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TMO Progress Report 42-135 November 15, 1998
T-PPM: A Novel Modulation Scheme for Optical
Communication Systems Impaired by
Pulse-Width Inaccuracies
K. Kiasaleh1and T.-Y. Yan2
M-ary pulse-position modulation (PPM) has received considerable attention for
direct-detection photon communications over unguided channels. The analysis gen-
erally assumes that the signaling set is orthogonal. However, the orthogonality of
the signaling set will be destroyed by the finite area and bandwidth of optical detec-
tors, resulting in severe intersymbol interference. This article presents the analysis
of a trellis-based pulse-position modulation (T-PPM) scheme for photon commu-
nications with non-rectangular pulses. The novelty of the scheme includes the use
of a set-partitioning methodology to increase the minimum distance using a simple
convolutional encoder. The Viterbi algorithm is used at the receiver to separate the
signaling set as part of the demodulation process. It has been shown that T-PPM
will restore performance losses due to reduced peak intensity during the detection
process. Furthermore, for a large range of background radiation levels, the average
number of photons per information bit for T-PPM is smaller than that of the regu-
lar PPM. Numerical examples show that for a symbol-error rate of 10−3when the
received pulses extend over 4 PPM slots, the average laser energy per symbol for
256-ary T-PPM could be reduced by as much as 2 dB.
I. Introduction
Pulse-position modulation (PPM) has received considerable attention as the modulation of choice for
direct-detection optical communications over unguided optical channels. There are several key aspects of
this modulation that are critical to the deployment of this modulation scheme for deep-space communica-
tions. First, the presence of a pulse in the symbol frame regardless of the transmitted symbol benefits the
clock recovery subsystem, whereas an on–off-keying (OOK) system may suffer a synchronization loss if a
sequence of 0’s is encountered. Also, if nonreturn-to-zero (NRZ) pulses are used, a sequence of 1’s also
can disrupt the synchronization subsystem of an OOK system. Unlike the OOK scenario, one does not
require a priori knowledge of the signal or background noise radiation levels to implement an optimum
PPM receiver. The other key requirement of systems considered for space applications is the peak laser
power level that must be large enough to survive huge deep-space losses. For this reason, Q-switched
lasers typically are employed for such applications. The current technology, however, does not support
a scenario where a Q-switched laser can be toggled between the “on” and “off” states at a high rate,
1Erik Jonsson School of Engineering and Computer Science, The University of Texas at Dallas, Richardson, Texas.
2Communications Systems and Research Section.
1
limiting severely the data rate that can be supported using an OOK scheme. Hence, an M-ary PPM
with large Mis more suitable than its OOK counterpart for deep-space applications.3Finally, it can
be shown that M-ary PPM for large Mis more energy efficient than its OOK counterpart, which is of
critical importance in a deep-space optical communication environment where energy consumption is a
key constraint of the channel.
For many deep-space applications, due to large distances and pointing inaccuracies, the signal level
received at the detector is significantly attenuated so that only a small number of photons per PPM slot
interval are typically observed. Since thermal noise is present at the receiver, the detection of signal pulses
in the above condition is significantly hampered. For this reason, avalanche photodetectors (APDs) are
utilized to boost the signal level over the additive noise level present at the receiver. An APD, in principle,
magnifies each incident photon to a large number of post-detection electrons. One major problem with
today’s APDs is the excess noise factor of the APD, which manifests itself as a random gain effect. That
is, the ratio of the number of emitted electrons in response to an incident photon is a random variable
with known statistics. Unfortunately, the complicated statistics of the APD response to incident photons
do not lead to a closed-form solution for the error rate of optical PPM communication systems.
One key approximation for the statistics of the number of emitted electrons when the received number
of photons obeys the Poisson statistics is due to Webb [1]. This approximation is shown to be fairly accu-
rate over a wide range of conditions, and for this reason, this approximation is widely used to assess the
error rates of various optical communication systems with APDs. When the number of incident photons
in a slot interval is fairly large (as compared with the APD noise figure) and the APD possesses a large
average gain, it has been shown that the APD’s statistic could be approximated by a Gaussian proba-
bility density function (pdf) [2,3]. This approximation leads to a closed-form bit-error-rate expression
for the PPM channels and usually is valid when large background radiation levels are present (daytime
operation). However, for a large number of cases when the background radiation level is small, the Gaus-
sian approximation fails to accurately predict the statistics of an APD detector. In fact, for a small
number of incident photons, the probability mass function (PMF) of the number of released electrons
significantly departs from a Gaussian form [2,3]. In that case, one has to resort to the Webb–McIntyre–
Conradi (WMC) pdf, which does not lead to a closed-form expression for the probability of error [1].
More significantly, the resulting expression requires a prohibitively large computing time.
In some recent studies, via extensive simulation, it has been demonstrated that when the thermal
noise level is non-negligible, the number of signal photons required to achieve a symbol-error rate in a
range of 10−2to 10−3using the Gaussian-approximated model and that predicted by the WMC model
are somewhat similar for background noise levels in excess of 1 photon per PPM slot. Since, for most
applications of interest, one observes in excess of 1 background photon per PPM slot, the Gaussian
model seems to be a reasonable model to exploit for the above scenario. This by no means implies that
the Gaussian assumption is a valid approximation for all symbol-error-rate levels. It only underscores
the negligible loss/gain in the signal photon count associated with the Gaussian assumption for symbol-
error rates in the range of 10−2to 10−3. It also is imperative to note that, for most applications, an
uncoded symbol-error rate of 10−2to 10−3is of interest. Obviously, with the inclusion of a forward error-
correcting code (FEC), a symbol-error rate of 10−6, which is required to support data communication,
can be achieved.
Another generic property of optical sensors for detecting narrow laser pulses is the requirement of
high-speed electronics to convert optical signals to electrical currents. In general, for deep-space direct-
detection applications, larger detector areas potentially could collect more photons, thus increasing the
3In an M-ary PPM scenario with large M, the laser remains in an “on” state for 1/M th of the symbol duration, allowing
for log2Mbits to be transmitted in a symbol interval. In an OOK scheme, given the same symbol duration (note that
there exists an upper limit on how fast the laser can be toggled between the two states), only 1 bit can be transmitted
over the same symbol interval.
2
number of photons per information bit. Unfortunately, larger detector sizes imply a lower bandwidth of
the detector, which tends to smear the observed narrow pulses over several PPM slots. The proposed
T-PPM excels in this area by allowing overlapping of the PPM signaling set. That is, as will be demon-
strated in this article, the proposed T-PPM could circumvent this effect without degrading the system
end-to-end performance. Furthermore, it will be shown in the ensuing sections that T-PPM can achieve a
performance similar to that of its PPM counterpart at a reduced number of photons per information bit.
In addition, T-PPM can accommodate lower peak laser power during slot detection by integrating ener-
gies over several slots. The lower peak power is of critical importance in energy-constrained, deep-space
communication systems.4
In this article, we are interested in assessing the impact of T-PPM on enhancing the end-to-end
performance of optical communication systems with pulse-width inaccuracies, which often are present
when high-power Q-switched lasers are employed. The presence of such inaccuracies results in a significant
degradation in the overall performance of conventional PPM systems.
II. System Model
For an M-ary PPM system, the transmitted laser power, P(t), may be described as
P(t)=Ps
∞
X
q=−∞
h(t−CqTslot −NqTs) (1)
where Psis the peak laser power in watts; h(t) is the pulse shape, which is confined to a PPM symbol
duration Tss; Tslot is the PPM slot duration in seconds; Cqis the PPM symbol taking on the set
{0,1,···,M −1}with equal probability, with Mdenoting the PPM alphabet size; and N≥1isa
parameter that will be defined shortly. Note that Tslot =Ts/M s. Ideally, the pulse shape h(t) is confined
to a PPM slot duration. In that case, one obtains an ideal PPM signaling scheme. In the scenario
considered here, however, it is assumed that h(t) extends beyond a slot interval, causing significant
degradation in performance. Also, note that, in the formulation shown above, Nis introduced to allow
for one to generate a “silent” period in between symbol durations in which a PPM pulse is transmitted.
This typically is a requirement for operating a high-power laser when the transmission of a symbol is
interrupted by silent periods of several-symbol duration to allow for the recharging of the laser.
Given an unguided optical channel, one can describe the intensity of the received optical signal, λr(t),
as
λr(t)=λb+λs
∞
X
q=−∞
h(t−CqTslot −NqTs+²qTslot) (2)
where λs=Prη/hν denotes the peak intensity of the received optical signal in photons/s with η,Pr,
h, and νdenoting the quantum efficiency of the detector, the received power collected by the optical
assembly, Planck’s constant, and the operating frequency of the laser in hertz, respectively. Moreover,
λbin Eq. (2) denotes the intensity of the background noise in photons/s, and ²qdenotes the timing
inaccuracy present due to imperfect time tracking at the receiver, which is assumed to be confined to
[−1/2,1/2].
4For energy efficiency, one must resort to a large alphabet size PPM scenario. In that event, and for a relatively high data
rate, one must employ fairly short laser pulses. Given the fixed required energy, this implies that the laser must generate
a large peak power. The large peak power then becomes a major constraint of the system.
3
Since an APD is employed at the receiver, the collected primary electrons cause the generation of
a large number of secondary electrons. In general, one can assume that for the mth detected primary
electrons, the APD generates Gmsecondary electrons. The cumulative generating function (CGF) of Gm
is implicitly given by [4]
s=µGm(s)−bln ha+(1−a)eµGm(s)i(3)
where µGm(s) denotes the CGF, a=[1+κ(¯g−1)]/¯g, and b=1/(1−κ), with ¯gand κdenoting the average
gain and the ionization factor of the APD, respectively. Let F=[E¡G2
m¢]/¯g2=κ¯g+(2−[1/¯g]) (1 −κ)
denote the APD noise figure. Then, it can be shown that the moment-generating function of Gmmay be
expressed explicitly as [5]
MGm(s)= F
(F−1)2h1−p1−2(F−1) ¯gsi−¯g
F−1s+ 1 (4)
Note that, by definition, MGm(s)=eµGm(s).
The response of the APD, in terms of the current in the load resistor of the APD, may be modeled as
follows:
x(t)=
∞
X
j=−∞
GjR(t−tj)+n(t) (5)
where tjdenotes the occurrence time of the jth primary electron, governed by a Poisson point process;
R(t) is the current response of the APD; and n(t) is the thermal noise present at the receiver. The n(t)
commonly is modeled as a zero-mean, additive white Gaussian noise (AWGN) with a two-sided power
spectrum density (PSD) level of N0/2=(2kT0)/RLA2/Hz with k,T0, and RLdenoting Boltzmann’s
constant, the effective temperature of the receiver in kelvins, and the APD load resistance in ohms (Ω).
At the receiver, the PPM detector performs Mdistinct integrations of the observed current over the
Mslots of the PPM signal. The symbol associated with the slot with a maximum integrated current is
declared as the demodulated symbol. This detection strategy is optimum when the integrated current
is governed by Gaussian statistics or when a shot-noise-limited (Poisson-statistics) scenario is considered
[2]. It also has been demonstrated that when the statistics of the APD are approximated by WMC
distribution and the observed signal is further corrupted by AWGN, the optimum detection strategy
is identical to that used for purely Gaussian or Poisson channels [6]. Therefore, we use the detection
mechanism outlined above to render a decision regarding the transmitted symbol.5
Let us define Xl,n as the outcome of integrating x(t) over the interval Il,n =[lTslot +NnTs,(l+1)Tslot
+NnTs]. Note that Il,n denotes the lth slot of the nth transmitted symbol.6We then have
Xl,n =
(l+1)Tslot+NnTs
Z
lTslot+NnTs
x(τ)dτ;l=0,1,···,M −1 (6)
5In the presence of timing error, the above method is no longer optimal. Since in the ensuing analysis we consider negligible
timing error as compared with the pulse-width inaccuracies, we resort to the aforementioned detection mechanism.
6In the ensuing analysis, and without loss of generality, we consider N= 4 (each transmitted symbol is followed by a
silent period of 3Tss). The selection of N= 4 is dictated by the requirements that currently are imposed on high-power
Q-switched lasers that are being considered for deep-space communication.
4
Given that the APD’s current response is such that it can be approximated with a delta function, we
arrive at
Xl,n =eK2(l, n)+νN(7)
where νNis a zero-mean Gaussian random variable with variance
σ2
ν=2kT0
RL
Tslot (8)
eis the charge of an electron in C, and K2(l, n) is the number of secondary electrons observed over
Il,n. The PMF of K2(l, n) in its exact form originally was discovered by McIntyre [7] and was verified
experimentally by Conradi [8]. This PMF is given by
Pr {K2(l, n)=k2|K1(l, n)=k1}=
k1Γ³k2
1−κ+1
´
k2(k2−k1)!Γ µκk2
1−κ+1+k1¶
×·1+κ(¯g−1)
¯g¸k1+κk2/(1−κ)·(1 −κ)(¯g−1)
¯g¸k2−k1
(9)
where K1(l, n) is the number of primary electrons observed over the interval Il,n. Given that the primary
electron statistics obey a Poisson model, Webb has demonstrated that the PMF of the secondary electron
count may be approximated by
Pr ©K2(l, n)=k2|¯
K1(l, n)=¯
k1ª=1
(2πC2
1)1/2
1
1+¡k2−¯
k1¯g¢
C1C2
3/2
exp
−¡k2−¯
k1¯g¢2
2C2
1"1+¡k2−¯
k1¯g¢
C1C2#
(10)
where ¯
K1(l, n) is the average number of primary electrons observed over Il,n,C2
1=¯g2¯
k1F, and C2
2=
¯
k1F/(F−1)2. Given the Poisson statistics, then
¯
K1(l, n)=
(l+1)Tslot+NnTs
Z
lTslot+NnTs
λr(τ)dτ =Kb+KsYl,n (11)
where
Yl,n =1
Tslot
(l+1)Tslot+NnTs
Z
lTslot+NnTs
∞
X
q=−∞
h(τ−CqTslot −NqTs+²qTslot)dτ
5
and Kb=λbTslot denotes the average number of primary electrons observed over a PPM slot interval
due to background noise. Moreover, Ks=λsTslot. Note that the above formulation implies that the
average number of primary electrons due to signal power observed over the lth slot of the nth symbol,
i.e., KsYl,n, decreases as the pulses smear in time, maintaining the average number of primary electrons
observed over a symbol interval (i.e., Ks) constant. That is, we assume that the energy observed over a
symbol interval remains constant as pulses spread over the adjacent slots, leading to less observed energy
per slot.
In the ensuing analysis, we assume that ²qis negligible (since it is a fraction of a slot interval), whereas
the intersymbol interference caused by the shape of the nonideal pulse shape, h(t), is assumed to be quite
significant. More precisely, we assume that h(t) extends beyond a slot boundary and perhaps extends
over 2 to 4 slot intervals. This causes severe degradation in performance in the absence of a strategy
to circumvent intersymbol interference (ISI). In what follows, we assume that the laser pulse obeys a
Gaussian shape. Namely,
h(t)= Tslot
p2πσ2
h
exp
−µt−Tslot
2¶2
2σ2
h
(12)
where we have assumed that the laser pulse is centered around the midpoint of the slot interval and that
the pulse has a standard deviation of σhs. Moreover, note that R∞
−∞ h(τ)dτ =Tslot. This assumption
is motivated by the fact that if h(t) is replaced with an ideal nonreturn-to-zero (NRZ) pulse of unit
amplitude, a similar result will be obtained. This, in turn, implies that regardless of the value of σ2
h, the
total laser energy over a symbol interval remains constant. That is, as the pulses are smeared in time, the
peak laser power is reduced, maintaining the constant-energy assumption that is critical to space-borne
optical communication systems. Considering that h(t) extends over a few slot intervals and that we have
a silent period of 3Mslots, and assuming that Cn=j,wehave
Zl,j =Yl,n |Cn=j=1
σh√2π
(l+1)Tslot
Z
lTslot
exp
−µτ−(2j+1)Tslot
2¶2
2σ2
h
dτ (13)
Note that the integral on the right-hand side of the above equation is a function of land jonly.
III. Performance Analysis—PPM With Imperfect Pulse Shape
Given the above model, one can obtain the performance of a PPM system with imperfect pulses. A
key obstacle, however, is the absence of a tractable model for the statistics of the APD signal when
corrupted by an AWGN. In a recent simulation study, it was demonstrated that the Ksrequired for
an APD-detected 256-ary PPM signal to achieve an error rate in the range of 10−2to 10−3and that
predicted using a Gaussian model differ only slightly over a wide range for Kb(1 ≤Kb≤1000), allowing
one to benefit from the Gaussian approximation for the above range of symbol-error rates. Furthermore,
we benefit from a union bound to establish an upper bound on the error rate of the uncoded PPM system
with imperfect pulse shapes. To that end,
P(u)
PPM ≤1
M
M−1
X
j=0
M−1
X
l=0;j6=l
Pr (El,j) (14)
6
where we have assumed equally likely PPM symbols. In the above equation, Pr (El,j ) is the pair-wise error
probability of making a decision in favor of the lth symbol when the jth symbol actually is transmitted
and P(u)
PPM is the symbol-error rate of the uncoded PPM system. This error event is possible only when
the integrated current of the lth slot exceeds that of the jth slot. As noted earlier, we resort to Gaussian
approximation for the range of error rates stated previously. If one allows for such an approximation,
then Xl,n is a Gaussian random variable when conditioned on Cq(assume ²q= 0). Hence, the mean and
variance of this random variable will be of interest to establish performance. To that end,
ml,j =E{Xl,n|Cn=j}=e¯g(Kb+KsZl,j)+Idc Tslot (15)
and
σ2
l,j =Var{Xl,n|Cn=j}=(e¯g)2F(Kb+KsZl,j)+σ2
ν+eIdcTslot (16)
where E{.|Cn=j}and Var{.|Cn=j}are the expected value and the variance of the random variable
Xl,n conditioned on {Cn=j}and Idc denotes the surface dark current of the APD in amperes, which is
assumed to be non-negligible here. In that event, and considering that an erroneous decision in favor of
the lth symbol is rendered if the integrated APD current over the lth slot exceeds that of the jth slot
(note that this is a comparison between Xl,n and Xj,n, which are a pair of conditionally [when conditioned
on {Cn=j}] independent Gaussian random variables), we have
Pr (El,j)=1
2erfc
mj,j −ml,j
r2³σ2
l,j +σ2
j,j ´
;l6=j(17)
where erfc (x) is the complementary error function. Given that the pulse shape follows a Gaussian pattern,
Zl,j and Zj,j may be obtained in terms of erfc (.) or erfc(.). However, we note that
mj,j −ml,j =e¯gKs(Zj,j −Zl,j) (18)
and
σ2
l,j +σ2
j,j =(e¯g)2F[2Kb+Ks(Zl,j +Zj,j)]+2¡σ2
ν+eIdcTslot ¢(19)
are functions of Zj,j −Zl,j and Zj,j +Zl,j only. Therefore, Zj,j −Zl,j and Zj,j +Zl,j are only of interest.
With some effort, it may be shown that
Zj,j −Zl,j =1−erfc µTslot
2√2σh¶−1
2erfc
−
Tslot µl−j+1
2¶
√2σh
+1
2erfc
−
Tslot µl−j−1
2¶
√2σh
(20)
and
7
Zj,j +Zl,j =1−erfc µTslot
2√2σh¶+1
2erfc
−
Tslot µl−j+1
2¶
√2σh
−1
2erfc
−
Tslot µl−j−1
2¶
√2σh
(21)
Now, substituting Eqs. (20) and (21) in Eqs. (18) and (19), respectively, and then substituting the
resulting expressions in Eq. (17), we arrive at a closed-form expression for the pair-wise error rate of a
PPM system with pulse-width inaccuracies. The remaining task is to compute the upper bound using
Eq. (14). We postpone a discussion on the performance of the system using Eq. (14) to Section VI of this
article. Instead, we proceed to describe the proposed T-PPM system in the next section.
IV. T-PPM
It is important to note that the use of trellis-coded modulation (TCM) and overlapping PPM (OPPM)
to enhance the capacity of PPM channels was originally discussed in [9]. There are several aspects of the
present analysis that set it apart from its predecessor. First, the analysis in [9] assumes square shape
pulses, whereas we will concern ourselves with Gaussian-type pulse shapes in this analysis. Second, the
analysis in [9] assumes either quantum-limited or shot-noise-limited scenarios, whereas a more realistic
situation (as is assumed in what follows) calls for the inclusion of additive Gaussian noise and APD excess
noise factors.
In the previous section, we described the performance of a PPM system impaired by imperfect pulse
shapes. Needless to say, the impact of an imperfect pulse shape on the performance of a PPM system
may be quite severe for pulse shapes extending over several slot intervals. Hence, the introduction of
TCM is necessary to overcome the impact of severe ISI caused by imperfect pulse shapes. Due to the
regular shape of the pulse, and from Eq. (17), it can readily be seen that Pr (El,j ) is a decreasing function
of |j−l|. That is, the pair-wise error rate is a decreasing function of the “distance” between symbols
jand l, where distance is defined in terms of the number of PPM slots that exit between the positions
(in time) of the jth and lth symbols.7In view of this observation, one can proceed with a set-partition
strategy for the generation of T-PPM signals that is aimed at increasing the minimum distance of the
PPM constellation.
We begin by introducing A={1,2,3,···,M}as a set containing the uncoded PPM symbols. Note
that the distance between a pair of symbols jand lselected from Ais simply |j−l|. This set is further
divided into sets B0={1,3,5,7,···} and B1={2,4,6,···}. Note that a pair of symbols selected
from either one of these sets is at a minimum distance of 2, whereas the symbols selected from Aare at a
minimum distance of 1. Next, we proceed to subdivide set B0to generate sets C0={1,5,9,···}and C2=
{3,7,11,···}. Set B1also can be subdivided to form C1={2,6,10,···}and C3={4,8,12,···}. Finally,
we perform one last partitioning of the previous sets to form D0={1,9,17,···},D4={5,13,21,···},
D2={3,11,19,···},D6={7,15,23,···},D1={2,10,18,···},D5={6,14,22,···},D3={4,12,20,···},
and D7={8,16,24}. The entire process is depicted in Fig. 1 for M= 256. Note that symbols selected
from Djfor any jare at a minimum distance of 8. Furthermore, symbols selected from Djand Diare at
a minimum distance of |i−j|from each other. Although it appears that one can continue this process
in hope of increasing the minimum distance beyond 8, the minimum distance between a pair of paths
through the trellis (which ultimately dictates the overall performance) cannot be increased indefinitely
with further set partitioning. Also, since we are interested only in eliminating the impact of ISI, a
minimum distance of 4 is sufficient to establish orthogonality among the symbols in the set. That is,
when the imperfect pulse h(t) stretches over 2 to 4 PPM slots, a minimum distance of 4 slots among the
symbols selected from Cj(for all j) ensures that there exists no overlap among the pulses in the set.
7Since the jth symbol is represented by a pulse in the jth slot, the distance between the jth and lth symbols is |j−l|.
8
d
min
= 4
d
min
= 1
A
= {1,2,3, ..., 256}
01
01
01
B
0 = {1,3,5,7, ...}
B
1 = {2,4,6, ...}
C
0 = {1,5,9, ...}
C
2 = {3,7,11, ...}
C
1 = {2,6,10, ...}
C
3 = {4,8,12, ...}
d
min
= 2
01
D
0 = {1,9,17,...}
D
4 = {5,13,21,...}
D
2 = {3,11,19,...}
D
6 = {7,15,23,...}
D
1 = {2,10,18,...}
D
5 = {6,14,22,...}
D
3 = {4,12,20,...}
D
7 = {8,16,24,...}
d
min
= 8
010101
Fig. 1. Set partitions for the 256-ary PPM signal set.
Without loss of generality, we concern ourselves with rate 1/2 and 2/3 convolutional encoders (CEs)
for the generation of T-PPM signals. As will be shown later, the rate 1/2 CE with a 4-state trellis is
quite suitable for the problem at hand, although the rate 2/3 CE with an 8-state trellis can offer a larger
minimum distance (dmin). We note also that the alphabet size of the PPM is 256 (8 bits), which allows
one to encode 7 data bits per T-PPM symbol (for both the rate 1/2 and rate 2/3 CEs, see Figs. 2 and 3).
Given that for high-power lasers used in deep-space communications the frame rates usually are kept
constant, this implies a loss in the data rate by a factor of 7/8. This, however, is a relatively small price
to pay to circumvent the substantial ISI caused by imperfect pulses. The rate 1/2 and 2/3 convolutional
encoders along with their respective trellises are depicted in Figs. 2 and 3, respectively. Note that, for
the rate 1/2 CE case, a 4-state trellis is used and that the 2 bits produced by the encoder are used to
select a set from the four possible C-type sets: C0,C1,C2, and C3. The remaining 6 bits are used to
select a signal from the selected set (note that there are 26signals in any of the four C-type sets). For
the rate 2/3 code, which, as shown in Fig. 3, corresponds to an 8-state trellis, the 3 bits produced by
the CE are used to select one of the eight possible D-type sets. Given that the code is a rate 2/3 CE,
then the remaining 5 bits are used to select a signal from the selected set. Once again, note that there
are 25signals in any of the eight D-type sets. The above arrangement, then, leads to the existence of a
number of parallel paths in the trellises depicted in Figs. 2 and 3. That is, for each path in the trellis
shown in Fig. 2, there exist 64 parallel paths (the size of the C-type sets). This is a common feature of
TCM systems that is due to the existence of uncoded bits used to select a signal from the constellation.
The number of parallel paths for the trellis in Fig. 3 is 32, which is identical to the size of any of the
D-type sets.
V. T-PPM Performance Analysis
In Figs. 2 and 3, we have depicted the trellises for the rate 1/2 and 2/3 T-PPM systems with 4 and
8 states, respectively. When the rate 1/2, 4-state trellis is used, one easily can conclude that the minimum
distance between the all-zero path and any path that departs from and re-emerges with the all-zero path
for the first time is 4. The path that leads to the minimum distance of 4 is due to a symbol selected
from C0. For the 8-state trellis, we also have depicted the path (dotted line) that is at a minimum
distance of 5 from the all-zero path. Note that parallel transitions in this trellis are selected from the
D-type sets, and, hence, a minimum distance of 8 exists between any pair of parallel paths in Fig. 3. As
noted earlier, although we have been able to increase the minimum distance between parallel paths in the
trellis via further set partitioning, the minimum distance between any path in the trellis and the all-zero
path has not increased dramatically. Due to the complexity of the trellis in Fig. 3 and the fact that a
9
d
C
0
C
2
C
2
C
0
C
1
C
3
C
3
C
1
d
= 0
d
= 1
(00)
(01)
(10)
(11)
C
1
C
1
C
3
C
0
C
0
C
2
C
2
C
0
C
1
C
3
C
3
C
1
Fig. 2. The rate 1/2 convolutional encoder along with its 4-state trellis that uses the set partitioning of Fig. 1 for the
generation of the rate 7/8 T-PPM signals.
D
0
D
0
D
0
D
0
D
4
D
2
D
6
D
0
D
1
D
2
d
min
= 5
D
0
D
4
D
2
D
6
D
1
D
5
D
3
D
7
D
4
D
0
D
6
D
2
D
5
D
1
D
7
D
3
D
2
D
6
D
0
D
4
D
3
D
7
D
1
D
5
D
6
D
2
D
4
D
0
D
7
D
3
D
5
D
1
d
0
d
1 = 11
01 10
00
d
0
d
1
Fig. 3. The rate 2/3 convolutional encoder along with its 8-state trellis that uses the set partitioning of Fig. 1 for the
generation of the rate 7/8 TCM-PPM signals.
10
minimum distance of 4 is sufficient to establish orthogonality among the symbols in the set, in what
follows, we limit our discussion to the rate 1/2, 4-state trellis depicted in Fig. 2. To that end, let us
consider the parallel paths in the trellis. Given that for any path there exist 63 other parallel paths for
M= 256 in Fig. 2 (see Fig. 1), an upper bound on the error rate may be obtained with the aid of a union
bound. This upper bound is given by
P(TCM)
PPM ≤63 Pr (El,l+4)
where Pr (El,l+4) denotes the pair-wise error rate for a pair of PPM symbols with imperfect pulse shapes
that are separated by 4 time slots. When h(t) extends over 2 to 4 times the duration of a slot, then
Pr (El,l+4) is not a function of l.
VI. Numerical Results
To underscore the impact of imperfect pulse shapes, a pair of adjacent Gaussian-type pulses are plotted
in Figs. 4 through 6. The laser energy (the area covered by the pulse) within one slot time is computed to
be 98, 78.8, and 59.5 percent, respectively, in Figs. 4 through 6. Note that, for a slot duration of 20 ns, for
instance, the case depicted in Fig. 6 corresponds to an 80-ns pulse scenario. It is immediately obvious that
ISI increases in an exponential fashion with σh. Since we are interested in a scenario where h(t) is limited
to 2 to 4 times the duration of a time slot, we limit our analysis to the cases where 0 ≤σh≤0.6Tslot s.
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-4-3-2-101234
t
/
T
slot
GAUSSIAN PULSE
Fig. 4. A pair of adjacent Gaussian-type
pulses with s
h
= 0.2
T
slot.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-4-3-2-101234
t
/
T
slot
GAUSSIAN PULSE
Fig. 5. A pair of adjacent Gaussian-type
pulses with s
h
= 0.4
T
slot.
0.7
0.6
0.5
0.4
0.3
0.2
-4-3-2-101234
t
/
T
slot
GAUSSIAN PULSE
Fig. 6. A pair of adjacent Gaussian-type
pulses with s
h
= 0.6
T
slot.
0.1
0.0
11
We first consider the case where a symbol-error rate of 10−2is of interest, and subsequently we obtain
the required Ksto achieve the desired error rate for a given Kb. Without loss of generality, we limit our
discussion to the following system parameters: M= 256, Tslot = 20 ns, ¯g= 40, κ=0.007, T0= 100 K
(cooled receiver), RL= 146.65 kΩ, and Idc = 2 nA. In Figs. 7 and 8, the Ksrequired to achieve a
symbol-error rate of 10−2is plotted versus Kbwhen the above set of parameters is used. We note that
the Gaussian assumption considered here does not lead to performance measures that agree with the
simulation results using WMC statistics for the entire range of Kbwhen an error rate of 10−2is of
interest. In particular, when Kb<50 (Fig. 7), the Gaussian assumption is less reliable than for the case
of Kb>50 (Fig. 8). Nonetheless, the loss in Ksusing the Gaussian approximation is not significant,
and hence we proceed with our analysis using the results shown above. Before doing so, however, it is
imperative to note that the simulation results for the required Ksusing WMC statistics (for the above set
of parameters) are shown to be in close agreement with those predicted using the Gaussian assumption
for a wide range of Kbwhen a symbol-error rate of 10−3is considered [10]. Since an error rate of 10−3
is typically of interest, we consider the numerical results shown below for a 10−3error rate to be a good
approximation of the results obtained using the more realistic WMC statistics for modeling the APD
output statistics.
The first observation that can be made from Figs. 7 and 8 is that, when a severe ISI is present (see the
curves associated with 80- and 60-ns pulses), the performance degrades substantially as compared with
the perfect pulse-shape case. The second significant observation is that, in the event of having perfect
pulses, the performance improves using T-PPM. This may be attributed to the coding gain. Note that
1 of the 7 information bits almost always will be detected correctly, since the major error is due to parallel
transitions in the set.
We also note that the gain in performance as a result of using T-PPM increases almost exponentially
with an increase in σh. In Figs. 9 and 10, a similar set of results is depicted when an error rate of 10−3
is of interest. We focus on these results, since an error rate of 10−3typically is needed to ensure a coded
performance8of 10−6.
140
UPPER BOUND ON REQUIRED
K
s
120
100
80
60
40
20
160
0 5 10 15 20 25 30 35 40 45 50
K
b
Fig. 7. The
K
s
required to achieve a symbol-error
rate of 10-2 for 256-ary PPM with imperfect pulses
when
K
b
is less than 50.
s
h
= 0
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
UNCODED PPM
T-PPM
s
h
= 0
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
UNCODED PPM
T-PPM
240
220
200
180
160
140
120
260
50 100 150 200
K
b
Fig. 8. The
K
s
required to achieve a symbol-error
rate of 10-2 for 256-ary PPM with imperfect pulses
when
K
b
is more than 50.
100
80
60
UPPER BOUND ON REQUIRED
K
s
8It is anticipated that T-PPM will be used in conjunction with other more powerful coding schemes. That is, the information
bits provided to T-PPM may be viewed as coded channel symbols provided by an FEC encoder (such as a Reed–Solomon
encoder).
12
200
UPPER BOUND ON REQUIRED
K
s
180
160
140
120
100
80
220
0 5 10 15 20 25 30 35 40 45 50
K
b
Fig. 9. The
K
s
required to achieve a symbol-error
rate of 10-3 for 256-ary PPM with imperfect pulses
when
K
b
is less than 50.
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
UNCODED PPM
T-PPM
60
40
s
h
= 0
s
h
= 0
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
300
UPPER BOUND ON REQUIRED
K
s
250
200
150
100
350
50 100 150 200
K
b
Fig. 10. The
K
s
required to achieve a symbol-error
rate of 10-3 for 256-ary PPM with imperfect pulses
when
K
b
is more than 50.
UNCODED PPM
T-PPM
50
Before discussing such results, it is imperative to note that the proposed T-PPM system transmits
7 information bits, as compared with the 8 bits that are transmitted over each symbol duration for the
uncoded system. Hence, it is appropriate to consider the required number of photons per information
bit in comparing the T-PPM scenario with the uncoded PPM case. We then proceed to use the average
number of required photons per information bit as a measure of efficiency of the modulation scheme.
Hence, in Figs. 11 through 14, we depict the average number of photons per information bit that is
required to achieve the desired error rate for a wide range of background radiation photon counts. We
note, however, that the loss of 1 bit in order to achieve the desired performance, given that there exists a
limit on laser energy, is a reasonable consideration. Hence, we discuss the performance of T-PPM using
both the average number of photons as well as the average number of photons per information bit as a
function of background radiation level in what follows.
From Figs. 9 and 10, it can be concluded that when Kb= 1 (night operation), one requires Ks= 133
to achieve an error rate of 10−3when 80-ns pulses are used (20-ns PPM slots). This number is reduced to
Ks= 79 with T-PPM, a reduction of about 2.2 dB in the required laser energy. However, as noted earlier,
T-PPM conveys only 7 bits of information. If one uses the number of photons per bit, in that case the
required number of photons per bit reduces from 133/8=16.6to79/7=11.2 using T-PPM (see Fig. 13).
This is a substantial gain in the overall system efficiency. For Kb= 100, the Ksrequired to achieve an
error rate of 10−3when 80-ns pulses are used reduces from 249 to 187 using T-PPM, a reduction of nearly
1.2 dB in the required laser energy. The number of photons per bit improves from 31.3 to 26.7 using
T-PPM (see Fig. 14). It is important to note that for small pulse spreading (60-ns or smaller pulses), the
gain in performance is not substantial and, hence, the gain achieved using T-PPM is noticeable only when
substantial pulse-width inaccuracies are present. In fact, for all cases considered, the system efficiency in
terms of the number of photons per bit increases or remains the same when the pulse width is equal to
or less than 60 ns. However, when the pulse width is increased to 80 ns, a substantial improvement in
system efficiency is observed (see Figs. 11 through 14).
When an error rate of 10−2is of interest and Kb= 200, from Fig. 8 one can conclude that Ks=
250 photons is needed9when 80-ns pulses are used. This number is reduced to 209 photons for a T-
PPM system, a savings of 0.77 dB in the required laser power. The number of photons per bit improves
from 31.2 to 29.8 (see Fig. 12), a gain smaller than that observed for the 10−3error-rate scenario.
9Note that, for this large background radiation level, the Gaussian assumption is fairly accurate.
13
REQUIRED NO. OF PHOTONS PER BIT
15
10
20
0510
15 20 25 30 35 40 45 50
K
b
Fig. 11. The average number of photons per
information bit required to achieve a symbol-error
rate of 10-2 for 256-ary PPM with imperfect pulses
when
K
b
is less than 50.
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
s
h
= 0
UNCODED PPM
T-PPM
5
REQUIRED NO. OF PHOTONS PER BIT
30
25
35
50 100 150 200
K
b
Fig. 12. The average number of photons per
information bit required to achieve a symbol-error
rate of 10-2 for 256-ary PPM with imperfect pulses
when
K
b
is more than 50.
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
s
h
= 0
UNCODED PPM
T-PPM
15
20
10
REQUIRED NO. OF PHOTONS PER BIT
24
22
26
051015 20 25 30 35 40 45 50
K
b
Fig. 13. The average number of photons per
information bit required to achieve a symbol-error
rate of 10-3 for 256-ary PPM with imperfect pulses
when
K
b
is less than 50.
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
s
h
= 0
UNCODED PPM
T-PPM
18
20
16
14
12
10
8
6
REQUIRED NO. OF PHOTONS PER BIT
30
25
35
50 100 150 200
K
b
Fig. 14. The average number of photons per
information bit required to achieve a symbol-error
rate of 10-3 for 256-ary PPM with imperfect pulses
when
K
b
is more than 50.
s
h
= 0.2
T
slot
s
h
= 0.4
T
slot
s
h
= 0.6
T
slot
s
h
= 0
UNCODED PPM
T-PPM
15
20
10
40
Finally, in Figs. 15 and 16, we examine the performance of T-PPM when the slot duration is reduced
with the remaining parameters kept fixed. Given that the contribution of thermal noise and APD dark
current to the output of the APD have variances that are directly proportional to the integration interval
(slot duration), as one decreases the slot duration, a gain in performance (in terms of a reduction in
the number of photons per information bit required to achieve a given error rate, 10−3) is observed.
More significantly, as one decreases the slot duration, for a fixed-background-radiation intensity level, the
average number of received photons per slot decreases accordingly. As seen from these figures, for the
2-ns slot scenario, a significant improvement in performance is observed. This result is not surprising.
To elaborate, as one decreases the slot duration with the total number of signal photons per slot kept
constant (i.e., when a higher-power laser is utilized), the performance approaches that of a quantum-
limited system, leading to the considerable gain in performance observed in Figs. 15 and 16.
14
REQUIRED NO. OF PHOTONS PER BIT
16
14
18
0.0 0.5 1.0 2.0
Fig. 15. The average number of photons per
information bit required to achieve a symbol-error
rate of 10-3 for T-PPM with imperfect pulses
(background intensity less than 2.5 109
photons/s). For all cases, s
h
= 0.6
T
slot.
T
slot = 2 ns
10
12
8
20
BACKGROUND INTENSITY, photons/s ( 109)
2.5
22
T
slot = 20 ns
T
slot = 16 ns
T
slot = 8 ns
1.5 8
REQUIRED NO. OF PHOTONS PER BIT
25
20
30
234 6
Fig. 16. The average number of photons per
information bit required to achieve a symbol-error
rate of 10-3 for T-PPM with imperfect pulses
(background intensity more than 2.5 109
photons/s). For all cases, s
h
= 0.6
T
slot.
T
slot = 2 ns
10
15
35
BACKGROUND INTENSITY, photons/s ( 109)
7
40
T
slot = 20 ns
T
slot = 16 ns
T
slot = 8 ns
5910
VII. Conclusions
This article introduced a robust trellis-based pulse-position modulation (T-PPM) as a technique for
deep-space optical communication and analyzed its performance using union bounds. The analysis as-
sumes the use of a maximal-likelihood receiver for demodulation while the signaling pulses are allowed to
extend over several PPM slots. It has been shown that, using a simple convolutional encoder at the trans-
mitter and a Viterbi algorithm at the receiver, T-PPM restores the performance losses due to reduced
intensity during the detection process.
Furthermore, using the average number of photons per information bit as a performance measure,
T-PPM requires less energy than its regular PPM counterpart by affording a smaller PPM slot width.
Numerical examples show that, for a symbol error of 10−3when the received pulses extend over 4 PPM
slots, the average laser energy per symbol for 255-ary T-PPM could be reduced by as much as 2 dB. In
addition, the increase in the transmitter efficiency could be more profound if the pulse-width duration
became narrower.
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