The optimum detection threshold in an on-off binary optical communications system operating in the presence of atmospheric turbulence was investigated assuming a poisson detection process and log normal scintillation. The dependence of the probability of bit error on log amplitude variance and received signal strength was analyzed and semi-emperical relationships to predict the optimum detection threshold derived. On the basis of this analysis a piecewise linear model for an adaptive threshold detection system is presented. Bit error probabilities for non-optimum threshold detection system were also investigated

Marino, J. T., Jr.

Webb, W. E.

English

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THE UNIVERSITY OF ALABAMA
COLLEGE OF ENGINEERING
BUREAU OF ENGINEERING RESEARCH
INTERIM REPORT
on Ir
Contract NAS8-25562
THRESHOLD DETECTION IN AN ON-OFF BINARY COMMUNICATIONS CHANNEL
WITH ATMOSPHERIC SCINTILLATION
-o
by
H 0 -4 U
William E. Webb az
Principal Investigator
U H
and u M
Joseph T. Marino, Jr. H He
Research Assistant P 4
Prepared for N Z
National Aeronautics and Space Administration E rm
George C. Marshall Space Flight Center
Marshall Space Flight Center, Alabama 35812 H
May 1974 V) 1
BER Report No. 172-70
U N I3V N VERSITY, ALABA MA 35486
https://ntrs.nasa.gov/search.jsp?R=19740023848 2020-03-23T04:06:16+00:00Z
INTERIM REPORT
on
Contract NAS8-25562
THRESHOLD DETECTION IN AN ON-OFF BINARY COMMUNICATIONS CHANNEL
WITH ATMOSPHERIC SCINTILLATION
by
William E. Webb
Principal Investigator
and
Joseph T. Marino, Jr.
Research Assistant
Prepared for
National Aeronautics and Space Administration
George C. Marshall Space Flight Center
Marshall Space Flight Center, Alabama 35812
May 1974
BER Report No. 172-70
THRESHOLD DETECTION IN AN
ON-OFF BINARY COMMUNICATIONS CHANNEL
WITH ATMOSPHERIC SCINTILLATION
William E. Webb and Joseph T. Marino, Jr.
Department of Electrical Engineering
University of Alabama
P.O. Box 6169
University, Alabama 35486
ABSTRACT
The optimum detection threshold in an on-off binary optical
communications system operating in the presence of atmospheric tur-
bulence has been investigated assuming a poisson detection process
and log normal scintillation. The dependence of the probability of
bit error on log amplitude variance and received signal strength has
been analyzed and semi-emperical relationships to predict the opti-
mum detection threshold derived. On the basis of this analysis a
piecewise linear model for an adaptive threshold detection system is
presented. The bit error probabilities for non-optimum threshold
detection systems have also been investigated.
-1-
2INTRODUCTION
It is well known that atmospheric scintillation not only in-
creases the bit error probability in a pulse code modulated optical
communications channel but also influences the decision level for
optimum threshold detection. The later effect is of considerable
practical importance to the design of efficient pulse code modulated
optical communications systems regardless of whether fixed or adap-
tive threshold detection is used. Fried and Schmelzer1 have consid-
ered the bit error rates in an optical communications channel by as-
suming gaussian detection statistics, an approximation which is valid
only for large numbers of signal photons. Titterton and Speck2 have
treated the problem using Poisson statistics so that their results
are valid for small numbers of signal photons also. Although both
previous investigators have recognized that the optimum detection
threshold changes in the presence of scintillation, neither has
studied the effect in depth.
In this paper we investigate the optimum detection threshold in
an on-off binary optical communications channel as a function of the
number of received photoelectrons and the strength of scintillation.
Poisson detection statistics and log normal scintillation are assumed
and, on the basis of this model, expressions to predict the optimum
detection threshold are derived. In addition, we have investigated
the bit error probabilities for sub-optimum choices of detection
3threshold. This is of importance to the design and analysis of
optical communications systems since exact optimization of the
threshold level is never possible, especially in the presence of
atmospheric effects. Application of these results to the design of
fixed and adaptive threshold optical receivers is discussed.
ERROR PROBABILITY IN THE OPTICAL CHANNEL.
We shall consider an on-off binary optical communications chan-
nel in which the laser transmitter may send either a '1', which cor-
responds to a pulse being transmitted, or a '0' which corresponds to
no pulse transmitted. Let S be the number of signal photoelectrons
per pulse received at the detector when a '1' is transmitted and eS
be the number of received signal photoelectrons when a '0' is trans-
mitted. Here c is the reciprocal of the modulator extinction ratio.
We will assume that in addition to the signal photoelectrons the
detector receives N noise photoelectrons per pulse, mostly due to
background. In a threshold detection system the receiver interprets
the received signal as either a '1' or a '0' depending on whether the
total number of received photoelectrons is greater or less than some
threshold level T.
In the presence of scintillation the probability of a detection
error is
PE = P(O)PFA + P(1)(1-PD) (1)
Where P(O) and P(1) are the apriori probabilities of sending '0' and a
'1' respectively, PFA is the false alarm probability, ie. the prob-
ability of a '1' being received given that a '0' was sent, and PD is
the detection probability, ie. the probability of a '1' being re-
ceived given that a '1' was sent. PD and PFA are given by
PFA exp [-(N+ES)] (N+ S) f(S)dS (2)jFA =t J!
PD J exp [-(N+S)] (N S) f(S)dS (3)j=t P
Assuming log normal scintillation and a symmetrical pulse code we
obtain
C 2T(T-1) 2_/ T t ' -8C ;Z12
1 e k * 1 (Z+4C (T-1/2))
E 8C k 4 C (TSOe )dZ1
ro_ 3 -8CZ;O *(T Z+4C (T-l/2))
+ ((SO-N)e+N) 
_e y (T,((SON)+N)e 0 )dZ
(4)
where y is the incomplete gamma function
y* (T,n) = e-nX (5)
n X=T 7- (5)
SO is the average number of signal photoelectrons per pulse and C
is the variance of In(S).
5The bit error probability has been evaluated by numerical in-
tegration of equation 4 on a UNIVAC 1108 computer. The threshold T
was initially taken to be the no scintillation optimum threshold
given by 3
= S(1-) + ln(P(O)/P(1))
O In((S+N)/(N+S)) (6)
An iterative procedure was then used to minimize the error probability
as a function of T, while holding SO, N, e and Ca constant. In this
way we were able to simultaneously determine the optimum threshold in
the presence of scintillation and study the effect of sub-optimum
threshold on the error rates. Calculations were repeated for values
of the parameters SO, c, N and Ck over the ranges likely to be
encountered in practice.
Figure 1 shows the probability of a bit error at optimum thresh-
old as a function of the number of received signal photoelectrons and
the log amplitude variance of scintillation with one background photo-
electron and an extinction ratio of 15 db. The computed values of
bit error probability agree with those previously published by
Titterton and Speck2
One feature of figure 1 which deserves note is the difference
in the slope of the curves for large and small values of the log
amplitude variance. When Ck is less than about 0.02 the error prob-
ability decreases rapidly with increasing number of signal photo-
6electrons, whereas for large values of CA the error probability tends
to become relatively independent of the number of signal photoelectrons
provided that SO is not too small. Thus there appears to be two re-
gions, one in which the error probability is determined primarily
by scintillation and the other in which it is mainly determined by
signal strength. This dicotomy, which we might call the signal
limited and'scintillation limited cases, will be seen more clearly
when we consider the optimum detection threshold in the presence of
scintillation.
An alternate method of displaying these results is in terms of
the transmitted power required to achieve a given bit error rate in
the presence of scintillation as compared to the power required when
there is no scintillation. In figure 2 the required power margin is
shown as a function of bit error rate and log amplitude variance.
These results are essentially independent of the choice of N and e
as long as both are relatively small. Fried and Schmeltzer1 have
developed a similar set of curves based on the assumption of gaussian
detection statistics. Comparison of our results with those of Fried
and Schmeltzer shows that the two models agree reasonably well in
predicting the required power margin for values of C less than about
0.02, but that the gaussian model may underestimate the necessary
power margin by as much as 8 db for Cz equal to 0.03 and 30 db for
CA equal to .05 at error rates of 10-6. For a log amplitude variance
of 0.1 the difference in the power margins predicted by the gaussian
7
and Poisson models may exceed 80 db. Thus the gaussian model is
reasonably accurate in the case of weak scintillation but gives an
extremely over-optimistic estimate of a communications system
performance in the presence of strong scintillation.
Since the Poisson distribution approaches a gaussian distri-
bution as the mean increases, it might be expected that our results
should reduce to those of Fried and Schmeltzer in the case of a large
average number of received signal photoelectrons. However, for the
system under consideration a large value of SO is not sufficient to
insure that the Poisson counting statistics can be adequately ap-
proximated by a gaussian distribution. The means of the Poisson
distributions in question are not SO but (N + S) and (N + ES), where
the variable S is averaged over all values from zero to infinity.
Thus if N is small, the mean of the Poisson distribution will be
small over part of the range of S, even though SO is large. Clearly,
when C. is large the contribution to the error probability from that
part of the range of S in which the gaussian approximation is invalid
will be greater. Physically this means that.during a deep fade there
are only a few photoelectrons per pulse; therefore gaussian statistics
can not be used. A second, and perhaps more significant, problem
with Fried and Schmeltzer's model is the assumption of additive noise
independent of the signal strength. This assumption is equivalent to
approximating the Poisson distribution by a gaussian distribution with
constant variance. Since the variance of a Poisson distribution is
8equal to the mean, this assumption is valid only if
N + S = N + eS = constant (7)
for all values of S with non negligible probability of occurance.
In particular, unless N is very large compared to SO, the noise
associated with a zero being received is different from the noise
associated with a one. Therefore Fried and Schmeltzer's ianalysis
can only be valid if the number of noise photoelectrons per pulse
is much greater than the number of signal photoelectrons. In this
case the two analyses should yield the same results.
As a check on our results we have calculated the error prob-
abilities assuming fourty noise photoelectrons per pulse, log
amplitude variances of 0.0 and 0.05 and error probabilities from
10-1 to 3x10l3. For these cases the computed power margin required
to compensate for scintillation agreed with that reported by Fried
and Schmeltzer to within one db. The residual error is due to the
fact that even for fourth noise photoelectrons the condition of
equation 7 is only approximately satisfied.
On the basis of our analysis we conclude that Fried and Schmeltzer's
results contain a fundamental inconsistancy; namely, the number of
signal photoelectrons required to obtain error probabilities less
than about 10-3 is so large that equation 7 can not be satisfied
unless the log amplitude variance is very small. Thus the assumption
of background limited operation is invalid except for the case of
weak scintillation or very high bit error rates.
9OPTIMUM THRESHOLD
The optimum threshold in the presence of scintillation is
shown in figure 3 as a function of the average number of received
signal photoelectrons per pulse. As is to be expected the optimum
threshold decreases with increasing scintillation. For SO greater
than about 10 the curves are very nearly straight lines whose slopes
are dependent on the strength of the scintillation. This linearity
can be understood in the no scintillation case by noting that when
cS is large compared to N equation 6 reduces to
T n(/) S. (8)
With scintillation we may replace equation 8 by a linear relation
of the form
T = a(C )S0 + B (9)
where the parameters a and a can be determined by linear least mean
square fits to the data of figure 3. To a reasonable approximation
we may take B to be 2.5, independent of C,. The slope a, on the
other hand, is strongly dependent on C. as shown in figure 4. The
dependence of the slope on Ck can be represented by the relation
7.5X103C 2
a(C,) = .145(e + 1) (10)
10
Perhaps the most striking feature of figure 4 is the distinct
knee that occurs for values of C near 0.02, and corresponds to the
transition from signal strength limited to scintillation limited
conditions. For Cz much larger than 0.02 the slope of the threshold
curve varies only slowly with increasing C. whereas for weak
scintillation the slope is a very strong function of C,. In fact
we can approximate a fairly well in these two regions by straight
lines as shown by the dashed curves in figures 4. Hence
TO = aiC S0 + b iS + C (11)
where the coefficients ai and b. take on either one of two values
depending upon whether C9 is greater or less than the break point.
A piecewise linear approximation of this sort is appealing from an
engineering point of view since it provides a convenient model for
implementing an adaptive threshold detection system. One could en-
vision, for example, a detection system in which the mean and variance
of the received signal power were continuously monitored and the
threshold set in accordance with equation 11. Such a system would
require only linear operations and a single discrete discontinuity
to control the threshold level.
As Titterton and Speck have pointed out (footnote 4 of reference
2) this analysis is applicable to systems in which the threshold is
varied on the basis of a long term estimate (ie. on the order of
seconds) of scintillation and is limited to systems with bit rates
of 107 or less. Tycz, Fitzmaurice and Premo [5] have considered a
perfectly adaptive threshold system for both the log normal and beta
channels.
SUB-OPTIMUM THRESHOLDS
In practice the detection threshold of an optical communications
system operating through the atmosphere will never be perfectly op-
timized. If the threshold is fixed then changing atmospheric condi-
tions will deoptimize the system and even if an adaptive threshold
is used the system will be incapable of precisely tracking changes
in signal strength and turbulence. In order to properly predict a
communication system's performance it is necessary to know the ex-
pected bit error probabilities for non-optimum detection thresholds.
Investigation of the performance of non-optimum threshold detection
is also necessary to the design of adaptive threshold systems.
Figure 5 shows the bit error probability as a function of de-
tection threshold with log amplitude variance as a parameter. The
data plotted is for the case of 40 signal photoelectrons and no
background photoelectrons per pulse and an extinction ratio of 15 db.
Other choices of parameters yielded curves which were essentially
similar.
Inspection of figure 5 shows that with a fixed:detection thresh-
old a decrease in scintillation will always result in an improvement
in system performance even though decreasing the scintillation de-
optimizes the system. That is to say that the bit error probability
12
decreases more rapidly with decreasing log amplitude variance than it
is increased by the corresponding deoptimization of the threshold
level. Thus if one selects the detection threshold that is optimum
for the strongest expected scintillation and predicts the error rate
on this basis, one is assured that the system performance will not
be worse under conditions of weaker scintillation. This strategy
might be appropriate if one wishes to insure that a given error rate
will be obtained under all expected conditions.
ACKNOWLEDGMENT
This research was supported by the National Aeronautics and
Space Administration under Contract NAS8-25562 with the Astrionics
Laboratory, George C. Marshall Space Flight Center, Huntsville,
Alabama.
13
REFERENCES
1. D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, p 1729-1737,
1967.
2. P. J. Titterton and J. P. Speck, Appl. Opt. 12, p 425-426,
1973.
3. This corresponds to Titterton and Speck's (op. cit.) equation
5. HoWever these authors' expression contains a typographical
error in that two pairs of parenthesis in the denominator
were omitted.
4. M. Tycz, M. W. Fitzmaurice and D. A. Premo, IEEE Trans. on
Communications, p 1069-1071, 1973.
Figure 1. Probability of bit error for an on-off binary PCM
optical communications system as a function of the number of
signal photoelectrons for log amplitude variances between 0,0
and O.. One background photoelectron per pulse and a modula-
tion extinction ratio of 15 db. was assumed.
Figure 2. The Loss Factor, defined as the additional trans-
mitted power required to compensate for atmospheric scintil-
lation, is given as a function of the number of received
photoelectrons per pulse and the log-amplitude variance of
scintillation. Other parameters are the same as the preceed-
ing figure.
Figure 3. The optimum detection threshold as a function of
number of signal photoelectrons per pulse and log amplitude
variance of scintillation. Other parameters are the same as
in the preceeding figures.
Figure 4. The slope of the linear portion of the detection
threshold curves (figure 3) as a function of the log amplitude
variance of scintillation. The dashed lines indicate a piece-
wise linear approximation discussed in the text.
Figure 5. Dependence of the bit error probability on detec-
tion threshold for non-optimum detection. Error probability
is plotted as a function of the normalized threshold, T/So,
for S0 = 40.0
-I3
CL
- C = .05
r 10 -
C:
LL._
.03
10-6
0.0 005 .01 .02
o- .10 I
30 60 90 120 150 180 210 240
SIGNAL STRENGTH So
102
101
I-
_c, = .05
oC-,
<c .03
CD .02
.01
10o
.005
10"k
10-7  106 10 10 10-3
BIT ERROR RATE PE
50 C, =0.0
.005 
.02
.01
o .05
C')
LU
-r-,
-
o 20
10
30 60 90 120 150 180 210 240
SIGNAL STRENGTH So
.25
.20-
,15 \
CL
.10
.05
.01 .02 .03 .04 .05 .06 .07 .08
LOG AMPLITUDE VARIANCE Cl
1i
l-1
10-3 .05
10
10
4-
.01
10-6 - .005
10 - 7 ~ c, = 0 .0
0,0 ,25 ,50 ,75 1,00 1,25
NORMALIZED THRESHOLD 1
So


Threshold detection in an on-off binary communications channel with atmospheric scintillation

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