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New hybrid reverse differential pulse
position width modulation scheme for
wireless optical communication
Renbo Liao
Hongzhan Liu
Yaojun Qiao
Downloaded From: http://spiedigitallibrary.org/ on 04/27/2015 Terms of Use: http://spiedl.org/terms
New hybrid reverse differential pulse position width
modulation scheme for wireless optical communication
Renbo Liao,aHongzhan Liu,a,b,*and Yaojun Qiaob
aSouth China Normal University, Laboratory of Photonic Information Technology, No. 378 Wai Huan Xi Road, Guangzhou Higher Education Mega
Center, Guangzhou 510006, China
bBeijing University of Posts and Telecommunications, State Key Laboratory of Information Photonics and Optical Communications,
No. 10, Xitucheng Road, Haidian District, Beijing 100876, China
Abstract. In order to improve the power efficiency and reduce the packet error rate of reverse differential pulse
position modulation (RDPPM) for wireless optical communication (WOC), a hybrid reverse differential pulse
position width modulation (RDPPWM) scheme is proposed, based on RDPPM and reverse pulse width modu-
lation. Subsequently, the symbol structure of RDPPWM is briefly analyzed, and its performance is compared with
that of other modulation schemes in terms of average transmitted power, bandwidth requirement, and packet
error rate over ideal additive white Gaussian noise (AWGN) channels. Based on the given model, the simulation
results show that the proposed modulation scheme has the advantages of improving the power efficiency and
reducing the bandwidth requirement. Moreover, in terms of error probability performance, RDPPWM can achieve
a much lower packet error rate than that of RDPPM. For example, at the same received signal power of
−28 dBm, the packet error rate of RDPPWM can decrease to 2.6×10−12, while that of RDPPM is
2.2×10−8. Furthermore, RDPPWM does not need symbol synchronization at the receiving end. These consid-
erations make RDPPWM a favorable candidate to select as the modulation scheme in the WOC systems. ©2014
Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.53.5.056112]
Keywords: wireless optical communication; reverse differential pulse position width modulation; average transmitted power; band-
width requirement; packet error rate.
Paper 131857 received Dec. 9, 2013; revised manuscript received Apr. 20, 2014; accepted for publication Apr. 30, 2014; published
online May 21, 2014.
1 Introduction
Wireless optical communications (WOC),1,2also known as
free space optical communications or atmosphere laser com-
munications, refers to the transmission of a modulated infra-
red beam through free space (atmosphere) to transmit data
between the transmitter end and the receiver end.3The theory
of WOC is essentially the same as that of optical fiber com-
munications, with the main difference being that the energy
of the beam is collimated and sent through the atmosphere
from the source to the destination and received by a photo-
detector.3,4In recent years, the WOC has gained significant
research and commercial attention, due to the need for cost-
effectiveness, large transmission capacity, high transmission
rate, high security, and good immunity to electromagnetic
interference.5,6Moreover, the WOC technology is an attrac-
tive complementary alternative to the radio frequency, wire-
less and wired for the “last mile”communications, as it does
not require right-of-way and digging trenches. Therefore, the
WOC becomes one of the most promising approaches for
addressing the emerging broadband access network.
One of the main factors in the implementation of a
high performance WOC system is the modulation scheme.
Current WOC systems typically use the intensity modulation
and direct detection7,8scheme on account of its simple imple-
mentation and low cost. The simplest approach is on-off
keying (OOK),9,10 in which a zero is represented by zero
intensity and a one by a positive intensity, but this scheme
shows degradation in terms of power efficiency. The pulse
position modulation (PPM)9–13 has been used widely in opti-
cal communication systems and has been adopted by the
IEEE 802.11 working group for the infrared physical layer
standard. However, the bandwidth efficiency of PPM is
low, and it needs symbol synchronization.1,9As a potential
alternative to PPM, the differential pulse position modulation
(DPPM)14–17 is a technique which offers a higher transmission
capacity, but its error probability performance is poor.15,16
Another candidate, the digital pulse interval modulation
(DPIM),18–20 has the advantages of shortening the symbol
length, high bandwidth efficiency, and not requiring symbol
synchronization, but it is too complex.14,19 Based on PPM
and DPPM, the reverse pulse position modulation (RPPM)
and the reverse differential pulse position modulation
(RDPPM) have been proposed in Ref. 21 considering that
the transmitted power of laser light source should be as low
as possible for a given bit error rate. RDPPM can achieve
high bandwidth efficiency and large transmission capacity.
However, it also has the disadvantages of low power effi-
ciency and poor packet error rate.
In order to improve the power efficiency and reduce the
packet error rate of RDPPM, in this paper, we propose a new
hybrid reverse differential pulse position width modulation
(RDPPWM) technique, based on combining RDPPM with
the reverse pulse width modulation (RPWM). The proposed
modulation scheme combines the advantages of RDPPM and
RPWM, and has the potential to offer an improved perfor-
mance. The performance of RDPPWM is analytically inves-
tigated and verified using simulated data, and compared with
*Address all correspondence to: Hongzhan Liu, E-mail: lhzscnu@163.com 0091-3286/2014/$25.00 © 2014 SPIE
Optical Engineering 056112-1 May 2014 •Vol. 53(5)
Optical Engineering 53(5), 056112 (May 2014)
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OOK, RDPPM, and RPWM. The rest of this paper is organ-
ized as follows. The symbol structures of RPWM and
RDPPWM are introduced in Sec. 2. The average transmitted
power and the bandwidth requirement of RDPPWM are ana-
lyzed in Sec. 3. The packet error rate of RDPPWM is calcu-
lated over ideal AWGN channels in Sec. 4. Finally, the
concluding remarks are presented in Sec. 5.
2 Symbol Properties
In the WOC, various modulation schemes have different per-
formance characteristics because they have different symbol
structures. Therefore, it is important to study the symbol
structure thoroughly before analyzing the performance of
different modulation schemes. In this section, we first briefly
introduce the symbol structures of RPWM and RDPPWM.
2.1 Symbol Structure of RPWM
In RPWM, a block of M¼log2Ldata bits is mapped into a
symbol that consists of Lslots. The symbol starts with kþ1
empty slots, followed by L−ðkþ1Þsuccessive pulses,
where kis equal to the decimal value of the binary input.
Table 1shows the mapping of all possible combinations of
aM¼3bits code word into RPWM and RDPPM symbols.
As shown in Table 1, RPWM reduces the average number
of pulses per symbol compared with RDPPM, thus resulting
in an increased power efficiency. Furthermore, the symbol
length of RPWM is fixed, which decreases the packet error
rate in the process of demodulation. However, the average
symbol length of RPWM is larger than that of RDPPM,
which may impair the bandwidth efficiency.
2.2 Symbol Structure of RDPPWM
In RDPPWM, the first rbits of the whole Mdata bits are
modulated by RDPPM, with the average duration of
Td¼2T∕ð1þ2rÞ, where Tis the overall duration of the
Mdata bits. Then, each slot is partitioned into a period of
time that is composed of 2M−rslots, which starts with kþ
1empty slots then continues with 2M−r−ðkþ1Þsuccessive
pulses, just as RPWM, whose duration is Tw¼Td∕2M−r.
Here, kis equal to the decimal value of the last M−r
bits. If M¼4,r¼2, taking one of the symbols, for exam-
ple, the coding process of RDPPWM is shown in Fig. 1.
Table 2shows the mapping of all possible combinations
of an M¼4bits code word into RPWM and RDPPM
symbols.
As shown in Table 2, RDPPWM reduces the average
number of pulses per symbol compared with RDPPM,
which can help to improve the power efficiency. Moreover,
the symbol length of RDPPWM is fixed in a given sequence
interval, but it is variable from one sequence interval to
another, which is determined by the information content
of the symbol rather than a predetermined clock period.
Furthermore, the average symbol length of RDPPWM is
similar to that of RDPPM as rincreases, which may increase
bandwidth efficiency.
3 Modulation Performance Analysis
In order to evaluate the performance of different modulation
schemes in WOC systems, several main performance char-
acteristics, such as the average transmitted power, the band-
width requirement, and the packet error rate, are generally
required.22 In most applications, particularly those using
portable transmitters and line-of-sight configuration, the
eye safety and the power emission set an upper limit on
the average transmitted power.21 In addition, an appropriate
modulation scheme should also reduce the bandwidth
requirement and decrease the packet error rate as much as
possible.
3.1 Average Transmitted Power
To simplify the analysis, we compare different modulation
schemes in terms of the average transmitted power when
sending the same symbol. It is assumed that different modu-
lation schemes have the same peak power Pt. If each optical
pulse carries the information bits 0 and 1 with equal prob-
ability, and sending the information bit 1 requires peak
power while sending the information bit 0 does not require
any power, the average transmitted power of OOK is
POOK ¼Pt∕2.9,11 It is evident that RDPPWM does not
display a regular periodic symbol structure, thus resulting
in a varied slot duration. For RDPPWM, the minimum
and the maximum symbol lengths are 2M−rand 2M,
respectively. Thus, the mean symbol length of RDPPWM
is given by
Lave;RDPPWM ¼2M−rð1þ2rÞ
2:(1)
Moreover, the number of slots for sending the information
bit 1 is equal to ð2M−1Þ∕2. Thus, the average transmitted
power of RDPPWM is obtained as
PRDPPWM ¼2M−1
2M−rð1þ2rÞPt:(2)
We can obtain the average transmitted power for the other
modulation schemes in the same way. Figure 2shows the
average transmitted power of RDPPM, RPWM, and
RDPPWM, normalized to OOK, versus the number of
bits per symbol, M. When r¼1, the average transmitted
power of RDPPWM slowly increases with increasing M,
and when r¼M−1, the average transmitted power also
increases as Mincreases. The main reason is that the prob-
ability of sending the information bit 1 per symbol increases
Table 1 Mapping of OOK code into 8-RDPPM and 8-RPWM
symbols.
Bits OOK 8-RDPPM 8-RPWM
000 000 0 01111111
001 001 10 00111111
010 010 110 00011111
011 011 1110 00001111
100 100 11110 00000111
101 101 111110 00000011
110 110 1111110 00000001
111 111 11111110 00000000
Optical Engineering 056112-2 May 2014 •Vol. 53(5)
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with increasing M, which improves the duty cycle so that it
makes the average transmitted power increase. Moreover, the
power efficiency of RDPPWM is better than that of RDPPM,
but not as good as that of RPWM. For instance, with M¼4,
RDPPWM (r¼1) requires an average transmitted power
that is 0.69 times that of RDPPM, and RDPPWM (r¼
M−1) requires an average power that is 0.94 times that
of RDPPM. The average powers of the two RDPPWM
types are lower than that of RDPPM. Therefore, RDPPWM
improves the power efficiency, and thus it solves the problem
of low power efficiency in RDPPM.
Figure 3shows the relationship between the average
transmitted power of RDPPWM and r. For a given value
of M, it is clear that the average transmitted power of
RDPPWM decreases as rdecreases, which means that the
power efficiency of RDPPWM increases as rdecreases.
When r¼1, the average transmitted power of RDPPWM
is the same as that of RPWM, and when r¼M, the average
transmitted power of RDPPWM is the same as that of
RDPPM. As the average transmitted power of RDPPWM
changes with r, the ideal transmitted power can be obtained
by setting an appropriate value of raccording to the signal
power requirement of the actual application.
3.2 Bandwidth Requirement
The bandwidth is typically estimated by the width of the
main lobe of the power spectral density in the respective opti-
cal communication systems. Since the pulse width of an
Fig. 1 Coding process of reverse differential pulse position width modulation (RDPPWM).
Table 2 Mapping of OOK code into 16-RDPPM and 16-RDPPWM
symbols.
Bits OOK 16-RDPPM 16-RDPPWM
0000 0000 0 0111
0001 0001 10 0011
0010 0010 110 0001
0011 0011 1110 0000
0100 0100 11110 11110111
0101 0101 111110 11110011
0110 0110 1111110 11110001
0111 0111 11111110 11110000
1000 1000 111111110 111111110111
1001 1001 1111111110 111111110011
1010 1010 11111111110 111111110001
1011 1011 111111111110 111111110000
1100 1100 1111111111110 1111111111110111
1101 1101 11111111111110 1111111111110011
1110 1110 111111111111110 1111111111110001
1111 1111 1111111111111110 1111111111110000
Fig. 2 Normalized average transmitted power of on-off keying (OOK),
reverse differential pulse position modulation (RDPPM), reverse pulse
width modulation (RPWM), and RDPPWM.
Optical Engineering 056112-3 May 2014 •Vol. 53(5)
Liao, Liu, and Qiao: New hybrid reverse differential pulse position width modulation scheme. . .
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optical signal is relatively narrow, in general, the bandwidth
is approximately equal to the inverse of the pulse width.11
Assuming that the transmitter conveys information bits at
the rate of Rb, the bandwidth of OOK is equal to the inverse
of the pulse width Tb¼1∕Rb, that is BOOK ¼1∕Tb¼
Rb.23,22 For RDPPWM, noting that the data rate is not a con-
stant, we have to use the average bit rate based on the average
symbol rate. Thus, the bandwidth requirement to support
communications at the same bit rate based on the average
symbol duration, relative to OOK, can be given as
BRDPPWM ¼2M−rð1þ2rÞ
2MRb:(3)
In the same way, we also can obtain the bandwidths of
OOK, RDPPM, and RPWM. According to the above analy-
sis, Fig. 4shows the bandwidth requirement versus the num-
ber of bits per symbol for RDPPM, RPWM, and RDPPWM.
As shown in Fig. 4, when r¼1and r¼M∕2, the
bandwidth of RDPPWM rapidly increases as Mincreases.
Moreover, the bandwidths of RDPPM and RDPPWM are
relatively small and approximately equal for a given value
of M¼2. However, the bandwidth requirement gradually
shows a great deal of variation with the increase of M.
The main reason is that the average symbol lengths of differ-
ent modulation schemes are relatively short, so that the num-
ber of corresponding slots are small when the modulation
order is lower, but the average symbol lengths of different
modulation schemes show a distinct difference as Min-
creases, which shows that the bandwidths of different modu-
lation schemes also have a considerable difference. More-
over, the bandwidth efficiency of RDPPWM is better than
that of RPWM, which means that applying RDPPWM can
solve the problem of low bandwidth efficiency in RPWM.
Figure 5shows the relationship between the bandwidth of
RDPPWM and r. It is obvious that the bandwidth of
RDPPWM decreases with increasing rwhen Mis a constant.
When r¼1, the bandwidth of RDPPWM is the same as that
of RPWM, and when r¼M, the bandwidth of RDPPWM is
the same as that of RDPPM. According to the signal band-
width requirement of the actual application, if rsuitably
increases, the bandwidth of RDPPWM can effectively solve
the problem of higher bandwidth requirement in RPWM. At
the same time, RDPPWM can improve the bandwidth effi-
ciency because it keeps the advantage of high bandwidth
efficiency available in RDPPM.
4 Error Performance Analysis and Simulation
4.1 Theoretical Analysis and Calculation
In WOC systems, the noise is mainly composed of shot noise
generated by a variety of background light sources such as
lightings and sunshine, hot noise resulting from resistance,
and other noises including the leakage current and the dark
noise produced by a photo-detector. Usually these noises can
be considered as Gaussian white noise that is independent of
the transmitting signals.22 Figure 6is the ideal AWGN chan-
nel modeled on the above assumption.
To simplify the analysis, it is assumed that the ideal
AWGN channels are distortion-free (the multipath dispersion
is ignored), and the attenuation factor of transmission path is
g.23,22 In Fig. 6, the input of the matched filter is ffiffiffiffiffiffiffi
gPt
pþnðtÞ
when the optical pulse exists, but the input is nðtÞwhen the
Fig. 3 Average transmitted power of RDPPWM changing with r.
Fig. 4 Normalized bandwidth of OOK, RDPPM, RPWM, and
RDPPWM. Fig. 5 Bandwidth of RDPPWM changing with r.
Optical Engineering 056112-4 May 2014 •Vol. 53(5)
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optical pulse does not exist, where Ptis the peak transmitted
power, while nðtÞis the Gaussian white noise with zero mean
and variance σ2
n¼N0B. Thus the output of the matched filter
at t¼Tswill be Epþn0ðTsÞwhen the optical pulse is trans-
mitted, but the output is n0ðTsÞwithout any optical pulse,
where Ep¼gPtTs, while n0ðTsÞis still the Gaussian white
noise with zero mean but now with the variance σ2¼
gPtT2
sN0B.24
Assuming that the detection threshold is kEpð0<k<1Þ,
and pe0 and pe1 are the probabilities of transmitting bit 0 but
receiving bit 1 and transmitting bit 1 but receiving bit 0,
respectively, then we can get23
pe0 ¼Zþ∞
kEp
1
ffiffiffiffiffi
2π
pσ
e−y2
2σ2dy¼QkEp
σ¼Q kffiffiffiffiffiffiffiffiffi
gPt
N0B
s!;
(4)
pe1 ¼ZkEp
−∞
1
ffiffiffiffiffi
2π
pσ
e−ðy−EpÞ2
2σ2dy¼Qð1−kÞEp
σ
¼Qð1−kÞffiffiffiffiffiffiffiffiffi
gPt
N0B
s;(5)
where QðxÞ¼∫þ∞
xð1∕ffiffiffiffiffi
2π
pÞe−t2∕2dt. The slot error rate25 is
pse ¼p0pe0 þp1pe1;(6)
where p0and p1are the probability of transmitting the infor-
mation bits 0 and 1, respectively.26
The concept of slot error rate has no meaning in aniso-
chronous schemes such as RDPPWM and RDPPM. This
is because the frames have varied symbol lengths; thus,
any erroneous slot not only affects the bits associated with
t, but also shifts the bits in the following frames.23 It is nec-
essary to discuss the packet error rate. For a given packet
length of Nbits, there are NLave∕Mslots after being modu-
lated by one scheme. The packet error rate is
ppe ¼1−ð1−pseÞNL
ave
M≈NLave pse
M;(7)
where Lave represents the average symbol length.24,26
It is assumed that different modulation schemes have the
same average transmitted power P. If each optical pulse car-
ries the information bits 0 and 1 with equal probability, the
peak power of OOK is Pt;OOK ¼2P.11 Then, the packet error
rate of OOK (where k¼1∕2) can be given by
ppe;OOK ¼NQffiffiffiffiffiffiffiffiffiffiffiffi
gP
2N0B
s:(8)
In the same way, the packet error rate of RDPPM is given
as
ppe;RDPPM ¼N
2M2Qkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s
þð2M−1ÞQð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s:(9)
Similarly, we can obtain the packet error rate of
RDPPWM as follows:
ppe;RDPPWM ¼N
2Mð2M−rþ1ÞQ
×kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
sþð2M−1Þ
×Qð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s:(10)
4.2 Simulation Results and Discussion
To evaluate the error performance, it is desirable to compare
the proposed modulation scheme with various other modu-
lation schemes via computer simulations. In this section, the
packet error rate of RDPPWM is calculated and compared
with RDPPM and OOK in WOC systems. Table 3lists
the parameters associated with the system model and the cor-
responding numerical values used for simulation.
As shown in Fig. 7(a), when the received signal power is
−28 dBm, the packet error rate of RDPPM is 2:2×10−8;
however, the corresponding packet error rate of RDPPWM
can decrease to 2.6 ×10−12, so that it offers a greatly
improved error performance. Therefore, for the same received
signal power, the packet error rate of RDPPWM is obviously
better than that of RDPPM, but marginally inferior to that of
OOK. One possible reason is that for a given value of M,the
symbol length of RDPPM is variable while that of OOK is
fixed. Thus the symbol length of RDPPWM is fixed in a
sequence interval, but it is variable when jumping from one
interval to another. For example, as shown in Table 2, the sym-
bol length of RDPPWM is 12 slots after input bit sequences
such as 1000, 1001, 1010, and 1011 are modulated, while the
Fig. 6 Ideal AWGN channel model.
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symbol length of RDPPWM is 16 slots after input bit
sequences such as 1100, 1101, 1110, and 1111 are modu-
lated. Therefore, the symbol length of RDPPWM is still rel-
atively fixed compared with that of RDPPM. In the process
of decision, when the symbol length is fixed, one slot error
does not influence the following symbols, but the slot error
of previous symbols can affect the judgment of the following
symbols if the symbol length is variable, which clearly
increases the packet error rate. Moreover, when the packet
error rate is equal to 10−10, the received signal power of
RDPPWM is −28.5 dBm, while that of RDPPM is
−27.1 dBm. Obviously, RDPPWM reduces the requirement
for received signal power so that it improves the system
receiver sensitivity.
In the same way, from Figs. 7(b)–7(d), we can obtain the
same conclusion, that the packet error rate of RDPPWM is
obviously better than that of RDPPM, but slightly worse than
that of OOK when they have the same received signal power.
That is to say, the effects of Mand Nare not important in
evaluating the advantages or disadvantages of the packet
error rate of RDPPWM compared with other modulation
schemes.
Figure 8shows the relationship between the packet error
rate of RDPPWM and the received signal power for different
M. It is obvious that, under the same received signal power
condition, the packet error rate of RDPPWM is the best when
M¼3, and that of RDPPWM is the worst when M¼6.In
Table 3 List of parameters over ideal AWGN channels.
Parameter Value
k0.5
g1
σ2
n1×10−8
Fig. 7 Packet error rate of OOK, RDPPM, and RDPPWM with different values of Mand N. (a) M¼4and
N¼1024, (b) M¼4and N¼2048, (c) M¼8and N¼1024, and (d) M¼8and N¼2048.
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other words, for the same received signal power, the packet
error rate of RDPPWM decreases with decreasing M, thus it
offers an improved error performance.
5 Conclusion
In this paper, first we propose a new RPWM scheme.
Subsequently, its symbol structure is briefly analyzed.
RPWM has the advantages of higher power efficiency and
low packet error rate, but poor bandwidth efficiency. In order
to make up for the disadvantages of RDPPM and RPWM, a
new hybrid scheme called RDPPWM is proposed in this
paper, based on both RDPPM and RPWM, and its symbol
structure is also analyzed. Then the new hybrid modulation
scheme is compared with other modulation schemes such as
OOK, RPWM, and RDPPM in terms of the average trans-
mitted power and the bandwidth requirement. The results
show that the power efficiency of RDPPWM is better than
that of RDPPM, and the bandwidth efficiency of RDPPWM
can be improved by appropriately increasing r. Furthermore,
the exact expression for the packet error rate of RDPPWM is
derived over ideal AWGN channels. The simulation results
show that the packet error rate of RDPPWM is obviously
better than that of RDPPM, but marginally inferior to that
of OOK. For example, when the received signal power is
−28 dBm, the packet error rate of RDPPM is 2:2×10−8;
however, the packet error rate of RDPPWM can decrease
to 2.6 ×10−12. Thus, RDPPWM is a promising scheme in
terms of improving the error performance in WOC systems.
Moreover, when the packet error rate is 10−10, the received
signal power of RDPPWM is −28.5 dBm while that of
RDPPM is −27.1 dBm. Therefore, RDPPWM reduces the
requirement for the received signal power, so that it improves
the system receiver sensitivity. In addition, RDPPWM does not
need symbol synchronization when receiving signals. From the
overall analysis, we conclude that RDPPWM has definite
potential application value in the future WOC systems.
Appendix A: Packet Error Rate of RDPPM
From Eq. (7), the packet error rate of RDPPM can be derived
as follows:
ppe;RDPPM ¼N
MLRDPPMðp0;RDPPM pe0;RDPPM
þp1;RDPPMpe1;RDPPM Þ:(11)
Then the peak power of RDPPM is obtained as21
Pt;RDPPM ¼2Mþ1
2M−1P: (12)
Substituting into Eqs. (4) and (5), we can obtain
pe0;RDPPM ¼Qkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s;(13)
pe1;RDPPM ¼Qð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s:(14)
Substituting Eqs. (12)–(14) into Eq. (11), the packet error
rate of RDPPM is given by
ppe;RDPPM ¼N
M×2Mþ1
2
×2
2Mþ1×Qkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
sþ2M−1
2Mþ1
×Qð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s
¼N
2M2Qkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s
þð2M−1ÞQð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2Mþ1
2M−1
gP
N0B
s;
where LRDPPM ¼ð2Mþ1Þ∕2,p0;RDPPM ¼2∕ð2Mþ1Þ,
and p1;RDPPM ¼ð2M−1Þ∕ð2Mþ1Þ.
Appendix B: Packet Error Rate of RDPPWM
From Eq. (7), the packet error rate of RDPPWM can be
derived as follows:
ppe;RDPPWM ¼N
MLRDPPWMðp0;RDPPWM pe0;RDPPWM
þp1;RDPPWMpe1;RDPPWM Þ:(15)
Substituting Eq. (2) into Eqs. (4) and (5), we can obtain
pe0;RDPPWM ¼Qkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s;(16)
pe1;RDPPWM ¼Qð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s:(17)
Fig. 8 Packet error rate of RDPPWM with different values of M.
Optical Engineering 056112-7 May 2014 •Vol. 53(5)
Liao, Liu, and Qiao: New hybrid reverse differential pulse position width modulation scheme. . .
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Substituting Eqs. (1), (16), and (17) into Eq. (15), the
packet error rate of RDPPWM is given as
ppe;RDPPWM ¼N
M×2M−rð1þ2rÞ
2
×2M−rþ1
2M−rð1þ2rÞ×Qkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s
þ2M−1
2M−rð1þ2rÞ×Qð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s
¼N
2Mð2M−rþ1ÞQkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s
þð2M−1ÞQð1−kÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2M−rð1þ2rÞ
2M−1
gP
N0B
s;
where p0;RDPPWM ¼ð2M−rþ1Þ∕½2M−rð1þ2rÞ and
p1;RDPPWM ¼ð2M−1Þ∕½2M−rð1þ2rÞ.
Acknowledgments
This work was supported by the National Basic Research
Program of China (Grant No. 2013CB29204).
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Renbo Liao is a master and mainly engaged in the research of modu-
lation technology for wireless optical communication. In addition, he is
participating in the project study of the key technology research on
wide spectrum wireless optical communications.
Hongzhan Liu received the PhD degree in optics engineering from
Shanghai Institute of Optics and Fine Mechanics, Chinese Academy
of Sciences, Shanghai, China, in 2006. He has also been a postdoc-
toral at the Beijing University of Posts and Telecommunications from
2010 to 2012. He is currently an associate professor with South China
Normal University. His current research interests include optical fiber,
sensing, and optical communications.
Yaojun Qiao obtained his PhD, MS, and BS degrees from Beijing
University of Posts and Telecommunications, Jilin University, and
Hebei Normal University in 2000, 1997, and 1994, respectively. He
worked for Lucent and Fujitsu from 2000 to 2007, and joined
Beijing University of Posts and Telecommunications in 2007. His
research interests include high-speed optical fiber communication
system and network.
Optical Engineering 056112-8 May 2014 •Vol. 53(5)
Liao, Liu, and Qiao: New hybrid reverse differential pulse position width modulation scheme. . .
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