Jing Li

Lehigh University, Bethlehem, Pennsylvania

Alan Hylton, James Budinger, Jennifer Nappier, Joseph Downey, and Daniel Raible

Glenn Research Center, Cleveland, Ohio

Dual-Pulse Pulse Position Modulation (DPPM) for 

Deep-Space Optical Communications: Performance 

and Practicality Analysis

NASA/TM—2012-216042

November 2012
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Jing Li

Lehigh University, Bethlehem, Pennsylvania

Alan Hylton, James Budinger, Jennifer Nappier, Joseph Downey, and Daniel Raible

Glenn Research Center, Cleveland, Ohio

Dual-Pulse Pulse Position Modulation (DPPM) for 

Deep-Space Optical Communications: Performance 

and Practicality Analysis

NASA/TM—2012-216042

November 2012

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 
Abstract—Due to its simplicity and robustness against wavefront distortion, pulse position modulation (PPM) with photon

counting detector has been seriously considered for long-haul

optical wireless systems. This paper evaluates the dual-pulse

case and compares it with the conventional single-pulse case.

Analytical expressions for symbol error rate and bit error rate are

first derived and numerically evaluated, for the strong, negativeexponential turbulent atmosphere; and bandwidth efficiency and

throughput are subsequently assessed. It is shown that, under

a set of practical constraints including pulse width and pulse

repetition frequency (PRF), dual-pulse PPM enables a better

channel utilization and hence a higher throughput than it singlepulse counterpart. This result is new and different from the

previous idealistic studies that showed multi-pulse PPM provided

no essential information-theoretic gains than single-pulse PPM.

I. INTRODUCTION

Optical wireless communications, or, free-space optics

(FSO), has been widely used in inter-satellite and deep space

communications, as well as in roof-top networks and as a

viable solution for the “last mile” problem. A number of viable

modulation schemes have been proposed for optical wireless

systems, including on-off keying (OOK), pulse position modulation (PPM), differential phase shift keying (DPSK), and

even pulse width modulation (PWM). Among them, PPM

with direct detection involving photon-counting receiver is

regarded by many as highly advantageous, especially for longhaul transmission such as inter-satellite, lunar, and deep-space

communications.

From the information-theoretic perspective, it has long

been known that under peak and average power constraints,

restricting the modulation to a binary, slotted scheme results in

only negligible capacity loss [3]. More recent analysis shows

that, under the peak and average power constraints typical of

a deep-space link, restricting the modulation to PPM results

in near-capacity performance also [10].

From the practical perspective, PPM signals are easy and

fast to modulate. They also allow for efficient photo-counting

receivers such as avalanche photodiode (APD) detectors. In

general, as optical signals transmit though a turbulent atmosphere, they encounter a large amount of wavefront distortion

in amplitude and especially in phase. To correct phase distortion is difficult, but is a must-have if single-mode operation

or diffraction-limited focusing is required. Since a photon

counting receiver does not need to amplify the optical signal,

the constraint for single-mode operation is relaxed, and, so

long as the photon-counting detector placed in the receiver

focal plane has adequate detection area, the second constraint

is also relaxed. Hence, PPM with photon counting receivers

is relatively robust against wavefront distortion due to atmospheric turbulence or scintillation. Additionally, PPM is also

particularly desirable for lunar or deep-space communication

systems, since the superb sensitivity of the downlink photocounting receiver can effectively reduce the required poweraperture product needed to operate the transmitter on the

satellite, which in term reduces the size, weight, and power of

the satellite.

There are two types of PPM schemes.

(1) In a conventional M-slot (single-pulse) PPM, a PPM

symbol interval is divided into M (even) time slots, and a

block of m = log2 M binary bits are mapped to one of the M

time slots, signified by an optical pulse in this very time slot.

The rate of the modulation, defined as the number of output

symbols vs the incoming bits is given as M/ log2 M = 2m/m.

(2) To improve the data rate and throughput, multi-pulse

PPM (MPPM) [1], also known as combinatorial PPM [2], has

been developed as an extension to the single-pulse PPM. Since

two or more pulses are allowed to convey information in each

M-slot symbol, there are potentially Ml

Δ

= M

l

 possible lpulse combinations, leading to a payload of up to log2(Ml)

bits per modulated symbol. Although the concept of MPPM

dates back to the late eighties (e.g. [1], [2]), the scheme has

recently received a revived interest, spurred in part by the ever

increasing data rate demand of today’s lunar and deep-space

communication systems (which now target a downlink data

rate of 600 Mbps to 1 Gbps), and in part by the emerging

technology of visible light communication (in which MPPM

can be used to, for example, simultaneously control brightness

and communicate data [5]).

For a propagation distance of less than 100m, the optical

signals tend to experience relatively weak turbulence, which

may be modeled as a log-normal distribution with good

accuracy. This generally corresponds to a scintillation index in

the range of [0, 0.75]. As the communication range increases,

NASA/TM—2012-216042 1

Dual-Pulse Pulse Position Modulation (DPPM) for Deep-Space 

Optical Communications: Performance and 

Practicality Analysis 

Jing Li 

Lehigh University 

Bethlehem, Pennsylvania 18015 

Alan Hylton, James Budinger, Jennifer Nappier, Joseph Downey, and Daniel Raible 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 
the accuracy of the log-normal distribution quickly fades.

Over the years, several statistical models have been developed

to describe FSO channels under a variety of atmospheric

turbulence conditions, including K distribution and I-K distribution [6], log-normal Rician distribution [7], gamma-gamma

distribution [8], [9], and the negative-exponential distribution

[10]. Although the gamma-gamma channel model, based on

the modified Rytov theory, captures a fairly wide range of

turbulent conditions, the negative-exponential channel model

remains one of the most popular models for strong scintillation

(when scintillation index is 1), due to its good accuracy and

analytical tractability.

The performance of single-pulse PPM, especially at relatively low order such as M = 2 and 4, have been extensively

studied in both weak and strong atmospheric scintillation.

Noteworthy studies include, for example, uncoded M-ary PPM

over log-normal and negative-exponential channels [10], ReedSolomon (RS) coded M-ary PPM over log-normal channels

[11], Reed-Solomon (RS) coded M-ary PPM over negativeexponential channels [12]. In comparison, the performance

of multi-pulse PPM is much less studied. Not considering

atmospheric scintillation, [4] performed a compressive study

of the error rate performance of dual-pulse PPM (DPPM)

by evaluating a pure Poisson counting process. However, the

performance of a scintillated DPPM FSO system has not

received adequate attention. This paper analyzes the error rate

performance of uncoded DPPM, as well as its bandwidth

efficiency and throughput, with comparison to single-pulse

PPM. Targeting lunar and deep space communications, we

consider strong atmospheric scintillation that is modeled as

the negative-exponential distribution.

Our main contributions include:

(1) We developer the analytical expressions for the symbol

error rate (SER) and the bit error rate (BER) of dual-pulse

PPM (and single-pulse PPM) over negative-exponential FSO

channels. We show that, for the same M = 2m-slot symbol,

DPPM incurs very marginal performance loss with but delivers

(m − 2) bits more per symbol.

(2) Using the Z-channel model as an abstraction of the

intensity-modulated directly-detected (IM/DD) FSO channel,

we evaluate the channel capacity. We show that for singlepulse PPM, smaller M delivers a better bandwidth efficiency

(in bits/Hz/s) and performs closer to the capacity limit (and

non-equal-probable OOK can achieve the capacity). We also

show that, from the information-theoretic perspective, M-slot

dual-pulse PPM is the equivalence to M/2-slot single-pulse

PPM, which suggests little need to adopt dual-pulse PPM.

(3) We further evaluate the system by considering the

practicality issues. The clock frequency sets a limit on the

minimum slot width, and the pulse repetition frequency (PRF)

sets a constraint on the minimum guard-time (between two

PPM symbols). Hence, in single-pulse PPM as well in dualpulse PPM, a smaller M does not necessarily lead to a better

bandwidth efficiency. For a given set of realistic constraints,

optimal values of M can be derived to maximize the throughput of either case. In general, carefully-designed dual-pulse

PPM can expect to reduce the amount of guard-time, improve

the channel utilization, and hence outperform single-pulse

PPM. For specific (realistic) system profiles, it is possible for

a carefully-designed dual-pulse PPM scheme to be 20% more

bandwidth efficient than the optimal single-pulse PPM, with

the same energy efficiency.

II. SYSTEM MODEL

Consider an FSO communication system that employs intensity modulation with direct detection. Light emitting diodes

(LED) or laser diodes (LD) are used as transmitters, and

avalanche photodiodes (APD) involving photon counting is

used as the receiver.

As a laser beam shoots through the turbulent atmosphere,

the intensity of the optical field experiences random fluctuations, commonly known as atmospheric scintillation or

atmospheric turbulence. The scintillation index is a rather

complicated function, which is mostly governed by the fluctuation of the index of fraction (due to temperature variations)

and the propagation distance, but is also related to the beam

parameters, and the heights of the transmitter and receiver.

A. Strong Atmospheric Scintillation

The log-normal fading model is rather accurate for clear-air

scintillation with a fairly short transmit range, but its accuracy

fades when the transmit distance increases beyond 100m and

when the scintillation indexes exceeds 0.75. Researchers have

shown that strong atmospheric scintillation may be modeled

as negative exponential distribution, which corresponds to a

scintillation index of 1.

The notations used in the discussion are listed as follows.

M the number of slots per PPM symbol, m =

log2 M

Ts the PPM symbol duration

τ the PPM slot time

Tθ the PPM symbol guard time

Ks the received photon count of a PPM “mark” slot

Kb the background-noise/radiation photon count per

PPM “space” slot

Γ(Ks) the equivalent signal-to-noise ratio (SNR) for the

optical detection process

Pe symbol error rate for single-pulse PPM (i.e.

misdetecting a mark)

Pdp

e symbol error rate for dual-pulse PPM

Pb bit error rate for single-pulse PPM

Pdp

b bit error rate for dual-pulse PPM

We have the following additional comments:

(1) In an ideal situation, the slot time τ relates to the symbol

duration Ts by Ts = M τ, where a slot may be either a signal

slot (“mark”) or an empty slot (“space”). In practice, however,

there may be constraints on pulse repetition frequency (PRF)

due to the physical limitation of the laser diodes, such that

two pulses may not be shoot immediately back to back. This

results in a a minimal time interval requirement between any

two consecutive marks, which necessitates the allocation of

NASA/TM—2012-216042 2
guard time Tg between two symbols. Hence, Ts = M τ + Tg.

In practice, the guard time Tg may be as small as one PPM

slot interval, or as large as 25% of the M-PPM slots: τ ≤

Tg ≤ Mτ/4.

The APD direct-detection receiver performs a photodetection process, which can be modeled fairly accurately as a

Poisson point process over M distinct and disjoint counting

intervals (M PPM slots) . The probability for each count

component is Poisson distribution with an average value

Kb = λbτ for the space slots and Ks + Kb = (λs + λb)τ for

the mark slots. For simplicity, the arrival rates λs and λb are

assumed constants over a PPM slot. λs is related to several

factors, including the quantum efficiency of the detector η,

the intensity of the received optical field I, the detector area

Ad, the wavelength of the light v, and Planck’s constant:

λs = ηAdI

hv .

(2) In a perfect “clear-air free-space” case, such as intersatellite communication outside of the earth atmosphere, I and

λs remains a constant throughput a transmission session, and

hence Ks follows a Poisson distribution. However, when the

signal travels through (part of) the atmosphere, the optical intensity I is no longer invariant, but becomes a random variable.

Specifically, under strong atmospheric scintillation, I may be

well-approximated by a negative exponential distribution, and

consequently, the photon count per PPM slot also follows a

negative exponential distribution, whose pdf is given by

f(Ks) =  1

λsτ exp 

− Ks

λsτ



, for Ks ≥ 0,

0, otherwise. (1)

where λsτ = E[Ks] is the average photon count per PPM

mark slot.

(3) The equivalent SNR Γ(Ks) is given by

Γ(Ks) = K2

s

FKs + Kn

(2)

Where Kn denotes the overall impact of thermal noise and

background photon radiation, and is a function of the receiver

temperature (in degrees Kelvin) T o, the Boltzmann constant

κ, the average APD gain g, the noise factor of the APD F,

the APD load resistance Rl, and the PPM slot duration τ and

the background photon count Kb:

Kn = 2FKb + 4κT oτ /(Rle2g2). (3)

III. ERROR RATE ANALYSIS

A. Symbol and Bit Error Rate for Single-Pulse PPM

Consider a PPM mark with Ks signal photons and thermal

noise. The APD performs a threshold detection, whose probability of miss-detecting a mark can be characterized by the

conventional Gaussian tail function [13]. Specifically, given

the instantaneous SNR Γ(Ks), the (conditional) probability of

misdetecting a PPM mark (i.e. missed detection) is given by

Pe(Ks) = Q

Γ(Ks)

, (4)

= 1

π

 π/2

0

exp 	− K2

s

2(FKs + Kn) sin2(θ)




dθ. (5)

where Q(·) is the Gaussian-Q function defined as:

Q(x) = 1

√2π

 ∞

x

e−t

2/2dt, (6)

= 1

π

 π/2

0

e−x2/(2 sin2(θ))dθ. (7)

Because of the atmospheric scintillation, the photon count

per PPM mark Ks follows some fading distribution. Specifically, under strong atmospheric scintillations, Ks follows the

negative exponential distribution as shown in (1), and the

probability of misdetecting a PPM mark can be evaluated by

averaging Pe(Ks) over Ks:

Pe =

 ∞

0

Pe(Ks)f(Ks)dKs. (8)

Substituting (5) and (1) in (8), we get

Pe = 1

λsπ

 π/2

0

 ∞

0

exp 	 −K2

s

2(FKs + Kn) sin2 θ

− Ks

λs




dKs

 
  A(θ)

dθ.

(9)

Let x

Δ

= FKs + Kn, we can simplify A(θ) to

A(θ) = 1

F

 ∞

0

exp 	 (x − Kn)2

2F2 sin2 θx − x − Kn

F λs




dx, (10)

= 1

F exp 	KnF

 1

λs

+

1

F sin2 θ




 ∞

0

exp 	 x

F λs




exp 	−ax − b

1

x




dx, (11)

where a = − 1

2F 2 sin2 θ , and b = − K2

n

2F 2 sin2 θ .

Using the Taylor expansion exp − x

F λs



=

∞

v=0

1

v!

 −x

F λs

v

, the expression in (11) becomes

A(θ) = 1

F exp 	KnF

 1

λs

+

1

F sin2 θ




∞

v=0

(−1)v

v!(F λs)v

 ∞

0

xvexp 	−ax − b

1

x




dx. (12)

Using the following equality [14],

 ∞

0

xvexp− ax − b

1

x



dx = 2 	 b

a


(v+1)/2

Kv+1

2

√

ab,

(13)

where Kv+1(·) is the modified Bessel function of the second

kind of order (v + 1), we get

A(θ) = 2

FKn

exp 	Kn

F

 1

λs

+

1

F sin2 θ




∞

v=0

1

v!

	

− Kn

F λs


v

Kv+1 	 Kn

F2 sin2 θ




. (14)

NASA/TM—2012-216042 3
Inserting (14) into (9), we get the symbol error probability

of M-slot PPM:

Pe = 2βeβ

K2

nπ

∞

v=0 

1

v!

(−β)

v

 π/2

0

exp (B(θ)) Kv+1 (B(θ)) dθ,

(15)

where β = Kn

F λs and B(θ) = Kn

F 2 sin2 θ .

In the case of negligible background radiation (known in

the optical jargon as the quantum-limited case), we have

Kn = 0. For high data-rate free-space optical communication

under the negative exponential fading, large Ks are typically

used to achieve a low probability of error. In such cases,

it is reasonable to assume zero or nearly-zero background

radiation/noise at the cost of a reduced effective signal strength

[10]. The symbol error rate Pe in (8) can then be significantly

simplified to [10]

Quantum-limited: Pe = 1

π

 π/2

0

sin2 θ

Ks

2F + sin2 θ

dθ, (16)

= 1

2



1 −

 Ks

Ks + 2F



. (17)

It is also worth noting that, in the quantum-limited case,

photons counts (above the detection threshold) can only occur

in the mark slot, such that the probability of detecting nonzero photon counts in the space slot is zero. In other words, the

APD detection error is highly asymmetric with a probability

Pe for misdetecting a mark, but zero probability of misdetecting a space. As shown in Fig. 1, the FSO channel therefore acts

like a Z-channel with a mark-to-space cross-over probability

of q = Pe. In general, for a properly-tuned practical FSO

system, even though there may be non-zero probability of

misdetecting a space, the probability of this type of error is

usually magnitudes lower than the probability of misdetecting

a mark, such that it can be safely ignored.

Space

Mark

Space

Mark

q

1−q

1

Fig. 1. Z-channel model for PPM FSO communications in the quantumlimited case, where the asymmetric cross-over probability q = Pe.

To compute the bit error rate, we consider the ensemble

average scenario, where the labeling for the PPM symbol is

arbitrary, and a misdetection could cause the demodulator to

output a random PPM symbol. As such, the bit error rate of

the M = 2m-slot PPM takes the form of:

Pb =

m

k=1 k

m

k



m2m Pe, (18)

≈ Pe/2. (19)

B. Symbol and Bit Error Rate for Dual-Pulse PPM

In the dual-pulse PPM case, a symbol error occurs as soon

as any one of the two marks is misdetected. Let Pe be the error

probability of misdetecting a mark in an M-slot PPM. Since

the probability of misdetecting a space is negligible compared

to Pe, we have the following expression for the dual-pulse

PPM SER:

Pdp

e = 1 − (1 − Pe)

2 = 2Pe − P2

e , (20)

where the superscript “dp” stands for dual-pulse.

We now evaluate the bit error rate for dual-pulse PPM

systems. Consider M = 2m PPM slots. There exist M(M −

1)/2=2m−1(2m −1) possible dual-pulse symbols, and hence

can convey log2(2m−1(2m − 1)) = 2m − 2 bits per dualpulse symbol. Clearly, the bit error rate of a dual-pulse PPM

modulation relates to the actual labeling scheme. We consider

two classes of dual-pulse labeling schemes:

• Class I: The set of 22m−2 valid dual-pulses are randomly

selected, and each dual-pulse combination is randomly

but uniquely mapped to a length-(2m − 2) bit sequence.

Such a dual-pulse modulation scheme represents the

ensemble average performance of all the dual-pulse PPM

schemes. An example of M = 16 is shown in Fig. 2(a).

• Class II: Class II labeling is a subset of Class I labeling,

by setting the constraint that one pulse must be picked

from, for example, the first M/2 half slots, and the other

must be picked from the second M/2 half slots. Each

pulse is separately labeled with log2(M/2) = m − 1

bits, so that the two pulses combined lead to an m-bit

sequence. The result is like two M/2-slot single-pulse

PPM symbols concatenated; see Fig. 2. This type of

labeling scheme would in general make the mapping table

less complex (than the more general case in Type I), as

well as reduce the number of bit errors (in the presence

of a symbol error).

Single−pulse

Dual−pulse (Type II)

Dual−pulse (Type I)

Fig. 2. Single-pulse PPM vs dual-pulse PPM. (M = 16 slots per symbol)

Class I: Due to the random selection of dual-pulses and

their random labels, if a dual-pulse symbol errs, it will incur,

on average, the same number of erroneous bits, regardless of

the detection error involves only one mark or both marks. The

average bit error rate can be computed by

Pdp

b =

2m−2

k=1 k

2m−2

k



(2m − 2)22m−2 Pdp

e , (21)

≈ Pe − P2

e /2. (22)

NASA/TM—2012-216042 4
Class II: Since each mark specifies (m−1) bits, independent

of the other, there is a difference between the one-mark error

and the two-mark error. The average number of bit errors

induced by an arbitrary one-mark error (with the other mark

correctly detected) is m−1

k=1 k

m−1

k



/2m−1, If both marks err,

then the number of bit errors gets doubled. Combining these

cases, we obtain the average bit error rate of Class-II dualpulse PPM

Pdp

b = 2Pe(1 − Pe)

m−1

k=1 k

m−1

k



(2m − 2)(2m−1) + P2

e

2

m−1

k=1 k

m−1

k



(2m − 2)(2m−1)

,

(23)

≈ P2

e /2 + Pe(2 − Pe)/2 = Pe/2. (24)

It should be noted that the SER and BER expressions for

the M = 2m-slot dual-pulse PPM in (20), (22) and (24) are

evaluated using the same mark energy (i.e Ks photons per

mark) as the single-pulse PPM. Since dual-pulse PPM has two

marks per M-slot symbol and carries (2m − 2) bits/symbol,

when comparing it to single-pulse PPM that has one mark and

carries m bits/symbol, an energy penalty of m/(m − 1) must

be added to the dual-pulse PPM.

C. Capacity of FSO Channels

Before numerically assessing the performance of singlepulse and dual-pulse PPM, let us analyze the capacity of the

Z-channel FSO communication systems depicted in Fig. 1.

Since the Z-channel is not a symmetric channel, the optimal

binary input distribution is therefore non-uniform. Let the

input X be a general Bernoulli distribution with the probability

of mark (“1”) being α. The output Y also follows a Bernoulli

distribution with the probability of mark being α(1 − q).

Following the definition of the capacity, we get

Cz(q) = max α I(X; Y ) = max α (H(Y ) − H(Y |X)), (25)

= max α



H(α(1 − q)) − αH(q)



, (26)

where q

Δ

= Pr(Y = 0|X = 1) = Pe is the crossover

probability, and H(x) = −x log2 x − (1 − x) log2(1 − x) is

the binary entropy function.

With some simplification, we can show that the maximum

value of CZ(q) is given by

Cz(q) = H

	 1

1+2H(q)/(1−q)




− H(q)/(1 − q)

1+2H(q)/(1−q) , (27)

= log2(1 + (1 − q)qq/(1−q)

), (28)

which is attained for α = 1/(1−q)

1+2H(q)/(1−q) . For very small q, the

capacity can be well approximated to

Cz(q) ≈ 1 − H(q)/2. (29)

IV. SINGLE-PULSE VS DUAL-PULSE PPM

To establish the true value of DPPM, we now provide a

multi-facet comparison between single-pulse and dual-pulse

PPM. We are aware of a couple of previous studies, which,

TABLE I

INFORMATION-THEORETIC PAYLOAD FOR l-PULSE PPM

M = 8 M = 16 M = 32

l # of sym bit/sym # of sym bit/sym # of sym bit/sym

1 8 3 16 4 32 5

2 28 4.81 120 6.91 496 8.95

3 56 4.81 560 9.13 4960 12.28

4 70 6.13 1820 10.83 35960 15.13

5 4368 12.09 201376 17.62

6 8008 12.97 906192 19.79

coming from fairly idealistic premises, show that going dualpulses does not really provide additional benefits over singlepulse (e.g. [4]). Our assessment below will consider both ideal

and practical scenarios.

From the information-theoretic perspective, an M-slot PPM

can support M single-pulse symbols, up to M

2

 dual-pulse

symbols, and up to M

l

 l-pulse symbols. Clearly, as long as

l does not increase beyond M/2, the larger the value of l, the

larger the number of l-pulse symbols that are available, and

the larger the payload there is to deliver (measured in terms of

the number of information bits per modulation symbol). This

speaks in favor of multi-pulse (many-pulse) PPM, and the gain

in the payload can be quite substantial especially for large M.

For example, Table I lists the number of available symbols

and the payload for M = 8, 16, 32 and l = 1, 2, 3, 4, 5, 6. As

we can see, with M = 32, single-pulse PPM can carry 5 bits

per symbol, whereas quad-pulse PPM can carry three times

more bits per symbol, and 6-pulse PPM can carry close to

four times as many bits per symbol.

It should be noted that the higher information-theoretic

payload (for the same M) comes at the cost of a (considerably) higher complexity as well as a (small) performance

degradation. An l-pulse PPM symbol has roughly l times as

high as a detection error as that of a single-pulse symbol. For

a fair evaluation between single-pulse and dual-pulse PPM, we

plot their error rate performance for different values of M, in

Figures 3-6. In all of the cases, APD noise factor is set to

F = 9, and the X-axis denotes the normalized energy (i.e.

photon count per information bit), measured in dB.

Fig. 3 demonstrates the BER and the SER for single-pulse

PPM, and Fig. 4 does dual-pulse PPM. In either case, BER

is slightly better than the respective SER, and for dual-pulse

PPM, Class-II BER is slightly better than that of the general

ensemble average (i.e. Class-I). It should be noted that, as M

increases, although the error rate performance also improves,

the gain is very small and practically negligible for high M.

Hence, it is fair to say that increasing M brings in only

marginal energy efficiency.

Now comparing the SER and the BER curves between

single-pulse PPM and dual-pulse PPM in Fig. 5 and Fig. 6, we

observe that, for the same value of M, (i) dual-pulse incurs

a slightly higher SER than single-pulse; and (ii) the average

BER of Class-II dual-pulse PPM is extremely close to that of

single-pulse PPM, and the average BER of Class-I dual-pulse

PPM is slightly higher.

NASA/TM—2012-216042 5
10 20 30 40 50 60 70

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Average photon count per data bit (dB)

Error rate for single−pulse PPM

SER and BER for single−pulse PPM, M=2,4,8,16,32,64

SER

BER

M=2

 4

 8

 16

 32

 64

Fig. 3. SER (red lines) and BER (blue lines) for single-pulse PPM. From

top down: M = 2, 4, 8, 16, 32, 64.

10 20 30 40 50 60 70

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Average photon count per data bit (dB)

Error rate for dual−pulse PPM

SER and BER for dual−pulse PPM, M=4,8,16,32,64

SER

BER − Class I

BER − Class II

M=4

 8

 16

 32

 64

Fig. 4. SER (red) and BER (blue for Class-I, and green for Class-II) for

dual-pulse PPM. From top down: M = 4, 8, 16, 32, 64.

It is then tempting to conclude that, disregard the higher

complexity, dual-pulse PPM is advantageous over singlepulse PPM in that it provides a considerably higher payload

with a minor performance degradation. However, on second

thought, the comparison has not been entirely fair. We caution

that unlike phase shift keying (PSK) and quadratic amplitude modulation (QAM), where a higher modulation order

implies a more efficient bandwidth utilization (measured in

bit/Hz/second), increasing the order of PPM also cause an

increase in the symbol interval (for the same slot width τ)

and hence a reduced bit/Hz/second. For example, 8-slot singlepulse PPM with slot width τ (seconds) and frequency band B

(Hz), delivers a bandwidth efficiency of 3

8

1

Bτ bit/Hz/s, whereas

going to M = 16 reduces the bandwidth efficiency to 1

4

1

Bτ

bit/Hz/s, and M = 32 further reduces it to 5

32

1

Bτ bit/Hz/s.

Hence, for single-pulse PPM, the most bandwidth efficient

10 15 20 25 30 35 40 45

10−5

10−4

10−3

10−2

10−1

Average photon count per data bit (dB)

Symbol error rate

SER for single−pulse and dual−pulse PPM, M=8,64

single−pulse

dual−pulse

M=8

M=64

Fig. 5. SER comparison between dual-pulse (blue) and single-pulse (red)

PPM. From top down: M = 8, 64.

10 15 20 25 30 35 40 45

10−5

10−4

10−3

10−2

10−1

Average photon count per data bit (dB)

Bit error rate

BER for single−pulse and dual−pulse PPM, M=8,64

single−pulse

dual−pulse −− Class I

dual−pulse −− Class II

M=8

M=64

Fig. 6. BER comparison between dual-pulse (green for Class-I, blue for

Class-II) and single-pulse (red) PPM. From top down: M = 8, 64.

case is M = 2 or 4, both delivering 1

2

1

Bτ bit/Hz/s. Now

consider the dual-pulse case. From Fig. 2 and all the performance curves, Type-II M-slot dual-pulse PPM is essentially

M/2-slot single-pulse PPM, where two symbols of the latter

concatenatively form one symbol of the former. Since the more

general dual-pulse case (i.e. Class-I) is even worse than ClassII performance-wise and complexity-wise, this leads to the

following immediate implications: first, there is no gain to go

from single-pulse to dual-pulse; and second, there is no need

to go for large M, and the 4-slot single-pulse PPM is both

simple and most bandwidth efficient.

To see this, Fig. 7 evaluates the “capacity” of different PPM

schemes. The X-axis denotes the power efficiency, measured

in photon count per information bit (dB), and the Y-axis

denotes the bandwidth efficiency, measured in normalized

NASA/TM—2012-216042 6
bit/Hz/second. The capacity of the Z-channel for both the

optimal input distribution and the uniform input distribution

are shown. Since here the crossover probability q is the

probability of misdetecting a mark, which is upper bounded

by 0.5 (i.e. random guess), the capacity can never drop to 0. In

fact, for optimal input, the limit is Cz(0.5) = 0.3219 bit/Hz/s,

and for uniform input, that is 0.3113 bit/Hz/s. The PPM

systems are evaluated at BER of 10−5, with uniform input.

We see that as M increases, the energy-efficiency increases

slightly, but the bandwidth-efficiency considerably decreases.

This clearly speaks for the preference to stay with low M, and

4-slot single-pulse PPM, or, 8-slot dual-pulse PPM, strikes the

best balance between bandwidth and energy efficiency.

−40 −20 0 20 40 60 80 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy efficiency (average photon count per data bit (dB)

Normalized bandwidth efficiency (bit/Hz/s)

Capacity of Z−channel, optimal input

Capacity of Z−channel, uniform input

M=4 M=2

M=32

M=64

M=8

M=16

Fig. 7. Bandwidth efficiency vs energy efficiency for singel-pulse PPM.

The previous conclusions would be all valid if there is no

requirement for guard time Tg. In practice, many laser emitting

systems have a pulse repetition frequency (PRF) constraint,

which specifies how frequently two laser pulses (of a decent

strength) can be generated, and which translates to the requirement for a minimum and a maximum time separation between

two pulses. This practicality issue immediately changes the

picture in two ways. First, in the case of single-pulse PPM, a

very small M would result in a very low channel utilization,

as a considerable amount of time is spent to guard between

two very short symbols. Hence, an appropriate value of M

should be selected in order to deliver the best throughput under

a given PRF constraint. Second, now there is benefit to be

gained by going from (M/2)-slot single-pulse PPM to Mslot dual-pulse PPM. As shown in Fig. 8, with the appropriate

system specifications, there exists an adequate pool of dualpulses meeting the separation requirement, such that explicit

guard time is needed only after every M slots. For example,

for M = 8, it is possible to design dual-pulse PPM with a

minimum of 12.5% separation between the two pulses (i.e.

1 slot separation); and for M ≥ 16, dual-pulse PPM with

a minimum of 25% separation (i.e. M/4 slots) is attainable

between the two pulses. On the other hand, single-pulse PPM

would need guard time after every M/2 slots. Clearly, dualpulse enhances the channel utilization and hence the data

throughput. When there is substantial guard time constraint,

such as a minimum of 16 slots, then dual-pulse PPM can

promise to be 20% more bandwidth efficient than single-pulse

PPM, while delivering the same energy efficiency.

single−pulse PPM

dual−pulse PPM

GT

GT GT

M PPM slots M PPM slots

M/2 slots M/2 slots M/2 slots

Fig. 8. Illustration of guard time (GT) and channel utilization for single-pulse

and dual-pulse PPM.

V. CONCLUSION

We have evaluated the error rate performance and the

bandwidth effiency for dual-pulse PPM as well as for singlepulse PPM under strong turbulent channels. We show that

without any guard time constraint, there is no benefit of

going from single-pulse to dual-pulse, and that 4-slot singlePPM strikes the best trade-off between energy efficiency and

bandwidth efficiency. When there is substantial guard time

constraint, then dual-pulse PPM can effectively improve the

channel utilization and hence the system throughput.

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