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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

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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.

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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.

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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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