2008, Graz, Austria 1/19

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 2/19

Outline

Aims and Objectives - Motivations

Introduction

DPIM and Convolutional Coded DPIM

Power Spectral Density of CC-DPIM

Results

Conclusions

2008, Graz, Austria 3/19

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 4/19

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 5/19

Modulation Techniques

Pulse Time Modulation

Analogue Digital

Isochronous Anisochronous Isochronous AnisochronousPIMPIWMPFMSWFM

PWMPPM

PPMMPPMDPWMPCM

DPIMDPIWMDH-PIMDPPM

2008, Graz, Austria 6/19

Digital Modulation Schemes

Information

Frame 4

1 1 1

Frame 3

1 1 0

Frame 2

0 1 0

Frame 1

0 0 0

DPIM

2008, Graz, Austria 7/19

Digital Pulse Interval Modulation

DPIM signal is defined :

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.

00

01

10

11

4-DPIM Symbols

NGS 1GS

SourceData

GSL

NGBLo

Lavg

1)3(5.0

)1(5.

n

sn nTtpatx

2008, Graz, Austria 8/19

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 9/19

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]

Z-1 Z-1

Output 1

Output 2

Data Sequence

(Ik)Z-1 Z-1

Output 1

Output 2

Data Sequence

(Ik)

2008, Graz, Austria 10/19

Convolutional Coded DPIM

][1

mL

Average symbol length of code data:

P[.] - probability function and },,,{ 110 LLLL

10 ][ LL

For L-DPIM

avg

L

LLLL

LL

2

1

2

)1(11

1

1

and

For CC-DPIM symbol length

)}2(2,,8,6{ L

Lave = L + 5

.

2008, Graz, Austria 11/19

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 12/19

DPIM - Decoder

Viterbi ‘Hard ‘ decision decoding The Chernoff upper bond on the error probability

is:

where Pse is the slot error probability of uncoded DPIM.

)1(4,1

),(

sese ppDII

IDTPb

It is also possible not use Viterbi algorithm instead one can use a simple look-up table.

2008, Graz, Austria 13/19

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 14/19

PSD of CC-DPIM

A DPIM pulse train may be expressed as [12]:

which is cyclostationary, where p(t) is the rectangular pulse shape, Ts is the slot duration and 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:

.

n

snc nTtpatx

}1,0{na

k

avg LLkp

][)( 01

2008, Graz, Austria 15/19

PSD of CC-DPIM

The general expression for the spectral distribution expressed by the spectral density is given as:

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)

mmmccvs ffTfFfTRfP

TfR )()()()(

1)(

2

2008, Graz, Austria 16/19

PSD of CC-DPIM (Contd.)

The continuous Spectrum of the CC-DPIM Sequence {an}is evaluated as:

Where z = ei2Πu, is the greatest common divisor.

The Discrete part of the spectrum is defined as:

Where

,])()([2)()()(2

zBzAzAzCuRc

muezzAuF mui

mmmcm ,,)()( 22

mzVzA )()(

)()()()( zUzzXmzVzB

)(][)()( 1 zVpdiagzVzC

2008, Graz, Austria 17/19

PSD of CC-DPIM (Contd.)

)(/)()( zgzhzX

1

0

)()(k

kzkpzh

0

)(k

kzpzg

0

)(k

kzzU

][ mLp

],,,1[)( 1 zzzV

,

2008, Graz, Austria 18/19

PSD of CC-DPIM - Simulation

8-CC-DPIM using (3-7), Pulse shape p(t) - rectangular with 100%

duty cycle.

2008, Graz, Austria 19/19

Results (1)

PSD of 8-CC-DPIM with 100% pulse duty cycle against the normalised frequency: (a) predicted, and (b) simulated

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-160

-140

-120

-100

-80

-60

-40

-20

0

20

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Normalised Frequency (fT)

Rvs

(fT

)/T

Clock (slot)DC level

2008, Graz, Austria 20/19

Results (2)

0 1 2 3 4 5 6 7 8 9 10-160

-140

-120

-100

-80

-60

-40

-20

0

20

Normalised Frequency (fT)

Rvs

(fT

)/T

(dB

)

PSD of 8-CC-DPIM with 50% pulse duty cycle against the normalised frequency: (a) predicted, and (b) simulated

1 2 3 4 5 6 7 8 9 10

Clock (slot)DC level

2008, Graz, Austria 21/19

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 22/19

Results (3)- Spectral Comparison

0 0.5 10

0.2

0.4

0.6

0.8

1

fT

8-CCDPIM

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

fT

8-DPIM

Rvs

(fT

)/T

High DC component

2008, Graz, Austria 23/19

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 24/19

Packet Error Rates

-2 -1 0 1 2 3 4 5 6 7 8Electrical SNR (dB)

Pro

babi

lity

of P

ack e

t err

or,

PE

R8,16,32-DPIM with one guard band @ R=100Mbps

Uncoded8-DPIM

Coded UpperBound 8-DPIMUncoded 32-DPIM

Coded UpperBound 32-DPIM

Uncoded16-DPIMCoded UpperBound 16-DPIM

10-10

10-8

10-6

10-4

10-12

2008, Graz, Austria 25/19

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 26/19

Thank You!