Amiri, Seyedeh Maryam

Ghassemlooy, Zabih

Hashemi, Seyyedkamal

English

Northumbria University Research Portal

Spectral Analysis of Convolutional Coded DPIM for Indoor Optical Wireless Communications    Z. Ghassemlooy, Fellow IET, Senior member IEEE, S. K. Hashemi and M. Amiri  Optical Communications Research Group, NCRLab Northumbria University, Newcastle upon Tyne, UK E-mail: fary.ghassemlooy@unn.ac.uk, seyyedkamal.hashemi@unn.ac.uk, maryam.amiri@unn.ac.uk   Abstract—Power spectral density (PSD) of the convolutional coded digital pulse interval modulation (CC-DPIM) is derived analytically based on the stationarization of variable-length word sequence (VLWS). To verify the theoretical model developed, CC-DPIM is simulated in Matlab and its spectrum is compared with the predicted results showing a close match. As in DPIM scheme, CC-DPIM spectrum also contains a distinct slot (clock) component which can be recovered at the receiver for both slot and symbol synchronization.  I. INTRODUCTION A number of modulation techniques have been proposed and studied for use in indoor as well as outdoor optical wireless communication links to meet the key requirements of power and bandwidth efficiencies. Equally powerful characteristics when selecting a modulation scheme in particular pulse modulation schemes is the power spectral density (PSD). Spectral evaluation of a data sequence relates to the statistical and stationary properties of the transmitted waveform. For a fixed length encoded messages such as the on-off keying (OOK) and the pulse position modulation (PPM), and the variable length digital pulse interval modulation (DPIM) the PSDs have been well studied and reported in the literature [1-6]. In [7] CC-DPIM scheme capable of detecting and correcting errors proposed for the diffuse indoor optical wireless links have been investigated. However, to the best of our knowledge no work has been reported on the spectral characteristics of the CC-DPIM scheme. In this paper for the first time, we theoretically investigate the PSD of the proposed CC-DPIM and verified the approach adopted by means of Matlab simulation.  Methods of evaluating the PSD of a signal include the periodogram technique to compute the spectral characteristics of dual header pulse interval modulation (DHPIM) as outlined in [8]. The autocorrelation function of a cyclostationary process for the DPPM scheme has been calculated to estimate the spectrum of the transmitted pulse stream in [9]. The method of stationarization is also evaluated in [10] and the theory of variable length random processes (or VLWS) has been developed in [11, 12] where stationarization of the vectors with random lengths is achieved by considering the vector length probabilities. The rest of this paper is organized as follows:  Section II outlines the basic concept of DPIM and CC-DPIM and shows their corresponding symbol structures.  Section III outlines the theoretical approached adapted to evaluate the spectral property of CC-DPIM. In section IV spectral plots and discusses are given, whereas in section V the concluding remarks are presented.  II. SYMBOL STRUCTURE     DPIM is a modulation technique in which each block of log2L data bits {dm} is mapped to one of L= 2M possible symbols }{ mD , each different in length with the mth symbol having the structure of ],,[110  mLmmmm DDDD  .  Each symbol of variable length Lm begins with a pulse of one slot duration (or “1”, i.e. 10 mD  ) followed by a number of empty slots (or “0”, i.e. mkm LkD  0,0 ) corresponding to the decimal value of the block of data bits being encoded. The mapping of data bits into 4-DPIM using no guard slot (GS) is depicted in Table I.    The average symbol length of the coded data   ][1 mL, where [.]  is the probability function and },,,{ 110  LLLL  .  For L-DPIM with no GS with identically independent distribution (i.i.d), 10 ][ LL , and avgLLLLLLL 212)1(1111 . The slot duration defined in terms of data bit duration is  avgs LLTT 2log .     With a code rate of ½ and a constraint length of 3, the convolutional coded DPIM with 2GSs is also shown in Table I, where each symbol start with a unique pattern of {11 10 11} followed by a number of zeros representing the input data as outlined in [7]. For CC-DPIM with ML 2  symbols, the symbol length is )}2(2,,8,6{  L  and the average symbol length with an i.i.d is L + 5. Hence, 1)5(  L . Table I illustrates the symbol structure for the DPIM and CC-DPIM schemes.   978-1-4244-1876-3/08/$25.00 ©2008 IEEECSNDSP08 - 102 - ProceedingsTable I. DPIM and CC-DPIM symbol structures for 2-bit input data   III. POWER SPECTRAL DENSITY A DPIM pulse train may be expressed as [12]:    nsnc nTtpatx     (1) which is cyclostationary, where p(t) is the rectangular pulse shape, Ts is the slot duration and }1,0{na  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: kavg LLkp][)( 01.    (2) The general expression for the spectral distribution expressed by the spectral density is given as: [11, 12]   mmmccvs ffTfFfTRfPTfR )()()()(1)(2  (3)  where T is the input period of the {an}, P(f) is the Fourier transform of p(t), and  )()( 222fTSincTfP  .      The continuous spectrum of }{ na  is defined as:  uicezzBzAzAzCuR22])()([2)()()(   (4)  and the discrete part of the spectrum (spectral lines) is given by:   muezzAuF muimmmcm ,,)()(22  (5)  and by considering equations (20) from [12], we can obtain for CC-DPIM:   5421)( zzzzzA          )()()()( zUzzXmzVzB (6) )1)(1()( 5421542   zzzzzzzzzC Where )(/)()( zgzhzX  ,     10)()(kkzkpzh ,    (7)  0)(kkzpzg ,  0)(kkzzU . ][   mLp ,  ],,,1[)( 1  zzzV  ,  and   is the greatest common divisor.    The mean vector  pLDm mm  ];[ , and   is the set of words with a length defined as:      110 ,, N ,    (8) where ],,,[ 110   nnnn   , }1,,1,0{  Nn  , where  N is the number of words with length  , where for CC-DPIM and DPIM, N  = 1. Since the symbol length is even then 2 .  IV. RESULTS AND DISCUSSION Figure 1(a) shows the predicted PSD of the 8-CC-DPIM using (3-7), assuming that pulse shape p(t) is rectangular with 100% duty cycle. The spectrum displays a sinc function profile with a distinctive clock (slot) frequency and its harmonics at odd multiples, and nulls at even multiples of the slot frequency. The nulls at normalized frequencies ,2,1)( 0 fT  are poles on the unit circle, followed by two symmetrically close poles on both sides at 15.0)( 0  fTfT . The spectral density (4) is a rational function of z and can be factored to two casual and stable functions of z  and 1z  as )()()( 1 zHzHzRc . To build )(zH one can derive poles and zeros of the PSD and extract those inside unit circle ( 1iz ). Having this information we can build filter )(zH which can be implemented as an Auto Regressive Moving Average (ARMA) filter.  The presence of the slot component suggest that a simple phase locked loop could be employed to recover the clock signal form the incoming CC-DPIM symbol stream for slot synchronization at the receiver. Unlike PPM, the PSD does not fall to zero at DC region similar to the DPIM. Thus, implying that it too would also be susceptible to the effect of baseline wander due to high-pass filtering of the ambient light at the receiver [13-15]. To confirm the theoretical analysis, the proposed scheme was simulated in Matlab and its estimated PSD is depicted in Figure 1(b), showing a close match to the predicted PSD.   M-bit Input data, {dm} 4-DPIM, Dm of length mL   NGS 2GS CC-DPIM (2GS) 00 01 10 11 1 1 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0   11 10 11 11 10 11 00 11 10 11 00 00 11 10 11 00 00 00  Proceedings - 103 - CSNDSP080 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-160-140-120-100-80-60-40-20020Rvs(fT)/T dBfT (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-160-140-120-100-80-60-40-20020Magnitude (dB)Normalized Frequency (b) Figure 1.  PSD of 8-CC-DPIM with 100% pulse duty cycle against the normalized frequency: (a) predicted, and (b) simulated  Figure 2 shows the PSD plots for the CC-DPIM with the header pulses having 50% duty cycle. The profile is the same as in Fig. 1 except for the first zero crossing being shifted to the right. In both figures, the slight difference in the profiles is due to the fact that in the theoretical analysis both the discrete and continuous components of the spectrum are separated while in the simulation these components are combined.  Figure 3 compares the spectrum of 8-CC-DPIM and 8- DPIM. For the normalized frequency u = 0, we can compute A(z) = 5, C(z) = 25 and B(z) = -1.8047. By incorporating these values into (3) the DC component  Rvs(0)/T0= 0.8307 is obtained as shown in Fig. 3. The DC component is 14.4219 times as high as the 8-DPIM DC value of 0.0576. Note the PPM scheme has no DC component, a desirable characteristics in optical wireless communications.  In most scenarios, the data sequence which generates the pulse stream is either wide sense stationary (WSS) or wide sense cyclostationary (WSCS). In both cases the timing jitter induced by the noise while propagating through the channel will introduce frequency offset, thus  0 1 2 3 4 5 6 7 8 9 10-160-140-120-100-80-60-40-20020Rvs(fT)/T dBfT (a) 0 1 2 3 4 5 6 7 8 9 10-160-140-120-100-80-60-40-20020Magnitude (dB)Normalized Frequency (b) Figure 2.  PSD of 8-CC-DPIM with 50% pulse duty cycle against the normalized frequency: (a) predicted, and (b) simulated affecting the continuous components of the spectrum. This will result in a more complex clock recovery scheme at the receiving end. The discrete components (delta functions (5)) corresponding to the mean of the sequence, are due to the data asymmetry. This can be reduced or eliminated if the mean is zero. For bipolar input data }1,1{ na the discrete components are no longer present in spectrum.   V. CONCLUSION Power spectral density of the convolutional coded digital pulse interval modulation was derived analytically based on the stationarization of variable-length word sequence. The predicted and simulated spectrums were compared showing a close match. The spectrum contains a slot (clock) component which could be extracted at the receiver for slot and symbol synchronizations, but it also has a DC component which is higher than that of DPIM and PPM. CSNDSP08 - 104 - Proceedings0 0.5 100.20.40.60.81Rvs(fT)/TfT8-CCDPIM0 0.5 1 1.5 200.050.10.150.20.25fT8-DPIM Figure 3.  Spectrum comparison between DPIM and CC-DPIM. APPENDIX DERIVATION OF (6)  The probability of two words }{ mD  has the distance of k digits in the symbol sequence }{ na  can be introduced by the cumulative length probabilities which satisfy the following difference equation:      0),()( kkypky                   (9) the sequence )( ky  converges to )(y  and the sequence )()()(  ykykx  is absolutely summable with z-Transform:   0 )()()]()([)(kkzgzhzykyzX                (10) The continuous spectrum of the sequence }{ na  is evaluated by the equation (4) where:    mzVzA )()(         )()()()( zUzzXmzVzB              (11) )(][)()( 1 zVpdiagzVzC    where ‘ is the transpose and diag is the diagonal matrix.  For CC-DPIM:  ]00110111[  , 51L , 2  Lp1   (All symbols are equally likely)  LpLDm mm1];[   ],,,1[)( 1  zzzV    Hence, by using these into (11) we can obtain (6).  REFERENCES [1] Z. Ghassemlooy, A. R. Hayes, N. L. Seed, and E. D. Kaluarachchi, “Digital pulse interval modulation for optical communications,” IEEE Commun. Mag., Vol. 36, No. 12, pp. 95-99, 1998. [2] D. F. Bishop, S. Million, T.M. Nguyen, M. K. Simon, “Power spectrum of unbalanced NRZ and biphase signals in thepresence of data asymmetry,” IEEE Trans. On Electromag.  Compatibility, Vol. 40, No. 1, pp. 55 – 61, 1998. [3] Ho Keang-Po, J. M. Kahn, “Spectrum of externally modulated optical signals,” J. Lightwave Technol., Vol. 22, No. 2, pp.658-663, 2004. [4] C. Pimentel, and V. C.  da Rocha, “ On the power spectral density of constrained sequences,“ IEEE Trans. Commun., vol. 55, no. 3, pp. 409-416, 2007. [5] M. Z. Win, “On the power spectral density of digital pulse streams generated by M-ary cyclostationary sequences in the presence of stationary timing jitter,” IEEE Trans. Commun., Vol. 46, No. 9, pp. 1135-1145, 1998. [6] Y. Han, “Theoretical sensitivity of directed-detection multilevel modulation formats for high spectral efficiency optical communications,” IEEE J. Select. Top. Quant. Electron., Vol. 12, No. 4, pp. 571-580, 2006. [7] S. Rajbhandari, Z. Ghassemlooy, N. M. Aldibbiat, M. Amiri, and  W. O. Popoola, “Convolutional coded DPIM for indoor non-diffuse optical wireless link, “ The 7th IASTED Inter. Conf. on Wireless and Optical Commun. (WOC 2007), Canada, pp. 286-290, 2007. [8] N. M. Aldibbiat, Z. Ghassemlooy and R. McLaughlin, "Spectral characteristics of Dual Header - Pulse Interval Modulation (DH-PIM)," IEE Proc. Commun., Vol. 148, No. 5, pp. 280-286, 2001. [9] D. Shiu and J. M. Kahn, “Differential pulse position modulation for power-efficient optical communication,” IEEE Trans. Commun., Vol. 47, No. 8, pp. 1201–1210, 1999. [10] H. L. Hurd, “Stationarizing properties of random shifts,” SIAM J. Appl. Math, Vol. 26, No. 1, pp. 203–212, 1975. [11] G. L. Cariolaro and G. L. Pierobon, “Stationary symbol sequences from variable-length word sequences,” IEEE Trans. Inform. Theory, Vol. IT-23, pp. 243–253, 1977. [12] G. Cariolaro, T. Erseghe, L. Vangelista, “Exact spectral evaluation of the family of digital pulse interval modulated signals,” IEEE Trans. Info. Theory, Vol. 47, No. 7, pp. 2983 – 2992, 2001. [13] .Z. Ghassemlooy, and N. M. Aldibbiat, “Baseline wander effect on indoor wireless infrared links operated by dual header pulse interval modulation”, The Mediterranean Journal of Electronics and Communications, Vol. 1, No. 1, Oct. 2005. [14] A. R. Hayes, Z. Ghassemlooy, and N. L. Seed, “Investigation of base line wander in DPIM”, IEE Proc. J., Vol. 147. No. 4, pp. 295-300, 2000. [15] Z. Ghassemlooy, “Investigation of the baseline wander effect on Indoor optical wireless system employing digital pulse interval modulation”, to appear in the IET Communications, Vol. 2, No. 1, pp. 53-60, 2008. Proceedings - 105 - CSNDSP08

Baseline wander effect on indoor wireless infrared links operated by dual header pulse interval modulation”,

Convolutional coded DPIM for indoor nondiffuse optical wireless link,

Differential pulse position modulation for power-efficient optical communication,”

Digital pulse interval modulation for optical communications,”

Exact spectral evaluation of the family of digital pulse interval modulated signals,”

Investigation of base line wander in DPIM”,

Investigation of the baseline wander effect on Indoor optical wireless system employing digital pulse interval modulation”, to appear in

On the power spectral density of constrained sequences,“

On the power spectral density of digital pulse streams generated by M-ary cyclostationary sequences in the presence of stationary timing jitter,”

Power spectrum of unbalanced NRZ and biphase signals in thepresence of data asymmetry,”

Spectral characteristics of Dual Header - Pulse Interval Modulation (DHPIM),&quot;

Spectrum of externally modulated optical signals,”

Stationarizing properties of random shifts,”

Stationary symbol sequences from variable-length word sequences,”

Theoretical sensitivity of directed-detection multilevel modulation formats for high spectral efficiency optical communications,”

Spectral analysis of convolutional coded DPIM for indoor optical wireless communications

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Citation: Ghassemlooy, Zabih, Hashemi, Seyyedkamal and Amiri, Seyedeh Maryam (2008) Spectral 
analysis of convolutional coded DPIM for indoor optical wireless communications. In: Proceedings of 
Communication Systems,  Networks  and Digital  Signal  Processing,  2008.  (CNSDSP 2008).  IEEE, 
Piscataway, NJ, pp. 102-104. ISBN 9781424418763 
Published by: IEEE
URL:  http://dx.doi.org/10.1109/CSNDSP.2008.4610820 
<http://dx.doi.org/10.1109/CSNDSP.2008.4610820>
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Spectral Analysis of Convolutional Coded DPIM 
for Indoor Optical Wireless Communications 
 
 
 
Z. Ghassemlooy, Fellow IET, Senior member IEEE, S. K. Hashemi and M. Amiri  
Optical Communications Research Group, NCRLab 
Northumbria University, Newcastle upon Tyne, UK 
E-mail: fary.ghassemlooy@unn.ac.uk, seyyedkamal.hashemi@unn.ac.uk, maryam.amiri@unn.ac.uk 
 
 
Abstract—Power spectral density (PSD) of the convolutional 
coded digital pulse interval modulation (CC-DPIM) is 
derived analytically based on the stationarization of 
variable-length word sequence (VLWS). To verify the 
theoretical model developed, CC-DPIM is simulated in 
Matlab and its spectrum is compared with the predicted 
results showing a close match. As in DPIM scheme, CC-
DPIM spectrum also contains a distinct slot (clock) 
component which can be recovered at the receiver for both 
slot and symbol synchronization.  
I. INTRODUCTION 
A number of modulation techniques have been 
proposed and studied for use in indoor as well as outdoor 
optical wireless communication links to meet the key 
requirements of power and bandwidth efficiencies. 
Equally powerful characteristics when selecting a 
modulation scheme in particular pulse modulation 
schemes is the power spectral density (PSD). Spectral 
evaluation of a data sequence relates to the statistical and 
stationary properties of the transmitted waveform. For a 
fixed length encoded messages such as the on-off keying 
(OOK) and the pulse position modulation (PPM), and the 
variable length digital pulse interval modulation (DPIM) 
the PSDs have been well studied and reported in the 
literature [1-6]. In [7] CC-DPIM scheme capable of 
detecting and correcting errors proposed for the diffuse 
indoor optical wireless links have been investigated. 
However, to the best of our knowledge no work has been 
reported on the spectral characteristics of the CC-DPIM 
scheme. In this paper for the first time, we theoretically 
investigate the PSD of the proposed CC-DPIM and 
verified the approach adopted by means of Matlab 
simulation.  
Methods of evaluating the PSD of a signal include the 
periodogram technique to compute the spectral 
characteristics of dual header pulse interval modulation 
(DHPIM) as outlined in [8]. The autocorrelation function 
of a cyclostationary process for the DPPM scheme has 
been calculated to estimate the spectrum of the 
transmitted pulse stream in [9]. The method of 
stationarization is also evaluated in [10] and the theory of 
variable length random processes (or VLWS) has been 
developed in [11, 12] where stationarization of the 
vectors with random lengths is achieved by considering 
the vector length probabilities. 
The rest of this paper is organized as follows:  Section 
II outlines the basic concept of DPIM and CC-DPIM and 
shows their corresponding symbol structures.  Section III 
outlines the theoretical approached adapted to evaluate 
the spectral property of CC-DPIM. In section IV spectral 
plots and discusses are given, whereas in section V the 
concluding remarks are presented. 
 
II. SYMBOL STRUCTURE 
    DPIM is a modulation technique in which each block 
of log2L data bits {dm} is mapped to one of L= 2
M
 
possible symbols }{ mD , each different in length with the 
m
th
 symbol having the structure of 
],,[
110  m
L
mmmm DDDD  .  Each symbol of variable 
length Lm begins with a pulse of one slot duration (or “1”, 
i.e. 10 mD  ) followed by a number of empty slots (or 
“0”, i.e. 
m
k
m LkD  0,0 ) corresponding to the 
decimal value of the block of data bits being encoded. 
The mapping of data bits into 4-DPIM using no guard 
slot (GS) is depicted in Table I. 
   The average symbol length of the coded data 
 

 ][1 mL
, where [.]  is the probability 
function and },,,{ 110  LLLL  .  
For L-DPIM with no GS with identically independent 
distribution (i.i.d), 1
0 ][
 LL , and 
avg
L
L
LLL
LL




 
2
1
2
)1(11
1
1 . The slot 
duration defined in terms of data bit duration is  
avgs LLTT 2log .  
   With a code rate of ½ and a constraint length of 3, the 
convolutional coded DPIM with 2GSs is also shown in 
Table I, where each symbol start with a unique pattern of 
{11 10 11} followed by a number of zeros representing 
the input data as outlined in [7]. For CC-DPIM with 
ML 2  symbols, the symbol length 
is )}2(2,,8,6{  L  and the average symbol 
length with an i.i.d is L + 5. Hence, 
1)5(  L . 
Table I illustrates the symbol structure for the DPIM 
and CC-DPIM schemes.  
 
978-1-4244-1876-3/08/$25.00 ©2008 IEEE
CSNDSP08 - 102 - Proceedings
Table I. 
DPIM and CC-DPIM symbol structures for 2-bit input data 
 
 
III. POWER SPECTRAL DENSITY 
A DPIM pulse train may be expressed as [12]: 
   



n
snc nTtpatx     (1) 
which is cyclostationary, where p(t) is the rectangular 
pulse shape, Ts is the slot duration and }1,0{na  for all 
n is a set of random variables that represent the presence 
or absence of a pulse in the n
th
 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: 




k
avg LLkp

][)( 0
1
.    (2) 
The general expression for the spectral distribution 
expressed by the spectral density is given as: [11, 12] 
 






 

m
mmccvs ffTfFfTRfP
T
fR )()()()(
1
)(
2  (3) 
 
where T is the input period of the {an}, P(f) is the Fourier 
transform of p(t), and  )()( 22
2
fTSincTfP  . 
 
    The continuous spectrum of }{ na  is defined as: 
 
ui
c
ez
zBzAzAzCuR
2
2
])()([2)()()(

   (4) 
 
and the discrete part of the spectrum (spectral lines) is 
given by: 


muezzAuF m
ui
mmmc
m ,,)()(
22  (5) 
 
and by considering equations (20) from [12], we can 
obtain for CC-DPIM:  
 
5421)( zzzzzA      
   



 )()()()( zUzzXmzVzB
 (6) 
)1)(1()( 5421542   zzzzzzzzzC
 
Where 
)(/)()( zgzhzX  ,   
 




 
1
0
)()(
k
kzkpzh ,    (7) 
 





0
)(
k
kzpzg ,  






0
)(
k
kzzU . 
][   mLp ,  
],,,1[)( 1  zzzV  , 
 
and   is the greatest common divisor. 
   The mean vector
  pLDm mm  ];[ , and   is 
the set of words with a length defined as: 
     110 ,, N ,    (8) 
where ],,,[ 110   nnnn   , }1,,1,0{  Nn  , 
where  
N is the number of words with length  , 
where for CC-DPIM and DPIM, 
N  = 1. Since the 
symbol length is even then 2 . 
 
IV. RESULTS AND DISCUSSION 
Figure 1(a) shows the predicted PSD of the 8-CC-
DPIM using (3-7), assuming that pulse shape p(t) is 
rectangular with 100% duty cycle. The spectrum displays 
a sinc function profile with a distinctive clock (slot) 
frequency and its harmonics at odd multiples, and nulls at 
even multiples of the slot frequency. The nulls at 
normalized frequencies ,2,1)( 0 fT  are poles on 
the unit circle, followed by two symmetrically close poles 
on both sides at 15.0)( 0  fTfT . The spectral density 
(4) is a rational function of z and can be factored to two 
casual and stable functions of z  and 1z  
as )()()( 1 zHzHzRc . To build )(zH one can derive 
poles and zeros of the PSD and extract those inside unit 
circle ( 1iz ). Having this information we can build 
filter )(zH which can be implemented as an Auto 
Regressive Moving Average (ARMA) filter. 
 
The presence of the slot component suggest that a 
simple phase locked loop could be employed to recover 
the clock signal form the incoming CC-DPIM symbol 
stream for slot synchronization at the receiver. Unlike 
PPM, the PSD does not fall to zero at DC region similar 
to the DPIM. Thus, implying that it too would also be 
susceptible to the effect of baseline wander due to high-
pass filtering of the ambient light at the receiver [13-15]. 
To confirm the theoretical analysis, the proposed scheme 
was simulated in Matlab and its estimated PSD is 
depicted in Figure 1(b), showing a close match to the 
predicted PSD.  
 
M-bit Input 
data, {dm} 
4-DPIM, Dm of length mL  
 NGS 2GS CC-DPIM (2GS) 
00 
01 
10 
11 
1 
1 0 
1 0 0 
1 0 0 0 
1 0 0 
1 0 0 0 
1 0 0 0 0 
1 0 0 0 0 0   
11 10 11 
11 10 11 00 
11 10 11 00 00 
11 10 11 00 00 00  
Proceedings - 103 - CSNDSP08
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
R
v
s
(f
T
)/
T
 d
B
fT
 
(a) 
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
M
a
g
n
it
u
d
e
 (
d
B
)
Normalized Frequency
 
(b) 
Figure 1.  PSD of 8-CC-DPIM with 100% pulse duty cycle against the 
normalized frequency: (a) predicted, and (b) simulated  
Figure 2 shows the PSD plots for the CC-DPIM with 
the header pulses having 50% duty cycle. The profile is 
the same as in Fig. 1 except for the first zero crossing 
being shifted to the right. In both figures, the slight 
difference in the profiles is due to the fact that in the 
theoretical analysis both the discrete and continuous 
components of the spectrum are separated while in the 
simulation these components are combined.  
Figure 3 compares the spectrum of 8-CC-DPIM and 8- 
DPIM. For the normalized frequency u = 0, we can 
compute A(z) = 5, C(z) = 25 and B(z) = -1.8047. By 
incorporating these values into (3) the DC component  
Rvs(0)/T0= 0.8307 is obtained as shown in Fig. 3. The DC 
component is 14.4219 times as high as the 8-DPIM DC 
value of 0.0576. Note the PPM scheme has no DC 
component, a desirable characteristics in optical wireless 
communications.  
In most scenarios, the data sequence which generates 
the pulse stream is either wide sense stationary (WSS) or 
wide sense cyclostationary (WSCS). In both cases the 
timing jitter induced by the noise while propagating 
through the channel will introduce frequency offset, thus  
0 1 2 3 4 5 6 7 8 9 10
-160
-140
-120
-100
-80
-60
-40
-20
0
20
R
v
s
(f
T
)/
T
 d
B
fT
 
(a) 
0 1 2 3 4 5 6 7 8 9 10
-160
-140
-120
-100
-80
-60
-40
-20
0
20
M
a
g
n
it
u
d
e
 (
d
B
)
Normalized Frequency
 
(b) 
Figure 2.  PSD of 8-CC-DPIM with 50% pulse duty cycle against the 
normalized frequency: (a) predicted, and (b) simulated 
affecting the continuous components of the spectrum. 
This will result in a more complex clock recovery scheme 
at the receiving end. 
The discrete components (delta functions (5)) 
corresponding to the mean of the sequence, are due to the 
data asymmetry. This can be reduced or eliminated if the 
mean is zero. For bipolar input data }1,1{ na the 
discrete components are no longer present in spectrum.   
V. CONCLUSION 
Power spectral density of the convolutional coded 
digital pulse interval modulation was derived analytically 
based on the stationarization of variable-length word 
sequence. The predicted and simulated spectrums were 
compared showing a close match. The spectrum contains 
a slot (clock) component which could be extracted at the 
receiver for slot and symbol synchronizations, but it also 
has a DC component which is higher than that of DPIM 
and PPM. 
CSNDSP08 - 104 - Proceedings
0 0.5 1
0
0.2
0.4
0.6
0.8
1
R
v
s
(f
T
)/
T
fT
8-CCDPIM
0 0.5 1 1.5 2
0
0.05
0.1
0.15
0.2
0.25
fT
8-DPIM
 
Figure 3.  Spectrum comparison between DPIM and CC-DPIM. 
APPENDIX 
DERIVATION OF (6) 
 
The probability of two words }{ mD  has the distance of 
k digits in the symbol sequence }{ na  can be introduced 
by the cumulative length probabilities which satisfy the 
following difference equation:  
 
 

  0),()( kkypky                   (9) 
the sequence )( ky  converges to )(y  and the 
sequence )()()(  ykykx  is absolutely summable 
with z-Transform: 
 



 
0 )(
)(
)]()([)(
k
k
zg
zh
zykyzX                (10) 
The continuous spectrum of the sequence }{ na  is 
evaluated by the equation (4) where: 
 
 

 mzVzA )()(
     
   



 )()()()( zUzzXmzVzB              (11) 
)(][)()( 1 zVpdiagzVzC 

   
where ‘ is the transpose and diag is the diagonal matrix. 
 
For CC-DPIM: 
 
]00110111[  , 
5
1


L
 , 2  
L
p
1
   (All symbols are equally likely) 
 
L
pLDm mm
1
];[   
],,,1[)( 1  zzzV   
 
Hence, by using these into (11) we can obtain (6). 
 
REFERENCES 
[1] Z. Ghassemlooy, A. R. Hayes, N. L. Seed, and E. D. 
Kaluarachchi, “Digital pulse interval modulation for optical 
communications,” IEEE Commun. Mag., Vol. 36, No. 12, pp. 95-
99, 1998. 
[2] D. F. Bishop, S. Million, T.M. Nguyen, M. K. Simon, “Power 
spectrum of unbalanced NRZ and biphase signals in thepresence 
of data asymmetry,” IEEE Trans. On Electromag.  Compatibility, 
Vol. 40, No. 1, pp. 55 – 61, 1998. 
[3] Ho Keang-Po, J. M. Kahn, “Spectrum of externally modulated 
optical signals,” J. Lightwave Technol., Vol. 22, No. 2, pp.658-
663, 2004. 
[4] C. Pimentel, and V. C.  da Rocha, “ On the power spectral density 
of constrained sequences,“ IEEE Trans. Commun., vol. 55, no. 3, 
pp. 409-416, 2007. 
[5] M. Z. Win, “On the power spectral density of digital pulse streams 
generated by M-ary cyclostationary sequences in the presence of 
stationary timing jitter,” IEEE Trans. Commun., Vol. 46, No. 9, 
pp. 1135-1145, 1998. 
[6] Y. Han, “Theoretical sensitivity of directed-detection multilevel 
modulation formats for high spectral efficiency optical 
communications,” IEEE J. Select. Top. Quant. Electron., Vol. 12, 
No. 4, pp. 571-580, 2006. 
[7] S. Rajbhandari, Z. Ghassemlooy, N. M. Aldibbiat, M. Amiri, and  
W. O. Popoola, “Convolutional coded DPIM for indoor non-
diffuse optical wireless link, “ The 7th IASTED Inter. Conf. on 
Wireless and Optical Commun. (WOC 2007), Canada, pp. 286-
290, 2007. 
[8] N. M. Aldibbiat, Z. Ghassemlooy and R. McLaughlin, "Spectral 
characteristics of Dual Header - Pulse Interval Modulation (DH-
PIM)," IEE Proc. Commun., Vol. 148, No. 5, pp. 280-286, 2001. 
[9] D. Shiu and J. M. Kahn, “Differential pulse position modulation 
for power-efficient optical communication,” IEEE Trans. 
Commun., Vol. 47, No. 8, pp. 1201–1210, 1999. 
[10] H. L. Hurd, “Stationarizing properties of random shifts,” SIAM J. 
Appl. Math, Vol. 26, No. 1, pp. 203–212, 1975. 
[11] G. L. Cariolaro and G. L. Pierobon, “Stationary symbol sequences 
from variable-length word sequences,” IEEE Trans. Inform. 
Theory, Vol. IT-23, pp. 243–253, 1977. 
[12] G. Cariolaro, T. Erseghe, L. Vangelista, “Exact spectral evaluation 
of the family of digital pulse interval modulated signals,” IEEE 
Trans. Info. Theory, Vol. 47, No. 7, pp. 2983 – 2992, 2001. 
[13] .Z. Ghassemlooy, and N. M. Aldibbiat, “Baseline wander effect on 
indoor wireless infrared links operated by dual header pulse 
interval modulation”, The Mediterranean Journal of Electronics 
and Communications, Vol. 1, No. 1, Oct. 2005. 
[14] A. R. Hayes, Z. Ghassemlooy, and N. L. Seed, “Investigation of 
base line wander in DPIM”, IEE Proc. J., Vol. 147. No. 4, pp. 
295-300, 2000. 
[15] Z. Ghassemlooy, “Investigation of the baseline wander effect on 
Indoor optical wireless system employing digital pulse interval 
modulation”, to appear in the IET Communications, Vol. 2, No. 1, 
pp. 53-60, 2008. 
Proceedings - 105 - CSNDSP08


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Spectral analysis of convolutional coded DPIM for indoor optical wireless communications

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