Power Spectral Density of
Convolutional Coded Pulse
Interval Modulation
Z. Ghassemlooy, S. K. Hashemi and M. Amiri
Optical Communications Research Group,
School of Computing, Engineering and Information Sciences,
Northumbria University,
Newcastle, U.K.
Web site: http://soe.unn.ac.uk/ocr
2008, Graz, Austria
1/19
Outline
 Aims and Objectives - Motivations
 Introduction
 DPIM and Convolutional Coded DPIM
 Power Spectral Density of CC-DPIM
 Results
 Conclusions
2008, Graz, Austria
2/19
Aims and Objective – Motivation
 Carry out analysis for the power spectral
density for the convolutional coded DPIM and
investigate:
 Bandwidth efficiency
 DC component.
 Compare the results with both the uncoded
and coded DPIM
2008, Graz, Austria
3/19
Indoor Optical Wireless
Communications
Definition:
 OWC is wireless transmission of light i.e. infrared radiation
through the medium of the air.
Some advantages are:






Higher bandwidth.
Unregulated bandwidth.
Immunity to electromagnetic interference.
High security compared with RF.
Absence of multipath fading (due to the use of IM/DD).
Complementary to RF.
2008, Graz, Austria
4/19
Modulation Techniques
Pulse Time Modulation
Digital
Analogue
Isochronous
PWM
PPM
Anisochronous
PIM
PIWM
PFM
SWFM
Isochronous
PPM
MPPM
DPWM
PCM
2008, Graz, Austria
Anisochronous
DPIM
DPIWM
DH-PIM
DPPM
5/19
Digital Modulation Schemes
Frame 2
Frame 1
0
0
0
0
1
Frame 3
0
1
1
Frame 4
0
1
1
1
Information
DPIM
2008, Graz, Austria
6/19
Digital Pulse Interval Modulation
 DPIM signal is defined :
xt  

a
np
t  nTs 
Source
Data
4-DPIM Symbols
NGS
1GS
n
 p(t) - rectangular pulse
shape,
 Ts - slot duration
 an - set of random variables
representing a pulse/no pulse
in the nth Ts
 L = 2M, hence for M = 2, L = 4
slots.
Lavg
o.5( L  1)


 0.5( L  3)

00
01
10
11
NGB
1GS
2008, Graz, Austria
7/19
DPIM - Convolutional Coding
 Linear block codes like Hamming code, Turbo
code and Trellis coding are difficult (if not
impossible ) to apply in PIM because of variable
symbol length.
 Hence, Convolutional coding
- since it acts on the serial input data rather than
the block.
2008, Graz, Austria
8/19
Convolutional Coding
 Defined as (n,k,K), where k
and n are the input (1) and
output bits (i.e. 2), and K is
the memory element.
 Code rate is defined as k/n
= 1/3.
 Constraint length (K)=3;
 The Generator Function:
 G0 = [111]
 G1 = [101]
Output 1
-1
Z
-1
Z
Data
Sequence
(Ik)
Output 2
2008, Graz, Austria
9/19
Convolutional Coded DPIM
Average symbol length of code data:
 1    [ Lm   ]

P[.] - probability function and
    {L0 , L1 ,, LL 1}
.
For L-DPIM
1
[ L0   ]  L
and

1
1 L
1 L( L  1) L  1
  

 Lavg
L 1
L
2
2
For CC-DPIM symbol length
    {6, 8,, 2( L  2)}
Lave = L + 5
2008, Graz, Austria
10/19
DPIM - Convolutional Coding
 2 empty slots / symbol - to ensure that the
memory is cleared after each symbol.
 Trellis path is limited to 2.
2008, Graz, Austria
11/19
DPIM - Decoder
 Viterbi ‘Hard ‘ decision decoding
 The Chernoff upper bond on the error probability
is:
T ( D, I )
Pb 
I
I  1, D  4 p se (1  p se )
where Pse is the slot error probability of uncoded
DPIM.
It is also possible not use Viterbi algorithm
instead one can use a simple look-up table.
2008, Graz, Austria
12/19
Power Spectral Density
 Generally signals can be divided into two models:
 Deterministic Model - No uncertainty about signal’s time
dependent behaviour at any instance of time.
 Random or Stochastic Model – Uncertain about signal’s
time-dependent behaviour at any instance of time. However
certain on the statistical behaviour of the signal on overall.
 Power of Random Signal
 Deterministic signals - Instantaneous power is x2(t).
 Random signals – There is no single number to associate
with the instantaneous power i.e. x2(t) is a random variable
for each time. The expected instantaneous power of x2(t)
need to obtained.
2008, Graz, Austria
13/19
PSD of CC-DPIM
 A DPIM pulse train may be expressed as [12]:
xc t  

 a pt  nT 
n  
n
s
which is cyclostationary, where p(t) is the rectangular pulse
shape, Ts is the slot duration and an  {0,1} for all n is a
set of random variables that represent the presence or
absence of a pulse in the nth time slot.
 xc(t) can be stationarized with the introduction of a
continuous variable  to give xs(t) = xc(t + ), where  is
equally distributed over [0, Ts] and is independent of an.
The distribution of stationarization depends on the length
probabilities given as:
1
 .
p(k )  Lavg  [ L0   ]
 k
2008, Graz, Austria
14/19
PSD of CC-DPIM

The general expression for the spectral distribution
expressed by the spectral density is given as:

1

2
Rvs ( f )  P( f )  Rc ( fT )   Fc ( f mT )( f  f m )
T
m  


Where



T is the input period of the {an} (the sequence !!),
P(f) is the Fourier transform of p(t), the rectangular pulse shape
|P(f)|2 = T2Sinc2(fT)
2008, Graz, Austria
15/19
PSD of CC-DPIM (Contd.)
 The continuous Spectrum of the CC-DPIM Sequence
{an}is evaluated as:
2

Rc (u )   C ( z )   A( z )  2[ A( z ) B( z ) ] ,


Where z = ei2Πu,  is the greatest common divisor.
 The Discrete part of the spectrum is defined as:
Fc (u m )  A( z m ) , z m  e i 2um , u m  m 
2
Where
A( z )  V ( z )m


B( z )  V ( z )m X ( z  ) z    U  ( z  )


C ( z )  V ( z )   diag [ p ] V ( z 1 )

2008, Graz, Austria
16/19
PSD of CC-DPIM (Contd.)
,
h( z )   p

  1
k
(



k
)
z

X ( z )  h( z ) / g ( z )
k 0

g ( z )   p  z  k

k 0
 
U  ( z )   z k
k 0
p  [ Lm   ]
V ( z )  [1, z , , z  1 ]
2008, Graz, Austria
17/19
PSD of CC-DPIM - Simulation
 8-CC-DPIM using (3-7),
 Pulse shape p(t) - rectangular with 100%
duty cycle.
2008, Graz, Austria
18/19
Results (1)
20
DC level
0
Clock (slot)
-20
Rvs (fT)/T
-40
-60
-80
-100
-120
-140
-160
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Normalised Frequency (fT)
PSD of 8-CC-DPIM with 100% pulse duty cycle against the normalised
frequency: (a) predicted, and (b) simulated
2008, Graz, Austria
19/19
5
Results (2)
20
DC level
Clock (slot)
0
Rvs (fT)/T (dB)
-20
-40
-60
-80
-100
-120
-140
-160
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
Normalised Frequency (fT)
PSD of 8-CC-DPIM with 50% pulse duty cycle against the normalised frequency: (a)
predicted, and (b) simulated
2008, Graz, Austria
20/19
10
Results (1&2) - Observation
 Slot (clock) component - Phase locked loop to recover it
at the receiver.
 The nulls at normalised frequencies (fT)0 = ±1, ±2,… are
poles on the unit circle.
 It is followed by two symmetrically close poles on both
sides at (fT)0 = ±1.5.
 With information on nulls and poles, filter H(z) can be
implemented as an Auto Regressive Moving Average
(ARMA) filter.
 DC level – may result in the baseline wander effect due to
high-pass filtering of the ambient light.
2008, Graz, Austria
21/19
Results (3)- Spectral Comparison
0.25
High DC component
1
0.2
Rvs (fT)/T
0.8
8-DPIM
0.15
8-CCDPIM
0.6
0.1
0.4
0.05
0.2
0
0
0.5
fT
1
0
0
0.5
1
1.5
2
fT
2008, Graz, Austria
22/19
Results (4) - Slot Error Rates
• Higher bit resolution
provides better
performance ( at the
expense of bandwidth)
• The code gain is 0.6
higher for bit
resolution of 5
compared to 3.
2008, Graz, Austria
23/19
Packet Error Rates
Probability of Packet error, PER
8,16,32-DPIM with one guard band@ R =100Mbps
Uncoded8-DPIM
Coded Upper
Bound 8-DPIM
Uncoded 32-DPIM
-4
10
Coded Upper
Bound 32-DPIM
Uncoded
16-DPIM
Coded Upper
Bound 16-DPIM
-6
10
-8
10
-10
10
-12
10
-2 -1 0 1 2 3 4 5
Electrical SNR (dB)
6
7
8
2008, Graz, Austria
24/19
Conclusions
 PSD of CC-DPIM has been derived analytically
based on the stationarisation of variable length
word sequence.
 Close match between predicted and simulated
results.
 Clock components can used for synchronisation.
 DCPIM > DCPPM, more susceptible to baseline
wander
 Convolutional coding has improved PER
performance of DPIM scheme.
2008, Graz, Austria
25/19
Thank You!
2008, Graz, Austria
26/19