (about 10)

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Power Spectra for Line Codes

• We can use the earlier results concerning power
  spectra to derive the PSD for PAM schemes
• Initially we will consider the case where the
  symbols are transmitted as weighted impulses.
• We will then generalise to arbitrary pulse shapes

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Signal TX as impulse train

• A PAM signal sent as a weighted impulse train is,

Where Ts is the symbol period. We will show that
the PSD of x(t) is given by,
and
Which is the discrete Autocorrelation function
(ACF). Also note that R(m) R(-m) for real
valued an
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Proof

• Using direct method, ie,

• The truncated signal xT(t) is,

• Now,

And substituting for xT(t) gives,
18
Proof
Now,
19
Proof

• Now,

20
Proof

• Let,

• Replace the outer sum over index n by 2N1,

21
Proof

• Now,

Where T(2N1)Ts, so
22
Proof
Remembering that R(m) R(-m) for real valued an
23
Proof

• Note that for R(m)0 for all m except zero, the
  PSD reduces to

e.g., for polar binary line coding.
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For Arbitrary Pulse Shapes

• Recall that a PAM signal with a desired pulse
  shape h(t) may be generated by filtering
  (actually convolving) the weighted impulse train
  with a filter whose impulse response is h(t)
• From the power spectra results we obtain,

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PSD of Specific Schemes

• Given a particular line coding scheme which
  generates symbols ak, the transmitted PSD is
  calculated as follows
• Determine the discrete ACF

Where,

• M is the number of possible values that akakm
  can take on
• Ri is the ith value of akakm
• pi is the probability that Ri occurs

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PSD of Specific Schemes

• Evaluate impulse train PSD
• Substitute R(m) into the PSD formula,

• Evaluate PSD with pulse shaping
• Multiply Sx(w) by H(w)2,

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Example- Polar Binary
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Example- Polar Binary

• So,

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Example- Polar Binary

• Now extend to pulses with a rectangular shape and
  duration Ts. To do this we convolve the impulse
  stream with a filter h(t) with a rectangular
  impulse response, i.e., from the E and I Data
  Book,

h(t)
FT
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Example- Polar Binary

• In our example, bTs and a1, so the filter
  frequency response is,

This response has an amplitude of Ts at w 0 and
zero crossings at multiples of 2p/Ts rad/s or
1/Ts Hz.
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Example- Polar Binary

• Now the PSD at the output of the filter H(w) is,

This response has an amplitude of Ts at w 0 and
zero crossings at multiples of 2p/Ts rad/s or
1/Ts Hz. This is consistent with the PSD plot for
polar binary signalling with rectangular pulses.
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Example- Polar Binary
PSDTs
PSDTs
f Ts
f Ts
Impulse Train PSD
Rectangular Pulse PSD
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Example- Bipolar (AMI)
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Example- Bipolar (AMI)

• So,

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Example- Bipolar (AMI)

• Note that if rectangular pulses are employed as
  in the previous Polar example, the resulting PSD
  is given by,

Where,
Note the zero crossings at multiples of 2p/Ts
rad/s or 1/Ts Hz and the lowering of the
frequency sidelobes evident in the PSD plot.
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Example- Bipolar (AMI)
PSDTs
PSDTs
f Ts
f Ts
Impulse Train PSD
Rectangular Pulse PSD
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ACF With Non-Zero Mean Data

• Sometimes the line coded data will have a
  non-zero mean value.
• This could be due to
• The line coding scheme, e.g., unipolar
• The probability distribution of the data
• Incorrect signalling voltages owing to faults
• The result is that the R(m) will have finite
  values for m in the range /- infinity

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ACF With Non-Zero Mean Data

• The result for Sx(w) is valid but is hard to
  interpret physically
• The solution is to express R(m) as the sum of two
  parts
• The first part has a constant value of R over the
  range of m from to - infinity
• The second part has non-zero values of R over a
  finite range of m

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ACF With Non-Zero Mean Data

• The advantage of this approach is that the first
  term can be represented in a format that is
  consistent with physical observations.
• We will now show that this representation
  comprises a sequence of spikes (impulses) in the
  frequency domain, occurring at multiples of the
  bit rate.

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ACF With Non-Zero Mean Data

• Suppose that R(m)R for all m, then we can
  express the PSD as follows

• Note the similarity in form to the Fourier series
  representation of an impulse train,

Note in time domain
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ACF With Non-Zero Mean Data

• Reminder, the Fourier Series (FS) for a periodic
  function is,

i.e., a weighted (complex) sum of phasors.

• We now wish to find the FS of the rectangular
  pulse train x(t),

t
x(t)
A
0
t
To
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ACF With Non-Zero Mean Data

• Now the coefficients are given by,

For the case where t goes to zero and At 1
(i.e., a unit impulse train),
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ACF With Non-Zero Mean Data

• So we know,

Remembering that ck1/T0 when x(t) is a sequence
of unit impulses we can write,
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ACF With Non-Zero Mean Data

• If we substitute, wt, mk and T02p/Ts, to get
  the equivalent relation in the frequency domain,

And hence we may express Sx(w) as a series of
impulses in the frequency domain,
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ACF With Non-Zero Mean Data

• The PSD due to a constant R for all m consists of
  spikes, known as line spectral components, at
  multiples of wo2p/Ts
• This result enables simplified calculation of
  PSDs when R(i) can be written in the form,

First calculate the PSD for R(i)C(i), then add
on the line spectrum for R(i)R
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Example- Unipolar Binary
47
Example- Unipolar Binary

• So,

The second term is known as a line spectrum
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Example- Unipolar Binary
PSDTs
PSDTs
f Ts
f Ts
Impulse Train PSD
Rectangular Pulse PSD
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Scrambling

• Some components of a transmission system work
  better if the bit sequence bk is random, ie,
  independent and uncorrelated
• Timing regeneration not possible with some line
  codes if long sequences of 0s or 1s occur
• Equalisers (see next section) rely on random bit
  sequences for successful operation
• A common solution is to randomise (or scramble)
  the input bit sequences (in a known manner) prior
  to line coding

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Scrambling

• The received signals then appear as if they come
  from a random source
• Unscrambling after detection restores the correct
  bit sequence
• A simple way to generate scramblers is to XOR the
  data with the output of a feedback shift register
  arrangement.

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Scrambling

• For example we may utilise the so called
  frame-synchronised scrambler (also called a
  cryptographic scrambler

PRBS- Pseudo Random Binary Sequence generator.
Usually a Maximal Length (ML) linear feedback
shift register arrangement.
PRBS at TX and RX must be synchronised
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Scrambling

• An alternative is the Self-Synchronised
  scrambler. Is susceptible to error propagation.

• Similar arrangements used for error rate testing.

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Summary

• In this section we have
• Examined the need for line codes and looked at
  some practical examples
• Seen how to calculate the power spectra for
  various line codes
• Briefly looked at scrambling to randomise data