604 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO.

4, APRIL 2006

Performance of Coherent DPSK Free-Space Optical

Communication Systems in K-Distributed Turbulence
Kamran Kiasaleh, Senior Member, IEEE

Abstract—Closed-form solutions for the bit-error rate of a free- can be modeled using K-related pdfs, which have been success-
space, heterodyne optical communication system is derived when fully used to model atmospheric turbulence deep into saturation
the optical beam is subjected to K-distributed optical turbulence. [8], [13]–[16]. In free-space optical communications over short
It is assumed that the scintillation index is confined to the range
(2,3) or that the number of scatterers in the propagation path is a ranges ( 1 km), it is conceivable that one may encounter the
random variable. conditions stated above. It is, therefore, of interest to charac-
terize the performance of such systems under a K-distributed
Index Terms—Coherent, free space, optical, turbulence.
regime.
It is noteworthy that in recent years, a number of models have
I. INTRODUCTION been introduced for the characterization of turbulence [17], [18].
These studies have not disputed the accuracy of K-distribution,

A TMOSPHERIC-induced scintillation is a major impair-

ment of free-space optical (FSO) communication links. In
particular, the impact of scintillation is to cause random vari-
but rather have offered alternative means of characterizing the
turbulent FSO channel which take into account the structure
constant and inner scale of the atmosphere. Interestingly, the
ations in signal intensity, leading to large signal fades. For the models presented in recent studies can be classified as gen-
most part, the studies presented in the literature focus on “clear- eralized K-distribution. The object of this letter is not to dis-
air” turbulence which results in a log-normal distributed signal pute the accuracy of the newly presented channel models. In-
intensity; for example, see [1]–[6]. Such a scenario is referred to stead, the objective, for the sake of completeness, is to provide
as the “weak turbulence” scenario, which is characterized by a the engineering and scientific communities with a closed-form
scintillation index less than 0.75. The log-normal model is also solution for the performance of coherent FSO systems under
valid for propagation distances less than 100 m, although some a K-distributed regime, which has been used widely to model
studies have shown the validity of this model for large propaga- turbulence.
tion distances [7]–[9]. Considering that the scintillation index is Although direct-detection reception has been the dominant
a function of the propagation distance, the log-normal assump- mode of detection in the FSO arena, due to harsh environmental
tion can only be satisfied for a small index of refraction structure conditions (such as high operating temperatures on rooftops),
constant and other beam parameters which lead to a small scin- thermal noise levels are nonnegligible. A means of overcoming
tillation index. thermal noise is to use heterodyne detection,2 and hence, an
However, when the normalized scintillation index [see (1)] investigation of the performance of this detection mechanism
is confined to the range (2, 3),1 or when moderate propaga- in turbulence is a first step toward providing a framework for
tion distances are encountered, the accuracy of the log-normal further investigations. It is also imperative to note that coherent
model has been challenged [10, pp. 178–182]. In fact, when optical communications over turbulent atmosphere has been
moderate propagation distances are encountered, the number of subject of many investigations, for example, see [10, p. 173]
scatterers in the channel is typically finite, and perhaps random, and [19].
due to unpredictable temperature fluctuations and varying wind We note that the correlation time of signal variations in tur-
velocity, which, in turn, gives rise to different statistics for the bulent atmosphere is a function of transversal wind velocity [1],
intensity of the received optical signal. To elaborate, a number and can be shown to be on the order of a few to hundreds of ms
of studies have considered a scattering model when the number [2], [4], [5]. Considering that we are considering hundreds of
of scatterers is governed by a negative binomial distribution, and megabit-per-second (Mb/s) to a few gigabit-per-second (Gb/s)
the phase disturbance associated with each scatterer is governed communication systems, one can assume a “frozen atmosphere”
by the von Mises probability density function (pdf) [11], [12]. model, where the characteristics of the channel remain constant
These assumptions lead to a situation where the signal intensity for at least two consecutive bit intervals. This condition is crit-
ical to a differentially encoded phase-shift keying (DPSK) com-
munication system that requires a constant channel condition for
Paper approved by R. Hui, the Editor for Optical Transmission and Switching
of the IEEE Communications Society. Manuscript received April 14, 2005; re-
at least two consecutive bit intervals to function properly. We
vised October 3, 2005. also note that regardless of the severity of turbulence, which may
The author is with the Erik Jonsson School of Engineering and Computer have a severe impact on the amplitude and phase of the signal
Science, The University of Texas at Dallas, Richardson TX 75083-0688 USA
(e-mail: kamran@utdallas.edu).
over time, the frozen atmosphere assumption (which is related
Digital Object Identifier 10.1109/TCOMM.2006.873067 to the transversal wind velocity) would limit the variation of the
1A large scintillation index may be due to several reasons, including a large 2Clearly, heterodyne detection is a more complicated detection mechanism,

index of refraction structure constant, which is dependent on the temperature of as compared with its direct-detection counterpart. However, an investigation of
the medium and is subject to large variations over the course of a day [1]. tradeoffs of coherent detection is beyond the scope of this letter.

0090-6778/$20.00 © 2006 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006 605

phase of the signal to a tolerable level over two consecutive bit the above, it is assumed that duration of a symbol time (typ-
intervals (which range from hundreds of picoseconds to a few ically in the nanoseconds range) is substantially smaller than
nanoseconds) [4]. Note that the variation in the phase, and not the coherence time of the channel (typically in the milliseconds
the absolute phase, is of importance here. Hence, under such range), which, in turn, leads to a constant amplitude scenario for
conditions, one may ignore the impact of fluctuations of phase the received signal. For this reason, time variations in over
on the system performance. a symbol time is assumed away. Given a slow-fading environ-
ment, the unconditional BER is given by
II. CHANNEL MODEL
Under the K-distribution regime, the signal intensity is mod- (4)
eled as follows [11], [12], [20]:

(5)

where denotes the optical signal intensity,3 denotes Making the change of variable , we arrive at
the pdf of , is the gamma function, ,
is the scintillation index defined as
(6)

(1) (7)

with denoting the expected value of the enclosed, is

the modified Bessel function of the second kind of order , and Let us consider the following identity [22, p. 738]:
is the mean intensity. Note that this pdf is defined in terms
of the first and second moments of intensity, which allows for
fitting the pdf to a specific set of observed data. As noted earlier,
we are concerned with the situation where exceeds 2 (note
that the above expression is invalid for ). Provided that
, then in this analysis.
where is the Whittaker function and
III. BIT-ERROR RATE
( denotes the real part of the enclosed). In
Assuming a shot-noise limited, heterodyne DPSK commu- the above case, and . Note that
nication system, the bit-error rate (BER) is given by (see [21, , and hence, the above condition is satisfied. may
p. 291]) now be expressed as

(2) (8)
where
where . Note that is the mean intensity
level, and hence SNR SNR may be viewed as the
SNR SNR (3) average SNR, where denotes the ensemble average of the
enclosed. Realizing that (note that )
and denotes the signal-intensity-dependent BER. Note
that the impact of phase noise is not considered in this anal- (9)
ysis. That is, we assume a negligible phase noise accumulation
over two consecutive bit intervals. In the above equation, de- (10)
notes the quantum efficiency of the detector, is the detector We note that [22, p. 1088]
area in , is the DPSK symbol duration in seconds, is
Planck’s constant in joules/Hertz, is the frequency of the re-
ceived optical signal in Hertz, and SNR denotes the signal
intensity-dependent signal-to-noise ratio (SNR). Note that (11)
denotes the received power over the detector area. In the pres-
ence of turbulence, takes on a random nature. In arriving at
with . Hence
3I denotes the instantaneous received power per unit area collected by the
receiver aperture. Although such a quantity is a function of time, we have sup-
pressed the variation in time. Note that under a frozen atmosphere assumption,
the signal intensity will not change over two consecutive symbol intervals, and
hence, one may assume that the signal intensity is not a function of time for the
(12)
purpose of this analysis.
606 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 4, APRIL 2006

to a scintillation index ranging from 2 to 3. It is important to

point out that the error rate presented here is for an uncoded
DPSK system, and hence, one may achieve a reasonable level
of performance using powerful coding schemes. An investiga-
tion of the error rate of a coded system is beyond the scope of
this letter, and is the subject of current investigation. It is impor-
tant to note that although an improvement in performance may
be achieved with the aid of powerful coding schemes, an error
rate of may not be achievable under the conditions stated
in this letter. This is not a surprising result, since it is widely
accepted that FSO systems under severe turbulence conditions
(similar to those considered in this letter) cannot achieve a re-
liability similar to that of fiber communications (i.e., error rate
of ). In such scenarios, one has to resort to RF backup sys-
tems, which typically operate at lower data rates, but provide
the desired error rate.
Finally, it would be of interest to show the performance as a
function of the average received optical power for a set of system
parameters. Note that in Fig. 1, the average SNR is limited to
20–27 dB. Let us assume Hz (1550 nm ra-
Fig. 1. P as a function of average SNR for :
= 1 2(3), :
= 1 4 (x),
diation), , joules/Hz, and
:
= 1 6 (square), and = 1:8 (o). 1 ns (bit rate of 1 Gb/s) for a typical FSO link. Provided that
the average power in watts is given by , one con-
cludes that the average received power in dBm is limited to the
Furthermore, when is not an integer, one may be able to following range for Fig. 1: dBm dBm.
express this BER in terms of the Kummer confluent hypergeo-
metric function, which is tabulated in [23, pp. 516–533]. That
ACKNOWLEDGMENT
is, first one may describe in terms of Whittaker
function as (see [23, pp. 505, eq. 13.1.34]) The author would like to thank all the referees for their
insightful comments which have enhanced the quality of this
letter.

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