COMMUNICATION
SCIENCES
AND
ENGINEERING
XV.
PROCESSING AND TRANSMISSION OF INFORMATION
Academic and Research Staff
Prof. R. N. Spann
Prof. R. D. Yates
J. Max
Prof. E. V. Hoversten
Prof. R. S. Kennedy
Prof. C. E. Shannon
Prof. P. Elias
Prof. R. G. Gallager
Prof. F. C. Hennie III
Graduate Students
L. Hatfield
H. M. Heggestad
Jane W-S. Liu
J. C. Moldon
P. Richter
Bucher
Buckman
Cohn
Greenspan
Halme
A.
J. H. Shapiro
M. A. Sirbu
M. A. Tamny
W. C. Wilder
J. S. Zaborowski, Jr.
ERROR BOUNDS FOR THE TURBULENT OPTICAL
CHANNEL (I)
Subject to some reasonable assumptions, the atmospheric optical channel can be
modeled1 as shown in Fig. XV-1.
of M orthogonal, equal-energy,
We suppose that this channel is used to transmit one
equi-probable,
(complex) waveforms S.(t); j = 1, .. -, M;
and that the receiver is to decide, with minimum error probability, which waveform was
transmitted.
It is known 2 that, in the notation of Fig. XV-1, the appropriate
receiver
evaluates the quantities
Lk =
ln
du p(u) I o (u y i k /N
o
) exp -u2Ek/No
}
for k = 1,... ,M
i=l
where p(u) is the lognormal density 3
+ nu)/2 /2u 2exp[-(
2 2 ) -1/2 exp[-(u 2 +1nu)
== (Z
p(u)p~u
2weu
Ek
k
k= 1,...,
dt Yi(t) Sk(t)
Yik =
Ac
2
Sk(t)
dt
k=l,
M
... , M.
and I ( -) is the modified Bessel function.
The transmitted waveform is then presumed
to be that one, say n, for which
L n > L kk
(5)
k= 1,... , M.
This work was supported principally by the National Aeronautics and Space Administration (Grant NsG-334), and in part by the Joint Services Electronics Programs
(U.S. Army, U.S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E).
QPR No. 89
193
A z
Ac z
nl(t)
n2()
2
x (t)
nD (t)
Ac zD
+
X
Path Quantities Are Independent And Identically Distributed.
n.(t) Are
Complex Zero-Mean Gaussian
Random Processes Whose Real And Imaginary Parts Are
Independent And Have
Power Density NoA c
z i. = e
vi
= e
xi+ji
, Where The y.i Are Com-
plex Gaussian Random Variables.
Fig.
QPR No. 89
XV-1.
Diversity representation of the turbulent
optical channel.
194
(XV.
PROCESSING AND TRANSMISSION OF INFORMATION)
Our objective is to establish the following bound to the error probability P[E]
<
P[E]
(6a)
2 -KE
where
(6b)
E (p) -
E =max
with
l+p
Sa
/(l+p)
ln
-
Eo(p)
dy e-Y
du p(u) I (Zu
0 y
p
) exp -u
ap
(6c)
P
where
(7a)
K = log 2 M
Ek
a
p
=-
(7b)
N
Da
p
(7 c)
- K
As a first step in the derivation, we note the following statistical properties of the
is transmitted. First, all of
yik conditioned upon the knowledge that the nth waveform
the Yik are statistically independent of each other. Second, for k t n, the yik are zeromean complex Gaussian random variables with
(8a)
(yik)Z = 0
and
(8b)
lyikl2 = 4EkNo.
Third, conditioned upon a knowledge of z i, yin is a complex Gaussian random variable
with
Jin
(9a)
= 27.
(Yin-Yi
1 n
(9b)
2 = 0
Yin-Yin)
and
QPR No. 89
195
(XV.
PROCESSING AND TRANSMISSION OF INFORMATION)
(9c)
Yin-YinJ 2 = 4EkNo .
The preceding properties imply that, conditioned upon the knowledge of the z. and of
the transmitted message, the Lk are statistically independent of each other. Moreover,
for k * n the L k are identically distributed. Finally, the distribution for the L k for k n
and also the distribution of Ln are independent of n.
Consequently, the error probabil-
ity conditioned upon the z. is
1
P[er ] = 1 -
dx po(xz)
L
dy pi(y)
,
(10)
where Po(x z) is the probability density of the random variable of Eq. 1
when the yik
are conditionally Gaussian with the moments of Eq. 9, and Pi(x) is the density of that
variable when the yik are conditionally Gaussian with the moments of Eq. 8.
To upper-bound P[E z], we first note that, for 0 < p < 1,
P[EIz]
<
dx Po(xZ)
L
-
(M-1)
00
dy pl(y)
(11a)
dy pl (y) dy.
(1l1b)
x
< dx po(xZ)
(M-1)
Or, upon introducing the Chernov bound,
dy pl(y)
< exp -[tx-yl(t)]
t
(1 2 a)
>0
with
y (t)= In
dy p(y) exp ty]
(12b)
we obtain
P[Elz] < MP
dx po(Xlz)
exp -[ptx-p-
l (t ) ] .
(13)
Averaging Eq. 13 over the random variables z i and defining
yo(s) = ln
p (iz) dz
dx po(xz) exp sx}
(14)
yields
P[E] < M P exp[pyl(t)+y o(-t)].
QPR No. 89
(15)
196
(XV.
PROCESSING AND TRANSMISSION OF INFORMATION)
To complete the derivation, we require more explicit expressions for yl (t) and yo(s).
These can be obtained from Eqs.
of the z..
1
1, 8,
9, 12,
and 14 in conjunction with the properties
The result is
Y1 (t) = D ln
fs0
dy e - y
c
du p(u) I (2uTay7) exp -u
(16a)
a
and
y (s) = D ln
dy e -
du p(u) Io(2u
o
We next set t = (l+p)-1 and combine Eqs.
P[E] < M P exp(l+p)
or, by virtue of Eqs.
p
y ) exp -u2a
.
(16b)
15 and 16 to obtain
(-)
16a and 6c,
P[E] < M P exp -KpE
(p).
Finally, we express M as 2 K , change from base e to base 2, and maximize the negative of the exponent to obtain the upper bound of Eq. 6a.
R. S. Kennedy, E. V.
Hoversten
References
I.
E. V. Hoversten and R. S. Kennedy, "A Lower Bound to the Error Probability for
the Atmospheric Optical Channel," Quarterly Progress Report No. 88, Research
Laboratory of Electronics, M. I. T., Cambridge, Mass., January 15, 1968, pp. 2 4 0246.
2.
R. S. Kennedy and E. V. Hoversten, "On the Atmosphere as an Optical Communication Channel" (submitted to IEEE Trans. on Information Theory).
3.
J. Aitchison and J. A. C. Brown, The Lognormal Distribution (Cambridge University
Press, London, 1963).
QPR No. 89
197